Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.

Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.

Table of contents

§

Title page

§

THE PREFACE TO THE READER.

To the READER.

§

To his learned and ingenious Friend Mr. John Taylor, in the deserved Praise of his Excellent Book intituled The saurarium Mathematicae.

To the Learned Authour my much respected Friend Mr. John Taylor on his Herculean labours in the Composure of this Excellent Mathematical Treasury.

An Acrostick on the Name of my much respected and ingenious Friend Mr. John Taylor.

ADVERTISEMENTS.

The Contents.

Arithmetick.

CHAP. I. Of ARITHMETICK.

SECTION I. The Explication of some Arithmetical Propositions.

PROPOSITION I. To three numbers given, to find a fourth in a Direct proportion.

PROP. II. To three numbers given, to find a fourth in an Inversed proportion.

PROP. III. To three numbers given, to find out a fourth in a Duplicate proportion.

PROP. IV. To three numbers given, to find a fourth in a Triplicate proportion.

PROP. V. To two numbers given, to find out a third, fourth, fifth, sixth, &c. Numbers in a continual proportion.

PROP. VI. Between two numbers given, to find out a mean Arithmetical proportional.

PROP. VII. Between two numbers given, to find out a mean Musical Proportional.

PROP. VIII. How to find the Square-Root of any whole number, or Fraction.

NOTE.

To Extract the Square-Root of a Vulgar or Decimal Fraction, and a Mixt-number.

PROP. IX. How to find the Cube-Root of any whole Number, or Fraction.

NOTE.

To Extract the Cube-Root, of any Vulgar or Decimal or Mixt Fraction consisting of a Whole Number and a Fraction.

CHAP. II. The Explication, and use of the Tables of LOGARITHMS.

SECT. I. The Explication of the Tables of the Logarithms, and of parts proportional.

PROP. I. Any Number given under 10000, or 100000, to find the Logarithm corresponding thereunto.

PROP. II.

PROP III. A Logarithm propounded to find the whole, or Mixt number, corresponding thereunto.

NOTE.

SECT. II. Of the Admirable use of the Logarithms in Arithmetick.

PROP. I. To Multiply one number by another.

PROP. II. To Divide one number by another.

PROP. III. To find the Square-Root of a Number.

NOTE.

PROP. IV. To find the Cube Root of any Number.

PROP. V. A Summ of Money being forborn for any number of years, to find how much it will amount unto, reckoning Interest on Interest, according to any Rate propounded.

PROP. VI. A Summ of Money being to be paid hereafter, to find what it is worth in •eady Money.

PROP. VII. A yearly rent, or Annuity to continue any number of years, to find what it is worth in ready Money, at any Rate of Interest propounded.

CHAP. III. The Explication of the SINES, TANGENTS, and SECANTS.

SECT. I. Of Right Signs, Tangents, Secants, Cosines, Tangents, and Secants: Of any Arch, or Angle of a Triangle.

PROP. I. To find the right Sine, or Tangent of any Arch or Angle of a Triangle containing any number of Degrees and Minutes.

PROP. II. To find the Co-Sine or Co-Tangent of any Arch, or Angle propounded.

PROP. III. To find the Secant of any Arch or Angle propounded.

CHAP. IV. Of GEOMETRY.

SECT. I. The Explication of some Geometrical Propositions.

PROP. I. To erect a perpendicular on any part of a line assigned.

PROP. II. To Erect a Perpendicular, on the End of a Line.

PROP. III. From a Point above to let fall a Perpendicular on a Line.

PROP. IV. To draw a right line Parallel to a right line, at any distance assigned.

PROP. V. To Protract an Angle of any Quantity of Degrees propounded.

PROP. VI. To measure an Angle already protracted.

PROP. VII. To divide an Angle into two Equal parts.

PROP. VIII. To divide a right line into any Number of Equal or Unequal parts; or like to any divided line propounded.

CASE II.

PROP. IX. How to Protract or lay down any of the Regular Figures, called Polygons.

CONSTRUCTION I.

CONSTRUCTION II.

CONSTRUCTION. III.

PROP. X. To divide a line according to any assigned proportion.

PROP. XI. To two lines given, to find a third proportional to each of them.

PROP. XII. To three lines given to find out a fourth proportional unto them.

PROP. XIII. To find a mean proportional Line between any two right lines given.

PROP. XIV. To find two mean proportional Lines between any two right Lines given.

PROP. XV. To make a Geometrical square equal to divers Geometrical squares.

PROP. XVI. To make a Circle equal to divers Circles propounded.

SECT. II. Of Planometry, or the way to measure any plain Superfice.

PROP. I. To find the superficial Content of a Geometrical square.

PROP. II. To find the superficial content of a Parallelogram, or long Square.

PROP. III. To find the superficial Content of any Right-lined Triangle.

PROP IV. To find the superficial Content of a Rhombus.

PROP. V. To find the superficial content of a Rhomboides.

PROP. VI. Te find the superficial Content of any Poligon, or many equal sided Superficies.

PROP. VII. To find the superficial Content of a Circle.

PROP. VIII. By the Diameter of a Circle given, to find the Circumference.

PROP. IX. By the Circumference of a Circle given, to find the Diameter.

PROP. X. By the Content of a Circle given, to find the Circumference.

PROP. XI. By the Content of a Circle given, to find the Diameter.

PROP. XII. By the Diameter of a Circle given to find the side of a square equal thereto.

PROP. XIII. By the Circumference of a Circle given, to find the side of a square equal to it.

PROP. XIV. By the Content of a Circle given to find the side a square equal to it.

PROP. XV. By the Diameter of a Circle given, to find the side of an Inscribed square.

PROP. XVI. By the Circumference of a Circle given, to find the side of an Inscribed Square.

PROP. XVII. •o find the Superficial Content of an Oval, or Elleipsis.

PROP. XVIII. To find the Superficial Content of any Section, or Portion of a Circle.

SECT. III. Of STEREOMETRY, or the way how to measure any Regular Solid.

PROP. I. To find the solid Content of a Cube.

PROP. II. To find the solid Content of a Parallelepipedon.

PROP. III. To find the solid Content of a Cylinder.

PROP. IV. To find the solid Content of a Pyramid.

PROP. V. To find the solid content of a Cone.

PROP. VI. By the Diameter of a Globe to find his solid Content.

PROP. VII. By the Circumference of a Sphere, or Globe, to find his solid Content.

PROP. VIII. By the Axis of a Globe, to make a Cube equal thereunto.

PROP. IX. By the Circumference of a Globe, to make a Cube equal thereunto.

PROP. X. By the solid Content of a Sphere or Globe, to make a Cube equal thereunto.

PROP. XI. A Segment of a Sphere being given to find the solid Content thereof.

CHAP. V. Of TRIGONOMETRY. Or the Doctrine of Triangles.

SECT. I. Some general Maxims, belonging to plain or Right-lined Triangles.

SECT. II. Of Plain Rectangled Triangles.

PROP. I. Two Angles and the Base of a Rectangled Triangle given, to find the other parts.

PROP. II. The Hypothenuse, Base, and one of the Angles Of a Rectangled Triangle given, to find the other parts thereof.

PROP. III. In a Rectangled Triangle, the Base, and Cathetus given to find the other parts thereof.

PROP. IV. The Base, and Hypothenuse, with the Angle between them given, to find the other parts of a Rect-angled Triangle.

SECT. III. Of Oblique-Angled Plain Triangles.

PROP. I. Two Angles, and a side opposite, in an Oblique-Angled Triangle given, to find the other parts thereof.

PROP. II. Two sides, and an Angle opposite to one of them in an Oblique-angled Triangle given, to find the other parts thereof.

PROP. III. Two Sides of an Oblique-angled Triangle, with the Angle included between them given, to find the other parts thereof.

PROP. IV. The three sides of an Oblique-angled Triangle given, to find the Angles.

I. In the Triangle CDB.

II. In the Triangle ADB.

SECT. IV. Of Spherical Rectangled Triangles.

PROP. I. Case 1. A Side and an Angle adjacent thereunto being given, to find the other Side.

PROP. II. Case 2. A Side and an Angle adjacent thereunto being given, to find the other Oblique-angle.

PROP. III. Case 3. A Side and an Angle adjacent thereunto being given, to find the Hypothenuse.

PROP. IV. Case 4. A Side and an Angle opposite thereunto being given, to find the other Oblique-angle.

PROP. V. Case 5. A Side and the opposite Angle given, to find the Hypothenuse.

PROP. VI. Case 6. A side and the opposite Angle given, to find the other side.

PROP. VII. Case 7. The Hypothenuse, and an Oblique Angle given, to find the side adjacent thereunto.

PROP. VIII. Case 8. The Hypothenuse, and an Oblique-angle given, to find the opposite Side.

PROP. IX. Case 9. The Hypothenuse, and an Oblique-angle given, to find the other Oblique-angle.

PROP. X. Case 10. The sides given, to find the Hypothenuse.

PROP. XI. Case 11. The sides given, to find an Angle.

PROP. XII. Case 12. The Hypothenuse, and a side given, to find the other side.

PROP. XIII. Case 13. The Hypothenuse, and a Side given, to find the contained Angle.

PROP. XIV. Case 14. The Hypothenuse, and a Side given, to find the opposite Angle.

PROP. XV. Case 15. The Oblique Angles given, to find either Side.

PROP. XVI. Case 16. The Oblique-angles given, to find the Hypothenuse.

SECT. V. Of Oblique-angled Spherical Triangles.

PROP. I. Case 1. Two Sides, and an Angle opposite to one of them given, to find the other opposite Angle.

PROP. II. Case 2. Two Angles and a Side opposite to one of them given, to find the Side opposite to the other.

PROP. III. Case 3. Two Sides and an Angle included between them being known, to find the other Angles.

PROP. IV. Case 4. Two Angles, and their Interjacent side being known, to find the other sides.

PROP. V. Case 5. Two Sides and an Angle opposite to one of them given, to find the third side.

PROP. VI. Case 6. Two Angles and a Side opposite to one of them given, to find the third Angle.

PROP. VII. Case 7. Two Sides and an Angle opposite to one of them given, to find the Included Angle.

PROP. VIII. Case 8. Two Angles and a Side opposite to one of them being known, to find the Interjacent Side.

PROP. IX. Case 9. Two Sides and their Included Angle being known, to find the third Side.

PROP. X. Case 10. Two Angles and their Interjacent Side known, to find the third Angle.

PROP. XI. Case 11. Three Sides given, to find an Angle.

PROP. XII. Case 12. Three Angles given, to find a Side.

CHAP. VI. Of ASTRONOMY.

SECT. I. Of Astronomical Definitions.

SECT. II. Of Astronomical Propositions.

PROP. I. The Distance of the Sun from the next Equinoctial point (either Aries or Libra) being known, to find his Declination.

PROP. II. The Sun's place given, to find his Right-Ascension.

PROP. III. To find the Sun's place or longitude from Aries, his Declination being given.

PROP. IV. By knowing the Suns Declination, to find his Right Ascension.

PROP. V. By knowing the Latitude of a Place, and the Suns Declination, to find the Ascensional Difference.

PROP. VI. To find the Suns Oblique Ascension or Descension.

PROP. VII. By knowing the Suns Declination, and the Latitude of a Place, to find the Suns Amplitude.

PROP. VIII. By knowing the Suns Declination and Amplitude, from the North part of the Horizon, to find the Latitude.

PROP. IX. By knowing the Latitude of a place, and the Sun's Declination, to find at what time the Sun will be on the true East or West Points.

PROP. X. By knowing the Sun's Declination, and Latitude of a place, to find his Altitude at six a Clock.

PROP. XI. By knowing the Latitude of a place, and the Sun's Declination, to find the Azimuth at six.

PROP. XII. By knowing the Latitude of a place, and the Sun's Declination, to find the Sun's Altitude when he i• on the true East or West points.

PROP. XIII. To find the Sun's Altitude at any time of the day.

PROP. XIV. By knowing the Latitude of a Place, with the Sun's Declination, and Altitude, to find the Hour of the Day.

PROP. XV. To find the Time of the Sun's Rising or Setting, and consequently the Length of the Day or Night.

PROP. XVI. The Sun's Declination, Altitude and Azimuth known, to find the Hour of the Day.

PROP. XVII. By knowing the Sun's Declination, Altitude, and Hour from Noon, to find the Azimuth.

PROP. XVIII. By knowing the Latitude of a place, the Altitude of the Sun, and the Hour from Noon, to find the Angle of the Sun's Position.

PROP. XIX. By knowing the Sun's Altitude, Declination, and Azimuth; to find the Latitude.

PROP. XX. To find the length of the Crepusculum, or Twilight

PROP. XXI. To find the Quantity of the Angles, which the Circles of the 12 Houses make with the Meridian.

PROP. XXII. To find the Right Ascension of the Point in the Equinoctial: and also the Point in the Ecliptick; called Medium Coeli or Cor Coeli.

PROP. XXIII. To find the Angle of the Ecliptick with the Meridian.

PROP. XXIV. To find the Angle of the Ecliptick with the Horizon.

PROP. XXV. To find the Amplitude Ortive of the Ascendent, or Horoscopus.

PROP. XXVI. To find the Ascendent degree of the Ecliptick, or the Cuspis of the first House.

PROP. XXVII. To find the Distance of the Cuspis of any House, from Med. Coeli.

PROP. XXVIII. To find the parts of the Angle of the Ecliptick with the Meridian, cut with an Arch perpendicular to the Circle of any of the Houses.

PROP. XXIX. To find the Pole's Altitude, above any of the Circles of the Houses.

PROP. XXX. By knowing the Latitude and Longitude of any fixed Star, to find his Right Ascension and Declination.

PROP. XXXI. By knowing the Pole's Altitude, to find when any fixed Star shall be due East or West.

PROP. XXXII. By knowing the Poles Altitude, to find the Elevation of any fixed Star above the Horizon, being due East or West.

PROP. XXXIII. To find out the Horizontal Parallax of the Moon.

PROP. XXXIV. The Horizontal Parallax of the Moon being known, to find her Parallax in any apparent Latitude.

PROP. XXXV. By knowing the Moon's Place in the Ecliptick, (having little or no Latitude) and her Parallax of Altitude, to find the Parallaxes of her Longitude and Latitude.

PROP. XXXVI. How by knowing the Refraction of a Star, to find his true Altitude.

EXAMPLE.

CHAP. VII. Of GEOGRAPHY.

SECT. I. Of GEOGRAPHICAL Definitions.

SECT. II. Geographical Descriptions of the Earth.

1

ASIA.

AFRICA.

AMERICA.

SECT. III. Of Geographical Propositions.

PROP. I. How to find the Distance of any two Cities or Places, which differ onely in Latitude.

PROP. II. To find the Distance of any two Places which differ only in Longitude.

PROP. III To find the Distance of any two Places, which differ both in Latitude, and in Longitude.

CHAP. VIII. of NAVIGATION.

SECT. I. Of Plain sailing, or sailing by the Plain Chart.

PROPI. The Rumb, and Distance sailed thereon being given, to find the Difference of Latitude, and the Departure from the Meridian.

PROP. II. By the Rumb and Difference of Latitude given, To find the Distance, and the Departure from the Meridian.

PROP. III. By knowing the Distance of the Meridians of two Places, and their Difference of Latitude, to find the Rumb, and Distance.

PROP. IV. Admit two Ships to set sail from one Port, one Ship sails W. S. W. 40', the other W. by N. so far untill she finds the first Ship to bear from her S. E. by E. I demand the second Ships distance from the Port, and their Distance asunder?

PROP. V. Two Ships sets sail from two Ports, which lie N. and South of each other, the one sails from the Northermost Port 72 29'/100, and then meets she other Ship, which came from the Southermost Port, on a N. W. Course, and had sailed from thence 56 80'/100 I demand the Rumb on which the first ship made her way, and also the Distance between the two Ports?

PROP. VI. Admit a Ship coming off the Main Ocean and I had sight of a Promontory or Cape, by which it is my desire to sail, I find it to bear from me S. S. E. and distant by Estimation 33', or Miles: But keeping still on my Course S. untill the Evening, having sailed 36' or Miles, I would then know how the Cape bears, and its distance from the Ship?

PROP. VII. Two Ports both lying in one Latitude, distant 64' or Miles, the Westermost of those Ports lieth opposite to an Island, more Northerly distant therefrom 47' or Miles, which Island is also distant from the Eastermost Port, 34' or Miles, I demand the Course from the Westermost Port to that Island?

SECT. II. Of sailing by the true Sea Chart, commonly called MERCATOR'S Chart.

PROP. I. To find by the Table, what Meridional parts are contained in any Difference of Latitude.

PROP. II By knowing the Latitudes, and the difference of Longitude of any two Places, to find the Rumb, and Distance.

PROP. III. By knowing the Latitudes, and distance of two Places, to find the Rumb, and Difference of Longitude.

PROP. IV. By knowing the Latitudes, and Rumb of two Places, to find their Distance, and Difference of Longitude.

PROP. V. By knowing the Rumb, Difference of Longitude, and one Latitude, to find the other Latitude, and the Distance.

PROP. VI. By knowing the Distance, one Latitude, and Rumb, to find the other Latitude, and Difference of Longitude.

SECT. III. Of Circular Sailing, or Sailing by the Arch of a Great Circle.

PROP. I. Two Places, the one under the Equinoctial, the other in any Latitude given; also their difference of Longitude given, to find.

PROP. II. Two Places proposed, the one lying under the Equinoctial, the other in any Latitude given; with their distance in a great Circle of the same Places being also known, to find.

PROP. III. Two Places lying in one Latitude given, their difference of Longitude being also known, to find.

PROP. IV. Two Places lying both in one Latitude given, and the nearest distance being also known, to find.

PROP. V. Two Places proposed lying in one Latitude, and the distance of those Places in their Parallel given; to find.

PROP. VI. By knowing the nearest Distance of two Places, their Difference of Longitude, and one of their Latitudes; to find the Direct Position thereof from the other.

PROP. VII. By knowing the Latitudes of two places, and likewise their Difference of Longitude; to find,

CHAP. IX. Of SURVEYING.

SECT. I. Of the use of the Protractor.

PROP. I. By the Protractor, to Protract an Angle of any quantity of degrees propounded.

PROP. II. By the Protractor given, to measure an Angle given.

SECT. II. Of the Manifold Use of the Semicircle, in taking the Plots of small Enclosures, Plains, Woods, or Mountains divers Ways.

PROP. I. How to take the Plot of a Field, by the Semicircle at one Station taken in any part thereof, from whence all the Angles may be seen, and measuring from the Station unto every Angle thereof.

PROP. II. How to delineate on Paper any Observation taken according to the Doctrine of the last Proposition.

PROP. III. How by the Semicircle to take the Plot of a Field at one Station in any Angle thereof, from whence you may view all the other Angles, by measuring from the Stationary-Angle, unto all the other Angles.

PROP. IV. How to delineate any Observation taken according to the Doctrine of the last Proposition.

PROP. V. How by the Semicircle to take the Plot of a Field at two Stations, by measuring from each Station to the visible Angles: the Field being so Irregular that from no one Place thereof, all the Angles can be seen.

PROP. VI. How to delineate any Observation taken according to the Doctrine of the last Proposition.

PROP. VII. How by the Semicircle, to take the Plot of a Field at t〈…〉 Stations, which lieth remote from you, when either by opposition of Enemies you may not, or by some other Impediment you cannot come into the same.

PROP. VIII. How to delineate any Observation taken according to the Doctrine of the last Proposition.

PROP. IX. How by the Semicircle, to take the Plot of a great Champain-Plain, Wood, or other overgrown Ground, by measuring round about the same, and making Observation at every Angle thereof.

PROP. X. How to delineate any Observation taken according unto the Doctrine of the last Proposition.

PROP. XI. How to take the Plot of any Field, by the help of the Chain only.

PROP. XII. How to delineate any Observation, taken according to the Doctrine of the last Proposition.

SECT. III. Of finding the Area or superficial Content of any Field, lying in any Regular or Irregular Form: by reducing the Irregular Fields into Regular Forms.

PROP. I. How to find the Area or superficial Content of a Trapezium.

PROP. II. To find the Area or superficial Content of a manysided Irregular Figure.

PROP. III. How to reduce any Number of Perches into Acres, and on the contrary, Acres into Perches.

SECT. IV. Of the Use of the Semicircle in taking Altitudes, Distances, &c.

PROP. I. How by the Semicircle to take an Accessible Altitude.

PROP. II. How by the Semicircle to take an Inaccessible Altitude, at two Stations.

PROP. III. How by the Semicircle to take an Inaccessible Distance at two Stations.

PROP. IV. How to find the Horizontal line of any Hill or Mountain, by the Semicircle.

SECT. V. How to find whether Water may be conveyed from a Spring-Head unto any appointed Place.

CHAP. X. Of MEASURING, Of Board, Glass, Tiling, Paving, Timber, Stone, and Irregular Solids, such as Geometry can give no Rule for the Measuring thereof.

SECT. I. Of the Measuring of Board, Glass, Paving, Tiling, &c.

PROP. I. To Measure a Piece of Board, Plank, Glass, &c.

PROP. II. To measure Tiling, Flooring, Roofing, and Partitioning-works.

PROP. III. To measure Paving, Plaistering, Wainscotting, and Painting-work.

SECT. II. Of the Measuring of Timber, Stone, and Irregular Solids.

PROP. I. How to Measure any kind of Timber, or Stone, whether Three-square, Four-square, Many-square, Round, or of any other fashion, provided it be streight and equal all along.

PROP. II. To measure Round Timber which is Hollow: or any other Hollow Body.

PROP. III. To Measure Tapering Timber, or Stone.

PROP. IV. How to find the Solid Content of any Solid Body, in any strange form, such as Geometry can given: no Rule for the measuring thereof.

CHAP. XI. Of GAUGING.

SECT. I. Of Gauging any Beer, Ale, or Wine-Cask, also any manner of Brewers Tuns.

PROP. I. To find the Solid Content in Inches of any Cask.

PROP. II. To find the Content of a Vessel in Wine, or Ale Gallons.

PROP. III. How to Gauge or Measure Brewers Tuns, &c.

SECT. II. Of Gauging or Measuring, and the Moulding of Ships.

PROP. I. To Gauge a Ship, thereby to find how many Tuns her burthen is.

PROP. II.

CHAP. XII. Of DIALLING.

SECT. I. Of the Delineation and Projection of sundry most usefull Dials.

PROP. I. How to draw the Hour-lines on an Equinoctial Plain.

PROP. II. How to draw the Hour-lines on a Polar Plane.

PROP. III. How to draw the Hour-lines on a Meridian Plane, which is an East, or West Dial.

PROP. IV. How to draw the Hour-lines on a direct South, and North Plane,

PROP. V. How to draw the Hour-lines on an Horizontal Plane.

PROP. VI. How to draw the Hour-lines, on an Erect declining Plane.

A Table of the Names and Latitudes of all the Principal Cities, Towns, and Islands, in and about Great Britain and Ireland.

CHAP. XIII. Of FORTIFICATION.

SECT. I. Of the Definitions of the Lines, and Angles, belonging to the Principal Ground work of any Regular Fortification.

SECT. II. Of General Maxims or Rules observed in Fortifications.

SECT. III. Of the Construction and making of the principal Ground-line of a Fort, according to the most Modern ways, used by the Italian, Dutch, French, or English Inginiers.

I. Of the Italian Fortifications.

PROP. I. To fortifie a Hexagon according to this Author's Proportion.

II. Of the French Fortifications.

PROP. II. To fortifie a Hexagon according to the Proportion of De la Mont.

PROP. III. To fortifie a Hexagon according to Manesson Mallet's Proportion.

III. Of the Dutch Fortifications.

PROP. IV. To fortifie a Hexagon according to the Emperour's Proportion.

IV. Of the English Fortifications.

PROP. V. How to fortifie a Hexagon according to Count Pagan's Proportion.

PROP. VI. To fortifie a Hexagon according to the way prescribed by His Majesty Carolus II.

PROP. VII. By the Semicircle to lay down on the Ground, any of the former Fortifications.

SECT. IV. Of the Dimensions, and Measures of the Rampires, Parapets, Mote, Coridor, or Covert-way, and its Esplanade, or Breastwork.

PROP. VIII. How to lay down the Profile of the Work, according to this Table.

SECT. V. Of the Dimensions and Construction of Pla〈…〉 forms, Caveleers and Cazemats in t〈…〉 Flanks.

SECT. VI. Of the Dimensions, and Constructions of those Out-Works, called Ravelins, Horn, Crownworks, &c.

SECT. VII. Of some Maxims or Rules necessary to be known in Irregular Fortification.

SECT. VIII. Of the Dimensions and Construction of small Forts, or Scones, which are built for the Defence of some Pass, River, or other place.

PROP. IX. To fortifie a Triangle, with Demi-Bastions.

PROP. X. To fortifie a Square with Demi-Bastions.

PROP. XI. To fortifie a Parallelogram with Demi-Bastions, and Tong.

PROP. XII. To fortifie a Star Redoubt of 4, 5, or 6 Points

PROP. XIII. To Delineate a Plain Redoubt

CHAP. XIV. Of Military Orders, or the Embattelling and Encamping of Souldiers.

SECT. I. Of the Embattelling and Ordering of Souldiers.

PROP. I. To Order any number of Souldiers into a Square Battail of Men.

PROP. II. To Order any number of Souldiers into a Double Battail.

PROP. III. To Order any number of Souldiers into a Battail of the Grand Front.

PROP. IV. Any number of Men, together with their distance in Rank and File, being propounded, to Order them into a Square Battail of Ground.

PROP. V. Any number of Souldiers propounded, to Order them in Rank and File, according to the reason of any two Numbers given.

SECT. II. Of Castermetation, or Quartering and Encamping of Souldiers.

CHAP. XV. Of GUNNERY.

SECT. I. Of the Names of the Principal Members of a Piece of Ordnance.

SECT. II. Of the Dimension of our Usal English Cannon, and other Ordnance, &c.

PROP. I. How to know the different Fortification of a Piece of Ordnance.

PROP. II. How to know how much Powder is fit for proof, and what for service, for any Piece of Ordnance.

PROP. III. To know what Bullet is fit to be used in any Piece of Ordnance.

PROP. IV. By knowing the proportion of Metals one to another, and by knowing the Weight of one Ball, to know what any other shall weigh.

SECT. III. Of the Qualification of an able Gunner, and necessary Operations before shooting, and in shooting.

PROP. I. How to Tertiate, Quadrate, and to Dispart a Piece of Ordinance.

PROP. II. To know how far any Piece of Ordnance will shoot, &c.

The Use of the Table of Randoms.

SECT. IV. Of Shooting in Mortar-Pieces.

PROP. I. To make Fuses for Bombs, &c.

§

PROP. II.

PROP. III. Finding that a Mortar of 300, being elevated 54°, sends his Bomb 760 88/100 Paces, what Degree of Elevation must that Mortar have, to shoot the Bomb 555 Paces?

§

A TABLE OF Logarithm Numbers, From One to Ten Thousand: Whereby the LOGARITHM OF ANY NUMBER Under Four Hundred Thousand may be readily discovered.

A TABLE OF PROPORTIONAL PARTS, WHEREBY The Intermediate Logarithms of all Numbers, AND The Numbers of all Logarithms from 10000 to 100000 may more readily be found out by the foregoing Table of Logarithms.

A TABLE OF ARTIFICIAL SINES AND TANGENTS To every DEGREE and MINUTE OF THE QUADRANT.

Some Books sold by W. Freeman at the Artichoke next St. Dunstan's Church in Fleet-street.

A Catalogue of Books.

§

[Page]

Vera Effigies Johannis Tayl [...]r

§

[Page]

[Page]

Thesaurarium Mathematicae, OR THE TREASURY OF THE MATHEMATICKS. CONTAINING Variety of usefull Practices in Arithmetick, Geometry, Trigonometry, Astronomy, Geography, Navigation and Surveying.

AS ALSO The Mensuration of Board, Glass, Tiling, Paving, Timber, Stone, and Irregular Solids.

LIKEWISE It teacheth the Art of Gauging, Dialling, Fortification, Military-Orders, and Gunnery; Explains the Logarithms, Sines, Tangents and Secants: Sheweth their use in Arithmetick, &c. To which is Annexed a Table of 10000 Logarithms, Log-Sines and Log-Tangents.

Illustrated with several Mathematical Sculptures on Copper Plates.

By JOHN TAYLOR, Gent.

— Deus regit Astra; feruntur

Illius arbitrio Sydera, Terra, Fretum.

LICENSED, June 26. 1686. Rob. Midgley.

LONDON, Printed by J. H. for W. Freeman at the Artichoke next St. Dunstan's Church in Fleetstreet. 1687.

[...]

[Page]

§

[Page]

[Page]

To the Right Honourable GEORGE Lord DARTMOUTH, Master of the Horse to K. James II.

Master General of His Majesty's Ordnance and Armories, One of His Majesty's most Honourable Privy Council, &c.

This small Mathematical Treasury is humbly Dedicated and presented by

My LORD,

Your Honour's most humble and obedient Servant John Taylor

THE PREFACE TO THE READER.

[Page]

[Page]

HOW admirably profitable the study of the Mathematicks hath been to these British Islands, and to all other parts of the Universe in which any kind of good Learning hath been esteemed and practised, is well known to all wise and judicious men. And indeed it is an undeniable truth, that among all humane Arts and Sciences whatsoever, the Noble Science Mathematical hath obtained the greatest evidence of certainty, as being the Queen of Truth that imposeth nothing on her Subjects but what she proves by most infallible Demonstrations.

Now this prerogative results from the verity and perspicuity of its Principles, which consist of Definitions, Postulats and Axioms. Hence comes it to pass that all Propositions that are proved by those most infallible Precepts are called certain [Page] demonstrative truths; for which cause it hath been the endeavour of sundry Philosophers to make the force of their Arguments (as far as the quality of their discourse woùld admit) amount unto Mathematical Demonstrations, as being the most convincing proof of a Proposition that by humane reasoning can be given.

Now having for divers years (amongst my other Studies) been conversant in the study of the Mathematicks, and for my own private use compiled this Treatise, never in the least intending it should have appeared in publick in this nice and critical Age; but it by chance falling into the hands of some of my Mathematical Friends and Acquaintance, I have at their requests condescended to publish it, though not without a great aversion in my own mind to expose my self in any publick thing.

But this difficulty being overcome, I shall give the Impartial Reader to understand that I have faithfully compiled this Treatise from the best of Authors (and my own Experience) that I have contracted their various Works into this little Cabinet or choice Compendium of the Mathematicks, in which thou shalt find the whole Subject clearly and intelligibly handled: I have used a plain and easie method: I have laboured to [Page] be as plain and perspicuous as possible: I have applied such Examples to each as may best demonstrate their Operation, be most easie for memory, and applicable to practice; here is indeed Multum in Parvo, the whole Marrow of the Mathematicks is in this Tract afforded thee, which is as a true and Golden Key to unlock the choicest mysteries in those Arts contained.

Thus, Reader, I have laid my labours before thee, and must intreat thee to use me as thou wouldst be done by, which is the Spontaneous act of every good man.

But if this shall chance to fall into the hands of any curious conceited person, who thinks himself wiser than the rest of the world, and so he beginneth enviously to carp hereat, and like a Countrey Cur bark at my backside, to him I shall in modesty onely say, — Facilius est unicuivis nostrum aliena curiosè observare: quam propria negotia rectè agere. 'Tis much easier for those captious Readers to carp than to copy.

If the Collections of Authors shall offend any, and so procure the same censure with the Jackdaw, as our learned Poet hath long since cautioned against —

Ne, si fortè suas repetitum venerit olim Grex avium plumas, moveat cornicularisum, Furtivis nudata coloribus.

[Page]

I would have such contentious Readers know that I have robbed no man of the honour of his Works, but have given to each his due; only I have borrowed some choice things of them, which is no more than what the most learned have always done.

Thus, Courteous and Impartial Reader, 'tis only for thee that I have taken these pains, and have submitted to the publication hereof, and it is to thee the future parts of my study shall be serviceable, hoping that thou wilt find success in all thy Studies according to thy desire and endeavour; which are and shall be the hearty wishes of him who is

London, July 12. 1686.

Thine and Urania's Servant, John Taylor.

To the READER.

[Page]

BEing desired to peruse this Mathematical Treasury, accordingly to gratifie the Request of my Friend I did, and I must confess with no small satisfaction to my self to see so much Practical Matter of usefull Mathematical Arts so neatly and compendiously digested into this Portable Volume; 'twill be usefull not only to Learners and meer Tyro's, but to others also who have made some considerable progress in these Studies. 'Tis well Methodiz'd, very Concise, yet Plain and Perspicuous, so that any person of a pregnant fancy, may without a Tutour (in some reasonable time) wade through the whole, or any part thereof, and such as would be more expeditious may take the assistance of a Teacher to instruct them.

The Author is wholly a stranger to me, but to give him his due, in my opinion he has discharg'd himself like a Master in these Arts, and an Ingenious Mathematician, to whom I return thanks for this his generous offer in presenting his Mathematical Treasury to the Publick, and remainFrom my House in Baldwin's Court in Baldwin's Gardens, over against the Old Hole in the Wall.

October 25. 1686.

A true Lover of the Mathematical Sciences, and all such that really delight in those pleasing (but usefull) Speculations, Henry Coley.

§

[Page]

Courteous Reader,

I Have perused this Treatise, and find that the Author has in every respect discharged himself like an Artist; the Work throughout the whole, is very plain and easie, nothing being omitted that might render it Intelligible to the meanest capacity: And indeed, I know not any Treatise of this nature extant that is more Practically handled, so that I doubt not but that it will be very serviceable to the Publick; and that thou in particular mayst find incouragement in the perusal thereof, is the hearty wish of him who is

Thine and Truths Servant, John Hawkins, Philomath.

Octob. 30. 1686. 2h. 40' P. M.

To his learned and ingenious Friend Mr. John Taylor, in the deserved Praise of his Excellent Book intituled The saurarium Mathematicae.

[Page]

ATlas and Hercules whom Poets feign,

The heavy load of the Earth to sustain:

If so? great Toyl and Labour then they took,

Yet not so much as thou hast in thy Book.

Not like to them thy Labours, fictions are,

Thy works so true ingenious and so rare,

That seldom yet such works from man did flow,

For thou by them dost teach us all to know

The secrets of all Sciences and Art,

Which freely unto us thou dost impart.

Thou shew'st us how Numbers to understand,

And how the Speech of Numbers to command.

Geometry, the Queen of truth, did cease

The Egyptian trouble, and did cause a peace;

When proudly Nile had overflown their ground,

And all their Bounds and Land-marks did confound:

By it, each man his proper Right did gain,

And Peace by it great Egypt did obtain.

This Art so conspicuous thou hast made,

That to thy Glory it can never fade.

By Sines, Tangents and Secants thou dost show

Us all the parts of Triangles to know.

Thy lofty Genius viewed the Stars on high,

So that full well thou know'st Astronomy.

[Page]

All motions of the Sun and Planets thou

Dost understand, thy works do shew it now.

For thou such Rules and Precepts dost apply,

In this thy Book unto Astronomy;

Demonstrated by rules so rare and plain,

That he's a Dunce that can't it now obtain.

Thus having view'd the Spheres of Heaven well,

Then on our Mother Earth thy Genius fell.

Thou viewd'st her round, ev'n by inspecting all

The known parts of the Cosometick Ball.

And here in this thy Book thou let'st us see,

How Nations all most disagreeing be.

The Seaman he adores thee as his Friend,

So liberal unto him thy Art doth lend;

And thou from him wilt not thy Talent hide,

Thy Book's a Light-house Mariners to guide.

Surveying thou dost teach and that so plain,

That any one that Art may well obtain:

And by that means Injustice to disband,

Attending Lord and Tenant of the Land.

To thee the Brooks and Springs do all submit,

And they will glide to that place thou think'st fit.

Thou shew'st Mechanick well to apprehend,

To measure Board or Glass, nay as a Friend,

Teachest them how both Timber round and square,

And Stones to measure of what kind so e're.

Therefore to thee they praises still will give,

And tho 'thy Body's dead thy Fame shall live.

The Art of Gauging thou dost plainly teach,

And farther far than worthy Oughtred reach

Into the Mystery of that curious part,

And noble Branch of Mathematick Art.

Thou measurest the course of times short stay,

Thus Dials shew us how time flies away;

[Page]

That thereby we may mind our fading breath,

And preparation make for certain death.

Thy Book's also prepar'd Mars to withstand,

In raising Forts for to defend the Land.

In ordering Armies in Battail aray,

And them Encamping when they make a stay.

The Gunner's Magazine lies in this Tract,

From whence directions he may have to sack,

Or storm a Town, or batter down a wall,

Or make a Breach and at the joyfull fall

Of Turrets high Huzza's to make for joy,

And entring in, his Enemies destroy.

These curious Arts with more than here are nam'd,

In this rich treasury so neatly fram'd,

Our friendly Authour doth to all impart,

Wishing success, and that with all his Heart.

But stop my Muse let us not be so rude,

We'll only wish him well and so conclude.

Maist thou, O Authour of this Treasury,

Reap to thy self profit and praise thereby;

And maist thou ever, ever happy be,

That we more of thy Learned works may see.

Live thou in splendid comfort to thy end,

So prays thy humble Servant, and thy Friend,

July 20. 1686.

Geo. Barrow.

To the Learned Authour my much respected Friend Mr. John Taylor on his Herculean labours in the Composure of this Excellent Mathematical Treasury.

[Page]

I ncrease our Muse, rouze up ye Sisters nine,

O n us bestow your Art that we may praise

H is works, his worth and his real design;

N ot honour vain, but skill aloft to raise.

Vain Glory's to him but a trifling Toy,

'Tis Art alone, 'Tis that which is his Joy.

T he Earth he hath trac'd, the Spheres of Heav'n view'd

A nd Stars and Seas whose billows loud do roar;

Y et is he not nor can he be so rude,

L ike many others to lock up his store,

O h he doth not! his Treasure ope 'doth stand:

R eceive it as a Jewel from his Hand.

Our noble Friend and Authour what's thy due?

Honour thou slight'st, Treasure's too vain for you.

Thy mind is fixt on Sciences above,

Thou art Urania's favourite and love.

Thou knowst her ways, her Art's at thy command,

She smiles upon thee, guides thee by the hand:

For which thy Name we will extoll and praise,

As far as Phoebus sends his golden raies.

Therefore in happiness let thy time run,

And rest in Peace when that thy Period's come.

Sept. 27. 1686.

Tho. Robinson.

An Acrostick on the Name of my much respected and ingenious Friend Mr. John Taylor.

[Page]

I f Mathematicks be the Art to teach,

O by thy Book the Learned then may reach

H eavens Poles, and Circles without doubt or fear,

Not to find out each Star it's Hemisphere.

T hough Archimedes hath much glory got

A mongst the Syracusians, why not,

Y ea Statues be erected to thy Name?

L et Eagles wings towre and soar thy fame.

O happy maist thou be, and this thy Book

R eaders instruct when e're they in it look.

London, July 29. 1686.

Fran. Pierce.

ADVERTISEMENTS.

A LL Gentlemen, or other Persons that shall have occasion for any sort of Mathematical Instruments, either for Sea or Land, may be furnished by John Worgan, Mathema [...]ical Instrument-maker; under St. Dunstan's Church [...]n Fleet-street, London.

At St. George's Church in Southwark are taught Writing, Arithmetick, Merchants Accounts, Geometry, Trigonometry, Astronomy, Navi [...]ation, Surveying, Dialling, Gauging and Gun [...]ery by John Hawkins, Philomath.

[Page]

Arts and Sciences Mathematical, profess'd and taught by HENRY COLEY, Philomath. at his House in Baldwins Court over against the Old Hole in the Wall, in Baldwins Gardens near Grays-Inn-Lane.

ARITHMETICK in
Whole Numbers and Vulgar Fractions.
Decimal, and by Logarithms.

GEOMETRY.
The Rudiments thereof, also the Demonstration and Practice, according to the best Authours.

ASTRONOMY.
The use of the Globes Coelestial, and Terrestrial.
To project the Sphere in Plano to any Latitude several ways. To calculate the Longitude and
Latitude of the Planets, with their Declination and Ascension. Also the true Time, Quantity, and Duration of Eclipses of the Luminaries for any time past or to come.

TRIGONOMETRY.
Or the Doctrine and Calculation of Triangles, both. —
Plain and Spherical.
With the Application of the several Cases thereof in the most useful Questions in. —
Geometry. Astronomy. Geography. Navigation. Dyalling, &c

NAVIGATION.
In either of the three principal kinds of Sayling, viz.
by the
Plain and
Mercator's Chart
Great Circle.

DYALLING.
Geometrically
Instrumentally
Arithmetically
by
The Sector, and other convenient Scales.
The Logarithms, Sines & Tan.

SURVEYING.
Several ready ways to measure a Plat, and divide Land, &c. also the taking of Altitudes, Profundities, Distances, &c. together with the Mensuration of all manner of
Superficies, as Boards, Glass and Pavement: also all Solids, viz. Timber, Stone, &c. Regular and Irregular.

GAGING.
To find the just quantity of Liquor in any Cask, whether full or partly empty. Also the content or solidity of Brewers Vessels, &c.
Tuns, Coppers, Backs, Coolers, &c.

ASTROLOGY.
In all its parts, and according to the best Authors, with several varieties therein, not known to every Professor. Non nobis nati sumus.

The Contents.

[Page]

CHAP. I.
OF Arithmetick | Page 1.
CHAP. II.
The Explanation and Use of the Table of Logarithms. | p. 18.
CHAP. III.
The Explanation of the Sines, Tangents and Secants. | p. 28.
CHAP. IV.
Of Geometry. | p. 32.
CHAP. V.
Of Trigonometry, or the Doctrine of Triangles. | p. 59.
CHAP. VI.
Of Astronomy. | p. 96.
CHAP. VII.
Of Geography, with a Geographical Description of the Earthly Globe. | p. 122.
CHAP. VIII
Of Navigation. | p. 186.
CHAP. IX.
Of Surveying. | p. 214.

[Page]

CHAP. X.
Of Measuring Boards, Glass, Tiling, Paving, Timber, Stones and Irregular Solids, such as Geometry can give no Rule for the Measuring thereof. | p. 242.
CHAP. XI.
Of Gauging. | p. 250.
CHAP. XII.
Of Dialling. | p. 255.
CHAP. XIII.
Of Fortification, according to the modern and best ways now used by the Italian, Dutch, French and English Inginiers. | p.279
CHAP. XIV.
Of Military Ordèrs, or the Embattelling and Encamping of Soldiers. | p. 301.
CHAP. XV.
Of Gunnery. | p. 306.
A Table of Logarithms. | p. 337.
A Table of Proportional Parts. | p. 401.
A Table of Artificial Sines and Tangents. | p. 417.

Arithmetick.

[Page]

CHAP. I. Of ARITHMETICK.

ARITHMETICK isOmnia quaeeunque à primaeva rer [...]m natura constructa sunt, Numcrorum vid [...]ntur ratione formata. Hoc enim fuit principale in animo conditor is exemplar. Boetiu [...] Arith. lib. 1. cap. 2. an Art of numbring well, for as magnitude, or greatness, is the subject of Geometry, so is multitude, or number, that of Arithmetick.

I shall not in this place trouble you with the first Rudiments of Arithmetick, as Numeration, Addition, Substraction, Multiplication, and Division: because they are already largely handled by many, as Mr. Leybourn, Mr. Wingate, and divers others, and also that then this Book would swell to too big a bulk for the Pocket, and so my design would be frustrated; I shall therefore only propose and operate [Page] some principal Propositions, that are of Special moment in Arithmetick, and which most immediately concern the other following parts of this Treatise.

SECTION I. The Explication of some Arithmetical Propositions.

PROPOSITION I. To three numbers given, to find a fourth in a Direct proportion.

To operate this proportion Multiply thē third term, by the second term, and their product divide by the first term, the Quotient shall be a fourth term required. Examp. 1. Admit the Circumference of a Circle whose Diameter is 14 parts be 44 parts, what is the Circumference of that Circle, whose Diameter is 21 parts? Now according to the Rule if you multiply the third term 21, by the second term 44, it produceth 924; which divided by the first Term 14, the Quotient is 66, and so the Circumference of the Circle, whose Diameter is 21, will be 66 parts, and so for any other in a direct proportion.

PROP. II. To three numbers given, to find a fourth in an Inversed proportion.

[Page]

To operate this proportion, Multiply the first term, by the second term, and their product divide by the third term, the Quotient is the fourth term required: Examp. Admit that 100 Pioneers, be able in 12 hours to cast a More of a certain length, breadth, and depth; in what time shall 60 Pioneers do the same? Now if according to the Rule, you Multiply the first term 100, by the second term 12, their product is 1200; which divided by the third term 60, the Quotient is 20, so I say that in 20 hours, 60 Pioneers shall do the same, and so for any other in an Inversed proportion.

PROP. III. To three numbers given, to find out a fourth in a Duplicate proportion.

The nature of this proposition is to discover the proportion of Lines, to Superficies, and Superficies, to Lines; for like Plains are in a duplicate Ratio; that is as the Quadret of their Homologal sides; therefore to Operate any Example in this proportion, Square the third term, and its square multiply by the second Term, their product divide by the square of the first Term, the Quotient is the 4th. term sought; Examp. Admit there be two Geometrical squares; now if the side of the greater [Page] square be 50 feet, and require 3000 Tiles to pave it; what number shall the lesser square require, whose side is 30 feet? To operate this according to the Rule, I square the third Term 30, whose square is 900: then I multiply it by the second Term 3000, its product is 2700000, which divided by 2500, the square of the first Term 50, the Quotient is 1080, and so many Tiles will pave the lesser square, whose side is 30 feet.

PROP. IV. To three numbers given, to find a fourth in a Triplicate proportion.

THE nature of this proposition is to discover the proportion of Lines to Solids, and Solids to Lines; for like Solids, are in a Triplicate Ratio, that is to the Cubes, of their Homologal sides: Therefore to operate any Question in this proportion, Cube the third Term, and his Cube multiply by the second Term, and their product divide by the Cube of the first Term; the Quotient is the fourth Term sought. Examp. Admit an Iron Bullet whose diameter is 4 Inches, weigh 9 pounds; what is the weight of that Bullet whose Diameter is 6 Inches? Now to operate this proportion; first according to the Rule I Cube the third Term 6 whose Cube is 216, then I multiply its Cube by the second Term 9, the product is, 1944, which divided by 64, the Cube of the first Term; the Quotient is 30 24/64 pounds which is equal unto 30l. 6 ounces: which is the weight [Page] of the propounded shot; and so for any other.

PROP. V. To two numbers given, to find out a third, fourth, fifth, sixth, &c. Numbers in a continual proportion.

To operate this proportion, you must multiply the second number by it self, and that product divide by the first Term, the Quotient is a third proportional: Again you must multiply the third Term by it self, and its Quadret divide by the second Term, the Quotient is a fourth proportional, and so after this manner a fifth, sixth; or as many more proportionals as you please may be found: Examp. Let it be required to find six numbers in a continual proportion to one another; as 4 to 8. To operate this first according to the Rule, I multiply the second Term 8 by it self the product is 64, which divided by the first Term 4, the Quotient is 16: so is 4, 8, and 16 in a continual proportion; And so observing the Rules prescribed, proceed in your operation untill you have found your six numbers in a continual proportion; which in this Example will be 4, 8, 16, [...]2, 64, and 128, and so will you have form'd six numbers in a continual proportion.

PROP. VI. Between two numbers given, to find out a mean Arithmetical proportional.

[Page]

THIS proposition might be performed without the help of the rule of proportion: nevertheless because it conduceth to the Resolution of the next ensuing proposition, I insert it in this place; To operate it this is the Rule: add half the difference of the given Terms, to the lesser Term, so that Agragate, is the Arithmetical mean required: Examp. Admit 20 and 50 to be the two numbers propounded: Now to operate this proposition, first according to the Rule, I find that the difference of the two given Terms 20, and 50, is 30, whose half is 15, which being added to the lesser Term 20, it makes 35, so is 35, a mean Arithmetical proportion betwixt 20, and 50, given.

PROP. VII. Between two numbers given, to find out a mean Musical Proportional.

BOETIUS hath this Rule for it, wherefore take his own words:In his second Book of his Arithmetick and the 38 Chapter; where he saith that this proportion hath, Magnam vim in Musici modulaminis temperamentis, & in Speculatione naturalium questionum: i. e. Great force in Musical composition, (or in the Composure of Musick) and in the discovery of the Secrets of Nature. saith he, ‘Differentiam terminorum in minorem terminum multiplica, & post junge terminos, & juxta cum qui inde confectus est; committe illum numerum, [Page] qui ex differentiis & termino minore productus est, cujus cum latitudinem inveneris, addas eam minori termino, & quod inde colligitur medium terminum pones.’ That is, Multiply the difference of the Terms, by the lesser term, and add likewise the same Terms together: this done if you divide the product, by the sum of the Terms, and to the Quotient thereof, add the lesser Term; the last Sum is the Musical mean desired: Examp. Admit the two numbers given be 6, and 12. I say that if the difference of the Terms which is 6, were Multiplied by the lesser Term 6, it would produce 36; then if you add the two terms 6, and 12, together: their sum would be 18, now if you divide 36, by 18, the Quotient is 2; lastly if to the Quotient 2, you add the lesser Term 6, the sum thereof will be 8, which is a Mean Musical proportional required.

PROP. VIII. How to find the Square-Root of any whole number, or Fraction.

Defin. To Extract the Root of any Square number propounded, is to find out another number, which being Multiplied by it self, produceth the Number propounded. Now for the more easie and ready Extraction of the Square-Root of any number given, This Table [Page] here under annexed will be usefull; which at first sight giveth all single Square numbers, with their respective Roots.

ROOT. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
SQUAR. | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81

The Explication of the Table.

In the uppermost rank of this Table, is placed the respective root of every single Squarenumber, and in the other the single Squarenumbers themselves; so that if the Root of 25 were demanded, the Answer would be 5, so the Square root of 49, is 7, of 81 is 9; and so for the Rest, and so contrarily the Square of the Root 5 is 25, of 7 is 49, of 9 is 81, &c.

Example: If the Square root of 20736, were required, first they being wrote down in order as you see, draw the Crooked-line,As you use to doe in Division to represent the Quotient. then to prepare this or any other number for Extraction, make a point over the place of Unites; and so on every other figure towards the Left-hand; as you see in the Margent. [Page] Then find the Root of [...] the first Square 2, which is 1; place it in the Quotient, and also under 2; then draw a line, and substract 1 from 2, there remains 1; which place under the line, then to the last remainder 1, bring down the next Square 07; and then there will be this number 107, which number I call a Resolvend: Then double the Root in the Quotient 1, whose double is 2, which 2 place under the place of tens in the Resolvend, under 0; so is this 2 called a Divisor; and 10 called a Dividend.

Then demand how often the Divisor 2, can be had in the Dividend 10, it permitteth but of 4, which place in the Quotient, and under 7 the place of Unites in the Resolvend, and there will appear this number 24; Then Multiply this 24, by 4, (the last Square placed in the Quotient) it produceth 96, which place orderly under 24, as you see, and this 96 is called a Ablatitium; (but some calleth it a Gnomon:) then draw a line under it, and substract 96, the Ablatitium, out of the Resolvend 107, there remains 11, which place orderly under the last drawn line, then thereunto bring down the next Square 36, so will there be a new Resolvend 1136; then double the whole Root 14 in the Quotient, whose double is 28; [Page] place it under the Resolvend 1136 as was afore directed; so shall 28 be a new Divisor, and 113 be a Dividend; then I find the Divisor 28 can be had in the Dividend 113, 4 times, which four place in the Quotient, and under the place of Unites in the Resolvend, so there appeareth this number 284, which number, multiplyed by 4, the last figure in the Quotient, produceth a new Ablatitium 1136; which place orderly under the Resolvend 1136, and then draw a line, then substract the Ablatitium 1136, from the Resolvend 1136; and the remainder is 00, or nothing: and thus the work of Extraction being finished, I find the Root of the Square number 20736, to be 144; and so must you have proceeded gradually step by step, if the number propounded, had consisted of some 4, 5, 6, or more Squares; still observing the aforegoing Rules and Directions.

NOTE.

BUT when a whole number, hath not a Root exactly expressible by any rational or true Number, then to find the fractional part of the Root very near; To the given whole number annex pairs of Cyphers, as 00, 0000, or 000000, then esteem the whole number, with the Cyphers both annexed thereunto, as one intire whole number: and Extract the Root thereof according to the foregoing Directions, then as many points as were placed over the Integers, so many of the first figures in the Quotient must be taken for Integers; and the remainder for the Roots fractional part in Decimal [Page] parts, and so you may proceed infinitely ne [...]r the true Root of a Number.

To Extract the Square-Root of a Vulgar or Decimal Fraction, and a Mixt-number.

First if the Fraction propounded be not in its least Ter [...], reduce it, and then by the Rules aforegoing, find the Root of the Numerator for a new Numerator; and of the Denominato [...] for a new Denominator; so shall this n [...]w Fraction be the Square-root of the Vulgar Fraction propounded, so the Square-root of 16/ [...] is 4/ [...]

But many times the Numerator and Denominator of a Vulgar Fraction hath not a perfect Square-root; to find whose Root infinitelyThe manner of Extracting the Square-root of a Decimal Fraction. near, you must reduce it into a Decimal Fraction, whose Numerator must consist of an equal number of places, to wit, 2, 4, 6, &c. Then Extra [...] the Square-root of that Decimal, as if i [...] were a whole number, and the Root that procee [...]eth from it is a Decimal Fraction, pre [...]ing the Square-root of the Fraction proposed, infinitely near: so the Root of 13/16 (whose De [...]ma is, 81250000) will be found to be [...] which is very near, for it wanteth not 1/10000 of an Unite of the exact Squareroot, of 13/16 propounded.

Now having a Mixt Number propounded whose Ro [...] is required, [...]o find which reduce itThe manner of Extracting the Square root of a M [...]x [...]-Number. into an improper Fraction, and then Extract the [Page] Root thereof as before. Suppose the Number propounded be 75 24/54; its improper Fraction is 679/9, whose Square-root I find to be 26/3. or 8 ⅔, very near, &c. But if it had not an Exact Square-root, then reduce the Fractional part of the given Mixt-number into a Decimal Fraction, of an even number of places, and then annex this Decimal to the Integers, and so Extract the same, as a whole number; and observe that so many points as were set over the Integers, so many of the first figures in the Quotient must be esteemed Integers; and the Remainder for the Roots Fractional part.

PROP. IX. How to find the Cube-Root of any whole Number, or Fraction.

Defin. To Extract the Cube-Root of any Number propounded, is to find out another Number, which being multiplied by it self, and that product by the number again, shall produce the number propounded; Now for the more easie and ready Extraction of the Cube-root of any number propounded, this Table hereafter annexed will be usefull, which at first sight giveth the Cube-root of any whole number under 1000; which are called single Cube-numbers.

[Page]

ROOT. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
CUBE. | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729

The Explication of the Table.

In the uppermost rank of the Table is placed the respective Roots of every single Cube, and in the other the respective single CubeNumbers; for if the Cube-root of 512 were desired, the Answer would be 8, of 64 is 4; and so of the rest: and if the Cube of the Root 7 were desired, it would be found 343; of 9 it would be 729, &c.

Examp. Admit the Cube root of the Number 262144, were required, first they being wrote down in order as you see, draw the Crooked-line.

Then place a point overAs you use to do in Division, to represent the Quotient. the place of Unites, and another over the place of Thousands; and so on still intermitting two places between every adjacent point; and observe that as many points, as in that order are placed over any number propounded, of so many figures doth the Root [Page] consist of: so that in this [...] Example, there being two points, therefore the Root consisteth of two places as you see in the Quotient; Now first find the Root of the first Cube 262; which permitteth but of 6, place 6 in the Quotient, and subscribe its Cube 216, under 262, and then draw a line under it, and substract 216, out of 262, and the remainder is 46, which place in order under the last drawn line as you see. Then to the Remainder 46, bring down the next Cube-number 144, so will there appear 46144, which I call a Resolvend: then draw a Line under it, and square the Number in the Quotient 6, whose square is 36; Then Triple it and it will be 108, Then subscribe this Triple square 108, under the Resolvend, so that the place of Unites in the Triple Square 8, may stand under 1 the place of Hundreds in the Resolvend: Then Triple the Root in the Quotient 6, whose Triple is 18, Then subscribe the Triple 18, under the Resolvend, so that the place of Unites 8 in the Triple, may stand under 4 the place of Tens in the Resolvend, and so draw a Line under neath it, and add the Triple Square 108, and the Triple 18 together in such order as they [Page] stand, their Sum is 1098, which may be called a Divisor, and the whole Resolvend 46144, except 4 the place of Unites a Dividend; then draw another line.

Then seek how many times 1098 the Divisor, can be had in 4614 the Dividend, it permitteth but of 4, which subscribe in the Quotient; Now Multiply the Triple square 108, by 4, it produceth 432, which in order subscribe under the Triple square 108: Then square 4, the figure last placed in the Quotient, whose square is 16; and Multiply it by 18 the Triple, it produceth 288, which subscribe under the Triple orderly, then subscribe the Cube of 4 (last placed in the Quotient) which is 64, in Order under the Resolvend. Then draw a [...]ine underneath it, then add the three numbers, viz. 432, 288, and 64, together in such order as they are placed, their sum is 46144: Then draw another line under the Work, subtracting the said total 46144, from the Resol [...]end 46144, there remains 00, or nothing, which remainder subscribe under the last drawn [...]ine, thus the work being finished I find the Cube root of 262144 the number propounded, to be 64: And thus you must have proceeded orderly step by step, if the number propounded [...]ad arisen to some 3, 4, 8, 10, or more places, observing the direction prescribed untill all had [...]bserved compleated.

NOTE.

BUT when a whole number, hath not a Cube-root expressible by any true or Rational [Page] number, then to proceed infinitely near the Exact truth annex to the number Tenaries of Cyphers as 000, 000000, 000000000, &c. then esteeming the whole number with the Cyphers annexed as one intire whole Number, Extract the root thereof, as is afore taught. Then as many points as were placed over the Whole Number, so many places of Integers will there be in the Root, and the rest expresseth the Root his Fractional part very near.

To Extract the Cube-Root, of any Vulgar or Decimal or Mixt Fraction consisting of a Whole Number and a Fraction.

To Extract the Cube-root of any Vulgar Fraction, you must first reduce it into his least terms, and then according to the former directions Extract the Cube-root of the Numerator, the Root found shall be a new Numerator so likewise the Root of the Denominator shall become a new Denominator; so shall this new Fraction be the Cube-root of the Fraction propounded, so I find the Cube-root of 8/125 to be 2/51 and so for any other Vulgar Fraction.

But many times the Numerator, and Denominator,The manner of Extracting the Cube-Root of a De [...]imal Fraction. hath not a true Root: Then to find the Root thereof infinitely near, you must reduce the Fraction given, into a Decimal, whose numerator is Tenaries of places, and then Extract the Root according to the former Directions, so shall the Root found, be a Decima [...] Fraction expressing near the Cube-root of th [...] [Page] Fraction propounded, so I find the Root of 8/12 or ⅔, whose Decimal is, 666666666, to be, 873/1000 very near the Root of 8/12 or ⅔ propounded.

Now having a Mixtnumber propounded,The manner of Extracting the Cube-Root of a Mixt-number. whose Root is required, first reduce it into an Improper Fraction, and then Extract the Cuberoot thereof, as is afore directed, so the Cuberoot of 12 10/27, Improper 343/27, will be found to be 7/3 or 2 ⅓.

But if it hath not an Exact Cube root, Then Reduce the Fractional part of the given Mixtnumber into a Decimal Fraction, which shall consist of Tenaries of places, Then to the whole number annex the Decimal Fraction, and Extract the Cube-root of the Whole, and observe that so many points as are over the Integers, so many of the first places in the Quotient must be Esteemed Integers, and the rest Expresseth the Fractional part of the Root in Decimal parts of a Fraction, so the Cube-root of 2 ⅜, Decimal 2, 375000000 &c. will be found to be 1, 334, or 1 334/1000, and is very near the true Root, and so for any other Mixt-number of this nature.

CHAP. II. The Explication, and use of the Tables of LOGARITHMS.

[Page]

SECT. I. The Explication of the Tables of the Logarithms, and of parts proportional.

THE Logarithms, were first invented,Defin. found out and framed, by that never to be forgotten and thrice Honourable Lord, the Lord Nepeir: which Numbers, so found out and framed by his diligent industry he was pleased to call Logarithms; which in the Greek signifies the Speech of Numbers, I shall not here trouble you with the manner or the Construction of those Tables of Logarithms but shall first lay down some brief and general Rules, that thereby the better you ma [...] Understand those Tables, and then I shall e [...] plain their manifold uses, in sundry Examples Arithmetical, &c.

PROP. I. Any Number given under 10000, or 100000, to find the Logarithm corresponding thereunto.

[Page]

1. If the number propounded consist of one place whose Logarithm is required to be found, as suppose (5,) look for 5, in the top of the left hand Column under the LetterSignifies the Number, or figure sought. N, and right against 5, and in the next Column under LOG.Signifies the Logarithm answering to the number Opposite. you will find this number or rank of figures, 0698970, which is the Logarithm of the number 5 required.

2. If the number consisteth of two places as if it were 57, look 57 under N, and opposite to it and under LOG. you will find this number 1. 755875, which is the Logarithm of 57, the number propounded.

3. If the number propounded consist of three places as 972, look for 972, under N, and opposite to 972, and under [...]o) the Column, you shall find this number 2. 987666, which is the Logarithm of 972, the number which was propounded.

4. But if the number consist of four places as 685, look the three first figures 168, under the Column N, and opposite to that, and un [...]er 5 at the top of the page, you will find this number 3. 226599, which is the Logarithm of 1685, the number propounded.

[Page]

5. But if the number☞ Note this Rule well, for this explains the use of the Table of proportional parts, printed at the end of this Book. given be above 10000, and under 100000, you may find its Logarithm by the Table of parts proportional, printed at the latter end of this Book. Thus if the Logarithm of 35786, be sought, first seek the Log. of 3578, which will be 553649, and the common Difference under D is 121; with this difference 121, Enter the Table of parts proportional, and finding 121 in the first Column under D, you may then lineally under 6, find the number 72, which add to the Log. of 3578, that is 553649, it produceth, 553712, which is the Log. of 35786 the number propounded: now because the number propounded 35786, ariseth to the place of X. M. therefore there must be the figure 4 prefixed before its Logarithm, and then it will be thus 4, 553712, which 4, is called the Index, as shall be hereafter shewed.

Now before we proceed to find numbers corresponding to Logarithms, it will be necessary to explain the meaning of the first figure to the left hand of any Logarithm placed, Mr. Briggs calleth it a CharacteristickDefinition. The Rule to find the Characteristick or Index appertaining to any Logarithm. or Index, which doth represent the distance of any the first figure of any whole number from Unity, whose Index is 0, a Cypher; so the Index o [...] 10 is 1, and so to 100 whose Index is 2, and s [...] to 1000 whose Index is 3, and so to 10000 whose Index is 4, and so if you persist furthe [...] the Characteristick is always one less in dignity [Page] than the places or figures os the number propounded.

PROP. II.

To find the Logarithm belonging to a Vulgar Fraction, and a Mixt number.

First as is before shewed if it be a Vulgar Fraction, find the Log. of the Numerator, and the Log: of the Denominator, then substract the Log: of the Numerator, from the Log: of the Denominator, the remainder [...] is the Log: of the Fraction propounded: Now if you would find the Logarithm of 5/7, do as is prescribed whose Log. I find to be 0. 146121, Now to find the Log. of a Mixt Number, reduce it into an Improper Fraction, and then do as before, so the Log of 15 ⅖, Improper 77/5;, is 1, 187, 52, and so do for any other Mixt number.

PROP III. A Logarithm propounded to find the whole, or Mixt number, corresponding thereunto.

For the more speedy finding the number, answering unto the Logarithm propounded, observe that if the Index be 0, then the Number sought may be found between 1 and 10; If 1, [Page] between 10 and 100; if 2, between 100, and 1000; if 3 between 1000 and 10000, and so on still observing the Rules of the Characteristick, or Index, therefore loo, in the Table untill you find the Logarithm proposed, and against it in the Margent according to the aforegoing directions under N, you shall find the number belonging thereunto. This Rule holds in force in Mixt Numbers also.

Thus.
0. 845098
1. 556302
2. 130334
3. 980276
Are the Logarithms of,
7
36
135
9556

NOTE.

But if you cannot find the Logarithm exactly in the Table, as in many operations it so hapneth, you must then take the nearest Logarithm Number to the Logarithm propounded, and so take the number belonging thereto [...]or the desired number.

SECT. II. Of the Admirable use of the Logarithms in Arithmetick.

[Page]

PROP. I. To Multiply one number by another.

Admit 90, be to be multiplied by 42, what is the product? To find which first find the Log. of the Multiplicand 90, whose Log. is 1. 95424: Then find the Log. of [...] the Multiplier 42, whose Log: is 1 62324, then add these two Log: together, viz. the Log: of the Multiplicand, and Multiplier, their sum is 3, 57748, which is the Log: of 3780, the product of 90; and 42, Multiplied together.

PROP. II. To Divide one number by another.

Admit the Dividend (or number to be divided) be 648, and the Divisor 72, what is the number that the Quotient shall consist off? To find which, first write down the Logarithm of the Dividend 648, which is 2. 81157 and also [Page] write down the Logarithm of the Divisor 72, which is 1. [...] 85733. Now substract the Log: of the Divisor, out of the Log: of the Dividend, the remainder is 0. 95424, which is the Logarithm of 9, so I conclude that the Divisor 72, is contained in the Dividend 648, 9 times, and so do for any other.

PROP. III. To find the Square-Root of a Number.

Admit it be required to Extract the SquareRoot of the Number 144, to perform which first write down the Log: of 144 which is 2. 15836. Then take the half thereof which is 1. 07918 which number 1. 07918, is the Log: of 12, the Root of 144 propounded, and so do for any other.

NOTE.

Now on the Contrary by doubling the Log. of any number, you have the Geometrical Square thereof.

PROP. IV. To find the Cube Root of any Number.

[Page]

Admit it be required to Extract the Cube Root of 1728, to perform which, First write down the Log of 1728 which is 3. 23754, then take the third part thereof which is 1. 07918, which is the Log. of 12; which is the Cuberoot of the Number propounded 1728, and so for any other. Note on the contrary if you multiply the Log. of any Number propounded by 3, it produceth the Log. of the Cube thereof.

PROP. V. A Summ of Money being forborn for any number of years, to find how much it will amount unto, reckoning Interest on Interest, according to any Rate propounded.

Admit 300 pounds Sterling, be put out for 4 years, for Compound Interest at 6 l. per Cent. what will it amount to when the four years are expired? To find which substract the Log of [...]00 l. the principal, whose Log. is 2. 477121, out of the Log. of 318 l. Principal and Interest for a year whose Log. is 2. 502427, the remainder is 0. 025306, which being multiplyed by 4, the number of years of its continuance, produceth 0. 101224, which added to the Log. of the principal 300l. to wit, to 2. 477121, makes [...], 578345, which is the Log. of 378 l. 14 s. 10d. 2q. very near, and so much will 300 l. amount to.

PROP. VI. A Summ of Money being to be paid hereafter, to find what it is worth in [...]eady Money.

[Page]

Admit 100 pounds Sterling, to be paid at 30 years end; I demand how much it is worth in ready Money? after the rate of Interest of 6 l. per Cent. To find which substract the Logarithm of 100 the principal, whose Log. is 2. 000000 from the Log. of 106 Principal and Interest, whose Log. is 2. 025306, the remainder is 0 025306, which Multiplyed by 30 the number of years to succeed, produceth 0. 759180, which substracted out of 2. 000000, leaveth 1. 240820, which is the Log. of 17 411/1000, which sheweth the said 100 l. is worth but 17 l. 8s. 2d 3q. fere.

PROP. VII. A yearly rent, or Annuity to continue any number of years, to find what it is worth in ready Money, at any Rate of Interest propounded.

What is 100 pound per annum to continue 30 years, worth in ready money at 6 l. per Cent. To find which first substract the Log of 100 l. the principal, which is 2. 000000 from the Log. of 106 l principal and interest for a year, whose Log. is 2. 025306 the remainder is 0. 025306: Then Multiply 0. 025306, by 30 the number of years of its continuance, it produceth the number 0. 759180; Then Divide 100 l. by 6 [Page] the rate of interest and the Quotient is 16 6667/10000, &c. which: 16 6667/10000, is the proportional parts of 100 l. the principal, then add the Log. thereof which is 1. 221829 to the former Log. 0. 759180 it produceth 1. 981009, which is the Log. of 95 7215/10000 parts the Arrearages with the said some for that Time, then from those Arrearages 95 7215/10000, substract the parts proportional of 100, to wit 16 6667/10000, the remainder is 79 [...]48/10000, which is the bare Arrearages for that proportional part; Then take the Log. of 79 0548/10000, which is 1. 897929, out of the which take the Log. found by Multiplication of years, to wit 0. 759180, there remains 1. 138749, which is the Log. of the value of the Arrearages in ready money, Then to the Log. 1. 138749, add the Log. of 100 l. principal, 2. 000000, it produceth this number 3. 138749; the Log. of 137 6 48/100, reduced is 1376 l. 9. sh. 7d. 80/100 or ⅘ fere: and so much is the said Annuity worth in ready money.

CHAP. III. The Explication of the SINES, TANGENTS, and SECANTS.

[Page]

SECT. I. Of Right Signs, Tangents, Secants, Cosines, Tangents, and Secants: Of any Arch, or Angle of a Triangle.

PROP. I. To find the right Sine, or Tangent of any Arch or Angle of a Triangle containing any number of Degrees and Minutes.

IF the Angle orAnd here'tis necessary to understand, that every Circle is supposed to be divided into 360 Equal parts which are called Degrees, and every of those Degrees into 60 Minutes, and every Minute into 60 Seconds, and every Second into 60 Thirds, &c. so that a Semi-Circle contains 180 Degrees, and a Quadrant 90 Degrees; Now an Arch or Angle of a Triangle, is the Intersection of its two sides, and the measure thereof, is an Arch of a Circle, which cutteth each of the two sides equidistant from the Angular point, (which is the Center.) Now the Logarithm Sine, or Tangof any such Arch of a Triangle, containing any Number of Degrees or Minutes of the Quadrant, may be found in the Tables, printed at the End of this Book, where they are plainly expressed, and are found as directed in the precedent Rules. Arch of the Triangle propounded beless than 45 Deg. the Sine, [Page] or Tangent belonging thereunto, is found in the Column under the Title SINE, or TANGENT, at the top of the Table; and if there be any Minutes annexēd unto the Degrees, you must find them out in the first Column under M. signifying Minutes, and opposite to those Minutes, and under the title aforesaid, you shall have the Logarithm of the Sine or Tangent, of the Arch or Angle required.

But if the Arch or Angle of a Triangle exceed 45 Degrees, you must then look for the Sine or Tangent belonging thereunto, in the bottom of the said Table, and if thereunto are Minutes annexed, you must look for them in the first Column to the Right hand under M. and so opposite to those Minutes in the Column above the Title, Sine, or Tang; there have you the Log. of the Sine, or Tangent, of the Arch or Angle, of the Triangle propounded.

[Page]

Examp. Suppose it were required to find the Log-Sine or Log-Tangent; of an Angle of 25 D. 37 M. whose Log Sine, whereof according to the former directions I find to be 9. 635833. and Tangent thereof to be 9. 680768. and so for any other under 45 degrees.

Again, suppose it were required to find the Log-Sine or Log-Tangent, of an Angle of 64D. 23M the Sine whereof, I find to be this number'9, 955065, and the Tangent thereof, 10. 319231, and so for any other Arch, or Angle of a Triangle, above 45 degrees.

PROP. II. To find the Co-Sine or Co-Tangent of any Arch, or Angle propounded.

Defin. The Co-sine or Co tangent, of an Angle or Arch, is the remaining part of the Angle propounded, to a Quadrent or 90 Degrees; and is by some called the Complement of an Angle, thus the Arch or Angle of 64D. 23M. taken out of 90D. leaves 25D. 27M. for its Complement, on the contrary if 25D. 37M. were taken out of 90 Degrees, there would remain 64D. 23M. for its Complement. So you see that these two Angles, are the Complements of each other, because they two are equal to a Quadrent or 90 Degrees.

Now the Logarithm of the Complement, may be exactly found with ease, for the Sines and Tangents of every degree, and Minute of the Quadrent in one Column is joyned with his Complement in the next Column, so that [Page] without substracting the Angle from 90D. you may readily find the Complement thereof either the Arch in Degrees and Minutes, or the Log. Sine, or Tangent thereof, as you have occasion: Thus the Log. of the Sines Complement before mentioned, to wit, 64D. 23M. Comp. is 25D. 37M. is 9. 635833, Tang. is 9. 680768; so 64D. 23M. is the others Compl. whose Sine is 9. 955065, and his Tang. is 10, 319231; so for any other.

PROP. III. To find the Secant of any Arch or Angle propounded.

In this little Book I have not room to set down the Tables of Artificial Secants at large, as I have done with the Sines and Tangents: Nevertheless I will not here omit to shew how they may be easily found out, by the Tables of Sines. The method is thus, substract the Logarithm Sine, of the Sines compl of an Angle, from the double Radius of the Tables, and the remainder shall be the Secant required: As if I desire the Secant of 25D. 37M. I find the Logarithm-sine of his complement to be 9. 955065, which substracted from the double Radius, that is 20. 000000: there remains 10, 044935 which is the Secant of it, and so the Secant of 64D. 23M. is 9. 955065; which is the Complement of the former, because they both are Equal to 20. 000000, the double Radius; and so may any other be found out.

CHAP. IV. Of GEOMETRY.

[Page]

Defin.THE End and Scope of Geometry is to measure well: for as Number or Multitude, is the Subject of Arithmetick: so is Magnitude that of Geometry: to measure well is therefore to consider the Nature of every thing that is to be measured; to compare such like things one with another: and to understand their Reason and proportion, and also their similitude: And this is the End and Scope of GeometryQuod quaeritur cognoscendi illius gratia, quod semper est, non & ejus quod oritur, quandoque & interit. Geometria, ejus quod est semper, Cognitio est. Ac tollet igitur (ô Generose vir) ad veritatem, animum: atque ita, ad Philosophandum prepar [...]vit cogitationem, ut ad supera convertamus: quae nunc, contra quàm decet, ad inferiora dejicimus, &c. Plato lib. 7. de Rap..

I shall not trouble you with the Definitions of Geometry, they being largely handled by many, and herein every one meanly conversant in the study of the Mathematicks is acquainted, [Page] but shall immediately fall in hand with the principal Propositions, which chiefly concern the other following parts of this treatise.

SECT. I. The Explication of some Geometrical Propositions.

PROP. I. To erect a perpendicular on any part of a line assigned.

LET the Line be A, B, and on the point D,Fig. 1, 'tis required to raise a perpendicular to A, B, To operate which first open your Compasses to any convenient distance, and placing one foot thereof in D, with the other make the two marks C, and E, equidistant from D; then open the Compasses to some other convenient distance, and set one foot in E, and describe the Arch FF; then likewise in C, describe the Arch GG, then through the Intersections of these two Arches, and to the point D, draw H D, perpendicular to A B; as was required.

PROP. II. To Erect a Perpendicular, on the End of a Line.

Let the given line be A B, and on the EndFig. 2, thereof at B, 'tis required to raise a Perpendicular [Page] line: To perform which open your Compasses to the distance B D, then on B as a Center, describe the Arch D, E, F, then from D, to E, place BD; then placing one foot in E, describe the Arch CF, then remove yourFig. 2. Compasses to F, and draw the Arch CE; Lastly through their Intersection draw C B, which is a Perpendicular to AB, on the end B; as required.

PROP. III. From a Point above to let fall a Perpendicular on a Line.

Let the line given be B A, and 'tis requiredFig. 3. from the point above at C; to let fall a Perpendicular to the said Line: To perform which place one foot of your Compasses in C, and open them beyond the given line A B, and describe the Arch EF; divide EF, into two parts in D; Lastly draw CD, which shall be perpendicular unto AB, falling from the point above at C, as was so required.

PROP. IV. To draw a right line Parallel to a right line, at any distance assigned.

Let the distance assigned be O E, and theFig. 4. Lime given be A B, and 'tis required to draw C D, Parallel to A B; at the distance O E: To perform which, take in your Compasses the distance O E, and on A, describe [Page] the Arch H, and on B, the Arch K; then draw C D, so as it may justly touch the two Arches, but cut them not, so shall C D; be parallel to A B, at the assigned distance O E, as was required.

PROP. V. To Protract an Angle of any Quantity of Degrees propounded.

Let it be required to Protract, or lay down an Angle, of 40 degrees: To perform which first draw a right line as A B, then open y [...]r Compasses to 60 degrees, in your line of Chords: and with that Distance on A, describeFig. 5. the Arch E F, then take 40 degrees in your Compasses out of your line of Chords, and place it on the Arch, from F, to H; Lastly through the point H, and from A draw A C; so shall the Angle CAB contain 40 degrees as required.

PROP. VI. To measure an Angle already protracted.

Let the Angle given be C A B, and 'tis [...]equired to find the Quantity thereof: To [...]erform which take in your Compasses 60 de [...]rees from your line of Chords; and on A,Fig. [...]. [...]escribe the Arch EF; then take in your Com [...]asses the Distance FH, and apply it to your [...]ne of Chords; and you will find the Angle, [...] AB to contain 40 degrees.

PROP. VII. To divide an Angle into two Equal parts.

[Page]

Let the Angle given be BAC, and 'tis required to divide it into two equal parts: To perform which do thus: first take in your Compasses any convenient distance, and placing one foot in A, describe the Arch FKHE,Fig. 6. then on H, describe the Arch KK, and on K, the Arch HH; lastly through the Intersections of these two Arches, draw the line AD, to the Angular point A; so shall the Angle BAC, be divided into two equal parts, viz. BA [...], and DAC; as required.

PROP. VIII. To divide a right line into any Number of Equal or Unequal parts; or like to any divided line propounded.

Let the line A B, be given to be divided into 5 equal parts; as the line CD. To perform which do thus: first on the point C, draw out a line making an Angle with CD at pleasure: then make CF, equal to AB; and joyn their Extremities FD, then draw Parallel linesFig. 7. to FD, through all the 5 points of CD, (by the 4 prop. aforegoing) which shall divide AB, into 5 equal parts; as required: This way is to be observed, when the line given to be divided, is greater than the divided line propounded.

CASE II.

[Page]

But if AB, be shorter than the given divided line CD; take the line AB, in your Compasses, and on D strike the Arch F, thenFig. 8. draw the Tangent CF, then take the nearest distance from the first division of CD, to the Tangent-line CF, which distance shall divide ABThose two propositions well understood, doth demonstrate many other propositions, and thereon is grounded the vse of the Sector. into 5 equal parts, as the given divided line CD; as required.

PROP. IX. How to Protract or lay down any of the Regular Figures, called Polygons.

To perform which divide 360 degrees, (the number of degrees in a Circle) by the number of the Poligon his sides: as if it be a Pentagon by 5, if a Hexagon by 6, &c. the Quotient is the Angle of the Center; its Complement to 180D. (or a Semi circle) is the Angle at the Figure, half whereof is the Angle of the Triangle at the Figure: Now I will shew how to delineate any Poligon three ways, viz. 1 by the Angle at the Center, 2. by the Angle at the Figure, 3. by the Angle of the Triangle at the Figure: I have hereunto annexed a Table, which gives at the first sight, (without the trouble of Division) 1. the quantity of the Angle at the Center; 2. the quantity of the Angle at the [Page] Figure; and 3 the Quantity of the Angle at the Triangle of the Figure, from a Triangle to a Decigon.

Names of the Poligons. | Sides | Angles at the Center | Angles at the Figure | Angles at the Trian.
D | M | D | M | D | M
Triangle | 3 | 120 | 00 | 60 | 00 | 30 | 00
Square | 4 | 90 | 00 | 90 | 00 | 45 | 00
Pentagon | 5 | 72 | 00 | 108 | 00 | 54 | 00
Hexagon | 6 | 60 | 00 | 120 | 00 | 60 | 00
Heptagon | 7 | 51 | 43½ | 128 | 34½ | 64 | 17¼
Octogon | 8 | 45 | 00 | 135 | 00 | 67 | 30
Nonigon | 9 | 40 | 00 | 140 | 00 | 70 | 00
Decigon | 10 | 36 | 00 | 144 | 00 | 72 | 00

CONSTRUCTION I.

First by the Angle at the Center, to delineate a Hexagon, whose Angle at the Center is 60 degrees, first lay down an Angle of 60 deg. [...] 9. (by prop. the 5. aforegoing) making its sides of a convenient length at pleasure, then take such a distance from O the Center of the figure, equally on both sides, as may make the third side equal to the side of the Poligon given; which here is 100 parts: But this in a Hexagon need not be done, because the 3 sides of the Triangle are equal, but in all other Poligons it must be done. Then divide the third side equally into two equal parts, and draw a line through it, from ☉ the Center: [Page] set each half of the side of the Poligon 100, to wit 50, on each from the middle of the third line.But if the third line do exceed or be short of the side of the Poligon propounded, then by parallels on each side, cut the sides of the Triangle, till you have found by those Intersections where to set the line proposed, in any Poligon, &c. thus having placed the side of the Hexagon PP, 100 parts, in order; describe the whole Hexagon PPPPPP, as was required.

CONSTRUCTION II.

Now by the Angle of the Figure, to delineate any regular Poligon, Let it be required to protract a Hexagon, whose side as afore is 100 parts; first I draw a line and make it 100 of those parts, then I sind in the precedent Table the Angle of a Hexagon at the figure to be 120 degrees: Then on each side of the drawn line, I lay down an Angle of 120 deg. (according to the 5 precedent propositions) and so work 6 times, (or as many times as your Poligon hath sides) making each side 100 parts,Fig. 10. and each Angle 120 degrees; so shall you have enclosed the Poligon PPPPPP, as required.

CONSTRUCTION. III.

To Protract or lay down a Hexagon, or any other regular Poligon, by the Angle of the Triangle, do thus; First draw the side of theFig. 11 Hexagon P P, make it 100 parts. I find in the precedent Table that the Angle of the Triangle is 60 deg; then at each end of the line P P, I [Page] lay down an Angle of 60 ☞ Observe these Rules well, for you will find them of infinite use in Fortification, &c. deg. (by prop. 5. precedent) and continue the two lines PO, and PO; untill they intersect eachFig. 11. other in O: then on O, as a Center (OP: being Radius) describe a Circle, and within it describe the Hexagon PPPPPP, as you see in the figure: and so may you delineate any other Poligon: whose Angels from a Triangle, to a Decigon, are all specified in the precedent Table.

PROP. X. To divide a line according to any assigned proportion.

Admit the right line given to be AB, and 'tis required to divide the same into two parts, bearing proportion the one to the other as the lines E, and F doth: To perform which, first draw the line CD, equal to the given line AB: Then draw the line HC, from C, to containFig. 12. an Angle at pleasure. Then from C to G, place the line F, and from G, to H, place the line E: Then draw the line HD. And lastly, draw GK parallel to HD, (by the 4 prop. precedent) so is the line DC, equal to AB, and divided into two parts, bearing such proportion to each other, as the two given lines E, and F, as was required.

PROP. XI. To two lines given, to find a third proportional to each of them.

[Page]

Admit the two given lines be A and B, andFig. 13. 'tis required to find a third proportional to A, as A, to B: First make an Angle at pleasure; as HIK. Then place the line B, from I, unto P; and the line A, from I, unto L; and draw PL. then also place the line A, from I unto M, and draw QM, parallel unto LP, (by 4 prop.) so shall the line IQ, be a third proportional unto the two given lines A, and B, as was required. For as B, is to A, so is A, unto the proportional found IQ.

PROP. XII. To three lines given to find out a fourth proportional unto them.

Admit the three given lines to be A, B, and C; and 'tis required to find a third proportional to them, which shall have such proportion unto A, as B, hath unto C. To perform which, first make an Angle at pleasure as DKG, now seeing the line C, hath such proportion to B, as the line A, unto the line sought: Therefore placeFig. 14. the line C, from K, unto H and B, from K, to F, and draw FH. Again, place the line A, from K, to I, and draw IE, parallel unto FH, (by 4 prop.) until it cutteth DK, in E; so have you the line KE, a fourth proportional, as was required. For as C, is unto B, so is A, unto the found line KE.

PROP. XIII. To find a mean proportional Line between any two right lines given.

[Page]

Let the two given lines be A, and B, between which it is required to find a mean proportional line. To perform which, first joyn the two lines A, and B together, so as they make the right line CED: Then describe thereon aFig. 15. Semicircle CFD. Then on the point E, erect the perpendicular EF, (by 1 prop.) to cut the limb of the Semi-circle in F, so shall EF, be a mean proportional line, between the two given lines A, and B, as required.

PROP. XIV. To find two mean proportional Lines between any two right Lines given.

Let the two given lines be A, and B; betweenFig. 16. which 'tis required to find two mean proportionals. To perform which, first make anAnd this of all other the Inventions of Plato, Apollonius, Sporus, Architas, Diocles, Nicomedus; & many other famous Geometricians and Philosophers, I like best for the ready performance of this Conclusion, whose several Methods I could here describe, but for brevity sake do omit them. Angle containing 90 deg. making the sides CD, and CE of a convenient length: then from C, place the line B, unto F, and the line A, from C, unto G; and draw FG, which divide equally in H, and describe the Semi-circle F K G. Then take the line B in your Compasses, and placeing one soot in G, with the other make a mark [Page] in the limb of the Semi-circle in K, then draw ST, in such sort that it may justly touch the Semi-circle in K, and may cut through the two sides of the Angle, equidistant from the Center of the Semi-circle H; so shall SF, and TG, be two mean proportionals, betwixt the two given lines A, and B, as required.

PROP. XV. To make a Geometrical square equal to divers Geometrical squares.

Let there be given the 5 sides of five Geometrical Squares, viz. A, B, C, D, E; and 'tis required to make one Geometrical Square, equal to the said five Sqares: To perform which first make a Right Angle as ABC, making its containedFig. 17. sides of a convenient length. Then from B, place A, to D, and from B, place B, to E, and draw Ed. Then place Ed, from B, to F, and C, from B, to G; and draw GF. Then place GF, from B, to H, and D, from B, to I; and draw [...]H. Lastly from B, unto K, place IH, and from B, unto L, place the line E; and draw LK. So shall LK, be the side of a Square, equal to the five Squares propounded.

PROP. XVI. To make a Circle equal to divers Circles propounded.

Let the two Circles propounded be A,Fig. 18. and B, and 'tis required to make a third Circle, [...]e [...]ual to the said Circles propounded. To perform [Page] which, first take the Diameter, of the lesser Circle A, and place it as a Tangent, on the Diameter of the greater Circle B, at right Angles;Fig. 18. as ECD. Then draw the Diagonal ED, which divide equally in F, on which as a Center describe the Circle K, making E D, the Diameter of which Circle K shall be equal unto the two given Circles A, and B, as requiredSo after the same manner, may divers Circles be added into one by the help of the former proposition well understood.

SECT. II. Of Planometry, or the way to measure any plain Superfice.

PLanometry is that part of the Mathematicks, derived from that Noble Science Geometry, by which the Superficies or Planes of things are measured, and by which their Superficial Content is found, which is done most commonly by the Squares of such Measures, Viz. a Square Inch, Square Foot, Square Yard, Square Pace, Square Perch, &c. That is whose side is an Inch, Foot, Yard, Pace, or Pearch Square. So that the Content of any Figure is said to be found, when you know how many such Inches, Feet, Yards, Paces, &c. are contained therein: Thus the End and Scope of Geometry is to measure well.

PROP. I. To find the superficial Content of a Geometrical square.

[Page]

Let the side of the Square AA be 4 Perch,Fig. 19. what is the Area, or superficial content thereof? To find which multiply its side 4, by its self, it produceth 16, which is the content of that Square AAAA, propounded.

PROP. II. To find the superficial content of a Parallelogram, or long Square.

Multiply the length in parts, by the breadth in parts; the product is the content thereof. So in the Parallelogram, or long Square ABCD, the length of the side AB, or CD is 20 Paces,Fig. 20. and the breadth AC, or BD is 10 paces, and his superficial content is required. I say therefore if according unto the Rule, you multiply the length 20, by the breadth 10, it produceth 200 Paces; which is the content of the Parallelogram or long Square ABCD.

PROP. III. To find the superficial Content of any Right-lined Triangle.

Although right-lined Triangles are of several kinds, and forms; as first in respect unto their Angles, they are either Right-angled, or [Page] Oblique-angled, i. e. Acute-angled, or Obtuseangled. Secondly in respect of their sides, they are either an Equilateral, Isosceles, or Scalenium Triangle: But now seeing they are all measur'd by one and the same manner, I shall therefore add but one Example for all; which take for a general Rule: which is,

Multiply the length of the Base, by theThe Rule. length of the Perpendicular, half their product is the Area or superficial content thereof. So if the content of the Triangle ABC, be required. To find which first from the Angle B, let fall the Perpendicular DB, on the Base AC, (by prop. 3. §. 1.) let therefore the length of theFig. 21. Perpendicular BD be 24, and the Base AC 44 parts. Now if the Base AC 44, were multiplyed by BD 24, the product is 1056, half whereof is 528, the Content of the Triangle ABC, propounded.

PROP IV. To find the superficial Content of a Rhombus.

First let fall a Perpendicular from one of the Obtuse-angles, unto its opposite side, (by prop. 3. §. 1.) and then Multiply the length of the side thereof, by the length of the Perpendicular, their product is the Content thereof.

So in the Rhombus ABCD, the side AC, or BD is 16 Inches, and the Perpendicular KC isFig. 22. 14 Inches, which multiplyed into the side 16, produceth 224 Inches; which is the Area, or superficial Content, of the Rhombus ABCD, propounded.

PROP. V. To find the superficial content of a Rhomboides.

[Page]

Frst let fall a Perpendicular, as in the former proposition, then the length thereof multiply by the length of the Perpendicular; the product is the Area, or superficial content thereof. For in the Rhomboides EDAH, whose length AH, or ED is 32 Feet, and the length of theFig. 23. Perpendicular HK is 16 Feet, which multiplyed together produceth 512 Feet, which is the Area or superficial content of the Rhomboides AHED, propounded.

PROP. VI. Te find the superficial Content of any Poligon, or many equal sided Superficies.

First from the Center unto the middle of either of the sides of the Poligon, let fall a Perpendicular, (by 3. prop §. 1.) Then multiply the length of half the Perifery, by the Perpendicular, the product shall be the Superficial Content of the Poligon.

Admit the Poligon to be an Hexagon AAAAFig. 24. AA, whose side AA is 22 Feet, and the Per [...]end [...]cular BE 19 Feet; now, if 66 half the Perifery, be multiplyed by 19 it produceth 1254 Feet; which is the Content of the Poligon AA, &c. as required.

PROP. VII. To find the superficial Content of a Circle.

[Page]

Multiply half the Circumference, by one half of the Diameter, their product is the superficial Content thereof.

Admit the Circumference of a Circle ACBD, be 44 Inches, what is the Area or Content thereof. (by the 9. prop. §. 2.) I find the DiameterFig. 25. to be 14 Inches, therefore I say if 22, half the Circumference, be multiplied by 7, half the Diameter, it shall produce 154 Inches; which is the superficial Content of the Circle ACDB, as required.

PROP. VIII. By the Diameter of a Circle given, to find the Circumference.

Suppose the Diameter be 14, what is the Circumference? The Analogy or Proportion holds thus, as 7, to 22, so is 14, unto 44, the Circumference required.

PROP. IX. By the Circumference of a Circle given, to find the Diameter.

Suppose the Circumference of a Circle be 44 what is the Diameter? the Analogy or Proportion is, as 22, to 7, so is 44, unto 14, the Diameter required.

[Page]

Now the proportion of the Diameter, unto the Circumference is as 7, unto-22; or as 113, to 355; or as 1, unto 3, 1415926, &c. so is the Diameter to the Circumference.

PROP. X. By the Content of a Circle given, to find the Circumference.

Suppose the Content of a Circle be 154, what is the Circumference, the Analogy or Proportion?

As 7, unto 4 times 22, which is 88, so is 154 the Content of the given Circle; to the square of the Circumference 1936, whose root being Extracted, as is taught (in prop. 8. §. 1. chap. 1.) gives the Circumference 44, as required.

PROP. XI. By the Content of a Circle given, to find the Diameter.

Suppose the Superficial Content of a Circle be 154 parts, what is the Diameter thereof? to find which this is the Analogy or Proportion.

As 22,

To 4 times 7, which is 28,

So is 154, the given Content,

To the Square of the Diameter 196, whose Root being Extracted (by 8 prop chap. 1. §. 1.) [...]iveth the Diameter 14, as required.

PROP. XII. By the Diameter of a Circle given to find the side of a square equal thereto.

[Page]

To find which this is the Analogy or Proportion.

As 1, 000000,

To 0, 886227.

So is the Diameter of the Circle propounded.

To the side of a Square, whose superficial Content, is equal unto the superficial Content, of the Circle propounded.

PROP. XIII. By the Circumference of a Circle given, to find the side of a square equal to it.

This is the Analogy or Proportion.

As 1. 000000,

To 0. 282093.

So is the Circumference of the Circle propounded, to the side of a Square equal to the Circle.

PROP. XIV. By the Content of a Circle given to find the side a square equal to it.

To do which, Extract the Square-Root o [...] the Content propounded, (by prop. 8 chap. [...] §. 1.) so is the Root, the side of a Geometrica [...] Square, equal thereunto.

PROP. XV. By the Diameter of a Circle given, to find the side of an Inscribed square.

[Page]

This is the Analogy or Proportion.

As 1. 000000,

To 0. 707107,

So is the Diameter of the Circle propounded, To the side of the inscribed Square.

PROP. XVI. By the Circumference of a Circle given, to find the side of an Inscribed Square.

This is the Analogy, or Proportion.

As 1. 000000,

To 0. 225079.

So is the Circumference of the Circle propounded, To the side of the inscribed Square.

PROP. XVII. [...]o find the Superficial Content of an Oval, or Elleipsis.

Let the Oval given be ABCD, and 'tis re [...]uired to find the Area or Superficial Content [...]ereof? To do which multiply the length A [...] 40 Inches, by the breadth CD 30 Inches, theFig. 26. [...] is 1200. Which divide by 1. 27324; [...]e Quotient is 942 48/100 parts. Which is the [...]ea or Superficial Content of the Oval ABCD [...]opounded

PROP. XVIII. To find the Superficial Content of any Section, or Portion of a Circle.

[Page]

Multiply half the Circute of the Section, by the Semidiameter of the whole Circle, and the product thence arising is the Area or superficial Content thereof.

Suppose there be a Circle whose Diameter isFig. 27. 14 parts, and the Circute of the Quadrent ABC is 11 parts, and the Content of the said Quadrent is desired? To find which multiply 5 ½ or, 5., 5 half the Circute of the Quadrent, by 7 the Semidiameter, the product is 38 5/10, which is the Content of the Quadrent ABC propounded.

SECT. III. Of STEREOMETRY, or the way how to measure any Regular Solid.

STereometry is that part of the Mathematicks, springing from Geometry, by which the Content of all Solid Bodies are discovered by two Multiplications, or three Dimention and is valued by the Cube of some famous Mea sure; as an Inch-Cube, a Foot-Cube, a Yard Cube, or a Perch-Cube, &c.

PROP. I. To find the solid Content of a Cube.

[Page]

Multiply the side into its self, and that product by its side again; their product is the solid Content thereof.

Suppose there be a Cube A, whose side isFig. 28. 2 Feet; and his solid Content is required? I say if his side 2, be multiplyed by its self, it produceth 4, which again multiplyed by 2, it produceth 8 Feet, which is the solid Content of the Cube propounded.

PROP. II. To find the solid Content of a Parallelepipedon.

First get the Superficial Content of the End, (by prop. 1, or 2, §. 2.) which multiply into the length, the product is the solid Content.

Suppose there be a Parallelepipedon B, whoseFig. 29. sides of the Base is 40, and 30 Inches, and length 120 Inches, and his Solid Content is demanded? I say if you multiply 30, by 40, the product is 1, 200, which is the superficial Content at the Base. Which multiplyed by the length 120 Inches produceth 144000 Inches, which is the solid Content of the Parallelepipedon B, propounded.

PROP. III. To find the solid Content of a Cylinder.

[Page]

First get the superficial Content of the Circle at the Base, (by prop. 7. §. 2.) and by it multiply its length, their product is the solid Content thereof.

Suppose there be a Cylinder as D, whoseFig. 30. Diameter of the Circle at the Base is 7 parts, and the length of the Cylinder is 14 parts, and 'tis required to find the solid Content thereof? First I find the superficial Content of the Base to be 38. 5, which multiplied into 14 the length, giveth 539 parts, which is the solid Content of the Cylinder propounded.

PROP. IV. To find the solid Content of a Pyramid.

First get the superficial Content of the Base of the Pyramid, (by some of the aforegoing propositions in Planometria) and then multiply that into ⅓ of his Altitude, the product is the solid Content thereof.

Suppose there be a Pyramid H, whose side of the Base is 4 ½ parts, or 4 5/10, and his Altitude 12 parts, and his solid Content is required? First I find, (by prop. 1. §. 2.) the superficial ContentFig. 31. of the Base to be 20 25/100 or 20 ¼, which multiplyed by 4, (which is ⅓ of the Altitude 12) produceth 81 parts, for the solid Content of the Pyramid propounded.

PROP. V. To find the solid content of a Cone.

[Page]

First find the superficial Content of the Circle at the Base, (by prop. 7. §. 2.) then multiply it by ⅓ of its Altitude or Heighth, the product is the solid Content thereof.

Suppose there be a Cone as B, whose DiameterFig. 32. of the Base is 7, and his Altitude or Heighth is 15 parts, and his solid Content is required? First I find the superficial Content of the Base to be 38½ or 38. 5; which multiplyed into 5, ⅓ of its Altitude or Heighth) produceth 192. 5, or ½, which is the solid Content of the Cone propounded.

PROP. VI. By the Diameter of a Globe to find his solid Content.

This is the Analogy or Proportion.

As 6 times 7, which is 42.

Is to 22,

So is the Cube of the Diameter of the Sphere, or Globe propounded.

To the solid Content thereof.

Suppose there be a Sphere or Globe, whose Diameter is 12 Inches; what is the solid Content thereof? say, (see the Globe R.)

As 42,

Is to 22,Fig. 33.

So is 1728, the Cube of the Diameter,

[Page]

To the solid Content 905 6/42 or 1/7 of the Globe, or Sphere propounded: This and all other such☞ Note that every Sphere is equal unto two Cones, whose Height and Diameter of the Base is the same with the Axis of the Sphere. And a Sphere is two thirds of a Cylinder, whose Height and Diameter of the Base is the same with the Axis of the Sphere; according unto the 9th. Manifestation of the first Book of Archimedes of the Sphere and Cylinder like Propositions, are performed by the help of the first Proposition, of the first Chapter of this Book.

PROP. VII. By the Circumference of a Sphere, or Globe, to find his solid Content.

This is the Analogy or Proportion.

As 1. 000000,

To 0. 016887,

So is the Cube of the Circumference of the Globe or Sphere propounded

To the solid Content thereof.

PROP. VIII. By the Axis of a Globe, to make a Cube equal thereunto.

This is the Analogy or Proportion.

As 1. 00000,

To 0. 80604,

So is the Axis of the Sphere propounded,

To the [...]u [...]-Root, which shall be equal to it.

PROP. IX. By the Circumference of a Globe, to make a Cube equal thereunto.

[Page]

This is the Analogy or Proportion.

As 1. 000000,

To 0 256556.

So is the Circumference of the Globe propounded,

To the Cube-Root, which shall be equal to the Sphere, or Globe, propounded.

PROP. X. By the solid Content of a Sphere or Globe, to make a Cube equal thereunto.

Extract the Cube-root of the solid Content of the Sphere or Globe, (by prop. 9. § 1. chap. 1.) so shall the Root, so found, be the side of a Cube, equal unto the Globe or Sphere propounded.

PROP. XI. A Segment of a Sphere being given to find the solid Content thereof.

To find which first say, As the Altitude of the other Segment, is to the Altitude of the Segment [Page] given: so is that Altitude of the other Segment increased by half the Axis, unto a fourth: Then say, As 1, to 1, 0472, so is the product of the Quadrant of half the Chord of the Circumference of that Segment, multiplyed by that fourth, To the solid Content of the Segment propounded.

CHAP. V. Of TRIGONOMETRY. Or the Doctrine of Triangles.

[Page]

SECT. I. Some general Maxims, belonging to plain or Right-lined Triangles.

TRIGONOMETRY is necessary in most parts of the Mathematicks, and herein indeed consisteth the most frequent use of the Logarithms, Sines, Tangents, and Secants: It is conversant in the measuring of Triangles, Plain or Spherical, comparing their Sides, and Angles together; according unto their known Analogies, or Proportions: So that any three parts of a Triangle being given, the other parts may be found out, and known: Now in the Doctrine of Right-lined Triangles, it will be necessary to know these Maxims following.

[Page]

1. That a Right-lined Triangle, is a Figure constituted, by the Conjunction, or Intersection, of the three Right, or Streight-lines thereof; in their Angles or Meeting-places. So that every Triangle hath six distinct parts, Viz. Three Sides, and three Angles.

2. That all Right-lined Triangles, are eitherFig. 34. Right-angled, That is, which hath one RightAngle, as ABC Fig. 34. Or Oblique-angled, whose three Angles are all Acute; that is, less than a Quadrant, or 90 deg; or else they have One Angle Obtuse, or greater than a Quadrent: So all Triangles, that have not one Right-angle,Fig. 36. are called Oblique-Triangles; as Fig. 36. to wit, the Triangle ABC.

3. That the three Angles, of any Right-lined Triangle, are equal unto two Right-angles; or 180 Degrees. So that any two of their Angles being known, the third Angle is also found, being the Complement of the other two; unto 180 Degrees: But this is more readily found in a Rectangled Triangle, for the Rectangle being a Quadrent, or 90 degrees, one of the acute Angles therefore being given, the other is readily known, being the Complement thereof unto 90 Degrees.

4 That the three sides, comprehending theFig. 35. Triangle, some call Leggs, others Sides, but in Rectangled Triangles, as in the Triangle ABC, I call AB, the Base, BC the Cathetus or Perpendicular; and AC the Hypothenuse.

5. That the Sines, of the Angles are proportional unto their opposite Sides; and their Sides, to their opposite Angles. So that if the Side of a Triangle were desired, put the Sine of the opposite [Page] Angle in the first place. Also if an Angle be required, put the Logarithm of his opposite side in the first place.

6. That the sides of any Rectangled Triangle may be measured by any Scale of equal parts, as Inches, Feet, Yards, Poles, Miles, Leagues, &c.

7. That if an Angle propounded, be greater than 90 deg. and so not to be found in the Tables, take the Complement thereof, unto 180 deg. and work by the Sine, or Tangent thereof, and the work will be the same.

And here for the more short, and speedy performance of these conclusions in Trigonometry; I have annexed, and used, these following Symbols; which I would have you take notice of.

= Equal, or Equal to.

+ More.

- Less.

× Multiplyed by.

° Degrees as 15°.

' Minutes as. 40'.

cr. A Side.

crs, Sides.

V An Angle.

VV Angles.

Z Sum.

X Difference.

S Sine.

Sc Co-sine.

T Tangent.

Tc Co-tangent.

Se Secant.

Sec Co-secant.

Co. Ar. Compl. Arithmetic.

R A Right-angle.

2R Two Right-angles.

Q Square.

SECT. II. Of Plain Rectangled Triangles.

[Page]

PROP. I. Two Angles and the Base of a Rectangled Triangle given, to find the other parts.

ADmit the Triangle given be ABC: NowFig. 34. the Angle at B, is an Angle of 90°, or a Right-angle; And the Angle at C is 57° 35', and the Base AB; is 736 parts.

Now first I find the Angle at A, to be 32° 25': it being the Complement of the Angle at C, unto 90°: Secondly, to find the Cathetus, or Perpendicular, this is the analogy or proportion. [...]

Add the Log. of the third and second Terms together, and from their Sum, deduct the Log. ofObserve this for a general Rule in Trigonometry. the first number, so is the Remainder, the Log. of [Page] the fourth Term, or Number sought, as you see in the aforegoing Example.

Thirdly to find the Hypothenuse AC, the Analogy or Proportion hold thus.

As S. V, C 57° 35',

To Log. Base AB 736 [...]arts.Fig. 34

So Radius or S. 90°,

To Log. Hypothenuse AC 871 8/10 parts required: Thus are the three required parts, of the given Triangle ABC found, viz. the Angle A to be 32° 25', the Cathetus BC to be 467 4/10 parts, and the Hypothenuse AC to be 871 8/10 parts, as was so required to be found.

PROP. II. The Hypothenuse, Base, and one of the Angles Of a Rectangled Triangle given, to find the other parts thereof.

In the Triangle ABC, the Hypothenuse AC is 871 8/10 parts, the Base AB is 736 parts, and the Angle at B, is known to be a Right-angle; or 90°: First to find the Angle at the Cathetus C, the analogy or proportion holds thus.

As Log. Hypothen: AC 871 8/10 partsFig. 34.

To Radius or S. 90°.

So Log. Base AB 736 parts,

To the S. V. at Cathetus C 57° 35'.

Secondly, now having found the Angle at the Cathetus C, to be 57° 35'; I say the Angle of the Base A is 32° 25', being the Compl. of the Angle C, unto 90°.

Thirdly to find the Cathetus BC, this is the [...]nalogy, or proportion.

[Page]

As Radius or S. 90°,

To Log. Hypothen. AC 871 8/10 parts,

So S. V. at Base A 32° 25'.Fig. 34.

To Log. Cathetus BC, 467 4/10 parts required. It may also be found, as in the former Proposition.

PROP. III. In a Rectangled Triangle, the Base, and Cathetus given to find the other parts thereof.

In the Triangle ABC, the Base AB is 736 parts, and the Cathetus BC is 467 4/10 parts, and the Angle B, between them is a right angle o [...] 90°: And here you may make either side of the Triangle, Radius, but I shall make BC the Cathetus Radius, and then to find the Angle at the Cathetus C, this is the Analogy or [...] portion.

As Log. Cathet, BC 467 4/10 parts,Fig. 35.

To Radius or S 90°.

So Log. Base AB 736 parts,

To T. V. Cathe C 57° 35', as required.

Secondly, I find the other Angle, at A to be [...] 32° 25', it being the Complement, to C 57 35', unto 90°.

Thirdly, To find out the Hypothenuse AC this is the analogy or proportion.

As S. V. Cathe C. 57° 35',

To Log. Base AB 736 parts,

So Radius or S. 90°,

To Log. Hypothenuse AC 871 8/10 parts, [...] quired. But making the Base AB Radius, yo [...] may find the Hypothenuse AC, by this anal [...] or proportion.

[Page]

[Page] [Page]

Plate 1 Page 65

[Page]

[Page]

As Radius or S. 90°,

To Log. Base AB 736 parts.

So Sc. V. Base A 32° 25',

To Log. Hypothenuse AE 871 8/10 parts required,Fig. 35. and thus you have all the parts of the Triangle propounded.

PROP. IV. The Base, and Hypothenuse, with the Angle between them given, to find the other parts of a Rect-angled Triangle.

In the Triangle ABC, the Base AB is 736 parts, and the Hypothenuse AC is 871 8/10 parts, and the Angle A included between them is 32° 25'. First to find the Angles, and first remember that the Angle B is a right Angle; or 90°. Secondly, that the Angle at C, is the Complement to the Angle at A 32° 25' unto 90°:Fig. 35. and therefore is 57° 35': Now these being known, you may find the Cathetus, by this analogy or proportion.

As S. V. Cathe. C. 57° 35',

To Log. Base AB 736 parts.

So S. V. Base A 32° 25',

To Log. Cathe. BC 467 4/10 parts required. Thus I have sufficiently explained all the Cases of Plain Rect-angle Triangles, for to these rules they may be all reduced.

SECT. III. Of Oblique-Angled Plain Triangles.

[Page]

PROP. I. Two Angles, and a side opposite, in an Oblique-Angled Triangle given, to find the other parts thereof.

IN the Triangle ABC, the Angleat A is 50°, and at C is 37°, and the side AB is 30 parts, and opposite to the Angle C

First, to find the Angle B, remember that (as 'tis said, in the third Maxim aforegoing) 'tis the Complement, to the Angles A 50°, and C 37°, to 180°, and therefore is the Angle at B 93°.

Secondly, having thus found the Angles, theFig. 36. two unknown sides, may be found by the proportion they bear to their opposite Angles, for that proportion holds also in these; thus to find the side BC, this is the analogy or proportion.

As S. V. C 37° 00',

To Log. side AB 30 parts.

So S. V. A. 50° 00',

To Log. side BC 38 19/100 parts required to be found.

But it may be more readily found, and performed in such case as this, where you have a Sine, or Tangent, in the first place, by the Arithmetical Complement thereof, and so save the Substraction.

[Page]

Now the readiest way to find the Arithmetical☞ The Rule to find the Complement Arithmetical, of any Logarithm Number. Complement is that of Mr. Norwood, in his Doctrine of Triangles; which is thus: begin with the first Figure towards the left hand of any Number and write down the Complement, or the remainder thereof, unto 9:9. 962398. And so do with all the0. 037602. rest of the Figures, as you see here done. Saying 9, wants of 9, 0: and again 9, wants 0: 6, wants 3; 2, wants 7: 3, wants 6; 9, wants 0: only when you come to the last Figure to the right hand, take it out of 10, so 8, wants 2; of 10: Thus you may readily find the Co-Ar. of any Sine, almost as soon as the Sine it self.

But if you want the Complement Arithmetical of any Tangent, you may take the Co-tang. which is exactly the Co-Arith. of the double Radius, so that the Tangent, and Co-tangent, of an Arch makes exactly 20. 000000.

Now if the Radius be in the first place, then there is no need of taking the Co-Arith. of the first Number, only you must cut off, the first I, to the left hand thus X, and you will have the Logarithm of the Number desired.

Thirdly, now to find the side AC, by the opposite Angle B; which is 93° 00': And see [...]ng the Angle B, exceeds 90°, you must work [...]y the Complement to 180°) as in the seventh [...]ork in page 61 is taught.

[Page]

Thus having found all the parts of the Triangle [ [...]] Fig. 36. propounded, Viz. The Angle B, to be 93° 00', the side AC to be 49 78/100 parts, and the side BC to be 38 19/100 parts, as was required to be found.

PROP. II. Two sides, and an Angle opposite to one of them in an Oblique-angled Triangle given, to find the other parts thereof.

In the Triangle ABC, the side AB is 30 parts, and the side AC, is 49 78/100 parts, and the opposite Angle C, is 37° 00'.

First, To find the Angle at B, this is the AnalogyFig. 36. or Proportion.

As Log. cr. AB 30 parts,

To S. V. at C 37° 00'.

So Log. cr. AC 49 78/100 parts,

To Sc. V. B 93° 00', as was required to be found.

Now seeing that the Angle C, is 37° 00', and the Angle B, is 93° 00', which makes 120° 00', therefore must the Angle A be 50° 00'; the [Page] Complement to 180°: so having found all the three Angles, you may find the other side CB,Fig. 36. 38 19/100 parts, as afore in the first proposition, by his opposite Angle.

PROP. III. Two Sides of an Oblique-angled Triangle, with the Angle included between them given, to find the other parts thereof.

In the Triangle ABC, the side AC is 49 78/100 parts, the side DB is 30 parts, and the Angle A between them is 50° 00'; and 'tis required to find the other parts of the Triangle propounded. To resolve this Conclusion, let fall a Perpendicular DB, from the Angle B, on the side AC; (by prop. 3. §. 1. chap. 4) and then proceed thus.

First, Seeing the Oblique-angled Triangle,Fig. 37. ABC is divided into two Rectangled Triangles, Viz. ADB, and BDC: Now I will begin with the Triangle ADB, in which is given the Angle A 50° 00', and the Angle D is a right Angle, or 90°, and the side AB 30 parts, and the sides AD, and DB, and the Angle at B, are required.

First to find the Angle at B, remember that it is the Complement unto the Angle A 50° 00', unto 90° 00', and therefore must the Angle B be 40° 00'; Now for to find the Cathetus BD, (as in prop. 1. and 2 §. 2. chap. 5.) by the Rule of opposition, the Analogy or Proportion holds thus.

[Page]

As Radius or S. 90°,

To Log. Hypoth. AB 30 parts.

So S. V. at A 50° 00',

To Log. Cath. BD 22 98/100 parts sought.

And AGAIN, say.

As Radius or S. 90°,

To Log. Hypoth. AB 30 parts.

So S. V. at B 40° 00',

To Log. Base AD 19 28/100 parts sought.

Thus in the Triangle ADB, you have found the Angle B, to be 40° 00', the Cathetus BD, to be 22 98/100 parts; and the Base AD to be 19 28/100 parts, as was so required.

Now for the other Triangle which is BDC, in which there is given the side BD, 22 98/100 parts, and the Angle at D, is a Right-angle, or 90°, and the sides DC, and CB, and the Angles B, and C, are required.

First to find the side DC, substract AD, 19 28/100Fig. 37. parts, out of AC, 49 78/100 parts; there remains the Base DC; 30 50/100 parts: Thus have you the two sides of the Triangle, to wit the Base DC, 30 50/100 parts, and the Cathetus BD, 22 98/100 parts, and the Angle D between them is a Right-angle or 90°. Now you may find the Angle at B, by the Tangent (as in prop. 3. §. 2. chap. 5.) thus.

As Log. Cath. BD, 22 98/100 parts,

To Radius or S. 90°.

So Log. Base CD 30 50/100 parts.

To T. V. B. 53° 00'.

Secondly, For the Angle C, remember 'tis the Complement of the Angle B, 53°, to 90°; and therefore is the Angle C, 37° 00', required.

[Page]

Thirdly, To find the Hypoth. BC, this is the Analogy or Proportion.

As S. V. B. 53° 00',

To Log. Base DC 30 50/100 parts.

So Radius or S. 90°,

To Log. Hypoth. BC 38 19/100 parts: Thus have you found all the required parts of the Triangle ABC propounded, viz. the Angle C to be 37° 00', the Angle B, to be 93° 00', That is equal unto the two Angles B 40° and B. 53° as afore in the former Proposition. and the Side BC, 38 19/10 [...] parts, as required to be found.

Another way to perform the same.

Take the Sum of the[ [...]] two sides, and the difference of the two sides; and work as followeth.

Now to find the twoFig. 37. Angles B, and C, this is the Manner, and by this Analogy or Proportion, they are found out and known.

As Log. Z. crs. AB, and CA, 79 78/100 parts,

To Log. X. crs. AB, and CA; 19 78/100 parts,

So T. of ½ VV unknown, 65° 00',

To T. ½X. of VV, 28° 00'.

This difference of Angles 28° 00', add unto 65° 00', (half the difference of the unknown Angles) and it shall produce 93° 00', which is the greater Angle, and substracted from it, leaves 37° 00', which is the lesser Angle C: so have you the required Angles.

PROP. IV. The three sides of an Oblique-angled Triangle given, to find the Angles.

[Page]

In the Triangle ABC, the side AC, is 49 78/100 parts, the side AB, is 30 parts, and the side BC, is 38 19/100 parts; and the three Angles of the Triangle are required.

The resolution of [...] this Conclusion is thus. Take the Summ and Differ.Fig. 37. of the two sides AB, and BC; And then work as follows: To find a Segment of the Base AC, to wit CE; say:

As Log. Base AC, 49 78/100 parts,

To Z. crs. AB, and BC; 68 19/100 parts,

So X. crs. AB, and BC; 8 19/100 parts,

To Log of a Segment of the Base AC, to wit C E 11 22/100 parts.

This Segment of the Base CE, 11 22/100 parts, being substracted from the whole Base AC, 49 78/100 parts, the remainder is EA 38 56/100 parts, in the middle of which as at D, the Perpendicular DB, will fall from the Angle B; and so divideFig. 37. it into two Rectangled Triangles, to wit, ADB, and CDB, whose Base DA is 19 28/100 parts, which taken from AC 49 78/100 parts, leaves the Base of the greater Triangle CD 30 50/100 parts.

Now having the two Bases of these two Triangles, and their Hypothenuses; to wit CD 30 50/100 parts, DA 19 28/100 parts, CB 38 19/100 parts, and BA 30 [Page] parts; you may find all their Angles, by the Rule of Opposite sides, to their Angles as afore.

I. In the Triangle CDB.

To find the Angles, this is the Analogy or Proportion.

As Log. BC 38 19/100 parts,

To Radius or S. 90°.

So Log. DC 30 50/100 parts,

To S. V. B 53° 00': whose Complement is theFig. 37. Angle at C 37° 00' unto 90: or a Quadrant.

II. In the Triangle ADB.

To find the Angles, this is the Analogy or Proportion.

As Log. AB, 30 parts,

To Radius or S. 90°.

So Log. AD 19 28/100 parts,

To S. V. B, 40° 00'.

The Complement whereof, unto 90° 00', is the Angle at A 50° 00'.

Now in the first Triangle CDB, there is found the Angle C, to be 37° 00', and the Angle B, to be 53° 00'.

In the second Triangle ADB, there is found the Angle A; to be 50° 00', and the Angle B, to be 40° 00'.

Now the two Angles at B, to wit 53° 00'; and 40° 00'; makes 93° 00', which is the Angle of the Oblique-angled Triangle ABC, at B: Thus the three Angles of the said given Triangle ABC, are found as was required, viz. the Angle A to be 50° 00', the Angle B to be 93° 00', and the Angle C to be 37° 00', as sought.

[Page]

Thus I have sufficiently, fully and plainly explained all the Cases of Plain Right-lined Triangles, both Right and Oblique-angled: I shall now fall in hand with Spherical Triangles, both Right and Oblique-angled.

SECT. IV. Of Spherical Rectangled Triangles.

And here first it will be necessary also to understand those few general Maxims or Rules, that are of special Moment, in the Doctrine of Spherical Triangles.

1. THat a Spherical Triangle is comprehended and formed, by the Conjunction and Intersection of three Arches of a Circle, described on the Surface of the Sphere or Globe.

2. That those Spherical Triangles, consisteth of six distinct parts, viz. three Sides and three Angles, any of which being known, the other is also found out and known.

3. That the three Sides of a Spherical Triangle, are parts or Arches of three great Circles of a Sphere, mutually intersection each other: and as plain or Right-lined Triangles, are measured by a Measure, or Scale of equal parts: So these are measured, by a Scale or Arch of equal Deg [...]ees.

[Page]

4. That a Great Circle is such a Circle that doth bessect the Sphere, dividing it into two equal parts; as the Equinoctial, the Ecliptick, the Meridians, the Horizon, &c.

5. That in a Right-angled Spherical Triangle, the Side subtending the Right-angle we call the Hypothenuse, the other two containing the Rightangle we may simply call the Sides, and for distinction either of them may be called the Base or Perpendicular.

6. That the Summ of the Sides of a Spherical Triangle are less than two Semicircles or 360°.

7. That if two Sides of a Spherical Triangle be equal to a Semicircle; then the two Angles at the Base shall be equal to two Right-angles; but if they be less, then the two Angles shall be less; but if greater, then shall the two Angles be greater than a Semicircle.

8. That the Summ of the Angles of a Spherical Triangle, is greater than two Right-angles.

9. That every spherical Triangle is either a Right, or Oblique-angled Triangle.

10. That the Sines of the Angles, are in proportion, unto the Sines of their opposite Sides; and the Sines of their opposite Sides, are in proportion unto the Sines of their opposite Angles.

11. That in a Right-angled Spherical Triangle, either of the Oblique-angles, is greater than the Complement of the other, but less than the Difference of the same Complement unto a Semicircle.

12. That a Perpendicular is part of the Arch of a great Circle, which, being let fall from any Angle of a spherical Triangle, cutteth the opposite Side of the Triangle at Right-angles, and so [Page] divideth the Triangle into two Right-angled Triangles, and these two parts (either of the Sides or Angles) so divided must be sometimes added together, and sometimes substracted from each other, according as the Perpendicular falls within or without the Triangle.

PROP. I. Case 1. A Side and an Angle adjacent thereunto being given, to find the other Side.

In the Triangle ABC, there is given the Side AB 27° 54'; and the Angle A 23° 30', and the Side BC is required, to find which this is the Analogy or Proportion.Fig. 38. [...]

PROP. II. Case 2. A Side and an Angle adjacent thereunto being given, to find the other Oblique-angle.

In the Triangle ABC, there is given the Side AB 27° 54', and the Angle A 23° 30',Fig. 38. and the Angle at C is required, to find which say by this Analogy or Proportion.

As the Radius or S 90° 00',

To Sc. of cr. AB 27, 54.

[Page]

So is S. V. at A 23, 30,

To Sc. V. at c 69, 22 required.

PROP. III. Case 3. A Side and an Angle adjacent thereunto being given, to find the Hypothenuse.

In the Triangle ABC, there is given the SideFig. 38. AB 27° 54', and the Angle at A 23° 30', and the Hypothenuse AC, is required; which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To Sc. of V. at A 23, 30.

So is Tc cr. AB, 27, 54.

To Tc. Hypothenuse AC, 30, 00 required.

PROP. IV. Case 4. A Side and an Angle opposite thereunto being given, to find the other Oblique-angle.

In the Triangle ABC, there is given the SideFig. 38. BC 11° 30', and the Angle A 23° 30', and the Angle C is required, to find which, say by this Analogy or Proportion.

As Sc. cr. BC, 11° 30',

To Radius or S. 90, 00.

So is Sc. V. at A, 23, 30,

To S. V. at C. 69, 22, as required.

PROP. V. Case 5. A Side and the opposite Angle given, to find the Hypothenuse.

[Page]

In the Triangle ABC, there is given theFig. 38. side BC 11° 30', and the Angle at A 23° 30', and the Hypothenuse AC, is required, which may be found by this Analogy or Proportion.

As S. V. at A 23° 30',

To Radius or S. 90, 00.

So is Ser. BC 11. 30,

To S. Hypothenuse AC 30, 00. as required.

PROP. VI. Case 6. A side and the opposite Angle given, to find the other side.

In the Triangle ABC, there is given the side BC 11° 30', and the Angle at A 23° 30', and the side AB is required, to find which this is the Analogy or Proportion.

As Radius or S 90° 00',Fig. 38.

To Tc. of V. at A. 23. 30,

So is T. cr. BC 11, 30,

To S. of cr. AB 27. 54 as was required.

PROP. VII. Case 7. The Hypothenuse, and an Oblique Angle given, to find the side adjacent thereunto.

In the Triangle ABC, there is given the [Page] Hypothenuse AC, 30° 00', and the Angle A 23° 30', and the side AB, is required, which is found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To Sc. V. at A, 23, 30.Fig. 38.

So is T. Hypoth. AC, 30, 00,

To T. cr. AB, 27, 54, as was required.

PROP. VIII. Case 8. The Hypothenuse, and an Oblique-angle given, to find the opposite Side.

In the Triangle ABC, there is given the Hypothenuse AC, 30° 00', and the Angle at A 23° 30', and the Side BC, is required, which is found by this Analogy or Proportion.Fig. 38.

As the Radius or S. 90° 00',

To S. Hypoth. AC, 30, 00.

So is S. V. at A, 23, 30,

To the S. cr. BC, 11, 30. which was required.

PROP. IX. Case 9. The Hypothenuse, and an Oblique-angle given, to find the other Oblique-angle.

In the Triangle ABC, there is given the Hypothenuse AC 30° 00', and the Angle A, 23° 30', now the Angle at C, is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To Sc. Hypoth. AC, 30, 00.

So is T. of V. at A, 23, 30,Fig. 38.

To Tc. of V. at C. 69, 22, as was required.

PROP. X. Case 10. The sides given, to find the Hypothenuse.

[Page]

In the Triangle ABC, there is given the side AB 27° 54', and the side BC 11° 30', and the Hypothenuse AC is required, to find which say by this Analogy or Proportion.

As the Radius or S. 90° 00',Fig. 38.

To Sc. cr. BC. 11, 30.

So is Sc. cr. AB 27, 54,

To Sc. Hypothenuse AC 30, 00. required.

PROP. XI. Case 11. The sides given, to find an Angle.

In the Triangle ABC, there is given, the side AB 27° 54', and the side BC 11° 30', and the Angle at A, is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',Fig. 38:

To S. cr. AB. 27, 54.

So is Tc. cr. BC. 11, 30,

To Tc. of V. at A. 23. 30. as required.

PROP. XII. Case 12. The Hypothenuse, and a side given, to find the other side.

In the Triangle ABC, there is given, the HypothenuseFig. 38. AC 30° 00', and the side AB 27° 54' and the side BC is required, which may be [Page] found by this Analogy or Proportion.

As Sc. cr. AB. 27° 54',

To Radius or S. 90 00.

So is Sc. Hypothenuse AC. 30° 00',

To Sc. cr. BC. 11° 30' as required.

PROP. XIII. Case 13. The Hypothenuse, and a Side given, to find the contained Angle.

In the Triangle ABC, there is given the HypothenuseFig. 38. AC 30° 00', and the side AB 27° 54', and the Angle at A is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To T. cr. AB. 27° 54'

So is Tc. Hypoth. AC 30° 00',

To Sc. of V. at A, 23° 30', as required.

PROP. XIV. Case 14. The Hypothenuse, and a Side given, to find the opposite Angle.

In the Triangle ABC, there is given the Hypothenuse AC 30° 00', and the side AB 27° 54', [...]ow the Angle C, is required, which may beFig. 38. [...]ound by this Analogy or Proportion.

As the S. Hypoth. C, 30° 00',

To Radius or S. 90° 00'.

So is S. of cr. AB, 27° 54',

To S of V. at C. 69 22, as required.

PROP. XV. Case 15. The Oblique Angles given, to find either Side.

[Page]

In the Triangle ABC, there is given the Angle A 23° 30', and the Angle at C 69° 22', and the side BC, is required, which may be found by this Analogy or Proportion.

As the S. of V. at C, 69° 22',Fig. 38.

To the Radius or S. 90° 00'.

So is the Sc. of V. at A, 23° 30',

To the Sc. of cr. BC, 11° 30', as required.

PROP. XVI. Case 16. The Oblique-angles given, to find the Hypothenuse.

In the Triangle ABC, there is given the Angle A 23° 30', the Angle C, 69° 22', and the Hypothenuse AC, is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',Fig. 38.

To Tc. of V. at C. 69°, 22',

So is Tc. of V. at A, 23 30,

To Sc. Hypoth. AC, 30 00, as required.

SECT. V. Of Oblique-angled Spherical Triangles.

[Page]

PROP. I. Case 1. Two Sides, and an Angle opposite to one of them given, to find the other opposite Angle.

IN the Triangle ADE, there is given the SideFig. 39. AE, 70° 00', the Side ED, 38° 30', and the Angle A, 30° 28', now the Angle at D, is required, to find which this is the Analogy or Proportion.

As S. cr. DE, 38° 30',

To S. V. at A, 30 28.

So is S. cr. AE, 70 00,

To S. V. at D, 130 03,Note that in thes [...] Operations, for the more facility of the learner, I omit Seconds, which doth belong unto the Angles, &c. required.

PROP. II. Case 2. Two Angles and a Side opposite to one of them given, to find the Side opposite to the other.

In the Triangle ADE, there is given the AngleFig. 39. at D, 130° 03', the Angle E, 31° 34', and the Side AE, 70° 00', now the Side AD, is required, which may be found by this Analogy or Proportion.

[Page]

As S. V. at D, 130° 03',

To S. cr. AE, 70 00.

So is S. V. at E, 31 34,

To S. cr. AD. 40 00, required.

PROP. III. Case 3. Two Sides and an Angle included between them being known, to find the other Angles.

In the Triangle ADE, there is given the SideFig. 39. AE, 70° 00', the Side AD, 40° 00', and the Angle A 30° 28', Now the Angles D, and E, are required, which is thus found: take the Sum and Difference of the two Sides, and work as followeth, saying. [...]

As S. ½ Z. crs. AE and AD, 55° co',

To S. ½ X. crs. AE and AD, 15 00.

So is Tc. ½ V. at A, 15 14,☞ And here observe that if the Sum of the two contained Sides exceed a Semicircle, then substract each side severally from 180°, and proceed with those Complements, as with the sides given, the Operation produceth the Complements of the Angles sought, unto a Semicircle or 180 Degrees.

To T. ½ X. VV. D and E. 49 1430".

[Page]

AGAIN.

As Sc. ½ Z. crs. AE and AD, 55° 00',

To Sc. ½ X. crs. AE and AD, 15° 00'.

So is Tc. ½ V. at A, 15° 14',

To T. ½ Z. VV. D and E, 80 48 30".

This difference of the Angles unknown D and E, 49° 14' 30", being added unto the half Sum of the Angles 80° 48' 30", (unknown) produceth the Greater Angle D 130° 03', and substracted from it, leaves the Lesser Angle E, to wit 31° 34'.

PROP. IV. Case 4. Two Angles, and their Interjacent side being known, to find the other sides.

In the Triangle ADE, there is given the Angl [...] Fig. 39. at A 30° 28', and the Angle at D 130° 03', and their Interjacent-side AD 40° 00', and the Sides DE and EA, are required: Which is thus found.

Take the Sum and Diffference of the two Angles,☞ And here observe that if the Sum of the two given Angles excede a Semicircle or 180°, substract them from a Semicircle, and proceed with the Residues, the Operation will produce each side's Complement to a Semicircle, or 180 Degrees. and work as followeth, saying.

[Page]

As S. ½ Z. of VV. A and D, 80° 15' 30", Fig. 39. [...]

To S. ½ X. of VV. A and D, 49 47 30.

So is T. ½ cr. AD, 20 00 00,

To T. ½ X. crs. DE and EA, 15 45 00.

AGAIN Say.

As Sc. ½ Z. of VV. A and D, 80° 15' 30",

To Sc. ½ X. of VV. A and D, 49 47 30.

So is T. ½ cr. AD, 20 00 00,

To T. ½ crs. Z. DE and AE. 54 15 00.

Add the half Difference of the Sides DE and AE, 15° 45', unto half the Sum of the Sides DE and AE, 54° 15'. It produceth the greater Side, the Side AE 70° 00', but if deducted from it, leaves the lesser Side ED, which is 38° 30', as was required.

PROP. V. Case 5. Two Sides and an Angle opposite to one of them given, to find the third side.

[Page]

In the Triangle ADE, there is given the SideFig. 39. AE 70° 00', the Side DE 38° 30', and the Angle A 30° 28', the Side AD is required.

First by Case 1. Prop. 1. I find the Angle at D to be 130° 03', and then proceed thus

First take the Sum and Difference of the two Angles; then also find the Difference of the two Sides given, and then work as followeth. [...] [...]

Now say,

As S. ½ X. VV. D and A, 49° 47' 30",

To S. ½ Z. VV. D and [...], 80 15 30.

So is T. ½ X. crs. AE and ED, 15 45 00,

To T. ½ cr. AD. 20° 00' 00": which doubled giveth the Side AD, 40 00 00, as was required.

PROP. VI. Case 6. Two Angles and a Side opposite to one of them given, to find the third Angle.

[Page]

In the Triangle ADE, there is given the AngleFig. 39. A 30° 28', the Angle D 130° 03', and his opposite Side AE 70° 00', and 'tis required to find the Angle at E.

First by Prop. 2. Case 2. I find the Side DE, opposed to the Angle A; to be 38° 30', then proceed thus.

Fi [...]st find the Sum and Difference of the Sides. Then find the Difference of the Angles. [...] [...]

Now say,

As S. ½ X. crs. DE and AE 15° 45',

To S. ½ Z. crs. EA and DE 54 15.

So is T. ½ X. VV. D and A 49 47 30",

To Tc. ½ V. at E 15° 47' 00". which doubled giveth the Angle at E 31 34, as required.

PROP. VII. Case 7. Two Sides and an Angle opposite to one of them given, to find the Included Angle.

[Page]

In the Triangle ADE, there is given the SideFig. 39. AE 70° 00', the Side ED 38° 30', and the Angle opposite thereunto at A 30° 28', and the Angle E is required.

First by Prop. 1. Case 1. I find the Angle D, opposite to AE, to be 130° 03', then proceed thus.

First find the Difference of the Angles, then find the Sum and Difference of the Sides. [...] [...]

Now say,

As S. ½ X. crs. AE and ED 15° 45',

To S ½ Z. crs. AE and ED 54 15.

So is T. ½ X. of VV. D and A 49 47 30",Fig. 39.

To Tc. ½ V. at E 15° 47'. Which doubled is the Angle at E 31° 34', as was required.

PROP. VIII. Case 8. Two Angles and a Side opposite to one of them being known, to find the Interjacent Side.

[Page]

In the Triangle ADE, there is given the Angle E 31° 34', the Angle D 130° 03', and his opposite Side AE 70° 00', Now the Side ED is required.

First by Prop. 2. Case 2. I find AD opposed to E, to be 40° 00', and then work thus.

Take the Sum and Difference of the Angles, then also find the Difference of the two Sides:Fig. 39. [...] [...]

Now say,

As S. ½ X. VV D and E 49° 14' 30",

To S. ½ Z VV D and E 80 48 30.

So is T ½ X crs. AD and AE 15 00 00",

To T. ½ crs. ED, 19° 15' 00", which being doubled is the Side ED 38° 30', as required.

PROP. IX. Case 9. Two Sides and their Included Angle being known, to find the third Side.

[Page]

In the Triangle APZ, there is given the SideFig. 40. ZP 38° 30', the Side PA 70°, and the Angle P, let be 31° 34, and the Side AZ is required.

The Resolution of this Case depends on the Catholike proposition of the Lord of Marchiston, by supposing the Oblique-Triangle to be divided (by a supposed Perpendicular falling either within or without the Triangle) into two Rectangulars.

Now in the Triangle AZP, let fall the Perperpendicular ZR; so is the Triangle AZP divided into two Rectangulars ARZ and ZRP. Now the Side AZ may be found at two Operations thus: say,

As the Radius or S. of 90° 00'

To Sc. of the included V, P. 31 34.

So is T. of the lesser Side PZ. 38 30,

To T. of a fourth Arch. 34 08.

If the contained Angle be less than 90°, take this fourth Arch from the greater Side; but if it be greater than 90°, from its Complement unto 180°, the Remainder is the Residual Arch: Now again say,

As Sc. of the fourth Arch. 34° 08'

To Sc. Residual Arch. 35 52

So Sc. of the lesser Side PZ. 38 30

To Sc. AZ the Side required. 40 00

[Page]

Note that if the Angles at the Base be both of one kind, that then the Perpendicular falls within the Triangle: if of diverse kinds, without the Triangle. But note that many times the Perpendicular will fall without the Triangle, as it dothFig. 41. now within; in such case the Sides of the Triangle must be continued, so will there be two Rectangulars, the one included within the other: as in the Triangle HIK, the Perpendicular let fall is KM, falling on the Side HE, and so the two Rectangulars found thereby will be IM K, and KMH, and so by the directions in the former proposition find out the Side IK, if required to be found.

PROP. X. Case 10. Two Angles and their Interjacent Side known, to find the third Angle.

In the Triangle AZP, there is given the Side ZP 38° 30', the Angle P 31° 34', and the Angle Z 130° 03', and the Angle at A is required.

First the Oblique-Triangle AZP, being reduced into two Rectangulars ARZ, and ZRP, byFig. 40. Case 9 aforegoing, I find the Angle RZP, to be 64° 19', (in the Triangle ZRP.) which taken out of Angle AZP 130° 03', leaves the Angle AZR 65° 44': Now the Angle A is found by this Analogy or Proportion.

As S. V. PZR, 64° 19',

To S. V. AZR 65 44,

So is Sc. V. at P 31 34,

To Sc. V. at A 30 28: which was required to be found out and known.

PROP. XI. Case 11. Three Sides given, to find an Angle.

[Page]

In the Triangle APZ, the Side AZ is 40° 00', the Side ZP is 38° 30', the Side AP is 70° 00', and the Angle Z is required. To find which do thus.

Add the three Sides: together, and from halfFig. 40. their Sum, deduct the Side opposite, to the required Angle: and then proceed as you see in the Operation following. [...] Fig. 40.

½Sum is 65° 07' 30", the Sc. ½ V. at Z which doubled is 130° 03' 12"; the Angle at Z required.

PROP. XII. Case 12. Three Angles given, to find a Side.

[Page]

In the Triangle AZP, the Angle A is 30° 28'Fig. 40. 11", the Angle Z 130° 03' 12", the Angle P is 31° 34' 26", and the Side AZ, opposite to P, is required.

This Case is likewise performed as the former Case or Proposition, the Angles being converted into Sides, and the Sides into Angles, by taking the Complement of the greatest Angle unto 180°: see the work. [...] Fig. 40. which being doubled, gives the Side AZ 40° 00 required to be found out and known

☞ But if the greater Side AP were required the Operation would produce the Complem [...] thereof unto a Semicircle or 180°; therfo [...] [Page] substract it from 180°, it leaves the remaining required Side sought.

Thus I have laid down all the Cases of Triangles, both Right-lined and Spherical; either Right, or Oblique-angled; I might hereunto have annexed many Varieties unto each Case, and some fundamental Axioms, which somewhat more would have Illustrated and Demonstrated those Cases, and Proportions; but because of the smallness of this Treatise, which is intended more for Practice than Theory, I have for brevity sake omitted them, and refer you for those things to larger Authors, who have largely discoursed thereon to good purpose.

CHAP. VI. Of ASTRONOMY.

[Page]

Defin. ASTRONOMY is an Art Mathematical, which measureth the distinct course of Times, Days, Years, &c. It sheweth the Distance, Magnitude, Natural Motions, Appearances and Passions, proper unto the Planets, and fixed Stars, for any time past, present and to come; by this we are certified of the Distance of the starry Sky, and of each Planet, from the Center of the Earth, and the Magnitude of any fixed Star or Planet, in respect of the Earth's Magnitude.

SECT. I. Of Astronomical Definitions.

Defin.

1. ASphere or Globe is a solid Body, containing onely one Superficies, in whose middle there is a point (called the [Page] Center,) from which all right or streight lines drawn unto the Circumference or Superficies, are Equal.

2. The Poles of the World, are two fixed points

Fig. 42.

in the Heavens Diametrically opposite the one to the other, the one called the Artick or NorthPole; noted in the Scheme by P. The other is called the Antartick, or South-Pole; as S. and is not to be seen of us, being in the lower Hemisphere.

3. The Axis of the World, is an imaginary

Fig. 42.

line drawn from the North-Pole, through the Center of the Earth, unto the South-Pole, about which the Diurnal motion is performed, from the East to the West; as the line PS.

4. The Meridians are great Circles, concurring

Fig. 42.

and intersecting one another, in the Poles of the World, as PES, and Pc S.

5. The Equinoctial, or Equator, is a great Circle, 90 deg. distant from the Poles of the World, cutting the Meridians at Right-angles, and divideth the World into two Equal parts, called

Fig. 42.

the Northern, and Southern Hemispheres, as E ♎ Q. in Scheme 42.

6. The Ecliptick is a great Circle, crossing the Equinoctial, in the two opposite points Aries and Libra, and maketh an Angle therewith (called, its Obliquity) of 23° 30', represented by ♋ ♎ ♑. This Circle is divided into 12 Sines, each containing 30° 00': As Aries

Fig. 42.

♈, Taurus ♉, Gemini ♊, Cancer ♋, Leo ♌, Virgo ♍, (which are called Northern Sines) Libra ♎; Scorpio ♏, Sagitarius ♐; Capricornus ♑, Aquarius ♒, and Pisces ♓; these are called Southern Sines.

7. The Zodiack is a Zone or Girdle, having 8 deg. of Latitude on either side the Ecliptick, in which space the Planets make their revolution. This Circle is a Circle which regulates the Years, Months, and Seasons,

Which is by the Greeks called [...], i e. bring Life; because the life of all Creatures depend on the cause of that Circle, for the Sun ascending in it and moving towards us, brings the Generation of things, and in descending the Corruption of all things sensible, and insensible, which are below the concavity of the Moon, &c.

and is distinguished

Fig. 42.

in the Scheme by the 12 Sines.

8. The Colures are two Meridians, dividing the Ecliptick, and the Equinoctial, into four equal parts; one of which passeth by the Equinoctial

Fig. 42.

points Aries, and Libra, and is called the Equinoctial Colure, as P ♎ S. The other by the beginning of Cancer, and Capricorn, and is called the Solstitial Colure, as P ♋, S ♑.

9. The Poles of the Ecliptick are two points, 23° 30' distant from the Poles of the World, as I and K.

Fig. 42.

10. The Tropicks are two small Circles, Parallel unto the Equinoctial, and distant therefrom 23° 30', limiting the Sun's greatest declination. The Northern Tropick passeth by the beginning of Cancer, and is therefore called the Tropick of Cancer, as ♋ a D. The Southern Tropick

Fig. 42

passeth by the beginning of Capricorn, and is therefore called the Tropick of Capricorn; as B b ♑.

11. The Polar Circles, are two small Circles parrallel to the Equinoctial, and distant therefrom 66° 30'; and from the Poles of the World [Page] 23°. 30'. That which is adjacent unto the North Pole, is called the Artick Circle, as G d I. and the other the Antartick Circle, as Kd M.

Fig. 42.

12. The Zenith, and the Nadir, are two points, Diametrically opposite the one to the other: the Zenith is the Vertical point, or the point over our heads, as Z, The Nadir, is opposite thereto as the point N.

13. The Azimuths or Vertical Circles are great Circles of the Sphere, concurring and intersecting each other, in the Zenith, and Nadir, as Z f N.

14 The Horizon, is a great Circle, 90 deg.

Fig. 42.

distant from the Zenith, and Nadir; cutting all the Azimuths, at Rightangles, and dividing the World into two equal parts, the upper and visible Hemisphere, and the lower and invisible Hemisphere, represented by H ♎ R.

15. The Meridian of a Place, is that Meridian, which passeth by the Zenith, and Nadir, of

Fig. 42.

the place as P Z S N.

16. The Alinicanthars, or Parallels of Altitude, are small Circles, parrallel unto the Horizon, (imagined to [...]pass through every degree and minute of the Meridian, between the Zenith, and Horizon, B a F.

17. Parallels of Latitude, or Declination, are small Circles parallel unto the Equinoctial; they

Fig. 42.

are called Parallels of Latitude, in respect to any place on the Earth, and Parallels of Declination, in respect of the Sun, or Stars, in the Heavens.

18. The Latitude of a place, is the height of the Pole above the, Horizon; or the distance between the Zenith and the Equinoctial.

19. The Latitude of a Star, is the Arch of a Circle, contained betwixt the Center of a Star, and the Ecliptick line: this Circle making Rightangles, with the Ecliptick, is accounted either Northward or Southward; according to the Scituation of the Star.

20. Longitude on Earth is measured by an Arch of the Equinoctial, contained between the Primary Meridian, (or Meridian of that place where Longitude is assigned to begin) and the Meridian of any other place, counted always Easterly.

21. The Longitude of a Star, is that part of the Ecliptick, which is contained between the Star's place in the Ecliptick, and the beginning of Aries, counting them according unto the succession of Sines.

22. The Altitude of the Sun or Stars, is the Arch of an Azimuth, contained betwixt the Center of the Sun, or Star, and the Horizon.

23. Ascension is the rising of any Star, or part of the Equinoctial, to any degree above the Horizon; and Descension is the setting of it.

24. Right Ascension, is the number of Degrees and Minutes of the Equinoctial; (i. e. from the beginning of Aries) which cometh unto the Meridian, with the Sun or Stars; or with any portion of the Ecliptick.

25. Oblique-Ascension, is an Arch of the Equinoctial, between the beginning of Aries, and that part of the Equinoctial which riseth with the Center of a Star; or with any portion of the Ecliptick in an Oblique Sphere: and Oblique Descention, is that part of the Equinoctial, tha [...] setteth therewith.

26. The Ascentional difference, is an Arch of the Equinoctial, being the difference betwixt the Right and Oblique-Ascension.

27. The Amplitude, of the Sun or Stars, is

Fig. 43.

the distance of the rising or setting thereof, from the East or West point of the Horizon.

28. The Parallax, is the difference between the true and apparent place of the Sun or Star; so the true place in respect of Altitude, is in the line ACE, or ADG, the Sun or Star being at C, or D, and the apparent place in the Line BCF, and BDH, so likewise the Angles of the Parallax are ACB, or ECF; and ADB, or GDB: also in the said Scheme, ABK representeth a

Fig. 43.

Quadrent (of the Globe or Earth,) on the Earth's Superficies: A the Center of the Earth, and B any point of the Earth's Surface.

29. The Refraction of a Star, is caused by the Atmosphere, or Vapourous thickness of the Air near the Earth's Superficies, whereby the Sun and Stars seem always to rise sooner, and and set later than really they do.

SECT. II. Of Astronomical Propositions.

[Page]

PROP. I. The Distance of the Sun from the next Equinoctial point (either Aries or Libra) being known, to find his Declination.

THE Analogy or Proportion.

As Radius or S. 90°,

To S. of the Sun's distance from the next Equinoctial point,

So it S. of the Sun's greatest Declination,

To the S. of the Sun's present Declination sought.

PROP. II. The Sun's place given, to find his Right-Ascension.

This is the Analogy or Proportion.

As Radius or S. 90°,

To T. of the Sun's Longitude from the next Equinoctial point,

So is the Sc. of his greatest Declination,

To T. of his Right-Ascension from the next Equinoctial point.

PROP. III. To find the Sun's place or longitude from Aries, his Declination being given.

[Page]

This is the Analogy or Proportion.

As S. of the Suns greatest Declination,

To Radius or S. 90° 00',

So is S. of his present Declination,

To S. of the Suns Place or Longitude from AriesIf the Sun's Declination be North, and increasing, this Proportion finds the Sun's distance from ♈; but if decreasing from ♎ in the northern Sines. But if the Sun's Declination be South, and increasing from ♎; if decreasing from ♈, among the Southern Sines.

PROP. IV. By knowing the Suns Declination, to find his Right Ascension.

This is the Analogy or Proportion.

As Radius or S. 90°,

To Tc. of the Suns greatest Declination,

So is T. of the Declination given,

To S. of the Suns right Ascension requiredFrom the next Equinoctial point either ♈, or ♎..

PROP. V. By knowing the Latitude of a Place, and the Suns Declination, to find the Ascensional Difference.

[...]

[Page]

[...]

[Page]

[Page]

This is the Analogy or Proportion.

As Radius or S. 90°,

To Tc. of the Latitude given,

So is T. of the Suns Declination given,

To the S. of the Ascensional difference required.

☞ Note that if you reduce the degrees, &c. of the Ascensional difference into hours, it will shew you how much the Sun riseth, or setteth before, or after six a Clock, in that Latitude.

PROP. VI. To find the Suns Oblique Ascension or Descension.

First find the Ascensional Difference by the 5th Proposition, and his Right-ascension by the fourth: Now if the Suns Declination be Northerly, deduct the Ascentional Difference out of his Right Ascension, from the beginning of ♈, (for the six Northern Signs ♈ ♉ ♊ ♋ ♌ ♍) it leaves the Oblique Ascension; and added unto the Rightascension, giveth the Oblique-descension.

But if the Suns Declination be Southerly, the Ascentional Difference, added to the Right-ascension, (for the six Southern Signs ♎ ♏ ♐ ♑ ♒ ♓) giveth the Right-ascension, and substracted there from leaves the Oblique-descension.

[Page] [Page] [Page]

Plate 11 Page 105

PROP. VII. By knowing the Suns Declination, and the Latitude of a Place, to find the Suns Amplitude.

[Page]

[Page]

This is the Analogy or Proportion.

As Sc. of the Latitude,

To the Radius or S. 90°.

So is the S. of the Suns Declination,

To the S. of the Amplitude from the East or West Points of the Horizon.

PROP. VIII. By knowing the Suns Declination and Amplitude, from the North part of the Horizon, to find the Latitude.

This is the Analogy or Proportion.

As Sc. of the Amplitude from the North,

To Radius or S. 90° 00'

So is S. of his Declination given,

To Sc. of the required Latitude.

PROP. IX. By knowing the Latitude of a place, and the Sun's Declination, to find at what time the Sun will be on the true East or West Points.

The Analogy or Proportion is.

As T. of the given Latitude,

To T. of the Sun's Declination propounded,

So is Radius or S. 90° 00',

To, Sc. of the Hour from Noon.

PROP. X. By knowing the Sun's Declination, and Latitude of a place, to find his Altitude at six a Clock.

[Page]

[...]

[Page]

[Page]

This is the Analogy or Proportion.

As Radius or S. 90° 00',

To S. of the Sun's Declination,

So is S. of the Latitude of the place,

To S. of the Sun's Altitude at six a Clock.

PROP. XI. By knowing the Latitude of a place, and the Sun's Declination, to find the Azimuth at six.

This is the Analogy or Proportion.

As Radius or S. 90° 00',

To the T. of the Sun's Declination,

So is Sc. of the Latitude of the place,

To the T. of the Azimuth sought.

PROP. XII. By knowing the Latitude of a place, and the Sun's Declination, to find the Sun's Altitude when he i [...] on the true East or West points.

This is the Analogy or Proportion.

As S. of the Latitude,

To the Radius or S. 90° 00',

So is the S. of the Declination,

To the S. of the Sun's Altitude being due Ea [...] or West.

PROP. XIII. To find the Sun's Altitude at any time of the day.

[Page]

The Analogy or Proportion is.

As Radius or S. 90° 00',

To Tc. of the Poles height,

So is S. of the Sun's Distance,

From the Hour of Six,

To the T. of an Arch: which being substracted from the Sun's Distance from the Pole; say,

As Sc. of the Arch found,

To Sc. of the remaining Arch of the Sun's Distance from the Pole,

So is S. of the Poles height,

To the S. of the Sun's Altitude at the Hour required.

PROP. XIV. By knowing the Latitude of a Place, with the Sun's Declination, and Altitude, to find the Hour of the Day.

To solve this Conclusion do thus: get the Sum of the Complements of the Latitude, Declination and Altitude givenAs in Case 11 of Oblique Spherical Triangles., Then find the Difference betwixt their half Sum, and the Complement of the Altitude; then say,

As Radius or S. 90° 00',

To Sc. of the Sun's Altitude,

So is Sc. of the Latitude of the Place,

[Page]

To a fourth Sine: then again say,

As the fourth S.

To the S. of ½ Z. of the Lat. Declin. and Alt.

So is the S. of X. of the Altitude to the ½ Z,

To a fifth S. unto which Sine, if you add the Radius or 90° 00', half that Sum shall be the Sine of an Arch, whose double Complement is the Hour from the Meridian.

PROP. XV. To find the Time of the Sun's Rising or Setting, and consequently the Length of the Day or Night.

To resolve this Conclusion, first by prop. the 5. find the Ascensional Difference, which reduced into Hours, and Minutes of Time, by allowing for every 15 Deg. one Hour, and for every Deg. less than 15°, 4', of Time, and for every 15 Min. one Minute of Time.

Secondly, If the Sun's Declination be Northerly, the Ascentional Difference added unto 6 Hours, gives the Time of Sun-setting, and substracted therefrom, leaves the Time of SunRising: On the contrary, if the Sun's Declination be Southerly, the Ascentional Difference added unto 6 Hours, gives the Time of Sun-Rising, and deducted therefrom, the Time of Sunsetting.

Thirdly, If you double the Time of SunRising, it gives you the length of the Night; and the Time of Sun-setting, the length of the Day.

PROP. XVI. The Sun's Declination, Altitude and Azimuth known, to find the Hour of the Day.

[Page]

The Analogy or Proportion is.

As the Sc. of the Sun's Declination,

To the S. of the Azimuth,

So is the Sc. of the Altitude,

To the S. of the Hour from Noon: which converted into Time, will shew the Hour of the Day.

PROP. XVII. By knowing the Sun's Declination, Altitude, and Hour from Noon, to find the Azimuth.

The Analogy or Proportion is.

As Sc. of the Sun's Altitude,

To S. of the Hour from Noon,

So is Sc. of the Sun's Declination,

To the S. of the Azimuth, required.

PROP. XVIII. By knowing the Latitude of a place, the Altitude of the Sun, and the Hour from Noon, to find the Angle of the Sun's Position.

This is the Analogy or Proportion.

As Sc. of the Sun's Altitude,

To S. of the Hour from Noon,

So is Sc. of the Latitude,

[Page]

To S. of the V. of the Sun's Position, at the time of the Question.

PROP. XIX. By knowing the Sun's Altitude, Declination, and Azimuth; to find the Latitude.

The Analogy or Proportion is.

As S. of the Sun's Azimuth,

To S. of his Distance from the North-pole,

So is S. of V of the Sun's Position,

To Sc. of the Latitude required.

PROP. XX. To find the length of the Crepusculum, or Twilight

The Crepusculum or Twilight, is nothing else but the Refraction of the Sun's Beams in the Density of the Air. Which the Learned Pet. Nonnius found the length of the Crepusculum (by his many strict observations Watched the Time after Sun-setting when the Twilight in the West was shut in, so that no more Twilight than in any other part of the Skie near the Horizon appeared there: then by one of the known fixed Stars, having found the true Hour of the Night, he found the length of the Twilight, to be as in the rule is mentioned.) to continue from the time of the S [...] passing below the Horizon of a place, untill the Sun had run below the said Horizon 18° 00', and then followed the shutting in of the Twilight, and untill the Sun was [Page] departed so low the Twilight continued. — To find which observe this Analogy or Proportion.

As Radius or S. 90°,

To Sc. of the Sun's Declination,

So is Sc. of the Poles-height,

To a fourth Sine: which keep.

Then out of the Sun's Distance from the South-Pole, subduct the Complement of the Pole; and of that remains and the degrees 62, being added to it, their Sum and Difference found, say again.

As the fourth Sine found,

To S. ½ Z of the remainder and 62° 00',

So is S. ½ X. of the remainder and 62° 00',

To a Number, which being multiplyed by the Radius is equal unto the Quadrat of the Sine of the ½ Angle of the Sun's Distance at the Ending of the Twilight, from Noon next ensuing.

Then from the Sun of the whole Angle converted into Hours, substract the Hour of the Sun's setting Or ½ Diurnal Arch., it gives you the length of the Crepusculum, or Twilight.

But the Sun being in the Winter Tropick, makes the Twilight longest of anyTo find the length of the least Crepusculum or Twilight. Twilight, the whole Winter half year: Now in a certain Parallel, betwixt that Tropick, and the Equinoctial is the shortest Crepusculum: the Declination of which Parallel, is thus found.

As the Tc. of the Pole,

To the S of the Pole,

So is the T. of 99° 00',

[Page]

To S. of the Declination of the Parallel, in which the Sun maketh the shortest Crepusculum of the Year.

But before the Crepusculum come to be shortest, there is another Parallel, in which the Crepusculum is equal to that of the Equinoctial: the Declination of which is found thus.

As the Radius or S. 90° 00',

To S. of the Poles Elevation or Altitude,

So i [...] S. of 18° 00',

To S. of the Declination of the Parallel, in which the Sun maketh the Crepusculum equal to that in the Equinoctial.

PROP. XXI. To find the Quantity of the Angles, which the Circles of the 12 Houses make with the Meridian.

This is the Analogy or Proportion.

As the Radius or S. 90°,

To T. of 60°: for the 11th, 9th, 5th and 3d House, Or to the T. 30° for the 12th, 8th, 6th, and 2d House,

So is the Sc. of the Pole,

To the Tc. of any House with the Meridian.

PROP. XXII. To find the Right Ascension of the Point in the Equinoctial: and also the Point in the Ecliptick; called Medium Coeli or Cor Coeli.

[Page]

First, To find the Right Ascension of the Point of the Equinoctial; called Medium Coeli, vel Cor Coeli, find out the Sun's Right Ascension, by prop. 2. Then reduce the whole Time from Noon last past into degrees, which add unto the right Ascension of the Sun, so shall their Agragat, be the right Ascension of the point, which in the Equinoctial, is called Medium Coeli, vel Cor Caeli, required to be found.

Secondly, By the 2 propositions aforegoing, you may find the right Ascension of the point in the Ecliptick Culminant in the Meridian, called Medium Coeli vel Cor Coeli, which is the Cuspis of the tenth House: and his Declination by prop. the first.

PROP. XXIII. To find the Angle of the Ecliptick with the Meridian.

The Analogy or Proportion is.

As the Radius or S. 90°,

To S. of the Sun's Greatest Declination,

So is Sc. of the Sun's right Ascension, from the next Equinoctial point,

To Sc. of the V. of the Ecliptick, with the Meridian.

PROP. XXIV. To find the Angle of the Ecliptick with the Horizon.

[Page]

The Analogy or Proportion is.

As Radius or S. 90°,

To Sc. of the Altitude of Cor Coeli,

So is S. of the V. Ecliptick with the Meridian,

To Sc. of the V. of the Ecliptick and Horizon sought.

PROP. XXV. To find the Amplitude Ortive of the Ascendent, or Horoscopus.

This is the Analogy or Proportion.

As Radius or S. 90°,

To S. of Altitude of Med. Coeli,

So is T. of V. Ecliptick with the Meridian,

To Tc. of the Amplitude Ortive of the Ascendent, or the distance of the Azimuth from the Meridian.

PROP. XXVI. To find the Ascendent degree of the Ecliptick, or the Cuspis of the first House.

The Amplitude Ortive of the Ascendent, is equal to the Distance of the Azimuth of 90°, from the Meridian, wherefore the Cuspis of the [Page] first House, or Ascendent Degree of the Ecliptick, may be found thus.

As Radius or S. 90°,

To Sc. of the V. Ecliptick with the Meridian,

So is Tc. of the Altitude of Med. Coeli,

To T. of the Distance of Med. Coeli, from the Ascendent Degrees.

PROP. XXVII. To find the Distance of the Cuspis of any House, from Med. Coeli.

This is the Analogy or Proportion.

As Sc. of the remaining part of V. of the Ecliptick with the Meridian, (found by prop. 28.)

To Sc of the former part of the V,

So is T. of the Altitude of Med. Coeli,

To T. of the Distance of the Cuspis of that House sought, from Med. Coeli.

PROP. XXVIII. To find the parts of the Angle of the Ecliptick with the Meridian, cut with an Arch perpendicular to the Circle of any of the Houses.

The Analogy or Proportion is:

As Radius or S. 90°,

To Sc. Altitude of Med. Coeli,

So is T. of the Circle of any House with the Meridian,

To Tc. of that part of that Angle which is next the Meridian:

Then substract that part found, out of the whole Angle, for the remaining or latter part

PROP. XXIX. To find the Pole's Altitude, above any of the Circles of the Houses.

[Page]

The Analogy or Proportion is.

As the Radius or S 90°,

To S. of V. of the Circle of the House with the Meridian: (found by the 21 prop.)

So is the S of the Poles Elevation, above the Horizon of the Place,

To S. of the Altitude of the Pole, above the Circle of Position.

PROP. XXX. By knowing the Latitude and Longitude of any fixed Star, to find his Right Ascension and Declination.

The Analogy or Proportion is.

1. As Radius or S. 90°,

To S. of the Stars Longitude from the next Equinoctial point,

So is Tc. of the Stars Latitude,

To T. of a fourth Arch.

Which compared with the Arch of Distance betwixt the Poles of the World and the Ecliptick 23°, 30'; And if the Latitude and Longitude of the Star be both of one Dignity, i e. when the Star hath North Latitude in the six Northern Sines, ♈, ♉, ♊, ♋, ♌, ♍, or South Latitude in the six Southern Sines, ♎, ♏, ♐, ♑, ♒, ♓: Then shall the difference between this found Arch, and the

[Page][Page][Page]

Page 216 Plate III

[Page] [Page] Distance of the Poles be your fifth Arch: But if the Latitude and Longitude of the Star be of contrary qualities, i. e the one Northern, and the other Southern, then add this fourth Arch to the Distance of the Poles 23° 30', and the Sum thereof shall be your fifth Arch; with which,

AGAIN, say.

2. As S. of the fourth Arch,

To S. of the fifth Arch,

So is T. of the Stars Longitude,

To T. of the Stars Right-ascension from the next Equinoctial point.

3. As Sc of the fourth Arch,

To Sc. of the fifth Arch,

So is S of the Stars Latitude,

To S. of the Stars Declination.

I might also shew how by having the Latitude and Longitude of any two fixed Stars, to find their Distance: but because 'tis the very same with finding the Distance of any two Places on Earth, I refer you to the Directions of Prop 1, 2 and 3. of Chap. 7, ensuing, where you will see the plain Demonstration thereof.

PROP. XXXI. By knowing the Pole's Altitude, to find when any fixed Star shall be due East or West.

This is the Analogy or Proportion.

As Radius or S. 90°,

[Page]

To T. of the Stars Declination,

So is Tc. of the Pole,

To Sc. of the Stars Horary Distance from the Meridian.

PROP. XXXII. By knowing the Poles Altitude, to find the Elevation of any fixed Star above the Horizon, being due East or West.

This is the Analogy or Proportion.

As S. of the Poles Altitude,

To Radius or S. 90°,

So is S. of the Stars Declination,

To S. of the Stars Elevation, above the Horizon, at due East or West.

PROP. XXXIII. To find out the Horizontal Parallax of the Moon.

The Analogy or Proportion.

As the Moons Distance from the Center of the Earth,

To the Earth's Semidiameter,

So is Radius or S. 90°,

So S. of the Moon's Horizontal Parallax in that Distance.

PROP. XXXIV. The Horizontal Parallax of the Moon being known, to find her Parallax in any apparent Latitude.

[Page]

This is the Analogy or Proportion.

As Radius or S. 90°,

To S. of the Moon's Altitude,

So is S. of the Moon's Horizontal Parallax,

To S. of the Parallax in that Altitude.

PROP. XXXV. By knowing the Moon's Place in the Ecliptick, (having little or no Latitude) and her Parallax of Altitude, to find the Parallaxes of her Longitude and Latitude.

First, If the Moon be in the 90° of the Ecliptick, she hath then no Parallax of Longitude, and the Parallax of the Latitude, is the very Parallax in that Altitude.

Secondly, But if the Moon be not in the 90th. Degree of the Ecliptick, to find the Parallaxes of the Latitude and Longitude, the Analogy or Proportion is,

1. As Radius or S. 90°,

To T. of the V. of the Ecliptick and Horizon,

So is Sc. of the Moon's Distance from the Ascendent, or Descendent deg. of the Ecliptick,

To Tc. of the Ecliptick's V, with the Azimuth of the Moon.

[Page]

AGAIN say,

2. As the Radius or S. 90°,

To S. of that V. found,

So is the Parallax of the Moon's Altitude,

To the Parallax of her Latitude sought.

LASTLY say,

3. As the Radius or S. 90°, 00'

To Sc. of the former V. found,

So is the Parallax of the Moon's Altitude,

To the Parallax of his Longitude sought, which being added to the true Motion of the Moon, if she be on the East part of the 90° of the Ecliptick. Or from it to be deducted if she be on the West part of the 90° of the Ecliptick.

PROP. XXXVI. How by knowing the Refraction of a Star, to find his true Altitude.

For the speedy performance of which I have annexed this Table of Refractions of the Stars observed by Tycho Brabe a Nobleman of Denmark, and a most famous Astronomer.

[Page]

A Table of the Refraction of the Stars observed by Tycho Brabe.

Altitude. | Refraction.
o° | 30' | 00"
1 | 21 | 30
2 | 15 | 30
3 | 12 | 30
4 | 11 | 00
5 | 10 | 00
6 | 9 | 00
7 | 8 | 15
8 | 6 | 45
9 | 6 | 00
10 | 5 | 30
11 | 5 | 00
12 | 4 | 30
13 | 4 | 00
14 | 3 | 30
15 | 3 | 00
16 | 2 | 30
17 | 2 | 00
18 | 1 | 15
19 | 0 | 30
20 | 0 | 00

The USE of which Table is thus.

EXAMPLE.

Suppose the Altitude of a Star were found by Observation to be 13°; the correspondent Refraction is 4' 00", which substracted from 13° leaves 12°, 56', which is the true Altitude

CHAP. VII. Of GEOGRAPHY.

[Page]

Defin. GEOGRAPHY is an art Mathematical, which sheweth how the Situations of Kingdoms, Provinces, Cities, Towns, Villages, Forts, Castles, Mountains, Woods, Havens, Rivers, Creeks, &c. being on the Surface of the Terrestrial Globe, may be described, and designed, in commensuration Analogical to Nature, and Verity: and most aptly to our view may be represented.

Ptolomy saith of Geography, [...]. That it is a description of all the known Earth, imitated by writing and delineation: with all other things belonging thereunto. Of all which I shall say somewhat, as to its Situation, Commodity, Customs, &c. concerning which Ovid saith, Met. lib. 2. [Page] Terra, viros, Urbesque gerit, frugesque, ferasque, Fluminaque; haec super est Caeli fulgentis imago.

In English Thus.

The Earth, Men, Towers, Fruits, Beasts, and Rivers bears,

And over these are place'd the Heavenly Spheres.

SECT. I. Of GEOGRAPHICAL Definitions.

Defin.

1. THE Globe of the Earth is a Spherical Body composed of Earth, and Water, and is divided into Continents, Islands and Seas.

2. A Continent is a great Quantity of Land not separated, interlaced or divided by the Sea, wherein are Kingdoms, Principalities and Nations, as EUROPE, ASIA and AFRICA, are one Continent: and AMERICA is another.

3. An Island is such a part of the Earth that is environed round with Water on every Side, as the Isle of Great Britain, Java, Wight, &c.

4. A Peninsula is such a Tract of Land which being almost cut off from the Main Land, and encompassed round with Water, yet nevertheless is joyned unto the firm Land, by some little Isthmus, as Peloponesus, Peruviana, Taurica, Cymtryca and Morea in the Levant.

5. An Isthmus is a little narrow Neck of Land which joyneth the Peninsula unto the Continent.

6. A Promontory is some high Mountain, which shooteth it self into the Sea, the utmost end of which is called a Cape: as Cape-boon, Esperance, Cape d'Verde, and Cape d'Coquibocao.

7. The Ocean is a general Collection of Waters, which environeth the World on every side, and produceth Seas, Straits, Bays, Lakes, and Rivers: Of which and other Waters Ovid thus speaks in his Metamorphosis.

Tum Freta diffudīt, rapidisque tumescere ventis

Jussit, & ambitae circundare littora terrae.

He spread the Seas, which then he did command

To swell with Winds, and compass round the Land.

8. The Sea is part of the Ocean, to which we cannot come but through some Strait, as the Mediterranean, or Baltick Sea.

9. A Strait is a part of the Ocean restrained within narrow bounds, yet openeth a way to the Sea, as the Straits of Gibralter, Helespont, &c

10. A Creek is a crooked Shoar thrusting, as. it were, two Armes forth to hold the Sea; as the Adriatick, Persian, and Corinthian Creeks: from whence are produced Rivers, Brooks and Fountains: which are engendred of Congealed Air in the Earths Concavity, and seconded by Sea-water creeping through the hidden Cranies of the Earth.

11. A Bay is a great Inlet of Land, as the Bay of Mexico, and Biscay.

12. A Gulph is a greater Inlet of Land and [Page] deeper than a Bay, as the Gulph of Venice, and Florida.

13. A Climate is a certain space of Earth and Sea, included within the space of two Parallels; and there have been anciently accounted these seven: viz. 1. Dia Meros, 2. Dia Syenes, 3. Dia Alexandria, 4. Dia Rhodes, 5. Dia Rhomes, 6. Dia Boristhenes, and 7. Dia Ripheos.

14. A Zone is a certain space of Earth contained betwixt certain Circles of the Sphere, of which there are five: viz. The Torrid or Burning Zone, two Temperate, and two Frigid or Frozen Zones.

The Torrid Zone is that which lieth on each side the Equinoctial, whose bounds are the two Tropicks of ♋ and ♑.

The two Temperate Zones are those which lieth betwixt the two Tropicks of ♋ and ♑, and the Palar Circles.

The two Frigid Zones lieth between the Artick and Antartick Circles, and their respective Poles: Of which Ovid thus speaks.

Metam. 1.

Utque duae dextrâ Coelum, totidemque sinistrâ

Parte secant Zonae, quinta est ardientior illis:

Sic onus inclusum munero distinxit eodem

Cura Dei, totidemque plaga tellure premuntur:

Quarum quae media est, non est habitalis aestu;

Nix tegit alta duas: totidem inter utram (que) locavit

Temperiemque dedit mistâ cum Frigore Flammâ.

SECT. II. Geographical Descriptions of the Earth.

[Page]

THE whole Earth is divided into four parts.

VIZ.

EUROPE,

ASIA,

AFRICA and

AMERICA.

1

EUROPE, the first part of the World, isEurope. divided from ASIA by the Mediterranean Sea; bounded on the West with the Western Ocean; East with the River Tanais. It is lesser than ASIA, or AFRICA, yet doth excell all the other parts, in Worthiness, Fame, Power, multitudes of well builded Cities, strong Fortifications, full of a Wity and Learned People, Courageous Wariours, and the knowledge of God, better than all the Riches of the World. It once had the dominion of ASIA and AFRICA, and in it were fourteen Mother Tongues, and doth contain these Provinces: Viz. Italy, Spain, Alps, France, Britain, Belgia, Germany, Denmark, Sweden, Russia, Poland, Hungary, S [...]lavonia, Dacia, and Greece, with its several Islands, which shall be mentioned in their due places.

[Page]

ItalyWhich Pliny hath adorned in these words (saith he) Italia terrarum omnium alumna, eadem & parens, numine Deûm electa, qua Coelum ipsum clariùs faceret, sparsa congregaret imperia, ritus molliret, tot populorum discordes linguas sermonis commercio ad Colloquia distraheret, & humanitati hominem daret, i. e. Italy (saith Pliny) is the Nurse and the Parent of all Religion, was elected by the Providence of the Gods, to make (if possible) the Heavens more famous; to gather the scattered Empires of the World into one Body, to temper the Barbarous rites of the Nations, to unite so many disagreeing Languages of Men by the benefit of one common Tongue: and in a word to restore Man to his Humanity. the Mother of Latine learning, is boundedEurope. East with the Adriatick and Tuscan Seas, West with France, North with Germany, and South with the River Varus, and the Alps. It hath had seven kinds of Governments: First Kings, Dictators, Consuls, Decimivires, Tribunes, Emperours, and lastly Popes. It far excelleth all the other Lands in EUROPE in fruitfulness and pleasantness. The Inhabitants are witty and frugal, yet hot and lascivious, and very jealous of their Wives; they are of the Popish Religion, and its chief Commodities are Rice, Silk, Velvets, Sattins, Taffeties, Grogerams, Arras, Gold and Silver, Threed, Venetian Glasses, &c.

Italy at this day contains the Kingdom of Naples, Sicily, Sardina, the Lands of the Pope, now Innocent the XI. the Dukedom of Tuscany, Urbin, the Republick of Venice, Genoa, and Luca. The Estates of Lumbardy, being the Dukedom of Millain, Mantua, Modena, Parma, Mountferrat, and the Principality of Piemont, of all which we shall treat in their order.

[Page]

The Kingdom of Naples is environed withEurope. the Adriatick, Ionian, and Tuscan Seas; except where it is joyned to the Lands of the Church, from which 'tis separated by a line drawn from the mouth of the River Tronto, falling into the Adriatick, and to the spring-head of Axofenus, taking in all the East of Italy, 1468 miles. It is very fertile, abounding with all things necessary for the life of Man, delight, and Physick: from hence come the Neopolitan Horses. It hath had 13 Princes, 24 Dukes, 25 Marquesses, 19 Earls, and 900 Baronets, and 26 Kings of several Countries of the Norman and Spanish race, whom 'tis now under: here the Disease called the French-Pox derived its Original: the Arms are Azure Seme of Flower-d'-lices, or a File of three Lables Gules; its revenues are 2500000 Crowns, 20000 of which belongs to the Pope, and the rest are imployed to maintain the Garisons against the Turks; so that scarcely 60000 Crowns falls to the King of Spain s share; it hath 20 Archbishops, and 124 Bishops Sees.

Sicily is situated under the fourth Climate, it shoots forth into the Seas with three Promontories; the Inhabitants are Eloquent, Ingenious, and Pleasant, but very unconstant, and Talkative; the first Inventors of Oratory. It's a fruitfull Soil, it yields Wine, Grain, Oyl, Hony, Gold, and Silver, Agats, Emeralds, Allom, Salt, Sugar, and Silks. Here is the Hill Aetna, supposed to be Hell, and by the Papist Purgatory, because of its vomiting Smoak and Fire: it hath many Cities, Rivers, Lakes, whose descriptions must here be omitted; it hath had eight Kings; the first were of the Arragon Family, and began [Page] Rule Anno 1281. But it's now united to the Crown of Spain; its Revenues are 800000, or a 1000000 of Ducats, which is disburst on theEurope. Account of the Vice-Roy, and Defence of the Countrey; the Arms are four Pallets Gules, Sable for Aragon, between two Flanches Argent, charged with as many Eagles Sable beaked Gules. It hath had seven Princes, four Dukes, thirteen Marquesses, fourteen Earls, one Viscount and fortyeight Barons, they are of the Romish Religion, and have three Archbishops, and nine Bishops.

The Kingdom and Isle of Sardina, lieth West from Sicily and Cap Bara, whose length is 180, and breadth 90 Miles; the People are low of Stature, and of a swarthy Complection, rude, slothfull, and rebellious, their diet mean, yet rich in their Apparel, they are of the Romish Religion; but have an ignorant and illiterate Clergy. It belongs to the King of Spain, and governed by a Vice-Roy, under whom are two Deputy Spaniards: but other inferiour Officers may be Natives. It hath neither Wolf, nor Serpent, nor venemous Beast, but the Fox only, and a little Spider, which cannot endure the light of the Sun; they are destitute of Water, and are therefore forced to keep the Rain that falls in Summer for their Use in Winter, the Air is unhealthfull and Pestilential; the Soyl Fertile, but ill manured; it hath plenty of Cattel, their Horses will last very long, the Natives ride on their Bullocks as we on our Horses, here is also a Beast called Mufrones, resembling a Stagg, whose Hide is used as Armour, and an Herb which eaten produces Death with excessive [...]aughter, it yields to the King of Spain but a [Page] small Revenue. The Arms are Or, a Cross Gules betwixt four Sarazens Heads, Sabled CurledEurope. Argent, it hath several Isles belonging thereunto, it hath three Archbishops, and fifteen Bishops.

The Lands of the Church or the Pope's Dominion in Italy, lieth West of Naples, extended North and South from the Adriatick to the Tuscan Seas, bounded on the North with the River Trontus; on the South-east with Axofenes: hy the Rivers Poe and Frore, separated from the Republique of Venice, on the South-west by the River Piscio; by which 'tis divided by the Modern Tuscany, it is in the middle of Italy, its breadth is 115, and length 300 Miles, it's most exceeding fruitfull, very Populous, there have been 15 Exarches of Ravena in Romandiola, 17 Dukes and Marquesses of Ferata; the Revenue thereof to the Pope is 250000 Crowns, there hath also been 6 Dukes of Urbin, its Revenue are 100000 Crowns, but the most splendid Glory of Italy is the City of Rome, sometimes the Empress of the World, and was the Seat of the past Popes, and the now present Pope Innocent the XI. the inferior spiritual Governours, are these, Viz. Cardinals, Friers of the Order of St. Basil, Austin, Jerome, Carmelites, Crouchedfriers, Dominicans, Benedictines, Franciscans, Jesuits and Oratorians; and of Nuns, the Order of St. Clear and Bridget, which to name wholly doth deserve a particular Treatise, here are 44 Archbishops, and 57 Bishops.

The Republique of Venice, lieth Northward of the Popes Dominions, from Romandiola to the Alps, limited on the South with the Territories [Page] of Ferrata, and Romandiola; on the West with the Dukedom of Millain; on the North with the Alps; and on the East with the Adriatiok Sea,Europe. and the River Arsia. It is a very fruitfull Countrey, well peopled, their Government Aristocratical, and popular, their Religion Popish, they baptize the Sea yearly; they have had a hundred Dukes, they have two Principal Orders of Knighthood of St. Mark the Patron of the famous City of which the Poet speaks.

Viderat Adriacis venetiam Neptunus inundis

Stare Urbem, & toto ponere jurae mari:

Nunc mihi Tarpeias, quantumvis Jupiter, arces

Objice, & illa tui moenia martis ait.

Sic Pelago Tibrim praefers, urbem aspice utram (que)

Illam bomines dices, hanc posuisse Deos.

Instituted 1330, and renewed 1562, they are to be all of Noble Blood: their Motto is Pax tibi Marce. The other is of the Glorious Virgin, iristituted 1222, their Duty is to be a refuge to Widows and Orphans, and to procure the peace of Italy; their Habit a White Surcoat over a Russet Cloak, representing Religion, as well as Belliarcity, there are two Patriarchs, and sixteen Bishops.

The Dukedom of Florence, being the Seat of the Great Duke of Tuscany, is bounded on the East by the River Pisca; on the West by the River Macra, and the Fort Sarzana; on the North by the Appenuine Hills; and on the South with the Tuscan Seas. Its length is 261 Miles, and breadth not known; the Order of Knighthood is that of St. Stephen, instituted 1567, they [Page] are to be Nobly born, and in lawfull Wedlock, without Insamy: their Robe is of White Chamblet, with a Red-Cross on the left Side ofEurope. their Midway Garment, their Number I cannot certainly know, the Grand Duke is their Sovereign; the Revenue of this Countrey is great, their Duke is also a Merchant, and receiveth Excise of all Commodities, the Arms is Or, five Tortecax Gules, two, two, and one, one, on chief Azure charged with three Flowerde-Luces, of the first. They are of the Popish Religion, and they have three Archbishops, and twenty six Bishops.

The Estate of Luca, lieth betwixt the Estate of the Grand Duke, and the Republique of Genoa: The Government is Aristocratical, and Democracy, their Principal Magistrate is called, Gon Fatinere, and is changed every second Month: being assisted by a certain Number of Citizens, which are changed every six Months, during which time they lie together in the Common Hall; their Protector is Elective from some Neighbouring Prince: they are a very generous People, good Merchants, they sell rich Cloths of Gold and Silver; the Revenues yearly are 80000 Crowns, it can raise for War 15000 Foot, and 3000 Horse, they are of the Popish Religion, and have two Bishops, and acknowledge the Bishop of Florence for their Metropolitan.

The Republique of Genoa, lieth West of Tuscany, from whence 'tis divided by the River Macra, it was formerly a large State but have now only Liguria, and the Isle of Corsica; the Inhabitants are good Warriours, Merchants, and subtle Userers; here the Women have the most [Page] [...]iberty of any in all Italy, so that they may convense with whom they will, either publiquely, or privately; from hence ariseth aEurope. Proverb, That Genoa is a Country of Mountains without Woods; Seas without Fish; Men without Faith; and Women without shame. They have a Duke, with Eight Assistants, all subject to the General Council of 400 Men; which hold but two years, they are of the Popish Religion, and have one Archbishop, and fourteen Bishops.

The Estates of Lumbardy is bounded on the East with Romandiola, and Ferrata; on the West and North with the Alps; and on the South with the Apenuine hills: Now as Italy is the Garden of EUROPE, so is Lumbardy of Italy, for its exceeding Fruitfulness.

The Dukedom of Millain hath on the East the State of Mantua, and Parma; on the West Piemont, and part of Switzerland; on the North Marca Trevigana; and on the South the Apenuine, parting it from Liguria: it was once the chief Dukedom in Christendom, and is now in the Spanish Territories; its Revenues are 8000 Ducat's, their Arms are Argent, a Serpent Azure Crown'd, Or, in his George an Infant Gules, their Religion is Popish, they have one Archbishop, and six Bishops.

The Dukedom of Mantua, is bounded West with Millain; East with Romandiola; North with Marca Triugiana; and South with the Dukedom of Parma. The Countrey yields good store of Corn, Fruit and Wine, the Inhabitants are rustick, foolish in their Apparel, it is a free state and hath had many Dukes, the Order of Knighthood is that of the Blood of Christ: instituted [Page] 1608, it consisteth of twenty Knights, the Mantuan Duke is their Sovereign; the Coller hath threads of Gold, layed on with Fire,Europe. with this Motto Domine probasti, to the Collar are pendent two Angels, supporting three drops of Blood, and circumscribed with this Motto, Nihil ista triste receptô. Its Revenue is 500000 Ducats; Its Arms are Argent, a Cross Patere Gules, between four Eagles, Sable membred of the second, under an Eschucheon in Fife, charged quarterly with Gules, a Lion Or, and Or, three barrs Sable: Their Religion is Popish, here is one Archbishop, with four Bishops.

I shall pass by the Dukedoms of Modena, Parma and Mountferrat, they being but small Estates of Italy, having but four Bishops: they are of the Popish Religion, the Arms of Modena and Parma are as Ferrata; and the Arms of Mountferrat, a chief Argent.

And here we should describe Piemont the last part of Italy, but being but part in Italy, and the Alps belonging to the Duke of Savoy, I shall defer it to the Alpian Descriptions.

Now Italy hath these most famous Cities, viz. Genoa, Milain, Venice, Florence, Rome, Bologne and Naples, the Rivers most famous are Arnus, Po and Tiber, and so much for Italy.

The Alps begin about the Ligustick Seas, and crosseth all along the Borders of France and Germany, and extend as far as the Gulph of Cornero; It hath these Provinces, viz. the Dukedom of Savoy (to which Piemont belongs) Geneva, Wallisland, Switzerland, and the Countrey of the Grizons, of all which I shall give a short and plain Description.

[Page]

Piemont is part of the Alps, situated at the Foot of the Mountains, bounded North withEurope. the Switzes; East with Millain and Mountferrat; West with Savoy; and on the South it runneth into a Narrow Vally to the Mediterranean, having Mountferrat on one side, and Province and part of the Alps on the other: it's very fruitfull compar'd with Savoy, but yet inferiour to any part of Italy: The Arms are Gules, a Cross Argent, charged with a Lebel of three points Azure.

Savoy is bounded East with Wallisland and Piemont; West and South with Daulphin, and LaBress; and North with Switzerland, and the Lake of Geneva: this is a Mountainous Countrey, very healthfull, but not very fruitfull. The Inhabitants are dull and slothfull, it hath had thirty Dukes and Earls, it is a place of Natural strength; its Revenues is yearly 1000000 of Crowns. The order of Knighthood is that of Anunciado, instituted 1480, their Coller hath 50 links, (to shew the Mystery of the Virgin) appendent to it is her Effigies, and instead of a Motto these Letters F. E. R. T. i. e. Fortitudo ejus Rhodum tenuit, which is engraven on each link of the Chain, interwoven like a True-loversknot. The Number fourteen, besides the Duke Soveraign of the Order, their Arms are G. a Cross A.

Geneva was a City of the Dukedom of Savoy, but now a free State: having both cast off the Duke and his Holiness the Pope, with all the Clergy. They are now Calvanist Protestants: their Government Presbyterial; their Language the worst of French, they are an industrious People, and good Merchants.

[Page]

Wallisland reacheth from the Mount De Burken, to the Town of St. Maurice, where the Hills do shut up the Valleys, so that a BridgeEurope. is lain from one Hill to 'tother, under which passeth the River Rosue, which Bridge is defended by a Castle and two strong Gates; on the other side 'tis surrounded with steep and horrid Mountains; covered with a Crust of Ice not passable by Armies; the Inhabitants are courteous to Strangers, but unnatural to each other: they are of the Romish Religion, and subject to the Bishop of Sion; the Deputies of the seven Resorts, have voices in his Election, and joyn with him in Diets, for chusing Magistrates; desiding Grievancies, and determining matters of State. The Valleys of this Countrey is very fruitfull in Saffron, Corn, Wine and Delicate Fruits, they have a Fountain of Salt, many hot Baths, and Spaw-Waters, they have plenty of Cattle, with a wild Stag footed as a Goat, and horned as a fallow Deer: who in Summer is blind with heat.

Switzerland is bounded East with the Grisons; West with Mount-Jove and the Lake of Geneva; North with Suevia; and South with Wallisland, and part of the Alps; this Land is a very Mountainous Countrey, but yet hath some rich Meddows. It is 240 in length and 180 Miles in breadth, the Inhabitants are rich, but rugged like their Soyl: good Souldiers: they are some Papists and some Protestants, others Zwinglians, yet have they toleration under a Popular Government.

The Countrey of the Grisons is bounded East with Tyrol, North with Switzerland, South [Page] with Suevia, Switzerland, and Lumbardy; it is a very Mountainous and Barren Land, their Religion Protestant, their GovernmentEurope. Popular; there are in this Alpin Provinces two Archbishops, and thirteen Bishops. Its chief Cities are Turin, Geneva, Basil and Zurich, in all of which are Universities.

France is bounded East with Germany; and South and East with the Mediterranean Seas and Alps; North with the British Seas; It hath been esteemed the worthiest Kingdom in Christendom, it yields plenty of Grains and Wines, wherewith it supporteth other Lands, it consisteth of many great Dukedoms and Provinces. It hath great and mighty Cities, the People are Ingenious and good Warriers, the Government is Monarchial, their Religion Popish, but intermixt with Protestants, which of late hath endured grievous Persecutions. Their Orders of Knighthood are that of St. Michael instituted 1409, consisting of 300 Persons, their Habit is a long Cloak of White Damask down to the Ground, with a Border interwoven with Cockleshels of Gold, interlaced and furred with Ermins, with a Hood of Crimson-velvet, and a long Tippit about their Necks, and a Coller woven with Cockleshells, with this Motto, Immensitremor Oceani, to it hangs appendent the Effigies of St. Michael conquering the Dragon. Their Seat is St. Michael's Mount in Normandy. 2dly the Order of the Holy Ghost instituted 1579, so that whosoever was admitted to the Order of St. Michael, must and was first dignified with this; proving their Nobility by three Descents; and be bound by Oath to maintain [Page] the Romish Religion; and persecute all Dissenters thereunto. Their Robe is a Black Velvet Mantle, portrayed with Lillies and Flames of Gold: theEurope. Coller of Flower-de-Luces, and Flowers of Gold, with a Dove and Cross appendent to it. The Arms of France, are Azure three Flower-deLuces, Or; It hath seventeen Archbishops, 107 Bishops, 132000 Parishes, and hath these Magnisicent Cities, viz. Amiens, Rouen, Paris, Troys, Nants, Orleans, Diion, Lyons, Burdeaux, Toulose, Marsailles, Grenoble and Anverse; the Rivers of most Note are the Loyre, Garone, Rhone and the Seyne.

The Pirenean-hills lyeth betwixt France and Spain, and are two Potent Kingdoms, esteemed 240 Miles long, the People are barbarous and scarce of no Religion at all.

Spain is separated from France by the Pirenean-hills; on all other sides environed with the Sea; this Land yieldeth all sorts of Wine, Oyl, Sugars, Grains, Metals, as Gold and Silver, and it is Fertile; the Inhabitants are Ambitious, Proud, Superstitious, Hypocrites and Lascivious, yet good Souldiers; by enduring Hunger, Thirst, Labour, &c. It containeth divers Kingdoms. 1. Goths. 2. Navars, it hath had 41 Kings. Their Arms are Gules, a Carbuncle Nowed Or, their Order of Knighthood was of the Lilly, their Blazon a Pot of Lillys, with the Effigies of the Virgin on it, their Duty is to defend the Faith, and daily to repeat a certain Number of Ave-Maries. 3dly Biscay and Empascon, hath had nineteen Lords, their Arms Argent, two Wolves Sable, each in his Mouth a Lamb of the second. 4ly Leon and Oviedo hath had thirty [Page] Kings, the Arms Argent, a Lion Passant crowned Or. 5ly. Galicia hath had ten Kings, the Arms Azure Sema of Cressels siched a ChaliceEurope. crowned Or. 6ly. Corduva hath had twenty Kings, the Arms Or, a Lion Gules armed and crowned of the first, a Border Azure charged with eight Towers Argent. 7ly. Granado hath had twenty Kings, the Arms Or, a Pomgranet slipped Vert. 8ly. Marcia. 9ly. Tolledo hath had eleven Morish Kings. Ioly. Castile hath had twenty Kings, the Order of Mercia is his chief Order, here the Armes is a Cross Argent, and four Beads, Gules in a Field Or, their Habit white, the Rule of their Order that of St. Augustine, they are to redeem Captives from Turky. 11ly. Portugal (the Native soyl of the most serene Catharine Queen Dowager) hath had 21 Kings, the Orders of Knight here is first Avis, wearing a Green Cross, 2dly of Christ instituted 1321, their Robe is a black Cassock under a white Surcoat with a Red Cross hanging in the midst a white Line, and their Duty is to expell Mores out of Boetica, the Arms are Argent, on five Escucheons Azure, as many Befants in saltire of the first pointed, Sable within a powder Gules, charged with seven Towers Or. 12ly. Majorica hath had four Kings. 13ly. Arragon hath had twenty Kings, their Order of Knighthood is of Mintesa, their Robe a red Cross on their Breast, the Arms Or, four Pallets Gules, all which Kingdoms are now united into one Monarchy, under the King of Spain, their Religion Popish: the King is not rich by reason of his great Expences to keep his Dominions, in which are eleven Archbishops, [Page] and 52 Bishops, and hath these most notable Cities, viz. Toledo, Madrid, Leon, Fax, Siville, Grenado, Mursy, Saragosa, Bracelon, Pamphelune,Europe. Bilbo, Priede, St. James of Compostella, and Lesbone, and Rivers famous are the Dower, Tagus, Gadian and Guadalguinr.

Great Britain consisteth of England and Scotland, and is the Biggest Isle in EUROPE, and the Glory thereof; it is a temperate Soyl, a sound Air, and yieldeth all manner of good things, 'tis environed all round with the Seas; I shall begin first with England.

England hath many pleasant Rivers well stored with Fish, excellent Havens, commodious Mines of Silver, Lead, Iron and Tinn, abundance of Woods, good Timber, plentifull in Cattle, good Wool of which is made fine Cloath, which serves not only themselves but vended into other Countreys, the chief City is London, in which are two of the Wonders of the World, viz. the Monument and Bridge over the Thames, the People are brave Warriers, both by Sea and Land, as Europe has felt and can testifie to their Grief, they are learned in all manner of noble Sciences; the Order of Knighthood is that of St. George, or the Garter, there are 26 Knights of it, whereof the King is the Soveraign, their Ensign is a blue Garter buckled on the left Leg, with this Motto — Hony Soit Qui Mall [...] Pense, and about their Necks they do wear a blue Ribbon, at the End of which hangs the Image of St. George, upon which day this Order is Celebrated: secondly of the Bath, instituted 1009, they use to be Created at the Coronation of Kings and Queens, and at the Enstalling [Page] of the Prince of Wales. The Knights thereof distinguished by a red Ribbon, which they wear about their Necks, their Duty is to defend Religion,Europe. Widows, Maids and Orphans, with the Kings right. Thirdly of Barronets and Hereditary Honour, the Arms are Mars, three Lions Passant, Gardant Sol, their Religion is the Protestant; they have two Archbishops and twenty Bishops. The length of England is 320, and breadth 250 Miles, it hath 857 famous Bridges, 325 Rivers, it's defended and invironed with Turbulent Seas; guarded by unaccessible Cleves and Rocks; and defended by a strong and Puissant Navy; so that of it may well be said,

Insula praedives, que toto vix eget Orbe;

Et cujus totus indiget Orbis ope.

Insula praedives, cujus miretur & optet

Delicias Solomon octavianus opes.

Its chief Cities are London, York, Bristol, and Rivers are the Thames, Severn, Humber, and the Ouze.

Wales is bounded on all sides with the Sea, except towards England on the East; it is a barren and mountainous Countrey: Its chief Commodities are their Freeze, and Cottons. The Inhabitants are faithfull in their promises to all men, but yet much enclined to Choler, and subject to Passion, which Aristotle calleth [...]. It contains 14 Shires, 13 Forests, 36 Parks, 230 Rivers, and 1016 Parishes. They are so resolute and valiant (saith Henry III. writing to Emanuel then Emperour of Constantinople) ‘That they dare encounter Naked with [Page] armed Men, being ready to spend their Blood for their Countrey, and pawn their Life for Praise.’ They are Protestants, and have four BishopsEurope. but no Towns of Note.

Scotland is the Northern part of Britain, environed all round with the Sea, unless where it is joyned to England. Polydore saith it is 480 in length, and 60 miles in breadth, divided into Highlands and Lowlands: the Highlands are Irish-Scots, and the Lowlands English-Scots. It is not so fruitfull as England; the chief City is Edenbrough. Its Commodities is course Wool, and Cloth, Malt, Hides, and Fish. The Order of Knighthood is that of St. Andrew, the Knights did wear about their Necks a Coller interlaced with Thistles, with the Picture of St. Andrew appendant thereunto, having this Motto — Nemo me impune lacessit. 2. Of Nova Scotia instituted by King James, Anno 1622, hereditary the Knight hereof distinguished by a Ribbond of Orange-tawny. The Arms of Scotland is Sol, a Lion Rampant, Mars within a double Tressure counterflowered; they are Protestants, and have 2 Archbishops and 12 Bishops. The Cities most Famous are Edenbrough, Sterlin, Aberdeen and St. Andrews: and they have the Famous Rivers Tay and Tweed.

Ireland is on all sides environed round with the Irish Seas, and St. Georges Channel. In length is 300 and breadth 120 miles. The Natives are strong and nimble, haughty, careless, hardy, bearing cold and hunger with patience and in a word, if they are bad you shall neve [...] find worse; but if good scarcely find better. The Wild Irish have a custom to kneel down to [Page] the New Moon, praying it to leave them in as good health as it found them. They received the Christian Faith 435. The Soil is fruitfullEurope. and it hath good Pasture, yet full of Boggs and Woods, and multitude of Fowls, and in it will dwell no venomous Creature. The Revenues yearly have been 40000 li. The Air is Temperate, cooler in Summer, and hotter in the Winter than in England. Their Arms are Azure an Harp or stringed Argent, they are some Protestants and Papists mixt: they have 4 Archbishops, and 19 Bishops; the chief City is Dublin.

The Islands belonging to Great Britain, are 1. Wight (the place where I first drew my Breath, and the Land of my Nativity) 2. Surlings, 3. Garnsey, 4. Jersey, 5. Anglesey, 6. Man, 7. Hebrides, 8. Orcades, 9. Portland, 10. Sunderland, 11. Holy Island. And thus I have done with the British Empire; all these Parts described belong to it, and are under the Royal Sceptre of his Sacred Majesty JAMES the Second (whom God long preserve.) Thus I have finished the description of Great Britain having this only to say — Quae Deus conjunxit nemo separet.

Belgia, or the Low Countreys, consisteth of several wealthy Provinces: viz. The Dukedom of Brabant, Guelderland, Lymburge, Flanders, Artois, Henault, Holland, Zeland, Mamen, Zukfen, the Marquisate of the Holy Empire, Freezeland, Michlen, Ouserisen, and Graving. All which Lands are very fertile and populous, having 208 Cities, and 6300 Villages, with Parish-Churches, Castles, and Forts; and is watered [Page] with the Rhine and the Mose, the Mara and the Sheld. It hath commodious Havens, the Inhabitants are brave Warriours, good Mechanicks,Europe. their chief Commodity is Rhenish Wine, Linnen and Woollen Cloth, Camericks, Lace of Gold and Silver, Silk, Taffatys, Velvet, Grogerams and Sayes, all manner of Twined threds, refined Sugars, Buff, Ox-hides, Spanish-leather, Pictures, Books, Cables, Ropes and Herrings. Now Belgia is bounded East with Westphalen, Gulick, Cleve, and the Isle of Triers; West with the Main Ocean; North with the River Ems; and South with Picardy and Champagne. The People are of the reformed Religion all except Flaunders and Artoise, and they have the Popish Tenents, here are three Archbishops, and fifteen Bishops. The Order of Knighthood is that of the Golden Fleece, instituted 1439. their Habit is a Coller of Gold, interlaced with Iron, Or. Ex ferre Flammam, at the end thereof hangs a Golden Fleece. Their chief Cities are Mentz, Antwerp, Amsterdam, Roterdam, and Rivers are the Sheild and Mosa.

Germany is the greatest Province in all EUROPE, and is bounded East with Russia, Poland and Hungary; West with France, Switzerland and Belgia; North with the Baltick Seas, and part of Denmark; and South with the Alps and parted from Italy: it contains Bohemia and Pragu, it is adorned with Magnificent Towers, strong Fortifications, Castles and Villages, very Popular; the Soyl is Fertile; many Navigable Rivers do to it belong, Good Spaws, Hot Baths, Mines of Gold and Silver, Tinn, Copper, Lead and Iron; they are some Papists, [Page] others Protestants, Zwinglians, Calvinists, and Lutherans. The Arms is Sol, an Eagle displayed with two Heads, Saturn armed, andEurope. Crowned Mars. There are six Archbishops, and 34 Bishops. They are a People much given to drinking; which made the Poet say —

Germani possunt cunctos tolerare labores,

O utinam possent tam bene ferre sitim.

The chief Cities of Germany are these, viz. Strasborough, Cologn, Munster, Norimbergh, Ausburg, Numick, Vienna, Prague, Dresda, Berlin, Stetin, Lubeck; Its chief Rivers the Rhine, Weser, Elbe, Oder, and the Danow, and Cities of Bohemia, are Cutenberg, and Budrozu.

Denmark and Norway, are two great Regions and bounded South with Germany; they have North Latitude 71° 30', toward the East they border on Sweden; and elsewhere environed with the Sea. Their Commodities are Oxen, Grain, Fish, Tallow, Sand, Nuts, Oyl, Hides, Goatskins, Fir-trees for Masts, Boards, &c. Pitch, Tarr and Brimstone: they are Lutherans. The Order of Knighthood is that of the Elephant, their Badge a Coller powdered with Elephants Towred, supporting the Kings Arms; having appendent the Effigies of the Virgin Mary; the Arms of the Land are Quarterly. 1. Or, three Lions passant Vert, Crowned of the first, for the Kingdom of Denmark. 2dly, Gules a Lion Rampant, Or, Crowned and Armed of the first, in his Paws a Dansk hatchet; Argent, for the Kingdom of Norway. There are two Archbishops, and 13 Bishops; its chief City is Coppenhagne.

[Page]

Sweden is a mighty Kingdom, is bounded East with Muscovia, West with the Dorfirin hills, North with the Frozen Ocean, and South withEurope. Denmark, Liesland and Mare Balticum; the Commodities are Copper, Iron, Lead, Furr, Buff, &c. They are brave Warriers, their Religion is Lutherans. The Arms Azure three Crowns Or, it hath two Archbishops, and eight Bishops.

Russia is bounded East with Tartaria, West with Livonia and Finland, North with the Frozen Ocean, and South with Lituania, and Mare Caspium, This Countrey is extreme cold: but yet Nature hath counterpoized it by supplying the Land plentifully with the best of Furrs, viz. Sable, White-fox, Martin, &c. It's subject to the Emperour of Russia; a vast Tract, and as wild a Government. The Inhabitants are Base and Ignorant, Contentious and Foolish, they deny the proceeding of the Holy-Ghost, they bury their Dead upright, with many other foolish Ceremonies; Muscovia is the Seat of the Empire. Its Commodities are Furrs, Flax, Ropes, Hides, Fish, and Whales-grease. The Arms are Sable, a Portal open of two Leaves, and as many degrees Or, they are of a mixt Romish Religion, not observing Learning as any thing: They have one Patriarch, two Archbishops, and eighteen Bishops. Its chief Cities are Mucon, Wolodimax, St. Michael, Cazan, and Astracan, it's chief Rivers are the Dwine, Volaga, and the Tana.

Poland is bounded South of Moldavia and Hungary; East with Moscovia, and Tartaria; West with Germany; and North with the Baltick Seas. The Commodities are Spruce-Beer, Amber, [Page] Wheat, Rye, Hony, Wax, Hemp, Flax, Pitch, Tarr; it hath Mines of Tinn and Copper; their Religion is partly Romish, and partly of the Greek-Church,Europe. and so there are of the Greek Church, two Archbishops, and six Bishops, and of the Romish Church three Archbishops, and nineteen Bishops: The Arms are Quarterly. 1. Gules an Eagle Argent Crowned and Armed Or, for Poland, and two Gules a Chevalier armed Cap-a-peid, advancing his Sword Argent, Mounted on a Barbed Course of the Second, for the Dukedom of Latuania. Its chief Cities are Cracovia, Warsovia, Damzerk, Vilna, Kion, Cameneca, and Smolensco; and Rivers are Vistula, Niemen, Dunae and the Boristhenes.

Hungary is bounded East with Transilvania, and Walachia, West with Stiria, Austria and Moravia, North with the Carpathian Mountains, and South with Sclavonia and part of Dacia. The People are valiant, and shew their Antiquity to be Scythians by their barbarous Manners, and neglect of Learning Their Sons equally inherit without Priviledge of Birthright, and their Daughters portion is only a New Attire. Its Commodities are Colours, Wheat, Beef, Salt, Wine and Fish, the German Emperour and Turk hath it between them. The Arms is eight Barrs Gules, and Argent, they are some of the Romish, and others Mahometans. There are two Archbishops, and thirteen Bishops, and its chief Cities are Transilvania, Valastia, Moldavia, Buda, Presbrough, Hermonstada, Tergoguis, Czuchan, Craffa and Bargos. Its Rivers are the Drin, Oxfeus, Peneus, Vardax, Marize and the Danubus.

[Page]

Sclavonia is bounded South with the Adriatick Seas, East with Greece, North with Hungary, and West with Carniola. It is fruitfull of allEurope. those Commodities found in Italy, and is under several Governments, viz. Turks, Venetians, Hungarians and Austrians. The Arms are Argent, a Cardinals Hat, the strings Pendant, and Pleated in a True-lovers-knot, meeting in the Base Gules. They are some Christians, and some Mahometans. There are four Archbishops, and twenty six Bishops. Its chiefest Cities are Nova, Zara, Nonigrad, Tinu, Sebenico, S. Nicolo, Trau, Spalato, Salona, Almisse, Starigrad, Vesicchio, Catara, and Doleigne.

Dacia is bounded East with the Euxine Seas, on the West with Hungary and Sclavonia, North with Podolia, and South with Thrace, and Macedonia. The Soyl is fruitfull in Corn and Wine. It yieldeth medicinal Plants, they have plenty of Fowls, both wild and tame, very Populous and of Nature like the Hungarians; they are all Mahometans; Its most famous Cities are Triste and Pedena.

Greece is bounded East with Propontick Hellespont, and Aegean Seas, West with the Adriatick, North with Mount Haemus, and South with the Ionian Seas. It was once the Mother [...] Arts and Sciences, but now the very Den [...] the Turkish Empire. The Soyl is very fruitfull [...] well manur'd, which made the Poet say —

Impius haec tam culta novalia miles habebit?

Barbarus has segetes? en queis consevimus arva

Its Commodities are Gold, Silver, Copper, [...] [Page] Wines, Velvets, Damask; here is the Mount of Parnassus: Here was the Temple of Delphos, consecrated to Apollo; where the DevilEurope. through the Oracle did deceive the People, but after the Crucifixion of Christ the Oracle ceased. Augustus (saith Suidas, in whose time Christ was born) consulting with the Oracle, received this Answer —

[...]

[...]

[...]

An Hebrew Child, whom the blest Gods adore,

Hath bid me leave these Shrines, and pack to Hell,

So that of Oracle I can no more:

In silence leave our Altar, and farewell.

Their Religion is mixt but they are chiefly Mahometans. The Arms of this Empire were Mars a Cross, Sol, between four Greek Beta's of the second; Bodin saith the four Beta's signified [...].

The most famous Cities in Greece are Buda, Salonique, Adrianopolis, Scutary, Durazzo, La Valone, L'Armiro, Brevezza, Larta, Lepanto, Setines or Athens, Thebes, Corinth, Patras, Misira or Lacaedomia.

I shall pass over the Islands of Sicily, Sardinia, Candia and Corsica: and thus we have finished the description of the first part of the World, called by the Name of EUROPE.

ASIA. Asia.

[Page]

ASIA the second part of the World is bounded on the North with the Northern Ocean; South, with the Red Sea; East, with the East Indian Ocean; and on the West with the Flood Tunais. It is bigger than EUROPE, or AFRICA, and is far more rich, Viz. in Pretious Stones, and Spices, and hath been renowned by the first and second Monarchs of the World. Here Man was Created, placed in Eden, seduced by Satan, and redeemed by our Blessed Saviour. In it was done most of the History mentioned in the Old Testament. It hath been Ruled by the Kings of China, of Persia, the Great Turk, and the Emperour of Rushia, and contains these Provinces. Viz. Anatolia, Cyprus, Syria, Palestine, Arabia, Chaldea, Assyria, Mesopotamia, Turcomania, Media, Persia, Tartaria, China, India and the Oriental Isles.

Anatolia is bounded West with the ThracianBosphorus, Helespont and the Aegean Seas; East with Euphrates; North with the Black Sea; and South with the Rhodian, Lydian and Pamphylian Seas. Its length is 630, and breadth 210 miles; the Air is sound, the Soil fruitfull, but in some places desolate: it is inhabited by Greeks and Turks. It hath these Cities of note. Viz. Anatolia, Bruce, Chiontai, Augoure, Trebisond, Sattalie: and Rivers are Alie, Jordan, Euphrates, and Tygris.

[Page]

Syprus is bounded all round with the Syrian and Sicilian Seas; whose length is 200, and compass 550 miles. It is stored with plenty ofAsia. all things, so that it wanteth no help of other Nations. Its Commodity is Wine, Oyl, Corn, Sugar, Cotton, Honey, &c. for which plenty of all things 'twas Consecrated to Venus, as Ovid saith:

Festa dies Veneris, tota celeberrima Cypro,

Venerat; Ipsa suis aderat Venus aurea festis.

The People are Warlike, Strong and Nimble, and very hospitable to Strangers Their Arms are Quarterly. 1. Argent, a Cross, Patent, betwixt 4 Crosses Or. 2. Cross-wise of 8 pieces Argent, and Azure, supporting a Lion Passant, Azure Crown'd Or. 3. A Lion Gules. 4 Argent a Lion Gules: they are of the Popish Religion, and have 2 Archbishops and 6 Bishops.

Syria is bounded East with Euphrates; West with the Mediterranean Seas; North with Cilicia; and South with Palestine and Arabia. Its length is 525 miles, and breadth 470. They are inhabited by Mahometans, Christians, and Pagans. They are a stout and warlike people. In this Countrey there are said to be Sheep whose Tails weigh some 30, and some 40 pounds; the People are also gluttenous; it is almost overrun by the Turks: Its most famous Cities are Aleppo, Te, Tripoly and Damal.

Palestine is bounded West with the Mediterranean Seas; East with the Arabian Desarts; North with the Anti-Libianus; and South with Arabia. The Inhabitants are of a middle stature, [Page] strong constitution, yet a stiffe necked and murmuring People and Idolaters. In this is the Land of Canaan, and the famous City Hierusalem, tho'Asia. now a Den of Idolatrous Mahometans. It abounds with all good things.

Arabia is bounded East with Chaldea and the Gulph of Persia; West with Palestine, Aegypt, and the Red Sea; North with Euphrates; and South with the Southern Ocean. The Inhabitants are Mahometans. Job's Habitation was here. It yields Frankincense, Pretious Stones, &c. It is now under the Great Turk's Sceptre. The most famous Cities are Herac, Ava, Medina, and Mectar: and it hath the famous River Cayban.

Chaldea is bounded East with Susiana; West with Arabia Deserta; North with Mesopotamia; and South with the Persian Bay. The Country is exceeding fruitfull; in it is supposed to have been the Garden of Eden; they were great Southsayers, and therefore flouted by the Satyrist. —

Chaldeis sed major erit fiducia, quicquid

Dixerit Astrologus, credent à fonte relatum

Ammonis, &c.

The Inhabitants are stout and valiant; they are Mahometans. Here Julian the Apostate breathed his Soul out to Satan, in these dying words, — Vicisti tandem Galileae: the chief Cities are Babylon, Bagdad, Balfora, and Sipparum, with the famous River Fazze.

Assyria is bounded East with Media; West with Mesopotamia; South with Susiana; and North with Turcomania and Chaldea. This is a [Page] very plain and level Countrey, and very fruitfull, having good Rivers: the Natives are brave stout Warriours, formal in their Habit.Asia. It is under the Turk's command, and governed by one of his Bassa's; who is able to bring into the Field at any time 100000 Souldiers: here are also a Sect of the Nestorians, and fifteen Christian Churches: its most famous Cities are Calach, Cittace, and Arbela.

Mesopotamia is bounded East with the River Tigris; West with Euphrates; North with Mount Taurus; and South with Chaldea, and Arabia Desertae. It aboundeth with all good things necessary for the life of Man; they are Mahometans, and a people unable to defend themselves but by the assistance of their Neighbours: It belongs to the Mahometan Empire. Its chief Cities are Edessa, and Cologenbar. I shall not describe Mount Taurus, because it is of no moment.

Turcomana is bounded East with Media and Mare Caspium; West with the Euxine Seas, Cappadocia, and Armenia major; North with Tartaria; and South with Mesopotamia and Assyria. It is a very mountainous Countrey; the people are handsome, stout and brave Warriours: the Women are good Archers. It hath Gold and Silver Mines: It yields Grain, Fruit and Wine; and in Colchis (a part thereof, and in Assyria) they sell their Children: The Arms are the Half Moon Or. It is inhabited by Mahometans, and under the Turkish Empire. Its chief Cities are Musol, Bagded, Batfora, Sanatopdy, and Derbent; with the famous River Arais.

[Page]

Media is bounded East with Parthia; West with Aremenia; North with Mare Caspium; and South with Persia. The Countrey is of aAsia. large extent and very different, even to a Miracle, for in the North part it is cold and barren, their Bread is dryed Almonds, and Drink the Juice of Herbs and Fruits. Their Food is Venison, and other Wild Beasts, which they catch by hunting. And in the South side the Country is of a rich Soil, plentifull in Corn, Wine, &c. They have been brave Warriours, and it was a custom with them to poyson their Arrows, in an Oyl called Oleum Mediacum: they are Mahometans.

Persia is bounded East with India; West with Media, Assyria, and Chaldea; North with Tartaria; and South with the Southern Ocean. This is a mighty rich Countrey governed by the Sophy, the people are strong and valiant, and though Mahometans, yet they War with the Turks for the Mahometan Religion in expounding the Alcoran. From hence comes Bezoars and other pretious Stones, Pearls, and Silk Works. It hath these famous Cities with Media: Viz Taurus, Gorgia, Cogsolama, Hysphan, Erat, Sus, Schiras, and Ormutz: and these Rivers Tiriditiri, and Bendimuz.

Tartary bounded East with China, the Oriental Ocean, and the Straits of Anian; West with Russia and Podolia; North with the Frozen Ocean; and South with China. Now the Tartarians are divided into certain Collonies, and differ in manners and Trade of Living, and are Men of a Square Stature, broad Faces, and look a Sq [...]int; they are hardy and valiant; they will [Page] eat either Horse-flesh or Man's Flesh. They drink Blood and Mares-milk; their Habit is very homely; they are some Mahometans, andAsia. some Pagans; their chief Commodity is rich Furrs, and they are governed by the great Cham of Tartary, and hath these famous Cities, viz. Zahasp, Samarcanda, Thibet, Cambalu and Tatur; and Rivers famous are Joniscoy, Oby, Chezel and Albiamu.

China is bounded East with the Oriental Ocean; West with India and Cathay; North with Altay and the Eastern Tartaries; and South with Canchin-China. It hath 591 Provinces, 1593 Walled Towns, 1154 Castles, 4200 unwalled Towns, and such an infinite Number of Villages, that the whole Countrey seems as one Town. It is reported that the Prince can bring into the Field 300000 Foot and 200000 Horse. The Land is fruitfull in Grain, full of wild and tame Beasts, it yields Silk, Pretious Stones, Gold, Copper, &c. The People are ingenious and great Artists, Witness their Wagon made to sail over the Land driven by the Wind: and Historians tells us, that the Art of Printing and of making Guns, is more Ancient with them than with us. They are Idolaters and worship the Sun, Moon and Stars, also they worship the Devil himself, that he may not hurt them. And it hath these most famous Cities, viz. Paguin, Quinjay, Caneun, Macao, Mancian and Magaia, with the great River Quinam.

India is bounded East with the Oriental Ocean, and part of China; West with the Persian Empire; North with Mount Taurus; and South with the Indian Ocean. This Countrey hath an [Page] Exact temperature of Air; two Summers and a double encrease, blest with all things necessary for the Life of Man. It hath Mines ofAsia. Gold and Silver, Pretious Stones, Spices, and Medicinal Druggs, abundance of Cattle, and Cammels, Apes, Dragons, Serpents, also multitude of Elephants; a Creature of a vast Bigness, some of which are said to be nine Cubits high, and as many long, and five Cubits thick. It is a Creature of wonderfull Sence: for 'tis reported of the Elephant on which King Phorus sate in the Warrs of Alexander, finding his Master strong and lusty, rushed boldly into the thickest of the Enemies Army: But when he once perceived him to be faint and weary, he withdrew himself out of the Battel, kneel'd down, and into his own Trunck received all the Arrows, directed at his Master. It also is of a most prodigious strength, for it is reported to carry a Wooden Tower on his Back, with thirty fighting men besides the Indian that Rules him. The Sea yields variety of Pearls and Fish; here is also the Leviathan or Whale, of which Pliny says there are some of 960 Foot long; here is the Rhinoceros also found: (such as hath of late been publickly shewed at the Bellsavage Inn on Ludgate-hill in London) a deadly and cruel Enemy to the Elephant, for though he be less, yet he will whet his horn against the Rocks, and then therewith strive to rip up the Elephants Belly, and is by many Naturalists supposed to be the Unicorn, for all the parts of his Body, especially his Horn, is a soveraign Antidote against Poyson. This Countrey is inhabited by Indians, Moors, Arabians, [Page] Jews, Tartars and Portugeze. The Natives are Tawny, tall and strong, and very punctual to their word. They eat no Fish nor Flesh, butAsia. live on things without life; being Pythagoreans. It is also reported that when the Husband dies, and is burning on the Funeral Pile, that then the Wife leaps into the Fire, and so the living and the dead burn together, which made the Poet say —

Et certamen habent lethi, quae viva sequatur,

Conjugium; pudor est non licuisse mori.

Ardent victrices, & praebent pectora flammae,

Imponuntque suis ora perusta viris.

In India these are the chief Cities, viz. Amedabur, Cambaia, Gouro, Diu, Bengala, Pangab, Agra, Goa, Calicut, Visnagor, Pegu, Arracan, Malaca, Camboge, and Faefo. The fairest Rivers are Indus, Ganges and Mecon.

The Oriental Islands are these, viz. 1. Japan, 2. The Phillepinae Isles, 3. The Maluccose, 4. Bantam, 5. The Selebes, 6. Borneo, 7. The Isles of Java, 8. Sumatra, 9. Zeiland, and other lesser Isles of which we shall not treat.

1. Japan is a rich Island abounding with Gold: So that Paulus Ventius saith, that in his time the King's Palace was covered therewith. It is a Mountainous Countrey, a healthfull Air; here the Wheat is ripe in May. It's full of Woods of tall Cedars, abundance of Beasts, Wild and Tame; and also Fowls. The Inhabitants are strong, and witty, and have but one Language. They are Christians, and Idolaters, and the chief Cities are these, viz. Bungo, Meaco and Sacay.

[Page]

The Phillipine Isles are in Number 40, called so in honour to Philip II. King of Spain, and are now inhabited by the Natives, and Spaniards,Asia. they are in a good Air and stored with rich Commodities; and in them are these Cities, Lusor, Manille, and Mindanao.

The Moluccoes Islands are many in Number, their Commodity is Cinnamon, (which grows in whole Woods; it is the Bark of a Tree, stript and laid in the Sun till it looks red; and in three years time the Tree receives his Bark again.) Ginger, Nutmegs, Mastick, Aloes, Pepper and Cloves: now the Clove groweth on a Tree like a Bay Tree; yielding blossoms first white, then green, and at last red and hard, and then are Cloves. In it is also found the Bird of Paradice, and no where else, which for the strangeness and fairness of Feathers exceeds all the Birds in the World. The People are Pagans. Here is a Mountain of a prodigious height, above the Clouds, and agreeing to the Element of Fire, which it seems to mount unto, through Flames, wherewith, a dreadfull Thunder, and a dark Smoak it sends forth continually.

The Isles of Bantam are in Number seven, one of which is continually burning, the Inhabi [...]ants are Barbarous, Weak of Bodies, Slothfull, Dull, and lying most confusedly together, without Rule, and are Mahometans. Its Commodities are Nutmegs, and both the yellow and white Saunderses. Now the Nutmeg grows on a Tree like a Peach Tree, the innermost part of whose Fruit is the Nutmeg, and is covered over with a Coat which ripe is called Mace; they yield their Fruit thrice in the Year, to wit, at April, August and December.

[Page]

The Selebes are a Number of Isles full of Barbarous People, and Man-Eaters, they have abundance of strange Birds: It yields Sugar,Asia. Cocanuts, Cloves, Oranges, &c. In some of these Isles they make Bread of the Pith, and Drink of the Juice of the Tree called Sagu: It hath these chief Towns, viz. Senderem and Macassar.

Borneo lieth West of the Celebes, and is in compass 2200 Miles, the Countrey yields Asses, Oxen, Herbs of Cattle and Horses. It yields Camphire, Agarick, and Mines of Adamants: They think the Sun and Moon to be Husband and Wife, and the Stars their Children, they reverently salute the Sun at his first rising. Their Affairs of State they Treat of in the Night, at which time the Councellor of State meets, and ascends some Tree, viewing the Heavens till the Moon ariseth, and then they go to their House of State. In it are these Towns, viz. Borneo, Taiopura, Tamaoratas, Malno and Sagadana. It is under the Government of the Kings of Borneo and Laus; the People are Idolaters.

Java Major, and Java Minor, are two Islands opposite to Borneo. They have plenty of Fruits, Grains, Beasts, Fish and Fowls, Gold and Pretious Stones. The Natives are of a middle Stature, broad faced and tawny, their Religion Mahometans, and they will eat their nearest of kin: the chief Town is Panarucan near a burning Hill, which in 1586 broke forth, and cast huge Stones into the City for three Days together, and destroyed much People. From the top of this vast high Mountain the Devil environed with a white and shining Cloud, doth sometimes shew himself unto his Worshippers, which live about those Hills.

[Page]

Sumatra lieth North of Java Major, betwixt it and the straits of Sincapura, its length is 900 Miles, and breadth 200; it is full of FennsAsia. and Rivers, with thick Woods, and hath a very hot Air; it is not fruitfull in Grain. Its Commodities are Ginger, Pepper, Agarick, Cassia, Wax, Honey, Silk, Cotten, Iron, Tinn and Sulphur. It hath also Mines of Gold, and is supposed to be Solomon's Ophyr. The King's Furniture of his House, and Trapping for his Elephants was beaten Gold, and he intituleth himself King of the Golden Mountains. Here is the notable Mountain Balalvanus, said to burn continually; out of which or not far off do arise two Fountains, the one is said to run pure Oyle, and the other Balsamum Sumatra; the People are Mahometans. The chief City and Seat of the King is Achen, beautifyed with the Royal Pallace, to which you pass through seven Gates one after another, with green Courts betwixt the two outermost; which are guarded with Women, that are expert at their Weapons, and use both Sword and Guns with great dexterity, and are the only Guard the King hath for his Person. The Government is Absolute and Arbitrary, merely at the King's pleasure.

Zeiland lies West of Sumatra, it is a good Soyl, and yields these Commodities, viz. Cinnamon, Oranges, Lemmons, most delicate fruit, Gold, Silver and Pretious Stones, it's full of wild and tame Beasts, Fish and Fowls, yet destitute of the Vine: the People are strong and tall, given to Ease and Pleasure, and are in general Mahometans. The chief Towns are Candia, Ventane, [Page] and Janasipata. They have Fish-shells passing currant for money, there are other lesser Isles which we do for brevity sake omit, and thusAsia. we have done with the description of the second part of the World called ASIA.

AFRICA.Africa.

AFRICA the third part of the World is bounded East with the Red-Sea; West with the Atlantick-Ocean; South with the Southern-Ocean; and North with the Mediterranean Sea; and contains these Provinces, viz. Egypt, Barbary, Numidia, Lybia, Terra-Nigritarum, Aethiopia-superior and Aethiopia-inferior, with the Islands thereto belonging. Its Commodities are Balm, Ivory, Ebony, Sugar, Ginger, Dates, Myrrh, Feathers, &c.

Egypt is bounded East with Idumea, and the Bay of Arabia; on the West with Barbary, Numidia, and Lybia; North with the Mediterranean Sea; and South with Aethiopia-superior. Its length is 562 Miles, and breadth 160. The Natives are of a Tawny Complection, their Wives are the Merchants, whilst the Husband attends the Houshold Affairs. They were the Inventers of Mathematical Sciences; they were also Magicians, and are still endued with a special Dexterity of Wit: They worship in every Town a particular God, but the God by them most adored was Apis. This Land is very fruitfull [Page] in all manner of Cattle, Cammels, and abundance of Goats; they have plenty of Fowls both wild and tame: It hath Metals and Pretious Stones,Africa. Good Wines and rare Fruits, as Oranges, Lemmons, Cittons, Pomgranets, Figgs, Cherries, &c. Here also groweth the Palm-Tree, which grow the Male and Female together; both put out Cods of Seeds, but the Female is not fruitfull unless she grow by the Male, and have her Seed mixt with his. The Pith of this Tree is good for Sallads, of the Wood they make Bedsteads, of the Leaves Baskets, Mats, and Fanns, of the outward husk of the fruit Cordage, of the inward brushes. Its fruit is the Dates, good for food, and finally 'tis said to produce all things necessary for the Life of Man, and its Branches are worn in token of Victory, as saith Horace.

— Palmaque nobilis,

Terrarum Dominos evebit ad Deos.

It hath many other Rarities which I am forc'd to omit. In it are these famous Cities, viz. Sabod, Cairo, Alexandria, Rascha, Damietta, Cosir and Surs, with the famous River Nilus, which by its overflowing makes the Land fertile, according to that of Lucia. —

Terra suis contenta bonis, non indiga mercis,

Aut Jovis; in solo tanta est fiducia Nilo.

Barbary is bounded East with Cyrenaica; West with the Atlantick-Ocean; North with the Mediterranean, the Straits of Gibralter, and part of the Atlantick-Ocean; and on the South by [Page] Mount Atlas. It is full of Hills and Woods, stored with Wild Beasts: as Lyons, Bears, &c. Large Herds of Cattle; it hath Dragons, Leopards,Africa. and Elephants; beautifull, swift, and strong Horses; it is the fruitfullest Countrey [...] the World in some parts of it; for [...]liny saith that not far from the City Tacape, you shall see a great Palm-Tree overshadowing an Olive; under that a Figg-Tree; under that a Pomgranat; under that a Vine; and under all Pease, Wheat and Herbs; all growing and flourishing at one time, which the Earth produceth of it self: Its length is 1500 Miles, and breadth 300 Miles, the Natives are of a Tawnyish Colour, rare Horsemen, crafty and unfaithfull, and above measure Jealous of their Wives. It contains these Kingdoms, Viz. Tunis, Algiers, Morocco and Feze, and it hath these Isles, Viz. Pantalaria, Carchana, Zerby, Gaulos and Malta, the two latter of which Isles are inhabited by Christians, and are of the Romish Religion; but for the other parts of Barbary, they are either Mahometans or Pagans. The most famous Cities are Morocco, Feze, Tangier, (which formerly was a Principal City of Barbary; but is now demolisht and lain level with the Ground, by the Command of His late Majesty Carolus II. of blessed Memory, and performed by the indefatigable skill and industry of the right Honourable George Lord Darmouth Anno, 1683.) Teleusin, Oran, Algi [...]r, Constantine, Tunis, Tripoly and Barca, with these famous Rivers, Ommiraby and Magrida.

Mount Atlas is a ridge of Hills of no small length, but of an exceeding heighth, above the Clouds, and is always covered with Snow, [Page] Summer and Winter, full of thick Woods, and against ASIA so fruitfull, that it affords excellent Fruit of it's natural growth; it receivedAfrica. it's Name from Atlas a King of Mauritania, fain. ed by the Poets to be turned into that Hill, by the Head of Medusa; he was seigned to be so high that his Head touched Heaven: The ground of this Fiction I suppose was from his extraordinary knowledge in Astronomy, which Virgil seems to intimate —

Jamque volans apicem & latera ardua cernit

Atlantis duri, Coelum qui vertice foluit.

N [...]vidia is bounded East with Egypt; West with the Atlantick-Ocean; North with Mount Atlas; and South with Libia Deserta. The Natives are a wandring and unstable People, for they spend their Lives in Hunting, and continue not above four or five Days in one place, but so long as it will graze their Camels. Here grow abundance of Dates, with which they feed themselves, and with the Stones fat their Goats. The Air here is so sound that it will cure the Fr [...]nch-Pox without any Course of Physick. They are Mahometans: its chief Provinces are Dara, Pescara, Fighig, Tegorarin and Biledulgerid; and its chief Cities are Taradath, Dara and Zev; they belong to the Scepter of M [...] rocco.

Lybia is bounded North with Numidia; East with Nuba; South with Terra-Nig [...] tarum; and West with Gualata. This is well termed a Desart, for in it may a man travel eight or ten Days and not see any Water, no [...] [Page] Trees, nor Grass. So that Merchants are forced to carry their Provision with them on Camels, which if it fails they kill their Camels,Africa. and drink the Juice of their Entrails It contains these Provinces, Viz. Zahaga, Zv [...]nz [...]ga, Targa, Lembta and Bordea. They are governed by the chief of the Clans, and are a People only differing from Brute-Beasts, by their Shape and their Speech.

Terra Nigritarum is bounded East with Ethiopia-superior; West with the Atlantick-Ocean; North with Lybia; and South with the Ethiopick-Ocean. The Countrey is under the Torrid-Zone, full of People, and most excessive hot; the soyl is exceeding fruitfull, brave Woods, Multitudes of Elephants and other Beasts: they have Mines both of Silver and Gold, very fine and pure; the Natives are ColeBlack, or very Tawny, and are now some of them Mahometans, but most of them Pagans. It hath now these Provinces, Viz. Ora, Anterosa, Gualata, Agadez, Cano, Ca [...]na, Sanaga, Gambra, Tombrutum, Melli, Gheneoa, G [...]ber, Gialofi, Guinea, Benin, Guangara, Bornum and Goaga, (in which groweth a Poyson, which if any eateth but the tenth part of a Grain it will end his Days) Bito, Temiano, Zegzeg, Zanfara, Gothan, Medra and Daum. And in it are these most remarkab'e Cities: Gue, Eata, Gueneha, Tomta, Agad [...]s, Cu [...]a, Tuta, Waver and Sanfara. The Rivers here that are most famous are Sernoga, Cambua and Ri [...]-Degrand.

Aethiopia-superior is bounded East with the Red-Sea, and Sinus Barbaricus; West with Lybia-inferior, Nubia, and Congo; North [Page] with Egypt, and Lybia Marmarica; and South with Monta-Luna. Now its length is said to beAfrica. 1500 Miles, and Circute 4300. It is under the Command of the Abassine Emperour: here the Air and Earth is so hot and pieircing, that if the Inhabitants go out of their Doors without Shoes they lose their Feet; here they also roast their Meat with the Sun: they have some grain, their Rivers are almost choaked up with Fish, their Woods stuffed with Deer, yet they will not trouble themselves to catch them. The Inhabitants are Lazy and destitute of all Learning, [...]hey are of an Olive Tawny: here is also a Fountain, that if a man drinks thereof he either falleth mad, or else for a long time is troubled with a continual Drowsiness, of which Ovid thus speaks —

Aethiopesque Lacus; quos si quis faucibus hausit,

Aut furit, aut patitur mirum gravitate saporem.

And it contains these Provinces, Viz. Guagere, Tigremaon, Angote, Amara, Damut, Gojamy, Bagamedrum, Barnagasse, Dancali, Dobas, Adel, Adea, Fatigar, Xoa and Barus. Now as for the Government of these Empires'tis merely Regal: here is the Order of St. Anthony, to which every Father that is a Gentleman, is to give one of his Sons: out of which they raise about 12000 Horse, which are to be a standing Guard of the Emperour's Person: their Oath is to defend the Frontiers of their Kingdom, to preserve Religion, and to root out the Enemies of their Faith; the Principals of their Religion are these. First, they circumcise their Children both Males [Page] and Females. Secondly, they Baptize the Males at 40 and Females at 80 Days after Circumcision. Thirdly, after the Eucharist they are notAfrica. to spit till Sun set. Fourthly, they profess but one Nature and one Will in Christ. Fifthly, they accept but of the three first general Councils. Sixthly, the Priests live by the own labour of their hands, and are not to beg. Seventhly, they baptize themselves every Epiphany in Lakes or Ponds, because that Day they say Christ was baptized by John in Jordan. Eighthly, they eat not of those Beasts which Moses pronounced unclean, keeping the Jews Sabbath, with the Lord's Day. Tenthly, they administer the Lord's Supper to Infants presently after Baptism. Eleventhly, they teach the Reasonable Soul of Man comes by Seminal Propagation. Twelfthly, that Infants dying unbaptized are saved, being sanctifyed by the Eucharist in the Womb, and finally they produce a Book of Eight Volumes, writ as they say by the Apostles at Jerulalem for that purpose, the Contents whereof they observe most solemnly, and thus they differ from the Papists.

Now the chief Cities in this Empire are these, Viz Barone, Caxumo, Amarar, Damont, [...] tes, Narre, Goyame, and Adeghena with the famous Rivers Zaire and Quilm [...]nei.

Aethiopia-inferior is bounded East with the Red-Sea; West with the Aethiolick Ocean; North with Terra-Negritarum, and the higher aethipy; and South with the Main Ocean And it contains these Provinces, Viz. Zanzibar, M [...]nomotapa, Cafravia, and Manigongo. The Natives are Black, with curled Hair, and are Pagans. [Page] In it are great Herds of Cattle, abundance of Deere, Antelopes, Baboons, Foxes, Hares, Ostriches, Pelicans and Herons, and in a Word whatAfrica. else is necessary for the Life of Man. In it are these most famous Cities, Viz. Banza, Loanga, S. Salvador, Cabazze, Sabula, Simbaos, Butua, and Tete. The Rivers are Cuama, Spiritus Sancto, and Dos Infante.

The Islands in AFRICA are these, Viz. the Aethiopick-Isles, Madagascar, Socofara, Mohelia, Mauritius, St. Helens, the Isles of Ascention, St. Thomas-Isles, the Princes-Isles, the Isles of Annibon, the Isles of Cape d'Verd, the Canaries, Madera, Holyport and the Hesperides. The Description of all which I am forced to omit because I have been so very large in the Description of the third part of the World called AFRICA.

AMERICA.

[Page]

America.

AMERICA, the fourth part of the World, was first discovered by Christopher Columbus, Anno 1492, but it hath received its Name from Americus Vesputius, who in the year of Christ 1597 did fail about it. Now this fourth part of the World is bounded East with the Atlantick Ocean; West with the West-Indian Ocean; South with the Magellanick Sea; and on the North with the Northern Ocean.

When first the Spaniards had entred on America they found the people without Apparel, and their Bread was made of the Jucca-Root, whose Juice is a strong poyson: but it being squeezed out and dried it makes Bread. They worshipped Devillish Spirits, which they call Zema; in remembrance of which they keep Images made of Cotton Wool, to which they did great reverence, supposing the Spirits of their Gods were there; and to blind them the more, the Devil would cause these Puppets to seem to move and to make a noise, so that they feared them so greatly that they durst not offend them; which if they did, then the Devil would come and destroy their Children. They were so ignorant that they thought the Spaniards to be immortal; but the doubt continued not long, for having taken some of them Prisoners, they put them under Water untill they were dead, and then they knew them to be mortal [Page] like other Men. They were quite destitute of all good Learning, reckoning their Time by a confused knowledge of the course of the Moon;America. they were honest and kind in their Entertainments, encouraged thereunto by an Opinion that there was a certain place to which the Souls of those that so lived, and dyed for the defence of their Countrey, should go to, and there be for ever happy. So natural is the knowledge of the Soul's Immortality, and of some Ubi, for its future reception, that we find some tract of it in the most Barbarous Nations. The Americans were of a fair and clear Complexion. This Countrey is very plentifull in Spices, and Fruits; and such Creatures which the other parts never knew: So fu l of Cows and Bulls, that the Spaniards kill thousands of them yearly only for the Hides and Tallow. Blest with abundance of Gold, that in some Mines they have found more Gold than Earth. They have Grey Lyons, their Dogs snowted like Foxes, neither can they bark; their Swine hath Talons sharp as Rozors, and their Navel on the ridges of their Backs; the Stags and Deer without Horns; their Sheep are so strong that they make them carry burthens of 150 pound weight; they have a Creature with the forepart as a Fox, and hinder as an Ape, except the Feet which are like a Man's; beneath their Belly is placed a Receptacle like a Purse, in which their young remains till they can shift for themselves, never coming thence but when they suck and then go in again. The Armadilla is like a barbed Horse, armed all over with Scales that seem to shut and open. The Vieugue [Page] resembling a Goat, but bigger, in whose Belly is found the Bezoar, good against Poyson. A Hare having a Tail like a Cat, under whoseAmerica. Skin nature hath placed a Bagg, which she useth as a Store-house: for having filled her self she putteth the residue of her provision therein. Pigritia, a little Beast that can go no further in fourteen days than a Man will cast a Stone. For their Birds they are of such variety of Colours and Notes, which are so rare and charming, that they surpass all other Birds in any other parts.

Now America is divided into two parts, viz. Mexicana, whose compass is said to be 13000 miles, and that other part called Peruana, whose Circumnavigation is esteemed 17000 miles. The Provinces of Mexicana are these: Viz Estotilant, Canada, Virginia, Florida, Califormia, Nova-Gallicia, Nova-Hispania, and Guatimala. Peruana contains these Provinces: viz. Castella-Aurea, Nova-Granado, Peru, Chile, paragnay, Brasila, Guiana, and Paria. To Peruana belongs these principal Isles: viz Hispaniola, Cuba, famaica, Porto-Rict, Barbadoes, the Charibe-Isles, Insula-Margaretta, Molaque-Isles, Remora, Insula Solamnis, and some other small Isles. But first of Mexicana.

Estotilant hath on the East the Main Ocean; South Canada; West Terra Incognita; and North Hudson's Bay. It comprehends Estotilant, so principally called, Terra Corterialis, New-foundland, and the Isles of Baccala [...]s. It is well stockt with all things necessary for the life of Man: the Natives are barbarous, fair, swift of Foot, and good Archers. They are Pagans.

[Page]

Canada is bounded North with Cortelialas; South with New England; East with the Main Ocean; and West with Terra Incognita. ItAmerica. contains these several Regions: viz. Nova Francia, Nova Scotia, Norumbegne, and four small Isles adjoyning thereto. The people when first discovered were very rude and barbarous, going Naked only a piece of Fishes Skin to cover their private parts, and had two or three Wives a piece, which never Marry after the death of their Husbands The Soil is fruitfull, and yields all manner of good things Here groweth the Sea Horse whose Teeth is an Antidote against Poyson It hath these principal Cities: viz. Hochelaga and Quebeque.

Virginia hath North Canada; South Florida; East Mare-del-Noo [...]; West with Terra Incognita. And it is now divided into New England, New Belgium, and Virginia strictly so called. It is in some parts (yea most parts) Mountainous, Wooddy and Barren, and full of Wild Beasts. It yields plenty of Cattle, wild and tame Fowls. Its Commodities are Furrs, Amber, Iron, Rop [...]s, Tobaco, Sturgecn, &c. The Natives are but few in number, and those very different both in Speech and Size, to a Miracle: those whom they call Sasques Honoxi, are to the English as Giants clad in Bears Skins; those whom they call Wig [...]ocomici, are as Dwarfs; for the most part without Beards; they hide their nakedness with a Skin, the rest of their Body they paint over in the figures of horrid Creatures The chief Towns are `fames's, and Plimouth, and Isle of Bermoodus, which I here omit.

[Page]

Florida is bounded North-East with Virginia; East with Mare-del-Noort; South with the Gulph of Mexico. It was first discovered by the English,America. Anno 1497. The Soil is very fertile in Grain and Fruit, Beast wild and tame, and so also for Fowls: It yields lofty Cedars, and Sassafras: It hath Gold and Silver Mines, and also Pearls.

The Natives are of an Olive-Colour, strong and fierce, and are clad like the former Natives of America. The Women when their Husbands are dead cut off their Hair, and cannot Marry till their Hair is grown out again. To it belongs these Islands: viz. the Isles of Tortugas, Martyres, and Lucaios: there are also about 24 small ones more which are insignificant. The Women here are most extreamly beautifull; the Natives are Pagans. Its chief Towns are St. H [...]elens, Ax Carolina, and Port-Royall.

Califormia is an Island having on the West New Spain, and New Gallicia; and so unto those undiscovered parts which lie furthest North, to the Straits of Anian; and 'tis divided into these four parts: viz. Quivira, Cibola, Califormia, specially so called, and Nova Albion. All which Countreys are indifferent fruitfull, full of Woods, and both wild and tame Beasts; plenty both of Fish and Fowl wild and tame: They worship the Sun as their chief God: They go naked both Men and Women in some parts, others are half way cloathed; and so very various that I cannot in this small Tract describe them. Its chief Town is Chichilticala. And here I cannot chuse but remark that in Quivira their Beasts are of strange forms, and are to them both Meat, Drink and Cloathes. [Page] For the Hides yields them Houses; their Bones and Hair, Bodkins and Threed; their Sinews, Cords; their Horns, Guts and Bladders, Vessels;America. their Dung, Fire; their Calveskins, Buckets to draw and keep Water in; their Blood, Drink; and their Flesh, Meat; and so much for Califormia.

Nova Gallicia is bounded East and South with Nova Hispania; West with the River Buena, Guia, and the Gulph Califormia; and North with Terra Incognita. It comprehendeth these Provinces: viz. Chialoa, Contiacan, Xalisco, Guadalajara, Zacatecas, New Biscay, and Nova Mexicana. In which Provinces the Air is indifferently temperate, yet sometimes given to Thunder, Storms, and Rain. It is full of Mountains, yields Brass, Iron, &c. They have plenty of Fish, Beast, Fowls, Fruit, and abundance of Honey. The Natives are wavering, crafty and lazy, given to singing and dancing. They go not naked: they are subject to the King of Spain. Its chief Cities are Guadalajara, and St. Johns.

Nova-Hispania is bounded East with the Gulph of Mexico, and the Bay of New-Spain; West with Nova Gallicia, and Mare-del-Zur; on the North with part of Nova Gallicia, and part of Florida; and on the South with the South Sea. It comprehendeth these Provinces: Viz. Mexicana, Mechoacan, Panuco, Trascala, Guaxata, Chiapa and Jucutan. In all which the Air is healthfull and temperate, rich in Mines of Gold and Silver, Cassia, Coccineel, which grows on a shrub called Tuna, yields grain, and delicate Fruit, Birds and Beasts both Wild and [Page] Tame: their Harvest is in October and in May.

The Natives are witty and hardy, yet so ignorant that they thought the Spanish-horse and ManAmerica. to have been but one Creature, and thought when the Horses Neighed they had spoken. The Spaniards whose Cruelties will never be forgotten, did in less than 17 years kill of the Natives 6000000; here is a Tree called Meto, it bears 40 kinds of Leaves, of which they make Conserves, Paper, Flax, Mantles, Matts, Shoes, Girdles; it yields a Juce like Syrup, which boyled becomes Hony, if purified Sugar; the Bark roasted is a good Emplaisture for Punctures or Contusions; and it yields a Gum Sovereign against Poyson: here is also a Burning Mountain called Propaeampeche, which sends forth two streams the one of Red and the other of Black Pitch: the Inhabitants are Pagans.

Guatimala is bounded North with Jacuta, and the Gulph Honduras; South with Mare-delZur; East with Castella-Aurea; and West with New Spain. The Soyl and People are as in Nova Hispania: it contains these Provinces, Viz. Chiapa, Verapaz, Guatimala, Hondarus, Niceragna and Teragna. And Towns of most Note are Cutrinidao and St. Michael's, the People are Pagans. And so much for Mexicana.

Peruana the Second Part of AMERICA, so called from Peru a Place of Note therein, and it doth contain these Provinces, Viz. CastellaAurea, Nova-Granada, Peru, Chile, Paragnay, Brasile, Guyana, and Paria and its Isles. But such Isles that fall not properly under some of these must be referred to the general Heads of the American Islands.

[Page]

Castella-del Oro, is bounded East and North with Mare-del-Noort; West with Mare-del-Zur; and South with Granada. And it containethAmerica. these Provinces, Viz. Panama, Darien, NovaAndaluzia, St. Martha and De-la-Hacha. In all which Provinces the Air is very hot and unhealthfull: the Soyl either Mountainous and Barren, or low and Miry: plenty of Beast and Fowls. Here is said to be a Tree which if one touch he is poysoned to death: the old Natives are now almost quite rooted out. Its chief City is Carthagena, which Sir Francis Drake in 1585 took by Assault. This Land hath abundance of Gold.

Nova-Granada is bounded North with Castella Aurea; West with Mare-del-Zur; East with Venez [...]la; and South with Terra Incognita. Its length is 390 Miles, and as much in breadth. It doth consist of these two parts, Viz. Granada, specially so called, and Popayan, both which hath a temperate Air, brave Woods, well stored with Cattle, and Fowls both wild and tame, plenty of Emeralds and Guacum: the People tall and strong; the Women handsome and better drest than their Neighbours: The chief Towns are S. Toy d'Bagota and Popayan.

Peru is bounded East with the Andes; West with Mare-del Zur; North with Popayan; and South with Chile. It is 2100 Miles in length, and its breadth is 300 Miles: it is a Mountainous Country: And here 'tis to be noted that in the Plains it never raineth; and that on the Hills it continually raineth from September to April, and then breaks up. In the Hilly Countreys the Summer begins in April, and endeth [Page] in September. In the Plains the Summer beginneth in October and endeth in April, So that a man may travel from Summer to Winter bothAmerica. in one Day; be frozen in the Morning when he setteth out, and scorched with heat before the dawning of the Day. It is not very plentifull of Corn nor Fruits, but they have a kind of Sheep which they call Pacos as bigg as an Ass, profitable both for fleece and burthen, but in tast as pleasant as our Mutten: So subtile that if it be overladen it will not for blows move a foot till the burthen be lessened, and it is a very hardy Creature. Here is a Figg-tree, the North part of which looketh towards the Mountains, and yieldeth its Fruit in Summer only, and the Part facing the Sea in Winter only. They have another Plant, that if put into the hands of the Sick and the Patient looks merry, they will recover; but if sad, die. It yieldeth also Multitudes of Rarities more. It's chief Commodities are Gold, Silver, Tobacco, Sarsaparilla and Balsamum d'Peru, and many other rich Drugs. The Natives are almost now rooted out of the Country. They are fierce and Barbarous. Now it contains these Provinces: viz. Quito, Los Quinxos, Lima, Cusco, Charcos and Colla [...].

Chile is bounded North with Deserta Alacama; West with Mare del Zuz; South with the Straits of Magellan; and East with Rio de la Plata. Its length is 1500 miles, and breadth uncertain. The Soil hereof in the Mid-land is mountainous and unfruitfull; towards the Seaside level and fertile; with products of Maize and Wheat, plenty of Gold and Silver, Cattle [Page] and Wine. The Natives are very tall and warlike, some of them affirmed to be eleven foot high; their Garments of the Skins of Beasts;America. they are of a white Complexion; their Armes Bows and Arrows. It is divided into Chile (especially so called) and Magellanica. Here Sir Walter Rawleigh planted two Collenies, who for want of timely Succors were either starved at home, or eaten by the Salvages, as they ranged the Countrey for food.

Paraguay is bounded South with Magellanica; East with the main Atlantick; North with Brazila; and West with Terra Incognita. It is said to be of a fruitfull Soyl, well stored with Sugar-Canes, Fraught with Mines of Gold, Brass, and Iron: great plenty of Amathyses, and Monkeys, Lyons and Tigers, the People are as the other Salvages, and it contains these Provinces, viz. Rio de la Plata, Tucaman and La Crux de Sierra, and it's chief Towns are Puenas Agrees, and Chividad.

Brazila is bounded East with Mare del Noort; West with Terra Incognita; North with Guiana; and South with Paraguay. Its said to be 1500 Mi'es long and 500 broad. The Countrey is full of Mountains, Rivers and Forests, the Air sound and healthfull; the Soyl is indifferent fruitfull: Its chief Commodities are Sugar and Brazele-wood. There is a Plant called Copiba which yields Balsam, soveraign for Poyson. An Herb called Viva, which if touched will shut up and not open till the Toucher is out of fight. A Creature which hath the Head of an Ape, the Foot of a Lyon, and the rest of a Man. The Ox-Fish with Arms, Fingers and Duggs, the [Page] rest as a Cow. So that it may be said of Brasila — Semper aliquid apportat novi.

The people are witty as appears by theirAmerica. sayings to the Christians (holding up a Wedge of Gold) say'd they, Behold your God oh ye Christians! on their Festival-days they go Naked, both Men and Women; and are able Swimmers, staying under water an hour and half: the Women are delivered without great pain: some of the Natives are all over Hairy, like Beasts: it containeth not Provinces, but these Captain-Ships: viz. Vincent, Rio de Juneiro, Holy Ghost, Porto-Seguro, Des Ilheos, Todos los Santos, Fernambuck, Tamaraca, Paraiba, Rio Grande, Saiara, Maragnon, and Para. Its chief Cities are, Meranhan, Tamaracai, and Olinda, and the great River Zoyal.

Guiana is bounded East with the Atlantick; West with Mount-Peru; North with the Flood Orenoque; and South with the Amazones. The Air here is indifferently good: near the Sea it is plain and level, up in the Countrey Mountainous; here the Trees keep their leaves all the year, with their fruit always ripe, and growing. The Inhabitants are under no settled Government: they punish only Murder, Theft, and Adultery; their Wives are their Slaves, and they may have as many as they please; they are without Religion or Notion of a Deity. It doth contain these Provinces: viz. Rio de las Amazones, Wiapoce, Orenoque, and the Isles of Guiana. Its Comodities are Sugar, and Cotton: in it are plenty of Beast, Fish, and Fowles; they are Swarthy in Complection, and great Idolaters; as for Cities it hath none of note.

[Page]

Paria is bounded on the East with Guiana; West with the Bay of Venezuela; North with the Atlantick Ocean; and South with Terra Incognita:America. and contains these Provinces: Viz. Cumana, Venezuela, S. Margarita, Cutagana, and its Isles. All which are not very fruitfull; it is well stored with Pearls; the People paint their Teeth and Bodies with Colour: The Women are trained up to ride, run, leap and swim: and also to Till the Land. In it are these most noted Cities: Viz. St. Jago, St. Michael de Nevery, and Mahanao.

As for the Descriptions of the American Isles I must beg the favour to omit: I shall therefore only name them having been so very large already; and they are these: Viz. Los. Ladrones, Fernandes, the Caribes; as Granada, S. Vincent, Barbados, Matinino, Dominica, Desrada; Guadalupe, Antego, S. Christopher, Nieves, Sancta Crux, and some lesser Isles belonging to them: As also Portorico, Monico Hispaniola, Cuba and Jamaica. Thus I have finished the Description of the known Earth.

Now the Names of the Seas are these: Viz. the Ocean Sea, Narrow Sea, Mediterranean Sea, Mare Major, Mare Pacificum, Mare Caspium, the East-Indian Sea, Perfian Sea, Red Sea, and Mare-del-Zuz, which are all the Principal-Seas.

Thus through the Blessing of God I have given you a brief, tho'true Description of all the known Earth and Seas, and have thus finished my Geographical Descriptions of the Division of the Earthly Globe.

[Page]

The Author on the Difficulties in the Collection of his [...], or little Description of the great World.

Oh thou Urania! Thou that hast now brought

Our Ship to Harbour sound, and richly fraught.

Tho'Aeolus his blustring Gales did send,

And foaming Billows high, the Skies did rend:

Tho'Blustring storms, and Thunder loud did roar,

And darkness Grim, opprest our Souls all'ore;

So that we could not view the Stars, nor Sky,

Nor Sun, nor Moon, nay Earth, could not espy.

Yet by thy Art, such safety we did find,

Safely to pass both raging Seas, and Wind.

And at the last a Harbour, safe did gain:

Rejecting fears; we quite cast off our pain.

When Seas are calm, and Winds more serene be,

Then we again will put our Ship to Sea;

That when refresht we farther may descry,

And search into this Noble Treasury.

[...]

SECT. III. Of Geographical Propositions.

[Page]

PROP. I. How to find the Distance of any two Cities or Places, which differ onely in Latitude.

IN this Proposition there are two Varieties which are these.

1. If both the Places lie under one and the same Meridian, and on one and the same side the Equinoctial, either on the North or South side thereof; then substract the lesser Latitude from the greater, and convert their difference into Miles (by allowing 60 Miles for a Degree) so have you the distance of the two Places propounded.

2. But if the two Places lie under one and the same Meridian, but the one on the South side of the Equinoctial, and the other on the North side, then add both their Latitudes together their Sum is their Distance.

PROP. II. To find the Distance of any two Places which differ only in Longitude.

There are also in this Proposition two Varieties.

1. The two Places may both lie under the Eqinoctial, and so have no Latitude: and if so, [Page] substract the lesser Longitude out of the greater, and convert the remainder into Miles; so have you the distance of any two Places so posuited.

2. But if the two Places differ only in Longitude, and lieth not under the Equinoctial, but under some other Intermediate Parallel of Latitude, between the Equinoctial, and one of the Poles: Then to find their distance, this is the Analogy or Proportion.

As Radius or S. 90°,

To Sc. of the Latitude:

So is S. of ½ X. of Longitude,

To S. of ½ their distance, which being doubled, and converted into Miles, giveth the required distance.

PROP. III To find the Distance of any two Places, which differ both in Latitude, and in Longitude.

In this Proposition three Varieties do present themselves to our View.

1. One of the Places may lie under the Equinoctial and have no Latitude, and the other under some Parallel of Latitude between the Equinoctial and one of the Poles. In such case observe this Analogy or Proportion.

As Radius or S. of 90°,

To Sc. of their X. of Longitude:

So is Sc. of their Latitude,

To Sc. of their Distance required.

2. But if both the Places proposed shall be without the Equinoctial, but on the one side either both towards the North, or both towards the South, then their Distance may be found, by this Analogy or Proportion.

As Radius or S. 90° 00',

To Sc. of their X. of Longitude:

So is To. of the greater Latitude,

To the T. of a fourth Arch, which substracted from the Complement of the lesser Latitude, the remainder must be the fifth Arch; Then say,

As Sc. of the fourth Arch,

To Sc. of the fifth Arch:

So is S. of the greater Latitudes,

To Sc. of the Distance of the two proposed Places.

3. The two Places propounded may be so situated, that one of them may lie on the North, and the other on the South side the Equinoctial: the Distance of Places so situated may be obtained, by this Analogy or Proportion.

As Radius or S. 90°,

To Sc. of X. of Longitude:

So is Tc. of the greater Latitude,

To T. of a fourth Arch, which being substracted out of the Summ of the other Latitude, and the Radius or 90° Deg. the remainder is a fifth Arch; Then say,

As Sc. of the fourth Arch,

To Sc. of the fifth Arch:

So is S. of the Latitude first taken,

To Sc. of the Distance required.

[Page]

These are all the Varieties of the Positions of Places on the Terrestrial Globe: For if the Distance of any two Places be required, they must fall under one or other of these Varieties, and may be obtained by one or other of the Proportions, mentioned in the three aforegoing Propositions.

Also if you know the Latitude and Longitude of any two fixed Stars, or their Right Ascension and Declination, then by these Rules their Distance may be found, which is of good use to Astronomy. It may also be applyed to Circular Sailing; of all other ways the most perfect: which is treated of in its due Place.

CHAP. VIII. of NAVIGATION.

[Page]

NAVIGATION So called from Navis a Ship, is an Art Mathematical, which sheweth how by the shortest good Way, by the aptest Direction, and in the shortest Time, to conduct a Ship from any one place unto any other place assigned: it hath been highly esteemed by the Ancients; it is the Glory, Beauty, Bullwark, Wall and Wealth of Britain, and the Bridge that joyns it to the Universe. Navigation is commonly divided into three sorts of Sailing: viz. Plain sailing, Mercator's sailing, and Circular sailing: Of all which three Parts I shall treat in their Order.

SECT. I. Of Plain sailing, or sailing by the Plain Chart.

[Page]

PLain Sailing, or sailing by the Plain Chart, is the most plainest way, and the Foundation of all the Rest: and although the Ground and Projection of the Plain Chart is erroneous, yet seeing it is more facile to the Learner, and may serve indifferently near the Equinoctial, because there the Degrees of Longitude, as well as the Degrees of Latitude, are Equal: Each Degree being divided into 60 Minutes, or Milles, though they are somewhat more than English Miles, Each Minute or Mile containing about 6000 Feet.

PROPI. The Rumb, and Distance sailed thereon being given,The Doctrine of Rightlined Triangles, both Right, and ObliqueAngled, applied to Propositions of Plain Sailing. to find the Difference of Latitude, and the Departure from the Meridian.

Admit a Ship sails N. W. by N. 372' or Miles, or 124 Leagues, I demand her Difference of Latitude and departure from the Meridian?

[...]

[Page]

[...]

[Page]

[Page]

In the Triangle ABC, the Hypothenuse AC representeth the distance sailed, or Rumb-line,Fig. 44. BC the departure from the Meridian, and AB the difference of Latitude.

1. To find which say,

As Radius or S. 90°,

To Log. distance sailed 372'.

So is Sc. V. A of the Course 56° 15',

To Log. cr. AB 309 3/10 Minutes, which being divided by 60' giveth 5°. 9'. 18" for the Difference of Latitude.

2. To find the Departure from the MeridianFig. 44. say,

As Radius or S 90°,

To Log. Rumb-line AC 372'.

So is S. of V. of the Course A 33° 45',

To Log. cr. BC 206 6/10 Minutes, the departure from the Meridian, which divided by 60 giveth 3°. 26'. 36" for the difference of Longitude.

Note that by this Proposition you may keep an Account how much you have sailed either East or West, North or South.

PROP. II. By the Rumb and Difference of Latitude given, To find the Distance, and the Departure from the Meridian.

Admit a Ship sail N. W. by W. untill herFig. 44. difference of Latitude be 309 3/10 Minutes, I demand her distance sailed, and her departure from the Meridian?

[Page]

1. To find the distance, say,

As Sc. of V. of the Course A 56° 15',

To Log. cr. AB the X. of Lat. 309 3/10 Minutes.

So is the Radius or S. 90°,

To Log. AC 372 the distance sailed.

2. For the Departure, say,Fig. 44.

As Sc. of A V. of the Course 56° 15",

To Log. cr. AB. X of Lat. 309 3/10 Minutes.

So is S. of V. of the Course A 33° 45',

To Log. cr. AB 206 6/10 Minutes, the Departure required.

By the help of this Proposition, when your Latitude by Observation doth not agree with your dead reckoning, (kept by the former Proposition) Then according to this Rule, you may make your way saild agree with your Observed Latitude, and so correct your Account or dead Reckoning.

PROP. III. By knowing the Distance of the Meridians of two Places, and their Difference of Latitude, to find the Rumb, and Distance.

Admit A, to represent the Lizard, AB the Parallel thereof, C. St. Mary's Islands, being one of the Azores, and CB the Meridian thereof.

In the Triangle ABC, there is given the side AB 816 Minutes, the Distance of the Lizard, from the Meridian of St. Marys, and the side CB their difference of Latitude 768 Minutes, IFig. 45. demand the Rumb: i.e. the Angle at C, and the Distance of the Lizard from St. Marys?

[Page]

1. For the Rumb or Angle at C, say,

As Log. cr. CB 768',

To radius [...] S 90°.

So is Log. cr. AB 816',

To T. of the B [...], or Angle at C 46° 44', and is from the Lizard unto St. Marys to the fourth Rumb of the Meridian, and 1° 44' more, Viz. S. W. and [...] 44'. Westerly, or from St. Marys, to the Lizard, N. E. and 1° 44' Easterly: and thus it shall be by the Plain Chart.

2. For their Distance AC, say,

As S. Rumb. or V. at C. 46° 44',

To Log. cr. AB 816' Minutes.

So is Radius or S. 90°,

To Log. Hypoth. AC 1120' 1/10 which is the DistanceFig. 45. of the Lizard, unto St. Marys Istand, and such should be the distance by the Plain Chart.

PROP. IV. Admit two Ships to set sail from one Port, one Ship sails W. S. W. 40', the other W. by N. so far untill she finds the first Ship to bear from her S. E. by E. I demand the second Ships distance from the Port, and their Distance asunder?

In the Triangle ADE, let A represent the Port, AD the W. S. W. course, and AE theFig. 46. Course W. by North.

1. To find the second Ships distance from the Port, say,

As S. of V. at E. 22° 30',

To Log. cr. AD 40' Minutes.

[Page]

So is S. of V. at D 123° 45',

To Log. cr. AE 86 98/100 Minutes, which is theFig. 46. distance required.

2. To find the two Ships their distance Asunder, say,

As S. of V. at E 22° 30',

To Log. cr. AD 40 Minutes.

So is S. of V. at A 33° 45',

To Log. cr. DE 58 12/100 Minutes, which is the Distance required.

PROP. V. Two Ships sets sail from two Ports, which lie N. and South of each other, the one sails from the Northermost Port 72 29'/100, and then meets she other Ship, which came from the Southermost Port, on a N. W. Course, and had sailed from thence 56 80'/100 I demand the Rumb on which the first ship made her way, and also the Distance between the two Ports?

In the Triangle ADE, let A be the SouthermostFig. 47. Port, AD the Course and way of the second Ship N. W. 56 80'/100, let E be the Northermost Port, ED the Course and Way of the other Ship 72 29'/100, and D the Place where they both meet.

1. To find the Rumb on which the first Ship sailed, say,

As Log. cr. DE 72 29/100 Minutes,

To S. of V. at A 45° 00'.

So is Log. cr. DA 56 80/100 Minutes,

[Page]

To S. of V. at E 33° 45', which sheweth the Course of the first Ship to be S. W. by South.

2. To find the Distance between the twoFig. 47. Ports A and E, say,

As S. of V. at A 45° 00',

To Log. cr. DE 72 29/100 Minutes,

So is S. of V. at D 101° 15',

To Log. cr. EA 100', which is the required Distance.

PROP. VI. Admit a Ship coming off the Main Ocean and I had sight of a Promontory or Cape, by which it is my desire to sail, I find it to bear from me S. S. E. and distant by Estimation 33', or Miles: But keeping still on my Course S. untill the Evening, having sailed 36' or Miles, I would then know how the Cape bears, and its distance from the Ship?

In the Triangle ADE, admit that at A, I do observe the Cape D, to bear from me S. S. E. 33', and having sail'd from A, to E 36' South, I desire to know its Distance, and bearing. In the Triangle, there is therefore given, AD 33',Fig. 48. AE. 36', and the Angle at A 22° 30'.

1. To find the Angle at E, say,

As Z. crs. AE, and AD 69',

To X. crs. AD, and AE 03'.

So is T. ½ VV unknown D and E 78° 45',

To the T. of 12° 20', which taken from 78° 45', leaves the Angle at E 66° 25', so that the [Page] Cape D then bears from me E. N. E. and 01° 05' Northerly.

2. To find the Distance of the Cape ED from the Ship, say,

As S. V. at E 66° 25',

To Log. cr. AD 33'.

So is S. V. at A 22° 30',

To Log. cr. ED 13 78/100 Miles distant, so that the Cape is then distant from the Ship 13 78/100 Miles.

PROP. VII. Two Ports both lying in one Latitude, distant 64' or Miles, the Westermost of those Ports lieth opposite to an Island, more Northerly distant therefrom 47' or Miles, which Island is also distant from the Eastermost Port, 34' or Miles, I demand the Course from the Westermost Port to that Island?

In the Triangle ADE, let A be the Westermost Port, and E, the Eastermost Port, distant Asunder 64'; and let D be the Island, distant from A 47', and from E 34': Then is the Angle atFig. 49'. A required, which is the Course or Rumb, from the Westermost Port, unto the Island: To find which, say,

As Log. cr. AE 64',

To Log. Z. crs. AD, and ED 81'.

So is Log. X. crs. AD, and ED 13',

To Log. os a certain line AO 16 454/ [...].

Which added to AE 64, is 80 454/1000,

The ½ whereof is AB, 40 227/1000

Then again say,

As Log. cr. AD 47', [Page] To Radius or S. 90°.

So is Log. AB 40 227/1000,

To Sc. V. at A, 58° 51', that is N. E. by E. 2° 36' Easterly, which is the Course from the Westermost Port A, unto the Island D.

SECT. II. Of sailing by the true Sea Chart, commonly called MERCATOR'S Chart.

THE true Sea Chart, commonly called MERCATOR'S ChartBut is indeed the Invention of our Learned Countryman Mr. Edw. Wright, although this Stranger hath almost got the Name and Praise thereof., performs the same Conclusions as the Plain Chart, and almost as speedily, but far more exactly: Because all Places may be laid down hereon, with the same truth as on the Globe it self: both to their Latitudes, Longitudes, Bearing and Distance from each other.

And here it will be necessary to have a Table of Meridional Ports, which I have extracted out of Mr. Wright's Tables, to every tenth Minute of Latitude; accounting it in single Miles, or Minutes of the Equinoctial, and have hereunto annexed the said Table.

[Page]

A Table of Meridional Miles

The Deg. of Lat. | The Minutes of each Degree. | The Difference.
0 | 10 | 20 | 30 | 40 | 50
The Meridional Miles.
0 | 0 | 10 | 20 | 30 | 40 | 50 | 10
1 | 60 | 70 | 80 | 90 | 100 | 110 | 10
2 | 120 | 130 | 140 | 150 | 160 | 170 | 10
3 | 180 | 190 | 200 | 210 | 220 | 230 | 10
4 | 240 | 250 | 260 | 270 | 280 | 290 | 10
5 | 300 | 310 | 320 | 330 | 340 | 350 | 10
6 | 360 | 370 | 380 | 390 | 400 | 410 | 10
7 | 421 | 431 | 441 | 451 | 461 | 471 | 10
8 | 481 | 491 | 501 | 511 | 521 | 532 | 10
9 | 542 | 552 | 562 | 572 | 582 | 592 | 10
10 | 603 | 613 | 623 | 633 | 643 | 653 | 10
11 | 664 | 674 | 684 | 694 | 704 | 715 | 10
12 | 725 | 735 | 745 | 755 | 766 | 776 | 10
13 | 786 | 797 | 807 | 817 | 827 | 838 | 10
14 | 848 | 858 | 869 | 879 | 889 | 900 | 10
15 | 910 | 920 | 931 | 941 | 951 | 962 | 10
16 | 972 | 983 | 993 | 1004 | 1014 | 1024 | 10
17 | 1035 | 1045 | 1056 | 1066 | 1077 | 1087 | 10
18 | 1098 | 1108 | 1119 | 1129 | 1140 | 1150 | 10
19 | 1161 | 1172 | 1182 | 1193 | 1203 | 1214 | 10
20 | 1225 | 1235 | 1246 | 1257 | 1267 | 1278 | 11
21 | 1289 | 1299 | 1310 | 1321 | 1332 | 1342 | 11
22 | 1353 | 1364 | 1375 | 1386 | 1396 | 1407 | 11
23 | 1418 | 1429 | 1440 | 1451 | 1462 | 1473 | 11
24 | 1484 | 1499 | 1505 | 1516 | 1527 | 1538 | 11
25 | 1549 | 1561 | 1572 | 1583 | 1594 | 1605 | 11
26 | 1616 | 1627 | 1638 | 1649 | 1661 | 1672 | 11
27 | 1683 | 1694 | 1705 | 1717 | 1728 | 1738 | 11
28 | 1751 | 1761 | 1773 | 1785 | 1796 | 1808 | 11
29 | 1819 | 1830 | 1842 | 1853 | 1865 | 1867 | 11
The Deg. of Lat. | The Minutes of each Degree. | The Difference.
0 | 10 | 20 | 30 | 40 | 50
The Meridional Miles.
30 | 1888 | 1899 | 1911 | 1923 | 1934 | 1946 | 12
31 | 1958 | 1969 | 1981 | 1993 | 2004 | 2016 | 12
32 | 2028 | 2040 | 2052 | 2063 | 2075 | 2087 | 12
33 | 2099 | 2111 | 2123 | 2135 | 2147 | 2159 | 12
34 | 2171 | 2183 | 2195 | 2207 | 2219 | 2231 | 12
35 | 2244 | 2256 | 2268 | 2281 | 2293 | 2305 | 12
36 | 2318 | 2330 | 2342 | 2355 | 2367 | 2380 | 12
37 | 2392 | 2405 | 2417 | 2430 | 2442 | 2455 | 12
38 | 2468 | 2481 | 2493 | 2506 | 2519 | 2532 | 13
39 | 2544 | 2557 | 2570 | 2583 | 2596 | 2609 | 13
40 | 2622 | 2635 | 2648 | 2662 | 2675 | 2668 | 13
41 | 2701 | 2714 | 2718 | 2741 | 2754 | 2768 | 13
42 | 2781 | 2795 | 2808 | 2822 | 2835 | 2849 | 13
43 | 2863 | 2876 | 2890 | 2904 | 2918 | 2931 | 14
44 | 2945 | 2959 | 2973 | 2987 | 3001 | 3015 | 14
45 | 3030 | 3044 | 3050 | 3072 | 3086 | 3101 | 14
46 | 3115 | 3130 | 3144 | 3159 | 3173 | 3188 | 14
47 | 3202 | 3217 | 3232 | 3247 | 3261 | 3276 | 15
48 | 3291 | 3306 | 3321 | 3336 | 3351 | 3366 | 15
49 | 3382 | 3397 | 3412 | 3428 | 3443 | 3459 | 15
50 | 3474 | 3490 | 3505 | 3521 | 3537 | 3553 | 16
51 | 3568 | 3584 | 3600 | 3616 | 3632 | 3649 | 16
52 | 3665 | 3681 | 3697 | 3714 | 3730 | 3747 | 16
53 | 3763 | 3780 | 3797 | 3814 | 3830 | 3847 | 17
54 | 3864 | 3881 | 3899 | 3616 | 3933 | 3950 | 17
55 | 3968 | 3985 | 4003 | 4020 | 4038 | 4056 | 18
56 | 4074 | 4092 | 4110 | 4128 | 4146 | 4164 | 19
57 | 4182 | 4201 | 4219 | 4238 | 4257 | 4275 | 19
58 | 4294 | 4313 | 4331 | 4351 | 4370 | 4390 | 20
59 | 4409 | 4428 | 4448 | 4468 | 4487 | 4507 | 20
The Deg. of Lat. | The Minutes of each Degree. | The Difference.
0 | 10 | 20 | 30 | 40 | 50
The Meridional Miles.
60 | 4527 | 4547 | 4567 | 4588 | 4608 | 4629 | 20
61 | 4643 | 4670 | 4691 | 4711 | 4733 | 4754 | 21
62 | 4775 | 4796 | 4818 | 4839 | 4861 | 4883 | 22
63 | 4905 | 4927 | 4949 | 4972 | 4994 | 5017 | 23
64 | 5039 | 5062 | 5085 | 5018 | 5132 | 5155 | 23
65 | 5179 | 5203 | 5226 | 5250 | 5275 | 5299 | 24
66 | 5324 | 5348 | 5373 | 5390 | 5423 | 5449 | 25
67 | 5474 | 5500 | 5520 | 5552 | 5678 | 5704 | 26
68 | 5631 | 5658 | 5685 | 5712 | 5739 | 5767 | 27
69 | 5795 | 5823 | 6021 | 5879 | 5908 | 5937 | 28
70 | 5966 | 5996 | 6125 | 6055 | 6085 | 6115 | 30
71 | [...] | 6177 | 6208 | 6239 | 6271 | 6303 | 31
72 | 6335 | 6368 | 6401 | 6431 | 6468 | 6501 | 33
73 | 6535 | 6570 | 6605 | 6640 | 6675 | 6718 | 35
74 | 6747 | 6783 | 6620 | 6857 | 6895 | 6933 | 37
75 | 6972 | 7010 | 7050 | [...] | 7130 | 7170 | 40
76 | 7211 | 7253 | 7295 | 7338 | 7381 | 7424 | 43
77 | 7469 | 7513 | 7559 | 7605 | 7651 | 7651 | 7698 | 46
78 | 7746 | 7795 | 7844 | 7894 | 7944 | 7996 | 50
79 | 8048 | 8100 | 8154 | 8209 | 8264 | 8320 | 55
80 | 8377 | 8435 | 8495 | 8555 | 8616 | 8678 | 60
81 | 8742 | 8806 | 8872 | 8939 | 9007 | 9077 | 68
82 | 9148 | 9221 | 9295 | 9371 | 9449 | 9523 | 77
83 | 9609 | 9692 | 9778 | 9865 | 9954 | [...] | 88
84 | 10141 | 10238 | [...] | 10441 | 10547 | 10656 | 105
85 | 10770 | 10887 | 11007 | 11133 | [...] | [...] | 128
86 | 11539 | 11686 | 11839 | 11999 | 12168 | 12344 | 165
87 | 12521 | 12718 | [...] | 13150 | 13388 | [...] | [...]
88 | 13920 | 14221 | 14550 | 14914 | 15321 | 15783 | 386
89 | 16318 | 16950 | 17726 | 18729 | 20152 | 22623

PROP. I. To find by the Table, what Meridional parts are contained in any Difference of Latitude.

[Page]

The Use of the Table is demonstrated by the several Examples following, after this Manner.

In this Proposition three Varieties present themselves unto our View.

1. When one Place is under the Equinoctial, the other having North, or South Latitude, his Meridional parts corresponding, is to be esteemed for the Meridional Difference of Latitude.

2. When both Places are towards one of the Poles, then the Meridional parts of the lesser, taken from the Meridional parts of the greater Latitude, the remainder is the Meridional difference required.

3. When one Place hath North, and the other South Latitude, their corresponding Meridional parts added together gives the Meridional difference of Latitude sought: thus having sound them out they may thus be applyed.

PROP. II By knowing the Latitudes, and the difference of Longitude of any two Places, to find the Rumb, and Distance.

[Page]

Admit there be a Port in the Latitude of 50° 00' North, and another in the Latitude of 13° 12' North, and their Difference of Longitude is 52° 57' West, I demand the Rumb and Distance?

In the Triangle A b c, let A b represent the proper difference of Latitude, bc the Departure, Ac the distance sailed, A the Angle of theFig. 50. Course, c the Complement of the Course.

In the Triangle ABC, AB is the Meridional difference of Latitude, BC the Difference of Longitude, A the Angle of the Rumb, C the Compl. of the Angle of the Rumb: These things being understood the work evidently appears to be the same as in Rightangled Plain Triangles.

There is then required first the Difference of Latitude, and this falls under the second Variety. [...]

1. To find the Rumb or Course say,

As Merid. X. Lat. 2676',

To Radius or S. 90°.

[Page]

So is X. of Longitude 3177',

To T. of the Rumb 49° 53', the Course there fore is S. W. ½ W, &c.

2. To find the Distance,Fig. 50. [...]

As Sc. Course 40° 07',

To proper X. of Lat. 2208'.

So is Radius or S. 90°,

To the Distance 3426 Minutes as required.

PROP. III. By knowing the Latitudes, and distance of two Places, to find the Rumb, and Difference of Longitude.

1. To find the Rumb or Course say,

As the Distance sailed,

To Radius or S. 90°.

So is the X. of Latitude,

To Sc. of the Rumb required.

2. To find the Difference of Longitude say,

As Radius or S. 90°,

To the X. of Latitude in Merid. Parts.

So is T. of the Rumb,

To the X. of Longitude required.

PROP. IV. By knowing the Latitudes, and Rumb of two Places, to find their Distance, and Difference of Longitude.

[Page]

1. To find the Distance say,

As Sc of the Rumb,

To the X of Latitude.

So is Radius or S. 90°,

To the Distance required.

2. To find the Difference of Longitude say,

As Radius or S. 90°,

To the X. of Latitude in M. Parts.

So is T. of the Rumb,

To the X. of Longitude required.

PROP. V. By knowing the Rumb, Difference of Longitude, and one Latitude, to find the other Latitude, and the Distance.

1. To find the other Latitude say,

As T. of the Rumb,

To the X of Longitude in parts.

So is Radius or S. 90°,

To the Merid. X. of Latitude required.

2. To find the Distance say,

As Sc. of the Rumb,

To the X. of Latitude.

So is Radius or S. 90°,

To the required Distance,

PROP. VI. By knowing the Distance, one Latitude, and Rumb, to find the other Latitude, and Difference of Longitude.

[Page]

1. To find the Difference of Latitude say,

As Radius or S. 90°,

To the Distance.

So is Sc. of the Rumb,

To the X. of Latitude required.

2. To find the Difference of Longitude say,

As Radius or S. 90°,

To the Merid. X. of Latitude.

So is T. of the Rumb,

To the X. of the Longitude required.

SECT. III. Of Circular Sailing, or Sailing by the Arch of a Great Circle.

THIS is of all other the most exact way of sailing, and above all other most perfect, shewing the nearest way, and distance between any two Places: and although it is hardly possible to keep close unto the Arch of a great Circle, yet it is of great advantage to keep conveniently near it, especially in an East or West [Page] Course: In the former Propositions of sailing, we used Meridians, Parallels and Rumbs, as the Sides of every Triangle, whether by the Plain or Mercator's Chart: but in Circular sailing the Rumbs are not used so, because they are Helispherical-lines, and not Circles; nor the Parallels, because they are not great Circles: Whereas the sides comprehending every Spherical Triangle are Arches of great Circles: Therefore here we use Arches of the Meridians, of the Equinoctial, and of other great Circles described, or so imagined to be described, from one Place unto another, on the Spherical Superficies of the Earth and Sea.

Therefore here ariseth two things observable: and,

1. If the two places lie under the Equinoctial, then is their Position East and West, and their distance is their Difference of Longitude, converted into Miles: or,

2. If the two Places proposed be in one and the same Meridian, then is their Position North and South, and their Distance is their Difference of Latitude converted into Miles.

And thus far doth Circular sailing agree with the former; their difference will evidently appear by these following Propositions.

PROP. I. Two Places, the one under the Equinoctial, the other in any Latitude given; also their difference of Longitude given, to find.

[Page]

1. Their Distance in the Arch of a great Circle.

2. The direct Position of the first place from the second.

3. And of the second Place from the first.

Here we call the Angles which the Arch makes with the Meridians of the places propounded, the Angles of the Direct Positions of those places one from the other: because the Arch of a great Circle. drawn between two places is the nearest distance; and the most direct way of the one, to the other Place. Now I shall not here demonstrate it by Schemes, as I have done in the other two Sections, but shall only lay down the proportions, whereby the required parts may be found; and so leave the ingenious Seamen to practice it with Schemes at his leasure: and,

1. To find the nearest distance from Place to place, in the Arch of a Great Circle: Say according to the 10 Case of Rectangled Spherical Triangles.

As the Radius,

To Sc. of X. of Longitude.

So is Sc. of X. of Latitude;

To Sc. of the Distance in the Arch required.

2. For the Direct Position, say by the 11 Case thus,

As the Radius,

To S. of X. of Latitude.

[Page]

So is Tc. of X. of Longitude,

To Tc. of V. of Position required.

3. For the Direct Position of the second Place from the first, say by the 11 Case thus,

As the Radius,

To S. of the X of Longitude.

So is Tc. of X. of Latitude,

To Tc. of V. of Position required.

PROP. II. Two Places proposed, the one lying under the Equinoctial, the other in any Latitude given; with their distance in a great Circle of the same Places being also known, to find.

1. Their Difference of Longitude.

2. The direct Position from the first to the second Place.

3. And from the second to the first Place.

1. For their Difference of Longitude, say by Case 12,

As Sc. of the Latitude,

To Radius.

So is Sc. of their Distance in the Arch,

To Sc. of their Difference of Longitude required.

2. Now to find the Direct Position from the first place to the second, say by the 13 Case; and thirdly, for the Direct Position from the second place to the first, say by the 14 Case of Rectangulars.

PROP. III. Two Places lying in one Latitude given, their difference of Longitude being also known, to find.

[Page]

1. The nearest distance of those two Places.

2. The direct Position of one Place from the other.

The Resolution of this Proposition depends on the 9 Case of Oblique Spherical Triangles: by supposing the Oblique Triangle, to be transfigured or converted into two Rectangulars, by a supposed Perpendicular: and then,

1. To find the nearest distance in the Arch, say by the 8 Case of Rectangulars.

As the Radius,

To Sc. of the Latitude.

So is S. of half X. of Longitude,

To S. of half the required distance, which doubled giveth the distance of the two places in the Arches, as sought.

2. For the Direct Position, say by the 9 Case.

As the Radius,

To S. of the Latitude.

So is T. of half X. of Longitude,

To Tc. of V. of Position required.

PROP. IV. Two Places lying both in one Latitude given, and the nearest distance being also known, to find.

[Page]

1. Their Difference of Longitude.

2. The direct Position of the one Place from the other.

The Resolution of this Proposition falls under the 11 Case of Oblique Spherical Triangles: for here you have the three sides of the Triangle given, viz. the Arch of Distance, and the other two sides (are both equal) being the Complements of the places Latitude: and here seeing the two sides are equal, therefore the two Angles of Position are also equal: now there is required the three Angles,

1. To find their Difference of Longitude, add the double of the Complement of Latitude to the Arch of Distance; then from half this Sum, deduct the Arch of Distance, and then proceed in all points as you see in Case the 11th. So shall their Difference of Longitude be obtained.

2. To find their direct Position:

First, to the double Complement of Latitude, add the Arch of Distance, then from half that agragate, deduct the Complement of Latitude, and then work as before, so shall the direct Position be attained.

PROP. V. Two Places proposed lying in one Latitude, and the distance of those Places in their Parallel given; to find.

[Page]

1. Their Difference of Longitude,

2. Their distance in the Arch of a great Circle,

3. The direct Position of the one from the other.

Now you must understand, that as the Semidiamiter of a Parallel, is in proportion to the Semidiamiter of the Equinoctial: so is any number of Miles in that Parallel, to the Minutes of Longitude answering to those Miles: fo that if we suppose the Semidiameter of the Equinoctial to be Radius, then the Semidiameter of any Parallel is the Sine of that Parallel's distance from the Pole, that is the Sc. of the Latitude of that Parallel: Therefore,

1. To find the Diff. of Longitude say,

As Sc. of the Latitude,

To the Radius,

So is the Distance in that Parallel,

To the Diff. of Longitude required.

2. Now the Difference of Latitude being obtained, the nearest distance may be found, as in the third proposition aforegoing: 3. so likewise may the Angles of Position also.

PROP. VI. By knowing the nearest Distance of two Places, their Difference of Longitude, and one of their Latitudes; to find the Direct Position thereof from the other.

[Page]

This Proposition falls under the first Case of Oblique Spherical Triangles, and is thus resolved: therefore,

As S. of the Distance of the two Places,

To S. of their X. of Longitude.

So is Sc. of the Latitude of the one Place given,

To S. of the Direct Position from the other as was so required.

PROP. VII. By knowing the Latitudes of two places, and likewise their Difference of Longitude; to find,

1. The distance in the Arch.

2. The direct Position from the first to the second place.

3. The direct Position from the second to the first place.

4. The Latitudes and Longitudes by which the Arch passeth.

5. The Course and Distance from Place to Place through those Latitudes and Longitudes according to Mercator.

I shall here make use of M. Norwood's example of a Voyage from the Summer-Islands, unto the [Page] Lizard: now because the work is various I have therefore illustrated it with a Scheme, and shall be as brief and facile as possible. Therefore,

In the Triangle ADE, let A be the SummerIslands, whose Latitude is 32° 25', AD the ComplementFig. 51. thereof 57° 35', let E represent the Lizard whose Latitude is 50° 00', and ED the Complement thereof 40° 00', and let their Difference of Longitude, namely the Angle ADE be 70° 00', now Drepresenteth the North-Pole, and AE an Arch of a great Circle passing by these two Places: now see the operation.

1. By having the Complements of the Latitudes of the two Places, viz. AD 57° 35', and ED 40° 00', and their Difference of Longitude, namely the Angle EDA 70° 00': you may find the nearest distance EA to be 53° 24'; by Case the 9. § 5. chap. 5.

2. Then having found the nearest distance inFig. 51. the Arch EA to be 53° 24', (or 3204 Miles) the Angle of Position from the Summer Islands to the Lizard, namely the Angle DAE, may be found by Case the 1. § 5. chap. 5. to be 48° 48', that is N. E. and 03° 48' Easterly.

3. And also by the same Case, may the Direct Position from the Lizard, to the Summer-Islands, namely the Angle AED befound to be 81° 10', that is W. by N. and 2° 25' Westerly.

4. In order to the finding the Latitudes and Longitudes by which the Arch passeth, first let fall the Perpendicular DB, so is the ObliqueFig. 51. Triangle ADE converted into two Rectangulars, viz. ABD, and DBE: secondly, by Case the 8. § 4. chap. 5. you may find the length of the [Page] Perpendicular DB to be 39° 26', whose Complement is 50° 34', which is the greatest Latitude by which the Arch ABE passeth, so the greatest ObliquityFig. 51.

BDc | 48° | 31
BDf | 38 | 31
BDg | 28 | 31
BDh | 18 | 31
BDj | 08 | 31

of the Equinoctial from that Circle is 50° 34'. — Thirdly, by Case the 9. § 4 chap. 5. you must find the vertical Angles, viz. ADB, and BDE, which will appear, the Angle ADB to be 58° 31', and EDB to be 11° 29': now these things being obtained, the Latitudes by which the Arch passeth at every tenth degree of Longitude from A, may be found by resolving the several Right-Angled Triangles, viz. BDc, BDf, &c. substracting 10° from ADB 58° 31', there remains BDc 48° 31', and so for the rest as in the Table. Now by knowing these Angles last found, and the Perpendicular BD before found to be 39° 26', you may by Case the 3. §. 4. chap. 5. find the Latitudes of the several points A. c. f. g. h. i. B. and E. to be as in the subsequent Table.

5. Thus havingFig. 51.

Latitude. | Longitude.
A. | 32° | 25' | 00 | 00'
c. | 38 | 51 | 10 | 00
f. | 43 | 34 | 20 | 00
g. | 46 | 54 | 30 | 00
h. | 49 | 04 | 40 | 00
i. | 50 | 15 | 50 | 00
B. | 50 | 34 | 60 | 00
E. | 50 | 00 | 70 | 00

found the Latitudes and Longitudes of the Arch, and the other required parts aforementioned, we now come to shew how the Course, and the Distance from place to place according to Mercator may be found. So to find, first the Course and Distance [Page] Ac. now there is given the Latitude of A 32° 25', and of c 38° 51', and their Difference of Longitude is 10° 00', now the Proper DifferenceFig. 51. of Latitude is 6° 26', or 386', and Meridional Difference of Latitude is 475'. Now knowing these things by proposition 2. § 2. chap. 8. yon may find the Course from A to c, to be N. E. 51° 38'; and the Distance Ac to be 622', and so those Rules prosecuted will shew the course and distance from c to f; from f to g; from g to h, &c. So of the rest, which for brevity sake I shall omit, and leave the Ingenious Seaman to Calculate at his Pleasure.

I might hereunto annex many more propositions of Circular Sailing, but because of the smallness of this Treatise, and that those Propositions already handled, being by the Ingenious Seaman well understood, will be sufficient to enable him to perform any other Conclusion in Circular Sailing whatsoever, I therefore here omit, and hasten forwards unto the other parts of this Mathematical Treasury.

[Page]

A Table of Angles, which every Rumb makethwith the Meridian.

These on this side the W. incline towards the N. end of the Meridian | Angles of Inclination with the Meridian. | These on this side the E. incline to the N. end of the Meridian.
Rumbs. | North | Rumbs.
N. by W. | 11° | 15' | N. by E.
N. N. W. | 22 | 30 | N. N. E.
N. W. by N | 33 | 45 | N. E. by N.
North West | 45 | 00 | North East
N. W. by W. | 56 | 15 | N. E. by E.
W. N. W. | 67 | 30 | E. N. E.
W. by N. | 78 | 45 | E. by N.
West | 90 | 00 | East
W. by S. | 78 | 45 | E. by S.
W. S. W. | 67 | 30 | E. S. E.
S. W. by W. | 56 | 15 | S. E. by E.
South West | 45 | 00 | South East
S. W. by S. | 33 | 45 | S. E. by S.
S. S. W. | 22 | 30 | S. S. E.
S. and by W. | 11 | 15 | S. and by E.
Rumbs | South | Rumbs
These on this side the W. incline unto the S. end of the Meridian. | These on this side the E. incline towards the S. end of the Meridian.
Note that if you account in quarter of Points, add for one quarter 2° 48', for one half 5° 37', for three quarters 8° 26', (not regarding the Seconds in Navigation.)

CHAP. IX. Of SURVEYING.

[Page]

IT hath been a custom among Modern Authors, that have treated on this Subject, that before they entred on the Work it self, to give the Description of the Instruments, used in; and chiefly appertaining to the Art of Surveying: viz. the Circumferentor, the Theodolite, the Plain-Table, and the Semicircle: concerning the descriptions of which Instruments I shall not here treat, but refer you unto those Authors that have largely and amply described them. I shall in this place onely demonstrate the Use of the Semicircle in taking the Plots of Enclosures, Champain-Plains, Woods and Mountains divers waysWhich Instrument, and the Plain Table, I esteem as the two aptest Instruments for Surveying of Land; i. e. the Plain Table for small Enclosures, and the Semicircle for Champain Plains, Woods, and Mountains.; and also in taking of Accessible, and Inaccessible Heights and Distances; and also I shall shew the use of a little Instrument called a Protractor, in the delineating on Paper the Plot of a Field, &c. which Instrument being so commonly known, and so generally used makes me omit the description thereof as superfluous.

[Page]

As for your Chain, I would have you, have it made of good round Wyre; to contain in length four Poles, or Perch, to be divided into an hundred equal parts called Links.

And here before we enter on the Work it self, it will be necessary to understand how by the Protractor to lay down an Angle of any quantity of degrees propounded, or to find the quantity of an Angle given.

SECT. I. Of the use of the Protractor.

PROP. I. By the Protractor, to Protract an Angle of any quantity of degrees propounded.

AN Angle may be laid down easily according to the directions of Prop. 5. §. 1. Ch. 4.Fig. 52. but because this is more usefull in Surveying, Know that if it be required to protract an Angle of 50 deg. having drawn the line A B at pleasure, place the Centre of your Protractor on C, and moving it by your Protracting Pinn, untill the Meridional line thereof be directly on the line A B, then make a Mark by the division of 50° on the limb of the Protractor as at D, and [...] [Page] [...] [Page] [Page] draw the line CD, so shall the Angle DCB, be an Angle of 50 degrees.

PROP. II. By the Protractor given, to measure an Angle given.

This is performed by the line of Chords also, according to prop. 6. §. 1. chap 4. and by the Protractor is found thus: Suppose DCB were an Angle whose Quantity were desired, to find which, first the Center of the Protractor applyedFig. 52. unto the Angular point C, and its Meridional line lying justly with CB; you shall perceive the Point D, to touch the limb of the Circle at 50 deg. Therefore I conclude the Measure of the Angle DCB, to be 50 degrees.

SECT. II. Of the Manifold Use of the Semicircle, in taking the Plots of small Enclosures, Plains, Woods, or Mountains divers Ways.

[Page]

PROP. I. How to take the Plot of a Field, by the Semicircle at one Station taken in any part thereof, from whence all the Angles may be seen, and measuring from the Station unto every Angle thereof.

SUppose ABCDEF were a Field, and 'tis required to take the Plot thereof: Having placed marks at all the Angles thereof, and made choice of your Station, which let be K; at which, place your Instrument, and turningFig. 53. it about untill the Needle hang over the Meridian Line of the Chart, there screw it fast: Then directing your sight to A, you'l find the Degree out by the Index to be 40° 15': Then measuring KA with your Chain it appears to be 5 Chains and 20 Links, which note down in your Field-book: and so do by all the rest untill you have found all the Angles and Distances from your Station K, to each respective Angle, which finished your work will stand thus.

[Page]

Angles. | D. | M. | C. | L.
A. | 40 | 15 | 5 | 20
B. | 88 | 00 | 6 | 10
C. | 130 | 00 | 5 | 50
D. | 200 | 00 | 7 | 00
E. | 250 | 00 | 5 | 00
F. | 310 | 00 | 5 | 20

PROP. II. How to delineate on Paper any Observation taken according to the Doctrine of the last Proposition.

Upon your Paper draw a Line to represent the Meridian line as M, H, then Placing the Center of your Protractor on the point K, laying the Meridian line of the Protractor on the Meridian line M, H, then seeing the Angle at A was 40° 15', make a Mark against 40° 15' of the Protractor, as at A, and so do with all theFig. 53. other Angles, as you find them in your Table: Then remove your Protractor, and draw the Lines KA, KB, &c. This done lay down on each line his respective Measure, as it appeareth in the Table. Lastly draw the Lines AB, BC, &c. So have you on the Paper the exact Figure of the Field.

PROP. III. How by the Semicircle to take the Plot of a Field at one Station in any Angle thereof, from whence you may view all the other Angles, by measuring from the Stationary-Angle, unto all the other Angles.

[Page]

Admit A, B, C, D, E, F, G, to be a Field, whose Plot is required: Place your Semicircle at G, and turning it about untill the Needle hang over the Meridian line of the Chart, and there screw it fast: Then direct your sights to the several Angles, viz. B, C, D, &c. in order one after the other, and so shall eace respectiveFig. 54. Angle be found, as in the subsequent Table: Then with your Chain, measure from your Stationary-Angle G, to all the other respective Angles, which done you have finished, and the work standeth thus.

Angles. | D. | M. | C. | L.
B. | 40 | 00 | 5 | 00
C. | 88 | 00 | 6 | 00
D. | 120 | 15 | 6 | 40
E. | 165 | 00 | 6 | 30
F. | 193 | 00 | 3 | 40
A. | 348 | 07 | 4 | 00

PROP. IV. How to delineate any Observation taken according to the Doctrine of the last Proposition.

[Page]

Upon your Paper draw a streight line as M, N, then take a point therein as G, to represent the Stationary-Angle, to which point apply the Center of your Protractor, (in all respects as isFig. 54. before taught) then according to the Notes in the Table, prick off all the Angles, viz. B, C, &c. according to their due quantity, then draw all the lines, viz. GB, GC, GD, &c. and on them place their respective measure (as appeareth in your Notes) lastly draw the lines AB, BC, CD, &c. So is there on the Paper the exact Figure of the Field, as was required.

PROP. V. How by the Semicircle to take the Plot of a Field at two Stations, by measuring from each Station to the visible Angles: the Field being so Irregular that from no one Place thereof, all the Angles can be seen.

Admit A, B, C, D, E, F, G, H, I, K, to be the Figure of a Field, whose Plot is required: having made choice of your two Stations, viz. Q, and P, and placed Marks in all the Angles: Then place your Semicircle at Q, and thereFig. 55. six it with the Needle hanging over the Meridian of the Chart, represented by R, Q, X, and direct your sights unto all the visible Angles, viz. [Page] A, B, C, D, E, and F, and note down the Quantity of each Angle in your Field-book: Then measure with your Chain from your Station Q, to the Angles A, B, C, D, E, and F, and their length so found, note down in your Field-book also.Fig. 55.

This done direct your sight unto your second Station P, and note down in your Field-book the degree of Declination, of your second-station P, from the Meridian. Then measure the Stationary Distance PQ with your Chain, and note it down in your Field-book also.

Then remove the Instrument unto P, your second-station, and there fix it with the Needle hanging over the Meridian line of the Chart represented by TPB, then direct your sights to the several visible Angles at this second Station, viz. F, G, H, I, and K, in order one after another, and note down the Quantity of each Angle in your Field-book: Then with your Chain measure from your Station P, to these several Angles G, H, I, and K, (in all respects as at the first station Q.) and their length so found note down in your Field-book likewise: So have you finished your Observation, and your work standeth thus.Fig. 55.

[Page]

The Observation taken at the first Station Q.

Angles.
D
M
C.
L

A
50
00
6
60

B
80
00
7
65

Fig. 55.

C
140
12
12
00

D
220
07
11
10

E
270
05
12
60

F
330
00
6
00

The Declination of the Station P, from the Meridian R Q X, is 30° 00', and the Stationary distance Q P is 9 Chains.

The Observation taken at the second Station P.

Angles.
D
M
C
L

F
227
11
00
00

G
297
00
12
00

H
347
16
9
90

I
60
00
6
00

K
90
00
6
26

Fig. 55.

☞ Note that the manner of taking the Plot of a large Champain Field, at many Stations, is almost the same with this Proposition; for he that can do the one, can also perform the other: therefore for brevity sake I here omit it as superfluous.

PROP. VI. How to delineate any Observation taken according to the Doctrine of the last Proposition.

[Page]

Upon your Paper draw the Meridian-line R Q X, then place the Center of your Protractor on Q, (representing your first Station) and its Meridional-line lay equal to R Q X, then prick off the Angles visible at your first Station Q, viz. A, B, C, D, E, and F, Of their due quantity, then draw Q A, Q B, &c. laying on them their corresponding measure, noted in your Field-book. Now because your second Station P, doth decline 30° 00', from the Meridian RQX, prick off 30° 00', and draw PQ,Fig. 55. making it 9 Chains as in your Field-book appeareth, so doth P represent your second Station. Then in all respects as before, place your Protractor at P your second Station, and draw the Meridian T P B parallel to R Q X, then prick off the several Angles, viz. F, G, H, I, and K, Of their due quantity, and then draw PF, PH, PI, &c. of their due length. Lastly draw the lines AB, BC, CD, &c. and so shall you have on your Paper the exact Figure of the Field as required.

PROP. VII. How by the Semicircle, to take the Plot of a Field at t [...] Stations, which lieth remote from you, when either by opposition of Enemies you may not, or by some other Impediment you cannot come into the same.

[Page]

Admit the Figure A, B, C, D, E, F, to be a Field into which by no means you can possibly enter, and yet of necessity the Plot thereof must be had, for the obtaining of which chuse any two Stations, it mattereth not whether near at hand or far off, so that all the Angles may be seen. Let your two Stations be H and L, (the full length of the Field if possible) then place your Instrument at H, and fixing it as isFig. 56. afore shewed, direct your sights to the several Angles of the Field, viz. A, B, C, &c. orderly one after another, observing their degrees as is afore taught, noting it down in your Field-book: then take up your Instrument, leaving a mark in its room at H, And measure with your Chain from Hunto L, your second Station, which note down in your Field-book; Then placing your Instrument at L, your second Station, and as is before taught, fixing it there, make the like Observation to the several Angles, viz. A, B, C, D, &c. as at the first Station H, and note it down in your Field-book also, And having so done you have finished, and your Work standeth thus.

[Page]

Observations at the first Station H, are

The Angle from H the first Station, unto L the second Station, is 180° 00', theFig. 56. Stationary distance HL, is 60 Chains.

Observations at the second Station L, are

1 Angles | D | M
A | 104 | 00
B | 88 | 07
C | 59 | 00
D | 48 | 00
E | 26 | 00
F | 21 | 30
2Angles | D | M
A | 16 | 00
B | 39 | 00
C | 50 | 09
D | 74 | 00
E | 100 | 00
F | 29 | 15

PROP. VIII. How to delineate any Observation taken according to the Doctrine of the last Proposition.

Upon your Paper draw a Line as HL, which make equal to 60 Chains, then placing the Center of your Protractor on H, your first Station, prick off all the Angles A, B, C, &c. asFig. 56. you find them in your Field-book, and draw HA, HB, HC, &c. at pleasure: then remove your Protractor unto your second Station L, placing it as before, and prick off all the Angles [Page] A, B, C, D, &c. as you find them in your Field notes; and draw the lines LA, LB, LC, &c. at length untill they intersect the former lines, HA, HB, &c. in the Points A, B, C, &c.Fig. 56. which Points of Intersection are the Angles of the Field. Lastly draw AB, BC, CD, &c. So shall you have on your Paper the Figure of your Field, required.

PROP. IX. How by the Semicircle, to take the Plot of a great Champain-Plain, Wood, or other overgrown Ground, by measuring round about the same, and making Observation at every Angle thereof.

Admit A, B, C, D, be the figure of a Large overgrown Champain-Field; whose Plot is required. First Place your Instrument at A, laying the Index on the Diameter; and turn it about, untill you espy the Angle at D, and there fix it fast: and direct your sights to B, and note the Degree cut by your Index, in your Field-book, (as afore is taught) then remove your Instrument to B, and there make the like observation, and so to C, and D, noting it down in your Field-book, as asore. Then with your Chain, measure the Sides AB, BC, CD, and DA, whoseFig. 57. length note down in your Field book, and so you have finished and your work standeth thus.

[Page]

Angles. | D | M | C | L
DAB | 100 | 00 | 12 | 20
ABC | 117 | 15 | 10 | 00
BCD | 71 | 30 | 19 | 20
CDA | 71 | 15 | 12 | 20

PROP. X. How to delineate any Observation taken according unto the Doctrine of the last Proposition.

Upon your Paper draw the line AB, at Pleasure,Fig. 57. and placing the Center of your Protractor on the Point A, prick off an Angle of 100°, and draw AD, setting on it, and also on AB, their corresponding measure, in your notes: Then on B, protract an Angle of 117° 15', draw BC of its due length: Then draw the line CD, so have you the exact figure of the Field, on your Paper.

PROP. XI. How to take the Plot of any Field, by the help of the Chain only.

Admit the Figure A, B, C, D, E, to represent a Field whose Plot is required. To obtain theFig. 58. which, first measure the sides CD, CB, and BD, and note their due length down in your Fieldbook, [Page] and then measure the Sides CA, and BA, and then note [...] down their Length in your Fieldbook. Then measure the sides BE, and ED, for the sides BC, andFig. 58. BD, were before known) which note down in your Fieldbook. So is your Field A, B, C, D, E, reduced into three Triangles, viz. CBD, CAB, and BED, the length of whose sides are all known, thus you have finished, and the works stands as you see.

PROP. XII. How to delineate any Observation, taken according to the Doctrine of the last Proposition.

Upon your Paper, draw a streight line, as CD, make it 5 Chains, 97/100, take CB in your Compasses, and strike an Obscure Arch; then take BD, and with that extent in D, cross the formerFig. 58. Arch in B, and draw BC, and BD. Then take in your Compasses BE, and on B, strike an Obscure Arch, then take DE, and also cross [Page] the former Arch in E, and draw BE, and ED. Lastly take the line CA, and on C strike an Obscure Arch, then take AB, and on B, intersectFig. 58. the former Arch in A, then draw CA, and AB, so have you on your Paper the exact figure of the Field A, B, C, D, E, as was required.

SECT. III. Of finding the Area or superficial Content of any Field, lying in any Regular or Irregular Form: by reducing the Irregular Fields into Regular Forms.

HAving already shewed how to take the Plot of any Field divers ways, by the Semicircle and Chain, and also by the Protractor how to delineate the Draught thereof on Paper, &c. I now come to shew how the Area or superficial Content of a Field may be attained, i. e. how many Acres, Roods and Perches are therein contained. To which end know; That a Statute Pole or Perch contains 16½ Feet; that 40 of those Perches in length, and 4 in breadth makes an Acre. So that an Acre contains 160 Perches, and a Rood 40 Perches; according to the Statute 33, of Edward the First.

[Page]

Now the Original of the Mensuration of Land, and all other Superficies, depends on the Mensuration of certain Geometrical Figures; as a Triangle, Square, &c. which may be measured according to the directions of §. 2. chap. 4 of Geometry: It would therefore here be superfluous to make a repetition of things already handled: I shall therefore omit it, and come to shew how any Field lying in any Irregular Form, may be measured by converting it into Regular Figures; for it seldom happeneth, but that the Plot of a Field, is either a TrapeziumIs a Quadrangle, whose sides are not Parallel, nor equal. Euclides postulat hant fabricam'Trapezium, tanquam mensulam vocari: & sanè nominis ejus ratio Geometrica nulla est: P. Rami lib. 14 pag. 94., or a many-sided Irregular Figure: therefore I shall first shew how to find the Content of a Trapezium. Secondly, of any many sided Irregular Figure; and thirdly, how to reduce any number of Perches into Acres, &c. and on the contrary any number of Acres, into Roods and Perches.

PROP. I. How to find the Area or superficial Content of a Trapezium.

Trapeziums are Quadrangles of sundry forms: yet take this as a general Rule, whereby their Content may be found. Admit it be required to find the Area or superficial Content of the Trapezium ABCD, to find which, first by drawing [Page] the Diagonal AD, you reduceth it into two Triangles, ABD, and ADC. Then by prop. 3. §. 1. of Chap. 4 let fall the two Perpendiculars on AD, from B, and C, Then by prop 3. §. 2 Ch.Fig. 59. 4. find the superficial Content of the two Trianangles ABD, and ADC, which added together, is the Content os the Trapezium; by which Rule the Content of the Trapezium, A, B, C, D, is found to be 630 Perches.

PROP. II. To find the Area or superficial Content of a manysided Irregular Figure.

Admit A, B, C, D, E, F, G, to be an Irregular many-sided Figure, representing a Field whose Content is required: now in regard the Field is Irregular, therefore reduce it into Triangles, viz. ABC, ACG, EDG, DEG, and DFG, andFig. 60. then find the Content of all the said Triangles, by prop. 3. §. 2. Chap. 4 and add their Contents together; so shall that Sum be the Content of the said Figure; and so do for any other.

PROP. III. How to reduce any Number of Perches into Acres, and on the contrary, Acres into Perches.

To find how many Acres are contained in any Number of Perches given, you must consider that 160 Perches do make a Statute Acre, therefore if you divide the Number of Perches [Page] propounded, by 160, the Quotient is the number of Acres contained therein; and if there be a remainder which exceed 40, then divide it by 40, the Quotient shall be Roods, and the remainder Perches.

But on the contrary, if it were required to find how many Perches are contained in a certain Number of Acres propounded. You must multiply the Number of Acres, by 160: the product shall be the Perches contained therein.

It may be here expected, that I should shew how to reduce customary Measure to statute Measure; and also that I should treat of the Division and Separation of Land. But because Mr. Rathborne, and of late Mr. Holwell, hath sufficiently explained the same, by many varieties, I shall for brevity sake omit it, and leave you to consult those Authors.

SECT. IV. Of the Use of the Semicircle in taking Altitudes, Distances, &c.

[Page]

PROP. I. How by the Semicircle to take an Accessible Altitude.

ADmit AB, be the Height of a Tower, which is required to be known. First placing your Semicircle at D, (with the Arch downwardsFig. 61. and the two sights fixed) place it HorizontalWhich to do is no more than thus; with a Thread and Plummet fastened at the Center of the Semicircle, so that it hath liberty to play, move the Semicircle until the Thread playeth against 90 deg. then screw it fast, and it is Horizontal. and screw it fast; Then move your Index, till through the sights thereof, you espy the top of the Tower at B, and observe what degree the lower part of the Index cutteth and that will be equal unto the Angle at D 50 deg Then measure the distance DA, which let be 299 Feet. Now the heighth of the Tower AB, is found, according to prop. 1. §. 2. Chap. 5. thus,

As Sc. V. at A 50° 00',

To Log. cr. DA 299 Feet.

So is S. V. at A 50 00,

To Log. AB 356 3/10 Feet the height of the Tower AB required.

PROP. II. How by the Semicircle to take an Inaccessible Altitude, at two Stations.

[Page]

Let AB be a Tower whose height is required; having placed your Instrument at E, as before direct your sights unto the Top of the Tower at B, and finding the Degree cut by the Index,Fig. 61. to be 23° 43', I say it is the Quantity of the Angle at E: Now by reason of Water, or such like Impediment, you can approach no nearer the Base of the Tower, than D, Therefore measure ED, which is found to be 512 Feet, then at D, make the like Observation, and the Angle at D, appeareth to be 50° 00', whose Complement is the Angle DBA, 40° 00', and the Complement of the Angle E 23° 43', is the Angle EBA 66° 17': Now if the lesser Angle at B, be taken out of the greater, the remainder is 26° 17', the Angle EBD: Now first to find the side BD, of the Trangle EBD, say according to prop. 1. §. 3. chap. 5. thus.

As S. of V. EBD, 26° 17',

To Log. cr. ED 512 Feet.

So is S. of V. at E 23° 43',

To Log. cr. BD 465 2/10 Feet required.

Now to find the Height of the Tower AB, say according to prop. 2. §. 2. chap. 5. thus.

As Radius or S. 90°,

To Log. cr. DB 465 2/10 Feet found.

So is S. of V. BDA 50° 00',

To Log. cr. BA 356 3/10 Feet, which is the heightFig. 61. of the Tower required.

[Page]

☞ Note that in taking any manner of Altitude the height of your Instrument must be added unto the height found, and that will give you the True Altitude required.

PROP. III. How by the Semicircle to take an Inaccessible Distance at two Stations.

Admit A, and B, be the two Stations, from either of which it is required to find the distance unto the Church at C; placing your Instrument at B, the Index lying on the Diameter, and direct your sights unto the Church at C, fasten your Instrument, and turn your sights about untill you see through your sights, your second Station at A, so will you find your Index to cutFig. 62. 30° 00', which is the Quantity of the Angle ABC. Then measure the distance AB, which is found to be 250 Yards, then with your Instrument at A, make the like Observation as before, and you will find the Angle BAC to contain 50° 00'. Now by the third Maxim of Plain Triangles §. 1. Chap. 5 you find also the Angle ACB, to be 100° 00': now to find the distance AC, and BC, you may by their opposite proportion according to prop. 1. §. 3. chap. 5. find the distance of AC, thus.

As S. of V. at C 100° 00',

To Log. cr. AB 250 yards.

So is S. of V. B 30° 00',

To Log. cr. AC 127 yards. Which is the distance of the Church from A.

[Page]

Now to find the distance BC, say,

As S. of V. at A 100° 00',

To Log. cr. AB 250 yards.

So S. is of V. at A 50° 00',

To Log. cr. BC 194 4/10 yards, which is the distance of the Station B, from the Church at C.

PROP. IV. How to find the Horizontal line of any Hill or Mountain, by the Semicircle.

Let Figure 63 be a Mountain, whose Horizontal-line AB is required to be found: to find which, place your Instrument at A, and having caused a Mark to be placed on the Top of the Mountain at C; (of the just height of your Instrument) then move your Index, untill through the sights thereof you espy the Mark at C, so will you find the Quantity of the Angle CAD, to be 50° 00', and by consequenceFig. 63 the Angle ACD to be 40° 00', then measure up the Hill AC, which is 346 yards. Now having obtained these several things, 'tis required to find the length of AD part of AB; to find which say,

As Radius or S. 90°,

To Log. cr. AC 346 Feet.

So is Sc. of V. at A 50° 00',

To Log. cr. AD 222 4/10 Feet.

Now seeing the Hill or Mountain descendeth on the other side, you must place your Instrument at C, and direct your sights unto the Bottom at B, and the Angle DCB will be found 50° 00', and the Angle CBD 40° 00'. Then measuring down the Mountain as CB, it appeareth

[Page] [Page] [Page]

Plate IV Page 237

4. 63. [...]. [Page] [Page] to be 415 Feet; then have you the Angles DCB, and CBD.

To find DB, part of AB, say,

As Radius or S. 90°,

To Log. cr. CB 415 Feet.

So is S. of V. BCD 50° 00',

To Log. cr. DB, 318 Feet: Now AD 222 4/10Fig. 63. Feet added thereunto produceth AB 540 4/10 Feet, which is the Horizontal line required of the Mountain ACBD.

☞ Note that when you come to delineate a Field wherein are Hills, you must protract the line AB, instead of the Hypothenusal Lines AC, and CB, and 'twill be necessary to distinguish those kind of Fields, by shadowing them off with Hills and Dales.

SECT. V. How to find whether Water may be conveyed from a Spring-Head unto any appointed Place.

THE Art of conveying of Water from a Spring-Head unto any appointed Place, hath a special respect unto measuring, and therefore I think it not amiss to assert it in this place, and enroll it under the Title of Surveying.

[Page]

In the performance of which we make use of a Water-level, the Construction and making whereof is sufficiently known to those who make Mathematical Instruments: Now if it were required to find whether Water may be conveyed in Pipes, &c. to any Place assigned: to perform which observe these Rules.

First at some 10, 20, 30, 40, 60, or 100 yards distant from the Spring-head in a rightline towards the Place unto which your Water is to be conveyed. Place your Water-level, being prepared of two Station Staves with moveable Vanes on each of them, graduated also after the usual Manner: Cause your first Assistant to set up one of them at the Spring-Head; Perpendicular unto the Horizon, and your second Assistant to erect another, as far from your Water-level towards the Place to which the Water is to be conveyed, as your Water-level is distant from the Spring-head: Now the Stationstaves in this order erected, and your Waterlevel placed precisely Horizontal, go unto the end of the Level, and looking through the sights, cause your first Assistant to move a Leaf of Paper, up or down your Station staff, untill through the sights you espy the very edge thereof, and then by some known sign or sound, intimate to your Assistant that the Paper is then in its true position, then let the first Assistant note against what Number of Feet, Inches, and parts of an Inch the edge of the Paper resteth; which he must note down in a Paper. Then your Water-level remaining immoveable, go to the other end thereof, and looking through the sights towards your other Station-staff, cause [Page] your second Assistant to move a Leaf of Paper along the Staff, till you see the very edge thereof through the sights, and then cause him by some known sign or sound, to take notice what number of Feet, &c. are cut by the said Paper, which let him keep, as your first Assistant did.

This done let your first Assistant bring his Station-staff from the Spring-head, and cause your second Assistant to take that Staff, and carry it forwards towards the Place, unto which the Water is to be conveyed; some 30, 40, 60, or 100 yards, and there to erect it Perpendicular as before, letting your second Assistant's staff stand immoveable, and your first Assistant to stand by it: Then in the Midway between your two Assistants, place your Water-level exactly Horizontal, and looking through the sights thereof, cause your first Assistant, and after that your second, to make their several observations in all respects as before.

In this manner you must go along from the Spring-head, to the place unto which you would have the Water conveyed, and if there be never so many several Stations, you must in all of them observe this manner of work precisely; so that by comparing the notes of your two Assistants together, you may easily know whether the Water may be conveyed from the Spring-head, or not, by calling your two Assistants together, and causing them to give in their notes of observation at each Station, which add together severally: Then if the Notes of the second Assistant, exceed the Notes of the first Assistant, take the lesser out of the greater, and [Page] the remainder will shew you how much the appointed Place, to which the Water is to be conveyed, is lower than the Spring-head.

The first Assistant's Note.

Station. | Feet. | Inch. | Parts.
1 | 15 | 3 | . 50
2 | 2 | 1 | . 25
3 | 1 | 6 | . 00
Sum | 18 | 10 | . 75

The second Assistant's Note.

Station. | Feet. | Inch. | Parts.
1 | 3 | 2 | . 75
2 | 14 | 0 | . 25
3 | 3 | 11 | . 00
Sum | 21 | 2 | . 00

By these two Tables you may perceive that the Notes of the first Assistant collected at his several Stations, being added together, amounts unto 18 Feet, 10 Inches, and 75/100 or ¾ of an Inch: and the Notes of your second Assistant collected at his several Stations, amounts unto 21 Feet, 2 Inches: So that the number of the first Assistant's Observations, being taken from the second's, there will remain 2 Feet, 3 Inches, and 25/100 or ¼ of an Inch. And so much is the place unto which the Water is to be brought, lower than the Spring-Head, according to the sleight Water-Level, and therefore the Water may easily be conveyed thither. And here observe these Notes.

[Page]

1. In your Passage between the Spring head, and the appointed Place, from Station to Station, you must observe this order, that your first Assistant at every Station must stand between the Spring-head, and your Water-level: otherwise great Errours will ensue.

2. That if the Notes of your first Assistant, exceed the Notes of the second Assistant, then 'tis impossible to bring the Water from that Spring-head unto the appointed place, but if their Notes are equal, it may be done, if the distance be but short.

3. That the most approved Authors concerning this particular do aver, that at every Mile's end there ought to be allowed 4½ Inches more than the Streight-level, for the current of the Water.

4. That if there be any Mountains lying in the way betwixt the Spring-head and the Place to which the Water is to be conveyed, you must then cut a Trench by the side of the Mountain, in which you must lay your Pipes equal with the streight Water-level, with the former allowance: and in case there be a Valley, you must then make a Trunk of strong wood, well under-propped with strong pieces of Timber, well Pitched, or Leaded, as is done in divers places between Ware and London.

5. That when the Spring will have too violent a Current, you must then convey your Water to the place assigned, by a Crooked or Winding line, and you also ought to lay the Pipes, the one up, and the other down, that thereby the Violence of the Current may be stopped.

CHAP. X. Of MEASURING, Of Board, Glass, Tiling, Paving, Timber, Stone, and Irregular Solids, such as Geometry can give no Rule for the Measuring thereof.

[Page]

SECT. I. Of the Measuring of Board, Glass, Paving, Tiling, &c.

I Have already in the fourth Chapter of this Book, and the second Section thereof, applyed Geometry to the finding out of the Superficial Content of all Regular Superficies. I have also in the ninth Chapter, and the third Section thereof, shewed how the Superficial Content of any Irregular Superficies may be found, by reducing [Page] them into Regular Forms: which I have explained amply in that Section, I shall therefore here be as plain and brief as is possible.

PROP. I. To Measure a Piece of Board, Plank, Glass, &c.

In Measuring of Board, Glass, &c. Carpenters and other Mechanicks measure by the Foot, 12 Inches unto the Foot; so that a Foot of Board, or Glass, contains 144 Square Inches.

Now if a Piece of Board, Plank, or Glass, be required to be measured, let it be either a Parallelogram, or Tapering Piece: first by the Rules aforegoing find the Content thereof in Inches, and that Product divide by 144, the Quotient is the Content of that Superficies in Feet.

PROP. II. To measure Tiling, Flooring, Roofing, and Partitioning-works.

In Tiling, Flooring, Roofing, and Partitioningwork, Carpenters, and other Workmen, reckon by the Square, which is 10 Feet every way; so that a Square containeth 100 Feet: Example.

There is a Roof 14 Feet broad, what length thereof shall make a Square? Divide 100 by 14, it yields 7 1/7 Feet.

[Page]

Now if you have any Number of Feet given, and the Number of Squares therein contained are required, divide that Number by 100, the product is Squares.

PROP. III. To measure Paving, Plaistering, Wainscotting, and Painting-work.

In Paving, Plaistering, Wainscotting, and Painting-work, Mechanicks reckon by the Yard Square, so each Yard is equal unto 9 Square Feet.

By the Rules aforegoing find the Superficial Content of the Court, Alley, &c. in Feet: which divide by 9, the Quotient is the Number of Yards in that work contained.

SECT. II. Of the Measuring of Timber, Stone, and Irregular Solids.

IN Superficial Measure a Superficial Foot contains 144 Square Inches; but in Solid Measure a Foot contains 1728 Cubick Inches. Now having already in the fourth Chapter of this Book, and the third Section thereof, largely applyed Geometry unto the Measuring of all [Page] Regular Solids, I shall therefore in this Place be as brief as possible, only I shall be somewhat larger in the Mensuration of Irregular Solids, which is of special Moment in sundry parts of the Mathematical Practices.

PROP. I. How to Measure any kind of Timber, or Stone, whether Three-square, Four-square, Many-square, Round, or of any other fashion, provided it be streight and equal all along.

To perform which first by the Rules aforegoing in Chap. 4. §. 2. get the Superficial Content at the End, and then say,

As 144, the Inches of the Superficial Content of the End of a Cubick Foot,

To a Cubick Foot containing 1000 parts;

So is the Superficial Content of the End of any piece of Timber,

To the Solid Content of one Foot length of the said piece of Timber.

According to which Mr. Phillips calculated the ensuing Table, which I have thought fit hereunto to annex.

Case 2 Or the solid Content in Feet, &c. may be found otherwise thus.

By the Rules aforegoing find the Content of the End of the piece of Timber in Inches, which Content multiply by the length of the said piece of Timber, or Stone in Inches, and that Product divide by 1728, it produceth the Solid [Page] Content of that Piece of Timber, or Stone, in Feet, and parts of a Foot.

A Table shewing the Solid Content of one Foot-length of any Piece of Timber, according to the Superficial Content at the End thereof.

Feet. | Parts. | Feet. | Parts.
The Inches of the Content at the End. | 1 | 0 | 007 | The Inches of the Content at the End. | 200 | 1 | 398
2 | 0 | 014 | 300 | 2 | 083
3 | 0 | 021 | 400 | 2 | 778
4 | 0 | 028 | 500 | 3 | 472
5 | 0 | 035 | 600 | 4 | 167
6 | 0 | 042 | 700 | 4 | 861
7 | 0 | 049 | 800 | 5 | 556
8 | 0 | 056 | 900 | 6 | 250
9 | 0 | 062 | 1000 | 6 | 944
10 | 0 | 069 | 2000 | 13 | 888
20 | 0 | 139 | 3000 | 20 | 833
30 | 0 | 208 | 4000 | 27 | 778
40 | 0 | 278 | 5000 | 34 | 722
50 | 0 | 347 | 6000 | 41 | 666
60 | 0 | 417 | 7000 | 48 | 711
70 | 0 | 485 | 8000 | 55 | 555
80 | 0 | 556 | 9000 | 62 | 500
90 | 0 | 625 | 10000 | 69 | 444
100 | 0 | 694 | 20000 | 138 | 888

PROP. II. To measure Round Timber which is Hollow: or any other Hollow Body.

[Page]

If Hollow Timber be to be measured, first measure the Stick as though it were not Hollow, then find the Solidity of the Concavity, as though it were Massie Timber, then substract this last found Content, out of the whole Content before found, the remainder is the Content of that Hollow Body.

PROP. III. To Measure Tapering Timber, or Stone.

Those Tapering Bodies are either Segments of Cones, or Pyramids: now the way to measure such bodies, is demonstrated in Prop. the 4. and 5. §. 3. Chap. 4: But now to find the Content of these Segments do thus: measure the Solidity of the whole Cone, or Pyramid, and then find the Content of the Top part thereof cut off, (as if it were a Cone, or Pyramid of it self) and the Content thereof, deduct from the Content of the whole Cone, or Pyramid: so shall the remainder be the Content of the Segment required: which reduced into Feet gives the Solid Content of that Piece of Timber in Feet. Now to find the length of the Top part cut off, from the Cone, or Pyramid, say,

As the Difference of the breadth of the two Ends, To the length between them:

[Page]

So is the breadth of the greater End, To the whole length of the Cone, or Pyramid.

PROP. IV. How to find the Solid Content of any Solid Body, in any strange form, such as Geometry can given: no Rule for the measuring thereof.

These strange forms are either Branches in Metal, Crowns, Cups, Bowles, Pots, Screws, or Twisted Ballisters,Whose Surface is bounded by a Line called by Proclus a Helicoides, but it may also be called a Helix, a Twist or Wreath, &c. or any other Irregular-Solid, that keep not in thickness one Quantity, but are thicker in one place, than in another, so that no man by Geometry, is possible to measure their Solidity.

Now for the finding the Content of any such like Irregular Body in Inches or Feet, do thus: Cause to be made a Hollow Cube, or Parallelepipedon, so that you may measure it with an InchRule without Difficulty, and so to know the true Content of the whole, or any part thereof at pleasure within the Concavity: Then take some other convenient Vessel, and put pure Springwater therein; then having filled the Vessel to a known Measure, make a Mark precisely round the very edge of the Water, then take the solid body and put it therein, then take out as much of the Water (as by means of the body put therein) is arisen above the Mark, untill the Water do justly touch at the Mark again: then put the Water taken forth into the [Page] Hollow Cube, and find the solid Content thereof (being transformed into a Cubick Body) in Feet, Inches, and parts of an Inch: Which Content is the just solidity of the Body put into the Water. (Archimedes by this Proposition found the deceit of the Crown of Gold which Gelo the Son of Hiero had vowed unto his Gods: now the Workmen had mixed Silver with the Gold, which Theft was discovered by the great skill of Archimedes)See Procl. lib. 2. cap. 3. & Viturvius lib. 9. cap 3. And herein you must be very curious not to spill any of the Water, or take out of the Vessel, or put into the Hollow Cube, any more than the just quantity arisen above the Mark, for if you do it will produce infinite Errours, and thus may the Solidity of any Irregular Body be found.

CHAP. XI. Of GAUGING.

[Page]

IN GAUGING there are two things chiefly necessary to be noted, yet both controverted. First, that seeing all manner of Casks, made to hold Liquor in, are for the most part the Trunk of a Sphereroid, cut off with two Circles, at Rightangles with the Base, and therefore Irregular, Therefore they must, first be reduced into a Regular Proportion. — And the second thing necessary to be noted, is to find the true quantity of an Ale, or Wine-Gallon in Cubick-Inches or parts of a Foot, that thereby the Content of the Vessel or Cask in Gallons may be known.

SECT. I. Of Gauging any Beer, Ale, or Wine-Cask, also any manner of Brewers Tuns.

[Page]

PROP. I. To find the Solid Content in Inches of any Cask.

I Shall follow Mr. Oughthred's method, which is, Take the Diameter of the Cask both at Head and Bung, by which find the Area's of their Circles, which done, then take two thirds of the Area of the Bung, and one third of the Area at the Head, which added together, shall be the Mean Area of the Cask; which multiplyed into the length of the Vessel, it will shew how many solid Inches are contained therein.

Example: Suppose the Diameter at the Head of a Vessel be 18, and at the Bung 32, and length is 40 Inches.

Now I find the Aggregate of the two Circles to be 620, and 989, Cubick Inches: which multiplyed by 40, the length, produceth 24839, 56/100 Cubick Inches, for the whole Content of that Cask in Cubick Inches.

PROP. II. To find the Content of a Vessel in Wine, or Ale Gallons.

[Page]

The Wine Gallon is established by the Consent of Artists, in these and other Nations, to contain 231 Cubick InchesSee Mr. Oughthred in his Book of the Circles of Proportion, page the 57. and Mr. Edm. Gunter in his Book of the Cross-staff, part 21. chap. the 4.. Yet Dr. Wybard affirms it to be somewhat less, to wit 225, at most: The Ale Gallon contains 282 Cubick Inches, according to the Establishment of Excise. Herein Artists differ somewhat in their Experiments.

Now having already shewed how to find the Content in Inches of any Cask, I now come to shew how to find the Content in Gallons, of any Beer, Ale, or Wine Cask, which is thus: Divide the Number of Inches given by 231, for Wine Measure, and 282, for Ale Measure. In the former Example I find the said Cask to contain 107, 53 Wine Gallons, and 88, 8, &c. Gallons in Ale Measure.

PROP. III. How to Gauge or Measure Brewers Tuns, &c.

Those Tuns are most commonly Segments of Cones or Pyramid, whose Basis is either a Square Parallelogram, Circle, or Oval; to measure which, [Page] let their form be what it will you must do thus. By the former Rules of Measuring such Segments or Bodies, you must find their Solid Content in Cubick Inches, (as in prop. 3. §. 2. chap. 10.) which Content divide by 282 Inches, (the Inches in one Gallon) it sheweth the Content in Gallons, and dividing the Gallons by 36, (the number of Gallons in a Barrel) it shews the Content in Barrels.

SECT. II. Of Gauging or Measuring, and the Moulding of Ships.

PROP. I. To Gauge a Ship, thereby to find how many Tuns her burthen is.

IN the Gauging or Measuring of Ships, Naupegers, or Ship-Wrights, observe these three Particular Rules: First, that if you measure the Ship within, you shall find the Content, or the Burthen the Ship will hold or take in. Secondly, if the Ship be measured on the outside, to her light mark as she swims being unladen, you shall have the Content of the Empty Ship. Thirdly, but if you measure from the light mark, to her full draught of Water being laden, you shall have the true Burthen of the Ship.

[Page]

Now to find the Content of the King's Royal Ships: Measure the length of the Keel, the breadth of the Mid ship Beam, and the depth of the Hold: which three multiply into one another, and divide their Product by 100; so shall you find how many Tuns her Burthen is.

But for Merchant's Ships, which give no allowance for Ordnance, Masts, Sails, Cables, Anchors, &c. which are all a Burthen, but no Tonnage, you must divide the product by 95, so shall their true Burthen be found.

PROP. II.

By knowing the Measure of a Ship, of one Burthen, to make another Ship, of the same Mould, which shall be double, or triple, or in any proportion, either more or less than the said Ship.

First you shall multiply the Keel Cubically; and in like manner every Beam; the Mid ship Beams multiply them Cubically; and also the Reaking of the Ship, both at Stem, and StemPost, multiply them Cubically; likewise the principal Timbers, that doth mould the Ship, multiply them Cubically; and the depth of the Hold, multiply it Cubically; and so consequently every Place, or Places, which doth lead any work, multiply them Cubically; then if it be required to have a Ship as big again, or thrice as big; double, or triple each respective Cubical number; then by prop. 9. §. 1. chap. 1: Or by prop. 4. §. 2. chap. 2. find the Cuberoots thereunto belonging; then according unto these respective Numbers, make your Keel, your Timbers, Beams, &c. which being done, you shall make a Ship of the Mould and Proportion desired.

CHAP. XII. Of DIALLING.

[Page]

HOROLOGIOGRAPHIA, or the Art of DIALLING, is an Art Mathematical, which demonstrateth the precise Distinction of Times, by the Sun, Moon and Stars, whereby the Time of the Day, or Night, may be knownWhich doth appear to have been in use above this 2400 Years, for King Achaz had a Dial: This Art requireth good skill in Geometry, and Astronomy: Now Cresibius that famous Philosopher measured the Hours and Times by the orderly running of Water. Then by Sand was the Hours measured. After that by Trochilike with Weights, and of late with Trochilike with Springs.. Now the Demonstrative delineation of Dials, consisteth chiefly in the finding out the Hour-lines, and their true distance one from the other: which lines are great Circles of a Sphere, which being projected on a plain Superficies, become streight-lines; which lines do continually vary, according as the Planes on which they are described, or projected, do lie situated in respect of the Horizon of the Place. [Page] Now a Dial may be made on any Plain Superficies, for all Plain Superficies are Posited either Perpendicular, Parallel, or Oblique, to the Horizon of the Place, in which the Plane is seated.

In the delineation of all which Dials in this Chapter described, (which are the most Eminent, and usefull Dials now used) I have used this Method: First, I have shewed how to delineate them by Geometrical Projection, by Scale, and Compass only: and secondly how they may be described by Arithmetical Calculation, of both which I have been very plain and large.

SECT. I. Of the Delineation and Projection of sundry most usefull Dials.

PROP. I. How to draw the Hour-lines on an Equinoctial Plain.

AN Equinoctial Plane, is such which lieth Parallel unto the Equinoctial, and is an Horizontal Plane, under the Pole. This is the first and plainest kind of Dials, and is made after this manner: First describe the Circle AE, W, E, R, for your Planes, then Cross it with the two Diameters EW, and AER. ThenFig. 64. divide the Semicircle E, W, R, into 12 equal parts in the points ☉, ☉, ☉, &c. Then from the Center Q, and through the said points [Page] draw streight lines, which shall be the true Hour-lines belonging unto this Equinoctial Plane. Now because these Planes are capable of receiving all the Hour-lines from Sun-rising unto the Sun-setting, in Summer; therefore the Hourlines of 4, and 5, in the Morning; and 7, and 8, in the Evening; must be delineated as you see done in the Figure: These Hours may be sub-divided into half Hours, and Quarters: TheFig. 64. Stile of this Dial, must be a streight Pin, or Wyre set Perpendicular, to the Plain, on the Center Q. and of any convenient length. This Dial may be made for any Latitude, and is of good use for Seamen, and others.

PROP. II. How to draw the Hour-lines on a Polar Plane.

A Polar Plane is one that lies Parallel unto the Pole, and under the Equinoctial is an Horizontal Dial: the way to make this Dial is thus. First draw the line AB, for the Horizontal line of the Plane; and cross it at the Middle at right angles, with the line 12, Q, 12, which is theFig. 65. Meridian or Hour line of 12; Then upon the line 12, Q 12, either above or below the point Q, assume any point as S, then setting one foot of your Compasses in S, describe the Semicircle CED, which divide into 12 Equal parts, in the points ☉, ☉, ☉, &c. Then lay a Ruler unto S, and unto the several points ☉, ☉, ☉, &c. and it will cross the line AB, in the points x, x, x, &c. Then through those points draw (by prop. 4. §. 1. chap. 4.) right lines all Parallel [Page] unto 12 Q 12, and so is your Dial finished. Then according unto the breadth of the Plane, you may proportionFig. 65. your Stile,Which may be either a Pin of the length of Q S, placed on Q, and Perpendicular unto the Plane, or it may be a piece of brass or elsewhat of the breadth of 12, to 3, or 9. Whose height must be equal to the distance between the two Hour-lines 12, and 9, or 12, and 3, and then will the shadow of the upper edge thereof shew the Hour of the day: The height of the Stile, is also found thus.

As the Tangent of the Hour-line 4 or 5,

To the Distance hereof from the Meridian.

So is the Radius,

To the Height of the Stile.

Then for the other Hour-line, say,

As the Radius,

To the Height of the Stile.

So is the Tangent of any other Hour-line,

To the Distance thereof from the Meridian line.

PROP. III. How to draw the Hour-lines on a Meridian Plane, which is an East, or West Dial.

A Meridian Plane stands upright directly in the Meridian, and hath two Faces, one towards the East, and the other towards the West.

Now admit it be required to make a direct East Dial, in the Latitude of 51° 32': let A, B,Fig. 66. C, D, be a Dial-plane, on which you would describe a Direct East Dial, on the point D, describe [Page] an obscure Arch HG, with the Radius of [...]our line of Chords, then take 38° 28', the Complement of your Latitude, place it from G to L; then draw DL quite through the Plane; Then to proportion your Stile unto your Plane, so that all the Hours may be placed thereon, from Sun-rising to 11 a Clock. Assume twoFig. 66. points in the line LD, as K, for 11; and I for the 6 a Clock Hour lines; then draw 6, 16, and 11, K 11, Perpendiculur to LD. This done, with the Radius of your line of Chords on L, strike the Arch OP, and from P, to O, place 15° 00'; and draw OK, to cut 6 I 6, in M, so shall IM be the height of the Stile proportioned unto this Plane; which may be a Plate of Brass, whose breadth must be equal to the distance between the Hour-lines of 6, and 9, which must be placed Perpendicular to the Plane, on the line 6, I 6, whose shadow of the upper edge, shall shew the Hour of the day. Now to draw the Hour-lines, with the Radius of your line of Chords, on M strike the Arch QN, which divide into 5 equal parts in the points •, •, •, &c. Then lay a Ruler from M unto each of those points, and it will cut the line JK in the points *, *, *, &c. through which points (by prop. 4. § 1. chap. 4.) draw Parallels to 6 I 6, as the lines 77, 88, &c. which shall be the true Hour-lines of an East Plane, from 6Fig. 66. in the Morning, till 11 before Noon. Then for the Hour-lines of 4, and 5, you must prick off 5 as far from 6, as 6 is from 7; and 4, as far as 6 is from 8; and draw the Hour-lines 55, and 44, as before. Thus is your Dial compleated, and in the forming of which, you have [...] [Page] [...] [Page] [Page] made both an East, and a West Dial; which is the same in all respects, only whereas the Arch H G, through which the Equinoctial passed in the East Dial, was described on the right hand of the Plane, in the West it must be drawn on the left hand, and the Hour-lines 4, 5, 6, 7, 8, 9, 10, and 11, in the Forenoon in the East Dial, must be 8, 7, 6, 5, 4, 3, 2, and 1, in the West inFig. 67. the Afternoon; as in the Figure plainly appeareth: Now you may find the distance of the Hour-lines from the Substile, by this Analogy or Proportion.

As the Radius,

To the Height of the Stile.

So is the Tangent of any Hours distance from 6,

To the distance thereof from the Substile.

PROP. IV. How to draw the Hour-lines on a direct South, and North Plane,

This Plane or Dial must stand upright, having his face or Plane, if it be a South Dial, directly opposite unto the South; but if a North Plane, directly opposite unto the North; now admit it be required to make a Direct South Dial,Fig. 68. for the Latitude of 51° 32': To make which first describe the Circle ABCD, to represent an E [...]ect direct South Plane, cross it with the Diameters CB, and AD, then out of your Line of Chords take 38° 28', the Complement of the Latitude, and set it from A, unto a, and from B, unto b, Then lay a Ruler from C unto a, and it will cut the Meridian ARD, in P, the Poles of

[Page] [Page] [Page]

Plate V Page 261

Fig. Fig. 68. Fig. 69. [Page] other hath the North Pole of the World elevated above it, and beholdeth the North part of the Meridian. The Hour-lines of 9, 10, 11, or 1, 2, and 3, is not expressed on this Plane, because 12, representeth 12, at Midnight; neither are the other said Hours expressed, because the Sun is never above the Horizon, at those Hours; Therefore the North Dial is capable only toFig. 69. receive these Hours, namely 4, 5, 6, 7, and 8, in the Morning; and 4, 5, 6, 7, and 8, at Night; as doth plainly appear in the Figure: Now the distance of the Hour-lines from the Meridian, may be found by this Analogy, or Proportion.

As Radius or S. 90°,

To Sc. of the Latitude.

So is T. of the Hour from Noon,

To T. of the Hour-line from the Meridian.

PROP. V. How to draw the Hour-lines on an Horizontal Plane.

This Horizontal Plane, or Dial, is one of the best and most usefull Dials in our Oblique Hemisphere: Admit it be required to make an Horizontal Dial, for the Latitude of 51° 32': To make which, first describe the Circle ABFig. 70. CD, which representeth your Horizontal Plane, Then cross it with the two Diameters ARC, and BRD, Then take 51° 32' out of your Line of Chords, and set it from B, to a, and from C, to b, Then lay a Ruler from A, unto a, and it will cut the Meridian BD, in P, the Pole of the [Page] World, Then lay a Ruler from A, unto b, and it will cut ABD the Meridian, in the point AE, where the Equinoctial cutteth the Meridian, then through the three points A, AE, and C, drawFig. 70. the Equinoctial Circle, whose Center is at H; (and found as in the former proposition) Then divide the Semicircle ADC into 12 equal parts, in the points •, •, •, &c. Then lay a Ruler to R the Center of the Plane, and on those points, so shall the Equinoctial Circle AAeC, be by it divided into 12 unequal parts in the points *, *, *, *, &c. Then a Ruler laid unto P the Pole of the World, and those Points, shall cut the Semicircle CDA in those Points I, I, I, &c. Lastly, from the Center R, and through those Points, let there be drawn right lines, which shall be the true Hour-lines of such an Horizontal Plane, from 6 in the Morning, untill 6 at Night; but for the Hours of 4 and 5 inFig. 70. the Morning; and 7 and 8 in the Evening; they are delineated by producing 4 and 5 in the Evening, through the Center R, and 7 and 8 in the Morning; extending them out, unto the other side of the Plane, so shall you have those Hour-lines also on your Plane delineated as you see in the Figure. The Stile of this Plane may be a thin Plate of Brass, cut exactly unto the Quantity of an Angle of 51° 32', and set Perpendicular on the Meridian line, for the forming of this Stile take out of your Line of Chords 51° 32', and set it from D, unto e, and draw Re, which shall be the Axis of the Stile, you may also prefix the Halves, and Quarters of Hours, in the very same manner as the Hours themselves were drawn.

[Page]

Now to find out the distance of the Hour-lines from the Meridian, say,

As the Radius or S. 90°,

To the S. of the Latitude.

So is the T. of the Hour from Noon,

To the T. of the Hour-line, from the Meridian Line.

These kinds of Dials being so frequently usedFig. 70. with us, in this Oblique Sphere, for the help of the speedy delineating of them, I have annexed hereunto the Table of Longomontanus, wherein the Hour-lines, for many Latitudes, are calculated.

[Page]

A Table shewing the Distance of the Hourlines from the Meridian, in these Degrees of Latitude.

An Horizontal Dial, Latitude. | The Hours from the Meridian. | A South Erect Dial, Latitude.
xi. | i. | x. | ii. | ix. | iii | viii. | iv. | vii | v. | vi.
D | M | D | M | D | M | D | M | D | M | D | M
30 | 7 | 38 | 16 | 6 | 26 | 34 | 40 | 54 | 61 | 49 | 90 | 00 | 60
31 | 7 | 51 | 16 | 34 | 27 | 14 | 41 | 42 | 62 | 28 | 90 | 00 | 59
32 | 8 | 4 | 17 | 1 | 27 | 53 | 42 | 30 | 63 | 6 | 90 | 00 | 58
33 | 8 | 17 | 17 | 27 | 28 | 34 | 43 | 17 | 63 | 45 | 90 | 00 | 57
34 | 8 | 30 | 17 | 54 | 29 | 13 | 44 | 5 | 64 | 42 | 90 | 00 | 56
35 | 8 | 43 | 18 | 20 | 29 | 49 | 44 | 46 | 64 | 56 | 90 | 00 | 55
36 | 8 | 56 | 18 | 45 | 30 | 25 | 45 | 28 | 65 | 27 | 90 | 00 | 54
37 | 9 | 9 | 19 | 9 | 31 | 1 | 46 | 9 | 65 | 58 | 90 | 00 | 53
38 | 9 | 21 | 19 | 34 | 31 | 37 | 46 | 50 | 66 | 29 | 90 | 00 | 52
39 | 9 | 33 | 19 | 57 | 32 | 9 | 47 | 26 | 66 | 55 | 90 | 00 | 51
40 | 9 | 46 | 20 | 20 | 32 | 40 | 48 | 1 | 67 | 20 | 90 | 00 | 50
41 | 9 | 58 | 20 | 43 | 33 | 14 | 48 | 37 | 67 | 45 | 90 | 00 | 49
42 | 10 | 10 | 21 | 7 | 33 | 47 | 49 | 13 | 68 | 11 | 90 | 00 | 48
43 | 10 | 22 | 21 | 29 | 34 | 17 | 49 | 44 | 68 | 32 | 90 | 00 | 47
44 | 10 | 24 | 21 | 50 | 34 | 46 | 50 | 14 | 68 | 52 | 90 | 00 | 46
45 | 10 | 43 | 22 | 12 | 35 | 15 | 50 | 45 | 69 | 14 | 90 | 00 | 45
46 | 10 | 54 | 22 | 33 | 35 | 44 | 51 | 16 | 69 | 37 | 90 | 00 | 44
47 | 11 | 5 | 22 | 33 | 36 | 10 | 51 | 43 | 69 | 53 | 90 | 00 | 43
48 | 11 | 16 | 23 | 12 | 36 | 35 | 52 | 9 | 70 | 10 | 90 | 00 | 42
49 | 11 | 26 | 23 | 32 | 37 | 1 | 52 | 35 | 70 | 28 | 90 | 00 | 41
50 | 11 | 36 | 23 | 51 | 37 | 27 | 53 | 1 | 70 | 43 | 90 | 00 | 40
51 | 11 | 46 | 24 | 9 | 37 | 50 | 53 | 24 | 70 | 58 | 90 | 00 | 39
52 | 11 | 56 | 24 | 26 | 38 | 13 | 53 | 46 | 71 | 12 | 90 | 00 | 38
53 | 12 | 5 | 24 | 44 | 38 | 36 | 54 | 8 | 71 | 27 | 90 | 00 | 37
54 | 12 | 14 | 25 | 2 | 38 | 59 | 54 | 30 | 71 | 41 | 90 | 00 | 36
55 | 12 | 23 | 25 | 18 | 39 | 18 | 54 | 50 | 71 | 53 | 90 | 00 | 35
56 | 12 | 32 | 25 | 33 | 39 | 38 | 55 | 9 | 72 | 4 | 90 | 00 | 34
57 | 12 | 46 | 25 | 49 | 39 | 58 | 55 | 28 | 72 | 16 | 90 | 00 | 33
58 | 12 | 48 | 26 | 5 | 40 | 18 | 55 | 46 | 72 | 27 | 90 | 00 | 32
59 | 13 | 56 | 26 | 19 | 40 | 36 | 56 | 1 | 72 | 38 | 90 | 00 | 31
60 | 13 | 58 | 26 | 30 | 40 | 53 | 56 | 15 | 72 | 47 | 90 | 00 | 30

PROP. VI. How to draw the Hour-lines, on an Erect declining Plane.

[Page]

These Planes are made to set on the sides of Houses, wherein the Meridian is always a Perpendicular,Fig. 71. drawn on the Plane, in whose top is the Center, where the Substile, and the Hourlines all meet.

Now before we can delineate the Hour-lines on any such Planes, two things must be given: As the Latitude of the Place, and the Planes Declination; by having which we must find these three things: viz. The Poles height above the Plane. The distance of the substile from the Meridian. And the Plane's difference of Longitude.

For the finding of which Requisites, by Geometrical Projection, we describe on the Dial Plane, these Circles of the Sphere, viz. The Horizon, Meridian, and Equinoctial, which being described in their true Position, on the Plane, we proceed thus.Fig. 71.

Admit it be required to make a Direct South Dial, on an Erect, Direct South Plane, Declining Westward 24° 20', in the Latitude of 51° 32'.

Now in order to find the requisites before mentioned, describe the Circle ZHNO, and cross it with the two Diameters ZQN, and H QO: now Z is the Zenith, N the Nadir, ZQN the Hour-line of 12, HQO the Horizon. Now seeing the Plane declines S. W. 34° 20': make Na, and Ob, each equal to 34 20: Then a Ruler layed from Z, to a, will cut the Horizon in S, the [Page] South point of the Horizon, through which draw the Meridian ZSN, whose Center is at Y, found as in the fourth Proposition aforegoing: Then a Ruler laid from Z to b, will cut the Horizon in W, the West point thereof. Now the Horizon and the Meridian being projected on the Plane, take out of your line of Chords 51° 32', which place from H, unto c, and from N, unto d; then lay a Ruler from W, unto c, and it cutteth the Meridian in P, the Pole of the World. Then through P and Q,Fig. 71. draw the line PQD, which representeth the Axis of the World, and the Substilar line of the Dial, then lay a Ruler from W, to d, it cutteth the Meridian in AE, so is W AE two points through which the Equinoctial must pass, whose Center is found as afore to be at M, (being always in the Axis of the World) so have you on your Plane the Horizon HQO, the Meridian ZPSAe N, and the Equinoctial LAeKWG, described on the Plane as required.

Now first to find the Poles height above the Plane, which in this Scheme is represented by BP, Lay a Ruler from G, unto P, and it shall cut the Plane in V, then measure the distance BV, on your line of Chords, and youFig. 71. will find it to contain 34° 33', which is the Poles height above the Plane.

Secondly, To find the distance of the Substile from the Meridian represented in the Scheme by the Arch ZB, or ND, which measured as afore will appear to be 18° 08', the distance of the Substile from the Meridian.

Thirdly, To find the Plane's Difference of Longitude, which in the Scheme is represented by [Page] the Angle AEPK, lay a Ruler from P, unto AE, and it cutteth the Plane in X, then measure the Arch DX, as afore, and so will you find the Planes Difference of Longitude, to be 30° 00': Thus by Geometrical Projection have we foundFig. 71. all the three Requisites: Now to find them by Arithmetical Calculation observe these Analogies or Proportions.

1. For the Poles height above the Plane, say,

As Radius or S. 90°,

To Sc. of the Latitude 38° 28'.

So is Sc. of the Declination 65° 40',

To S. of the Poles height above the Plane 34° 33'.

2. For the Distance of the Substile, from the Meridian, say,

As the Radius or S. 90° 00',

To the S. of the Plane's Declination 24° 20'.

So is Tc. of the Latitude 38° 28',

To the T. of the Substilar Distance from the Meridian 18° 10'.

3. For the Plane's Difference of Longitude, say,

As the Sc. of the Latitude 38° 28',Fig. 71.

To the Radius or S. 90° 00'.

So is S. of the Substilar Distance 18° 10',

To the S. of the Difference of Longitude 30 Deg.

Or, it may be found thus.

As the S. of the Latitude,

To the Radius.

So is the T. of the Declination,

To the T. of the Difference of Longitude required.

[Page]

These things found, we come now to shew how the Hour-lines may be projected. To project which observe, First, to lay a Ruler from P the Pole of the World, to AE the Intersection ofFig. 71. the Equinoctial with the Meridian, and it will cut the Plane in x, where begin to divide the Semicircle L x G, into 12 Equal parts in the Points •, •, •, •, &c. Then lay a Ruler from Q, to every of those parts, and it shall cut the Equinoctial; and divide it into 12 unequal parts, in the points *, *, *, *, &c. Then a Ruler laid from P the Pole of the World unto each of these points, it will divide the Plane into 12 unequal parts in the Points I, I, I, I, &c. Then by a Ruler laid from the Center Q, to those points, draw right lines, which shall be the true Hour-lines proper unto such a Declining Plane, as you see plainly demonstrated by the Scheme.

Now the Substilar line falleth in this Dial, just on the Hour-line of 2, in the Afternoon, because the Plane declineth Westerly. The Angle of the Stile is DQR 34° 33'. which may be either a Plate or Wyre, brought into such an Angle, which must be placed Perpendicular to the Plane, and directly over the Substilar line QD 2.

Now the distance of the Hour-lines, from the Substilar line, may also be found by this Analogy or Proportion.

As the Radius,Fig. 71.

To the S. height of the Pole above the Plane.

So is the T. of the Hour-line from the Meridian of the Plane,

To the T. of the Hour-line from the Substile.

[Page]

Thus have you compleated your Dial; as you see in the Scheme, and here you may take notice that having finished a West Decliner, you have also made an East Decliner; if you only convert the Hour-lines of the West Decliner, in such manner as you see in Fig. 72. on the EastFig. 72. Decliner, and compleat all as you see in that Scheme.

Thus I have explained the making and delineating of the best and most usefull Dials both by Geometrical Projection, and also by Arithmetical Calculations, in as brief and compendious a manner as possible. There are sundry other kind of Dials, as Incliners, Decliners, and Recliners, which being not so usefull, for brevity sake, they are here omitted: As for Instrumental Dials, as Quadrants, Rings, Cylinders, &c. Which depend on the Sun's height, I refer you to Mr. Edm. Gunter's Book, wherein they are largely described.

As for the Beautifying and Adorning of those Dials, &c. by describing on them the Equinoctial, Tropicks, Parallels of Declination, Parallels of the Sun's Place, Length of Days, the Sun's Rising and Setting, Jewish, Italian, and Babylonish Hours, Almicanthars, Azimuths, Circles of Position, the Signs Right Ascending, Descending, Culminating, &c. I do advise you to consult Mr. Gunter, Mr. Foster, Mr. Wells, and Mr. Holwel's Works, all which Authors have very learnedly shewed the describing of them, by several large Schemes, and Figures, for the plainer Illustration thereof.

Now seeing the Latitude of a Place must be first known, before a Dial can be made to it,

[Page] [Page] [Page]

Plate VI P [...]

Fig. 72. [Page] [Page] I have therefore hereunto annexed a Table of the Latitudes of all the principal Cities, Towns, and Islands, in and about Great Britain and Ireland; so that if you are to make a Dial, for any of those parts, you may have recourse to this Table, and make your Dial to the Latitude of that place, which you find to be the nearest to the Place, for which you are to make your Dial.

A Table of the Names and Latitudes of all the Principal Cities, Towns, and Islands, in and about Great Britain and Ireland.

ENGLAND.

D. | M.
ARundel | 51 | 00
Bedford | 52 | 15
Barwick | 55 | 54
Bristol | 51 | 35
Buckingham | 52 | 10
Cambridge | 52 | 20
Canterbury | 51 | 25
Carlisle | 55 | 20
Chichester | 50 | 48
Chester | 53 | 18
Colchester | 52 | 08
Dover | 51 | 20
Derby | 53 | 00
Dorchester | 50 | 50
Durham | 54 | 56
Exeter | 50 | 48
D. | M.
Falmouth | 55 | 22
Glocester | 51 | 57
Guilford | 51 | 12
Hartford | 51 | 54
Hereford | 52 | 17
Huntington | 52 | 30
Ipswich | 52 | 20
London | 51 | 30
Lincoln | 53 | 20
Leicester | 52 | 45
Lancaster | 54 | 15
Northampton | 52 | 24
Norwich | 52 | 45
Nottingham | 53 | 00
Newcastle | 55 | 12
Oxford | 51 | 50
Portsmouth | 51 | 08
Plimouth | 50 | 36
Reding | 51 | 40
D. | M.
Salsbury | 51 | 12
Stafford | 52 | 50
Stanford | 54 | 44
Shrewsbury | 52 | 50
Truero | 50 | 30
Winchester | 51 | 03
Worcester | 52 | 25
Warwick | 52 | 30
York | 54 | 00

WALES.

D. | M.
ANglesey | 53 | 28
Barmonth | 52 | 50
Brecknock | 52 | 01
Cardigan | 52 | 12
Caermarthen | 51 | 56
Carnarvan | 53 | 16
Denbigh | 53 | 13
Flint | 53 | 17
Landaffe | 51 | 35
Monmouth | 51 | 51
Montgomery | 51 | 56
Pembroke | 51 | 46
Radnor | 52 | 19
St. David | 52 | 00

SCOTLAND.

D. | M.
ABerdeen | 57 | 30
Dunblain | 56 | 21
Dunkel | 56 | 48
Edenbrough | 56 | 00
Glascow | 55 | 58
Kinsaile | 57 | 44
Orkney | 60 | 06
D. | M.
St. Andrews | 56 | 40
Skyrassin | 58 | 38
Sterling | 56 | 12

IRELAND.

D. | M.
ANtrim | 54 | 38
Arglas | 54 | 10
Armagh | 54 | 14
Carterlagh | 52 | 41
Clare | 52 | 34
Corke | 51 | 55
Droghedagh | 53 | 58
Dublin | 53 | 55
Dundalke | 53 | 52
Galloway | 53 | 02
Kenney | 52 | 30
Kildare | 53 | 00
Kings Town | 53 | 08
Knockfergns | 54 | 40
Kynsale | 51 | 41
Lymerick | 52 | 30
Queens Town | 52 | 52
Waterford | 52 | 09
Wexford | 52 | 18
Youhall | 51 | 53

ISLANDS.

D. | M.
WIght | 50 | 48
Portland | 50 | 30
Man | 54 | 24
Limdey | 51 | 22
Jerzey | 49 | 12
Garnzey | 49 | [...]

CHAP. XIII. Of FORTIFICATION.

[Page]

THE Utility of this Mathematical Art called Fortification, or Military Architecture, is so well known, that it needs not my commendation, and therefore to speak any thing thereto, were but to light a Candle before the Sun.

In the handling of this part of the Mathematicks, I shall be as brief as possible, yet as plain as can be desired: In the prosecution of which, I shall use this Method. As First, I shall give you the most principal Definitions or Terms belonging to this Art. Secondly, I shall prescribe the most conducing Maxims or Rules herein observed. Thirdly, I shall shew how to delineate the Ground-line of any Fortification, according to the several Proportions, used by the best and most experienc'd Inginiers of Italy, France, Holland and England; Fourthly, I shall describe the Construction of the chief and principal Outworks now in use; and Lastly, lay down some general Maxims or Rules, by most Modern Authors observed in Irregular Fortifications.

SECT. I. Of the Definitions of the Lines, and Angles, belonging to the Principal Ground work of any Regular Fortification.

[Page]

1. THE Exterior or outward Line, which

Defin.

boundeth the Rampart, at the Foot next the Ditch, is the principal and only Line to be regarded [...] all Regular, or Irregular Fortifications, being the Basis on which all the other Lines, and parts of the Fortification doth depend.

2. The Exterior Polygon, is the outward side of any Regular Figure, as in the Hexagon (which Figure I shall make use of through this Tract) the side AA, is the Exterior Polygon.

Fig. 73.

3. The Interior Polygon, is the inward side of any Regular Figure, as in the Hexagon is noted by any of the sides between P and P.

4. The Bastion or Bulwork, is that great work of any Fort, that advanceth its self towards the Campaigne, and here are six all marked with B, the lines which terminate them, are two Gorges, two Flanks, and two Faces.

Fig. 73.

5. The Demi-Gorge or Gorge-line PC, is half the Entrance into the Bastion, and terminates the point C, whereby the Flank shall be raised.

6 The Flank is another Out-line of the Bastion as CF, which terminateth the Curtain, and Face.

7. The Face is the utmost line of the Bastion, as FA, two lines thereof doth form the Angle of the Bastion A, or the Flanked Angle.

8. The line forming the Flank FF, is a pricked line, made use of by the Dutch Inginiers, and others.

9. The Capital is AP, part of the line coming

Fig. 73.

from the Center ☉, terminated at the point of the Bastion A.

10. The Curtain is that part of the Interior Polygon CC, which lieth betwixt the two Bastions B, and B.

11. The line of Defence is AC, passing from A, the point of the Bastion, to C the Angle of

Fig. 73.

the Flank, and Curtain, and ought never to exceed 800 English Feet

Because the length of the part of a Musket doth not much exceed that Mèasure.

.

12. The line Stringent, is the line coming from the point of the Bastion A, and prolonged on the Face AF, to the Curtain D, which sheweth that DC, the part of the Curtain, (by some called the second Flank will scour the Face.

13. The Diameter of the Interior Polygon, is the line ☉P, coming from the Center thereof ☉.

14. The shortest line from the Center unto the Curtain, is ☉ m. These are the Definitions of the principal lines, appertaining to the Groundwork of any Regular Fortification, the Angles followeth.

15. The Angle of the Center of the Polygon

Fig. 73.

is P☉P

16. The Angle of the Polygon PPP, is always the Complement of the Angle at the Center, or remainer unto 180 Degrees.

17. The Angle of the Triangle PPO is always the one half of the Angle of the Polygon PPP.

18. The Angle of the Bastion, or the Flanked Angle FAF, is exposed unto the Batteries of the

Fig. 73.

Besiegers, and formed by the two Faces, FA, and FA, which ought never to be less than 60, nor much above 100 Degrees.

19. The Angle of the Espaule, or Shoulder, is formed by the Face, and Flank, as AFC.

20. The Angle of the Flank CCF, is formed by the Curtain, and the Flank, and is most commonly a Right Angle, but by some later Inginiers, is made Obtuse, or more than a Right Angle, or 90 Degrees.

21. The Angle made by the two lines Sitchant, At A is called the Angle of the Tenaile.

22. The Angle forming the Flank, is CPF, which Angle is made use of by most of the Dutch Inginiers.

Fig. 73.

SECT. II. Of General Maxims or Rules observed in Fortifications.

1. THat all the parts of the Place, be of Cannon Proof flanked, i. e. defended from another place, which place is no farther [Page] distant than the reach of a Musket-shoot, from the place to be Flanked or defended.

Because the Defence ought to be easie, quick, certain, and of little charge, all which qualities the Musket hath and the Cannon hath not, therefore the Defence of Fortification ought to be measured by the Port of a Musket, and not by that of a Cannon.

2. That in all the Place, there may be no part of the Wall, or outside of the Rampire, that is not seen from the top to the bottome of the Mote, or Ditch.

3. That the Bastions are large, and full of Earth, and not empty; the bigger they are, they are the more to be esteemed, there being the more room to intrench, in case of necessity: whose Gorge let be at least 35 fadoms, and their Flank at least 18 fadoms.

4. That the Angle of the Bastion, or Flanked Angle, be not much above 90, nor much less than 60 Degrees, for in the former it would lie too very Obtuse, and open, at the Point; and in the latter it would be too slender, and so easily to be battered down, by the Enemies Cannon.

5. That the Angle of the Flank may be somewhat Obtuse; neither is there any more virtue in a Right-angle, than in any other, for the defence of the Fort.

6. That the length of an extended Curtain be not above 135 Fadoms, nor the single above 80 Fadoms, nor be less than 40 Fadoms, to be well defended from two Flanks.

7. That the Rampire be so wide, that so a Parapet of Earth Cannon proof may be erected thereon, and a Teraplane left, full wide for the Ordnance to be recoiled.

8. That the Mote or Ditch be at least 20 Fadoms broad, and as deep as possible. Now dry Motes in great Cities are to be preferred before others, that are full of Water, to facilitate the Sallies, the relief, and retreat of the Besieged; and in small Fortifications the Motes full of Water are the most Esteemable, because in such Sallies are not necessary, and Surprises are very much to be feared.

9. that the Parts that are most remote from the Center, be commanded by those which are nearest to it.

10. That the Defence of a Face is much stronger, when the Angle made by the Face, and Exterior Polygon is a great Angle; this Maxim is so very essential, that it will try the goodness of any Fortification whatsoever: Thus I have described the 10 chiefest Maxims, necessary for good Fortifications.

SECT. III. Of the Construction and making of the principal Ground-line of a Fort, according to the most Modern ways, used by the Italian, Dutch, French, or English Inginiers.

[Page]

I. Of the Italian Fortifications.

GEnnaro Maria, Mathematician to the Catholick King, wrote at Florence, his Elements of Military Architecture entituled, Breve Trattato delle Moderne Fortificazioni. This Italian Author was a very Learned and Skilfull Mathematician, and famous in his Nation. In his said Book Printed 1665, he makes the Interior Polygon 800, and not less than 600 Feet, his Demi-Gorge, he makes ⅛ of it, and so for the Flank of the Quadrangle. But for the Pentagon, and all Figures above, he makes the Flanks 1/10 part of the Gorge more, and he placeth his Flank at Right Angles with the Curtain.

Supposing his Interior Polygon 1000 parts, his Gorges will be 125, and in the Quadrangle the Flanks will be 125, but of the Pentagon, and all above, 138 parts. For the Faces, he makes them to fall on the third part of the Curtain, unless in the Square, which he allows no second Flank.

PROP. I. To fortifie a Hexagon according to this Author's Proportion.

[Page]

First describe the Hexagon PPP, &c, thenFig. 74. divide the Interior Polygon PP, into 1000 equal parts, take 125 for the Gorges, and set it from P to C. Then on C raise a Perpendicular, make it equal to 138 parts, for your Flanks CF, then draw the Face AF, falling on the third part of the Curtain CC, at D, and so do on every Bastion, untill the work is compleated.

II. Of the French Fortifications.

Monsieur De la Mont, in his Fortifications Offensive, and Defensive, printed 1671: And Monsieur Manesson Mallet in his late work, intituled Travaux de Mars, printed 1672, assigneth these proportions for the laying down the Ground-line of a Fort.

Both these Authors make the Interior Polygon 768 English Feet, which they divide into 5 parts, and taking one for the Gorge 153½ Feet. Both divides it into 3 parts, and takes one for the Capital, that is 256 Feet.

Now our first Author De la Mont, makes theFig. 74. Flank to stand at Right-angles and takes 115½ Feet for it, which is ¼ of the Curtain, and so draws the Bastions, in all save the Quadrangle, and Pentagon, which he makes to have no second Flank.

PROP. II. To fortifie a Hexagon according to the Proportion of De la Mont.

[Page]

First describe your Hexagon P, P, P, &c. Now supposing your Interior Polygon PP, 1000 parts, the Capital 333, the Gorge 200, and the Flank 150 parts, take out of your Triangular Scale Fig.Fig. 74. 75, (which is made for the more speedy delineation according to this proportion of De la Mont) PA for the Capital, and prick it off from PA, on all the Bastions. Then take PC, and prick off all the Gorges from P to C. Then take FC and prick it off at Right Angles, from C to F. Lastly draw all the Faces AF, AF, &c. so is your Hexagon compleat, as required.

PROP. III. To fortifie a Hexagon according to Manesson Mallet's Proportion.

Now our Authour Monsieur Manesson Mallet, in his Works intituled Travaux de Mars, deviates from our former Authour, only in this: that as De la Mont did place his Flanks at Right Angles, he places them at 98 Degrees with the Curtains, and leaves no second Flank in all his Fortifications.

Therefore having described the Polygon PP,Fig. 74. &c. divide PP into 1000 parts, prick off the Capitals PA 333, and the Gorges PC 200, then lay off the Flanks CF, 150 parts, at an Angle [Page] of 98 deg. with the Curtain CC (by prop 5. §. 1. chap. 4.) and draw all the Faces, AF, AF, &c. Falling on C the point of the Flank and Curtain, so shall your Hexagon be fortified as was required.

III. Of the Dutch Fortifications.

The Emperour Ferdinand III. hath learnedlyFig. 74. altered the Method of Fritach, Dogen, Goldman, and Faulhaberus, all which were Dutch Inginiers, and wrote large Volumes on this Subject; in his Works intituled Amussis Ferdinandea, published 1654; by turning their way of working by Angles, into working by Sides.

Thus he setteth down a Catholick way of delineating the Sides, or Lines of any Fort by his 60 prop. thus, the Interior Polygon to be 66, the Capital 24, the Gorge 15, and the Flank 12. Or in making the Interior Polygon 22, the Capital 8, the Gorge 5, and the Flank 4. Or yet making the Interior Polygon 1000, the Capital 363, the Gorge 227, and the Flank 181, this is an Epitome of all the Dutch Fortifications, and is general excepting for the Square, which must have no second Flank.Fig. 74.

PROP. IV. To fortifie a Hexagon according to the Emperour's Proportion.

First describe the Polygon PPP, &c. divide P P, &c. into 22 parts, take 8 for the Capitals [Page] PA, which prick off all round from P to A, take 5 for the Gorges; which prick off all round from C to P, then take 4 for the Flank CF,Fig. 74. which prick off all round at Right-angles from C to F, lastly draw the Faces AF, AF, AF, &c. So is the Hexagon compleated as was required.

IV. Of the English Fortifications.

His late Majesty of Great Britain Carolus II. of ever blessed Memory, hath much facilitated the Method of Count Pagan, who in his Fortifications printed at Paris 1645, did place the Flanks at Right-angles with the Line of Defence, and he works by the Exterior Polygon. Now His Majesty places the Flank, at Right-angles with the line of defence of the Interior Polygon, and works after another manner: Count Pagan makes the proportion of the Grand Royal Fort. Supposing the Exterior Polygon to be 1000 parts, will make the Perpendicular MT to be 150, andFig. 74. the Complement of the line of Defence TC to be 185, which may serve for a general proportion be the length what it will, only in a Square the proportions must thus be altered in the Grand Royal Fort, the Perpendicular MT must be 162, in the Mean R 144, and in the Petty Royal 126, the Complement of the Line of Defence for the Grand Royal Fort is 228, and for the Mean Royal Fort 198, also for the Petty Royal Fort 198.

PROP. V. How to fortifie a Hexagon according to Count Pagan's Proportion.

[Page]

To delineate this Work draw a line, about the middle whereof as at M, set off MA, the half of the Exterior Polygon 500 parts, which makes the Exterior Polygon 1000, then on M (by prop. 1. §. 1. chap. 4.) raise the PerpendicularFig. 74. Mm, which make Mt, MT Equal to 150, then draw ATC, and ATC, then take 185, and place it from T to C, and to C, and draw CC for the Curtain, then on the points C raise Perpendiculars CF, to the line of defence CA, for the Flanks, so have you also the Faces FA. Then on the Points A set off half the Angle of the Figure, to wit 60° (as you see in the Table in page 38) and draw the lines OA and O A, so shall O be the Center of the Figure, and PC the Gorge, and AP the Capitals: then finishFig. 74. each Bastion at your own discretion, and the Work is finished as required.

PROP. VI. To fortifie a Hexagon according to the way prescribed by His Majesty Carolus II.

His late Majesty C. II. hath much facilitated this Work, as will appear in this following Example, by making the line of Defence, stand at Right-angles with the Flank of the Interior Polygon, by this Table, which supposes the InteriorFig. 74. Polygon to be 1000. Then

[Page]

Polygons | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Straitlines.
Capital | 398 | 437 | 367 | 333 | 312 | 300 | 291 | 233
Gorge-line | 155 | 196 | 203 | 242 | 252 | 260 | 263 | 300

Now describe the Hexagon PP, &c. Then divide the Interior Poligon PP, into 1000 parts, take 367 and prick off all the Capitals PA; Then take 203 and prick off all the Gorges from P to C. Now draw the lines of defence AC andFig. 74. AC, &c. Then at C, set the Flanks at Right Angles with the line of Defence AC, so shall FC be the Flank, and FA the Faces, then finish every Bastion, and your Hexagon is fortified as was required.

☞ Thus have I set down the several Ways and Rules, for laying the fundamental Groundline, from the most considerable Inginiers ofObserve this for a general Rule in Regular Fortification. this last Age, out of all which it's most agreeable to those Authors, and to practice, to take ⅓ of the Interior Polygon for the Capital, ⅕ for the Gorge, and Flank, which leaves 6/10 for the Curtain, and let this be taken for a general Rule, where the Flank, and Curtain, stand at Right Angles.

PROP. VII. By the Semicircle to lay down on the Ground, any of the former Fortifications.

[Page]

Having drawn the Plot of your Fort on ImperialFig. 76. paper, or Vellom, and if it be a Regular Fort you need not describe it but two half Bastions from the Center, for that will be sufficient. Having such a Plate whose length is set down on each respective line, and all proper Angles expressed, will not only be usefull for laying down the Work, but for finding the Solidity of the Ramparts, Parapets, and the other Earth Works See Fig 76.

If it be in such a Place, that from the CenterCase 1. of the Fort, all the Angles may be seen, place your Semicircle at Z, and lay off all the Angles of the Center, which here is 60°; then mark out the Diametrical lines, and making them their due length, as by your Plate they appear to be, set Piquets, on all the P, P's upright with the Plane, Then take up your Instrument and place a Piquet at Z. Then lock-spit out all the Polygons PP. Then mark out the Gorges CP, then set out the Flanks CF, either at RightFig. 76. Angles, or as otherwise required. Then lockspit out the Flanks CF, and the Faces AF, having first set off the Capital PA, so is the Fort lined out for the Ground-line.

But if there be Houses and Obstacles in theCase 2. way, that from the Center all may not be seen, then must you mark out any one side and measure it, and at each End set off the Angles of [Page] the Polygon, (which here is 120°) and draw side after side, untill all be finished: Then finish the Bastions as before, and here great care must be had, or else you will run into infinite Errours.

☞ But you have liberty Experimentally to alter any of the former proportions, as you have occasion, and as will best serve the Place; as you see by the fortifying a streight lined Figure: Fig. 77. wherein CountFig. 77. Pagan's or in Manesson's way it may not be allowed without some alterations.

SECT. IV. Of the Dimensions, and Measures of the Rampires, Parapets, Mote, Coridor, or Covert-way, and its Esplanade, or Breastwork.

THE Rampire's thickness and height, must receive its Determination from the Judgment of the Inginiers, and Purse of the Prince.

The Height TS, must not exceed 18 Feet,Fig. 78. not be less than Ten; the thickness may be from 50, to 80 RA, in all Royal Works, and according as Earth is to be had. The slope of the inward side of the Rampire TR, is commonly a foot for a foot, therefore RS, the Talu, will be equal to the Height TS, so if TS be 18, RS will be 18, if 15 then 15 feet.

[Page]

The outward Slope QA, is generally proportioned ½ a Foot for a Foot, so if the Height OQ be 18, the Talu OA, will be 9 Feet, &c.

The Height of the Parapet ZD, must alwaysFig. 78. be 6 Feet, the Exterior Height PM must be 4 Feet, the thickness of the Parapet DQ, in light Earth must be 20 Feet, in stiff Earth 16, and in Solid Rough Clay 14 Feet; suppose it be 18 Feet, PM will be 4 Feet, MQ 2 Feet, LD 1 Foot, so will the lower thickness LQ be 21 Feet.

The Height of the Banquet VX is 1½ Foot, and thickness VL, 3 Feet.

The Lizier must be made so wide, as to support the Rampire from slipping into the Ditch, and is taken from 3, to 10 Feet; the Mote or Ditch may be from 70, to 130 Feet broad, that is, from E to G, and the depth IF may be 8, 10, or 12 Foot deep, the little Ditch at the bottom of the Mote represented by c q g, must be as large and deep as the Earth and Work will give leave.

The Coridor and the Esplanade or Breast-workFig. 78. on it, is left about 18 Feet wide, from G to C; on which is placed a Parapet, and Banquet, like that on the Rampire, which Parapet or Esplanade, must slope so into the Campaigne, that a streight line drawn from Z, the Top of the Rampire, may terminate OFd, the Slope thereof.

PROP. VIII. How to lay down the Profile of the Work, according to this Table.

[Page]

Feet.

The Base of the Rampart RA
70

Height TS and QO
16

Interior Talu RS
16

Exterior Talu OA
8

Base of the Parapet LQ
21

Interior Height ZD
6

Exterior Height MP
4

Exterior Talu MQ
2

Interior Talu QD
1

Breadth of the Banquet V, L
3

Height of it V, X
1½

The Terra Plana TV
25

The Lizier AE
3

The Mote's breadth EG
112

Fig. 78.

The Depth of it IF
12

Breadth at its bottom FH
88

The Talus EI, or KG
12

The Breadth of the little Ditch c g.
18

The Depth of it
5

The Coridor GC
18

The Seat of the Esplanade
60

The Height CF
6

Now to lay down this Profile draw a line of a convenient length as RSOACGD for the level or Ground-line, then by your Scale of 20, o [...] 30, at most in an Inch, representing Feet. Take out of it 70 for RA, 16 for R S, 8 [Page] for OA, 3 for AE, 112 for EG, 18 for GC, and 60 for CD, and mark them off on your Paper (as in Fig. 78.) at S, O, I, K, C, raise orFig. 78. let fall Perpendiculars (by prop. 1, 2, or 3. §. 1. chap. 4.) then take 16 for ST, and OQ, 12 for IF and KH, and 6 for Cf, and draw RT, TQ, QA, EF, HG, cf, fd: Then from Q set off QL 12, LV 3, QM 2, and LD 1, and raise the Perpendiculars MP 4, DZ6, and VX 1½ Then draw VX, XY, YZ, ZP, and PQ, and make the little Ditch by its measure, so is th [...] Profile perfected: as for the Faus-Bray, they ar [...] now out of use, therefore I omit them.

The Solid Content of those Earth-Works may easily be attained by the former Rules which Content being got in Feet, divide that product by 324, the Quotient shall be the Soli [...] Flores contained therein, a Flore being 18 foo [...] square and 1 Foot deep.

SECT. V. Of the Dimensions and Construction of Pla [...] forms, Caveleers and Cazemats in t [...] Flanks.

1. PLatforms are Plantations where G [...] are to be placed, and are common [...] made of Plank, and Sleepers, there neede [...] for one Gun, to be but one Platform, whi [...] [Page] must be 8 Feet broad next the Parapet, and 14 Feet wide at the other End, and their length should be 18 Feet.

2. An Embrasure is the Port-Hole made in the Parapet, which towards the Gun must be 4 Feet wide, and towards the Campaigne 8 Feet wide, whose height must be proportioned unto the Wheel of the Carriage; and are 16, 18 or 20 Feet assunder.

3. Cavaleers or Mounts are Massy pieces of Earth raised on or near the Rampart, above the Parapet, on which Ordnance and small shot may be planted. As to their Construction I shall follow the Method of Manesson, who places them in the Gorge of the Bastion, and gives this Rule for it, [saith he]. ‘Lengthen outFig. 79. the line of defence to E, untill it cut the Capital, the Center of your Cavaleer shall be the middle point betwixt P and E, to wit at F, then with the distance of 84 Feet on the Center F strike a Circle, which shall be the Base of your Cavaleer:’ Now its height ought to be at least 20 Feet; and if the Work be to be faced with Stone, or Brick it needs not not have a Talu above 3½ Feet, so that the Diameter at the top will be about 153 Feet, whereon, set a Parapet of 20 Seat, and high, and other Demensions as aforesaid in the Rampire, and there will be a Terra-plana at the top of above 100 Feet, whereon six pieces of Ordnance may be planted, making Embrasures and Platforms as was last directed.

4. Cazemats are made in the Corners of the Flanks, and are several Platforms for Guns to be planted on, thereby to be hid from the [...] [Page] [...] [Page] [Page] Battery of the Enemy: As to the Construction I shall follow Manesson's Directions, first as to the form, and also to the measure: [saith he] ‘The Caremate shall take up one half of the Flank, and no more; The Grand Caremate D B is about 7, 8 or 9 Feet from the Level of the Plane of the Fort, and hath a passage into it from within the Fort A, C is its Parapet of 20 or 22 Foot Seat, and in it let there be 3 or 4 Embrasures; D is the part thereof most hid from the Enemies Cannon; F is the MagazineFig. 79. for this Battery; H is the second Caremate, G the Ladder, and L the Magazine, and M the Parapet; this is to hold but one Gun; M is the third Caremate on the level of the Bast. which let be all firm, in which let there be no void place.’ The Dimensions and Construction according to the Method of this our Authour are thus [saith he] ‘Lengthen the Line of Defence from C to G some 40 Feet, then draw CD, parallel to Cf, (by prop. 4. §. 1. chap. 4.) let CF be half of cf, so that cF may be equal to Ff, then from the middle of the Face opposed, draw KF, and let it cut GD, in I, then make IFig. 79. L, and FM equal to 6 Feet, then make MN 66 Feet, and draw NO parallel to the Flank, which let be 24 Feet: Lastly [saith he] for the Orillon or Blind, prolong the Face FT 36 Feet, and also FV 36 Feet, then joyn TV, and make that part all solid: So is your Caremate finished: Let the height of the lower Cazemat, be 6 Feet as before, and let all the rest be compleated as you see in the Figure.’

SECT. VI. Of the Dimensions, and Constructions of those Out-Works, called Ravelins, Horn, Crownworks, &c.

[Page]

THE Ravelin is a certain Work lying beyond the Mote, or Ditch, for the covering the Curtain, Bridge, and Gate; the Angle of the Ravelin must not be less than 60, nor much above 100 deg. the manner of delineating it is thus. Lengthen out the middle line of the Curtain OM unto a convenient length,Fig. 80. then take with your Compasses the length of the Curtain CC, and setting on Foot in F, the point of the Face and Flank, cross the middle line in q; then laying your Ruler at q, and to the points F, draw the lines of the Ravelin q R and qS, which shall be the Ground-lines of the Ravelin: The M [...]te surrounding it must be half the breadth of the Great Mote; the Rampert may be 30 Feet thick, and some 6, 7 or 8 Feet high, on which may stand a Parapet equal to that of the Rampire.

Now if from the points F you raise streight lines into the Campaign, at Right Angles to the Curtain, and from the points F set off FE, andFig. 81. FE 720 or 750 Feet, then may you joyn EE either with, A;

Single Tenaile: which is done by joyning EE, and dividing it into four equal parts, take one and place from D to N, and so draw EN [Page] and EN, so have you a Single Tenail IENEL, which must have a Mote Rampire, and Parapet like the Ravelin. Secondly it may if occasion require be fortified with, A;

Horn-Work: which is done by joyning the points E E, and fortifying the Exterior Polygon EE as is afore taught: Or divide EE into threeFig. 83. parts; make ME, and EN equal to MO; then draw N M, which divide so likewise at O and P; then draw E O and E P; then at P and O raise Perpendiculars O Q and P R, so shall M, E, Q, O, P, R, E, N, be the Horn-Work which was desired: which must likewise have small Rampires and Parapets, as afore.

For the Crown-Work: From the Center of the Fort O draw O M B of a convenient length, then from the middle of the Ravelin set off 1000 or 800 Feet to B, then on q, strike the Arch D B E, set off the Curtain, and DemiGorge P C C, from B to F, and G both ways, then draw C F and C E, to terminate the points I and H on the Counterscerp; then tokeFig. 84. ⅓ part of B F or B E, and set it from B to M; and srom F to L, and from E to M; then draw L M, and M N; then for your DemiBastions make N P and L O equal to N E, &c. Then for the Demi-Gorges of the whole Bastion in the middle, let them be equal to ⅕ of the Interiour Poligon L M or M N, viz. M Y or M X; then finish the Bastions by drawing the lines of Defence, and raising Perpendiculars, or making Angles of 98° at O, X, Y, and P, then the Crown work is finished as desired. You may make Ravelins and other Works (beforementioned) before these Curtains if occasion require. [Page] There are some other Works which are used; as Half Moons, Bonnets, Double Tenails, Countergards, Horseshoes, Priests-Caps, &c. which would be superfluous to speak of in this place.

5. Cittadels, are Castles or Forts of the least sort, and are the Outworks lastly used, which areBuilt to bridle the Town or the Place, left the Burghers should be rebellious, and to be the last refuge or place of retreat. commonly of 4 or 5 Bastions, and are placed in such Order, that there may be two Faces, and a Curtain towards the Town: the Construction whereof is after this manner. Lengthen out the line OM, and therein find the Center of the Cittadel, the Interior Polygon of the Pentagon may be ¾ of the Curtain adjoyning, or a little more; the Center of the Square may be on P the point of the Interior Polygon, the Center of the Hexagon may be near the outward point of the Bastion of the Town, taken away to make the Cittadel in, which may be delineated as afore: The Motes and other Works in proportion accordingly, and the Rampires as high as those of the City or Town.

SECT. VII. Of some Maxims or Rules necessary to be known in Irregular Fortification.

[Page]

IRregular Fortifications is when any Town or Place is to be fortified, which lieth in an Irregular form; i. e. whose Sides and Angles are unequal in the fortifying of Irregular FiguresThe Inginier must first form a Map of the Town or Place, with all the Ways, Passages, Old Walls, Rivers, Pools, Enclosures, and all other matters fit to be known in the draught, and then he is to design what Works he findeth most agreeing to the place to be Fortified.. I shall here say very little, only I shall lay down some Precepts that are of immediate concern in fortifying of Irregular Figures, and shall refer you to peruse Marlois, Dogen, Fritach, Taurnier, Dilichius, &c. which will greatly satisfie and help you: To this end know,

1. That the same Laws and Maxims for Regular Fortifications stand and be in force for Irregular; i. e. that the line of Defence must not exceed the Port of a Musquet, nor the Angles of the Bastion be less than 60°, nor much above 90°, &c.

2. That no inward Angle of the Place be less than 90°, if it be so it must be altered, and that point may be made the outward point of a Bastion.

[Page]

3. That between Regular and Irregular Fortifications, there is no other difference, but by rectifying the sides that are too short, or too long, and altering the Angles that are too little; as for the sides, if they be above 500, and under 1000 Feet, they may be fortified by Bastions placed according to the usual manner, at the extreme points thereof; But if the sides be between 1000 and 1700 Feet, then in the midst you may place a Plat Bastion, and at the Extreme Points, place two Bastions, as before: But if the line be less than 500 Feet, you may lengthen it, by producing it into the Plane: As for the Angles, they are made greater or lesser according as occasion requireth. For the Raising the Rampires, Parapets, and other Outworks, they are to be as in the Regular, and the Out-work may be placed before the Curtains as was before mentioned.

4. That the Capital, in any Regular or Irregular Bastion, is found by dividing the Angle of the Polygon into two equal parts (by prop. 7. §. 1. chap. 4.) and by producing the line of Angular Division or Separation, on which the due length of the Capital must be placed, which observe for a general Rule.

SECT. VIII. Of the Dimensions and Construction of small Forts, or Scones, which are built for the Defence of some Pass, River, or other place.

[Page]

WHEN they are made Regular, of 4, 5, or 6 Bastions, then they may be fortified by the precedent Rules, but there are others of smaller Dimensions fit for the same purpose: viz. Triangle with Demi-Bastions, Square with Demi-Bastions, Parallelograms with Demi-Bastions and Tong, Star Redoubts of four, five or six points, and Plain Redoubts.

PROP. IX. To fortifie a Triangle, with Demi-Bastions.

This Triangle may consist and be comprehended of three equal or unequal sides in this Example: let it be an Equilaterial Triangle PPPFig. 84. Now divide PP into three parts, then take 1, and prick off the Capitals PA, &c. and the Gorges make equal thereunto, as PC, PC, &c. then make the Flanks FC to stand at Right Angles, and to be ½ of PC or PA, then draw the Faces AF, AF, &c. and the Work is finished as required.

PROP. X. To fortifie a Square with Demi-Bastions.

[Page]

The sides of the Square may be from 100 to 200 Feet, let PP be 180 Feet, which divide into 3 parts, take one for the Gorges PC, and forFig. 85. the Capitals PA, and prick them off all round as you see, then take ⅙ of PP, and at Right Angles prick off the Flanks CF, then draw the Faces AF, AF, &c. and the Figure is compleated.

PROP. XI. To fortifie a Parallelogram with Demi-Bastions, and Tong.

First describe the Parallelogram, or LongSquare, PPPP, then divide PP into 6 parts (the side on which the Tong, or Tenaile, is placed) and make MC equal unto ⅙ thereof, and also MG, and MH. Draw CG, GC, and CH, HC,Fig. 86. then finish the Demi-Bastions as before, so shall the Work be compleated as was required. A Long Square may also be fortified as Fig. the 77.

PROP. XII. To fortifie a Star Redoubt of 4, 5, or 6 Points

1. A Star Redoubt of four points may have his side from 40 to 60 Feet: First describe the Square PPPP, then divide PP into two parts at M, take ¼ of PM, (and by prop. 1. §. 1. ch. 4.)

Fig. 87.

[Page] raise Perpendiculars round at M, make MA equal to ¼ of PM, and draw all as in the Figure.

2. A Star Redoubt of five points is thus fortified. Describe the Pentagon PP, &c. then divide PP into halves at M, raise the Perpendiculars MA, make MA equal to ⅓ of PM, and draw the Fort

Fig. 88.

in all respects as the Figure representeth.

3. A Star Redoubt of six points is thus fortified. Describe the Hexagon PPP, &c. divide PP into two equal parts at M, then raise Perpendiculars at the M's, then make MA equal to ½ of PM,

Fig. 89.

or ¼ of PP, and draw every respective line as you see in the Figure.

PROP. XIII. To Delineate a Plain Redoubt

Plain Redoubts are called Grand Redoubts, which are used as Batteries in Approaches, whose side may be from 60 to 80 Feet, or Petit Redoubt,Fig. 90. which are used for a Court of Guards in the Trenches, and may be from 20 to 50 Feet, and are framed and delineated in all respects as you see in Fig 90.

The Profile's to be set on these several Works, and the Motes, are alterable and uncertain, for they being sometimes used in Approaches; then they do require the Breast-work at the Bottom to be 7 or 8 Foot wide, and the Interior Height 6, and the Exterior 5 Feet, and the Mote to be either 8, 10 or 12 Feet, sometimes 14 or 20 Feet wide at the bottom, and the height of 7, 8 or 9 Feet, to have two, or three [Page] ascents to rise to the Parapet. There are many other things belonging to this Art, which the limitation I am bound to, will not permit here to be treated of.

CHAP. XIV. Of Military Orders, or the Embattelling and Encamping of Souldiers.

SECT. I. Of the Embattelling and Ordering of Souldiers.

BATTAILS are considered either in respect of the number of Men, or in respect of the form of Ground. In the respect of the number of Men, it is either a Square Battail, a Double Battail, a Battail of the Grand Front, or a Battail of any proportion, of the number in Rank to the number in File. In respect of the form of the Ground, the Battail is either a Geometrical Square of Ground, or a long Square of Ground. For the Distance, or Order of Souldiers, [Page] martialled in Array, is distinguished either into Open Order, which is when the Centers of their places are 7 Feet distant assuunder, both in Rank and File, or Order; which is when the Centers of the places are 3 ½ Feet distant both in Rank and File; or else 3 ½ Feet in Rank, and 7 Feet in File.

PROP. I. To Order any number of Souldiers into a Square Battail of Men.

Admit it were required to Martial into a Square Battail 16129 Men: To doe which extract the Square Root of 16129 (by prop. 8. §. 1. chap. 1.) which is 127, therefore you are to place 127 Men in Rank, and also in File.

PROP. II. To Order any number of Souldiers into a Double Battail.

Admit 16928 Men were to be Martialled into a Double Battail, extract the Root of half the number of Men; i. e. of 8464, whose Root is 92, therefore I say that 92 Men must be placed in File, and 184 in Rank, to order that number of Men propounded into a Double Battail.

[Page] [Page] [Page]

Plate VII Page 302

PROP. III. To Order any number of Souldiers into a Battail of the Grand Front.

[Page]

[Page]

Admit 16900 Souldiers were to be Martialled into a Battail of the Grand Front, that is Quadruple. Extract the Square Root of 4225 (that is ¼ of the Men) the Root is 65; therefore I say 65 must be placed in File, and 260 in Rank, to form a Battail of the Grand Front.

PROP. IV. Any number of Men, together with their distance in Rank and File, being propounded, to Order them into a Square Battail of Ground.

Admit 2500 Souldiers were to be Martialled into a Square Battail of Ground, in such sort that their distance in File should be 7 feet, and in Rank 3 feet, and 'tis required to know how many Men must be placed in Rank and in File to draw up 2500 Men into Square Battail of the Ground. According to prop. 1. §. 1. ch. 1. say, As — 7 to 3, So is 2500 to 1071, &c. whose Square Root is 32, &c. Therefore I say 32 Men are to be placed in File. Now to find how many Men are to be placed in Rank, divide 2500 by 32, the Quotient is 78, which are the number of Men to be placed in Rank, and 4 Men to be disposed elsewhere.

PROP. V. Any number of Souldiers propounded, to Order them in Rank and File, according to the reason of any two Numbers given.

[Page]

Admit 6400 Souldiers are to be Martialled into Array, in such Order that the number of Men placed in File, shall bear such proportion to the number in Rank as 7 to 13; (according to prop. 1. §. 1. chap. 1.) say as 7 to 13, so is 6400 to 11885, &c. whose Square Root is 109, &c. the number of Men to be placed in Rank, by which divide 6400, it produces, 58, &c. the number of Men to be placed in File, and 78 Men to be employed elsewhere.

SECT. II. Of Castermetation, or Quartering and Encamping of Souldiers.

IN Quartering and Encamping of Souldiers, it is requisite, the Quarter-Master General, and all other under Quarter-Masters, be skilled at Foot measure, that so they may lay out their Quarters as directed.

The common allowance for the depth of Ground, that a Regiment of Horse or Foot will take up, the wideness must be answerable to [Page] the Number of Men 200 Feet for the Huts in length, and 100 for the Commanders, and Sutlers, before them; every two Souldiers to a Hut, 8 Feet broad, and 8 Feet deep, 2 Feet Hut, from Hut, so that there may stand 20 Huts in the 200 Feet, the Ally betwixt Hut, and Hut, may be 8 Feet, that is 16 Feet in width, and 200 in length for 40 Men, which is 3200 Feet, and for the 100 Feet more, 1600 Feet, in all 4800 Feet, and there must be 25 Rows of Huts, for 1000 Men; so that for a Regiment of Foot containing 1000 Men, with Officers, and Sutlers, will take up 120000 Feet, which is 2 Acres and 3 Roods, which because of Ways may be allowed 3 Acres of Ground, for every Regiment, which may be 350 Feet deep, and 370 Feet wide, or near 360 Feet square: Now if 1000 Men, Officers, Sutlers, High-ways and all take up a Square of 360 Feet, how many Feet shall the Side of a Square be wherein 10000 Footmen, &c. may be encamped? say (by prop. 1. chap. 1.) as 1000, to 10000, so is the Square of 360, viz. 129600, to 1296000, the Square of 1138 Feet, which is very near 30 Acres of Ground.

For the Quartering of Horse, you must keep the same depth of 300 Feet for all, and take 200 Feet for the Huts, the Horse Huts must be 10 Feet deep, and 4 wide; so that 12 Horses may stand in one Hut together, which is 48 Feet long, and 10 wide, and 6 Feet a Street; The Huts for the Troops, will be 6, for 12 Troops; now conceive a Regiment to consist of 8 Troops, 50 to a Troop, it will take up leaving 20 Feet Streets, and Cross-ways, very near as much Ground as a Regiment of Foot, Ways and all must be allowed [Page] 3 Acres, near 360 Feet square, so that 10 Regiments of Horse will take up 30 Acres: Moreover, it will be needfull and you may very well allow, as much ground as both Horse and Foot will take, for the Train of Artillery, Victuallers, Parade Places, &c. From these considerations the young beginner, nay even the better practised Souldier may receive help, and thereby be enabled to Encamp an Army if required.

CHAP. XV. Of GUNNERY.

SECT. I. Of the Names of the Principal Members of a Piece of Ordnance.

1. Defin.ACANNON is a long round Body, either of Brass, or Iron, formed and made hollow by Art, and proportion, to offend afar off, with a Ball of Iron, Stone, or any Artificial Substance, charged with Gun-Powder, in its charged Cilinder, which being fired, in an instant performs [Page] its desired Effect. This Machine was invented byThis Military Engine Bombarda, Gun, Cannon, &c. So called from Bombo, a resounding Noise, Cannone, or Cannon, from the likeness it holds with his Canna, Bore, or Concavity; Artigleria, from Artiglio, the Talons, or Claws of Ravenous Fowls, because its shot flying afar off tears and defaces all that it doth meet; from whence some Natures of this Machine are called Smeriglii, long winged Hawks, Falconi, Falconets; Passa volanti, swift flying Arrows, &c. an Englishman, and first put in practice by the Venetians against the Genoveses at Chiezza, Anno 1376.

2. The Superficies of the Mettal, is the outside round about the Piece.

3. The Body is the Substance of the whole Mass of Mettal.

4. The Chase is the Concavity of the Piece, in which they put the Charge.

5. the Muzzel is the Extremity of the Chase by which you load, and unload the Piece.

6. The Calibre is AB the Diameter of theFig. 91. Muzzel or Mouth.

7. The Touch-hole, is that little vent, which passeth from the Convex Superficies, to the very Chamber of the Piece, made to give fire to the Powder within as C, that which encloseth the Extremity of the Chase about the Touch-hole is called the Breech or Coyl.

8. The Cascabel is the Pammel at the Breech or Coyl as D.

The Trunnions, are pieces of Metal fixed unto the Exterior Superficies of the Gun on which he moves in the Carriage as E, E.

[Page]

The Body of the Piece, is that which is comprehended betwixt the Center of the Trunnions and the Cascable EG.Fig. 91.

The Vacant Cylinder, is comprehended betwixt the Cent: of the Trunnions & the Muzzel as EB.

The Frees, or Muzzel Ring is that thick Cornish which, incompasseth the Convex Superficies of the Piece at I, The Base Ring is KLG, The Reinforced Ring is M, The Trunnion Ring is N, and the Cornish Ring is O.

The Line of the Cylinder, is a direct line imagined to be described along the Chase Parallel unto the middle of the Chase as XZ.

The Line of Metal, is a line touching bothFig. 91. Cornishes, as MNI.

The Dispart line of the Piece, is the difference betwixt the Semidiameter of the Muzzel, and Base Ring as the line IH.

The Vent of the Piece is the difference betwixt the Diameter of the Shot, and the Mouth of the Piece, as e d.

The Chamber, or Charged Cylinder, is that part of the Chase towards the Touch-hole equally large, nor narrower in one place than in another, and doth contain the Powder and Ball.

SECT. II. Of the Dimension of our Usal English Cannon, and other Ordnance, &c.

IN the following Table I have set down the length and weight of our most usual English Ordnance, the Diameters and Weight of their Bullets, the length and breadth of their Ladles, the Weight of Powder to Charge them, &c.

[Page]

The Names of the several Pieces of Ordnance. | Guns length | Guns weight | Guns bore | Bullets diamet. | Bullets weight | Ladles length | Ladles breadth | Powder weight | Shoots Level | Utmost Random
Feet | Inches | Pounds | Inches | 8 parts | Inches | 8 parts | Pounds | Ounces | Inches | 8 parts | Inches | 8 parts | Pounds | Ounces | Paces | Paces
A Base. | 4 | 6 | 200 | 1 | 2 | 1 | 1 | 0 | 5 | 4 | 0 | 2 | 0 | 0 | 8 | 60 | 600
A Rabinet. | 5 | 6 | 300 | 1 | 4 | 1 | 3 | 0 | 8 | 4 | 1 | 2 | 4 | 0 | 2 | 70 | 700
A Falconet. | 6 | 0 | 400 | 2 | 2 | 2 | 1 | 1 | 5 | 7 | 4 | 4 | 0 | 1 | 4 | 90 | 900
A Falconi | 7 | 0 | 750 | 2 | 6 | 2 | 5 | 2 | 8 | 8 | 2 | 4 | 4 | 2 | 4 | 130 | 1300
Minion ordinary. | 7 | 0 | 800 | 3 | 0 | 3 | 7 | 3 | 4 | 8 | 4 | 5 | 0 | 2 | 8 | 120 | 1200
Minion largest. | 8 | 0 | 1000 | 3 | 2 | 3 | 0 | 3 | 12 | 9 | 0 | 5 | 0 | 3 | 4 | 125 | 1250
Saker leaft. | 8 | 0 | 1400 | 3 | 4 | 3 | 2 | 4 | 12 | 9 | 6 | 6 | 4 | 3 | 6 | 150 | 1500
Saker ordinary. | 9 | 0 | 1500 | 3 | 6 | 3 | 4 | 6 | 0 | 10 | 4 | 6 | 6 | 4 | 0 | 160 | 1600
Saker old sort. | 10 | 0 | 1800 | 4 | 0 | 3 | 6 | 7 | 5 | 11 | 0 | 7 | 2 | 5 | 0 | 163 | 1630
Demiculver least. | 10 | 0 | 2000 | 4 | 2 | 4 | 0 | 9 | 0 | 12 | 0 | 8 | 0 | 6 | 4 | 174 | 1740
Demiculver ordinary. | 11 | 0 | 2700 | 4 | 4 | 4 | 2 | 10 | 11 | 12 | 6 | 8 | 0 | 7 | 4 | 175 | 1750
Demiculver old sort. | 11 | 0 | 3000 | 4 | 6 | 4 | 4 | 12 | 11 | 13 | 4 | 8 | 4 | 8 | 8 | 178 | 1780
Culverin least. | 11 | 0 | 4000 | 5 | 0 | 4 | 6 | 15 | 0 | 14 | 2 | 9 | 0 | 10 | 0 | 180 | 1800
Culverin ordinary. | 12 | 0 | 4500 | 5 | 2 | 5 | 0 | 17 | 5 | 16 | 0 | 9 | 4 | 11 | 6 | 181 | 1810
Culverin largest. | 12 | 0 | 4800 | 5 | 4 | 5 | 2 | 20 | 0 | 16 | 0 | 10 | 0 | 11 | 8 | 183 | 1830
Demicannon least | 11 | 0 | 5400 | 6 | 2 | 6 | 0 | 30 | 0 | 20 | 0 | 11 | 4 | 14 | 0 | 156 | 1560
Demicannon ordin. | 12 | 0 | 5600 | 6 | 4 | 6 | 1 | 32 | 0 | 22 | 0 | 12 | 0 | 17 | 8 | 162 | 1620
Demicannon large | 12 | 0 | 6000 | 6 | 6 | 6 | 3 | 36 | 0 | 22 | 6 | 12 | 6 | 18 | 0 | 180 | 1800
Cannon Royal | 12 | 0 | 8000 | 8 | 0 | 7 | 4 | 58 | 0 | 24 | 0 | 14 | 0 | 32 | 8 | 185 | 1850

PROP. I. How to know the different Fortification of a Piece of Ordnance.

[Page]

In fortifying any Piece of Ordnance there are three degrees observed, as first Legitimate Pieces, which are those that are ordinarily fortified; secondly Bastard Pieces, which are such whose Fortification is lessened; thirdly Double fortified Pieces, or extraordinary Pieces.

The Fortification of any Piece of Ordnance, is accounted by the thickness of the Metal at the Touch-hole, Trunnions, and at the Muzzel, in proportion to the Diameter of the Bore.

The Legitimate Pieces, or the ordinary fortified Cannons, have ⅞ at the Touch hole, ⅝ at the Trunnions, and ⅜ at the Muzzel of the thickness of the bore, in thickness of Metal. Bastard Cannons, or lesned Cannons, have ¾ at their Touch-hole, or 12/16, and 9/16 at their Trunnions, and 7/16 at their Muzzel: the Double fortified Cannons have full one Diameter of the Bore in thickness of Metal at the Touch-hole, and 11/16 at the Trunnions; and 7/16 at their Muzzel. Now all double fortified Culverins, &c. are 1 ⅛ at the Touch-hole, 15/16 at the Trunnions, and 9/16 at the Muzzel, and the Ordinary fortified Culverins, are fortified every way as double fortified Cannons, and lesned Culverins as Ordinary Cannons in all respects.

PROP. II. How to know how much Powder is fit for proof, and what for service, for any Piece of Ordnance.

[Page]

For Cannons take ⅘ of the weight of their Iron Bullet of good Corn Powder for Proof, and for service ½ the weight of the Iron Bullet is sufficient, especially for Iron Ordnance, which will not endure so much Powder, as Brass ones will receive by ¼ in Weight, for Culverins allow the whole Weight of the Shoot for Proof, and ⅔ for Service. For Sakers, and Falcons, take ⅘ of the Weight of the Shoot, and for lesser Pieces the whole weight may be used in service, untill they grow hot, but then there must be some abatement made at discretion, and take 1 ⅓ of the weight of their Iron Bullet for Proof.

PROP. III. To know what Bullet is fit to be used in any Piece of Ordnance.

The Bullet must be somewhat less than the Bore of the Gun, that so it may have vent in the discharge, some Authors affirm ¼ of an Inch less than the Bore will serve, all Ordnance, but this vent is too much for a Falcon, &c. and too little for a Cannon: therefore I approve them not, but commend Mr. Phillipes's proportionIn his Mathematical Manual. to your Use, which is to divide the Bore of the Gun into 20 equal parts; and let the Diameter of the Bullet be 19/20 thereof, according to whichpage 165. proportion the precedent Table is calculated.

PROP. IV. By knowing the proportion of Metals one to another, and by knowing the Weight of one Ball, to know what any other shall weigh.

[Page]

The common received proportions for Metals are these.

Lead is to Iron as 2, to 3.

Lead is to Brass as 24, to 19.

Lead is to Stone as 4, to 1.

Iron is to Lead as 3, to 2.

Iron is to Brass as 16, to 18.

Iron is to Stone as 3, to 8.

The more exact proportion betwixt Metals are thus known. Admita Cube, or Ball of Gold, weigh 100 l. A Cube of any of those Metals ensuing of the same bigness, shall bear such proportion, as followeth, to the said Cube of Gold.

li. | pts. | li. | pts.
Gold. | 100 | 00 | Iron. | 42 | 10
Quicksilver. | 71 | 43 | Tinn. | 38 | 95
Lead. | 60 | 53 | Stone. | 15 | 80
Silver. | 54 | 39 | Water. | 05 | 68
Brass. | 47 | 37

It is the opinion of Dr. Wybard in his Tactometria, that a Bullet of Cast Iron, whose Diameter is 4 Inches, doth weigh 9 l. Averdupoize weight.

Now to find what any other Bullet, or Cube shall weigh; say (as in prop. 4. chap. 1.) As the Cube of the Bullet propounded, is to his weight, so is the Cube of another Bullet given, to his weight, and so observe still this proportion.

SECT. III. Of the Qualification of an able Gunner, and necessary Operations before shooting, and in shooting.

[Page]

A Gunner ought to be a Man of Courage, Experience, and Vigilant; he ought to have good skill in Arithmetick, to know the Extraction of the Roots, &c. He ought to have skill in Geometry, to take heights, distances, &c. to know the Divisions and Use of his Circle, Quadrant, and Quadret; to know how to level, and to lay Platforms, and to raise Batteries. He must know the Names of all sorts of Ordnance, their Weight, the Height of their Bore, the Height and Weight of their Shot, the length and breadth of their Ladles, how much Powder to use for proof, and action; The Shoots Level, and the Shoots Random; He must know the Names of all the Members of a Piece of Ordnance, he must also know the length, thickness and breadth of all manner of Carriages, and must know all the parts thereof: Viz. the Cheeks or Sides, the Axtree, Spokes, Nave, Hoops, Transomes, Bolts, Plates, Drawing-Hooks, the Clout, the Hole for the Linspin, the Shafts, the Thill and Thill-bolt, the Fore-lock, and Fore-lockkeys, Capsquares, the Fore-lock-pins and Chain, the Pintle and Bolt-hole, Fellows, Nayles, Fellow-bars, Stirropes, the Ruts of the Wheel, Dowledges, Beds, Coines, Leveres, Hand screws, &c. He must also know how to make his Ladles, Spunges, Cartridges, [Page] whether of Paper or Canvas, and to have by him Formers of all sorts, Sheep-skins undrest to make Spunges, Powder, Shot, Needles, Thread, Paste and Starch, Marlin, Twine, Nails, Handspikes, Crows of Iron, Granado-shells, and Materials for Composition, Fasces, Budg Barrels, Cannon-Baskets, &c. These being general things he is to know, and at all times to have ready by him, and he is more particularly to know these following parts of his Art: As,

PROP. I. How to Tertiate, Quadrate, and to Dispart a Piece of Ordinance.

1. To Tertiate a Piece, is to find whether it hath its due thickness at the Trunnions, Touchhole, and Neck; if the Trunnions, and the Neck are in its due order, and the Chase streight.

2. To Quadrate a Piece mounted, is to see whether it be directly placed, and equally poized in the Carriage; which is known by finding in the Convex Superficies of the Base, and Muzzel Ring; the point which is Perpendicular, over the Soul of the Piece which may be found by the Gunners Instrument, called a Level; an Instrument whose use is so vulgarly known, that it needeth not my Explanation.

3. To Dispart a Piece, is to fix, or elevate on the Convex point of the Muzzel Ring, a Mark, as far distant from the Cylinder, or Soul of the Piece, as is the point of the Base-Ring; to the end, that the Visail-ray which passeth by these marks, may be Parallel to the Chase, Soul, or [Page] Cylinder of the Piece. Now the Dispart, i. e. the difference of the Semidiameters of the Cornishes, may be by a pair of Calliper Compasses attained. Which found, place on the Top of the Cornish-Ring, near the Muzzel, over the middle of the Inferior Cylinder.

PROP. II. To know how far any Piece of Ordnance will shoot, &c.

As to the several shootings in Artillery, Authors differ much in their Judgments, and Opinions, but they all unanimously agree that the Ball being shot forth flies through the Air, with a Violent, Mixt and Natural motion; describing a Parabolical line, in whose beginning and ending are lines sensibly streight, and in the middle carved: In the beginning the ImprestFig. 91. force driving forward by the Fire, the Natural gravity of the Ball doth describe a Right-line, called the Direct line, or Rangs of the Ball's Circute.

In the middle that force diminisheth, and the Natural Gravity prevaileth, so that it describeth a curved line, called the Ball's middle Helical or Conical ArchSee Mr. Diggs in his Pantometria, page 179.; In the End the Natural Gravity overcoming the Imprest violence, (which becomes altogether weak and faint) describes a new right line, called the Ball's declining line, in which the Ball tends towards the Center of the Earth, as towards a Place natural unto all heavy bodies: See Figure the 92. These motions are somewhat longer, according as the Piece is [Page] mounted from the level unto the Angle of 45 deg. which is called the Utmost Random: The Elevation of which, is regulated by the Gunners Quadrant, the Use of which Instrument is so generally known, and by so many Authors fully explained, that I here crave leave to omit it: But take these for General Rules.

1. That a Shoot at Right Angle, strikes more

General Rules to be observed in the battering down of a Place, or making of Breaches.

violent and furiously than at Oblique Angles, therefore Gunners use when they are to bàtter down a Tower, Wall, or Earth-work, to shoot point blank at the Object, Tire by Tire; by discharging all the Pieces in Battery against the self same. Object, in the same; Instant, holding it for a Maxim, that ten Cannons discharged together, do fan more Execution than discharged one after another. Now at Oblique Angles they shoot either Cross ways or by rebounding.

2. That the speediest way to make a B [...]a [...]h in a Wall, &c. Is by shooting at the Object from two Batteries, which ruins for more speedily than by striking the Object, with one Battery, at Right Angles, although that one Battery, hath as many Cannon as the other two hath.

3. That if you were to batter a Flank, covered with an Orillion, (which because you cannot possibly batter it right forward) you must therefore of necessity batter it Obliquely, by way of Rebounding, thus: Chuse a fit place in the Curtain to be your Object, on which you may play with your Battery obliquely, so that by a rebound the shoot may leap [...] the Flanks, holding for a Maxim, in this operation,

According to learned D'Chales, on the 4th Prop. of the first Book of Euclid.

That the Angles of Incidence and Reflection are Equal.

Now we come to shew the length of the [Page] Right Range, of all our Common English Ordnance, which is set down in the precedent Table, in which the Cannon exceed not 185 Paces, &c. Esteeming the Pace 5 English Feet, nor his utmost Random above 1850 Paces, which Table so sheweth for all other Natures.

As for the Ranges, and Randoms, to the several Degrees and points of Mounture of the Quadrant, I have hereunto annexed the Tables, calculated by the Experiments of sundry most Eminent Artists, whose Works will perpetuate their Worth and Name to succeding Generations.

A Table of Ranges, and Randoms, to the several Degrees of Mounture of the Quadrent.

A TABLE OF
Right Ranges or Points Blanks. | Randoms or the First Graze.
The Degrees of the Pieces Mounture. | 0 | The Right Range in Paces of 5 Feet. | 192 | The Degrees of Mounture. | 0 | The Paces of the Random 5 Feet a Pace. | 192
1 | 209 | 1 | 298
2 | 227 | 2 | 404
3 | 249 | 3 | 510
4 | 261 | 4 | 610
5 | 278 | 5 | 722
6 | 285 | 6 | 828
7 | 302 | 7 | 934
8 | 320 | 8 | 1044
9 | 337 | 9 | 1129
10 | 354 | 10 | 1214
20 | 454 | 20 | 1917
30 | 693 | 30 | 2185
40 | 855 | 40 | 2289
50 | 1000 | 50 | 2283
60 | 1140 | 60 | 1792
70 | 1220 | 70 | 1214
80 | 1300 | 80 | 1000
90 | 1350 | 90

The Use of the Table of Randoms.

[Page]

This Table is most agreeing to Cannons, and Culverins; and the greatest sort of Ordnance, the Use thereof is thus.

Admit a Saker to be mounted to 3 deg. shoots the Bullet 323 Paces, how far will it shoot being mounted unto 7 deg. Say (by prop. 1. chap. 1.)

As 510 the Tabular distance for 3 deg. of Mounture, to 323, the distance found,

So is 934 the Tabular distance for 7 deg. of Mounture, to 591 272/510, the distance required, which the Saker according to this Experiment shall shoot at 7 deg. of Mounture.

Mr. NYE in his Book of Gunnery printed Anno, 1647, saith he made an Experiment by a Saker of 8 Feet long, which he loaded with three pounds of Powder, of an exact weight, both Powder and Wad at every charge, every time ramming it down with three equal stroaks, as near as possible; but on the Bullet he put no Wad, because the Saker was mounted; And thus he made four Shoots, each of them half an Hour after the other, that so the Piece might be of equal temper, and mounted his Piece to these 4 degrees of Mounture, viz. 1 deg. 5 deg. 7 deg. 10 deg. and found these Randoms.

At 1 Deg. the Random was 225 Paces.

At 5 Deg. the Random was 416 Paces.

At 7 Deg. the Random was 505 Paces.

At 10 Deg. the Random was 630 Paces.

[Page]

According to which Experiment, he framed this Table of Randoms.

Deg. | Paces | Deg. | Paces
0 | 206 | 6 | 461
1 | 225 | 7 | 505
2 | 274 | 8 | 548
3 | 323 | 9 | 589
4 | 370 | 10 | 630
5 | 416

Captain HEXAM in his Book of Gunnery, shews how by finding out the Random of a Cannon, for the first Degree of Mounture, thereby to find the Random for every Degree to 45 deg. or utmost Random, and this is his Rule to perform it. First find how many Paces the Cannon will shoot being laid level by the Metal, (which by him is accounted 1 deg.) Then divide the distance found, by 50, then multiply the Quotient by 11, so shall the product be the greatest Digression, or Difference betwixt Rangs, and Rang; which being divided by 44, the Quotient giveth the Number of Paces, which the Bullet will lose in the other Rangs, from Degree, unto Degree; according to this Rule, this Table is calculated.

[Page]

A Table of Randoms to 45 Degrees, accounting a Pace 2 ½ Foot.

D. Moun | Paces. | Diff. | D. Moun. | Paces. | Diff.
0 | 0775 | 225 | 23 | 4685 | 110
1 | 1000 | 220 | 24 | 4795 | 105
2 | 1220 | 215 | 25 | 4900 | 100
3 | 1435 | 210 | 26 | 5000 | 95
4 | 1645 | 205 | 27 | 5095 | 90
5 | 1850 | 200 | 28 | 5185 | 85
6 | 2050 | 195 | 29 | 5270 | 80
7 | 2245 | 190 | 30 | 5350 | 75
8 | 2435 | 185 | 31 | 5425 | 70
9 | 2620 | 180 | 32 | 5595 | 65
10 | 2800 | 175 | 33 | 5560 | 60
11 | 2975 | 170 | 34 | 5620 | 55
12 | 3145 | 165 | 35 | 5675 | 50
13 | 3310 | 160 | 36 | 5725 | 45
14 | 3470 | 155 | 37 | 5770 | 40
15 | 3625 | 150 | 38 | 5810 | 35
16 | 3775 | 145 | 39 | 5845 | 30
17 | 3920 | 140 | 40 | 5875 | 25
18 | 4060 | 135 | 41 | 5900 | 20
19 | 4595 | 130 | 42 | 5920 | 15
20 | 4325 | 125 | 43 | 5935 | 10
21 | 4450 | 120 | 44 | 5945 | 5
22 | 4570 | 115 | 45 | 5950

[Page]

I have hereunto also annexed the Table calculated by Alexander Bianco, for all sort of Ordnance, (which Table I account one of the best that was ever yet found Extant) In his Work printed 1648.

A Table of Randoms for the first six Points of the Gunner's Quadrant.

Points. | 1 | 2 | 3 | 4 | 5 | 6
Falconet. | 375 | 637 | 795 | 885 | 892 | 900
Falcon. | 550 | 935 | 1166 | 1254 | 1309 | 1320
Minion. | 450 | 765 | 954 | 1026 | 1071 | 1080
Saker. | 625 | 1062 | 1325 | 1125 | 1487 | 1500
Demi-culver. | 725 | 1232 | 1537 | 1653 | 1725 | 1740
Culverin. | 750 | 1275 | 1590 | 1710 | 1785 | 1800
Demi-cannon. | 625 | 1062 | 1325 | 1425 | 1487 | 1500
Cannon of 7. | 675 | 1147 | 1431 | 1489 | 1606 | 1620
Double Cannon. | 750 | 1275 | 1590 | 1710 | 1785 | 1800

SECT. IV. Of Shooting in Mortar-Pieces.

A Mortar-Piece is a short Piece, with which they shoot Bombs, Granado-Shells, StoneBalls, &c. not by a Right line but from a Curved, from on high; so that it may fall where it [Page] should be desired: Now this Mortar is placed in the Carriage, in all respects as you see in Fig. 93. in which A signifies the Carriage, B the Mortar,Fig. 93. C the Course the shoot flies, and D the Place on which it falls.

Bombs are great hollow Balls of Iron, or Brass, in which are put fine Sifted Gun-Powder, which by a Fuse, they proportion to them a due Fire, that so they may break assoon as they fall amongst the Enemies. These Fuses are small Trunks of Wood, Tinn, or Iron, filled with a prepared Composition for that purpose. Granadoes are of the same form with Bombs, only smaller, and many times are cast by hand, and are made of Iron, Brass, Glass, or Earth.

Now in Order to the well shooting in those kind of Machines called Mortars, 'tis requisite to observe these following Rules: as,

1. That before you make a shoot at any Place, you find the distance thereof from your Mortar, which may be obtained by Prop. 3. §. 4. Chap. 9.

2. That the Bombs, or other Bodies that are to be shot, be of equal weight, otherwise the shoots will vary

3 That the Carriage in breadth be always on a Level, and without any descent, that so it may not leap in discharging.

4. That the Powder with which the Mortar is loaded, be always of the same force and weight.

5. That the Charge of the Mortar, as well in Powder as in Wadding, be always rammed in with blows equally heavy, and of equal number.

6 That the Wadds be always either of Wood, or Tampeons, or else of Okam, for the strongest drives it farthest.

7. That the Fuses be newly made, in those days that they are to be used, and that they be made of a Composition proportionable to the Range that the shoot shall make in the Air, so that the Bomb may break in the very moment of its fall; which Composition must be such, that though it fall in the Water, yet not to extinguish, but the Bomb there to break. Now before we proceed any farther, I think it necessary, to shew how to compose your Ingredients for your Fuse.

PROP. I. To make Fuses for Bombs, &c.

The Composition for Bombs must be of a slow motion, that so time enough may be given to throw either Bombs, Granadoes, Fire-balls, Thundring-Barrels, &c. They are compounded of these Ingredients, thus: Take a pound of Gun-Powder, 4/16 of Sulphur 4/16 of Salt-Peter, well beaten, dry, and sifted separately, then mix it, and make up your Fuse thereof: Or take Powder of Benjamin, and Small-Coles, all well beaten and mixed together with some Oyl of Piter, and so fill your Fuse therewith.

Now the use of Mortar-Pieces, being for the most part to shoot up at Random, therefore the Randoms of these Pieces is very necessary to be known: Therefore hereunto I have annexed the Tables of Randoms, calculated by the Experience of the best of Authors, which have wrote on this Subject; most of which do agree in their Randoms, although they are in a several dress.

[Page]

Diego-Uffano-Zutphen in his Works printed 1621, hath calculated these two following Tables, the one for the 12 points of the Quadrant, the other for every Degree, taking the one Half of each Number, and so 'tis reduced into our English Paces of 5 Feet, which Tables were esteemed and made use of, both by Captain Hexam, and Mr. Norton, and are as followeth.

§

A Table of Randoms for Mortar-Pieces, to the 12 Points of the Gunner's Quadrant, calculated by Diego-Uffano-Zutphen.

583 | 570 | 534 | 468 | 377 | 248 | 100
6 | 5 | 4 | 3 | 2 | 1 | 0
. | . | . | . | . | . | . | ☉
6 | 7 | 8 | 9 | 10 | 11 | 12
583 | 570 | 534 | 468 | 377 | 248 | 000

Now suppose the Mortar to be placed at ☉, the Pricks in the middle line representeth the several Randoms, numbred with the Degrees of the Quadrant, forward and backward, unto which the several Randoms are set; so you see that the Mortar being levelled point blank, throweth the Bomb 100 Paces, if the Mortar be mounted one Point, it throws the Bomb 248 Paces, &c. untill 'tis mounted to the 6th. point, [Page] 583 Paces, which is the utmost Random: Now if the Mortar be mounted higher to 7, 8, 9, &c. Points, the Randoms decrease again as before they did increase: as you see in the Table.

But in those latter Randoms there lieth a great mistake, as shall be made palpably appear. For if as they are distant from the sixth Point you make them equal to one another, then the Random of the 12 points, must be equal to the Random of 0 point, or the Level Random, which is 100 Paces from the Mortar. Now it is contrary to all Art and Reason, to think that if the Mortar be elevated to the 12th. point, i. e. bolt upright, it should shoot the Bomb 100 Paces from the Mortar; no, it cannot be; but according to all Reason the Bomb must fall down either on, or near the Mortar, and not 100 Paces distant, as is most erroneously conceived; the like errour is in the following Table of our said Author; but because Mr. Phillipps in his Mathematical Manual hath amply demonstrated their Errours, I therefore shall say no more to the Errours that have been a long time generally conceived and embraced as a truth, but now are removed.

[Page]

A Table of Randoms for Mortar-Pieces, to every Degree of the Quadrant.

The Degrees of Mounture. | 0 | The Paces of the Random. | 100 | The Degrees of Mounture. | 89 | The Degrees of Mounture. | 23 | The Paces of the Random. | 480 | The Degrees of Mounture. | 66
1 | 122 | 88 | 24 | 490 | 65
2 | 143 | 87 | 25 | 500 | 64
3 | 164 | 86 | 26 | 510 | 63
4 | 185 | 85 | 27 | 518 | 62
5 | 204 | 84 | 28 | 525 | 61
6 | 224 | 83 | 29 | 531 | 60
7 | 243 | 82 | 30 | 536 | 59
8 | 263 | 81 | 31 | 540 | 58
9 | 280 | 80 | 32 | 543 | 57
10 | 297 | 79 | 33 | 549 | 56
11 | 315 | 78 | 34 | 552 | 55
12 | 331 | 77 | 35 | 558 | 54
13 | 347 | 76 | 36 | 562 | 53
14 | 362 | 75 | 37 | 568 | 52
15 | 377 | 74 | 38 | 573 | 51
16 | 393 | 73 | 39 | 577 | 50
17 | 406 | 72 | 40 | 580 | 49
18 | 419 | 71 | 41 | 581 | 48
19 | 432 | 70 | 42 | 582 | 47
20 | 445 | 69 | 43 | 583 | 46
21 | 457 | 68 | 44 | 584
22 | 460 | 67 | 45 | 585

The most exact Tables of Randoms for the Mortar, that I have seen or can find in any Ancient, or Modern Author, is this following Table, calculated by the experience and trial of that Famous Inginier Tomaso Moretii of Brescia, Inginier to the most serene Republique of Venice, in his Works Intituled, Trattatu delle Artiglieria, printed 1665. Where he supposeth the utmost [Page] Random, equal to 10000, according to which proportion he framed this following Table.

A Table of the several Randoms of each Degree of the Quadrant, the greatest Equal to 10000.

Elev. | Elev. | Elev. | Elev.
1° | 349 | 89° | 23° | 7193 | 67°
2 | 698 | 88 | 24 | 7431 | 66
3 | 1045 | 87 | 25 | 7660 | 65
4 | 1392 | 86 | 26 | 7880 | 64
5 | 1736 | 85 | 27 | 8090 | 63
6 | 2079 | 84 | 28 | 8290 | 62
7 | 2419 | 83 | 29 | 8480 | 61
8 | 2756 | 82 | 30 | 8660 | 60
9 | 3090 | 81 | 31 | 8829 | 59
10 | 3420 | 80 | 32 | 8988 | 58
11 | 3746 | 79 | 33 | 9135 | 57
12 | 4067 | 78 | 34 | 9272 | 56
13 | 4384 | 77 | 35 | 9397 | 55
14 | 4695 | 76 | 36 | 9511 | 54
15 | 5000 | 75 | 37 | 9613 | 53
16 | 5299 | 74 | 38 | 9703 | 52
17 | 5592 | 73 | 39 | 9781 | 51
18 | 5870 | 72 | 40 | 9848 | 50
19 | 6157 | 71 | 41 | 9903 | 49
20 | 6428 | 70 | 42 | 9945 | 48
21 | 6691 | 69 | 43 | 9976 | 47
22 | 6947 | 68 | 44 | 9994 | 46
45 | 10000 | 45

The Use of the Precedent Table is explained by these following Propositions.

PROP. II.

Finding that a Mortar of 300, with a Tampeon of Wood, being elevated 45°, or 6 Points of the Quadrant, sends a Bomb 800 Paces, how many Paces shall the same shoot, at the Elevation of 54°?

Look at the said 54° of the Table, and you Demonstration. [Page] will find thē proportional Number 9511, to correspond thereunto. Now you find the proportional Number belonging to 45° is 10000, then by Prop. 1. Chap. 1. Say as 10000, to 800, so is 9511, to 760 88/100, which are the Paces, the Mortar will send the Bomb at the Elevation of 54 Degrees.

PROP. III. Finding that a Mortar of 300, being elevated 54°, sends his Bomb 760 88/100 Paces, what Degree of Elevation must that Mortar have, to shoot the Bomb 555 Paces?

This is but the Converse of the former, therefore (according to Prop. 1. Chap. 1.) say, as 760 88/160 Paces, gives the proportional part or number 9511; so doth 555 Paces, give the proportional part 6945. Which number sought among the proportional Numbers, in the Table, you will find 68 Degrees to correspond to that proportional Number 6945, so that the Mortar must be elevated to 68 Degrees to shoot the Bomb 555 Paces, which was required to be known. These Rules and Precepts here delivered, I esteem necessary to be known by every Gunner, who intends to be serviceable for his Prince and Countrey.

Vive, vale: Siquid novisti rectius istis, Candidus imperti: Si non his utere mecum. Hora. lib. 1. Epist

FINIS.

[Page]

[Page]

§

[Page]

Plate VIII

A TABLE OF Logarithm Numbers, From One to Ten Thousand: Whereby the LOGARITHM OF ANY NUMBER Under Four Hundred Thousand may be readily discovered.

[Page]

[Page]

LONDON, Printed by J. Heptinstall for W. Freeman, at the Artichoke next St. Dunstan's Church in Fleet street. MDCLXXXVII.

[Page]

[Page]

N | Log. | N | Log. | N | Log.
1 | 0. 000000 | 34 | 1. 531479 | 67 | 1. 826075
2 | 0. 301030 | 35 | 1. 544068 | 68 | 1. 832509
3 | 0. 477121 | 36 | 1. 556303 | 69 | 1. 838849
4 | 0. 602060 | 37 | 1. 568202 | 70 | 1. 845098
5 | 0. 698970 | 38 | 1. 579783 | 71 | 1. 851258
6 | 0. 778151 | 39 | 1. 591064 | 72 | 1. 857332
7 | 0. 845098 | 40 | 1. 602060 | 73 | 1. 863323
8 | 0. 903090 | 41 | 1. 612784 | 74 | 1. 869232
9 | 0. 954242 | 42 | 1. 623249 | 75 | 1. 875061
10 | 1. 000000 | 43 | 1. 633468 | 76 | 1. 880813 | 11 | 1. 041393 | 44 | 1. 643452 | 77 | 1. 886491 | 12 | 1. 079181 | 45 | 1. 653212 | 78 | 1. 892094
13 | 1. 113943 | 46 | 1. 662758 | 79 | 1. 897627
14 | 1. 146128 | 47 | 1. 672098 | 80 | 1. 903090
15 | 1. 176091 | 48 | 1. 681241 | 81 | 1. 908485
16 | 1. 204120 | 49 | 1. 690196 | 82 | 1. 913814
17 | 1. 230449 | 50 | 1 698970 | 83 | 1. 919078
18 | 1. 255272 | 51 | 1. 707570 | 84 | 1. 924279
19 | 1. 278753 | 52 | 1. 716003 | 85 | 1. 929419
20 | 1. 301030 | 53 | 1. 724276 | 86 | 1. 934498
21 | 1. 322219 | 54 | 1. 732394 | 87 | 1. 939519
22 | 1. 342422 | 55 | 1. 740362 | 88 | 1. 944482
23 | 1. 361728 | 56 | 1. 748188 | 89 | 1. 949390
24 | 1. 380211 | 57 | 1. 755875 | 90 | 1. 954242
25 | 1. 397940 | 58 | 1. 763428 | 91 | 1. 959041
26 | 1. 414973 | 59 | 1. 770852 | 92 | 1. 963788
27 | 1. 431364 | 60 | 1. 778151 | 93 | 1. 968483
28 | 1. 447158 | 61 | 1. 785330 | 94 | 1. 973128
29 | 1. 462398 | 62 | 1. 792391 | 95 | 1. 977723
30 | 1. 477121 | 63 | 1. 799340 | 96 | 1. 982271
31 | 1. 491361 | 64 | 1. 806180 | 97 | 1. 986772
32 | 1. 505150 | 65 | 1. 812913 | 98 | 1 991226
33 | 1. 518514 | 66 | 1 819544 | 99 | 1. 995635
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
100 | 000000 | 000434 | 000868 | 001301 | 001734 | 002166 | 002598 | 003029 | 003461 | 003891 | 432
101 | 004321 | 004751 | 005181 | 005609 | 006038 | 006466 | 006894 | 007321 | 007748 | 008174 | 428
102 | 008600 | 009026 | 009451 | 009876 | 010299 | 010724 | 011147 | 011570 | 011993 | 012415 | 424
103 | 012837 | 013259 | 013679 | 014100 | 014521 | 014940 | 015359 | 015779 | 016197 | 016616 | 416
104 | 017033 | 017451 | 017898 | 018284 | 018700 | 019116 | 019532 | 019947 | 020361 | 020775 | 416
105 | 021189 | 021603 | 022016 | 022428 | 022841 | 023252 | 023664 | 024075 | 024486 | 024896 | 412
106 | 025306 | 025715 | 026125 | 026533 | 026942 | 027349 | 027757 | 028164 | 028371 | 028978 | 408
107 | 029384 | 029789 | 030195 | 030599 | 031004 | 031408 | 031812 | 032216 | 032619 | 033021 | 404
108 | 033424 | 033826 | 034227 | 034628 | 035029 | 035429 | 035829 | 036229 | 036629 | 037028 | 400
109 | 037426 | 037825 | 038223 | 038620 | 039017 | 039414 | 039811 | 040207 | 040602 | 040998 | 396
110 | 041393 | 041787 | 042182 | 042576 | 042969 | 043362 | 043755 | 044148 | 044539 | 044932 | 393
111 | 045323 | 045714 | 046105 | 046495 | 046885 | 047275 | 047664 | 048053 | 048442 | 048830 | 389
112 | 049218 | 049603 | 049993 | 050379 | 050766 | 051153 | 051538 | [...] | 052309 | 052694 | 386
113 | 053078 | 053463 | 053846 | 054229 | 054613 | 054996 | 055378 | 055760 | 056142 | 056524 | 382
114 | 056905 | 057286 | 057666 | 058046 | 058426 | 058805 | 059185 | 059563 | 059942 | 060320 | 379
115 | 060698 | 061075 | 061452 | 061829 | 062206 | 062582 | 062958 | 063333 | 063709 | 064083 | 376
116 | 064458 | 064832 | 065206 | 065579 | 065953 | 066326 | 066699 | 067071 | 067443 | 067815 | 372
117 | 068186 | 068557 | 068928 | 069298 | 069668 | 070038 | 070407 | 070776 | 071145 | 071514 | 369
118 | 071882 | 072249 | 072617 | 072985 | 073352 | 073718 | 074085 | 074451 | 074816 | 075182 | 366
119 | 075547 | 075912 | 076276 | 076640 | 077004 | 077368 | 077731 | 078094 | 078457 | 078819 | 363
120 | 079181 | 079543 | 079904 | 080266 | 080626 | 080987 | 081347 | 081707 | 082067 | 082426 | 360
121 | 082785 | 083144 | 083503 | 083861 | 084219 | 084576 | 084934 | 085291 | 085647 | 086004 | 357
122 | 086359 | 086716 | 087071 | 087426 | 087781 | 088136 | 088490 | 088845 | 089198 | 089552 | 355
123 | 089905 | 090258 | 090610 | 090963 | 091315 | 091667 | 092018 | 092369 | 092721 | 093071 | 351
124 | 093422 | 093772 | 094122 | 094471 | 094820 | 095169 | 095518 | 095866 | 096215 | 096562 | 349
125 | 096910 | 097257 | 097604 | 097951 | 098298 | 098644 | 098989 | 099335 | 099681 | 100026 | 346
126 | 100371 | 100715 | 101059 | 101403 | 101747 | 102091 | 102434 | 102777 | 103119 | 103462 | 343
127 | 103804 | 104146 | 104487 | 104828 | 105169 | 105510 | 105851 | 106191 | 106531 | 106871 | 340
128 | 107209 | 107549 | 107888 | 108227 | 108565 | 108903 | 109241 | 109579 | 109916 | 110253 | 338
129 | 100589 | 100926 | 111263 | 111599 | 111934 | 112269 | 112605 | 112939 | 113275 | 113609 | 335
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
130 | 113943 | 114277 | 114611 | 114944 | 115278 | 115611 | 115943 | 116276 | 116608 | 116939 | 333
131 | 117271 | 117603 | 117934 | 118265 | 118595 | 118926 | 119256 | 119586 | 119915 | 120245 | 330
132 | 120574 | 120903 | 121231 | 121559 | 121888 | 122216 | 122544 | 122871 | 123198 | 123525 | 328
133 | 123852 | 124178 | 124504 | 124830 | 125156 | 125481 | 125806 | 126131 | 126456 | 126781 | 325
134 | 127105 | 127429 | 127753 | 128076 | 128399 | 128722 | 129045 | 129368 | 129689 | 130012 | 323
135 | 130334 | 130655 | 130977 | 131298 | 131619 | 131939 | 132259 | 132579 | 132899 | 133219 | 321
136 | 133539 | 133858 | 134177 | 134496 | 134814 | 135133 | 135451 | 135769 | 136086 | 136403 | 318
137 | 136721 | 137037 | 137354 | 137671 | 137987 | 138303 | 138618 | 138934 | 139249 | 139564 | 315
138 | 139879 | 140194 | 140508 | 140822 | 141136 | 141449 | 141763 | 142076 | 142389 | 142702 | 314
139 | 143015 | 143327 | 143639 | 143951 | 144263 | 144574 | 144885 | 145196 | 145507 | 145818 | 311
140 | 146128 | 146438 | 146748 | 147058 | 147367 | 147676 | 147985 | 148294 | 148603 | 148911 | 309
141 | 149219 | 149527 | 149835 | 150142 | 150449 | 150756 | 151063 | 151369 | 151676 | 151982 | 307
142 | 152288 | 152594 | 152899 | 153205 | 153509 | 153815 | 154119 | 154423 | 154728 | 155032 | 305
143 | 155336 | 155639 | 155943 | 156246 | 156549 | 156852 | 157154 | 157457 | 157759 | 158061 | 303
144 | 158362 | 158664 | 158965 | 159266 | 159567 | 159868 | 160168 | 160469 | 160769 | 161068 | 301
145 | 161368 | 161667 | 161967 | 162266 | 162564 | 162863 | 163161 | 163459 | 163758 | 164055 | 299
146 | 164353 | 164650 | 164947 | 165244 | 165541 | 165838 | 166134 | 166430 | 166726 | 167022 | 297
147 | 167317 | 167613 | 167908 | 168203 | 168497 | 168792 | 169086 | 169380 | 169674 | 169968 | 295
148 | 170262 | 170555 | 170848 | 171141 | 171434 | 171726 | 172019 | 172311 | 172603 | 172895 | 293
149 | 173186 | 173478 | 173769 | 174059 | 174351 | 174641 | 174932 | 175222 | 175512 | 175802 | 291
150 | 176091 | 176381 | 176669 | 176959 | 177248 | 177536 | 177825 | 178113 | 178401 | 178689 | 289
151 | 178977 | 179264 | 179552 | 179839 | 180126 | 180413 | 180699 | 180986 | 181272 | 181558 | 287
152 | 181844 | 182129 | 182415 | 182699 | 182985 | 183269 | 183555 | 183839 | 184123 | 184407 | 285
153 | 184691 | 184975 | 185259 | 185542 | 185825 | 186108 | 186391 | 186674 | 186956 | 187239 | 283
154 | 187521 | 187803 | 188084 | 188316 | 188647 | 188928 | 189209 | 189490 | 189771 | 190051 | 281
155 | 190332 | 190612 | 190892 | 191171 | 191451 | 191730 | 192009 | 192289 | 192567 | 192846 | 279
156 | 193125 | 193403 | 193681 | 193959 | 194237 | 194514 | 194792 | 195069 | 195346 | 195623 | 278
157 | 195899 | 196176 | 196453 | 196729 | 197005 | 197281 | 197556 | 197832 | 198107 | 198382 | 276
158 | 198657 | 198932 | 199206 | 199481 | 199755 | 200029 | 200303 | 200577 | 200850 | 201124 | 274
159 | 201397 | 201670 | 201943 | 202216 | 202488 | 202761 | 203033 | 203303 | 203577 | 203848 | 272
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
160 | 204119 | 204391 | 204663 | 204934 | 205204 | 205475 | 205746 | 206016 | 206286 | 206556 | 271
161 | 206826 | 207096 | 207365 | 207364 | 207904 | 208173 | 208441 | 208710 | 208978 | 209247 | 269
162 | 209515 | 209783 | 210051 | 210319 | 210586 | 210853 | 211121 | 211388 | 211654 | 211921 | 267
163 | 212187 | 212454 | 212720 | 212986 | 213252 | 213518 | 213783 | 214049 | 214314 | 214579 | 266
164 | 214844 | 215109 | 215373 | 215638 | 215902 | 216166 | 216429 | 216694 | 216957 | 217221 | 264
165 | 217484 | 217747 | 218010 | 218273 | 218536 | 218798 | 219060 | 219323 | 219585 | 219846 | 262
166 | 220108 | 220369 | 220631 | 220892 | 221153 | 221414 | 221675 | 221936 | 222196 | 222456 | 261
167 | 222716 | 222676 | 223236 | 223496 | 223755 | 224015 | 224274 | 224533 | 224791 | 225051 | 259
168 | 225309 | 225568 | 225827 | 226084 | 226342 | 226599 | 226858 | 227115 | 221372 | 227629 | 258
169 | 227887 | 228142 | 228400 | 228657 | 228913 | 229169 | 229426 | 229682 | 229938 | 230193 | 256
170 | 230449 | 230704 | 230959 | 231215 | 231469 | 231724 | 231979 | 232234 | 232488 | 232742 | 254
171 | 232996 | 233250 | 233504 | 233752 | 234011 | 234264 | 234517 | 234770 | 235023 | 235276 | 253
172 | 235528 | 235781 | 236033 | 236285 | 236537 | 236789 | 237041 | 237292 | 237544 | 237795 | 252
173 | 238046 | 238297 | 238548 | 238799 | 239049 | 239299 | 239549 | 239799 | 240049 | 240299 | 250
174 | 240549 | 240799 | 241048 | 241297 | 241546 | 241795 | 242044 | 242293 | 242541 | 242789 | 249
175 | 243038 | 243286 | 243534 | 243782 | 244029 | 244177 | 244525 | 244772 | 245019 | 245266 | 248
176 | 245513 | 245759 | 246006 | 246252 | 246499 | 246745 | 246991 | 247237 | 247482 | 247728 | 246
177 | 247973 | 248219 | 248464 | 248709 | 248954 | 245198 | 249443 | 249687 | 249932 | 250176 | 245
178 | 250420 | 250664 | 250908 | 251151 | 251395 | 251638 | 251881 | 252125 | 252368 | 252610 | 243
179 | 252853 | 253096 | 253334 | 253580 | 253822 | 254064 | 254306 | 254548 | 254789 | 255031 | 242
180 | 255273 | 255514 | 255755 | 255996 | 256237 | 256477 | 256718 | 256958 | 257198 | 257438 | 241
181 | 257679 | 257918 | 258158 | 258398 | 258637 | 258877 | 259116 | 250355 | 259594 | 259833 | 239
182 | 260071 | 260309 | 260548 | 260787 | 261025 | 261263 | 261501 | 261739 | 261976 | 262214 | 238
183 | 262451 | 262688 | 262925 | 263162 | 263399 | 263636 | 263873 | 264109 | 264346 | 264582 | 237
184 | 264818 | 265054 | 265289 | 265525 | 265761 | 265996 | 266232 | 266467 | 266702 | 266937 | 235
185 | 267172 | 267406 | 267641 | 267875 | 268109 | 268344 | 268578 | 268812 | 269046 | 269279 | 234
186 | 269513 | 269746 | 269979 | 270213 | 270446 | 270679 | 270912 | 271144 | 271377 | 271609 | 233
187 | 271842 | 272074 | 272306 | 272538 | 272769 | 273001 | 273233 | 273464 | 273696 | 273927 | 232
188 | 274158 | 274389 | 274619 | 274850 | 275081 | 275311 | 275542 | 275772 | 276002 | 276232 | 230
189 | 276462 | 276692 | 296921 | 277151 | 277379 | 277609 | 277838 | 278067 | 278296 | 278525 | 229
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
190 | 278754 | 278982 | 279211 | 279439 | 276667 | 279895 | 280123 | 280351 | 280578 | 280806 | 228
191 | 281033 | 281261 | 281488 | 281714 | 281942 | 282169 | 282396 | 282622 | 282849 | 283075 | 227
192 | 283301 | 283527 | 283753 | 283979 | 284205 | 284431 | 284656 | 284882 | 285107 | 285332 | 226
193 | 285557 | 285782 | 286007 | 286232 | 286456 | 286681 | 286905 | 287129 | 287354 | 287578 | 225
194 | 287802 | 288026 | 288249 | 288473 | 288696 | 288919 | 289143 | 289366 | 289589 | 289812 | 223
195 | 290035 | 290257 | 290479 | 290702 | 290925 | 291147 | 291369 | 291591 | 291813 | 292034 | 222
196 | 292256 | 292478 | 292699 | 292920 | 293141 | 293362 | 293584 | 293804 | 294025 | 294246 | 221
197 | 294466 | 294687 | 294907 | 295127 | 295347 | 295567 | 295787 | 296007 | 296226 | 296446 | 220
198 | 296665 | 296884 | 297104 | 297323 | 297542 | 297761 | 297979 | 298198 | 298416 | 298635 | 219
199 | 298853 | 299071 | 299289 | 299507 | 299725 | 299943 | 200161 | 300378 | 200595 | 300813 | 218
200 | 301030 | 301247 | 301464 | 301681 | 301898 | 302114 | 302331 | 302547 | 302764 | 302979 | 217
201 | 303196 | 303412 | 303628 | 303844 | 304059 | 304275 | 304491 | 304706 | 304921 | 305136 | 216
202 | 305351 | 305566 | 305781 | 305996 | 306211 | 306425 | 306639 | 306854 | 307068 | 307282 | 215
203 | 307496 | 307709 | 307924 | 308137 | 308351 | 308564 | 308778 | 308991 | 309204 | 309417 | 213
204 | 309630 | 309843 | 310056 | 310268 | 310481 | 310693 | 310906 | 311118 | 311329 | 311542 | 212
205 | 311754 | 311966 | 312177 | 312389 | 312600 | 312812 | 313023 | 313234 | 313445 | 313656 | 211
206 | 313867 | 314078 | 314289 | 314499 | 314709 | 314920 | 315130 | 315340 | 315551 | 315760 | 210
207 | 315970 | 316180 | 316389 | 316599 | 316809 | 317018 | 317227 | 317436 | 317646 | 317854 | 209
208 | 318063 | 318272 | 318481 | 318689 | 318898 | 319106 | 319314 | 319522 | 319730 | 319938 | 208
209 | 320146 | 320354 | 320562 | 320769 | 320977 | 321184 | 321391 | 321598 | 321805 | 322012 | 207
210 | 322219 | 322426 | 322633 | 323839 | 323046 | 323252 | 323458 | 323665 | 323871 | 324077 | 206
211 | 324282 | 324488 | 324694 | 324899 | 325105 | 325310 | 325516 | 325721 | 325926 | 326131 | 205
212 | 326336 | 326541 | 326745 | 326949 | 327155 | 327359 | 327563 | 327767 | 327972 | 328176 | 204
213 | 328379 | 328583 | 328787 | 328991 | 329194 | 329398 | 329601 | 329805 | 330008 | 330211 | 203
214 | 330414 | 330617 | 330819 | 331022 | 331225 | 331427 | 331629 | 331832 | 332034 | 332236 | 202
215 | 332438 | 332640 | 332842 | 333044 | 333246 | 333447 | 333649 | 333859 | 334051 | 334253 | 202
216 | 334454 | 334655 | 334856 | 335057 | 335257 | 335458 | 335658 | 335859 | 336059 | 336259 | 201
217 | 336459 | 336659 | 336859 | 337059 | 337259 | 337459 | 337659 | 337859 | 338058 | 338257 | 200
218 | 338456 | 338656 | 338856 | 339054 | 339253 | 339453 | 339650 | 339849 | 340047 | 340246 | 199
219 | 340444 | 340642 | 340841 | 341039 | 341237 | 341435 | 341632 | 341830 | 342028 | 342225 | 198
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
220 | 342227 | 342620 | 342817 | 343014 | 343212 | 343409 | 343606 | 343802 | 343999 | 344196 | 197
221 | 344392 | 344589 | 344785 | 344981 | 345178 | 345373 | 345569 | 345766 | 345962 | 346157 | 196
222 | 346353 | 346549 | 346744 | 346939 | 347135 | 347330 | 347525 | 347720 | 347915 | 348110 | 195
223 | 348305 | 348499 | 348694 | 348889 | 349083 | 349278 | 349472 | 349659 | 349860 | 350054 | 194
224 | 350248 | 350442 | 350636 | 350829 | 351023 | 351216 | 351409 | 351603 | 351796 | 351989 | 193
225 | 352183 | 352375 | 352568 | 352761 | 352954 | 353147 | 353339 | 353532 | 353724 | 353916 | 193
226 | 354108 | 354301 | 354493 | 354685 | 354876 | 355068 | 355239 | 355452 | 345643 | 355834 | 192
227 | 356026 | 356217 | 356408 | 356599 | 356790 | 356981 | 357172 | 357363 | 357554 | 357744 | 191
228 | 357935 | 358125 | 358316 | 358506 | 358696 | 258886 | 359076 | 359266 | 359456 | 359646 | 190
229 | 359835 | 360025 | 360215 | 360404 | 360593 | 360783 | 360972 | 361161 | 361350 | 361539 | 189
230 | 361728 | 361917 | 362105 | 362294 | 362482 | 362671 | 362859 | 363048 | 363236 | 363424 | 188
231 | 363612 | 363799 | 363988 | 364176 | 364363 | 364551 | 364739 | 364926 | 365113 | 365301 | 188
232 | 365488 | 365675 | 365862 | 366049 | 366236 | 366423 | 366609 | 366796 | 366983 | 367169 | 187
233 | 367356 | 367542 | 367729 | 367915 | 368101 | 368287 | 368473 | 368659 | 368845 | 369030 | 186
234 | 369216 | 369401 | 369587 | 369772 | 369958 | 370143 | 370328 | 370513 | 370698 | 370882 | 185
235 | 371068 | 371253 | 371437 | 371622 | 371806 | 371991 | 372175 | 372359 | 372544 | 372728 | 184
236 | 372912 | 373096 | 373279 | 373464 | 373647 | 373831 | 374015 | 374198 | 374382 | 374565 | 184
237 | 374748 | 374932 | 375115 | 375298 | 375481 | 375664 | 375846 | 376029 | 376212 | 376394 | 183
238 | 376577 | 376159 | 376942 | 377124 | 377306 | 377488 | 377670 | 377852 | 378034 | 378216 | 182
239 | 378398 | 378579 | 378761 | 378943 | 379124 | 379306 | 379487 | 379668 | 379849 | 380030 | 181
240 | 380211 | 380392 | 380573 | 380754 | 380934 | 381115 | 381296 | 381476 | 381656 | 381837 | 181
241 | 382017 | 382197 | 382377 | 382557 | 382737 | 382917 | 383097 | 383277 | 383456 | 383636 | 180
242 | 383815 | 383995 | 384174 | 384353 | 384533 | 384712 | 384891 | 385069 | 385249 | 385428 | 179
243 | 385606 | 385785 | 385964 | 386142 | 386321 | 389499 | 386677 | 386856 | 387034 | 387212 | 178
244 | 387389 | 387568 | 387746 | 387923 | 388101 | 388279 | 388456 | 388634 | 388811 | 388989 | 178
245 | 389166 | 389343 | 389520 | 389698 | 389875 | 390051 | 390228 | 390405 | 390582 | 390759 | 177
246 | 390935 | 391112 | 391288 | 391464 | 391641 | 391817 | 391993 | 392169 | 392345 | 392521 | 176
247 | 392697 | 392873 | 393048 | 393224 | 393399 | 393575 | 393751 | 393926 | 394101 | 394277 | 176
248 | 394452 | 394627 | 394802 | 394977 | 395152 | 395326 | 395501 | 395676 | 395850 | 396025 | 175
249 | 396199 | 396374 | 396548 | 396722 | 396896 | 397071 | 397245 | 397419 | 397592 | 397766 | 174
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
250 | 397940 | 398114 | 398287 | 398461 | 398634 | 398808 | 398981 | 399154 | 399328 | 399501 | 173
251 | 399674 | 399847 | 400019 | 400192 | 400365 | 400538 | 400711 | 400883 | 401056 | 401228 | 173
252 | 401401 | 401573 | 401745 | 401917 | 402089 | 402261 | 402433 | 402605 | 402777 | 402949 | 172
253 | 403121 | 403292 | 403464 | 403635 | 403807 | 403978 | 404149 | 404320 | 404492 | 404663 | 171
254 | 404834 | 405005 | 405176 | 405346 | 405517 | 405688 | 405858 | 406029 | 406199 | 406369 | 171
255 | 406540 | 406710 | 406881 | 407051 | 407221 | 407391 | 407561 | 407731 | 407901 | 408070 | 170
256 | 408239 | 408409 | 408579 | 408749 | 408918 | 409087 | 409257 | 409426 | 409595 | 409764 | 169
257 | 409933 | 410102 | 410271 | 410439 | 410609 | 410777 | 410946 | 411114 | 411283 | 411451 | 169
258 | 411619 | 411788 | 411956 | 412124 | 412293 | 412461 | 412629 | 412796 | 412964 | 413132 | 168
259 | 413299 | 413467 | 413635 | 413803 | 413969 | 414137 | 413405 | 414472 | 414639 | 414806 | 167
260 | 414973 | 415140 | 415307 | 415474 | 415641 | 415808 | 415974 | 416141 | 416308 | 416474 | 167
261 | 416641 | 416807 | 416973 | 417139 | 417306 | 417472 | 417638 | 417804 | 417969 | 418135 | 166
262 | 418301 | 418467 | 418633 | 418798 | 418964 | 419129 | 419295 | 419460 | 419625 | 419791 | 165
263 | 419956 | 420121 | 420289 | 420451 | 420616 | 420781 | 420945 | 421110 | 421275 | 421439 | 165
264 | 421604 | 421768 | 421933 | 422097 | 422261 | 422426 | 422589 | 422754 | 422918 | 423082 | 164
265 | 423246 | 423409 | 423574 | 423737 | 423901 | 424065 | 424228 | 424392 | 424555 | 424718 | 164
266 | 424882 | 425045 | 425208 | 425371 | 425534 | 425697 | 425860 | 426023 | 426186 | 426349 | 163
267 | 426511 | 426674 | 426836 | 426999 | 427161 | 427324 | 427486 | 427648 | 427811 | 427973 | 162
268 | 428135 | 428297 | 428459 | 428621 | 428783 | 428944 | 429106 | 429268 | 429429 | 429591 | 162
269 | 429752 | 429914 | 430075 | 430236 | 430398 | 430559 | 430719 | 430881 | 431042 | 431203 | 161
270 | 431369 | 431525 | 431685 | 431846 | 432007 | 432167 | 432328 | 432488 | 432649 | 432809 | 161
271 | 432969 | 433129 | 433289 | 433449 | 433609 | 433769 | 433929 | 434089 | 434249 | 434409 | 160
272 | 434569 | 434729 | 434888 | 435048 | 435207 | 435366 | 435526 | 435685 | 435844 | 436004 | 159
273 | 436163 | 436322 | 436481 | 436639 | 436799 | 436957 | 437116 | 437275 | 437433 | 437592 | 159
274 | 437751 | 437909 | 438067 | 438226 | 438384 | 438542 | 438701 | 438859 | 439175 | 439165 | 158
275 | 439333 | 439491 | 439648 | 439806 | 439964 | 440122 | 440279 | 440437 | 440594 | 440752 | 158
276 | 440909 | 441066 | 441224 | 441381 | 441538 | 441695 | 441852 | 442009 | 442166 | 442323 | 157
277 | 442479 | 442637 | 442793 | 442949 | 443106 | 443263 | 443419 | 443576 | 443732 | 443889 | 157
278 | 444045 | 444201 | 444357 | 444513 | 444669 | 444825 | 444981 | 445137 | 445293 | 445449 | 156
279 | 445604 | 445759 | 445915 | 446071 | 446226 | 446382 | 446537 | 446692 | 446848 | 447003 | 155
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
280 | 447158 | 447313 | 447468 | 447623 | 447778 | 447932 | 448088 | 448242 | 448397 | 448552 | 155
281 | 448706 | 448861 | 449015 | 449169 | 449324 | 449478 | 449633 | 449787 | 459941 | 450095 | 154
282 | 452049 | 450403 | 450557 | 450711 | 450865 | 451018 | 451172 | 451326 | 451479 | 451633 | 154
283 | 451786 | 451939 | 452093 | 452247 | 452399 | 452553 | 452706 | 452859 | 453012 | 453165 | 153
284 | 453318 | 453471 | 453624 | 453777 | 453929 | 454082 | 454235 | 454387 | 454539 | 454692 | 153
285 | 454845 | 454997 | 455149 | 455302 | 455454 | 455606 | 455758 | 455910 | 456062 | 456214 | 152
286 | 456366 | 456518 | 456669 | 456821 | 456973 | 457125 | 457276 | 457428 | 457579 | 457731 | 152
287 | 457889 | 458033 | 458184 | 458336 | 458487 | 458638 | 458789 | 458939 | 459091 | 459242 | 151
288 | 459392 | 459543 | 459694 | 459845 | 459995 | 460146 | 460296 | 460447 | 460597 | 460748 | 151
289 | 460898 | 461048 | 461198 | 461348 | 461499 | 461649 | 461799 | 461948 | 462098 | 462248 | 150
290 | 462398 | 462548 | 462647 | 462847 | 462997 | 463146 | 463296 | 463445 | 463594 | 463744 | 150
291 | 463893 | 464042 | 464191 | 464340 | 464489 | 464639 | 464788 | 464936 | 465085 | 465234 | 149
292 | 465383 | 465532 | 465680 | 465829 | 465977 | 466126 | 466274 | 466423 | 466571 | 466719 | 149
293 | 466868 | 467016 | 467164 | 467312 | 467460 | 467608 | 467756 | 467904 | 468052 | 468199 | 148
294 | 468347 | 468495 | 468643 | 468790 | 468938 | 969085 | 469233 | 469380 | 469527 | 469675 | 147
295 | 469822 | 469969 | 470116 | 470263 | 470410 | 470557 | 470704 | 470851 | 470998 | 471145 | 147
296 | 471292 | 471438 | 471585 | 471732 | 471878 | 472025 | 472171 | 472318 | 472464 | 472610 | 146
297 | 472756 | 472903 | 473049 | 473195 | 473341 | 473487 | 473633 | 473779 | 473925 | 474071 | 146
298 | 474216 | 474362 | 474508 | 474653 | 474799 | 474944 | 475089 | 475235 | 475381 | 475526 | 145
299 | 475671 | 475816 | 475962 | 476107 | 476252 | 476397 | 476542 | 476687 | 479832 | 476976 | 145
300 | 477121 | 477266 | 477411 | 477555 | 477699 | 477844 | 477989 | 478133 | 478278 | 478422 | 145
301 | 478566 | 478711 | 478855 | 478999 | 479143 | 479287 | 479431 | 479575 | 479719 | 479863 | 144
302 | 480007 | 480151 | 480294 | 480438 | 480582 | 480725 | 480869 | 481012 | 481156 | 481299 | 144
303 | 481443 | 481586 | 481729 | 481872 | 482016 | 482159 | 482302 | 482445 | 482588 | 482731 | 143
304 | 482874 | 483016 | 483159 | 483302 | 483445 | 483587 | 483729 | 483872 | 484015 | 484157 | 143
305 | 484299 | 484442 | 484585 | 484727 | 484869 | 485011 | 485153 | 485295 | 485437 | 485579 | 142
306 | 485721 | 485863 | 486005 | 486147 | 486289 | 486430 | 486572 | 486714 | 486855 | 486997 | 142
307 | 487138 | 487279 | 487421 | 487563 | 487704 | 487845 | 487986 | 488127 | 488269 | 488409 | 141
308 | 488551 | 488692 | 488333 | 488974 | 489114 | 489255 | 489396 | 489537 | 489677 | 489818 | 141
309 | 489958 | 490099 | 490239 | 490379 | 490520 | 490661 | 490801 | 490941 | 491081 | 491222 | 140
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
310 | 491362 | 491502 | 491642 | 491782 | 491922 | 492062 | 492201 | 492341 | 492481 | 492621 | 140
311 | 492760 | 492900 | 493039 | 493179 | 493319 | 493458 | 493597 | 493737 | 493876 | 494015 | 139
312 | 494155 | 494294 | 494433 | 494572 | 494711 | 494850 | 494989 | 495128 | 495267 | 495406 | 139
313 | 495544 | 495683 | 495822 | 495960 | 496099 | 496238 | 496376 | 496515 | 596653 | 496791 | 139
314 | 496929 | 497068 | 497206 | 497344 | 497483 | 497921 | 497759 | 497897 | 498035 | 498173 | 138
315 | 498311 | 498448 | 498586 | 498724 | 498862 | 498999 | 495137 | 499275 | 499412 | 499549 | 138
316 | 499687 | 499824 | 499962 | 500099 | 500236 | 500374 | 500510 | 500648 | 500785 | 500922 | 137
317 | 501059 | 501196 | 501333 | 501470 | 501607 | 501744 | 501880 | 502017 | 502154 | 502291 | 137
318 | 502427 | 502564 | 502700 | 502837 | 502973 | 503109 | 503246 | 503382 | 503518 | 503655 | 136
319 | 503791 | 503927 | 504063 | 504199 | 504335 | 504471 | 504607 | 504743 | 504878 | 505014 | 136
320 | 505149 | 505286 | 505421 | 505557 | 505193 | 505828 | 505964 | 506099 | 506234 | 506369 | 136
321 | 506505 | 506640 | 506776 | 506911 | 507046 | 507181 | 507316 | 507451 | 507586 | 507721 | 135
322 | 507856 | 507991 | 508126 | 508260 | 508395 | 508529 | 508664 | 508799 | 508934 | 509068 | 135
323 | 509203 | 509337 | 509471 | 509606 | 509740 | 509894 | 510009 | 510143 | 510277 | 510411 | 134
324 | 510545 | 510679 | 510813 | 510947 | 511081 | 511215 | 511349 | 511482 | 511616 | 511749 | 134
325 | 511883 | 512017 | 512151 | 512284 | 512418 | 512551 | 512684 | 512818 | 512951 | 513084 | 133
326 | 513218 | 513351 | 513485 | 513617 | 513750 | 513883 | 514016 | 514149 | 514282 | 514415 | 133
327 | 514548 | 514681 | 514813 | 514946 | 515079 | 515211 | 515344 | 515476 | 515609 | 515741 | 133
328 | 515874 | 516006 | 516139 | 516271 | 516403 | 516535 | 516668 | 516796 | 516932 | 517064 | 132
329 | 517196 | 517328 | 517459 | 517592 | 517724 | 517855 | 517987 | 518119 | 518251 | 518382 | 132
330 | 518514 | 518646 | 518777 | 518909 | 519040 | 519171 | 519303 | 519434 | 519566 | 519697 | 131
331 | 519828 | 519959 | 520090 | 520221 | 520353 | 520484 | 520615 | 520745 | 520876 | 521007 | 131
332 | 521138 | 521269 | 521399 | 521530 | 521661 | 521792 | 521922 | 522053 | 522183 | 522314 | 131
333 | 522454 | 522575 | 522705 | 522835 | 522966 | 523096 | 523226 | 523356 | 523486 | 523616 | 130
334 | 523746 | 523876 | 524006 | 524136 | 524266 | 524396 | 524526 | 524656 | 524785 | 424915 | 130
335 | 520545 | 525174 | 525304 | 525434 | 525563 | 525693 | 525822 | 525951 | 526081 | 526210 | 129
336 | 526339 | 526469 | 526598 | 526727 | 526856 | 526985 | 527114 | 527243 | 527372 | 527501 | 129
337 | 527629 | 527759 | 527888 | 528016 | 528145 | 528274 | 528402 | 528531 | 528659 | 528788 | 129
338 | 528916 | 529045 | 529174 | 529302 | 529430 | 525559 | 529687 | 529815 | 529943 | 530072 | 128
339 | 530199 | 530328 | 530456 | 530584 | 530712 | 530839 | 530968 | 531096 | 531223 | 531351 | 128
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
340 | 531479 | 531607 | 531734 | 531862 | 531989 | 532117 | 532245 | 532372 | 532499 | 532627 | 128
341 | 532754 | 532882 | 533009 | 533136 | 533264 | 533391 | 533518 | 533645 | 533772 | 533899 | 127
342 | 534026 | 534153 | 534280 | 534407 | 534534 | 534661 | 534787 | 534914 | 535041 | 535167 | 127
343 | 535294 | 535421 | 535547 | 535674 | 535800 | 535927 | 536053 | 536179 | 536304 | 536432 | 126
344 | 536558 | 536685 | 536811 | 536937 | 537063 | 537189 | 537315 | 537441 | 537567 | 537693 | 126
345 | 537819 | 537945 | 538071 | 538197 | 538322 | 538448 | 538574 | 538699 | 538825 | 538951 | 126
346 | 539076 | 539202 | 539327 | 539452 | 539578 | 539703 | 539829 | 539954 | 540079 | 540204 | 125
347 | 540329 | 540455 | 540579 | 540705 | 540829 | 540955 | 541079 | 541205 | 541329 | 541454 | 125
348 | 541579 | 541704 | 541829 | 541953 | 542078 | 542203 | 542327 | 542452 | 542576 | 542701 | 125
349 | 542825 | 542949 | 543074 | 543199 | 543323 | 543447 | 543571 | 543696 | 543819 | 543944 | 124
350 | 545008 | 544192 | 544316 | 544440 | 544564 | 544688 | 544812 | 544934 | 545059 | 545183 | 124
351 | 545307 | 545431 | 545555 | 545678 | 545802 | 545925 | 546049 | 546172 | 546296 | 546419 | 124
352 | 546543 | 546666 | 546789 | 546913 | 547036 | 547159 | 547282 | 547405 | 547529 | 547652 | 123
353 | 547775 | 547898 | 548021 | 548144 | 548267 | 548389 | 548512 | 548635 | 548758 | 548881 | 123
354 | 549003 | 549126 | 549249 | 549371 | 549494 | 549616 | 549739 | 549861 | 549984 | 550106 | 123
355 | 550228 | 550351 | 550473 | 550595 | 550717 | 550839 | 550962 | 551084 | 551206 | 551328 | 122
356 | 551449 | 551572 | 551694 | 551816 | 551938 | 552059 | 552181 | 552303 | 552425 | 552547 | 122
357 | 552668 | 552789 | 552911 | 553033 | 553155 | 553276 | 553398 | 553519 | 553640 | 553762 | 121
358 | 553883 | 554004 | 554126 | 554457 | 554368 | 554489 | 554610 | 554731 | 554852 | 554973 | 121
359 | 555094 | 555215 | 555336 | 554247 | 555578 | 555699 | 555819 | 555940 | 556061 | 556182 | 121
360 | 556303 | 556423 | 556544 | 556664 | 556785 | 556905 | 557026 | 557146 | 557267 | 557387 | 120
361 | 557057 | 557627 | 557748 | 557868 | 557988 | 558108 | 558228 | 558349 | 558469 | 558589 | 120
362 | 558709 | 558829 | 558948 | 559068 | 559188 | 559308 | 559428 | 559548 | 559667 | 559787 | 120
363 | 559907 | 560026 | 560146 | 560265 | 560385 | 560504 | 560624 | 560743 | 560863 | 560982 | 119
364 | 561101 | 561221 | 561339 | 561459 | 561578 | 561698 | 561817 | 561936 | 562055 | 562174 | 119
365 | 562293 | 562412 | 562531 | 562649 | 562769 | 562887 | 563006 | 563125 | 563244 | 563362 | 119
366 | 563481 | 563599 | 563718 | 563836 | 563955 | 564074 | 564192 | 564311 | 564429 | 564548 | 119
367 | 564666 | 564784 | 564903 | 565021 | 565139 | 565257 | 565376 | 565494 | 565612 | 565729 | 118
368 | 565848 | 565966 | 566084 | 566202 | 566319 | 566437 | 566555 | 566673 | 566791 | 566909 | 118
369 | 567026 | 567144 | 567262 | 567379 | 567497 | 567614 | 567732 | 567849 | 567967 | 568084 | 118
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
370 | 568202 | 568319 | 568436 | 568554 | 568671 | 568788 | 568905 | 569023 | 596139 | 569257 | 117
371 | 569374 | 569491 | 569608 | 569725 | 569842 | 569959 | 570076 | 570193 | 570309 | 570426 | 117
372 | 570543 | 570659 | 570776 | 570893 | 571009 | 571126 | 571243 | 571359 | 571476 | 571592 | 117
373 | 571709 | 571825 | 571942 | 572058 | 572274 | 572291 | 572407 | 572523 | 572639 | 572755 | 116
374 | 572872 | 572988 | 573104 | 573219 | 573336 | 573452 | 573568 | 573684 | 573799 | 573915 | 116
375 | 574031 | 574147 | 574263 | 574379 | 574494 | 574609 | 574726 | 574841 | 574957 | 575072 | 116
376 | 575188 | 575303 | 575419 | 575534 | 575649 | 575765 | 575880 | 575996 | 576111 | 576226 | 115
377 | 576341 | 576457 | 576572 | 576687 | 576802 | 576917 | 577032 | 577147 | 577262 | 577377 | 115
378 | 577492 | 577607 | 577722 | 577836 | 577951 | 578066 | 578181 | 578295 | 578409 | 578525 | 115
379 | 578639 | 578754 | 578868 | 578983 | 579097 | 579212 | 579326 | 579441 | 579555 | 579669 | 114
380 | 579784 | 579898 | 580012 | 580126 | 580241 | 580355 | 580469 | 580583 | 580697 | 580811 | 114
381 | 580925 | 581039 | 581153 | 581267 | 581381 | 581495 | 581608 | 581722 | 581836 | 581949 | 114
382 | 582063 | 582177 | 582291 | 582404 | 582518 | 582631 | 582745 | 582858 | 582972 | 583085 | 114
383 | 583199 | 583312 | 583426 | 583539 | 583652 | 583765 | 583879 | 583992 | 584105 | 584218 | 113
384 | 584331 | 584444 | 584587 | 584670 | 584783 | 584896 | 585009 | 585122 | 585235 | 585348 | 113
385 | 585461 | 585574 | 585686 | 585799 | 585912 | 586024 | 586137 | 586249 | 586362 | 586475 | 113
386 | 586587 | 586699 | 586812 | 586925 | 587037 | 587149 | 587262 | 587374 | 587486 | 587599 | 112
387 | 587712 | 587823 | 587935 | 588047 | 588159 | 588272 | 588384 | 588496 | 588608 | 588719 | 112
388 | 588832 | 588944 | 589056 | 589167 | 589279 | 589391 | 589503 | 589615 | 589726 | 589838 | 112
389 | 589949 | 599061 | 590173 | 590284 | 590396 | 590507 | 590619 | 590730 | 590842 | 590953 | 112
390 | 591065 | 591176 | 591287 | 591399 | 591509 | 591621 | 591732 | 591843 | 591955 | 592066 | 111
391 | 592177 | 592288 | 592399 | 592509 | 592621 | 592732 | 592843 | 592954 | 593064 | 593175 | 111
392 | 593286 | 593397 | 593508 | 593618 | 593729 | 593839 | 593950 | 594061 | 594171 | 594282 | 111
393 | 594393 | 594503 | 594614 | 594724 | 594834 | 594945 | 595055 | 595165 | 595276 | 595386 | 110
394 | 595496 | 595606 | 595717 | 595827 | 595937 | 596047 | 596157 | 596267 | 596377 | 596487 | 110
395 | 596597 | 596707 | 596817 | 596927 | 597037 | 597146 | 597256 | 597366 | 597476 | 597586 | 110
396 | 597695 | 597805 | 597914 | 598024 | 598134 | 598243 | 598353 | 598462 | 598572 | 598681 | 110
397 | 598790 | 598899 | 599009 | 599119 | 599228 | 599337 | 599446 | 599556 | 599665 | 599774 | 109
398 | 599883 | 599992 | 600101 | 600210 | 600319 | 600428 | 600537 | 600646 | 600755 | 600864 | 109
399 | 690073 | 601082 | 601191 | 601299 | 601408 | 601517 | 601625 | 601734 | 601843 | 601951 | 109
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
400 | 602059 | 602169 | 602277 | 602386 | 602494 | 602603 | 602817 | 602819 | 602928 | 603036 | 108
401 | 603144 | 603253 | 603361 | 603469 | 603577 | 603686 | 603794 | 603902 | 604009 | 604118 | 108
402 | 604226 | 604334 | 604442 | 604550 | 604658 | 604766 | 604874 | 604982 | 605089 | 605197 | 108
403 | 605305 | 605413 | 605521 | 605628 | 605736 | 605844 | 605951 | 606059 | 606166 | 606274 | 108
404 | 606381 | 606489 | 606596 | 606704 | 606811 | 606919 | 607026 | 607133 | 607241 | 607348 | 107
405 | 607455 | 607562 | 607669 | 607777 | 607884 | 607991 | 608098 | 608205 | 608312 | 608419 | 107
406 | 608526 | 608633 | 608739 | 608847 | 608954 | 609061 | 609167 | 609274 | 609381 | 609488 | 107
407 | 609594 | 609701 | 609808 | 609914 | 610021 | 610128 | 610234 | 610341 | 610447 | 610555 | 107
408 | 610660 | 610767 | 610873 | 610979 | [...] | 611192 | 611298 | 611405 | 611511 | 611617 | 106
409 | 611723 | 611829 | 611936 | 612042 | 612148 | 612254 | 612359 | 612466 | 612572 | 612678 | 106
410 | 612784 | 612889 | 612996 | 613102 | 613207 | 613313 | 613419 | 613525 | 613630 | 613736 | 106
411 | 613842 | 613947 | 614053 | 614159 | 614264 | 614369 | 614475 | 614581 | 614686 | 614792 | 106
412 | 614897 | 615003 | 615108 | 615213 | 615319 | 615424 | 615529 | 615634 | 615739 | 615845 | 105
413 | 615950 | 616055 | 616160 | 616265 | 616370 | 616476 | 616581 | 616686 | 616790 | 616895 | 105
414 | 617000 | 617105 | 617210 | 617315 | 617419 | 617525 | 617629 | 617734 | 617839 | 617943 | 105
415 | 618048 | 618153 | 618257 | 618362 | 618466 | 616571 | 618676 | 618780 | 618889 | 618989 | 105
416 | 619093 | 619198 | 619302 | 619406 | 619511 | 619615 | 619719 | 619824 | 619928 | 620032 | 104
417 | 620136 | 620240 | 620344 | 620448 | 620552 | 620656 | 620760 | 620864 | 620968 | 621072 | 104
418 | 621176 | 621280 | 621384 | 621448 | 621592 | 621695 | 621799 | 621902 | 622007 | 622110 | 104
419 | 622214 | 622318 | 622421 | 622525 | 622628 | 622722 | 622835 | 622939 | 623042 | 623146 | 104
420 | 623249 | 623353 | 623456 | 623559 | 623663 | 623766 | 623869 | 623973 | 624076 | 624179 | 103
421 | 624282 | 624385 | 624488 | 625591 | 624695 | 624798 | 624901 | 625004 | 625107 | 625209 | 103
422 | 625312 | 625415 | 625518 | 625621 | 625724 | 625827 | 625929 | 626032 | 626135 | 626237 | 103
423 | 626340 | 626443 | 626546 | 626648 | 626751 | 626853 | 626956 | 627058 | 627161 | 627263 | 103
424 | 627366 | 627468 | 627571 | 627673 | 627775 | 627878 | 627979 | 628082 | 628185 | 628287 | 102
425 | 628389 | 628491 | 628593 | 628695 | 628797 | 628899 | 629002 | 629104 | 629206 | 629308 | 102
426 | 629409 | 629512 | 629613 | 629715 | 629817 | 629919 | 620021 | 630123 | 630224 | 630326 | 102
427 | 630428 | 630529 | 630631 | 630733 | 630835 | 630936 | 631038 | 631139 | 631241 | 631342 | 102
428 | 631444 | 631545 | 631647 | 631746 | 631849 | 631951 | 632051 | 632153 | 632255 | 632356 | 101
429 | 632457 | 632559 | 632659 | 632761 | 632862 | 632963 | 633064 | 633165 | 633266 | 633367 | 101
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
430 | 633468 | 633569 | 633670 | 633771 | 633872 | 633973 | 634075 | 634075 | 634276 | 634376 | 100
431 | 634477 | 634578 | 634679 | 634779 | 634880 | 634981 | 635081 | 635182 | 635283 | 635383 | 100
432 | 635484 | 635584 | 635685 | 635785 | 635886 | 635986 | 636087 | 636187 | 636288 | 636388 | 100
433 | 636488 | 636588 | 636688 | 636789 | 636889 | 636989 | 637089 | 637189 | 637289 | 637389 | 100
434 | 637489 | 637589 | 637689 | 637789 | 637889 | 637989 | 638089 | 638189 | 638289 | 638389 | 99
435 | 638489 | 638589 | 638689 | 638789 | 638888 | 638988 | 639088 | 639188 | 639287 | 639387 | 99
436 | 639486 | 639586 | 639686 | 639785 | 639885 | 639984 | 640084 | 640183 | 640283 | 640382 | 99
437 | 640481 | 640581 | 640680 | 640779 | 640879 | 640978 | 641077 | 641177 | 641276 | 641375 | 99
438 | 641475 | 641573 | 641672 | 641771 | 641871 | 641969 | 642069 | 642168 | 642267 | 642366 | 99
439 | 642465 | 642563 | 642662 | 642761 | 642860 | 642959 | 643058 | 643156 | 643255 | 643354 | 99
440 | 643453 | 643551 | 643650 | 643749 | 643847 | 643946 | 644044 | 644143 | 644242 | 644340 | 98
441 | 644439 | 644537 | 644636 | 644734 | 644833 | 644931 | 645029 | 645127 | 645226 | 645324 | 98
442 | 645422 | 645521 | 645619 | 645717 | 645815 | 645913 | 646011 | 646109 | 646208 | 646306 | 98
443 | 646404 | 646502 | 646599 | 646698 | 646796 | 646894 | 646992 | 647089 | 647187 | 647285 | 98
444 | 647383 | 647481 | 647579 | 647676 | 647774 | 647872 | 647969 | 648067 | 648165 | 648262 | 98
445 | 648360 | 648458 | 648555 | 648653 | 648750 | 648848 | 648945 | 649043 | 649140 | 649237 | 97
446 | 649335 | 649432 | 649529 | 649627 | 649724 | 649821 | 649919 | 650016 | 650113 | 650210 | 97
447 | 650308 | 650405 | 650502 | 650599 | 650696 | 650793 | 650890 | 650987 | 651084 | 651181 | 97
448 | 651278 | 651375 | 651472 | 651569 | 651666 | 651762 | 651859 | 651956 | 652053 | 652149 | 97
449 | 652246 | 652343 | 652439 | 652536 | 652633 | 652729 | 652826 | 652923 | 653019 | 653116 | 97
450 | 653213 | 653309 | 653405 | 653502 | 653598 | 653695 | 653791 | 653888 | 653984 | 654080 | 96
451 | 654177 | 654273 | 654369 | 654465 | 654562 | 654658 | 654754 | 654850 | 654946 | 655042 | 96
452 | 655138 | 655235 | 655331 | 655427 | 655523 | 655619 | 655715 | 655810 | 655906 | 656002 | 96
453 | 656098 | 656194 | 656289 | 656386 | 656482 | 656577 | 656673 | 656769 | 656864 | 656960 | 96
454 | 657056 | 657152 | 657247 | 657343 | 657438 | 657534 | 657629 | 657725 | 657820 | 657916 | 96
455 | 658011 | 658107 | 658202 | 658298 | 658393 | 658488 | 658584 | 658679 | 658774 | 658869 | 95
456 | 658965 | 659060 | 659155 | 659250 | 659346 | 659441 | 659536 | 659630 | 659726 | 659821 | 95
457 | 659916 | 660011 | 660106 | 660201 | 660296 | 660391 | 660486 | 660581 | 660676 | 660771 | 95
458 | 660865 | 660960 | 661055 | 661149 | 661245 | 661339 | 661434 | 661529 | 661623 | 661718 | 95
459 | 661813 | 661907 | 662002 | 662096 | 662191 | 662289 | 662380 | 662475 | 662569 | 662663 | 95
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
460 | 662758 | 662852 | 662947 | 663041 | 663135 | 663229 | 663324 | 663418 | 663512 | 663607 | 94
461 | 663701 | 663795 | 663889 | 663983 | 664078 | 664172 | 664266 | 664359 | 664454 | 664548 | 94
462 | 664642 | 664736 | 664829 | 664924 | 665018 | 665112 | 665206 | 665299 | 665393 | 665487 | 94
463 | 665581 | 665675 | 665769 | 665862 | 665956 | 666049 | 666143 | 666237 | 666331 | 666424 | 94
464 | 666518 | 666611 | 666705 | 666799 | 666892 | 666986 | 667079 | 667173 | 667266 | 667359 | 94
465 | 667453 | 667546 | 667639 | 667733 | 667826 | 667919 | 668013 | 668106 | 668199 | 668293 | 93
466 | 668386 | 668479 | 668572 | 668665 | 668759 | 668852 | 668945 | 669038 | 669131 | 669224 | 93
467 | 669317 | 669409 | 669503 | 669596 | 669689 | 669782 | 669875 | 669967 | 670060 | 670153 | 93
468 | 670246 | 670339 | 670431 | 670524 | 670617 | 670709 | 670802 | 670895 | 670988 | 671080 | 93
469 | 671173 | 671265 | 671358 | 671451 | 671543 | 671636 | 671728 | 671821 | 671913 | 672005 | 93
470 | 672098 | 672190 | 672283 | 672375 | 672467 | 672559 | 672652 | 672744 | 672836 | 672929 | 92
471 | 673021 | 673113 | 673205 | 673297 | 673389 | 673482 | 673574 | 673666 | 673758 | 673849 | 92
472 | 673942 | 674034 | 674126 | 674218 | 674309 | 674402 | 674494 | 674586 | 674677 | 674769 | 92
473 | 674861 | 674953 | 675045 | 675137 | 675228 | 675319 | 675412 | 675503 | 675595 | 675687 | 92
474 | 675778 | 675869 | 675962 | 676053 | 676145 | 676236 | 676328 | 676419 | 676511 | 676602 | 92
475 | 676694 | 676785 | 676876 | 676968 | 677059 | 677151 | 677242 | 677333 | 677424 | 677516 | 91
476 | 677607 | 677698 | 677789 | 677881 | 677972 | 678063 | 678154 | 678245 | 678335 | 678427 | 91
477 | 678518 | 678609 | 678700 | 678791 | 678882 | 678973 | 679064 | 679155 | 679246 | 679337 | 91
478 | 679428 | 679519 | 679609 | 679700 | 679791 | 679882 | 679972 | 680063 | 680154 | 680245 | 91
479 | 680336 | 680426 | 680517 | 680607 | 680698 | 680789 | 680879 | 680969 | 681060 | 681151 | 91
480 | 681241 | 681332 | 681422 | 681513 | 681603 | 681693 | 681784 | 681874 | 681964 | 682055 | 90
481 | 682145 | 682235 | 682326 | 682416 | 682506 | 682596 | 682686 | 682777 | 682867 | 682957 | 90
482 | 683047 | 683137 | 683227 | 683317 | 683407 | 683497 | 683587 | 683677 | 683767 | 683857 | 90
483 | 683947 | 684037 | 684127 | 684217 | 684307 | 684396 | 684486 | 684576 | 684666 | 684756 | 90
484 | 684845 | 684935 | 685025 | 685114 | 685204 | 685294 | 685383 | 685473 | 685563 | 685652 | 90
485 | 685742 | 685831 | 685921 | 686010 | 686099 | 682189 | 686279 | 686368 | 686458 | 686547 | 89
486 | 686636 | 686326 | 686815 | 686904 | 686994 | 687083 | 687172 | 687211 | 687351 | 687439 | 89
487 | 687529 | 687618 | 687707 | 687796 | 687885 | 687975 | 688054 | 688153 | 688242 | 688331 | 89
488 | 688419 | 688509 | 688598 | 688687 | 688776 | 688865 | 688953 | 689042 | 689131 | 689220 | 89
489 | 689309 | 689398 | 689486 | 689575 | 687664 | 689753 | 689841 | 689930 | 690019 | 690007 | 89
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
490 | 690196 | 690285 | 690373 | 690462 | 690550 | 690639 | 690728 | 690816 | 690905 | 690993 | 89
491 | 691081 | 691169 | 691258 | 691347 | 691435 | 691524 | 691612 | 691700 | 691789 | 691877 | 88
492 | 691965 | 692053 | 692142 | 692229 | 692318 | 692406 | 692494 | 692583 | 692671 | 692759 | 88
493 | 692847 | 692935 | 693023 | 693111 | 693199 | 693287 | 693375 | 693463 | 693551 | 693639 | 88
494 | 693727 | 693815 | 693903 | 693991 | 694078 | 694166 | 694254 | 694342 | 694429 | 694517 | 88
495 | 694605 | 694693 | 694781 | 694868 | 694956 | 695044 | 695131 | 695219 | 695307 | 695394 | 88
496 | 695482 | 695569 | 695657 | 695744 | 695832 | 695919 | 696007 | 696094 | 696182 | 696269 | 87
497 | 696356 | 696444 | 696531 | 696618 | 696706 | 696793 | 696880 | 696968 | 697055 | 697142 | 87
498 | 697229 | 697317 | 697404 | 697491 | 697578 | 697665 | 697752 | 697839 | 697926 | 698014 | 87
499 | 698101 | 698188 | 698275 | 698362 | 698449 | 698535 | 698622 | 698709 | 698706 | 698883 | 87
500 | 698970 | 699057 | 699144 | 699281 | 699317 | 699404 | 699491 | 699528 | 699664 | 699751 | 87
501 | 699838 | 699924 | 700011 | 700098 | 700184 | 700271 | 700358 | 700444 | 705031 | 700617 | 87
502 | 700704 | 700790 | 700877 | 700963 | 701049 | 701136 | 701222 | 701309 | 701395 | 701482 | 86
503 | 701568 | 701654 | 701741 | 701827 | 701913 | 701999 | 702086 | 702172 | 702258 | 702344 | 86
504 | 702430 | 702517 | 702603 | 702689 | 702775 | 702861 | 702947 | 703033 | 703119 | 703205 | 86
505 | 703291 | 793377 | 703463 | 703549 | 703635 | 703721 | 703807 | 703893 | 703979 | 704065 | 86
506 | 704151 | 704236 | 704322 | 704408 | 704494 | 704579 | 704665 | 704751 | 704837 | 704922 | 86
507 | 705008 | 705094 | 705179 | 705265 | 705350 | 705436 | 705522 | 705607 | 705693 | 705778 | 86
508 | 705863 | 705949 | 706035 | 706130 | 706206 | 706291 | 706376 | 706462 | 706547 | 706632 | 85
509 | 706718 | 706803 | 706888 | 706974 | 707059 | 707144 | 707219 | 707315 | 707399 | 707485 | 85
510 | 707570 | 707655 | 707740 | 707826 | 707911 | 707996 | 708081 | 708166 | 708251 | 708336 | 85
511 | 708421 | 708506 | 708591 | 708676 | 708761 | 708846 | 708931 | 709015 | 709100 | 709185 | 85
512 | 709269 | 709355 | 709439 | 709524 | 709609 | 709694 | 709789 | 709863 | 709948 | 710033 | 85
513 | 710117 | 710202 | 710287 | 710371 | 710456 | 710540 | 710625 | 710709 | 710794 | 710879 | 85
514 | 710963 | 711048 | 711132 | 711217 | 711301 | 711385 | 711469 | 711554 | 711639 | 711723 | 84
515 | 711807 | 711891 | 711976 | 712060 | 712144 | 712229 | 712313 | 712397 | 712481 | 712566 | 84
516 | 712649 | 712734 | 712818 | 712902 | 712986 | 713070 | 713154 | 713238 | 713223 | 713407 | 84
517 | 713491 | 713575 | 713659 | 713742 | 713826 | 713910 | 713994 | 714078 | 714162 | 714246 | 84
518 | 714329 | 714414 | 714497 | 714581 | 714665 | 714749 | 714833 | 714916 | 714999 | 715084 | 84
519 | 715167 | 715251 | 715335 | 715418 | 715501 | 715586 | 715669 | 715753 | 715836 | 715919 | 84
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
520 | 716003 | 716087 | 716170 | 716254 | 716337 | 716421 | 716504 | 716588 | 716671 | 716754 | 83
521 | 716838 | 716921 | 717004 | 717088 | 717171 | 717254 | 717338 | 717421 | 717509 | 717587 | 83
522 | 717671 | 717754 | 717837 | 717920 | 718003 | 718086 | 718169 | 718253 | 718336 | 718419 | 83
523 | 718502 | 718585 | 718668 | 718751 | 718834 | 718917 | 718999 | 719083 | 719165 | 719248 | 83
524 | 719331 | 719414 | 719497 | 719579 | 719663 | 719745 | 719828 | 719911 | 719994 | 720077 | 83
525 | 720159 | 720242 | 720325 | 720407 | 720490 | 720573 | 720655 | 720738 | 720821 | 720903 | 83
526 | 720986 | 721068 | 721151 | 721233 | 721316 | 721398 | 721481 | 721563 | 721646 | 721728 | 82
527 | 721811 | 721893 | 721975 | 722058 | 722140 | 722222 | 722305 | 722387 | 722469 | 722552 | 82
528 | 722634 | 722716 | 722798 | 722881 | 722963 | 723045 | 723127 | 723209 | 723291 | 723374 | 82
529 | 723456 | 723538 | 723619 | 723702 | 723784 | 723866 | 723948 | 724029 | 724112 | 724194 | 82
530 | 724276 | 724358 | 724439 | 724522 | 724604 | 724685 | 724767 | 724849 | 724931 | 725013 | 82
531 | 725095 | 725176 | 725258 | 725339 | 725422 | 725503 | 725585 | 725667 | 725748 | 725829 | 82
532 | 725912 | 725993 | 726075 | 726156 | 726238 | 726319 | 726401 | 726483 | 726564 | 726646 | 82
533 | 726727 | 726809 | 726890 | 726972 | 727053 | 727134 | 727216 | 727297 | 727379 | 727459 | 81
534 | 727541 | 727623 | 727704 | 727785 | 727866 | 727948 | 728029 | 728110 | 728191 | 728273 | 81
535 | 728354 | 728435 | 728516 | 728597 | 728678 | 728759 | 728841 | 728922 | 729003 | 729084 | 81
536 | 729165 | 729246 | 729327 | 729408 | 729489 | 729569 | 729651 | 729732 | 729813 | [...] | 81
537 | 729974 | 730055 | 730136 | 730117 | 730298 | 730378 | 730459 | 730540 | 730621 | 730702 | 81
538 | 730782 | 730863 | 730944 | 731024 | 731105 | 731186 | 731266 | 731347 | 731423 | 731508 | 81
539 | 731589 | 731669 | 731749 | 731830 | 731911 | 731991 | 732072 | 732152 | 732233 | 732313 | 81
540 | 732394 | 732474 | 732555 | 732635 | 732715 | 732796 | 732876 | 732956 | 733037 | 733117 | 80
541 | 733197 | 733278 | 733358 | 733438 | 733518 | 733598 | 733679 | 733759 | 733839 | 733919 | 80
542 | 733999 | 734079 | 734159 | 734239 | 734319 | 734399 | 734479 | 734559 | 734639 | 734719 | 80
543 | 734799 | 734879 | 734959 | 735039 | 735119 | 735199 | 735279 | 735359 | 735439 | 735519 | 80
544 | 735599 | 735679 | 735759 | 735838 | 735918 | 735998 | 736078 | 736157 | 736237 | 736317 | 80
545 | 736397 | 736476 | 736556 | 736635 | 736715 | 736795 | 736874 | 736954 | 737034 | 737113 | 80
546 | 737191 | 737272 | 737352 | 737431 | 737511 | 737590 | 737669 | 737749 | 737829 | 737908 | 79
547 | 737987 | 738067 | 738146 | 738225 | 738305 | 738384 | 738463 | 738543 | 738622 | 738701 | 79
548 | 738781 | 738859 | 738939 | 739018 | 739097 | 739177 | 739259 | 739335 | 739414 | 739493 | 79
549 | 739572 | 739651 | 739731 | 739809 | 739889 | 739968 | 740047 | 740126 | 740205 | 740284 | 79
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
550 | 740363 | 740442 | 740521 | 740599 | 740678 | 740757 | 740836 | 740915 | 740994 | 741073 | 79
551 | 741152 | 741230 | 741309 | 741388 | 741467 | 741546 | 741624 | 741703 | 741782 | 741860 | 79
552 | 741939 | 742018 | 742096 | 742175 | 742254 | 742332 | 742411 | 742489 | 742568 | 742647 | 79
553 | 742725 | 742802 | 742882 | 742961 | 743039 | 743118 | 743196 | 743275 | 743353 | 743431 | 78
554 | 743509 | 743588 | 743667 | 743745 | 743823 | 743902 | 743979 | 744058 | 744136 | 744215 | 78
555 | 744293 | 744371 | 744449 | 744528 | 744606 | 744684 | 744764 | 744840 | 744919 | 744997 | 78
556 | 745075 | 745153 | 745231 | 745309 | 745387 | 745465 | 745543 | 745621 | 745699 | 745777 | 78
557 | 745855 | 745933 | 746011 | 746089 | 746167 | 746245 | 746323 | 746401 | 746479 | 746556 | 78
558 | 746634 | 746712 | 746789 | 746868 | 746945 | 747023 | 747101 | 747179 | 747256 | 747334 | 78
559 | 747412 | 747489 | 747567 | 747645 | 747722 | 747800 | 747878 | 747955 | 748033 | 748110 | 78
560 | 748188 | 748266 | 748343 | 748421 | 748498 | 748576 | 748653 | 748731 | 748808 | 748885 | 77
561 | 748963 | 749040 | 749118 | 749195 | 749272 | 749349 | 749427 | 749504 | 749582 | 749659 | 77
562 | 749736 | 749814 | 749891 | 749968 | 750045 | 750123 | 750199 | 750277 | 750354 | 750431 | 77
563 | 750508 | 750586 | 750663 | 750739 | 750817 | 750894 | 750971 | 751048 | 751125 | 751202 | 77
564 | 751279 | 751356 | 751433 | 751510 | 751587 | 751664 | 751741 | 751818 | 751895 | 751972 | 77
565 | 752048 | 752125 | 752202 | 752279 | 752356 | 752433 | 752509 | 752586 | 752663 | 752739 | 77
566 | [...] | 752893 | 752969 | 753047 | 753123 | 753199 | 753277 | 753353 | 753429 | 753506 | 77
567 | 753583 | 753659 | 753736 | 753813 | 753889 | 753966 | 754042 | 754119 | 754195 | 754272 | 77
568 | 754348 | 755425 | 754501 | 754578 | 754654 | 754730 | 754800 | 754883 | 754959 | 755036 | 76
569 | 755112 | 755189 | 755265 | 755341 | 755417 | 755494 | 755569 | 755646 | 755722 | 755799 | 76
570 | 755875 | 755951 | 756027 | 756103 | 756179 | 756256 | 756332 | 756408 | 756484 | 756560 | 76
571 | 756636 | 756712 | 756788 | 756864 | 756940 | 757016 | 757092 | 757167 | 757244 | 757320 | 76
572 | 757396 | 757472 | 757548 | 757627 | 757699 | 757775 | 757851 | 757927 | 758003 | 758079 | 76
573 | 758155 | 758230 | 758306 | 758382 | 758458 | 758533 | 758609 | 758685 | 758761 | 758836 | 76
574 | 758912 | 758988 | 759063 | 759139 | 759214 | 759290 | 759366 | 759411 | 759517 | 759592 | 76
575 | 759668 | 759743 | 759819 | 759894 | 759969 | 760045 | 760121 | 760196 | 760272 | 760347 | 75
576 | 760422 | 760498 | 760573 | 760649 | 760723 | 760799 | 760875 | 760949 | 761025 | [...] | 75
577 | 761176 | 761251 | 761326 | 761402 | 761477 | 761552 | 761627 | 761702 | 761778 | 761853 | 75
578 | 761928 | 762003 | 762078 | 762153 | 762228 | 762303 | 762378 | 762453 | 762529 | 762604 | 75
579 | 762679 | 762754 | 762829 | 762904 | 762978 | 763053 | 763128 | 763203 | 763279 | 763353 | 75
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
580 | 763428 | 763503 | 763578 | 763653 | 763727 | 763802 | 763877 | 763952 | 764027 | 764101 | 75
581 | 764176 | 764251 | 764326 | 764400 | 764475 | 764549 | 764624 | 764699 | 764774 | 764848 | 75
582 | 764923 | 764998 | 765072 | 765147 | 765221 | 765296 | 765370 | 765445 | 765519 | 765594 | 75
583 | 765669 | 765743 | 765818 | 765892 | 765966 | 766041 | 766115 | 766189 | 766264 | 766338 | 74
584 | 766413 | 766487 | 766562 | 766636 | 766710 | 766785 | 766859 | 766933 | 767007 | 767082 | 74
585 | 767156 | 767230 | 767304 | 767379 | 767453 | 767527 | 767601 | 767675 | 767749 | 767823 | 74
586 | 767898 | 767972 | 768046 | 768119 | 768194 | 768268 | 768342 | 768416 | 768490 | [...] | 74
587 | 768638 | 768712 | 768786 | 768860 | 768934 | 769008 | 769082 | 769156 | 769229 | 769303 | 74
588 | 769377 | 769451 | 769525 | 769599 | 769673 | 769746 | 799820 | 769894 | 769968 | 770042 | 74
589 | 770115 | 770189 | 770263 | 770336 | 770410 | 770484 | 770557 | 770631 | 770705 | 770778 | 74
590 | 770852 | 770926 | 770999 | 771073 | 771146 | 771219 | 771293 | 771367 | 771440 | 771514 | 74
591 | 771587 | 761661 | 771734 | 771808 | 771881 | 771955 | 772028 | 772102 | 772175 | 772248 | 73
592 | 772322 | 772395 | 772468 | 772542 | 772615 | 772688 | 772762 | 772835 | 772908 | 772981 | 73
593 | 773055 | 773128 | 773201 | 773274 | 773348 | 773421 | 773494 | 773567 | 773640 | 773713 | 73
594 | 773786 | 773859 | 773933 | 774006 | 774079 | 774152 | 774225 | 774298 | 774371 | 774444 | 73
595 | 774517 | 774589 | 774663 | 774736 | 774809 | 774882 | 774955 | 775028 | 775100 | 775173 | 73
596 | 775246 | 775319 | 775392 | 775465 | 775538 | 775610 | 775683 | 775756 | 775829 | 775902 | 73
597 | 775974 | 776047 | 776119 | 776193 | 776265 | 776338 | 776411 | 776483 | 776556 | 776629 | 73
598 | 776701 | 776774 | 776846 | 776919 | 776992 | 777064 | 777137 | 777209 | 777282 | 777354 | 73
599 | 777427 | 777499 | 777572 | 777644 | 777717 | 777789 | 777862 | 777934 | 778006 | 778079 | 72
600 | 778151 | 778224 | 778296 | 778368 | 778441 | 778513 | 778585 | 778658 | 778729 | 778802 | 72
601 | 778874 | 778947 | 779019 | 779091 | 779163 | 779236 | 779308 | 779380 | 779452 | 779524 | 72
602 | 779596 | 779669 | 779741 | 779813 | 779884 | 779957 | 780029 | 780101 | 780173 | 780245 | 72
603 | 780317 | 780389 | 780461 | 780533 | 780805 | 780677 | 780749 | 780821 | 780893 | 780965 | 72
604 | 781037 | 781109 | 781181 | 781253 | 781324 | 781396 | 781468 | 781539 | 781612 | 781684 | 72
605 | 781755 | 781827 | 781899 | 781971 | 782042 | 782114 | 782186 | 782258 | 782329 | 782401 | 72
606 | 782473 | 782544 | 782616 | 782688 | 782759 | 782831 | 782902 | 782974 | 783046 | 783117 | 72
607 | 783189 | 783260 | 783332 | 783403 | 783475 | 783546 | 783618 | 783689 | 783761 | 783832 | 71
608 | 783904 | 783975 | 784046 | 784118 | 784189 | 784261 | 784332 | 784403 | 784475 | 784546 | 71
609 | 784617 | 784689 | 784759 | 784831 | 784902 | 784974 | 785045 | 785116 | 785187 | 785259 | 71
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
610 | 785329 | 785401 | 785472 | 785543 | 785615 | 785686 | 785757 | 785828 | 785899 | 785970 | 71
611 | 786041 | 786112 | 786183 | 786254 | 786325 | 786396 | 786467 | 786538 | 786609 | 786680 | 71
612 | 786751 | 786822 | 786893 | 786964 | 787035 | 787106 | 787177 | 787248 | 787319 | 787389 | 71
613 | 787460 | 787531 | 787602 | 787673 | 787744 | 787815 | 787885 | 787956 | 788027 | 788098 | 71
614 | 788164 | 788239 | 788309 | 788381 | 788451 | 788522 | 788593 | 788663 | 788734 | 788804 | 71
615 | 788875 | 788946 | 789016 | 789087 | 789157 | 789228 | 789299 | 789369 | 789439 | 789510 | 71
616 | 789581 | 789651 | 789722 | 789792 | 789863 | 789933 | 790004 | 790074 | 790144 | 790215 | 70
617 | 790285 | 790356 | 790426 | 790496 | 790567 | 790637 | 790707 | 790778 | 790848 | 790918 | 70
618 | 790988 | 791059 | 791129 | 791199 | 791269 | 791339 | 791409 | 791480 | 791550 | 791620 | 70
619 | 791691 | 791761 | 791831 | 791901 | 791971 | 792041 | 792111 | 792181 | 792252 | 792322 | 70
620 | 792392 | 792462 | 792532 | 792602 | 792672 | 792742 | 792812 | 792882 | 792952 | 793022 | 70
621 | 793092 | 793162 | 793231 | 793301 | 793371 | 793441 | 793511 | 793581 | 793651 | 793721 | 70
622 | 793791 | 793860 | 793930 | 793999 | 794069 | 794139 | 794209 | 794279 | 794349 | 794418 | 70
623 | 794488 | 794558 | 794627 | 794697 | 794767 | 794836 | 794906 | [...] | 795045 | 795115 | 70
624 | 795185 | 795254 | 795324 | 795393 | 795463 | 795532 | 795602 | 795672 | 795741 | 795810 | 70
625 | 795880 | 795949 | 796019 | 796088 | 796158 | 796227 | 796297 | 796366 | 796436 | 796505 | 69
626 | 796574 | 796644 | 796713 | 796782 | 796852 | 796921 | 796990 | 797059 | 797129 | 797198 | 69
627 | 797268 | 797337 | 797406 | 797475 | 797545 | 797614 | 797683 | 797752 | 797821 | 797890 | 69
628 | 797959 | 798029 | 798098 | 798167 | 798236 | 798305 | 798374 | 798443 | 798513 | 798582 | 69
629 | 798651 | 798719 | 798789 | 798858 | 798927 | 798996 | 799065 | 799134 | 799203 | 799272 | 69
630 | 799341 | 799409 | 799478 | 799547 | 799616 | 799685 | 799754 | 799823 | 799892 | 799961 | 69
631 | 800029 | 800098 | 800167 | 800236 | 800305 | 800373 | 800442 | 800511 | 800579 | 800648 | 69
632 | 800717 | 800786 | 800854 | 800923 | 800992 | 801061 | 801129 | 801198 | 801266 | 801335 | 69
633 | 801404 | 801472 | 801541 | 801609 | 801678 | 801747 | 801815 | 801884 | 801952 | 802021 | 69
634 | 802089 | 802158 | 802226 | 802295 | 802363 | 802432 | 802500 | 802568 | 802637 | 802705 | 69
635 | 802774 | 802842 | 802910 | 802979 | 803047 | 803116 | 803184 | 803252 | 803321 | 803389 | 68
636 | 803457 | 803525 | 803594 | 803662 | 803730 | 803798 | 803867 | 803935 | 804003 | 804071 | 68
637 | 804139 | 804208 | 804276 | 804344 | 804412 | 804480 | 804548 | 804616 | 804685 | 804753 | 68
638 | 804821 | 804889 | 804957 | 805025 | 805093 | 805161 | 805229 | 805297 | 805365 | 805433 | 68
639 | 805501 | 805569 | 805637 | 805705 | 805773 | 805841 | 805908 | 805976 | 806044 | 806112 | 68
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
640 | 806179 | 806248 | 806316 | 806384 | 806451 | 806519 | 806587 | 806655 | 806723 | 806790 | 68
641 | 806858 | 806926 | 806994 | 807061 | 807129 | 807197 | 807264 | 807332 | 807399 | 807467 | 68
642 | 807535 | 807603 | 807670 | 807738 | 807806 | 807873 | 807941 | 808008 | 808078 | 808143 | 68
643 | 808211 | 808279 | 808346 | 808414 | 808481 | 808549 | [...] | 808684 | 808751 | 808818 | 67
644 | 808886 | 808953 | 809021 | 809088 | 809156 | 809223 | 809298 | 809358 | 809425 | 809492 | 67
645 | 809559 | 809627 | 809694 | 809762 | 809829 | 809866 | 809964 | 810031 | 810098 | [...] | 67
646 | 810233 | 810299 | 810367 | 810434 | 810501 | 810569 | 810636 | 810703 | 810778 | 810837 | 67
647 | 810904 | 810971 | 811039 | 811106 | 811173 | 811239 | 811307 | 811374 | 811441 | 811508 | 67
648 | 811575 | 811642 | 811709 | 811776 | 811843 | 811909 | 811977 | 812044 | 812111 | 812178 | 67
649 | 812245 | 812312 | 812379 | 812445 | 812512 | 812579 | 812646 | 812713 | 812779 | 812847 | 67
650 | 812913 | 812980 | 813047 | 813114 | 813181 | 813247 | 813314 | 813381 | 813448 | 813514 | 67
651 | 813581 | 813648 | 813714 | 813781 | 813848 | 813914 | 813981 | 814048 | 814114 | 814181 | 67
652 | 814248 | 814314 | 814381 | 814447 | 814514 | 814581 | 814647 | 814714 | 814780 | 814847 | 67
653 | 814913 | 814979 | 815046 | 815113 | 815129 | [...] | 815312 | 815378 | 815445 | 815511 | 66
654 | 815578 | 815644 | 815711 | 815777 | 815843 | 815909 | 815978 | 816042 | 816109 | 816175 | 66
655 | 816241 | 816308 | 816374 | 816440 | 816506 | 816573 | 816639 | 816705 | 816771 | 816838 | 66
656 | 816904 | 816910 | 817036 | 817102 | 817169 | 817235 | 817301 | 817367 | 817433 | 817499 | 66
657 | 817565 | 817631 | 817698 | 817764 | 817829 | 817896 | 817962 | 818028 | 818094 | 818159 | 66
658 | 818226 | 818292 | 818358 | 818424 | 818489 | 818556 | 818622 | 818688 | 818754 | 818819 | 66
659 | 818885 | 818951 | 819017 | 819083 | 819149 | 819215 | 819281 | 819346 | 819412 | 819478 | 66
660 | 819543 | 819609 | 819676 | 819741 | 819807 | 819873 | 819939 | 820004 | 820070 | 820136 | 66
661 | 820201 | 820267 | 820333 | 820399 | 820464 | 820529 | 820595 | 820661 | 820727 | 820792 | 66
662 | 820858 | 820924 | 820989 | 821055 | 821120 | 821186 | 821251 | 821317 | 821382 | [...] | 66
663 | 821514 | 821509 | 821645 | 821709 | 821775 | 821841 | 821906 | 821972 | 822037 | 822103 | 65
664 | 822168 | 822233 | 822299 | 822364 | 822429 | 822495 | 822560 | 822626 | 822691 | 822756 | 65
665 | 822822 | 822887 | 822952 | 823018 | 823083 | 823148 | 823213 | 823279 | 823344 | 823409 | 65
666 | 823474 | 823539 | 823605 | 823669 | 823735 | 823800 | 823865 | 823930 | 823996 | 824061 | 65
667 | 824126 | 824191 | 824256 | 824321 | 824386 | 824451 | 824516 | 824581 | 824646 | 824711 | 65
668 | 824776 | 824841 | 824906 | 824971 | 825036 | 825101 | 825166 | 825231 | 825296 | 825361 | 65
669 | 825426 | 825491 | 825556 | 825021 | 825686 | 825751 | 825815 | 825880 | 825945 | 826009 | 65
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
670 | 826075 | 826139 | 826204 | 826269 | 826334 | 826399 | 826464 | 826528 | 826593 | 826658 | 65
671 | 826723 | 826787 | 826852 | 826917 | 826981 | 827046 | 827111 | 827175 | 827239 | 827305 | 65
672 | 827369 | 827434 | 827499 | 827563 | 827628 | 827692 | 827757 | 827822 | 827886 | 827951 | 65
673 | 828015 | 828079 | 828144 | 828209 | 828273 | 828338 | 828402 | 828467 | 828531 | 828595 | 64
674 | 828659 | 828724 | 828789 | 828853 | 828918 | 828982 | 829046 | 829111 | 829175 | 829239 | 64
675 | 829304 | 829368 | 829432 | 829497 | 829561 | 829625 | 829689 | 829754 | 829818 | 829882 | 64
676 | 829947 | 830011 | 830075 | 830139 | 830204 | 830268 | 830332 | 830396 | 830460 | 830525 | 64
677 | 830589 | 830653 | 830717 | 830781 | 830845 | 830909 | 830973 | 831037 | 831102 | 831166 | 64
678 | 831229 | 831294 | 831358 | 831422 | 831486 | 831549 | 831614 | 831678 | 831742 | 831806 | 64
679 | 831869 | 831934 | 831998 | 832062 | 832126 | 833189 | 832253 | 832317 | 832381 | 832445 | 64
680 | 832509 | 832573 | 832637 | 832700 | 832764 | 832828 | 832892 | 832956 | 833019 | 833083 | 64
681 | 833147 | 833211 | 833275 | 833338 | 833402 | 833466 | 833529 | 833593 | 833657 | 833721 | 64
682 | 833784 | 833848 | 833912 | 833975 | 834039 | 834103 | 834166 | 834229 | 834294 | 834357 | 64
683 | 834421 | 834484 | 834548 | 834611 | 834675 | 834739 | 834802 | 834866 | 834929 | 834993 | 64
684 | 835056 | 835119 | 835183 | 835247 | 835310 | 835373 | 835437 | 835500 | 835564 | 835627 | 63
685 | 835691 | 835754 | 835817 | 835881 | 835944 | 836007 | 836071 | 836134 | 836197 | 836261 | 63
686 | 836324 | 836387 | 836451 | 836514 | 836577 | 836641 | 836704 | 836767 | 836830 | 836894 | 63
687 | 836957 | 837019 | 837083 | 837146 | 837209 | 837273 | 837339 | 837399 | 837462 | 837525 | 63
688 | 837588 | 837652 | 837715 | 837777 | 837841 | 837904 | 837967 | 838030 | 838093 | 838156 | 63
689 | 838219 | 838282 | 838345 | 838408 | 838471 | 838534 | 838597 | 838660 | 838723 | 838786 | 63
690 | 838849 | 838912 | 838975 | 839038 | 839101 | 839164 | 839227 | 839289 | 839352 | 839415 | 63
691 | 839478 | 839541 | 839604 | 839667 | 839729 | 839792 | 839855 | 839918 | 839981 | 840043 | 63
692 | 840106 | 840169 | 840232 | 840394 | 840357 | 840419 | 840482 | 840545 | 840608 | 840671 | 63
693 | 840733 | 840796 | 840859 | 840921 | 840984 | 841049 | 841109 | 841172 | 841234 | 841297 | 63
694 | 841359 | 841423 | 841485 | 841547 | 841609 | 841672 | 841735 | 841797 | 841859 | 841922 | 63
695 | 841985 | 842047 | 842109 | 842172 | 842235 | 842297 | 842359 | 842432 | 842484 | 842547 | 62
696 | 842609 | 842672 | 842734 | 842796 | 842859 | 842921 | 842983 | 843046 | 843108 | 843170 | 62
697 | 843233 | 843295 | 843357 | 843419 | 843482 | 843544 | 843606 | 843669 | 843731 | 843293 | 62
698 | 843855 | 843918 | 843979 | 844042 | 844104 | 844166 | 844229 | 844291 | 844353 | 844415 | 62
699 | 844477 | 844539 | 844601 | 844664 | 844726 | 844788 | 844849 | 844912 | 844974 | 845036 | 62
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
700 | 845098 | 845160 | 845222 | 845284 | 845346 | 845408 | 845470 | 845532 | 845594 | 845656 | 62
701 | 845718 | 845779 | 845842 | 845904 | 845666 | 846028 | 846089 | 846151 | 846213 | 846275 | 62
702 | 846337 | 846399 | 846461 | 846523 | 846585 | 846646 | 846708 | 846769 | 846832 | 846894 | 62
703 | 846955 | 847017 | 847079 | 847141 | 847202 | 847264 | 847326 | 847388 | 847449 | 847511 | 62
704 | 847573 | 847634 | 847696 | 847758 | 847819 | 847881 | 847943 | 848004 | 848067 | 848128 | 62
705 | 848189 | 848251 | 848312 | 848374 | 848435 | 848497 | 848559 | 848620 | 848682 | 848743 | 62
706 | 848805 | 848866 | 848928 | 848989 | 849051 | 849112 | 849174 | 849235 | 849297 | 849358 | 61
707 | 849419 | 849481 | 849542 | 849604 | 849665 | 849726 | 849788 | 849849 | 849911 | 849972 | 61
708 | 850033 | 850095 | 850156 | 850217 | 850279 | 850339 | 850401 | 850462 | 850524 | 850585 | 61
709 | 850646 | 850707 | 850769 | 850829 | 850891 | 850952 | 851014 | 851075 | 851136 | 851197 | 61
710 | 851258 | 851319 | 851381 | 851442 | 851503 | 851564 | 851625 | 851686 | 851747 | 851809 | 61
711 | 851869 | 851931 | 851992 | 851053 | 852114 | 852175 | 852236 | 852297 | 852358 | 852419 | 61
712 | 852479 | 852541 | 852602 | 852663 | 852724 | 852785 | 852846 | 852907 | 852968 | 853029 | 61
713 | 853089 | 853150 | 853211 | 853272 | 853339 | 853394 | 853455 | 853516 | 853577 | 853637 | 61
714 | 853698 | 853759 | 853819 | 853881 | 853941 | 854002 | 854063 | 854124 | 854185 | 851245 | 61
715 | 854306 | 854367 | 854428 | 854488 | 854549 | 854609 | 854670 | 854731 | 854792 | 854852 | 61
716 | 853913 | 854974 | 855034 | 855095 | 855156 | 855216 | 855277 | 855337 | 855398 | 855459 | 61
717 | 855519 | 855579 | 855640 | 855701 | 855761 | 855822 | 855882 | 855943 | 856003 | 856064 | 61
718 | 856124 | 856185 | 856245 | 856306 | 856366 | 856427 | 856487 | 856548 | 856608 | 856668 | 60
719 | 856729 | 856789 | 856849 | 856910 | 856970 | 857031 | 857091 | 857152 | 857212 | 857272 | 60
720 | 857332 | 857393 | 857453 | 857513 | 857574 | 857634 | 857694 | 857755 | 857815 | 857875 | 60
721 | 857935 | 857995 | 858056 | 858116 | 858176 | 858236 | 858297 | 858357 | 858417 | 858477 | 60
722 | 858537 | 858597 | 858657 | 858718 | 858778 | 858838 | 858898 | 858958 | 859018 | 859078 | 60
723 | 859138 | 859198 | 859258 | 859318 | 859379 | 859439 | 859499 | 859559 | 859619 | 859679 | 60
724 | 859739 | 859799 | 859859 | 859918 | 859978 | 860038 | 860098 | 860158 | 860218 | 860278 | 60
725 | 860338 | 860398 | 860458 | 860518 | 860578 | 860637 | 860697 | 860757 | 860817 | 860877 | 60
726 | 860037 | 860996 | 861056 | 861116 | 861176 | 861236 | 861295 | 860355 | 861415 | 861475 | 60
727 | 861534 | 861594 | 861654 | 861714 | 861773 | 861833 | 861893 | 861952 | 862012 | 862072 | 60
728 | 862131 | 862191 | 862251 | 862310 | 862369 | 862429 | 862489 | 862549 | 862608 | 862668 | 60
729 | 862728 | 862787 | 862847 | 862906 | 862966 | 863025 | 863085 | 863144 | 863204 | 863623 | 60
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
730 | 863323 | 863382 | 863442 | 863501 | 863561 | 863620 | 863679 | 863739 | 863799 | 863858 | 59
731 | 863917 | 863977 | 864036 | 864096 | 864155 | 864214 | 864274 | 864333 | 864392 | 864452 | 59
732 | 864511 | 864570 | 864629 | 864689 | 864748 | 864808 | 864867 | 864926 | 864985 | 864045 | 59
733 | 865104 | 865163 | 865222 | 865282 | 865341 | 865400 | 865459 | 865519 | 865578 | 865637 | 59
734 | 865696 | 865755 | 865814 | 865874 | 865933 | 865992 | 866051 | 866110 | 866169 | 866228 | 59
735 | 866187 | 866346 | 866405 | 866465 | 866524 | 866583 | 866642 | 866701 | 866759 | 866819 | 59
736 | 866878 | 866937 | 866996 | 867055 | 867114 | 867173 | 867232 | 867291 | 867349 | 867409 | 59
737 | 867467 | 867526 | 867585 | 867644 | 867703 | 867762 | 867821 | 867879 | 867939 | 867998 | 59
738 | 868056 | 868115 | 868174 | 868233 | 868292 | 868350 | 868409 | 868468 | 868527 | 868586 | 59
739 | 868643 | 868703 | 868762 | 868821 | 868879 | 868938 | 868997 | 869056 | 869114 | 869173 | 59
740 | 869232 | 869290 | 869349 | 869408 | 869466 | 869525 | 869584 | 869642 | 869701 | 869759 | 59
741 | 869818 | 869877 | 869935 | 869994 | 870053 | 870111 | 870169 | 870228 | 870287 | 870345 | 59
742 | 870404 | 870462 | 870521 | 870579 | 870638 | 870696 | 870755 | 870813 | 870872 | 870930 | 59
743 | 870989 | 871047 | 871106 | 871164 | 871223 | 871281 | 871339 | 871398 | 871456 | 871515 | 58
744 | 871573 | 871631 | 871689 | 871748 | 871806 | 871865 | 871923 | 871981 | 871039 | 872098 | 58
745 | 872156 | 872215 | 872273 | 872331 | 872389 | 872448 | 872506 | 872564 | 872622 | 872681 | 58
746 | 872739 | 872797 | 872855 | 872913 | 872972 | 873029 | 873088 | 873146 | 873204 | 873262 | 58
747 | 873321 | 873379 | 873437 | 873495 | 873553 | 873611 | 873669 | 873727 | 873785 | 873844 | 58
748 | 873902 | 873959 | 874018 | 874076 | 874134 | 874192 | 874249 | 874308 | 874366 | 874424 | 58
749 | 874482 | 874539 | 874598 | 874656 | 874714 | 874772 | 874829 | 874888 | 874945 | 875003 | 58
750 | 875061 | 875119 | 875177 | 875235 | 875293 | 875351 | 875409 | 875466 | 875524 | 875582 | 58
751 | 875639 | 875698 | 875756 | 875813 | 875871 | 875929 | 875987 | 876045 | 876102 | 876160 | 58
752 | 876218 | 876276 | 876333 | 876391 | 876449 | 876507 | 876564 | 876622 | 876679 | 876737 | 58
753 | 876795 | 876853 | 876910 | 876968 | 877026 | 877083 | 877141 | 877199 | 877256 | 877314 | 58
754 | 877371 | 877429 | 877487 | 877544 | 877602 | 877659 | 877717 | 877774 | 877832 | 877889 | 58
755 | 877947 | 878004 | 878062 | 878119 | 878177 | 878234 | 878292 | 878349 | 878407 | 878464 | 57
756 | 878522 | 878579 | 878637 | 878694 | 878752 | 878808 | 878866 | 878924 | 878981 | 879039 | 57
757 | 879096 | 879153 | 879211 | 879268 | 879325 | 879382 | 879459 | 879497 | 879555 | 879612 | 57
758 | 879669 | 879726 | 879784 | 879841 | 879898 | 879955 | 880013 | 880070 | 880127 | 880185 | 57
759 | 870242 | 880299 | 880356 | 880413 | 880471 | 880527 | 880585 | 880642 | 880699 | 880756 | 57
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
760 | 880814 | 880871 | 880928 | 880985 | 881042 | 881099 | 881156 | 881213 | 881271 | 881328 | 57
761 | 881385 | 881442 | 881499 | 881556 | 881613 | 881669 | 881727 | 881784 | 881841 | 881898 | 57
762 | 881955 | 882012 | 882069 | 882126 | 882103 | 882239 | 882297 | 882354 | 882411 | 882468 | 57
763 | 882525 | 882581 | 882638 | 882695 | 882752 | 882809 | 882866 | 882923 | 882979 | 883037 | 57
764 | 883093 | 883050 | 883207 | 883264 | 883321 | 883377 | 883434 | 883491 | 883548 | 883605 | 57
765 | 883661 | 883718 | 883775 | 883832 | 883888 | 883945 | 884002 | 884059 | 884115 | 884172 | 57
766 | 884229 | 884285 | 884342 | 884399 | [...] | 884512 | 884569 | 884625 | 884682 | 884739 | 57
767 | 884795 | 884852 | 884909 | 884965 | 885022 | 885078 | 885135 | 885192 | 885248 | 885305 | 57
768 | 885361 | 885418 | 885474 | 885531 | 885587 | 885644 | 885700 | 885757 | 885813 | 885869 | 57
769 | 885926 | 885983 | 886039 | 886096 | 886152 | 886209 | 886265 | 886321 | 886378 | 886434 | 56
770 | 886491 | 886547 | 886604 | 886659 | [...] | 886773 | 886829 | 886885 | 886941 | 886998 | 56
771 | 887054 | 887111 | 887167 | 887223 | 887279 | 887336 | 887392 | 887449 | 887505 | 887561 | 56
772 | 887617 | 887674 | 887720 | 887786 | 887842 | 887898 | 887955 | 888011 | 888067 | 888123 | 56
773 | 888179 | 888236 | 888292 | 888348 | 888404 | 888460 | 888516 | 888573 | 888629 | 888685 | 56
774 | 888741 | 888797 | 888853 | 888909 | 888965 | 889021 | 889077 | 889134 | 889189 | 889246 | 56
775 | 889302 | 889358 | 889414 | 889469 | 889523 | 889582 | 889638 | 889694 | 889749 | 889806 | 56
776 | 889862 | 889918 | 889974 | 890029 | 890036 | 890141 | 890197 | 890253 | 890309 | 890365 | 56
777 | 890421 | 890477 | 890533 | 890589 | 890645 | 890700 | 890756 | 890812 | 890868 | 890924 | 56
778 | 890979 | 891035 | 891091 | 891147 | 891203 | 891259 | 891314 | 891370 | 891426 | 891482 | 56
779 | 891537 | 891593 | 891649 | 891705 | 891760 | 891816 | 891872 | 891928 | 891983 | 892039 | 56
780 | 892095 | 892150 | 892206 | 892262 | 892317 | 892373 | 892429 | 892484 | 892539 | 892595 | 56
781 | 898651 | 892707 | 892762 | 892818 | 892818 | 892929 | 892985 | 893040 | 893096 | 893151 | 56
782 | 893207 | 893262 | 893318 | 893373 | 893429 | 893484 | 893539 | 893595 | 893651 | 893706 | 56
783 | 893762 | 893817 | 893873 | 893928 | 893984 | 894039 | 894094 | 894149 | 894205 | 894261 | 55
784 | 894316 | 894371 | 894427 | 894482 | 894538 | 894593 | 894648 | 894704 | 894759 | 894814 | 55
785 | 894869 | 894925 | 894980 | 895036 | 895091 | 895146 | 895201 | 895257 | 895312 | 895367 | 55
786 | 895423 | 895478 | 895533 | 895588 | 895644 | 895699 | 895754 | 895809 | 895864 | 895919 | 55
787 | 895975 | 896029 | 896085 | 896140 | 896195 | 896251 | 896306 | 896361 | 896416 | 896471 | 55
788 | 896526 | 896581 | 896636 | 896692 | 896747 | 896802 | 896857 | 896912 | 896967 | 897022 | 55
789 | 897077 | 897132 | 897184 | 897242 | 897297 | 897352 | 897407 | 897462 | 897517 | 897572 | 55
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
790 | 897627 | 897682 | 897737 | 897792 | 897847 | 897902 | 897957 | 898012 | 898067 | 898122 | 55
791 | 898176 | 898231 | 898286 | 898341 | 898396 | 898451 | 898506 | 898561 | 898615 | 898670 | 55
792 | 898725 | 898780 | 898835 | 898889 | 898944 | 898999 | 899054 | 899109 | 899164 | 899218 | 55
793 | 899273 | 899328 | 899383 | 899437 | 899492 | 899547 | 899602 | 899656 | 899711 | 899766 | 55
794 | 899821 | 899875 | 899929 | 899985 | 900039 | 900094 | 900149 | 900203 | 900258 | 900312 | 55
795 | 900367 | 900422 | 900476 | 900531 | 900586 | 900640 | 900695 | 900749 | 900804 | 900859 | 55
796 | 900913 | 900968 | 901022 | 901077 | 901131 | 901186 | 901240 | 901295 | 901349 | 901404 | 55
797 | 901458 | 901513 | 901567 | 901622 | 901676 | 901731 | 901785 | 901839 | [...] | 901948 | 54
798 | 902003 | 902057 | 902112 | 902166 | 902221 | 902275 | 902329 | 902384 | 902438 | 902492 | 54
799 | 902547 | 902601 | 902655 | 902709 | 902764 | 902818 | 902873 | 902927 | 902981 | 903036 | 54
800 | 903089 | 903144 | [...] | 903253 | 903307 | 903361 | 903416 | 903469 | 903524 | 903578 | 54
801 | 903633 | 903687 | 903741 | 903795 | 903849 | 903904 | 903956 | 904012 | 904066 | 904120 | 54
802 | 904174 | 904229 | 904283 | 904337 | 904391 | 904445 | 904499 | 904553 | 904607 | 904661 | 54
803 | 904716 | 904769 | 904824 | 904878 | 904932 | 904986 | 905039 | 905094 | 905148 | 905202 | 54
804 | 905256 | 905310 | 905364 | 905418 | 905472 | 905526 | 905580 | 905634 | 905688 | 905742 | 54
805 | 905796 | [...] | 905904 | 905958 | 906012 | 906066 | 906119 | 906173 | 906227 | 906281 | 54
806 | 906335 | 906389 | 906443 | 906497 | 906551 | 906604 | 906658 | 906712 | 906766 | 906819 | 54
807 | [...] | 906927 | 907981 | 907035 | 907089 | 907143 | 907196 | 907250 | 907304 | 907358 | 54
808 | 907411 | 907465 | 907519 | 907573 | 907626 | 907680 | 907734 | 907787 | 907841 | 907895 | 54
809 | 907949 | 908002 | 908056 | 908109 | 908163 | 908217 | 908270 | 908324 | 908378 | 908431 | 54
810 | 908485 | 908539 | 908592 | 908646 | 908699 | 908753 | 908807 | 908860 | 908914 | 908967 | 54
811 | 909021 | 909074 | 909128 | 909181 | 909235 | 909289 | 909341 | 909396 | 909449 | 909503 | 54
812 | 905556 | 909609 | 909663 | 909716 | 909769 | 909823 | 909877 | 909930 | 909984 | 910037 | 53
813 | 910091 | 910144 | 910197 | 910251 | 910304 | 910358 | 910411 | 910464 | 910518 | 910571 | 53
814 | 910624 | 910678 | 910731 | 910784 | 910838 | 910891 | 910944 | 910998 | 911051 | 911104 | 53
815 | 911158 | 911211 | 911263 | 911317 | 911371 | 911424 | 911477 | 911530 | 911584 | 911637 | 53
816 | 911690 | 911743 | 911797 | 911849 | 911903 | 911956 | 912009 | 912063 | 912116 | 912169 | 53
817 | 912222 | 912275 | 912323 | 912381 | 912435 | 912488 | 912541 | 912594 | 912647 | 912700 | 53
818 | 912753 | 912806 | 912859 | 912913 | 912966 | 913019 | 913072 | 913125 | 913178 | 913231 | 53
819 | 913284 | 913337 | 913380 | 913443 | 913496 | 913549 | 913602 | 913655 | 913708 | 913761 | 53
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
820 | 913814 | 913867 | 913919 | 913973 | 914026 | 914079 | 914132 | 914184 | 914237 | 914290 | 53
821 | 914343 | 914396 | 914449 | 914502 | 914555 | 914608 | 914660 | 914713 | 914766 | 914819 | 53
822 | 914872 | 914925 | 914977 | 915030 | 915083 | 915136 | 915189 | 915241 | 915294 | 915347 | 53
823 | 915399 | 915453 | 915505 | 915558 | 915611 | 915664 | 915716 | 915769 | 915822 | 915875 | 53
824 | 915927 | 915979 | 916033 | 916085 | 916138 | 916191 | 916243 | 916296 | 916349 | 916401 | 53
825 | 916454 | 916507 | 916559 | 916612 | 916664 | 916717 | 916769 | 916822 | 916875 | 916927 | 53
826 | 916980 | 917033 | 917085 | 917138 | 917190 | 917243 | 917295 | 917348 | 917400 | 917453 | 53
827 | 917506 | 917558 | 917611 | 917663 | 917716 | 917768 | 917820 | 917873 | 917925 | 917978 | 52
828 | 918030 | 918083 | 918135 | 918188 | 918240 | 918293 | 918345 | 918397 | 918449 | 918502 | 52
829 | 918555 | 918607 | 918659 | 918712 | 918764 | 918816 | 918869 | 918921 | 918973 | 919026 | 52
830 | 919078 | 919130 | 919183 | 919235 | 919287 | 919339 | 919392 | 919444 | 919496 | 919549 | 52
831 | 919601 | 919653 | 919706 | 919758 | 919810 | 919862 | 919914 | 919967 | 920019 | 920071 | 52
832 | 920123 | 920176 | 920228 | 920279 | 920332 | 920384 | 920436 | 920489 | 920541 | 920593 | 52
833 | 920645 | 920697 | 920749 | 920801 | 920853 | 920906 | 920958 | 921009 | 921062 | 921114 | 52
834 | 921166 | 921218 | 921270 | 921322 | 921374 | 921426 | 921478 | 921530 | 921582 | 921634 | 52
835 | 921686 | 921738 | 921790 | 921842 | 921894 | 921946 | 921998 | 922050 | 922102 | 922154 | 52
836 | 922206 | 922258 | 922310 | 922362 | 922414 | 922466 | 922518 | 922569 | 922622 | 922674 | 52
837 | 922725 | 922777 | 922829 | 922881 | 922933 | 922985 | 923037 | 923089 | 923140 | 923192 | 52
838 | 923244 | 923296 | 923348 | 923399 | 923451 | 923503 | 923555 | 923607 | 923658 | 923710 | 52
839 | 923762 | 923814 | 923865 | 923917 | 923969 | 924021 | 924072 | 924124 | 924176 | 924228 | 52
840 | 924279 | 924331 | 924383 | 924434 | 924486 | 924538 | 924589 | 924641 | 924693 | 924744 | 52
841 | 924796 | 924848 | 924899 | 924951 | 925003 | 925054 | 925106 | 925157 | 925209 | 925261 | 52
842 | 925312 | 925364 | 925415 | 925461 | 925518 | 925569 | 925621 | 925673 | 925725 | 925776 | 52
843 | 925828 | 925879 | 925931 | 925982 | 926034 | 926085 | 926137 | 926188 | 926239 | 926291 | 51
844 | 926342 | 926394 | 926445 | 926497 | 926548 | 926599 | 926651 | 926702 | 926754 | [...] | 51
845 | 926857 | 926908 | 926959 | 927011 | 927062 | 927114 | 927165 | 927216 | 927268 | 927319 | 51
846 | 927370 | 927422 | 927473 | 927524 | 927576 | 927627 | 927678 | 927729 | 927781 | 927832 | 51
847 | 927883 | 927935 | 927986 | 928037 | 928088 | 928139 | 928191 | 928242 | 928293 | 928345 | 51
848 | 928396 | 928447 | 928498 | 928549 | 928601 | 928652 | 928703 | 928754 | 928805 | 928857 | 51
849 | 928908 | 928959 | 929009 | 929061 | 929112 | 929163 | 929215 | 929266 | 929317 | 929368 | 51
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
850 | 929419 | 929470 | 929521 | 929572 | 929623 | 929674 | 929725 | 929776 | 929827 | 929879 | 51
851 | 929929 | 929981 | 930032 | 930083 | 930134 | 930185 | 930236 | 930287 | 930338 | 930389 | 51
852 | 930439 | 930491 | 930542 | 930592 | 930643 | 930694 | 930745 | 930796 | 930847 | 930898 | 51
853 | 930949 | 930999 | 931051 | 931102 | 931153 | 931204 | 931254 | 931305 | 931356 | 931407 | 51
854 | 931458 | 931509 | 931559 | 931610 | 931661 | 931712 | 931763 | 931814 | 931865 | 931915 | 51
855 | 931966 | 932017 | 932068 | 932118 | 932169 | 932220 | 932271 | 932322 | 932372 | 932423 | 51
856 | 932474 | 932524 | 932575 | 932626 | 932677 | 932727 | 932778 | 932829 | 932879 | 932930 | 51
857 | 932981 | 933031 | 933082 | 933133 | 933183 | 933234 | 933284 | 933335 | 933386 | 933437 | 51
858 | 933487 | 933538 | 933589 | 933639 | 933689 | 933740 | 933791 | 933841 | 933892 | 933943 | 51
859 | 933993 | 934044 | 934094 | 934145 | 934195 | 934246 | 934296 | 934347 | 934397 | 934448 | 51
860 | 934498 | 934549 | 934599 | 934649 | 934700 | 934751 | 934801 | 934852 | 934902 | 934953 | 50
861 | 935003 | 935056 | 935104 | 935154 | 935205 | 935255 | 935306 | 935356 | 935406 | 935457 | 50
862 | 935507 | 935558 | 935608 | 935658 | 935709 | 935759 | 935809 | 935859 | 935910 | 935960 | 50
863 | 936011 | 936061 | 936111 | 936162 | 936212 | 936262 | 936313 | 936363 | [...] | [...] | 50
864 | 936514 | 936564 | 936614 | 936665 | 936715 | 936765 | 936815 | 636865 | 936916 | 936966 | 50
865 | 937016 | 937066 | 937117 | 937167 | 937217 | 937267 | 937317 | 937367 | 937418 | 937468 | 50
866 | 937518 | 937568 | 937618 | 937668 | 937718 | 937769 | 937819 | 937869 | 937919 | 937969 | 50
867 | 938019 | 938069 | 938119 | 938169 | 938219 | 938269 | 938319 | 938369 | 938419 | 938469 | 50
868 | 938519 | 938569 | 938619 | 938669 | 938719 | 938769 | 938819 | 938869 | 938919 | 938969 | 50
869 | 939019 | 939069 | 939119 | 939169 | 939219 | 939269 | 939319 | 939369 | 939419 | 939469 | 50
870 | 939519 | 939569 | 939619 | 939669 | 939719 | 939769 | 939819 | 939869 | 939918 | 939968 | 50
871 | 940018 | 940068 | 940118 | 940168 | 940218 | 940267 | 940317 | 940367 | 940417 | 940467 | 50
872 | 940516 | 940566 | 940616 | 940666 | 940716 | 940765 | 940815 | 940865 | 940915 | 940964 | 50
873 | 941014 | 941064 | 941114 | 941163 | 941213 | 941263 | 941313 | 941362 | 941412 | 941462 | 50
874 | 941511 | 941561 | 941611 | 941660 | 941710 | 941759 | 941809 | 946859 | 941909 | 941958 | 50
875 | 942008 | 942058 | 942107 | 942157 | 942207 | 942256 | 942306 | 942355 | 942405 | 942455 | 50
876 | 942504 | 942554 | 942603 | 942653 | 942702 | 942752 | 942801 | 942851 | 942901 | 942950 | 50
877 | 942999 | 943049 | 943099 | 943148 | 943198 | 943247 | 943297 | 943346 | 943496 | 943445 | 49
878 | 943495 | 943544 | 943594 | 943643 | 943692 | 943742 | 943791 | 943841 | 943890 | 943939 | 49
879 | 943989 | 944038 | 944088 | 944137 | 944186 | 944236 | 944285 | 944335 | 944384 | 944433 | 49
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
880 | 944483 | 944532 | 944581 | 944631 | 944680 | 944729 | 944779 | 944828 | 944877 | 944927 | 49
881 | 944976 | 945025 | 945074 | 945124 | 945873 | 945222 | 945272 | 945321 | 945370 | 945419 | 49
882 | 945468 | 945518 | 945567 | 945616 | 945665 | 945715 | 945764 | 945813 | 945862 | 945912 | 49
883 | 945961 | 946009 | 946059 | 946108 | 946157 | 946207 | 946256 | 946305 | 946354 | 946403 | 49
884 | 946452 | 946501 | 946551 | 946599 | 946649 | 946698 | 946747 | 946796 | 946845 | 946894 | 49
885 | 946943 | 946992 | 947041 | 947090 | 947139 | 947189 | 947238 | 947287 | 947336 | 947385 | 49
886 | 947434 | 947483 | 947532 | 947581 | 947629 | 947679 | 947728 | 947777 | 947826 | 947875 | 49
887 | 947924 | 947973 | 948022 | 948070 | 948119 | 948168 | 948217 | 948266 | 948315 | 948364 | 49
888 | 948413 | 948462 | 948511 | 948559 | 948609 | 948657 | 948706 | 948755 | 948804 | 948853 | 49
889 | 948902 | 948951 | 948999 | 949048 | 949097 | 949146 | 949195 | 949244 | 949292 | 949341 | 49
890 | 949390 | 949439 | 949488 | 949536 | 949585 | 949633 | 949683 | 949731 | 949780 | 949829 | 49
891 | 949878 | 949926 | 949975 | 950024 | 950073 | 950121 | 950170 | 950219 | 950267 | 950316 | 49
892 | 950365 | 950414 | 950462 | 950511 | 950559 | 950608 | 950657 | 950706 | 950754 | 950803 | 49
893 | 950851 | 950900 | 950949 | 950997 | 951046 | 951095 | 951143 | 951192 | 951240 | 951289 | 49
894 | 951338 | 951386 | 951435 | 951483 | 951532 | 951580 | 951629 | 951677 | 951729 | 951775 | 49
895 | 951823 | 951872 | 951920 | 951969 | 952017 | 952066 | 952114 | 952163 | 952210 | 952259 | 49
896 | 952308 | 952356 | 952405 | 952453 | 952502 | 952550 | 952399 | 952647 | 952696 | 952744 | 48
897 | 952792 | 952841 | 952889 | 952938 | 952986 | 953034 | 953083 | 953131 | 953179 | 953228 | 48
898 | 953276 | 953325 | 953373 | 953421 | 953469 | 953518 | 953566 | 953615 | 953663 | 953711 | 48
899 | 953759 | 953808 | 953856 | 953905 | 953953 | 954001 | 954049 | 954099 | 954146 | 954194 | 48
900 | 954243 | 954292 | 954339 | 954387 | 954435 | 954484 | 954532 | 954580 | 954628 | 954677 | 48
901 | 954725 | 954773 | 954821 | 954869 | 954918 | 954966 | 955014 | 955062 | 955110 | 955158 | 48
902 | 955207 | 955255 | 955303 | 955351 | 955399 | 955447 | 955495 | 955543 | 955592 | 955639 | 48
903 | 955688 | 955736 | 955784 | 955832 | 955880 | 955928 | 955976 | 956024 | 956075 | 956120 | 48
904 | 956168 | 956216 | 956265 | 956313 | 956361 | 956409 | 956457 | 956505 | 956533 | 956601 | 48
905 | 956649 | 956697 | 956745 | 956793 | 956840 | 956888 | 956936 | 956984 | 957032 | 957080 | 48
906 | 957128 | 957176 | 957224 | 955272 | 957319 | 957368 | 957416 | 957464 | 957512 | 957559 | 48
907 | 957607 | 957655 | 957703 | 957751 | 957799 | 957847 | 957894 | 957942 | 957900 | 958038 | 48
908 | 958086 | 958134 | 958181 | 958229 | 958277 | 958325 | 958373 | 958421 | 958468 | 958516 | 48
909 | 958564 | 958612 | 958659 | 958707 | 958755 | 958803 | 958850 | 958898 | 958946 | 958994 | 48
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
910 | 959041 | 959089 | 959137 | 959185 | 959231 | 959279 | 959328 | 959375 | 959423 | 959471 | 48
911 | 959518 | 959566 | 959614 | 959661 | 959709 | 959757 | 959804 | 959852 | 959899 | 959947 | 48
912 | 959995 | 960042 | 960090 | 960138 | 960185 | 960233 | 960280 | 960328 | 960376 | 960423 | 48
913 | 960471 | 960518 | 960566 | 960613 | 960661 | 960709 | 960756 | 960804 | 960851 | 960899 | 48
914 | 960946 | 960994 | 961041 | 961089 | 961136 | 961184 | 961231 | 961279 | 961326 | 961374 | 47
915 | 961421 | 961469 | 961516 | 961563 | 961611 | 961658 | 961706 | 961753 | 961801 | 961848 | 47
916 | 961895 | 961943 | 961990 | 962038 | 962085 | 962132 | 962179 | 962227 | 962275 | 962322 | 47
917 | 962369 | 962417 | 962464 | 962511 | 962559 | 962606 | 962653 | 962701 | 969748 | 962795 | 47
918 | 962842 | 962886 | 962937 | 962985 | 963032 | 963079 | 963126 | 963174 | 963221 | 963268 | 47
919 | 963315 | 963363 | 963410 | 963457 | 963504 | 963552 | 963599 | 963646 | 963693 | 963741 | 47
920 | 963788 | 963835 | 963882 | 963929 | 963977 | 964024 | 964071 | 964118 | 964165 | 964212 | 47
921 | 964259 | 964307 | 964354 | 964401 | 964448 | 964495 | 964542 | 964589 | 964637 | 964684 | 47
922 | 964731 | 964778 | 964825 | 964872 | 964919 | 964966 | 965013 | 965061 | 965108 | 965155 | 47
923 | 965202 | 965249 | 965296 | 965343 | 965389 | 965437 | 965484 | 965531 | 965578 | 965624 | 47
924 | 965672 | 965719 | 965766 | 965813 | 965859 | 965906 | 965954 | 966001 | 966048 | 966095 | 47
925 | 966142 | 966189 | 966239 | 966283 | 966329 | 966376 | 966423 | 966470 | 966517 | 966564 | 47
926 | 966611 | 966658 | 966705 | 966752 | 966799 | 966845 | 966892 | 966939 | 966986 | 967033 | 47
927 | 967079 | 967127 | 967173 | 967220 | 967267 | 967314 | 967361 | 967408 | 967454 | 967501 | 47
928 | 967548 | 967595 | 967642 | 967688 | 967735 | 967782 | 967829 | 967875 | 967922 | 967969 | 47
929 | 968016 | 968062 | 968109 | 968156 | 968202 | 968249 | 968296 | 968343 | 968389 | 968436 | 47
930 | 968483 | 968529 | 968576 | 968623 | 968669 | 968716 | 968763 | 968809 | 968856 | 968902 | 47
931 | 968949 | 968996 | 969043 | 969089 | 969136 | 969183 | 969229 | 969276 | 969323 | 969369 | 47
932 | 969416 | 969463 | 969509 | 969556 | 969602 | 969649 | 969695 | 969741 | 969789 | 969835 | 47
933 | 969882 | 969928 | 969975 | 970021 | 970068 | 970114 | 970161 | 970207 | 970254 | 970300 | 47
934 | 970347 | 970393 | 970439 | 970486 | 970533 | 970579 | 970626 | 970672 | 970719 | 970765 | 46
935 | 970812 | 970858 | 970904 | 970951 | 970997 | 971044 | 971090 | 971137 | 971183 | 971229 | 46
936 | 971286 | 971322 | 971369 | 971415 | 971461 | 971508 | 971554 | 971601 | 971647 | 971693 | 46
937 | 971739 | 971786 | 971832 | 971879 | 971925 | 971971 | 972018 | 972064 | 972110 | 972157 | 46
938 | 972203 | 972249 | 972295 | 972342 | 972388 | 972434 | 972481 | 972527 | 972573 | 972619 | 46
939 | 972666 | 972712 | 972758 | 972804 | 972851 | 972897 | 972943 | 972989 | 973035 | 973082 | 46
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
940 | 973128 | 973174 | 973220 | 973266 | 973313 | 973359 | 973405 | 973451 | 973497 | 973543 | 46
941 | 973589 | 973636 | 973682 | 973728 | 973774 | 973820 | 973866 | 973913 | 973959 | 974005 | 46
942 | 974050 | 974097 | 974143 | 974189 | 974235 | 974281 | 974327 | 974374 | 974419 | 974466 | 46
943 | 974512 | 974558 | 974604 | 974649 | 974695 | 974742 | 974788 | 974834 | 974819 | 974926 | 46
944 | 974972 | 975018 | 975064 | 975109 | 975156 | 975202 | 975248 | 975294 | 975339 | 975386 | 46
945 | 975432 | 975478 | 975524 | 975569 | 975616 | 975662 | 975707 | 975753 | 975799 | 975845 | 46
946 | 975891 | 975937 | 975983 | 976029 | 976075 | 976121 | 976167 | 976212 | 976258 | 976304 | 46
947 | 976349 | 976396 | 976442 | 976488 | 976533 | 976579 | 976625 | 976671 | 976717 | 976763 | 46
948 | 976808 | 976854 | 976899 | 976946 | 976992 | 977037 | 977083 | 977129 | 977175 | 977220 | 46
949 | 977266 | 977312 | 977358 | 977403 | 977449 | 977495 | 977541 | 977586 | 977632 | 977678 | 46
950 | 977724 | 977769 | 977815 | 977861 | 977906 | 977952 | 977998 | 978042 | 978089 | 978135 | 46
951 | 978181 | 978226 | 978272 | 978317 | 978363 | 978409 | 978454 | 978500 | 978546 | 978591 | 46
952 | 978637 | 978683 | 978728 | 978774 | 978819 | 978865 | 978911 | 978956 | 979002 | 979047 | 46
953 | 979093 | 979138 | 979184 | 979229 | 979275 | 979321 | 979366 | 979412 | 979457 | 979503 | 46
954 | 979548 | 979594 | 979639 | 979685 | 979730 | 979776 | 979821 | 979867 | 979912 | 979958 | 46
955 | 980003 | 980049 | 980094 | 980139 | 980185 | 980231 | 980276 | 980322 | 980367 | 980412 | 45
956 | 980458 | 980503 | 980549 | 980594 | 980639 | 980685 | 980730 | 980776 | 980821 | 980867 | 45
957 | 980912 | 980957 | 981003 | 981048 | 981093 | 981139 | 981184 | 981229 | 981275 | 981320 | 45
958 | 981366 | 981411 | 981456 | 981501 | 981547 | 981592 | 981637 | 981683 | 981728 | 981773 | 45
959 | 981819 | 981864 | 981909 | 981954 | 981999 | 982045 | 982090 | 982135 | 982181 | 982226 | 45
960 | 982271 | 982316 | 982362 | 982407 | 982452 | 982497 | 982543 | 982588 | 982633 | 982678 | 45
961 | 982723 | 982769 | 982814 | 982859 | 982904 | 982949 | 982994 | 983039 | 983085 | 983129 | 45
962 | 983175 | 983220 | 983265 | 983310 | 983356 | 983401 | 983446 | 983490 | 983536 | 983581 | 45
963 | 983626 | 983671 | 983716 | 983762 | 983807 | 983852 | 983897 | 983942 | 983987 | 984032 | 45
964 | 984077 | 984122 | 984167 | 984212 | 984257 | 984302 | 984347 | 984392 | 984437 | 984482 | 45
965 | 984527 | 984572 | 984617 | 984662 | 984707 | 984752 | 984797 | 984842 | 984887 | 984932 | 45
966 | 984977 | 985022 | 985067 | 985112 | 985157 | 985202 | 985247 | 985292 | 985337 | 985382 | 45
967 | 985426 | 985471 | 985516 | 985561 | 985606 | 985651 | 985696 | 985741 | 985786 | 985830 | 45
968 | 985875 | 985920 | 985965 | 986009 | 986055 | 986099 | 986144 | 986189 | 986234 | 986279 | 45
969 | 986324 | 986369 | 986413 | 986458 | 986504 | 986548 | 986593 | 986637 | 986682 | 986727 | 45
N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D
970 | 986772 | 986817 | 986861 | 986906 | 986951 | 986996 | 987040 | 987085 | 987129 | 987175 | 45
971 | 987219 | 987264 | 987309 | 987353 | 987398 | 987443 | 987488 | 987532 | 987577 | 987622 | 45
972 | 987666 | 987711 | 987756 | 987800 | 987845 | 987889 | 987934 | 987979 | 988024 | 988068 | 45
973 | 988113 | 988157 | 988202 | 988247 | 988291 | 988336 | 988381 | 988425 | 988469 | 988514 | 45
974 | 988559 | 988604 | 988748 | 988693 | 988737 | 988782 | 988826 | 988871 | 988916 | 988960 | 45
975 | 989005 | 989049 | 989094 | 989138 | 989183 | 989227 | 989272 | 989316 | 989361 | 989405 | 45
976 | 989449 | 989494 | 989539 | 989584 | 989628 | 989672 | 989717 | 989761 | 989806 | 989850 | 44
977 | 989895 | 989939 | 989983 | 990028 | 990072 | 990117 | 990161 | 990206 | 980150 | 990294 | 44
978 | 990339 | 990383 | 990428 | 990472 | 990516 | 990561 | 990605 | 990649 | 990694 | 990738 | 44
979 | 990783 | 990827 | 990871 | 990916 | 990960 | 991004 | 991049 | 991093 | 991137 | 991182 | 44
980 | 991226 | 991270 | 991315 | 991359 | 991403 | 991448 | 991492 | 991536 | 991580 | 991625 | 44
981 | 991669 | 991713 | 991758 | 991802 | 991846 | 991890 | 991935 | 991979 | 992023 | 992067 | 44
982 | 992111 | 992156 | 992199 | 992244 | 992288 | 992333 | 992377 | 992421 | 992465 | 992509 | 44
983 | 992554 | 992598 | 992642 | 992686 | 992730 | 992774 | 992819 | 992863 | 992907 | 992951 | 44
984 | 992995 | 993039 | 993083 | 993127 | 993172 | 993216 | 993259 | 993304 | 993348 | 993392 | 44
985 | 993436 | 993480 | 993524 | 993568 | 993613 | 993657 | 993701 | 993745 | 993789 | 993833 | 44
986 | 993877 | 993921 | 993965 | 994009 | 994053 | 994097 | 994141 | 994185 | 994229 | 994273 | 44
987 | 994317 | 994361 | 994405 | 994449 | 994493 | 994537 | 994581 | 994625 | 994669 | 994713 | 44
988 | 994756 | 994801 | 994845 | 994889 | 994933 | 994977 | 995021 | 995065 | 995108 | 995152 | 44
989 | 995196 | 995240 | 995284 | 995328 | 995372 | 995416 | 995459 | 995504 | 995547 | 995591 | 44
990 | 995635 | 995679 | 995723 | 995764 | 995811 | 995854 | 995898 | 995942 | 995986 | 996029 | 44
991 | 996074 | 996117 | 996161 | 996205 | 996249 | 996293 | 996337 | 996380 | 996424 | 996468 | 44
992 | 996512 | 996555 | 996599 | 996643 | 996687 | 996731 | 996774 | 996818 | 996862 | 996906 | 44
993 | 996949 | 996993 | 997037 | 997080 | 997124 | 997168 | 997212 | 997255 | 997299 | 997343 | 44
994 | 997386 | 997430 | 997474 | 997517 | 997561 | 997605 | 997648 | 997692 | 997736 | 997779 | 44
995 | 997823 | 997867 | 997910 | 997954 | 997998 | 998041 | 998085 | 998129 | 998170 | 998216 | 44
996 | 998259 | 998303 | 998347 | 998390 | 998434 | 998477 | 998521 | 998564 | 998608 | 998652 | 44
997 | 998695 | 998739 | 998783 | 998826 | 998869 | 998913 | 998956 | 998999 | 999043 | 999087 | 44
998 | 999133 | 999174 | 999218 | 999261 | 999305 | 999348 | 999392 | 999435 | 999479 | 999522 | 44
999 | 999565 | 999609 | 999652 | 999696 | 999739 | 999783 | 999826 | 999869 | 999913 | 999957 | 43

A TABLE OF PROPORTIONAL PARTS, WHEREBY The Intermediate Logarithms of all Numbers, AND The Numbers of all Logarithms from 10000 to 100000 may more readily be found out by the foregoing Table of Logarithms.

[Page]

[Page]

LONDON, Printed by J. Heptinstall for W. Freeman, at the Artichoke next St. Dunstan's Church in Fleetstreet. MDCLXXXVII.

[Page]

[Page]

A TABLE OF Proportional Parts.
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
43 | 4 | 8 | 12 | 17 | 21 | 25 | 30 | 34 | 38
44 | 4 | 8 | 13 | 17 | 22 | 26 | 30 | 35 | 39
45 | 4 | 9 | 13 | 18 | 22 | 27 | 31 | 36 | 40
46 | 4 | 9 | 13 | 18 | 23 | 27 | 32 | 36 | 41
47 | 4 | 9 | 14 | 18 | 23 | 28 | 32 | 37 | 42
48 | 4 | 9 | 14 | 19 | 24 | 28 | 33 | 38 | 43
49 | 4 | 9 | 14 | 19 | 24 | 29 | 34 | 39 | 44
50 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45
51 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45
52 | 5 | 10 | 15 | 20 | 26 | 31 | 36 | 41 | 46
53 | 5 | 10 | 15 | 21 | 26 | 31 | 37 | 42 | 47
54 | 5 | 10 | 16 | 21 | 27 | 32 | 37 | 43 | 48
55 | 5 | 11 | 16 | 22 | 27 | 33 | 38 | 44 | 49
56 | 5 | 11 | 16 | 22 | 28 | 33 | 39 | 44 | 50
57 | 5 | 11 | 17 | 22 | 28 | 34 | 39 | 45 | 51
58 | 5 | 11 | 17 | 23 | 29 | 34 | 40 | 46 | 52
59 | 5 | 11 | 17 | 23 | 29 | 35 | 41 | 47 | 53
60 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54
61 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54
62 | 6 | 12 | 18 | 24 | 31 | 37 | 43 | 49 | 55
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
63 | 6 | 12 | 18 | 25 | 31 | 37 | 44 | 50 | 56
64 | 6 | 12 | 19 | 25 | 32 | 38 | 44 | 51 | 57
65 | 6 | 13 | 19 | 26 | 32 | 39 | 45 | 52 | 58
66 | 6 | 13 | 19 | 26 | 33 | 39 | 46 | 52 | 59
67 | 6 | 13 | 20 | 26 | 33 | 40 | 46 | 53 | 60
68 | 6 | 13 | 20 | 27 | 34 | 40 | 47 | 54 | 61
69 | 6 | 13 | 20 | 27 | 34 | 41 | 48 | 55 | 62
70 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63
71 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63
72 | 7 | 14 | 21 | 28 | 36 | 43 | 50 | 57 | 64
73 | 7 | 14 | 21 | 29 | 36 | 43 | 51 | 58 | 65
74 | 7 | 14 | 22 | 29 | 37 | 44 | 51 | 59 | 66
75 | 7 | 15 | 22 | 30 | 37 | 45 | 52 | 60 | 67
76 | 7 | 15 | 22 | 30 | 38 | 45 | 53 | 60 | 68
77 | 7 | 15 | 23 | 30 | 38 | 46 | 53 | 61 | 69
78 | 7 | 15 | 23 | 31 | 39 | 46 | 54 | 62 | 70
79 | 7 | 15 | 23 | 31 | 39 | 47 | 55 | 63 | 71
80 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72
81 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72
82 | 8 | 16 | 24 | 32 | 41 | 49 | 57 | 65 | 73
83 | 8 | 16 | 24 | 33 | 41 | 49 | 58 | 66 | 74
84 | 8 | 16 | 25 | 33 | 42 | 50 | 58 | 67 | 75
85 | 8 | 17 | 25 | 34 | 42 | 51 | 59 | 68 | 76
86 | 8 | 17 | 25 | 34 | 43 | 51 | 60 | 68 | 77
87 | 8 | 17 | 26 | 34 | 43 | 52 | 60 | 69 | 78
88 | 8 | 17 | 26 | 35 | 44 | 52 | 61 | 70 | 79
89 | 8 | 17 | 26 | 35 | 44 | 53 | 62 | 71 | 80
90 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81
91 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81
92 | 9 | 18 | 27 | 36 | 46 | 55 | 64 | 73 | 82
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
93 | 9 | 18 | 27 | 37 | 46 | 55 | 65 | 74 | 83
94 | 9 | 18 | 28 | 37 | 47 | 56 | [...] | 75 | 84
95 | 9 | 19 | 28 | 38 | 47 | 57 | 66 | 76 | 85
96 | 9 | 19 | 28 | 38 | 48 | 57 | 67 | 76 | 86
97 | 9 | 19 | 29 | 38 | 48 | 58 | 67 | 77 | 87
98 | 9 | 19 | 29 | 39 | 49 | 58 | 68 | 78 | 88
99 | 9 | 19 | 29 | 39 | 49 | 59 | 69 | 79 | 89
100 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90
101 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90
102 | 10 | 20 | 30 | 40 | 51 | 61 | 71 | 81 | 91
103 | 10 | 20 | 30 | 41 | 51 | 61 | 72 | 82 | 92
104 | 10 | 20 | 31 | 41 | 52 | 62 | 72 | 83 | 93
105 | 10 | 21 | 31 | 42 | 52 | 63 | 73 | 84 | 94
106 | 10 | 21 | 31 | 42 | 53 | 63 | 74 | 84 | 95
107 | 10 | 21 | 32 | 42 | 53 | 64 | 74 | 85 | 96
108 | 10 | 21 | 32 | 43 | 54 | 64 | 75 | 86 | 97
109 | 10 | 21 | 32 | 43 | 54 | 65 | 76 | 87 | 98
110 | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99
111 | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99
112 | 11 | 22 | 33 | 44 | 56 | 67 | 78 | 89 | 100
113 | 11 | 22 | 33 | 45 | 57 | 67 | 78 | 90 | 101
114 | 11 | 22 | 34 | 45 | 57 | 68 | 79 | 91 | 102
115 | 11 | 23 | 34 | 46 | 57 | 69 | 80 | 92 | 103
116 | 11 | 23 | 34 | 46 | 58 | 69 | 81 | 92 | 104
117 | 11 | 23 | 35 | 46 | 58 | 70 | 81 | 73 | 105
118 | 11 | 23 | 35 | 47 | 59 | 70 | 82 | 94 | 106
119 | 11 | 23 | 35 | 47 | 59 | 71 | 83 | 95 | 107
120 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108
121 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108
122 | 12 | 24 | 36 | 48 | 61 | 73 | 85 | 97 | 109
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
123 | 12 | 24 | 36 | 48 | 61 | 73 | 86 | 98 | 110
124 | 12 | 24 | 37 | 49 | 62 | 74 | 86 | 99 | 111
125 | 12 | 25 | 37 | 50 | 62 | 75 | 87 | 100 | 112
126 | 12 | 25 | 37 | 50 | 63 | 75 | 88 | 100 | 113
127 | 12 | 25 | 38 | 50 | 63 | 76 | 88 | 101 | 114
128 | 12 | 25 | 38 | 51 | 64 | 76 | 89 | 102 | 115
129 | 12 | 25 | 38 | 51 | 64 | 77 | 90 | 103 | 116
130 | 13 | 26 | 39 | 52 | 65 | 78 | 91 | 104 | 117
131 | 13 | 26 | 39 | 52 | 65 | 78 | 91 | 104 | 117
132 | 13 | 26 | 39 | 52 | 66 | 79 | 92 | 105 | 118
133 | 13 | 26 | 39 | 53 | 66 | 79 | 93 | 106 | 119
134 | 13 | 26 | 40 | 53 | 67 | 80 | 93 | 107 | 120
135 | 13 | 27 | 40 | 54 | 67 | 81 | 94 | 108 | 121
136 | 13 | 27 | 40 | 54 | 68 | 81 | 95 | 108 | 122
137 | 13 | 27 | 41 | 54 | 68 | 82 | 95 | 109 | 123
[...] | 13 | 27 | 41 | 55 | 69 | 82 | 96 | 110 | 124
139 | 13 | 27 | 41 | 55 | 69 | 83 | 97 | 111 | 125
140 | 14 | 28 | 42 | 56 | 70 | 84 | 98 | 112 | 126
141 | 14 | 28 | 42 | 56 | 70 | 84 | 99 | 112 | 126
142 | 14 | 28 | 42 | 56 | 71 | 85 | 99 | 113 | 127
143 | 14 | 28 | 42 | 57 | 71 | 85 | 100 | 114 | 128
144 | 14 | 28 | 43 | 57 | 72 | 86 | 100 | 115 | 129
145 | 14 | 28 | 43 | 58 | 72 | 87 | 101 | 116 | 130
146 | 14 | 29 | 43 | 58 | 73 | 87 | 102 | 116 | 131
147 | 14 | 29 | 44 | 58 | 73 | 88 | 102 | 117 | 132
148 | 14 | 29 | 44 | 59 | 74 | 88 | 103 | 118 | 133
149 | 14 | 29 | 44 | 59 | 74 | 89 | 104 | 119 | 134
150 | 15 | 30 | 45 | 60 | 75 | 90 | 105 | 120 | 135
151 | 15 | 30 | 45 | 60 | 75 | 90 | 105 | 120 | 135
152 | 15 | 30 | 45 | 60 | 76 | 91 | 106 | 121 | 136
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
153 | 15 | 30 | 45 | 60 | 76 | 91 | 107 | 122 | 137
154 | 15 | 30 | 46 | 61 | 77 | 92 | 107 | 123 | 138
155 | 15 | 31 | 46 | 62 | 77 | 93 | 108 | 124 | 139
156 | 15 | 31 | 46 | 62 | 78 | 93 | 109 | 124 | 140
157 | 15 | 31 | 47 | 62 | 78 | 94 | 109 | 125 | 141
158 | 15 | 31 | 47 | 63 | 79 | 94 | 110 | 126 | 142
159 | 15 | 31 | 47 | 63 | 79 | 95 | 111 | 127 | 143
160 | 16 | 32 | 48 | 64 | 80 | 96 | 112 | 128 | 144
161 | 16 | 32 | 48 | 64 | 80 | 96 | 112 | 128 | 144
162 | 16 | 32 | 48 | 64 | 81 | 97 | 113 | 129 | 145
163 | 16 | 32 | 48 | 65 | 82 | 98 | 114 | 130 | 146
164 | 16 | 32 | 49 | 66 | 82 | 98 | 114 | 131 | 147
165 | 16 | 33 | 49 | 66 | 82 | 99 | 115 | 132 | 148
166 | 16 | 33 | 49 | 66 | 83 | 99 | 116 | 132 | 149
167 | 16 | 33 | 50 | 66 | 83 | 100 | 116 | 133 | 150
168 | 16 | 33 | 50 | 67 | 84 | 100 | [...] | 134 | 151
169 | 17 | 33 | 50 | 67 | 84 | 101 | 118 | 135 | 152
170 | 17 | 34 | 51 | 68 | 85 | 102 | 119 | 136 | 153
171 | 17 | 34 | 51 | 68 | 85 | 102 | 119 | 136 | 153
172 | 17 | 34 | 51 | 68 | 86 | 103 | 120 | 137 | 154
173 | 17 | 34 | 51 | 69 | 86 | 103 | 121 | 138 | 155
174 | 17 | 34 | 52 | 69 | 87 | 104 | 121 | 139 | 156
175 | 17 | 34 | 52 | 70 | 87 | 105 | 122 | 140 | 157
176 | 17 | 35 | 52 | 70 | 88 | 105 | 123 | 140 | 158
177 | 17 | 35 | 53 | 70 | 88 | 106 | 123 | 141 | 159
178 | 17 | 35 | 53 | 71 | 89 | 106 | 124 | 142 | 160
179 | 17 | 35 | 53 | 71 | 89 | 107 | 125 | 143 | 161
180 | 18 | 36 | 54 | 72 | 90 | 108 | 126 | 144 | 162
181 | 18 | 36 | 54 | 72 | 90 | 108 | 126 | 144 | 162
182 | 18 | 36 | 54 | 72 | 91 | 109 | 127 | 145 | 163
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
183 | 18 | 36 | 54 | 73 | 91 | 109 | 128 | 146 | 164
184 | 18 | 36 | 55 | 73 | 92 | 110 | 128 | 147 | 165
185 | 18 | 37 | 55 | 74 | 92 | 111 | 129 | 148 | 166
186 | 18 | 37 | 55 | 74 | 93 | 111 | 130 | 148 | 167
187 | 18 | 37 | 56 | 74 | 83 | 112 | 130 | 149 | 168
188 | 18 | 37 | 56 | 75 | 94 | 112 | 131 | 150 | 169
189 | 18 | 37 | 56 | 75 | 94 | 113 | 132 | 151 | 170
190 | 19 | 38 | 57 | 76 | 95 | 114 | 133 | 152 | 171
191 | 19 | 38 | 57 | 76 | 95 | 114 | 133 | 152 | 171
192 | 19 | 38 | 57 | 76 | 96 | 115 | 134 | 153 | 172
193 | 19 | 38 | 57 | 77 | 96 | 115 | 135 | 154 | 173
194 | 19 | 38 | 58 | 77 | 97 | 116 | 135 | 155 | 174
195 | 19 | 39 | 58 | 78 | 97 | 117 | 136 | 156 | 175
196 | 19 | 39 | 59 | 78 | 98 | 117 | 136 | 156 | 176
197 | 19 | 39 | 59 | 78 | 98 | 118 | 137 | 157 | 177
198 | 19 | 39 | 59 | 79 | 99 | 118 | 138 | 158 | 178
199 | 19 | 39 | 59 | 79 | 99 | 119 | 139 | 159 | 179
200 | 20 | 40 | 60 | 80 | 100 | 120 | 140 | 160 | 180
201 | 20 | 40 | 60 | 80 | 100 | 120 | 140 | 160 | 180
202 | 20 | 40 | 60 | 80 | 101 | 121 | 141 | 161 | 181
203 | 20 | 40 | 60 | 81 | 101 | 121 | 142 | 162 | 182
204 | 20 | 40 | 61 | 81 | 102 | 122 | 142 | 163 | 183
205 | 20 | 41 | 61 | 82 | 102 | 123 | 143 | 164 | 184
206 | 20 | 41 | 61 | 82 | 103 | 123 | 144 | 164 | 185
207 | 20 | 41 | 62 | 82 | 103 | 124 | 144 | 165 | 186
208 | 20 | 41 | 62 | 83 | 104 | 124 | 145 | 166 | 187
209 | 20 | 41 | 62 | 83 | 104 | 125 | 146 | 167 | 188
210 | 21 | 42 | 63 | 84 | 105 | 126 | 147 | 168 | 189
211 | 21 | 42 | 63 | 84 | 105 | 126 | 147 | 168 | 189
212 | 21 | 42 | 63 | 84 | 106 | 127 | 148 | 169 | 190
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
213 | 21 | 42 | 63 | 85 | 106 | 127 | 149 | 170 | 191
214 | 21 | 42 | 64 | 85 | 107 | 128 | 149 | 171 | 192
215 | 21 | 43 | 64 | 86 | 107 | 129 | 150 | 172 | 193
216 | 21 | 43 | 64 | 86 | 108 | 129 | 151 | 172 | 194
217 | 21 | 43 | 65 | 86 | 108 | 130 | 151 | 173 | 195
218 | 21 | 43 | 65 | 87 | 109 | 130 | 152 | 174 | 196
219 | 21 | 43 | 65 | 87 | 109 | 131 | 153 | 175 | 197
220 | 22 | 44 | 66 | 88 | 110 | 132 | 154 | 176 | 198
221 | 22 | 44 | 66 | 88 | 110 | 132 | 154 | 176 | 198
222 | 22 | 44 | 66 | 88 | 111 | 133 | 155 | 177 | 199
223 | 22 | 44 | 66 | 89 | 111 | 133 | 156 | 178 | 200
224 | 22 | 44 | 67 | 89 | 112 | 134 | 156 | 179 | 201
225 | 22 | 45 | 67 | 90 | 112 | 135 | 157 | 180 | 202
226 | 22 | 45 | 67 | 90 | 113 | 135 | 158 | 180 | 203
227 | 22 | 45 | 68 | 90 | 113 | 136 | 158 | 181 | 204
228 | 22 | 45 | 68 | 91 | 114 | 136 | 159 | 182 | 205
229 | 22 | 45 | 68 | 91 | 114 | 137 | 160 | 183 | 206
230 | 23 | 46 | 69 | 92 | 115 | 138 | 161 | 184 | 207
231 | 23 | 46 | 69 | 92 | 115 | 138 | 161 | 184 | 207
232 | 23 | 46 | 69 | 92 | 116 | 139 | 162 | 185 | 208
233 | 23 | 46 | 69 | 93 | 116 | 139 | 163 | 186 | 209
234 | 23 | 46 | 70 | 93 | 117 | 140 | 163 | 187 | 210
235 | 23 | 47 | 70 | 94 | 117 | 141 | 164 | 188 | 211
236 | 23 | 47 | 70 | 94 | 118 | 141 | 165 | 188 | 212
237 | 23 | 47 | 71 | 94 | 118 | 142 | 165 | 189 | 213
238 | 23 | 47 | 71 | 95 | 119 | 142 | 166 | 190 | 214
239 | 23 | 47 | 71 | 95 | 119 | 143 | 167 | 191 | 215
240 | 24 | 48 | 72 | 96 | 120 | 144 | 168 | 192 | 216
241 | 24 | 48 | 72 | 96 | 120 | 144 | 168 | 192 | 216
242 | 24 | 48 | 72 | 96 | 121 | 145 | 169 | 193 | 217
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
243 | 24 | 48 | 72 | 97 | 121 | 145 | 170 | 194 | 218
244 | 24 | 48 | 73 | 97 | 122 | 146 | 170 | 195 | 219
245 | 24 | 49 | 73 | 98 | 122 | 147 | 171 | 196 | 220
246 | 24 | 49 | 73 | 98 | 123 | 147 | 172 | 196 | 221
247 | 24 | 49 | 74 | 98 | 123 | 148 | 172 | 197 | 222
248 | 24 | 49 | 74 | 99 | 124 | 148 | 173 | 198 | 223
249 | 24 | 49 | 74 | 99 | 124 | 149 | 174 | 199 | 224
250 | 25 | 50 | 75 | 100 | 125 | 150 | 175 | 200 | 225
251 | 25 | 50 | 75 | 100 | 125 | 150 | 175 | 200 | 225
252 | 25 | 50 | 75 | 100 | 126 | 151 | 176 | 201 | 226
253 | 25 | 50 | 75 | 101 | 126 | 151 | 177 | 202 | 227
254 | 25 | 50 | 76 | 101 | 127 | 152 | 177 | 203 | 228
255 | 25 | 50 | 76 | 102 | 127 | 153 | 178 | 204 | 229
256 | 25 | 51 | 76 | 102 | 128 | 153 | 179 | 204 | 230
257 | 25 | 51 | 77 | 102 | 128 | 154 | 179 | 205 | 231
258 | 25 | 51 | 77 | 103 | 129 | 154 | 180 | 206 | 232
259 | 25 | 51 | 77 | 103 | 129 | 155 | 181 | 207 | 233
260 | 26 | 52 | 78 | 104 | 130 | 156 | 182 | 208 | 234
261 | 26 | 52 | 78 | 104 | 130 | 156 | 182 | 208 | 234
262 | 26 | 52 | 78 | 104 | 131 | 156 | 183 | 209 | 235
263 | 26 | 52 | 78 | 105 | 131 | 157 | 184 | 210 | 236
264 | 26 | 52 | 79 | 105 | 132 | 158 | 184 | 211 | 237
265 | 26 | 53 | 79 | 106 | 132 | 159 | 185 | 212 | 238
266 | 26 | 53 | 79 | 106 | 133 | 159 | 186 | 212 | 239
267 | 26 | 53 | 80 | 106 | 133 | 160 | 186 | 213 | 240
268 | 26 | 53 | 80 | 107 | 134 | 160 | 187 | 214 | 241
269 | 26 | 53 | 80 | 107 | 134 | 161 | 188 | 215 | 242
270 | 27 | 54 | 81 | 108 | 135 | 162 | 189 | 216 | 243
271 | 27 | 54 | 81 | 108 | 135 | 162 | 189 | 216 | 243
272 | 27 | 54 | 81 | 108 | 136 | 163 | 190 | 217 | 244
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
273 | 27 | 54 | 81 | 109 | 136 | 163 | 191 | 218 | 245
274 | 27 | 54 | 82 | 109 | 137 | 164 | 191 | 219 | 246
275 | 27 | 55 | 82 | 110 | 137 | 165 | 192 | 220 | 247
276 | 27 | 55 | 82 | 110 | 138 | 165 | 193 | 220 | 248
277 | 27 | 55 | 83 | 110 | 138 | 166 | 193 | 221 | 249
278 | 27 | 55 | 83 | 111 | 139 | 166 | 194 | 222 | 250
279 | 27 | 55 | 83 | 111 | 139 | 167 | 195 | 223 | 251
280 | 28 | 56 | 84 | 112 | 140 | 168 | 196 | 224 | 252
281 | 28 | 56 | 84 | 112 | 140 | 168 | 196 | 224 | 252
282 | 28 | 56 | 84 | 112 | 141 | 169 | 197 | 225 | 253
283 | 28 | 56 | 84 | 113 | 141 | 169 | 198 | 226 | 254
284 | 28 | 56 | 85 | 113 | 142 | 170 | 198 | 227 | 255
285 | 28 | 57 | 85 | 114 | 142 | 171 | 199 | 228 | 256
286 | 28 | 57 | 85 | 114 | 143 | 171 | 200 | 228 | 257
287 | 28 | 57 | 86 | 114 | 143 | 172 | 200 | 229 | 258
288 | 28 | 57 | 86 | 115 | 144 | 172 | 201 | 230 | 259
289 | 28 | 57 | 86 | 115 | 144 | 173 | 202 | 231 | 260
290 | 29 | 58 | 87 | 116 | 145 | 174 | 203 | 232 | 261
291 | 29 | 58 | 87 | 116 | 145 | 174 | 203 | 232 | 261
292 | 29 | 58 | 87 | 116 | 146 | 175 | 204 | 233 | 262
293 | 29 | 58 | 87 | 117 | 146 | 175 | 205 | 234 | 263
294 | 29 | 58 | 88 | 117 | 147 | 176 | 205 | 235 | 264
295 | 29 | 59 | 88 | 118 | 147 | 177 | 206 | 236 | 265
296 | 29 | 59 | 88 | 118 | 148 | 177 | 207 | 236 | 266
297 | 29 | 59 | 88 | 118 | 148 | 178 | 207 | 237 | 267
298 | 29 | 59 | 89 | 119 | 149 | 178 | 208 | 238 | 268
299 | 29 | 59 | 89 | 119 | 149 | 179 | 209 | 239 | 269
300 | 30 | 60 | 90 | 120 | 150 | 180 | 210 | 240 | 270
301 | 30 | 60 | 90 | 120 | 150 | 180 | 210 | 240 | 270
302 | 30 | 60 | 90 | 120 | 151 | 181 | 211 | 241 | 271
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
303 | 30 | 60 | 90 | 121 | 151 | 181 | 212 | 242 | 272
304 | 30 | 60 | 91 | 121 | 152 | 182 | 212 | 243 | 273
305 | 30 | 61 | 91 | 122 | 152 | 183 | 213 | 244 | 274
306 | 30 | 61 | 91 | 122 | 153 | 183 | 214 | 244 | 275
307 | 30 | 61 | 92 | 122 | 153 | 184 | 214 | 245 | 276
308 | 30 | 61 | 92 | 123 | 154 | 184 | 215 | 246 | 277
309 | 30 | 61 | 92 | 123 | 154 | 185 | 216 | 247 | 278
310 | 31 | 62 | 93 | 124 | 155 | 186 | 217 | 248 | 279
311 | 31 | 62 | 93 | 124 | 155 | 186 | 217 | 248 | 279
312 | 31 | 62 | 93 | 124 | 156 | 187 | 218 | 249 | 280
313 | 31 | 62 | 93 | 125 | 156 | 187 | 219 | 250 | 281
314 | 31 | 62 | 94 | 125 | 157 | 183 | 219 | 251 | 282
315 | 31 | 63 | 94 | 126 | 157 | 189 | 220 | 252 | 283
316 | 31 | 63 | 94 | 126 | 158 | 189 | 221 | 252 | 284
317 | 31 | 63 | 95 | 126 | 158 | 190 | 221 | 253 | 285
318 | 31 | 63 | 95 | 127 | 159 | 190 | 222 | 254 | 286
319 | 31 | 63 | 95 | 127 | 159 | 191 | 223 | 255 | 287
320 | 32 | 64 | 96 | 128 | 160 | 192 | 224 | 256 | 288
321 | 32 | 64 | 96 | 128 | 160 | 192 | 224 | 256 | 288
322 | 32 | 64 | 96 | 128 | 161 | 193 | 225 | 257 | 289
323 | 32 | 64 | 96 | 129 | 161 | 193 | 226 | 258 | 290
324 | 32 | 64 | 97 | 129 | 162 | 194 | 226 | 259 | 291
325 | 32 | 65 | 97 | 130 | 162 | 195 | 227 | 260 | 292
326 | 32 | 65 | 97 | 130 | 163 | 195 | 228 | 260 | 293
327 | 32 | 65 | 98 | 130 | 163 | 196 | 228 | 261 | 294
328 | 32 | 65 | 98 | 131 | 163 | 196 | 229 | 262 | 295
329 | 32 | 65 | 98 | 131 | 164 | 197 | 230 | 263 | 296
330 | 33 | 66 | 99 | 132 | 165 | 198 | 231 | 264 | 297
331 | 33 | 66 | 99 | 132 | 165 | 198 | 231 | 264 | 297
332 | 33 | 66 | 99 | 132 | 166 | 199 | 232 | 265 | 298
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
333 | 33 | 66 | 99 | 133 | 166 | 199 | 233 | 266 | 299
334 | 33 | 66 | 100 | 133 | 167 | 200 | 233 | 267 | 300
335 | 33 | 67 | 100 | 134 | 167 | 201 | 234 | 268 | 301
336 | 33 | 67 | 100 | 134 | 168 | 201 | 235 | 268 | 302
337 | 33 | 67 | 101 | 134 | 168 | 202 | 235 | 269 | 303
338 | 33 | 67 | 101 | 135 | 169 | 202 | 236 | 270 | 304
339 | 33 | 67 | 101 | 135 | 169 | 203 | 237 | 271 | 305
340 | 34 | 68 | 102 | 136 | 170 | 204 | 238 | 272 | 306
341 | 34 | 68 | 102 | 136 | 170 | 204 | 238 | 272 | 306
342 | 34 | 68 | 102 | 136 | 171 | 205 | 239 | 273 | 307
343 | 34 | 68 | 102 | 137 | 171 | 205 | 240 | 274 | 308
344 | 34 | 68 | 103 | 137 | 172 | 206 | 240 | 275 | 309
345 | 34 | 69 | 103 | 138 | 172 | 207 | 241 | 276 | 310
346 | 34 | 69 | 103 | 138 | 173 | 207 | 242 | 276 | 311
347 | 34 | 69 | 104 | 138 | 173 | 208 | 242 | 277 | 312
348 | 34 | 69 | 104 | 139 | 174 | 208 | 243 | 278 | 313
349 | 34 | 69 | 104 | 139 | 174 | 209 | 244 | 279 | 314
350 | 35 | 70 | 105 | 140 | 175 | 210 | 245 | 280 | 315
351 | 35 | 70 | 105 | 140 | 175 | 210 | 245 | 280 | 315
352 | 35 | 70 | 105 | 140 | 176 | 211 | 246 | 281 | 316
353 | 35 | 70 | 105 | 141 | 176 | 211 | 247 | 282 | 317
354 | 35 | 70 | 106 | 141 | 177 | 212 | 247 | 283 | 318
355 | 35 | 71 | 106 | 142 | 177 | 213 | 248 | 284 | 319
356 | 35 | 71 | 106 | 142 | 178 | 213 | 249 | 284 | 320
357 | 35 | 71 | 107 | 142 | 178 | 214 | 249 | 285 | 321
358 | 35 | 71 | 107 | 143 | 179 | 214 | 250 | 286 | 322
359 | 35 | 71 | 107 | 143 | 179 | 215 | 251 | 287 | 323
360 | 36 | 72 | 108 | 144 | 180 | 216 | 252 | 288 | 324
361 | 36 | 72 | 108 | 144 | 180 | 216 | 252 | 288 | 324
362 | 36 | 72 | 108 | 144 | 181 | 217 | 253 | 289 | 325
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
363 | 36 | 72 | 108 | 145 | 181 | 217 | 254 | 290 | 326
364 | 36 | 72 | 109 | 145 | 182 | 218 | 254 | 291 | 327
365 | 36 | 73 | 109 | 146 | 182 | 219 | 255 | 292 | 328
366 | 36 | 73 | 109 | 146 | 182 | 219 | 256 | 292 | 329
367 | 36 | 73 | 110 | 146 | 183 | 220 | 256 | 293 | 330
368 | 36 | 73 | 110 | 147 | 184 | 220 | 257 | 294 | 331
369 | 36 | 73 | 110 | 147 | 184 | 221 | 258 | 295 | 332
370 | 37 | 74 | 111 | 148 | 185 | 222 | 259 | 296 | 333
371 | 37 | 74 | 111 | 148 | 185 | 222 | 259 | 296 | 333
372 | 37 | 74 | 111 | 148 | 186 | 223 | 260 | 297 | 334
373 | 37 | 74 | 111 | 149 | 186 | 223 | 261 | 298 | 335
374 | 37 | 74 | 112 | 149 | 187 | 224 | 261 | 299 | 336
375 | 37 | 75 | 112 | 150 | 187 | 225 | 262 | 300 | 337
376 | 37 | 75 | 112 | 150 | 188 | 225 | 263 | 300 | 338
377 | 37 | 75 | 113 | 150 | 188 | 226 | 263 | 301 | 339
378 | 37 | 75 | 113 | 151 | 189 | 226 | 264 | 302 | 340
379 | 37 | 75 | 113 | 151 | 189 | 227 | 265 | 303 | 341
380 | 38 | 76 | 114 | 152 | 190 | 228 | 266 | 304 | 342
381 | 38 | 76 | 114 | 152 | 190 | 228 | 266 | 304 | 342
382 | 38 | 76 | 114 | 152 | 191 | 229 | 267 | 305 | 343
383 | 38 | 76 | 114 | 153 | 191 | 229 | 268 | 306 | 344
384 | 38 | 76 | 115 | 153 | 192 | 230 | 268 | 307 | 345
385 | 38 | 77 | 115 | 154 | 192 | 231 | 269 | 308 | 346
386 | 38 | 77 | 115 | 154 | 193 | 231 | 270 | 308 | 347
387 | 38 | 77 | 116 | 154 | 193 | 232 | 270 | 309 | 348
388 | 38 | 77 | 116 | 155 | 194 | 232 | 271 | 310 | 349
389 | 38 | 77 | 116 | 155 | 194 | 233 | 272 | 311 | 350
390 | 39 | 78 | 117 | 156 | 195 | 233 | 273 | 312 | 351
391 | 39 | 78 | 117 | 156 | 195 | 233 | 273 | 312 | 351
392 | 39 | 78 | 117 | 156 | 196 | 234 | 274 | 313 | 352
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
393 | 39 | 78 | 117 | 157 | 196 | 235 | 275 | 314 | 353
394 | 39 | 78 | 118 | 157 | 197 | 236 | 275 | 315 | 354
395 | 39 | 79 | 118 | 158 | 197 | 237 | 276 | 316 | 355
396 | 39 | 79 | 118 | 158 | 198 | 237 | 277 | 316 | 356
397 | 39 | 79 | 119 | 158 | 198 | 238 | 277 | 317 | 357
398 | 39 | 79 | 119 | 159 | 199 | 238 | 278 | 318 | 358
399 | 39 | 79 | 119 | 159 | 199 | 239 | 279 | 319 | 359
400 | 40 | 80 | 120 | 160 | 200 | 240 | 280 | 320 | 360
401 | 40 | 80 | 120 | 160 | 200 | 240 | 280 | 320 | 360
402 | 40 | 80 | 120 | 160 | 201 | 241 | 281 | 321 | 361
403 | 40 | 80 | 120 | 161 | 201 | 241 | 282 | 322 | 362
404 | 40 | 80 | 121 | 161 | 202 | 242 | 282 | 323 | 363
405 | 40 | 81 | 121 | 162 | 202 | 243 | 283 | 324 | 364
406 | 40 | 81 | 121 | 162 | 203 | 243 | 284 | 324 | 365
407 | 40 | 81 | 122 | 162 | 203 | 244 | 284 | 325 | 366
408 | 40 | 81 | 122 | 163 | 204 | 244 | 285 | 326 | 367
409 | 40 | 81 | 122 | 163 | 204 | 245 | 286 | 327 | 368
410 | 41 | 82 | 123 | 164 | 205 | 246 | 287 | 328 | 369
411 | 41 | 82 | 123 | 164 | 205 | 246 | 287 | 328 | 369
412 | 41 | 82 | 123 | 164 | 206 | 247 | 288 | 329 | 370
413 | 41 | 82 | 123 | 165 | 206 | 247 | 289 | 330 | 371
414 | 41 | 82 | 124 | 165 | 207 | 248 | 289 | 331 | 372
415 | 41 | 83 | 124 | 166 | 207 | 249 | 290 | 332 | 373
416 | 41 | 83 | 124 | 166 | 208 | 249 | 291 | 332 | 374
417 | 41 | 83 | 125 | 166 | 208 | 250 | 291 | 333 | 375
418 | 41 | 83 | 125 | 167 | 209 | 250 | 292 | 334 | 376
419 | 41 | 83 | 125 | 167 | 209 | 251 | 293 | 335 | 377
420 | 42 | 84 | 126 | 168 | 210 | 252 | 294 | 336 | 378
421 | 42 | 84 | 126 | 168 | 210 | 252 | 294 | 336 | 378
422 | 42 | 84 | 126 | 168 | 211 | 253 | 295 | 337 | 379
D | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
423 | 42 | 84 | 126 | 169 | 211 | 253 | 296 | 338 | 380
424 | 42 | 84 | 127 | 169 | 212 | 254 | 296 | 339 | 381
425 | 42 | 85 | 127 | 170 | 212 | 255 | 297 | 340 | 382
426 | 42 | 85 | 127 | 170 | 213 | 255 | 298 | 340 | 383
427 | 42 | 85 | 128 | 170 | 213 | 256 | 298 | 341 | 384
428 | 42 | 85 | 128 | 171 | 214 | 256 | 299 | 342 | 385
429 | 42 | 85 | 128 | 171 | 214 | 257 | 300 | 343 | 386
430 | 43 | 86 | 129 | 172 | 215 | 258 | 301 | 344 | 387
431 | 43 | 86 | 129 | 172 | 215 | 258 | 301 | 344 | 387
432 | 43 | 86 | 129 | 172 | 216 | 259 | 302 | 345 | 388
433 | 43 | 86 | 129 | 173 | 216 | 259 | 303 | 346 | 389
434 | 43 | 86 | 130 | 173 | 217 | 260 | 304 | 347 | 390
435 | 43 | 87 | 130 | 174 | 217 | 261 | 304 | 348 | 391

A TABLE OF ARTIFICIAL SINES AND TANGENTS To every DEGREE and MINUTE OF THE QUADRANT.

[Page]

LONDON, Printed by J. Heptinstall for W. Freeman, at the Artichoke next St. Dunstan'S Church in Fleet street. MDCLXXXVII.

[Page]

Degree 0.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 0. 000000 | 10. 000000 | 0. 000000 | Infinita. | 60
1 | 6. 463726 | 9. 999999 | 6. 463726 | 13. 536274 | 59
2 | 6. 764756 | 9. 999999 | 6. 764756 | 13. 235244 | 58
3 | 6. 940847 | 9. 999999 | 6. 940847 | 13. 059153 | 57
4 | 7. 065786 | 9. 999999 | 7. 065786 | 12. 934214 | 56
5 | 7. 162696 | 9. 999999 | 7. 162696 | 12. 837304 | 55
6 | 7. 241877 | 9. 999999 | 7. 241878 | 12. 758122 | 54
7 | 7. 308824 | 9. 999999 | 7. 308825 | 12. 691175 | 53
8 | 7. 366816 | 9. 999999 | 7. 366817 | 12. 633183 | 52
9 | 7. [...] | 9. 999999 | 7. 417970 | 12. 582030 | 51
10 | 7. 463726 | 9. 999998 | 7. 463727 | 12. 536273 | 50
11 | 7. 505118 | 9. 999998 | 7. 505120 | 12. 494880 | 49
12 | 7. 542906 | 9. 999997 | 7. 542909 | 12. 457091 | 48
13 | 7. 577668 | 9. 999997 | 7. 577272 | 12. 422328 | 47
14 | 7. 609853 | 9. 999996 | 7. 609857 | 12. 390143 | 46
15 | 7. 639816 | 9. 999996 | 7. 639826 | 12. 360180 | 45
16 | 7. 667844 | 9. 999995 | 7. 667849 | 12. 332151 | 44
17 | 7. 694173 | 9. 999995 | 7. 694179 | 12. 305821 | 43
18 | 7. 718977 | 9. 999994 | 7. 719003 | 12. 281997 | 42
19 | 7. 742477 | 9. 999993 | 7. 742484 | 12. 257516 | 41
20 | 7. 764754 | 9. 999993 | 7. 764761 | 12. 235239 | 40
21 | 7. 785943 | 9. 999992 | 7. 785951 | 12. 214049 | 39
22 | 7. 806146 | [...] 999991 | 7. 806145 | 12. 193845 | 38
23 | 7. 825451 | 9. 999990 | 7. 825460 | 12. 174540 | 37
24 | 7. 843034 | 9. 999989 | 7. 843944 | 12. 156056 | 36
25 | 7. 861662 | 9. 999989 | 7. 861674 | 12. 138326 | 35
26 | 7. 878695 | 9. 999988 | 7. 878708 | 12. 121292 | 34
27 | 7. 895085 | 9. 999987 | 7. 895099 | 12. 104901 | 33
28 | 7. 910879 | 9. 999986 | 7. 910894 | 12. 089106 | 32
29 | 7. 926119 | 9. 999985 | 7. 926134 | 12. 073866 | 31
30 | 7. 940842 | 9. 999983 | 7. 940858 | 12. 059142 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 89.
Degree 0.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 7. 940842 | 9. 999983 | 7. 940858 | 12. 059142 | 30
31 | 7. 955082 | 9. 999982 | 7. 955100 | 12. 044900 | 29
32 | 7. 968870 | 9. 999981 | 7. 968889 | 12. 031111 | 28
33 | 7. 982233 | 9. 999980 | 7. 982253 | 12. 017747 | 27
34 | 7. 995198 | 9. 999978 | 7. 995215 | 12. 004781 | 26
35 | 8. 007787 | 9. 999978 | 8. 007810 | 11. 992191 | 25
36 | 8. 020021 | 9. 999976 | 8. 020044 | 11. 979956 | 24
37 | 8. 031919 | 9. 999975 | 8. 031945 | 11. 968055 | 23
38 | 8. 043601 | 9. 999973 | 8. 043527 | 11. 956473 | 22
39 | 8. 054781 | 9. 999972 | 8. 054809 | 11. 945181 | 21
40 | 8. 065776 | 9. 999971 | 8. 065806 | 11. 934194 | 20
41 | 8. 076500 | 9. 999969 | 8. 076531 | 11. 923469 | 19
42 | 8. 086965 | 9. 999968 | 8. 086997 | 11. 913003 | 18
43 | 8. 097183 | 9. 999966 | 8. 097217 | 11. 902783 | 17
44 | 8. 107167 | 9. 999964 | 8. 107203 | 11. 892797 | 16
45 | 8. 116926 | 9. 999963 | 8. 116963 | 11. 883037 | 15
46 | 8. 126471 | 9. 999961 | 8. 126510 | 11. 873490 | 14
47 | 8. 135810 | 9. 999959 | 8. 135851 | 11. 864149 | 13
48 | 8. 144953 | 9. 999958 | 8. 144996 | 11. 855004 | 12
49 | 8. 153907 | 9. 999956 | 8. 153952 | 11. 846048 | 11
50 | 8. 162681 | 9. 999954 | 8. 162737 | 11. 837273 | 10
51 | 8. 171280 | 9. 999952 | 8. 171328 | 11. 828672 | 9
52 | 8. 179713 | 9. 999950 | 8. 179763 | 11. 820237 | 8
53 | 8. 187985 | 9. 999948 | 8. 188036 | 11. 811964 | 7
54 | 8. 196102 | 9. 999946 | 8. 196156 | 11. 803844 | 6
55 | 8. 204070 | 9. 999944 | 8. 204126 | 11. 795674 | 5
56 | 8. 211895 | 9. 999942 | 8. 211953 | 11. 788047 | 4
57 | 8. 219581 | 9. 999940 | 8. 219641 | 11. 780359 | 3
58 | 8. 227134 | 9. 999938 | 8. 227195 | 11. 772805 | 2
59 | 8. 234557 | 9. 999936 | 8. 234621 | 11. 765379 | 1
60 | 8. 241855 | 9. 999934 | 8. 241921 | 11. 758079 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 89.
Degree 1.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 8. 241855 | 9. 999934 | 8. 241921 | 11. 758079 | 60
1 | 8. 249033 | 9. 999932 | 8. 249102 | 11. 750898 | 59
2 | 8. 256094 | 9. 999929 | 8. 256165 | 11. 743835 | 58
3 | 8. 263042 | 9. 999927 | 8. 263115 | 11. 736885 | 57
4 | 8. 269881 | 9. 999925 | 8. 269956 | 11. 730044 | 56
5 | 8. 276614 | 9. 999922 | 8. 276691 | 11. 723309 | 55
6 | 8. 283243 | 9. 999920 | 8. 283323 | 11. 716677 | 54
7 | 8. 289773 | 9. 999918 | 8. 289856 | 11. 716144 | 53
8 | 8. 296207 | 9. 999915 | 8. 296292 | 11. 703708 | 52
9 | 8. 302546 | 9. 999913 | 8. 302634 | 11. 697366 | 51
10 | 8. 308794 | 9. 999910 | 8. 308884 | 11. 691116 | 50
11 | 8. 314954 | 9. 999907 | 8. 315046 | 11. 684954 | 49
12 | 8. 321027 | 9. 999905 | 8. 321122 | 11. 678878 | 48
13 | 8. 327016 | 9. 999902 | 8. 327114 | 11. 672886 | 47
14 | 8. 332924 | 9. 999899 | 8. 333025 | 11. 666975 | 46
15 | 8. 338753 | 9. 999897 | 8. 338856 | 11. 661144 | 45
16 | 8. 344504 | 9. 999894 | 8. 344610 | 11. 655390 | 44
17 | 8. 350180 | 9. 999891 | 8. 350289 | 11. 649711 | 43
18 | 8. 355783 | 9. 999888 | 8. 355895 | 11. 644105 | 42
19 | 8. 361315 | 9. 999885 | 8. 361430 | 11. 638570 | 41
20 | 8. 366777 | 9. 999882 | 8. 366895 | 11. 633105 | 40
21 | 8. 372171 | 9. 999879 | 8. 372292 | 11. 627708 | 39
22 | 8. 377499 | 9. 999876 | 8. 377622 | 11. 622378 | 38
23 | 8. 387762 | 9. 999873 | 8. 382889 | 11. 617111 | 37
24 | 8. 387962 | 9. 999870 | 8. 388092 | 11. 611908 | 36
25 | 8. 393101 | 9. 999867 | 8. 393234 | 11. 606766 | 35
26 | 8. 398179 | 9. 999864 | 8. 398315 | 11. 601685 | 34
27 | 8. 403199 | 9. 999861 | 8. 403338 | 11. 596662 | 33
28 | 8. 408161 | 9. 999858 | 8. 408304 | 11. 591696 | 32
29 | 8. 413068 | 9. 999854 | 8. 413213 | 11. 586787 | 31
30 | [...] | 9. 999851 | 8. 418068 | 11. 581932 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 88.
Degree 1.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 8. 417919 | 9. 999851 | 8. 418068 | 11. 581932 | 30
31 | 8. 422717 | 9. 999848 | 8. 422869 | 11. 577131 | 29
32 | 8. 427462 | 9. 999844 | 8. 427618 | 11. 572382 | 28
33 | 8. 432156 | 9. 999841 | 8. 432315 | 11. 567685 | 27
34 | 8. 436800 | 9. 999838 | 8. 436962 | 11. 563038 | 26
35 | 8. 441394 | 9. 999834 | 8. 441560 | 11. 558440 | 25
36 | 8. 445941 | 9. 999831 | 8. 446110 | 11. 553990 | 24
37 | 8. 450440 | 9. 999827 | 8. 450613 | 11. 549387 | 23
38 | 8. 454893 | 9. 999824 | 8. 455070 | 11. 544930 | 22
39 | 8. 459301 | 9. 999820 | 8. 459481 | 11. 540519 | 21
40 | 8. 463665 | 9. 999816 | 8. 463849 | 11. 536151 | 20
41 | 8. 467985 | 9. 999812 | 8. 468172 | 11. 531828 | 19
42 | 8. 472263 | 9. 999809 | 8. 472454 | 11. 527546 | 18
43 | 8. 476498 | 9. 999805 | 8. 476693 | 11. 523307 | 17
44 | 8. 480693 | 9. 999801 | 8. 480892 | 11. 519108 | 16
45 | 8. 484848 | 9. 999797 | 8. 485050 | 11. 514950 | 15
46 | 8. 488963 | 9. 999794 | 8. 486170 | 11. 510830 | 14
47 | 8. 493040 | 9. 999790 | 8. 483250 | 11. 506750 | 13
48 | 8. 497078 | 9. 999786 | 8. 497293 | 11. 502707 | 12
49 | 8. 501080 | 9. 999782 | 8. 501298 | 11. 498702 | 11
50 | 8. 505045 | 9. 999778 | 8. 505267 | 11. 494733 | 10
51 | 8. 508974 | 9. 999774 | 8. 509200 | 11. 490800 | 9
52 | 8. 512867 | 9. 999769 | 8. 513098 | 11. 486902 | 8
53 | 8. 516726 | 9. 999765 | 8. 516961 | 11. 483039 | 7
54 | 8. 520551 | 9. 999761 | 8. 520790 | 11. 479210 | 6
55 | 8. 524343 | 9. 999756 | 8. 524586 | 11. 475414 | 5
56 | 8. 528102 | 9. 999753 | 8. 528349 | 11. 471651 | 4
57 | 8. 531828 | 9. 999748 | 8. 532080 | 11. 467620 | 3
58 | 8. 535523 | 9. 999744 | 8. 535779 | 11. 464221 | 2
59 | 8. 539186 | 9. 999740 | 8. 539447 | 11. 460553 | 1
60 | 8. 542819 | 9. 999735 | 8. 543084 | 11. 456916 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 88.
Degree 2.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 8. 542819 | 9. 999735 | 8. 543084 | 11. 456916 | 60
1 | 8. 546422 | 9. 999731 | 8. 546691 | 11. 453309 | 59
2 | 8. 549995 | 9. 999726 | 8. 550268 | 11. 449732 | 58
3 | 8. 553558 | 9. 999722 | 8. 553817 | 11. 446183 | 57
4 | 8. 557054 | 9. 999717 | 8. 557336 | 11. 442664 | 56
5 | 8. 560540 | 9. 999713 | 8. 560827 | 11. 439172 | 55
6 | 8. 563999 | 9. 999708 | 8. 564291 | 11. 435709 | 54
7 | 8. 567431 | 9. 999703 | 8. 567727 | 11. 432272 | 53
8 | 8. 570836 | 9. 999699 | 8. 571137 | 11. 428863 | 52
9 | 8. 574214 | 9. 999694 | 8. 574520 | 11. 425480 | 51
10 | 8. 577566 | 9. 999689 | 8. 577877 | 11. 422123 | 50
11 | 8. 580892 | 9. 999685 | 8. 581208 | 11. 418792 | 49
12 | 8. 584193 | 9. 999680 | 8. 584514 | 11. 415486 | 48
13 | 8. 587469 | 9. 999675 | 8. 587795 | 11. 412205 | 47
14 | 8. 590721 | 9. 999670 | 8. 591051 | 11. 408949 | 46
15 | 8. 593948 | 9. 999665 | 8. 594283 | 11. 405717 | 45
16 | 8. 597152 | 9. 999660 | 8. 597492 | 11. 402508 | 44
17 | 8. 600332 | 9. 999655 | 8. 600677 | 11. 399323 | 43
18 | 8. 603488 | 9. 999650 | 8. 603838 | 11. 396161 | 42
19 | 8. 606622 | 9. 999645 | 8. 606978 | 11. 393022 | 41
20 | 8. 609734 | 9. 999640 | 8. 610094 | 11. 389906 | 40
21 | 8. 612823 | 9. 999635 | 8. 613189 | 11. 386811 | 39
22 | 8. 615891 | 9. 999629 | 8. 616262 | 11. 383738 | 38
23 | 8. 618937 | 9. 999624 | 8. 619313 | 11. 380687 | 37
24 | 8. 621967 | 9. 999619 | 8. 622343 | 11. 377657 | 36
25 | 8. 624965 | 9. 999614 | 8. 625352 | 11. 374648 | 35
26 | 8. 627948 | 9. 999608 | 8. 628340 | 11. 371660 | 34
27 | 8. 630911 | 9. 999603 | 8. 631308 | 11. 368692 | 33
28 | 8. 633854 | 9. 999597 | 8. 634456 | 11. 365744 | 32
29 | 8. 636776 | 9. 999592 | 8. 637184 | 11. 362816 | 31
30 | 8. 639679 | 9. 999586 | 8. 640093 | 11. 359907 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 87.
Degree 2.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 8. 639679 | 9. 999586 | 8. 640093 | 11. 359907 | 30
31 | 8. 642563 | 9. 999581 | 8. 642982 | 11. 357017 | 29
32 | 8. 645428 | 9. 999575 | 8. 645853 | 11. 354147 | 28
33 | 8. 648274 | 9. 999570 | 8. 648704 | 11. 351296 | 27
34 | 8. 651102 | 9. 999564 | 8. 651538 | 11. 348463 | 26
35 | 8. 653911 | 9. 999558 | 8. 654352 | 11. 345648 | 25
36 | 8. 656702 | 9. 999553 | 8. 657149 | 11. 342851 | 24
37 | 8. 659475 | 9. 999547 | 8. 659928 | 11. 340072 | 23
38 | 8. 662230 | 9. 999541 | 8. 662689 | 11. 337311 | 22
39 | 8. 664968 | 9. 999535 | 8. 665433 | 11. 334567 | 21
40 | 8. 667689 | 9. 999529 | 8. 668160 | 11. 331840 | 20
41 | 8. 670393 | 9. 999523 | 8. 670869 | 11. 329130 | 19
42 | 8. 673080 | 9. 999518 | 8. 673563 | 11. 326437 | 18
43 | 8. 675751 | 9. 999512 | 8. 676239 | 11. 323761 | 17
44 | 8. 678405 | 9. 999506 | 8. 678899 | 11. 321100 | 16
45 | 8. 681043 | 9. 999499 | 8. 681544 | 11. 318456 | 15
46 | 8. 683665 | 9. 999493 | 8. 684172 | 11. 315828 | 14
47 | 8. 686272 | 9. 999487 | 8. 686784 | 11. 313216 | 13
48 | 8. 688892 | 9. 999481 | 8. 689381 | 11. 310619 | 12
49 | 8. 691438 | 9. 999475 | 8. 691963 | 11. 308037 | 11
50 | 8. 693998 | 9. 999469 | 8. 694529 | 11. 305471 | 10
51 | 8. 696543 | 9. 999462 | 8. 697081 | 11. 302919 | 9
52 | 8. 699073 | 9. 999456 | 8. 699617 | 11. 300383 | 8
53 | 8. 701589 | 9. 999450 | 8. 702139 | 11. 297861 | 7
54 | 8. 704090 | 9. 999443 | 8. 704646 | 11. 295354 | 6
55 | 8. 706576 | 9. 999437 | 8. 707130 | 11. 292860 | 5
56 | 8. 709049 | 9. 999431 | 8. 709618 | 11. 290381 | 4
57 | 8. 711507 | 9. 999424 | 8. 712083 | 11. 287917 | 3
58 | 8. 713952 | 9. 999418 | 8. 714534 | 11. 285466 | 2
59 | 8. 716383 | 9. 999411 | 8. 716972 | 11. 283028 | 1
60 | 8. 718800 | 9. 999404 | 8. 719396 | 11. 280604 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 87.

[Page]

Degree 3.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 8. 718800 | 9. 999404 | 8. 719396 | 11. 280604 | 60
1 | 8. 721204 | 9. 999398 | 8. 721806 | 11. 278194 | 59
2 | 8. 723595 | 9. 999391 | 8. 724254 | 11. 275796 | 58
3 | 8. 725972 | 9. 999384 | 8. 726588 | 11. 273412 | 57
4 | 8. 728336 | 9. 999378 | 8. 728959 | 11. 271041 | 56
5 | 8. 730688 | 9. 999371 | 8. 731317 | 11. 268683 | 55
6 | 8. 733027 | 9. 999364 | 8. 733663 | 11. 266337 | 54
7 | 8. 735354 | 9. 999357 | 8. 735996 | 11. 264004 | 53
8 | 8. 737667 | 9 999350 | 8. 738317 | 11. 261683 | 52
9 | 8. 739969 | 9. 999343 | 8. 740626 | 11. 259374 | 51
10 | 8. 742259 | 9. 999336 | 8. 742922 | 11. 257078 | 50
11 | 8. 744536 | 9. 999329 | 8. 745007 | 11. 254793 | 49
12 | 8. 746801 | 9. 999322 | 8 747479 | 11. 252521 | 48
13 | 8. 745955 | 9. 999315 | 8. 749740 | 11. 250240 | 47
14 | 8. 751297 | 9. 999308 | 8. 751989 | 11. 248011 | 46
15 | 8. 753528 | 9. 999301 | 8. 754227 | 11. 245773 | 45
16 | 8. 755747 | 9. 999294 | 8. 756453 | 11. 243542 | 44
17 | 8. 757955 | 9. 999286 | 8. 758668 | 11. 241332 | 43
18 | 8. 760151 | 9. 999279 | 8. 760872 | 11. 239128 | 42
19 | 8. 762337 | 9. 999272 | 8. 763065 | 11. 236935 | 41
20 | 8. 764511 | 9. 999265 | 8. 765246 | 11. 234754 | 40
21 | 8. 766675 | 9. 999257 | 8. 767417 | 11. 232583 | 39
22 | 8. 768828 | 9. 999250 | 8. 769578 | 11. 230422 | 38
23 | 8. 770970 | 9. 999242 | 8. 771727 | 11. 228273 | 37
24 | 8. 773101 | 9. 999235 | 8. 773866 | 11. 229134 | 36
25 | 8. 775223 | 9. 999227 | 8. 775995 | 11. 224005 | 35
26 | 8. 777333 | 9. 999220 | 8. 778114 | 11. 221886 | 34
27 | 8. 779434 | 9. 999212 | 8. 783222 | 11. 219778 | 33
28 | 8. 781524 | 9. 999204 | 8. 782320 | 11. 217680 | 32
29 | 8. 783605 | 9. 999197 | 8. 784404 | 11. 215592 | 31
30 | 8. 785675 | 9. 999189 | 8. 786486 | 11. 213514 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 86.
Degree 3.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 8. 785675 | 9. 999189 | 8. 786486 | 11. 213514 | 30
31 | 8. 787736 | 9. 999181 | 8. 788554 | 11. 211446 | 29
32 | 8. 789787 | 9. 999174 | 8. 790613 | 11. 209387 | 28
33 | 8. 791828 | 9. 999166 | 8. 792662 | 11. 207338 | 27
34 | 8. 793859 | 9 999158 | 8. 794701 | 11. 205299 | 26
35 | 8. 795881 | 9. 999150 | 8. 796731 | 11. 203269 | 25
36 | 8. 797894 | 9. 999142 | 8 798752 | 11 201248 | 24
37 | 8. 799897 | 9. 999134 | 8. 800763 | 11 199237 | 23
38 | 8. 801891 | 9. 999126 | 8. 802765 | 11. 197235 | 22
39 | 8 803876 | 9. 999118 | 8. 807458 | 11. 195242 | 21
40 | 8. 805852 | 9 999110 | 8. 806742 | 11. 193258 | 20
41 | 8. 807819 | 9. 999102 | 8. 808717 | 11. 191283 | 19
42 | 8. 809777 | 9. 999094 | 8. 812683 | 11. 189317 | 18
43 | 8. 811726 | 9 999086 | 8. 812641 | 11. 187359 | 17
44 | 8. 813667 | 9. 999077 | 8. 814589 | 11. 185411 | 16
45 | 8. 815598 | 9. 999069 | 8. 816529 | 11. 183471 | 15
46 | 8. 817522 | 9. 999061 | 8. 818461 | 11. 181539 | 14
47 | 8. 819436 | 9. 999052 | 8. [...] | [...] | 13
48 | 8. 821342 | 9. 999044 | 8. 822298 | 11. 177702 | 12
49 | 8. 823240 | 9. 999036 | 8. 824205 | 11. 175795 | 11
50 | 8. 825130 | 9. 999027 | 8 826103 | 11. 173897 | 10
51 | 8. 827011 | 9. 999019 | 8. 827992 | 11. 172003 | 9
52 | 8. 828884 | 9. 999010 | 8. 829874 | 11. 170126 | 8
53 | 8 830749 | 9. 999002 | 8. 831748 | 11. 168252 | 7
54 | 8. 832106 | 9. 998993 | 8. 833613 | 11. 166387 | 6
55 | 8. 834456 | 9. 998984 | 8. 835471 | 11. 164529 | 5
56 | 8. 836297 | 9. 998976 | 8. 837321 | 11. 162679 | 4
57 | 8. 838130 | 9. 998967 | 8. 839163 | 11. 160837 | 3
58 | 8. 839956 | 9 998958 | 8. 840998 | 11. 159002 | 2
59 | 8. 841774 | 9 998940 | 8. 842825 | 11. 157175 | 1
60 | 8. 843585 | 9. 998941 | 8. 844644 | 11. 155356 | 0
Co-sine | Sine | Co tang. | Tangent. | M
Degree 86.

[Page]

Degree 4.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 8. 843584 | 9. 998941 | 8. 844644 | 11. 155356 | 60
1 | 8. 845387 | 9. 998931 | 8. 846455 | 11. 153545 | 59
2 | 8. 847183 | 9 998923 | 8. 848240 | 11. 151740 | 58
3 | 8. 848971 | 9. 998914 | 8. 850057 | 11. 149943 | 57
4 | 8. 850751 | 9. 998905 | 8. 851846 | 11. 148154 | 56
5 | 8. 852525 | 9. 998896 | 8. 853628 | 11. 146372 | 55
6 | 8. 854291 | 9. 998887 | 8. 855403 | 11. 144597 | 54
7 | 8. 856049 | 9. 998878 | 8. 857171 | 11. 142829 | 53
8 | 8. 857801 | 9. 998869 | 8. 858932 | 11. 141068 | 52
9 | 8. 859546 | 9. 998860 | 8. 860686 | 11. 139314 | 51
10 | 8. 861283 | 9. 998851 | 8. 862433 | 11. 137567 | 50
11 | 8. 863014 | 9. 998841 | 8. 864173 | 11. 135827 | 49
12 | 8. 864738 | 9. 998832 | 8. 865906 | 11. 134094 | 48
13 | 8. 866454 | 9. 998823 | 8. 867632 | 11. 132368 | 47
14 | 8 868165 | 9. 998813 | 8. 869351 | 11. 130649 | 46
15 | 8. 869868 | 9. 998804 | 8. 871064 | 11. 128936 | 45
16 | 8. 871565 | 9. 998795 | 8. 872750 | 11. 127230 | 44
17 | 8. 873255 | 9. 998785 | 8. 874469 | 11. 125531 | 43
18 | 8. 874938 | 9. 998776 | 8. 876162 | 11. 123838 | 42
19 | 8. 876615 | 9. 998766 | 8. 897849 | 11. 122151 | 41
20 | 8. 878285 | 9 998757 | 8. 879529 | 11. 120471 | 40
21 | 8. 879949 | 9. 998747 | 8. 881202 | 11. 118798 | 39
22 | 8. 881607 | 9. 998738 | 8. 882869 | 11. 117131 | 38
23 | 8. 883258 | 9. 998728 | 8. 884530 | 11. 115470 | 37
24 | 8. 884903 | 9. 998718 | 8. 886185 | 11. 113815 | 36
25 | 8. 886542 | 9. 998708 | 8. 887833 | 11. 112167 | 35
26 | 8. 888174 | 9. 998699 | 8. 889476 | 11. 110524 | 34
27 | 8. 889801 | 9. 998689 | 8. 891112 | 11. 108888 | 33
28 | 8. 891421 | 9. 998679 | 8. 892742 | 11. 107258 | 32
29 | 8. 893035 | 9. 998669 | 8. 894366 | 11. 105634 | 31
30 | 8. 894643 | 9. 998659 | 8. 845984 | 11. 104016 | 30
Co-sine. | Sine | Co-tang. | Tangent. | M
Degree 85
Degree 4
M | Sine | Co-sine | Tangent | Co-tang.
30 | 8. 894643 | 9. 998659 | 8. 895984 | 11. 104016 | 30
31 | 8. 896246 | 9. 998649 | 8. 897596 | 11. 102404 | 29
32 | 8. 897842 | 9. 998639 | 8. 899203 | 11. 100797 | 28
33 | 8. 899432 | 9. 998629 | 8. 900803 | 11. 099197 | 27
34 | 8. 901017 | 9. 998619 | 8. 902398 | 11. 097602 | 26
35 | 8. 902596 | 9. 998609 | 8. 903987 | 11. 096013 | 25
36 | 8. 904169 | 9. 998599 | 8. 905570 | 11. 094430 | 24
37 | 8. 905736 | 9. 998589 | 8. 907147 | 11. 092853 | 23
38 | 8. 907297 | 9. 998577 | 8. 908719 | 11. 091281 | 22
39 | 8. 908853 | 9. 998568 | 8. 910285 | 11. 089715 | 21
40 | 8. 910404 | 9. 998558 | 8. 911846 | 11. 088154 | 20
41 | 8. 911949 | 9. 998548 | 8. 913401 | 11. 086599 | 19
42 | 8. 913488 | 9. 998537 | 8. 914951 | 11. 085049 | 18
43 | 8. 915022 | 9. 998527 | 8. 916495 | 11. 083505 | 17
44 | 8. 916550 | 9. 998516 | 8. 918034 | 11. 081966 | 16
45 | 8. 918073 | 9. 998506 | 8. 919568 | 11. 080432 | 15
46 | 8. 919591 | 9. 998495 | 8. 921096 | 11. 078904 | 14
47 | 8. 921103 | 9. 998485 | 8. 922619 | 11. 077381 | 13
48 | 8. 922610 | 9. 998474 | 8. 924136 | 11. 075864 | 12
49 | 8. 924112 | 9. 998464 | 8. 925649 | 11. 074351 | 11
50 | 8. 925609 | 9. 998453 | 8. 927156 | 11. 072844 | 10
51 | 8. 927100 | 9 998442 | 8. 928658 | 11. 071342 | 9
52 | 8. 928587 | 9. 998431 | 8. 930155 | 11. 069845 | 8
53 | 8. 930068 | 9. 998421 | 8. 931647 | 11. 068353 | 7
54 | 8. 931544 | 9. 998410 | 8. 933134 | 11. 066866 | 6
55 | 8. 933015 | 9. 998399 | 8. 934616 | 11. 065384 | 5
56 | 8. 934481 | 9. 998388 | 8. 936093 | 11. 063907 | 4
57 | 8. 935942 | 9. 998377 | 8. 937565 | 11. 062435 | 3
58 | 8. 937398 | 9. 998366 | 8. 939032 | 11. 060968 | 2
59 | 8. 938850 | 9. 998355 | 8. 940494 | 11. 059506 | 1
60 | 8. 940296 | 9. 998344 | 8. 941952 | 11. 058048 | 0
Co-sine | Sine | Co-tang. | Tangent. | M
Degree 85.

[Page]

Degree 5.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 8. 940296 | 9. 998344 | 8. 941952 | 11. 058048 | 60
1 | 8. 941738 | 9. 998333 | 8. 943404 | 11. 056596 | 59
2 | 8. 943174 | 9. 998322 | 8. 944852 | 11. 055148 | 58
3 | 8. 944606 | 9. 998311 | 8. 946295 | 11. 053705 | 57
4 | 8. 946034 | 9. 998300 | 8. 947734 | 11. 052266 | 56
5 | 8. 957456 | 9. 998289 | 8. 949168 | 11. 050832 | 55
6 | 8. 958814 | 9. 998277 | 8. 950597 | 11. 049403 | 54
7 | 8. 950287 | 9. 998266 | 8. 952021 | 11. 047979 | 53
8 | 8. 951696 | 9. 998255 | 8. 953441 | 11. 046559 | 52
9 | 8. 953099 | 9. 998243 | 8. 954856 | 11. 045144 | 51
10 | 8. 954499 | 9. 998232 | 8. 956267 | 11. 043733 | 50
11 | 8. 955894 | 9. 998220 | 8. 957674 | 11. 042326 | 49
12 | 8. 957284 | 9. 998209 | 8. 959075 | 11. 040925 | 48
13 | 8. 958670 | 9. 998197 | 8. 960473 | 11. 039527 | 47
14 | 8. 960052 | 9. 998186 | 8. 961866 | 11. 038134 | 46
15 | 8. 961429 | 9. 998174 | 8. 963254 | 11. 036746 | 45
16 | 8. 962801 | 9. 998163 | 8. 964639 | 11. 035361 | 44
17 | 8. 964170 | 9. 998151 | 8 966019 | 11. 033981 | 43
18 | 8. 965534 | 9. 998139 | 8. 967394 | 11. 032606 | 42
19 | 8. 966893 | 9. 998128 | 8. 968766 | 11. 031234 | 41
20 | 8. 968249 | 9. 998106 | 8. 970133 | 11. 029867 | 40
21 | 8. 969600 | 9. 998104 | 8. 971495 | 11. 028505 | 39
22 | 8. 970947 | 9 998092 | 8 972855 | 11. 027145 | 38
23 | 8. 972289 | 9. 998080 | 8. 974209 | 11. 025791 | 37
24 | 8. 973626 | 9. 998068 | 8. 975560 | 11. 024440 | 36
25 | 8. 974962 | 9. 998056 | 8. 976906 | 11. 023094 | 35
26 | 8. 976293 | 9. 998044 | 8. 978248 | 11. 021752 | 34
27 | 8. 977619 | 9. 998032 | 8. 979586 | 11 020414 | 33
28 | 8 978941 | 9. 998020 | 8. 980921 | 11. 019079 | 32
29 | 8. 980259 | 9. 998008 | 8 982251 | 11. 017749 | 31
30 | 8. 981573 | 9. 997996 | 8. 983577 | 11. 016423 | 30
Co-sine. | Sine | Co-tang. | Tangent. | M
Degree 84.
Degree 5.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 8. 981573 | 9. 997996 | 8. 983577 | 11. 016423 | 30
31 | 8. 982883 | 9. 997984 | 8. 984899 | 11. 015101 | 29
32 | 8. 984189 | 9. 997971 | 8. 986217 | 11. 013783 | 28
33 | 8. 985491 | 9. 997959 | 8 987532 | 11. 012468 | 27
34 | 8. 986789 | 9. 997947 | 8. 988842 | 11. 011158 | 26
35 | 8. 988083 | 9. 997935 | 8. 990149 | 11. 009851 | 25
36 | 8. 989374 | 9. 997922 | 8. 991451 | 11. 008549 | 24
37 | 8. 990660 | 9. 997910 | 8 992750 | 11. 007250 | 23
38 | 8. 991943 | 9 997897 | 8 994045 | 11. 005955 | 22
39 | 8. 993228 | 9. 997885 | 8. 995337 | 11. 004663 | 21
40 | 8. 994497 | 9. 997873 | 8. 996624 | 11. 003376 | 20
41 | 8. 995768 | 9. 997860 | 8 997908 | 11. 002092 | 19
42 | 8. 997036 | 9 997847 | 8. 999188 | 11. 000812 | 18
43 | 8. 998299 | 9. 997835 | 9. 000465 | 10. 999535 | 17
44 | 8. 999560 | 9. 997822 | 9. 001738 | 10. 998262 | 16
45 | 9. 000816 | 9. 997809 | 9. 003007 | 10. 996993 | 15
46 | 9. 002069 | 9. 997797 | 9. 004272 | 10. 995728 | 14
47 | 9. 003318 | 9. 997784 | 9. 005534 | 10. 994466 | 13
48 | 9. 004563 | 9. 997771 | 9. 006792 | 10. 993208 | 12
49 | 9. 005805 | 9. 997758 | 9. 008047 | 10. 991953 | 11
50 | 9. 007044 | 9. 997730 | 9. 009298 | 10. 990708 | 10
51 | 9. 008278 | 9. 997732 | 9. 010546 | 10. 989454 | 9
52 | 9. 009510 | 9. 997719 | 9 011790 | 10. 988210 | 8
53 | 9. 010737 | 9. 997706 | 9. 013031 | 10. 986969 | 7
54 | 9. 011962 | 9. 997693 | 9. 014268 | 10. 985732 | 6
55 | 9. 013182 | 9. 997680 | 9. 015502 | 10. 984498 | 5
56 | 9. 014399 | 9. 997667 | 9. 016732 | 10. 983268 | 4
57 | 9. 015613 | 9. 997654 | 9. 017959 | 10. 982041 | 3
58 | 9. 016824 | 9. 997641 | 9. 019183 | 10. 980817 | 2
59 | 9. 018031 | 9. 997628 | 9. 020403 | 10. 979597 | 1
60 | 9. 019235 | 9. 997612 | 9. 021620 | 10. 978380 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 84.

[Page]

Degree 6
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 019235 | 9. 997614 | 9. 021620 | 10. 978380 | 60
1 | 9. 020435 | 9. 997601 | 9. 022834 | 10. 977166 | 59
2 | 9. 021632 | 9. 997588 | 9. 024044 | 10. 975956 | 58
3 | 9. 022825 | 9. 997574 | 9. 025251 | 10. 974749 | 57
4 | 9. 024016 | 9. 997561 | 9. 026455 | 10. 973545 | 56
5 | 9. 025203 | 9. 997548 | 9. 027655 | 10. 972345 | 55
6 | 9. 026386 | 9. 997534 | 9. 028852 | 10. 971148 | 54
7 | 9. 027567 | 9. 997520 | 9. 030046 | 10. 969954 | 53
8 | 9. 028744 | 9. 997507 | 9. 031237 | 10. 968763 | 52
9 | 9. 029918 | 9. 997493 | 9. 032425 | 10. 967575 | 51
10 | 9. 031089 | 9. 997480 | 9. 033609 | 10. 966391 | 50
11 | 9. 032257 | 9. 997466 | 9. 034791 | 10. 965209 | 49
12 | 9. 033421 | 9. 997452 | 9. 035969 | 10. 964031 | 48
13 | 9. 034582 | 9. 997439 | 9. 036144 | 10. 962856 | 47
14 | 9. 035741 | 9. 997425 | 9. 038316 | 10. 961684 | 46
15 | 9. 036896 | 9. 997411 | 9. 039485 | 10. 960515 | 45
16 | 9. 038048 | 9. 997397 | 9. 040651 | 10. 959349 | 44
17 | 9. 039197 | 9. 997383 | 9. 041813 | 10. 958187 | 43
18 | 9. 040342 | 9. 997369 | 9. 042973 | 10. 957027 | 42
19 | 9. 041485 | 9. 997355 | 9. 044130 | 10. 955870 | 41
20 | 9. 042625 | 9. 997341 | 9. 045284 | 10. 954716 | 40
21 | 9. 043762 | 9. 997327 | 9. 046434 | 10. 953566 | 39
22 | 9. 044895 | 9. 997313 | 9. 047582 | 10. 952418 | 38
23 | 9. 046026 | 9. 997299 | 9. 048727 | 10. 951273 | 37
24 | 9. 047154 | 9. 997285 | 9. 049869 | 10. 950131 | 36
25 | 9. 049279 | 9. 997271 | 9. 051008 | 10. 948992 | 35
26 | 9. 049400 | 9. 997256 | 9. 052144 | 10. 947856 | 34
27 | 9. 050519 | 9. 997242 | 9. 043277 | 10. 946723 | 33
28 | 9. 051635 | 9. 997228 | 9. 054408 | 10. 945592 | 32
29 | 9. 052749 | 9. 997214 | 9. 055535 | 10. 944465 | 31
30 | 9. 053859 | 9. 997199 | 9. 056640 | 10. 943340 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 83.
Degree 6.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 053859 | 9. 997199 | 9. 056640 | 10. 943340 | 30
31 | 9. 054966 | 9. 997185 | 9. 057781 | 10. 942219 | 29
32 | 9. 056071 | 9. 997170 | 9. 058900 | 10. 941100 | 28
33 | 9. 057172 | 9. 997156 | 9. 060016 | 10. 939984 | 27
34 | 9. 058271 | 9. 997141 | 9. 061130 | 10. 938870 | 26
35 | 9. 059367 | 9. 997127 | 9. 062240 | 10. 937760 | 25
36 | 9. 060460 | 9. 997112 | 9. 063348 | 10. 936652 | 24
37 | 9. 061551 | 9. 997098 | 9. 064453 | 10. 935547 | 23
38 | 9. 062638 | 9. 997083 | 9. 065556 | 10. 934444 | 22
39 | 9. 063723 | 9. 997068 | 9. 066655 | 10. 933345 | 21
40 | 9. 064806 | 9. 997053 | 9. 067752 | 10. 932248 | 20
41 | 9. 065885 | 9. 997039 | 9. 068847 | 10. 931153 | 19
42 | 9. 066962 | 9. 997024 | 9. 069938 | 10. 930062 | 18
43 | 9. 063036 | 9. 997009 | 9. 071027 | 10. 928973 | 17
44 | 9. 069107 | 9. 996994 | 9. 072113 | 10. 927887 | 16
45 | 9. 070176 | 9. 996979 | 9. 073197 | 10. 926803 | 15
46 | 9. 071242 | 9. 996964 | 9. 074278 | 10. 925722 | 14
47 | 9. 072306 | 9. 996949 | 9. 075356 | 10. 924644 | 13
48 | 9. 073366 | 9. 996934 | 9. 076432 | 10. 923568 | 12
49 | 9. 074424 | 9. 996919 | 9. 077505 | 10. 922495 | 11
50 | 9. 075480 | 9. 996904 | 9. 078576 | 10. 921424 | 10
51 | 9. 076533 | 9. 996889 | 9. 079644 | 10. 920356 | 9
52 | 9. 077583 | 9. 996874 | 9. 080710 | 10. 919290 | 8
53 | 9. 078631 | 9. 996858 | 9. 081773 | 10. 918227 | 7
54 | 9. 079676 | 9. 996843 | 9. 082833 | 10. 917167 | 6
55 | 9. 080719 | 9. 996828 | 9. 083891 | 10. 916109 | 5
56 | 9. 081759 | 9. 996812 | 9. 084947 | 10. 915053 | 4
57 | 9. 082797 | 9. 996797 | 9. 085999 | 10. 914100 | 3
58 | 9. 083832 | 9. 996782 | 9. 087050 | 10. 912950 | 2
59 | 9. 084864 | 9. 996766 | 9. 088098 | 10. 911902 | 1
60 | 9. 085894 | 9. 996751 | 9. 089144 | 10. 910856 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 83.

[Page]

Degree 7.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 085894 | 9. 996751 | 9. 089144 | 10. 910856 | 60
1 | 9. 086922 | 9. 996735 | 9. 090187 | 10. 909813 | 59
2 | 9. 087947 | 9. 996720 | 9. 091228 | 10. 908772 | 58
3 | 9. 088970 | 9. 996704 | 9. 092266 | 10. 907734 | 57
4 | 9. 089990 | 9. 996688 | 9. 093302 | 10. 906698 | 56
5 | 9. 091088 | 9. 996673 | 9. 094336 | 10. 905664 | 55
6 | 9. 092024 | 9. 996657 | 9. 095367 | 10. 904633 | 54
7 | 9. 093037 | 9. 996641 | 9. 096395 | 10. 903604 | 53
8 | 9. 094047 | 9. 996625 | 9. 097422 | 10. 902578 | 52
9 | 9. 095056 | 9. 996610 | 9. 098446 | 10. 901554 | 51
10 | 9. 096062 | 9. 996594 | 9. 099468 | 10. 900532 | 50
11 | 9. 097065 | 9. 996578 | 9. 100487 | 10. 899513 | 49
12 | 9 098066 | 9 996562 | 9 101504 | 10. 898496 | 48
13 | 9. 099065 | 9. 996546 | 9. 102519 | 10. 897481 | 47
14 | 9. 100062 | 9. 996530 | 9 103532 | 10. 896468 | 46
15 | 9. 101056 | 9. 996514 | 9. 104542 | 10. 895458 | 45
16 | 9102048 | 9 996498 | 9. 105550 | 10. 894450 | 44
17 | 9 103037 | 9. 996482 | 9. 106556 | 10. 893444 | 43
18 | 9. 104025 | 9 996465 | 9. 107559 | 10. 892441 | 42
19 | 9. 105010 | 9 996449 | 9. 108560 | 10. 891440 | 41
20 | 9. 105992 | 9. 996433 | 9. 109559 | 10. 890441 | 40
21 | 9. 106973 | 9. 996417 | 9. 110556 | 10. 889444 | 39
22 | 9. 107951 | 9. 996400 | 9. 111551 | 10. 888449 | 38
23 | 9. 108927 | 9. 996384 | 9. 112543 | 10. 887457 | 37
24 | 9. 109901 | 9. 996368 | 9. 113533 | 10. 886467 | 36
25 | 9. 110873 | 9. 996351 | 9. 114521 | 10. 885478 | 35
26 | 9. 111842 | 9. 996335 | 9. 115507 | 10. 884493 | 34
27 | 9. 112809 | 9. 996318 | 9. 116491 | 10. 883509 | 33
28 | 9. 113774 | 9 996302 | 9. 117472 | 10. 882528 | 32
29 | 9. 114737 | 9. 996235 | 9. 118452 | 10. 881548 | 31
30 | 9. 115698 | 9. 996269 | 9. 119429 | 10. 880571 | 30
Co sine | Sine | Co-tang. | Tangent | M
Degree 82.
Degree 7.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 115698 | 9. 996269 | 9. 119429 | 10. 880571 | 30
31 | 9. 116656 | 9. 996252 | 9. 120404 | 10. 879596 | 29
32 | 9. 117612 | 9. 996235 | 9. 121377 | 10. 878623 | 28
33 | 9. 118567 | 9. 996218 | 9. 122348 | 10. 877652 | 27
34 | 9. 119519 | 9. 996202 | 9. 123317 | 10. 876683 | 26
35 | 9. 120469 | 9. 996185 | 9. 124284 | 10. 875716 | 25
36 | 9. 121417 | 9. 996168 | 9. 125248 | 10. 874751 | 24
37 | 9. 122362 | 9. 996151 | 9. 126211 | 10. 873789 | 23
38 | 9. 123306 | 9. 996134 | 9. 127172 | 10. 872828 | 22
39 | 9. 124248 | 9. 996117 | 9. 128130 | 10. 871870 | 21
40 | 9. 125187 | 9. 996100 | 9. 129087 | 10. 870913 | 20
41 | 9. 126125 | 9. 996083 | 9. 130041 | 10. 869959 | 19
42 | 9. 127060 | 9. 996066 | 9. 130994 | 10. 869006 | 18
43 | 9. 127993 | 9. 996049 | 9. 131944 | 10. 868056 | 17
44 | 9. 128925 | 9. 996032 | 9. 132893 | 10. 867107 | 16
45 | 9. 129854 | 9. 996015 | 9. 133839 | 10. 869161 | 15
46 | 9. 130781 | 9. 995998 | 9. 134784 | 10. 865216 | 14
47 | 9. 131706 | 9 995980 | 9. 135726 | 10. 864274 | 13
48 | 9. 132630 | 9. 995963 | 9. 136666 | 10. 863334 | 12
49 | 9. 133551 | 9 995946 | 9. 137605 | 10. 862395 | 11
50 | 9. 134470 | 9. 995928 | 9. 138542 | 10. 861458 | 10
51 | 9. 135387 | 9. 995911 | 9. 139476 | 10. 860524 | 9
52 | 9. 136303 | 9. 995894 | 9. 140409 | 10. 859591 | 8
53 | 9. 137216 | 9. 995876 | 9. 141340 | 10. 858660 | 7
54 | 9. 138127 | 9. 995859 | 9. 142269 | 10. 857731 | 6
55 | 9. 139037 | 9. 995841 | 9. 143196 | 10. 856804 | 5
56 | 9. 139944 | 9. 995825 | 9. 144121 | 10. 855879 | 4
57 | 9. 140850 | 9. 995806 | 9. 145044 | 10. 854956 | 3
58 | 9. 141754 | 9. 995788 | 9. 145965 | 10. 854035 | 2
59 | 9 142655 | 9. 995770 | 9. 146885 | 10. 853115 | 1
60 | 9. 142555 | 9. 995753 | 9. 147803 | 10. 852197 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 82.

[Page]

Degree 8.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 143555 | 9. 995753 | 9. 147803 | 10. 852197 | 60
1 | 9. 144453 | 9. 995735 | 9. 148718 | 10. 851282 | 59
2 | 9. 145349 | 9. 995717 | 9. 149632 | 10. 850368 | 58
3 | 9. 146243 | 9. 995699 | 9. 159544 | 10. 849456 | 57
4 | 9. 147136 | 9. 995681 | 9. 151454 | 10. 848546 | 56
5 | 9. 148026 | 9. 995664 | 9. 152363 | 10. 847637 | 55
6 | 9. 148919 | 9 995646 | 9. 153269 | 10. 846731 | 54
7 | 9. 149881 | 9. 995628 | 9. 154174 | 10. 845825 | 53
8 | 9. 150686 | 9. 995610 | 9. 155077 | 10. 844923 | 52
9 | 9. 151569 | 9. 995591 | 9. 155978 | 10. 844022 | 51
10 | 9. 152451 | 9. 995573 | 9. 156877 | 10. 843123 | 50
11 | 9. 153330 | 9. 995555 | 9. 157775 | 10. 842225 | 49
12 | 9. 154208 | 9. 995537 | 9. 158671 | 10. 841329 | 48
13 | 9. 155082 | 9. 995519 | 9. 159565 | 10. 840435 | 47
14 | 9. 155957 | 9. 995501 | 9 160457 | 10. 839543 | 46
15 | 9. 156830 | 9. 995482 | 9. 161347 | 10. 838653 | 45
16 | 9. 157700 | 9. 995464 | 9. 162236 | 10. 837764 | 44
17 | 9. 158569 | 9. 995446 | 9. 163123 | 10. 836877 | 43
18 | 9. 159436 | 9. 995427 | 9. 164008 | 10. 835992 | 42
19 | 9. 160301 | 9. 995409 | 9. 164892 | 10. 835108 | 41
20 | 9. 161164 | 9. 995390 | 9. 165773 | 10. 834226 | 40
21 | 9. 162025 | 9. 995372 | 9. 166654 | 10. 833346 | 39
22 | 9. 162885 | 9. 995353 | 9. 167532 | 10. 832468 | 38
23 | 9. 163743 | 9. 995334 | 9. 168409 | 10. 831591 | 37
24 | 9. 164600 | 9. 995316 | 9. 169284 | 10. 830716 | 36
25 | 9. 165454 | 9. 995297 | 9. 160157 | 10. 829843 | 35
26 | 9. 166307 | 9. 995278 | 9. 171029 | 10. 828971 | 34
27 | 9. 167158 | 9. 995260 | 9. 171899 | 10. 828101 | 33
28 | 9. 168008 | 9. 995241 | 9. 172767 | 10. 827233 | 32
29 | 9. 168856 | 9 995222 | 9 173634 | 10. 826366 | 31
30 | 9. 169702 | 9. 995203 | 9. 174499 | 10. 825501 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 81.
Degree 8.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 169702 | 9. 995203 | 9. 174499 | 10. 825501 | 30
31 | 9. 170546 | 9. 995184 | 9. 175362 | 10. 824638 | 29
32 | 9. 171389 | 9. 995165 | 9. 176224 | 10. 823776 | 28
33 | 9. 172230 | 9. 995146 | 9. 177084 | 10. 822916 | 27
34 | 9. 173070 | 9. 995127 | 9. 177942 | 10. 822057 | 26
35 | 9. 173908 | 9. 995108 | 9. 178799 | 10. 821201 | 25
36 | 9. 174744 | 9. 995089 | 9. 179655 | 10. 820345 | 24
37 | 9. 175578 | 9. 995070 | 9. 180508 | 10. 819492 | 23
38 | 9. 176411 | 9. 995061 | 9. 181360 | 10. 818640 | 22
39 | 9. 177242 | 9. 995032 | 9. 182211 | 10. 817789 | 21
40 | 9. 178072 | 9. 995012 | 9. 183060 | 10. 816940 | 20
41 | 9. 178900 | 9. 994993 | 9. 183907 | 10. 816093 | 19
42 | 9. 179726 | 9. 994974 | 9. 184752 | 10. 815248 | 18
43 | 9. 180551 | 9. 994955 | 9. 185597 | 10. 814403 | 17
44 | 9. 181374 | 9. 994935 | 9. 186439 | 10. 813561 | 16
45 | 9. 182196 | 9. 994916 | 9. 187280 | 10. 812720 | 15
46 | 9. 183016 | 9. 994896 | 9. 188120 | 10. 811880 | 14
47 | 9. 183834 | 9. 994876 | 9. 188957 | 10. 811042 | 13
48 | 9. 184651 | 9. 994857 | 9. 189794 | 10. 810206 | 12
49 | 9. 185466 | 9 994838 | 9. 190629 | 10. 809371 | 11
50 | 9. 186280 | 9. 994818 | 9. 191462 | 10. 808538 | 10
51 | 9. 187092 | 9. 994798 | 9. 192294 | 10. 807706 | 9
52 | 9. 187903 | 9. 994779 | 9. 193124 | 10. 806876 | 8
53 | 9. 188712 | 9. 994759 | 9. 193953 | 10. 806047 | 7
54 | 9. 189519 | 9. 994739 | 9. 194780 | 10. 805220 | 6
55 | 9. 190325 | 9. 994719 | 9. 195606 | 10. 804394 | 5
56 | 9. 191130 | 9. 994699 | 9. 196440 | 10. 803569 | 4
57 | 9. 191933 | 9. 994680 | 9. 197253 | 10. 802747 | 3
58 | 9. 192734 | 9. 994660 | 9. 198674 | 10. 801926 | 2
59 | 9. 193534 | 9. 994640 | 9. 198894 | 10. 801106 | 1
60 | 9. 194332 | 9. 994620 | 9. 199712 | 10. 800287 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 81.

[Page]

Degree 9.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 194332 | 9. 994620 | 9. 199712 | 10. 800287 | 60
1 | 9. 195129 | 9. 994600 | 9. 200529 | 10. 799470 | 59
2 | 9. 195925 | 9. 994580 | 9. 201345 | 10. 798655 | 58
3 | 9. 196718 | 9. 994560 | 9. 202159 | 10. 797841 | 57
4 | 9. 197511 | 9. 994540 | 9. 202971 | 10. 797029 | 56
5 | 9. 198302 | 9. 994519 | 9. 203782 | 10. 796218 | 55
6 | 9. 199091 | 9. 994499 | 9. 204592 | 10. 795408 | 54
7 | 9. 199879 | 9. 994479 | 9. 205400 | 10. 794600 | 53
8 | 9. 200666 | 9. 994459 | 9. 206207 | 10. 793793 | 52
9 | 9. 201451 | 9. 994438 | 9. 207013 | 10. 792987 | 51
10 | 9. 202234 | 9. 994418 | 9. 207817 | 10. 792183 | 50
11 | 9. 203017 | 9. 994398 | 9. 208619 | 10. 791381 | 49
12 | 9 203797 | 9. 994377 | 9. 209420 | 10. 790580 | 48
13 | 9. 204577 | 9. 994357 | 9. 210220 | 10. 789780 | 47
14 | 9. 205354 | 9. 994336 | 9. 211018 | 10. 788982 | 46
15 | 9. 206131 | 9. 994316 | 9. 211815 | 10. 788185 | 45
16 | 9. 206906 | 9. 994195 | 9. 212611 | 10. 787385 | 44
17 | 9. 207679 | 9. 994174 | 9. 213405 | 10. 786595 | 43
18 | 9. 208452 | 9. 994154 | 9. 214198 | 10. 785802 | 42
19 | 9. 209222 | 9. 994133 | 9. 214989 | 10. 785011 | 41
20 | 9. 209992 | 9. 994112 | 9. 215780 | 10. 784220 | 40
21 | 9. 210760 | 9. 994191 | 9. 216568 | 10. 783432 | 39
22 | 9. 211526 | 9. 994171 | 9. 217356 | 10. 782644 | 38
23 | 9. 212291 | 9. 994150 | 9. 218142 | 10. 781858 | 37
24 | 9. 213055 | 9. 994129 | 9. 218926 | 10. 781070 | 36
25 | 9. 213818 | 9. 994108 | 9. 219710 | 10. 780294 | 35
26 | 9. 214579 | 9. 994087 | 9. 220491 | 10. 779508 | 34
27 | 9. 215338 | 9. 994066 | 9. 221272 | 10. 778728 | 33
28 | 9. 216097 | 9. 994044 | 9. 222052 | 10. 777948 | 32
29 | 9. 216854 | 9. 994024 | 9. 222830 | 10. 777170 | 31
30 | 9. 217609 | 9. 994003 | 9. 223607 | 10. 776393 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 80.
Degree 9.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 217609 | 9. 994003 | 9. 223607 | 10. 716393 | 30
31 | 9. 218363 | 9. 993982 | 9. 224382 | 10. 775618 | 29
32 | 9. 219116 | 9. 993960 | 9. 225156 | 10. 774844 | 28
33 | 9. 219868 | 9. 993939 | 9. 225929 | 10. 774071 | 27
34 | 9. 220618 | 9. 993918 | 9. 226704 | 10. 773300 | 26
35 | 9. 221367 | 9. 993897 | 9 227471 | 10. 772529 | 25
36 | 9. 222115 | 9 993875 | 9 228240 | 10. 771760 | 24
37 | 9. 222861 | 9. 993854 | 9. 229007 | 10. 770993 | 23
38 | 9. 223606 | 9. 993832 | 9. 229774 | 10. 770226 | 22
39 | 9. 224349 | 9. 993811 | 9. 230539 | 10. 769461 | 21
40 | 9. 225092 | 9. 993789 | 9. 231302 | 10. 768698 | 20
41 | 9. 225833 | 9. 993768 | 9. 232065 | 10. 767935 | 19
42 | 9. 226573 | 9. 993746 | 9. 232826 | 10. 767174 | 18
43 | 9 227311 | 9. 993725 | 9. 233586 | 10. 766414 | 17
44 | 9. 228048 | 9. 993703 | 9. 234345 | 10. 765655 | 16
45 | 9 228784 | 9. 993681 | 9 235103 | 10. 764897 | 15
46 | 9. 239518 | 9. 993660 | 9. 235859 | 10. 764141 | 14
47 | 9. 230252 | 9. 993638 | 9. 236614 | 10. 763386 | 13
48 | 9. 230984 | 9. 993616 | 9. 237368 | 10. 762632 | 12
49 | 9. 231715 | 9. 993594 | 9. 238120 | 10. 761880 | 11
50 | 9. 232444 | 9. 993572 | 9. 238872 | 10. 761128 | 10
51 | 9. 233172 | 9. 993550 | 9. 239622 | 10. 760378 | 9
52 | 9. 233899 | 9. 993528 | 9. 240371 | 10. 759629 | 8
53 | 9 234625 | 9. 993506 | 9. 241118 | 10. 758882 | 7
54 | 9. 235349 | 9. 993484 | 9. 241865 | 10. 758135 | 6
55 | 9. 236073 | 9. 993462 | 9 242610 | 10 757390 | 5
56 | 9. 236795 | 9. 993440 | 9. 243354 | 10. 756646 | 4
57 | 9. 237515 | 9. 993418 | 9. 244097 | 10. 755903 | 3
58 | 9. 238835 | 9. 993396 | 9. 244839 | 10. 755161 | 2
59 | 9. 238952 | 9. 993374 | 9. 245579 | 10. 754421 | 1
60 | 9. 239670 | 9. 993351 | 9. 246319 | 10. 753681 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 80.

[Page]

Degree 10.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 239670 | 9. 993351 | 9. 246319 | 10. 753681 | 60
1 | 9. 240386 | 9. 993329 | 9. 247057 | 10. 752943 | 59
2 | 9. 241101 | 9. 993307 | 9. 247794 | 10. 752206 | 58
3 | 9. 241814 | 9. 993284 | 9. 248530 | 10. 751470 | 57
4 | 9. 242526 | 9. 993262 | 9. 249264 | 10. 750736 | 56
5 | 9. 243237 | 9. 993240 | 9. 249998 | 10. 750002 | 55
6 | 9. 243947 | 9. 993117 | 9. 250730 | 10. 749270 | 54
7 | 9 244656 | 9. 993195 | 9. 251461 | 10. 748539 | 53
8 | 9. 245363 | 9. 993172 | 9. 252191 | 10. 747809 | 52
9 | 9. 246070 | 9. 993149 | 9. 252920 | 10. 747080 | 51
10 | 9. 246775 | 9 993127 | 9. 253648 | 10. 746352 | 50
11 | 9. 247478 | 9. 993104 | 9. 254374 | 10. 745626 | 49
12 | 9. 248181 | 9. 993011 | 9. 255200 | 10. 744900 | 48
13 | 9. 248883 | 9. 993059 | 9. 255824 | 10. 744176 | 47
14 | 9. 249583 | 9. 993036 | 9. 256547 | 10. 743453 | 46
15 | 9. 250282 | 9. 993013 | 9. 257269 | 10. 742731 | 45
16 | 9. 250980 | 9. 992990 | 9. 257990 | 10. 742010 | 44
17 | 9. 251677 | 9. 992967 | 9. 258710 | 10. 741290 | 43
18 | 9. 252373 | 9. 992944 | 9. 259429 | 10. 740571 | 42
19 | 9. 253067 | 9. 992921 | 9. 260146 | 10. 739854 | 41
20 | 9. 253761 | 9. 992898 | 9. 260863 | 10. 739137 | 40
21 | 9 254453 | 9. 992875 | 9. 261578 | 10. 738422 | 39
22 | 9. 255144 | 9. 992852 | 9. 262292 | 10. 737708 | 38
23 | 9. 255834 | 9. 992829 | 9. 263005 | 10. 736995 | 37
24 | 9. 256523 | 9. 992806 | 9. 263717 | 10. 736283 | 36
25 | 9. 257211 | 9. 992783 | 9. 264428 | 10. 735572 | 35
26 | 9. 257898 | 9. 992759 | 9. 265138 | 10. 734862 | 34
27 | 9. 258583 | 9. 992736 | 9. 265847 | 10. 734153 | 33
28 | 9. 259268 | 9. 992713 | 9. 266555 | 10. 733445 | 32
29 | 9. 259951 | 9 992690 | 9. 267261 | 10. 732739 | 31
30 | 9 260633 | 9. 992666 | 9. 267967 | 10. 732033 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 79.
Degree 10.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 260633 | 9. 992666 | 9. 267967 | 10. 732033 | 30
31 | 9. 261314 | 9. 992643 | 9. 268671 | 10. 731329 | 29
32 | 9. 261994 | 9. 992619 | 9. 269375 | 10. 730625 | 28
33 | 9. 262673 | 9. 992596 | 9. 270778 | 10. 729923 | 27
34 | 9. 263351 | 9. 992572 | 9. 271479 | 10. 729221 | 26
35 | 9. 264027 | 9. 992549 | 9. 271479 | 10. 728521 | 25
36 | 9. 264703 | 9. 992525 | 9. 272178 | 10. 727822 | 24
37 | 9. 265378 | 9. 992501 | 9. [...] | 10. 727124 | 23
38 | 9. 266051 | 9. 992478 | 9. 273573 | 10. 726427 | 22
39 | 9. 266723 | 9. 992454 | 9. 274269 | 10. 725731 | 21
40 | 9. 267395 | 9. 992430 | 9. 274964 | 10. 725036 | 20
41 | 9. 268065 | 9. 992406 | 9. 275658 | 10. 724342 | 19
42 | 9. 268734 | 9. 992382 | 9. 276351 | 10. 723649 | 18
43 | 9. 269402 | 9. 992362 | 9. 277043 | 10. 722957 | 17
44 | 9. 270069 | 9. 992335 | 9. 277734 | 10. 722267 | 16
45 | 9. 270735 | 9. 992311 | 9. 278424 | 10. 721576 | 15
46 | 9. 271400 | 9. 992287 | 9. 279113 | 10. 720887 | 14
47 | 9. 272063 | 9. 992263 | 9. 279801 | 10. 720199 | 13
48 | 9. 272726 | 9. 992239 | 9 280488 | 10. 719512 | 12
49 | 9. 273388 | 9. 992214 | 9. 281174 | 10. 718826 | 11
50 | 9. 274049 | 9. 992190 | 9. 281858 | 10. 718142 | 10
51 | 9. 274708 | 9 992166 | 9. 282542 | 10. 717458 | 9
52 | 9. 275367 | 9. 992142 | 9. 283225 | 10. 716775 | 8
53 | 9. 276025 | 9. 992118 | 9283907 | 10. 716093 | 7
54 | 9. 276681 | 9. 992093 | 9. 284588 | 10. 715412 | 6
55 | 9. 277337 | 9. 992069 | 9. 285268 | 10. 714732 | 5
56 | 9. 277991 | 9. 992045 | 9. 285946 | 10. 714053 | 4
57 | 9. 278685 | 9. 992020 | 9 286624 | 10. 713376 | 3
58 | 9. 279297 | 9. 991996 | 9. 287301 | 10. 712699 | 2
59 | 9 279948 | 9. 991971 | 9. 287977 | 10. 712023 | 1
60 | 9. 280599 | 9. 991947 | 9. 288652 | 10. 711348 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 79.

[Page]

Degree 11.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 280599 | 9. 991947 | 9. 288652 | 10. 711348 | 60
1 | 9. 281229 | 9. 991922 | 9. 289326 | 10. 710674 | 59
2 | 9. 281897 | 9. 991897 | 9. 289999 | 10. 710001 | 58
3 | 9. 282544 | 9. 991873 | 9. 290671 | 10. 709329 | 57
4 | 9. 283190 | 9. 991848 | 9. 291342 | 10. 708658 | 56
5 | 9. 283836 | 9. 991823 | 9 292013 | 10. 707987 | 55
6 | 9. 284480 | 9. 991799 | 9. 292682 | 10. 707318 | 54
7 | 9. 285124 | 9. 991774 | 9. 293350 | 10. 706650 | 53
8 | 9. 285766 | 9. 991749 | 9. 294017 | 10. 705983 | 52
9 | 9. 286408 | 9. 991724 | 9. 294684 | 10. 705316 | 51
10 | 9. 287048 | 9. 991699 | 9. 295349 | 10. 704651 | 50
11 | 9. 287688 | 9. 991674 | 9. 296013 | 10. 703987 | 49
12 | 9. 288326 | 9. 991649 | 9. 296677 | 10. 703323 | 48
13 | 9. 288964 | 9. 991624 | 9. 297339 | 10. 702661 | 47
14 | 9. 289600 | 9. 991599 | 9. 298001 | 10. 701999 | 46
15 | 9. 290236 | 9. 991574 | 9. 298662 | 10. 701338 | 45
16 | 9. 290870 | 9. 991549 | 9. 299322 | 10. 700678 | 44
17 | 9. 291504 | 9. 991524 | 9. 299980 | 10. 700020 | 43
18 | 9. 292137 | 9. 991498 | 9. 300638 | 10. 699362 | 42
19 | 9. 292768 | 9. 991473 | 9. 301295 | 10. 698705 | 41
20 | 9. 293399 | 9. 991448 | 9. 301951 | 10. 698049 | 40
21 | 9. 294029 | 9. 991422 | 9. 302607 | 10. 697393 | 39
22 | 9. 294658 | 9. 991397 | 9. 303261 | 10. 696739 | 38
23 | 9. 295286 | 9. 991372 | 9. 303914 | 10. 696086 | 37
24 | 9. 295913 | 9. 991346 | 9. 304567 | 10. 695433 | 36
25 | 9. 296539 | 9. 991321 | 9. 305218 | 10. 694782 | 35
26 | 9. 297164 | 9. 991295 | 9. 305867 | 10. 694131 | 34
27 | 9. 297788 | 9. 991270 | 9. 306519 | 10. 693481 | 33
28 | 9. 298412 | 9. 991244 | 9. 307168 | 10. 692832 | 32
29 | 9. 299034 | 9. 991218 | 9. 307816 | 10. 692184 | 31
30 | 9. 299655 | 9 991193 | 9. 308463 | 10. 691537 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 78.
Degree 11.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 299655 | 9 991193 | 9. 308463 | 10. 691537 | 30
31 | 9. 300276 | 9. 991167 | 9. 309109 | 10. 690891 | 29
32 | 9. 300895 | 9. 991141 | 9. 309754 | 10. 690246 | 28
33 | 9. 301514 | 9. 991115 | 9. 310399 | 10. 689601 | 27
34 | 9. 302132 | 9. 991090 | 9. 311042 | 10. 688958 | 26
35 | 9. 302749 | 9. 991064 | 9. 311685 | 10. 688315 | 25
36 | 9. 303364 | 9. 991038 | 9. 312327 | 10. 687673 | 24
37 | 9. 303979 | 9. 991012 | 9. 312968 | 10. 687032 | 23
38 | 9. 304593 | 9. 990986 | 9. 313608 | 10. 686392 | 22
39 | 9. 305207 | 9. 990960 | 9. 314247 | 10. 685753 | 21
40 | 9. 305819 | 9. 990934 | 9. 314885 | 10. 685115 | 20
41 | 9. 306430 | 9. 990908 | 9. 315523 | 10. 684477 | 19
42 | 9. 307041 | 9. 990882 | 9 316159 | 10. 683841 | 18
43 | 9. 307650 | 9. 990855 | 9. 316795 | 10. 683205 | 17
44 | 9. 308259 | 9. 990829 | 9. 317430 | 10. 682570 | 16
45 | 9. 308867 | 9. 990803 | 9. 318064 | 10. 681936 | 15
46 | 9. 309474 | 9. 990777 | 9. 318647 | 10. 681303 | 14
47 | 9. 310080 | 9. 990750 | 9. 319330 | 10. 680670 | 13
48 | 9. 310685 | 9. 990724 | 9. 319961 | 10 680039 | 12
49 | 9. 311289 | 9. 990697 | 9. 320592 | 10. 679408 | 11
50 | 9. 311899 | 9. 990671 | 9. 321222 | 10. 678778 | 10
51 | 9. 312495 | 9. 990645 | 9. 321851 | 10. 678149 | 9
52 | 9. 313097 | 9. 990618 | 9. 322479 | 10. 677521 | 8
53 | 9. 313698 | 9. 990591 | 9 323106 | 10. 676894 | 7
54 | 9. 314297 | 9. 990565 | 9. 323733 | 10. 676267 | 6
55 | 9. 314897 | 9. 990538 | 9. 324358 | 10. 675642 | 5
56 | 9. 315495 | 9. 990512 | 9. 324983 | 10. 675017 | 4
57 | 9. 316092 | 9. 990485 | 9. 325607 | 10. 674393 | 3
58 | 9. 316689 | 9. 990458 | 9. 326231 | 10. 673769 | 2
59 | 9. 317284 | 9. 990431 | 9. 326853 | 10. 673147 | 1
60 | 9. 317879 | 9. 990404 | 9. 327475 | 10. 672525 | 0
Co-sine. | Sine | Co-tang. | Tangent | M
Degree 78.

[Page]

Degree 12.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 317879 | 9. 990404 | 9. 327475 | 10. 672525 | 60
1 | 9. 318473 | 9. 990377 | 9. 328095 | 10. 671905 | 59
2 | 9. 319066 | 9. 990351 | 9. 328715 | 10. 671285 | 58
3 | 9. 319658 | 9. 990324 | 9. 329334 | 10. 670666 | 57
4 | 9. 320250 | 9. 990297 | 9. 329953 | 10. 670047 | 56
5 | 9. 320840 | 9. 990270 | 9. 320570 | 10. 669430 | 55
6 | 9. 321430 | 9. 990242 | 9. 331187 | 10. 668813 | 54
7 | 9. 322019 | 9. 990215 | 9. 331803 | 10 668197 | 53
8 | 9. 322607 | 9. 990188 | 9. 332418 | 10. 667582 | 52
9 | 9. 323194 | 9. 990161 | 9. 333033 | 10. 666967 | 51
10 | 9. 323780 | 9. 990134 | 9. 333646 | 10. 666354 | 50
11 | 9. 324366 | 9. 990107 | 9. 334259 | 10. 665741 | 49
12 | 9. 324950 | 9. 990079 | 9. 334871 | 10. 665129 | 48
13 | 9. 325534 | 9. 990052 | 9. 335482 | 10. 664518 | 47
14 | 9. 326117 | 9. 990025 | 9. 336093 | 10. 663907 | 46
15 | 9. 326699 | 9. 989997 | 9. 336700 | 10. 663298 | 45
16 | 9. 327281 | 9 989970 | 9. 337311 | 10. 662689 | 44
17 | 9. 327862 | 9. 989942 | 9. 337919 | 10. 662081 | 43
18 | 9. 328441 | 9. 989915 | 9. 338527 | 10. 661473 | 42
19 | 9. 329020 | 9. 989887 | 9. 339133 | 10. 660867 | 41
20 | 9. 329599 | 9. 989860 | 9. 339739 | 10. 660261 | 40
21 | 9. 330176 | 9. 989832 | 9. 340344 | 10. 659656 | 39
22 | 9. 330753 | 9. 989804 | 9. 340948 | 10. 659052 | 38
23 | 9. 331328 | 9. 989777 | 9. 341552 | 10. 658448 | 37
24 | 9. 331903 | 9. 989749 | 9. 342155 | 10. 657845 | 36
25 | 9. 332478 | 9 989721 | 9. 342757 | 10. 657243 | 35
26 | 9 333051 | 9. 989693 | 9. 343358 | 10. 656642 | 34
27 | 9. 333624 | 9. 989665 | 9. 343958 | 10. 656042 | 33
28 | 9. 334195 | 9. 989637 | 9. 344558 | 10. 655442 | 32
29 | 9. 334766 | 9 989609 | 9. 345157 | 10. 654843 | 31
30 | 9. 335337 | 9. 989581 | 9 345755 | 10. 654245 | 30
Co-sine | Sine | Co-tang. | Tangent. | M
Degree 77.
Degree 12.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 335337 | 9. 989581 | 9. 345755 | 10. 654245 | 30
31 | 9. 335906 | 9. 989553 | 9. 346353 | 10. 653647 | 29
32 | 9. 336475 | 9. 989525 | 9. 346949 | 10. 653051 | 28
33 | 9. 337043 | 9. 989597 | 9. 347545 | 10. 652455 | 27
34 | 9. 337610 | 9. 989469 | 9. 348141 | 10. 651859 | 26
35 | 9. 338176 | 9. 989441 | 9. 348735 | 10. 651265 | 25
36 | 9. 338742 | 9. 989413 | 9. 349329 | 10. 650671 | 24
37 | 9. 339306 | 9. 989384 | 9. 349922 | 10. 650078 | 23
38 | 9. 339870 | 9. 989356 | 9. 350514 | 10. 649486 | 22
39 | 9 340434 | 9. 989328 | 9. 351106 | 10. 648894 | 21
40 | 9. 340996 | 9. 989299 | 9. 351697 | 10. 648303 | 20
41 | 9. 341558 | 9. 989271 | 9. 352287 | 10. 647713 | 19
42 | 9. 342119 | 9. 989243 | 9. 352876 | 10. 647124 | 18
43 | 9. 342679 | 9. 989214 | 9. 353465 | 10. 646535 | 17
44 | 9. 343239 | 9. 989186 | 9. 354053 | 10. 645947 | 16
45 | 9. 343797 | 9. 989157 | 9. 354640 | 10. 645360 | 15
46 | 9. 344355 | 9. 989128 | 9. 355227 | 10. 644773 | 14
47 | 9. 344912 | 9. 989100 | 9 355812 | 10. 644187 | 13
48 | 9. 345469 | 9. 989071 | 9. 356398 | 10. 643602 | 12
49 | 9. 346024 | 9. 989042 | 9. 356982 | 10. 643018 | 11
50 | 9. 346579 | 9. 989014 | 9. 357566 | 10. 642434 | 10
51 | 9. 347134 | 9. 988985 | 9. 358149 | 10. 641851 | 9
52 | 9. 347687 | 9. 988956 | 9. 358731 | 10. 641269 | 8
53 | 9. 348240 | 9. 988927 | 9. 359313 | 10. 640687 | 7
54 | 9. 348792 | 9. 988898 | 9. 359893 | 10. 640107 | 6
55 | 9 349343 | 9. 988869 | 9. 360474 | 10. 639526 | 5
56 | 9. 349893 | 9. 988840 | 9. 361053 | 10. 638947 | 4
57 | 9. 350443 | 9. 988811 | 9. 361632 | 10. 638368 | 3
58 | 9. 350992 | 9. 988782 | 9. 362210 | 10. 637790 | 2
59 | 9. 351540 | 9. 988754 | 9. 362787 | 10. 637213 | 1
60 | 9. 352088 | 9. 988724 | 9. 363364 | 10. 636636 | 0
Co-sine. | Sine | Co-tang. | Tangent. | M
Degree 77.

[Page]

Degree 13.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 352088 | 9. 988724 | 9. 363364 | 10. 636636 | 60
1 | 9. 352635 | 9. 988695 | 9. 363940 | 10. 636060 | 59
2 | 9. 353181 | 9. 988666 | 9. 364515 | 10. 635485 | 58
3 | 9. 353726 | 9. 988636 | 9. 365090 | 10. 634910 | 57
4 | 9. 354271 | 9. 988607 | 9. 365664 | 10. 634336 | 56
5 | 9. 354185 | 9. 988578 | 9. 366237 | 10. 633763 | 55
6 | 9. 355358 | 9. 988548 | 9. 366810 | 10. 633190 | 54
7 | 9. 355901 | 9. 988519 | 9. 367382 | 10. 632618 | 53
8 | 9 356443 | 9. 988489 | 9. 367953 | 10. 632047 | 52
9 | 9. 356984 | 9. 988460 | 9. 368524 | 10. 631476 | 51
10 | 9. 357524 | 9. 988430 | 9. 369094 | 10. 630906 | 50
11 | 9. 358064 | 9. 988401 | 9. 369663 | 10. 630337 | 49
12 | 9. 358603 | 9. 988371 | 9. 370232 | 10. 629768 | 48
13 | 9. 359141 | 9. 988341 | 9. 370799 | 10. 629201 | 47
14 | 9. 359679 | 9. 988312 | 9. 371367 | 10. 628633 | 46
15 | 9. 350215 | 9 988282 | 9. 371933 | 10. 628067 | 45
16 | 9. 360752 | 9 988252 | 9. 372499 | 10. 627501 | 44
17 | 9. 361287 | 9. 988223 | 9. 373064 | 10. 626936 | 43
18 | 9. 361822 | 9. 988193 | 9 373629 | 10. 626371 | 42
19 | 9 362356 | 9. 988163 | 9. 374193 | 10. 625807 | 41
20 | 9. 362889 | 9. 988133 | 9. 374756 | 10. 625244 | 40
21 | 9. 363422 | 9. 988103 | 9. 375319 | 10. 624681 | 39
22 | 9 363954 | 9. 988073 | 9 375881 | 10. 624119 | 38
23 | 9. 364485 | 9. 988043 | 9. 376442 | 10. 623558 | 37
24 | 9. 365016 | 9. 988013 | 9 377003 | 10. 622997 | 36
25 | 9 365546 | 9. 987983 | 9 377563 | 10. 622437 | 35
26 | 9 366075 | 9. 987953 | 9. 378122 | 10. 621878 | 34
27 | 9. 366604 | 9. 987922 | 9. 378681 | 10. 621319 | 33
28 | 9 367132 | 9. 987892 | 9. 379239 | 10. 620761 | 32
29 | 9. 367659 | 9. 987862 | 9. 379797 | 10. 620203 | 31
30 | 9. 368185 | 9. 987832 | 9. 380354 | 10. 619646 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 76.
Degree 13.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 368185 | 9. 987832 | 9. 380354 | 10. 619646 | 30
31 | 9. 368711 | 9. 987801 | 9. 380910 | 10. 619090 | 29
32 | 9. 369236 | 9. 987771 | 9. 381466 | 10. 618534 | 28
33 | 9. 369761 | 9. 987740 | 9. 382021 | 10. 617980 | 27
34 | 9. 370285 | 9. 987710 | 9. 382575 | 10. 617425 | 26
35 | 9. 370808 | 9. 987679 | 9. 383129 | 10. 616871 | 25
36 | 9. 371330 | 9. 987649 | 9. 383682 | 10. 616318 | 24
37 | 9. 371852 | 9. 987618 | 9. 384234 | 10. 615766 | 23
38 | 9. 372373 | 9. 987588 | 9. 384786 | 10. 615214 | 22
39 | 9. 372894 | 9. 987557 | 9. 385337 | 10. 614663 | 21
40 | 9. 373414 | 9. 987526 | 9. 385888 | 10. 614112 | 20
41 | 9. 373933 | 9. 987496 | 9. 386438 | 10. 613562 | 19
42 | 9. 374452 | 9. 987465 | 9. 386987 | 10. 613013 | 18
43 | 9. 374970 | 9. 987434 | 9. 387536 | 10. 612464 | 17
44 | 9. 375487 | 9. 987403 | 9. 388084 | 10. 611916 | 16
45 | 9. 376003 | 9. 987372 | 9. 388631 | 10. 611369 | 15
46 | 9. 376519 | 9. 987341 | 9. 389178 | 10. 610822 | 14
47 | 9. 377035 | 9. 987310 | 9. 389724 | 10. 610276 | 13
48 | 9. 377549 | 9. 987279 | 9. 390270 | 10. 609730 | 12
49 | 9. 378063 | 9. 987248 | 9. 390815 | 10. 609185 | 11
50 | 9. 378577 | 9. 987217 | 9. 391360 | 10. 608640 | 10
51 | 9. 379089 | 9. 987186 | 9. 391907 | 10. 608097 | 9
52 | 9. 379601 | 9. 987155 | 9. 392467 | 10. 607553 | 8
53 | 9. 380113 | 9. 987124 | 9. 392989 | 10. 607011 | 7
54 | 9. 380624 | 9. 987092 | 9. 393531 | 10. 606469 | 6
55 | 9. 381134 | 9. 987061 | 9 394074 | 10. 605927 | 5
56 | 9. 381643 | 9. 987030 | 9. 394614 | 10. 605386 | 4
57 | 9. 382152 | 9. 986998 | 9. 395154 | 10. 604846 | 3
58 | 9 382661 | 9. 986967 | 9. 395694 | 10. 604306 | 2
59 | 9. 383168 | 9. 986936 | 9. 396233 | 10. 603767 | 1
60 | 9. 383675 | 9. 986904 | 9. 396770 | 10. 603229 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 76.

[Page]

Degree 14
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 383675 | 9 986904 | 9. 396771 | 10. 603229 | 60
1 | 9. 384181 | 9. 986873 | 9. 397309 | 10. 602694 | 59
2 | 9. 384687 | 9. 986841 | 9. 397846 | 10. 602154 | 58
3 | 9. 385192 | 9. 986809 | 9. 398383 | 10. 601617 | 57
4 | 9. 385697 | 9 986778 | 9. 398919 | 10. 601081 | 56
5 | 9. 386201 | 9. 986746 | 9. 399455 | 10. 600545 | 55
6 | 9. 386704 | 9. 986714 | 9. 399990 | 10. 600010 | 54
7 | 9. 387207 | 9. 986683 | 9. 400524 | 10. 599476 | 53
8 | 9. 387709 | 9. 986651 | 9. 401058 | 10. 598942 | 52
9 | 9. 388210 | 9. 986619 | 9. 401591 | 10. 598409 | 51
10 | 9. 388711 | 9. 986587 | 9. 402124 | 10. 597876 | 50
11 | 9. 389211 | 9. 986555 | 9. 402656 | 10. 597344 | 49
12 | 9. 389711 | 9. 986523 | 9. 403187 | 10. 596813 | 48
13 | 9. 390210 | 9. 986491 | 9. 403718 | 10. 596282 | 47
14 | 9. 390708 | 9. 986459 | 9. 404249 | 10. 595751 | 46
15 | 9. 391206 | 9. 986427 | 9. 404778 | 10. 595222 | 45
16 | 9. 391703 | 9. 986395 | 9. 405306 | 10. 594692 | 44
17 | 9. 392199 | 9. 986363 | 9. 405836 | 10. 594164 | 43
18 | 9. 392695 | 9. 986331 | 9. 406364 | 10. 593636 | 42
19 | 9. 393190 | 9. 986299 | 9. 406892 | 10. 593608 | 41
20 | 9. 393685 | 9. 986266 | 9. 407419 | 10. 592581 | 40
21 | 9. 394179 | 9. 986234 | 9. 407945 | 10. 592055 | 39
22 | 9. 394673 | 9. 986201 | 9. 408471 | 10. 591529 | 38
23 | 9. 395166 | 9. 986869 | 9. 408996 | 10. 591001 | 37
24 | 9. 395654 | 9. 986137 | 9. 409521 | 10. 590479 | 36
25 | 9. 396150 | 9. 986104 | 9. 410045 | 10. 589954 | 35
26 | 9. 396641 | 9. 986072 | 9. 410569 | 10. 589431 | 34
27 | 9. 397131 | 9. 986039 | 9. 411097 | 10. 588908 | 33
28 | 9. 397621 | 9. 986007 | 9. 411615 | 10. 588385 | 32
29 | 9. 398111 | 9. 985974 | 9. 412137 | 10. 587863 | 31
30 | 9. 398600 | 9. 986942 | 9. 412658 | 10. 587342 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 75.
Degree 14.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 398600 | 9. 985942 | 9. 412658 | 10. 587342 | 30
31 | 9. 399087 | 9. 985909 | 9. 413179 | 10. 586821 | 29
32 | 9. 399575 | 9. 985876 | 9. 413699 | 10. 586301 | 28
33 | 9. 400062 | 9. 985843 | 9. 414219 | 10. 585781 | 27
34 | 9. 400549 | 9. 985811 | 9. 414738 | 10. 585262 | 26
35 | 9. 401035 | 9. 985778 | 9. 415257 | 10. 584742 | 25
36 | 9. 401520 | 9. 985745 | 9. 415775 | 10. 584225 | 24
37 | 9. 402005 | 9. 985712 | 9. 416293 | 10. 583707 | 23
38 | 9. 402489 | 9. 985679 | 9. 416810 | 10. 583190 | 22
39 | 9. 402972 | 9. 985646 | 9. 417326 | 10. 582674 | 21
40 | 9. 403455 | 9. 985613 | 9. 417842 | 10. 582157 | 20
41 | 9. 403938 | 9. 985580 | 9. 418357 | 10 581642 | 19
42 | 9. 404420 | 9. 985547 | 9. 418873 | 10. 581127 | 18
43 | 9. 404901 | 9. 985513 | 9. 419387 | 10 580613 | 17
44 | 9. 405382 | 9. 985480 | 9. 419901 | 10. 580099 | 16
45 | 9. 405862 | 9. 985447 | 9. 420415 | 10. 579585 | 15
46 | 9. 406341 | 9. 985414 | 9. 420927 | 10. 579072 | 14
47 | 9. 406820 | 9. 985380 | 9. 421440 | 10. 578560 | 13
48 | 9. 407299 | 9. 985347 | 9. 421951 | 10. 578048 | 12
49 | 9. 407776 | 9. 985314 | 9. 422463 | 10. 577537 | 11
50 | 9. 408254 | 9. 985280 | 9. 422973 | 10. 577026 | 10
51 | 9. 408731 | 9. 985247 | 9. 423484 | 10. 576516 | 9
52 | 9. 409207 | 9. 985213 | 9. 423993 | 10. 576007 | 8
53 | 9. 409682 | 9. 985180 | 9. 424503 | 10. 575497 | 7
54 | 9. 410157 | 9. 985146 | 9. 425011 | 10. 574989 | 6
55 | 9. 410632 | 9. 985112 | 9. 425518 | 10. 574480 | 5
56 | 9. 411106 | 9. 985079 | 9. 426027 | 10. 573973 | 4
57 | 9. 411579 | 9. 985045 | 9. 426534 | 10. 573466 | 3
58 | 9. 412052 | 9. 985011 | 9. 427041 | 10. 572959 | 2
59 | 9. 412524 | 9. 984977 | 9. 427547 | 10. 572453 | 1
60 | 9. 412996 | 9. 984943 | 9. 428052 | 10. 571947 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 75.

[Page]

Degree 15.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 412996 | 9. 984944 | 9. 428052 | 10. 571947 | 60
1 | 9. 413467 | 9. 984910 | 9. 428557 | 10. 571442 | 59
2 | 9. 413938 | 9. 984876 | 9. 429067 | 10. 570938 | 58
3 | 9. 414408 | 9. 984842 | 9. 429566 | 10. 570434 | 57
4 | 9. 414878 | 9. 984808 | 9. 430070 | 10. 569930 | 56
5 | 9. 415347 | 9. 984774 | 9. 430573 | 10. 569427 | 55
6 | 9. 415815 | 9. 984740 | 9. 431075 | 10. 568925 | 54
7 | 9. 416283 | 9. 984706 | 9. 431577 | 10. 568423 | 53
8 | 9. 416850 | 9 984672 | 9. 432079 | 10. 567921 | 52
9 | 9. 417217 | 9. 984637 | 9. 432580 | 10. 567420 | 51
10 | 9. 417684 | 9. 984603 | 9. 433080 | 10. 566920 | 50
11 | 9. 418149 | 9. 984569 | 9. 433580 | 10. 566419 | 49
12 | 9. 418615 | 9. 984535 | 9. 434080 | 10. 565920 | 48
13 | 9. 419079 | 9. 984500 | 9. 434579 | 10. 565421 | 47
14 | 9. 419544 | 9. 984466 | 9. 435078 | 10. 564922 | 46
15 | 9. 420007 | 9. 984431 | 9. 435576 | 10. 564424 | 45
16 | 9. 420470 | 9. 984397 | 9. 436073 | 10. 563927 | 44
17 | 9. 420933 | 9. 984363 | 9. 436570 | 10. 563430 | 43
18 | 9. 421395 | 9. 984328 | 9. 437067 | 10. 562933 | 42
19 | 9. 421856 | 9. 984293 | 9. 437563 | 10. 562437 | 41
20 | 9. 422317 | 9. 984259 | 9. 438059 | 10. 561941 | 40
21 | 9. 422778 | 9. 984224 | 9. 438554 | 10. 561446 | 39
22 | 9. 423238 | 9. 984189 | 9. 439548 | 10. 560952 | 38
23 | 9 423697 | 9. 984155 | 9. 439543 | 10. 560457 | 37
24 | 9. 424156 | 9. 984120 | 9. 440036 | 10. 559964 | 36
25 | 9. 424615 | 9. 984085 | 9. 440529 | 10. 559471 | 35
26 | 9. 425072 | 9. 984050 | 9. 441022 | 10. 558978 | 34
27 | 9 425530 | 9 984015 | 9. 441514 | 10. 558486 | 33
28 | 9. 425987 | 9. 983980 | 9. 442006 | 10. 557994 | 32
29 | 9. 426443 | 9. 983945 | 9. 442497 | 10. 557503 | 31
30 | 9. 426899 | 9. 983910 | 9. 442988 | 10. 557011 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 74.
Degree 15.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 426899 | 9. 983910 | 9. 442988 | 10. 557011 | 30
31 | 9. 427354 | 9. 983875 | 9. 443479 | 10. 556521 | 29
32 | 9. 427809 | 9. 983840 | 9. 443968 | 10. 556031 | 28
33 | 9. 428264 | 9. 983805 | 9. 444458 | 10. 555542 | 27
34 | 9. 428717 | 9. 983770 | 9. 444947 | 10. 555035 | 26
35 | 9. 429170 | 9. 983735 | 9. 445435 | 10. 554565 | 25
36 | 9. 429623 | 9. 983699 | 9. 445923 | 10. 554077 | 24
37 | 9. 430075 | 9. 983664 | 9. 446411 | 10. 553589 | 23
38 | 9. 430507 | 9 983629 | 9. 446898 | 10. 553102 | 22
39 | 9. 430978 | 9. 983593 | 9. 447384 | 10. 552616 | 21
40 | 9. 431429 | 9. 983558 | 9. 447870 | 10. 552129 | 20
41 | 9. 431879 | 9. 983523 | 9. 448356 | 10. 551644 | 19
42 | 9. 432328 | 9. 983487 | 9. 448841 | 10. 551159 | 18
43 | 9. 432778 | 9. 983452 | 9. 449326 | 10. 550674 | 17
44 | 9. 433206 | 9. 983416 | 9. 449810 | 10. 550181 | 16
45 | 9. 433674 | 9. 983380 | 9. 450294 | 10. 559706 | 15
46 | 9. 434122 | 9. 983345 | 9. 450777 | 10. 549223 | 14
47 | 9. 434569 | 9. 983309 | 9. 451260 | 10. 548740 | 13
48 | 9. 435016 | 9. 983273 | 9. 451743 | 10. 548257 | 12
49 | 9. 435462 | 9. 983238 | 9. 452225 | 10. 547775 | 11
50 | 9. 435918 | 9. 983202 | 9. 452706 | 10. 547294 | 10
51 | 9. 436353 | 9. 983166 | 9. 453187 | 10. 546813 | 9
52 | 9. 436798 | 9. 983130 | 9. 453668 | 10. 546332 | 8
53 | 9. 437242 | 9. 983094 | 9. 454148 | 10. 545852 | 7
54 | 9. 437686 | 9. 983058 | 9 454629 | 10. 545372 | 6
55 | 9. 438129 | 9. 983022 | 9 455107 | 10. 544893 | 5
56 | 9. 438572 | 9. 982986 | 9. 455586 | 10. 544414 | 4
57 | 9. 439014 | 9. 982950 | 9. 456064 | 10. 543936 | 3
58 | 9. 439456 | 9. 982914 | 9. 456542 | 10. 543458 | 2
59 | 9. 439897 | 9. 982878 | 9. 457019 | 10. 542980 | 1
60 | 9. 440338 | 9. 982842 | 9. 457496 | 10. 542503 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 74.

[Page]

Degree 16.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 440338 | 9. 982842 | 9. 457496 | 10. 542503 | 60
1 | 9. 440778 | 9. 982805 | 9. 457973 | 10. 542027 | 59
2 | 9. 441218 | 9. 982769 | 9 458449 | 10. 541551 | 58
3 | 9. 441658 | 9. 982733 | 9. 458925 | 10. 541075 | 57
4 | 9. 442096 | 9. 982696 | 9. 459400 | 10. 540600 | 56
5 | 9. 442535 | 9 982660 | 9. 459875 | 10. 540125 | 55
6 | 9. 442973 | 9. 982623 | 9. 460349 | 10. 539651 | 54
7 | 9. 443416 | 9. 982587 | 9 460829 | 10. 539177 | 53
8 | 9. 443848 | 9. 982550 | 9. 461297 | 10. 538703 | 52
9 | 9. 444284 | 9. 982514 | 9. 461770 | 10. 538230 | 51
10 | 9. 444720 | 9. 982477 | 9. 462242 | 10. 537758 | 50
11 | 9. 445155 | 9. 982441 | 9. 462714 | 10. 537285 | 49
12 | 9. 445590 | 9. 982404 | 9. 463186 | 10. 536814 | 48
13 | 9. 446025 | 9. 982367 | 9. 463658 | 10. 536342 | 47
14 | 9. 446459 | 9. 982330 | 9. 464129 | 10. 535871 | 46
15 | 9. 446893 | 9. 982294 | 9 464599 | 10. 535401 | 45
16 | 9. 447326 | 9. 982257 | 9 465069 | 10. 534931 | 44
17 | 9. 447759 | 9. 982220 | 9. 465539 | 10. 534461 | 43
18 | 9. 448191 | 9. 982183 | 9. 466008 | 10. 533992 | 42
19 | 9. 448623 | 9. 982146 | 9. 466476 | 10. 533523 | 41
20 | 9. 449054 | 9. 982109 | 9. 466945 | 10. 533055 | 40
21 | 9. 449485 | 9. 982072 | 9. 467413 | 10. 532587 | 39
22 | 9. 449915 | 9. 982035 | 9. 467880 | 10. 532120 | 38
23 | 9. 450345 | 9. 981998 | 9. 468347 | 10. 531653 | 37
24 | 9. 450775 | 9. 981961 | 9. 468814 | 10. 531186 | 36
25 | 9. 451203 | 9 981923 | 9. 469280 | 10. 530720 | 35
26 | 9. 451632 | 9 981886 | 9. 469746 | 10. 530254 | 34
27 | 9. 452060 | 9. 981849 | 9. 470211 | 10. 529789 | 33
28 | 9. 452488 | 9. 981812 | 9. 470676 | 10. 529324 | 32
29 | 9. 452915 | 9. 981774 | 9. 471141 | 10. 528859 | 31
30 | 9. 453342 | 9. 981737 | 9. 471605 | 10. 528395 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 73.
Degree 16.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 453342 | 9. 981737 | 9. 471605 | 10. 528395 | 30
31 | 9. 453768 | 9. 981699 | 9. 472068 | 10. 527931 | 29
32 | 9. 454194 | 9. 981662 | 9. 472532 | 10. 527468 | 28
33 | 9. 454619 | 9. 981624 | 9 472995 | 10. 527005 | 27
34 | 9 455044 | 9. 981587 | 9. 473457 | 10. 526543 | 26
35 | 9. 455469 | 9. 981549 | 9. 473919 | 10. 526081 | 25
36 | 9. 455892 | 9. 981512 | 9. 474381 | 10. 525619 | 24
37 | 9. 456316 | 9. 981474 | 9 474842 | 10. 525158 | 23
38 | 9. 456739 | 9. 981436 | 9. 475303 | 10. 524695 | 22
39 | 9. 457162 | 9. 981398 | 9. 475763 | 10. 524237 | 21
40 | 9. 457584 | 9. 981361 | 9. 476223 | 10. 523777 | 20
41 | 9 458006 | 9. 981323 | 9. 476683 | 10. 523317 | 19
42 | 9. 458427 | 9 981285 | 9. 477142 | 10. 522858 | 18
43 | 9. 458848 | 9. 981247 | 9. 477601 | 10. 522399 | 17
44 | 9. 459268 | 9. 981209 | 9. 478059 | 10. 521941 | 16
45 | 9. 459684 | 9. 981171 | 9. 478517 | 10. 521483 | 15
46 | 9. 460108 | 9. 981133 | 9. 478975 | 10. 521025 | 14
47 | 9. 460527 | 9. 981095 | 9. 479432 | 10. 520168 | 13
48 | 9. 460946 | 9. 981057 | 9. 479886 | 10. 520111 | 12
49 | 9. 461364 | 9. 981019 | 9. 480345 | 10. 519655 | 11
50 | 9. 461782 | 9. 980980 | 9. 480801 | 10. 519199 | 10
51 | 9. 462199 | 9. 980942 | 9. 481257 | 10. 518743 | 9
52 | 9. 462616 | 9. 980904 | 9. 481712 | 10. 518288 | 8
53 | 9. 463032 | 9. 980866 | 9 482167 | 10. 517833 | 7
54 | 9. 463448 | 9. 980827 | 9. 482621 | 10. 517379 | 6
55 | 9 463864 | 9. 980789 | 9. 483075 | 10. 516925 | 5
56 | 9. 464279 | 9. 980750 | 9 483528 | 10. 516471 | 4
57 | 9. 464694 | 9. 980712 | 9. 483982 | 10. 516018 | 3
58 | 9 465108 | 9. 980672 | 9. 484434 | 10. 515565 | 2
59 | 9. 465522 | 9. 980635 | 9. 484887 | 10. 515113 | 1
60 | 9. 465935 | 9. 980596 | 9. 485339 | 10. 514661 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 73.

[Page]

Degree 17.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9 465935 | 9. 980596 | 9. 485339 | 10. 514661 | 60
1 | 9. 466348 | 9. 980558 | 9. 485791 | 10. 514209 | 59
2 | 9. 466761 | 9. 980519 | 9. 486272 | 10. 513758 | 58
3 | 9 467173 | 9. 980480 | 9. 486693 | 10. 513307 | 57
4 | 9. 467585 | 9. 980441 | 9. 487143 | 10. 512857 | 56
5 | 9. 467996 | 9. 980403 | 9. 487593 | 10. 512407 | 55
6 | 9 468407 | 9. 980364 | 9. 488043 | 10. 511957 | 54
7 | 9. 468817 | 9. 980325 | 9. 488493 | 10. 511507 | 53
8 | 9. 469227 | 9. 980286 | 9 488941 | 10. 511059 | 52
9 | 9. 469637 | 9. 980247 | 9. 489390 | 10. 510610 | 51
10 | 9. 460446 | 9. 980208 | 9. 489838 | 10. 510162 | 50
11. | 9 470455 | 9. 980169 | 9. 490286 | 10. 509714 | 49
12 | 9. 471863 | 9. 980130 | 9. 490733 | 10. 509267 | 48
13 | 9. 471071 | 9. 980091 | 9. 491180 | 10. 508820 | 47
14 | 9. 471678 | 9. 980052 | 9. 491627 | 10. 508373 | 46
15 | 9. 472086 | 9. 980012 | 9. 492073 | 10. 507928 | 45
16 | 9. 472492 | 9. 979973 | 9. 492519 | 10. 507481 | 44
17 | 9. 472898 | 9. 979934 | 9. 492964 | 10. 507035 | 43
18 | 9. 473304 | 9. 979894 | 9. 493410 | 10. 506590 | 42
19 | 9. 473710 | 9. 979855 | 9. 493854 | 10. 506145 | 41
20 | 9. 474115 | 9. 979816 | 9. 494299 | 10. 505701 | 40
21 | 9. 474519 | 9. 979776 | 9 494743 | 10. 505257 | 39
22 | 9. 474923 | 9. 979737 | 9. 495186 | 10. 504813 | 38
23 | 9. 475327 | 9. 979697 | 9. 495630 | 10. 504370 | 37
24 | 9. 475730 | 9. 979658 | 9. 496073 | 10. 503928 | 36
25 | 9. 476133 | 9. 979618 | 9. 496515 | 10. 503485 | 35
26 | 9. 476539 | 9. 979578 | 9. 496957 | 10. 503043 | 34
27 | 9 476938 | 9. 979539 | 9. 497399 | 10. 502601 | 33
28 | 9. 477340 | 9. 979499 | 9. 497840 | 10. 502160 | 32
29 | 9. 477741 | 9 979459 | 9. [...] | 10. 501718 | 31
30 | 9. 478142 | 9. 979419 | 9. 498722 | 10. 501278 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 72.
Degree 17.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 478142 | 9. 979419 | 9. 498722 | 10. 501278 | 30
31 | 9. 478542 | 9. 979380 | 9. 499163 | 10. 500837 | 29
32 | 9 478942 | 9. 979340 | 9. 499602 | 10. 500398 | 28
33 | 9. 479342 | 9 979300 | 9. 500042 | 10. 499958 | 27
34 | 9. 479741 | 9. 979260 | 9. 500481 | 10. 499519 | 26
35 | 9. 480140 | 9. 979220 | 9. 500920 | 10. 499080 | 25
36 | 9. 480538 | 9. 979180 | 9. 501359 | 10. 498641 | 24
37 | 9. 480936 | 9. 979140 | 9. 501797 | 10. 498203 | 23
38 | 9. 481334 | 9. 979099 | 9. 502234 | 10. 497765 | 22
39 | 9. 481731 | 9. 979059 | 9. 502672 | 10. 497328 | 21
40 | 9. 482128 | 9 979019 | 9. 503109 | 10. 496891 | 20
41 | 9. 482525 | 9. 978980 | 9. 503546 | 10. 496454 | 19
42 | 9. 482921 | 9. 978939 | 9. 503982 | 10. 496018 | 18
43 | 9. 483316 | 9. 978898 | 9. 504418 | 10. 495582 | 17
44 | 9. 483711 | 9. 978858 | 9. 504854 | 10. 495146 | 16
45 | 9. 484106 | 9. 978817 | 9. 505289 | 10. 494711 | 15
46 | 9. 484501 | 9. 978777 | 9. 505724 | 10. 494216 | 14
47 | 9. 484895 | 9. 978736 | 9. 506158 | 10. 493841 | 13
48 | 9. 485289 | 9. 978696 | 9. 506593 | 10. 493407 | 12
49 | 9. 485682 | 9. 978655 | 9. 507026 | 10. 492973 | 11
50 | 9. 486075 | 9 978615 | 9. 507459 | 10. 492540 | 10
51 | 9. 486467 | 9. 978574 | 9. 507892 | 10. 492107 | 9
52 | 9. 486859 | 9. 978533 | 9. 508326 | 10. 491674 | 8
53 | 9. 487251 | 9. 978493 | 9. 508759 | 10. 491241 | 7
54 | 9. 487642 | 9. 978452 | 9. 509181 | 10. 490809 | 6
55 | 9. 488033 | 9. 978411 | 9. 509622 | 10 490377 | 5
56 | 9. 488424 | 9. 978370 | 9. 510044 | 10 489916 | 4
57 | 9. 488814 | 9. 978329 | 9. 510486 | 10. 489515 | 3
58 | 9. 489204 | 9. 978288 | 9. 510916 | 10. 489084 | 2
59 | 9. 489593 | 9. 978247 | 9. 511346 | 10 488654 | 1
60 | 9. 489982 | 9. 978206 | 9. 511776 | 10. 488225 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 72.

[Page]

Degree 18.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 489982 | 9. 978206 | 9. 511776 | 10. 488224 | 60
1 | 9. 490371 | 9. 978165 | 9. 512206 | 10. 487794 | 59
2 | 9. 490759 | 9. 978124 | 9. 512635 | 10. 487365 | 58
3 | 9. 491147 | 9. 978083 | 9. 513064 | 10. 486936 | 57
4 | 9. 491534 | 9. 978042 | 9. 513493 | 10. 486507 | 56
5 | 9. 491922 | 9. 978000 | 9. 513921 | 10. 486079 | 55
6 | 9. 492308 | 9. 977959 | 9. 514349 | 10. 485651 | 54
7 | 9. 492695 | 9. 977918 | 9 514777 | 10. 485223 | 53
8 | 9. 493080 | 9 977877 | 9. 515204 | 10. 484796 | 52
9 | 9. 493466 | 9. 977835 | 9. 515631 | 10. 484369 | 51
10 | 9. 493851 | 9. 977794 | 9. 516057 | 10 483942 | 50
11 | 9. 494236 | 9 977752 | 9. 516484 | 10. 483516 | 49
12 | 9. 494620 | 9. 977711 | 9. 516910 | 10. 483090 | 48
13 | 9. 495005 | 9. 977669 | 9. 517335 | 40. 482665 | 47
14 | 9. 495388 | 9. 977628 | 9. 517761 | 10. 482239 | 46
15 | 9. 495771 | 9. 977586 | 9. 518185 | 10. 481814 | 45
16 | 9. 496154 | 9. 977544 | 9. 518610 | 10. 481390 | 44
17 | 9. 496537 | 9. 977503 | 9. 519034 | 10. 480966 | 43
18 | 9. 496919 | 9. 977461 | 9. 519458 | 10. 480542 | 42
19 | 9. 497301 | 9. 977419 | 9. 519882 | 10. 480118 | 41
20 | 9. 497682 | 9. 977377 | 9. 520305 | 10. 489695 | 40
21 | 9. 498063 | 9. 977335 | 9. 520728 | 10. 479272 | 39
22 | 9. 498444 | 9 977293 | 9. 521151 | 10. 478849 | 38
23 | 9. 498824 | 9 977251 | 9. 521573 | 10. 478427 | 37
24 | 9. 499204 | 9. 977209 | 9. 521995 | 10. 478005 | 36
25 | 9. 499584 | 9. 977167 | 9. 522417 | 10. 477583 | 35
26 | 9. 499963 | 9. 977125 | 9. 522838 | 10. 477162 | 34
27 | 9. 500342 | 9 977083 | 9. 523259 | 10. 476741 | 33
28 | 9. 500720 | 9. 977041 | 9. 523679 | 10. 476320 | 32
29 | 9. 501099 | 9 977999 | 9. 524109 | 10. 475900 | 31
30 | 9. 501476 | 9. 977956 | 9. 524520 | 10. 475480 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 71.
Degree 18.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 501476 | 9. 976956 | 9. 524520 | 10. 475480 | 30
31 | 9. 501854 | 9. 976914 | 9. 524939 | 10. 475060 | 29
32 | 9. 502231 | 9. 976872 | 9. 525359 | 10. 474641 | 28
33 | 9. 502607 | 9. 976830 | 9. 525778 | 10. 474222 | 27
34 | 9. 502984 | 9. 976787 | 9. 526197 | 10. 473803 | 26
35 | 9. 503360 | 9. 976745 | 9. 526615 | 10. 473385 | 25
36 | 9. 503735 | 9. 976702 | 9. 527033 | 10. 472967 | 24
37 | 9. 504110 | 9. 976660 | 9. 527451 | 10. 472549 | 23
38 | 9. 504485 | 9. 976617 | 9. 527868 | 10. 472132 | 22
39 | 9. 504840 | 9. 976574 | 9. 528285 | 10. 471715 | 21
40 | 9. 505234 | 9. 976532 | 9. 528702 | 10. 471298 | 20
41 | 9. 505608 | 9. 976489 | 9. 529118 | 10. 470881 | 19
42 | 9. 505981 | 9. 976446 | 9. 529535 | 10. 470465 | 18
43 | 9. 506354 | 9. 976404 | 9. 529950 | 10. 470049 | 17
44 | 9. 506727 | 9. 976361 | 9. 530366 | 10. 469634 | 16
45 | 9. 507099 | 9. 976318 | 9. 530781 | 10. 469219 | 15
46 | 9. 507471 | 9. 976275 | 9. 531196 | 10 468804 | 14
47 | 9. 507843 | 9. 976232 | 9. 531611 | 10. 468389 | 13
48 | 9. 508214 | 9. 976185 | 9. 532025 | 10. 467975 | 12
49 | 9. 508585 | 9. 976146 | 9. 532436 | 10. 467561 | 11
50 | 9. 508955 | 9. 976103 | 9. 532852 | 10. 467147 | 10
51 | 9. 509326 | 9. 976060 | 9. 533266 | 10. 466734 | 9
52 | 9. 509696 | 9. 976017 | 9. 533679 | 10. 466321 | 8
53 | 9. 510065 | 9. 975973 | 9. 534092 | 10. 465908 | 7
54 | 9. 510434 | 9. 975930 | 9. 534504 | 10. 465496 | 6
55 | 9. 510803 | 9. 975887 | 9. 534916 | 10. 465084 | 5
56 | 9. 511171 | 9. 975844 | 9. 535328 | 10. 464672 | 4
57 | 9. 511540 | 9. 975800 | 9. 535739 | 10. 464261 | 3
58 | 9. 511907 | 9. 975757 | 9. 536150 | 10. 463849 | 2
59 | 9. 512275 | 9. 975713 | 9. 536561 | 10. 463439 | 1
60 | 9. 512642 | 9. 975670 | 9. 536972 | 10. 463028 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 71.

[Page]

Degree 19.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 512642 | 9. 975670 | 9. 536972 | 10. 463028 | 60
1 | 9. 513009 | 9. 975626 | 9. 537382 | 10. 462618 | 59
2 | 9. 513375 | 9. 975583 | 9. 537792 | 10. 462208 | 58
3 | 9. 513741 | 9. 975539 | 9. 538202 | 10. 461798 | 57
4 | 9. 514107 | 9. 975496 | 9. 538610 | 10. 461389 | 56
5 | 9. 514472 | 9. 975452 | 9. 539020 | 10. 460980 | 55
6 | 9. 514837 | 9. 975408 | 9. 539429 | 10. 460571 | 54
7 | 9. 515202 | 9. 975364 | 9. 539837 | 10. 460163 | 53
8 | 9. 515566 | 9. 975321 | 9. 540245 | 10. 459755 | 52
9 | 9. 515930 | 9. 975277 | 9. 540653 | 10. 459347 | 51
10 | 9. 516294 | 9. 975233 | 9. 541061 | 10. 458939 | 50
11 | 9. 516657 | 9. 975189 | 9. 541468 | 10. 458532 | 49
12 | 9. 517020 | 9. 975145 | 9. 541875 | 10. 458125 | 48
13 | 9. 517382 | 9. 975101 | 9. 542281 | 10. 457719 | 47
14 | 9. 517745 | 9. 975057 | 9. 542688 | 10. 457312 | 46
15 | 9. 518107 | 9. 975013 | 9. 543094 | 10. 456906 | 45
16 | 9. 518468 | 9. 974969 | 9. 543499 | 10. 456501 | 44
17 | 9. 518829 | 9. 974925 | 9. 543905 | 10. 456095 | 43
18 | 9. 519190 | 9. 974880 | 9. 544310 | 10. 455690 | 42
19 | 9. 519551 | 9. 974836 | 9. 544715 | 10. 455285 | 41
20 | 9. 519911 | 9. 974792 | 9. 545119 | 10. 454881 | 40
21 | 9. 520271 | 9. 974747 | 9. 545524 | 10. 454476 | 39
22 | 9. 520631 | 9. 974703 | 9. 545927 | 10. 454072 | 38
23 | 9. 520990 | 9. 974659 | 9. 546331 | 10. 453669 | 37
24 | 9. 521349 | 9. 974614 | 9. 546735 | 10. 453265 | 36
25 | 9. 521707 | 9. 974570 | 9. 547138 | 10. 452862 | 35
26 | 9. 522065 | 9. 974525 | 9. 547540 | 10. 452459 | 34
27 | 9. 522423 | 9. 974480 | 9. 547943 | 10. 452057 | 33
28 | 9. 522781 | 9. 974436 | 9. 548345 | 10. 451655 | 32
29 | 9. 523138 | 9. 974391 | 9. 548747 | 10. 451253 | 31
30 | 9. 523495 | 9. 974346 | 9. 549149 | 10. 450851 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 70.
Degree 19.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9 523495 | 9. 974346 | 9. 549149 | 10. 450851 | 30
31 | 9. 523851 | 9. 974302 | 9. 549550 | 10. 450450 | 29
32 | 9. 524208 | 9. 974257 | 9. 549951 | 10. 450049 | 28
33 | 9. 524564 | 9. 974212 | 9. 550352 | 10. 449648 | 27
34 | 9. 524920 | 9. 974167 | 9. 550752 | 10. 449248 | 26
35 | 9. 525275 | 9. 974122 | 9. 551152 | 10. 448848 | 25
36 | 9. 525630 | 9. 974077 | 9. 551552 | 10. 448448 | 24
37 | 9. 525984 | 9. 974032 | 9. 551952 | 10. 448048 | 23
38 | 9. 526339 | 9. 973987 | 9. 552351 | 10. 447649 | 22
39 | 9. 526693 | 9. 973942 | 9. 552750 | 10. 447250 | 21
40 | 9. 527046 | 9. 973897 | 9. 553149 | 10. 446851 | 20
41 | 9. 527400 | 9. 973852 | 9. 553548 | 10. 446452 | 19
42 | 9. 527753 | 9. 973807 | 9. 553946 | 10. 446054 | 18
43 | 9. 528105 | 9. 973761 | 9. 554344 | 10. 445656 | 17
44 | 9. 528458 | 9. 973716 | 9 554741 | 10. 445259 | 16
45 | 9. 528810 | 9. 973671 | 9. 555139 | 10. 444861 | 15
46 | 9. 529161 | 9. 973615 | 9. 555536 | 10. 444464 | 14
47 | 9. 529513 | 9. 973580 | 9. 555932 | 10. 444068 | 13
48 | 9. 529864 | 9. 973535 | 9. 556329 | 10. 443671 | 12
49 | 9. 530214 | 9 973489 | 9. 556727 | 10. 443275 | 11
50 | 9. 530565 | 9. 973443 | 9. 557121 | 10. 442879 | 10
51 | 9. 530915 | 9. 973398 | 9. 557517 | 10. 442483 | 9
52 | 9. 531265 | 9. 973352 | 9. 557912 | 10. 442088 | 8
53 | 9. 531614 | 9. 973307 | 9. 558308 | 10. 441693 | 7
54 | 9. 531963 | 9. 973261 | 9. 558702 | 10. 441298 | 6
55 | 9. 532312 | 9. 973215 | 9. 559097 | 10. 440903 | 5
56 | 9. 532661 | 9. 973169 | 9. 559491 | 10. 440509 | 4
57 | 9. 533009 | 9. 973123 | 9. 559885 | 10. 440115 | 3
58 | 9. 533357 | 9. 973078 | 9 560279 | 10. 439721 | 2
59 | 9. 533704 | 9. 973032 | 9. 560673 | 10. 439327 | 1
60 | 9. 534052 | 9. 972986 | 9. 561066 | 10. 438934 | 0
Co-sine. | Sine | Co-tang. | Tangent | M
Degree 70.

[Page]

Degree 20.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 534052 | 9. 972986 | 9. 561066 | 10. 438934 | 60
1 | 9. 534399 | 9. 972940 | 9. 561459 | 10. 438541 | 59
2 | 9. 534746 | 9. 972894 | 9. 561851 | 10. 438148 | 58
3 | 9. 535091 | 9. 972848 | 9. 562244 | 10. 437756 | 57
4 | 9. 535437 | 9. 972801 | 9. 562636 | 10. 437364 | 56
5 | 9. 535782 | 9. 972755 | 9. 563028 | 10. 436972 | 55
6 | 9. 536129 | 9. 972709 | 9. 563419 | 10. 436580 | 54
7 | 9 536474 | 9. 972663 | 9. 563811 | 10. 436189 | 53
8 | 9. 536818 | 9. 972617 | 9. 564202 | 10. 435798 | 52
9 | 9. 537163 | 9. 972570 | 9. 564592 | 10. 435407 | 51
10 | 9. 537507 | 9. 972524 | 9. 564983 | 10. 435017 | 50
11 | 9. 537851 | 9. 972477 | 9. 565373 | 10. 434627 | 49
12 | 9. 538194 | 9. 972431 | 9. 565763 | 10. 434237 | 48
13 | 9. 538537 | 9. 972384 | 9 566153 | 10. 433847 | 47
14 | 9. 538880 | 9. 972338 | 9. 566542 | 10. 433457 | 46
15 | 9. 539222 | 9. 972291 | 9. 566932 | 10. 433068 | 45
16 | 9. 539565 | 9. 972245 | 9. 567320 | 10. 432679 | 44
17 | 9. 539907 | 9. 972198 | 9. 567709 | 10. 432291 | 43
18 | 9. 540249 | 9. 972151 | 9. 568097 | 10. 431902 | 42
19 | 9. 540590 | 9. 972105 | 9. 568486 | 10. 431514 | 41
20 | 9. 540931 | 9. 972058 | 9. 569873 | 10. 431126 | 40
21 | 9. 541272 | 9. 972011 | 9. 569261 | 10. 430739 | 39
22 | 9. 541612 | 9 971964 | 9. 569648 | 10. 430351 | 38
23 | 9. 541953 | 9. 971917 | 9. 570035 | 10. 429964 | 37
24 | 9. 542292 | 9. 971870 | 9 570422 | 10. 429578 | 36
25 | 9. 542632 | 9. 971823 | 9. 570809 | 10. 429191 | 35
26 | 9. 542971 | 9. 971776 | 9. 571195 | 10. 428805 | 34
27 | 9. 543310 | 9. 971729 | 9. 571581 | 10. 428419 | 33
28 | 9. 543649 | 9. 971682 | 9. 571967 | 10. 428033 | 32
29 | 9 543987 | 9. 971635 | 9. 572352 | 10. 427648 | 31
30 | 9. 544325 | 9. 971588 | 9. 572738 | 10. 427262 | 30
Co-sine. | Sine | Co-tang. | Tangent | M
Degree 69.
Degree 20.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 544325 | 9. 971588 | 9. 572738 | 10. 427262 | 30
31 | 9. 544663 | 9. 971540 | 9. 573123 | 10. 426877 | 29
32 | 9. 545000 | 9. 971493 | 9. 573507 | 10. 426492 | 28
33 | 9. 545338 | 9. 971446 | 9. 573892 | 10. 426108 | 27
34 | 9. 545674 | 9. 971398 | 9. 574276 | 10. 425724 | 26
35 | 9. 546011 | 9. 971351 | 9. 574660 | 10. 425340 | 25
36 | 9. 546347 | 9. 971303 | 9. 575044 | 10. 424956 | 24
37 | 9. 546683 | 9. 971256 | 9. 575427 | 10. 424573 | 23
38 | 9. 547019 | 9. 971208 | 9. 575810 | 10. 424189 | 22
39 | 9. 547354 | 9. 971161 | 9. 576193 | 10. 423807 | 21
40 | 9. 547689 | 9. 971112 | 9. 576576 | 10. 423424 | 20
41 | 9. 548024 | 9. 971065 | 9. 576958 | 10. 423041 | 19
42 | 9. 548358 | 9. 971018 | 9. 577341 | 10. 422659 | 18
43 | 9. 548693 | 9. 970970 | 9. 577723 | 10. 422277 | 17
44 | 9. 549026 | 9. 970922 | 9. 578104 | 10. 421896 | 16
45 | 9. 549360 | 9. 970874 | 9. 578486 | 10. 421514 | 15
46 | 9. 549693 | 9. 970826 | 9. 578867 | 10. 421133 | 14
47 | 9. 550026 | 9. 970779 | 9. 579248 | 10. 420752 | 13
48 | 9. 550359 | 9. 970731 | 9. 579628 | 10. 420371 | 12
49 | 9. 550692 | 9. 970683 | 9. 580009 | 10. 419991 | 11
50 | 9. 551024 | 9. 970634 | 9. 580389 | 10. 419611 | 10
51 | 9. 551355 | 9. 970586 | 9. 580769 | 10. 419231 | 9
52 | 9. 551687 | 9. 970538 | 9. 581149 | 10. 418851 | 8
53 | 9. 552018 | 9. 970490 | 9 581528 | 10. 418472 | 7
54 | 9. 552349 | 9. 970442 | 9. 581907 | 10. 418092 | 6
55 | 9. 552680 | 9. 970394 | 9. 582286 | 10. 417713 | 5
56 | 9. 553010 | 9. 970345 | 9. 582665 | 10. 417335 | 4
57 | 9. 553340 | 9. 970297 | 9. 583043 | 10. 416956 | 3
58 | 9. 553670 | 9. 970249 | 9. 583422 | 10. 416578 | 2
59 | 9. 554000 | 9. 970200 | 9. 583800 | 10. 416200 | 1
60 | 9. 554329 | 9. 970152 | 9. 584177 | 10. 415823 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 69.

[Page]

Degree 21.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 554329 | 9. 970152 | 9. 584177 | 10. 415822 | 60
1 | 9. 554658 | 9. 970103 | 9. 584555 | 10. 415445 | 59
2 | 9. 554987 | 9. 970055 | 9. 584932 | 10. 415068 | 58
3 | 9. 555315 | 9. 970006 | 9. 585308 | 10. 414691 | 57
4 | 9. 555643 | 9. 969957 | 9 585686 | 10. 414314 | 56
5 | 9. 555971 | 9. 969909 | 9. 586062 | 10. 413938 | 55
6 | 9. 556299 | 9. 969860 | 9. 586439 | 10. 413561 | 54
7 | 9. 556626 | 9. 969811 | 9. 586815 | 10. 413185 | 53
8 | 9. 556953 | 9. 969762 | 9. 587190 | 10 412800 | 52
9 | 9. 557279 | 9. 969713 | 9. 587566 | 10. 412434 | 51
10 | 9. 557606 | 9. 969665 | 9. 587941 | 10. 412059 | 50
11 | 9. 557932 | 9. 969616 | 9. 588316 | 10. 411684 | 49
12 | 9. 558258 | 9. 969567 | 9. 588691 | 10. 411309 | 48
13 | 9. 558583 | 9. 969518 | 9. 589066 | 10. 410934 | 47
14 | 9. 558909 | 9. 969469 | 9. 589440 | 10. 410560 | 46
15 | 9. 559234 | 9. 969419 | 9. 589814 | 10. 410185 | 45
16 | 9. 559558 | 9. 969370 | 9. 590188 | 10. 409812 | 44
17 | 9. 559883 | 9. 969321 | 9. 590561 | 10. 409438 | 43
18 | 9. 560207 | 9. 969272 | 9. 590935 | 10. 409065 | 42
19 | 9. 560531 | 9. 969223 | 9. 591308 | 10. 408692 | 41
20 | 9. 560855 | 9. 969173 | 9. 591681 | 10. 408319 | 40
21 | 9. 561178 | 9. 969124 | 9. 592054 | 10. 407946 | 39
22 | 9. 561501 | 9. 969075 | 9. 592426 | 10. 407574 | 38
23 | 9. 561824 | 9. 969025 | 9. 592798 | 10. 407201 | 37
24 | 9. 562146 | 9. 968976 | 9. 593170 | 10. 406829 | 36
25 | 9. 562468 | 9. 968926 | 9. 593542 | 10. 406457 | 35
26 | 9. 562790 | 9. 968877 | 9. 593914 | 10. 406086 | 34
27 | 9. 563112 | 9. 968827 | 9. 594285 | 10. 405715 | 33
28 | 9. 563433 | 9. 968777 | 9. 594656 | 10. 405344 | 32
29 | 9. 563754 | 9. 968728 | 9. 595027 | 10. 405073 | 31
30 | 9. 564075 | 9. 968678 | 9. 595397 | 10. 404602 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 68.
Degree 21.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 564075 | 9. 968678 | 9. 595397 | 10. 404602 | 30
31 | 9. 564396 | 9. 968628 | 9. 595768 | 10. 404232 | 29
32 | 9. 564716 | 9. 698578 | 9. 596138 | 10. 403862 | 28
33 | 9. 565036 | 9. 968528 | 9. 596508 | 10. 403492 | 27
34 | 9. 565356 | 9. 968478 | 9. 596878 | 10. 403122 | 26
35 | 9. 565675 | 6. 968428 | 9. 597247 | 10. 402753 | 25
36 | 9. 565995 | 9. 968378 | 9. 597616 | 10. 402384 | 24
37 | 9. 566314 | 9. 968328 | 9. 597985 | 10. 402015 | 23
38 | 9. 566632 | 9. 968278 | 9. 598354 | 10. 401646 | 22
39 | 9. 566951 | 9. 968228 | 9. 598722 | 10. 401277 | 21
40 | 9. 567269 | 9. 968178 | 9. 599091 | 10. 400909 | 20
41 | 9. 567587 | 9. 968128 | 9. 599459 | 10 400541 | 19
42 | 9. 567904 | 9. 968078 | 9. 599827 | 10. 400173 | 18
43 | 9. 568222 | 9. 968027 | 9. 600194 | 10. 399806 | 17
44 | 9. 568539 | 9. 967977 | 9. 600562 | 10. 399438 | 16
45 | 9. 568855 | 9. 967927 | 9. 600929 | 10. 399071 | 15
46 | 9. 569172 | 9. 967876 | 9. 601296 | 10. 398704 | 14
47 | 9. 569488 | 9. 967826 | 9. 601662 | 10. 398337 | 13
48 | 9. 569804 | 9 967775 | 9. 602029 | 10. 397971 | 12
49 | 9. 570120 | 9. 967725 | 9. 602395 | 10. 397605 | 11
50 | 9. 570435 | 9. 967674 | 9. 602761 | 10. 397239 | 10
51 | 9. 570751 | 9. 967623 | 9. 603127 | 10. 396873 | 9
52 | 9. 571065 | 9. 967573 | 9. 603493 | 10. 396507 | 8
53 | 9. 571380 | 9. 967522 | 9. 603858 | 10. 396142 | 7
54 | 9. 571695 | 9. 967471 | 9. 604223 | 10. 395777 | 6
55 | 9. 572009 | 9. 967420 | 9. 604588 | 10. 395412 | 5
56 | 9. 572322 | 9. 967370 | 9. 604953 | 10. 395047 | 4
57 | 9. 572636 | 9. 967319 | 9. 605317 | 10. 394683 | 3
58 | 9. 572949 | 9. 967268 | 9. 605681 | 10. 394318 | 2
59 | 9. 573263 | 9. 967217 | 9. 606046 | 10. 393954 | 1
60 | 9. 573575 | 9. 967166 | 9. 606409 | 10. 393590 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 68.

[Page]

Degree 22.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 573575 | 9. 967166 | 9. 606409 | 10. 393590 | 60
1 | 9. 573888 | 9. 967115 | 9. 606773 | 10. 393227 | 59
2 | 9. 574200 | 9. 967064 | 9. 607136 | 10. 392863 | 58
3 | 9. 574512 | 9. 967012 | 9. 607500 | 10. 392500 | 57
4 | 9. 574824 | 9. 966961 | 9. 607862 | 10. 392137 | 56
5 | 9. 575135 | 9. 966910 | 9. 608225 | 10. 391774 | 55
6 | 9. 575447 | 9. 966859 | 9. 608588 | 10. 391412 | 54
7 | 9. 575758 | 9. 966807 | 9. 608950 | 10. 391050 | 53
8 | 9. 576068 | 9. 966756 | 9. 609312 | 10. 390688 | 52
9 | 9. 576379 | 9. 966705 | 9. 609674 | 10. 390326 | 51
10 | 9. 576689 | 9. 966653 | 9. 600036 | 10. 389964 | 50
11 | 9. 576999 | 9. 966602 | 9. 610397 | 10. 389603 | 49
12 | 9. 577309 | 9. 966550 | 9. 610758 | 10. 389241 | 48
13 | 9. 577618 | 9. 966499 | 9. 611119 | 10. 388880 | 47
14 | 9. 577927 | 9 966447 | 9. 611480 | 10. 388520 | 46
15 | 9. 578236 | 9. 966395 | 9. 611841 | 10. 388159 | 45
16 | 9. 578545 | 9. 966344 | 9. 612201 | 10. 387799 | 44
17 | 9. 578853 | 9. 966292 | 9. 612561 | 10. 387438 | 43
18 | 9. 579161 | 9. 966240 | 9. 612921 | 10. 387078 | 42
19 | 9. 579469 | 9. 966188 | 9. 613281 | 10. 386719 | 41
20 | 9. 579777 | 9. 966136 | 9. 613641 | 10. 386359 | 40
21 | 9. 580084 | 9. 966084 | 9. 614000 | 10. 386000 | 39
22 | 9. 580392 | 9. 966032 | 9. 614359 | 10. 385641 | 38
23 | 9 580698 | 9. 965980 | 9. 614718 | 10. 385282 | 37
24 | 9. 581005 | 9. 965928 | 9 615077 | 10. 384923 | 36
25 | 9. [...] | 9. 965876 | 9. 615435 | 10. 384565 | 35
26 | 9. 581618 | 9 965824 | 9. 615793 | 10. 384207 | 34
27 | 9 581923 | 9. 965772 | 9. 616151 | 10. 383448 | 33
28 | 9. 582229 | 9. 965720 | 9. 616509 | 10. 383491 | 32
29 | 9. 582534 | 9. 965668 | 9. 616867 | 10. 383133 | 31
30 | 9. 582840 | 9. 965615 | 9. 617224 | 10. 382776 | 30
Co sine | Sine | Co-tang. | Tangent | M
Degree 67.
Degree 22.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 582840 | 9. 965615 | 9. 617224 | 10. 382776 | 30
31 | 9. 583144 | 9. 965563 | 9. 617581 | 10. 382418 | 29
32 | 9. 583449 | 9. 965511 | 9. 617938 | 10. 382061 | 28
33 | 9. 583753 | 9. 965458 | 9. 618295 | 10. 381705 | 27
34 | 9. 584058 | 9. 965406 | 9. 618652 | 10. 381348 | 26
35 | 9. 584361 | 9. 965353 | 9. 619008 | 10. 380992 | 25
36 | 9. 584665 | 9. 965301 | 9. 619364 | 10. 380635 | 24
37 | 9. 584968 | 9. 965248 | 9. 619720 | 10. 380279 | 23
38 | 9. 585271 | 9. 965195 | 9. 620076 | 10. 379924 | 22
39 | 9. 585574 | 9. 965143 | 9. 620432 | 10. 379568 | 21
40 | 9. 585877 | 9. 965090 | 9. 620787 | 10. 379213 | 20
41 | 9. 586179 | 9. 965037 | 9. 621142 | 10. 378858 | 19
42 | 9. 586481 | 9. 964984 | 9. 621497 | 10. 378503 | 18
43 | 9. 586783 | 9. 964931 | 9. 621852 | 10. 378148 | 17
44 | 9. 587085 | 9. 964878 | 9. 622206 | 10. 377793 | 16
45 | 9. 587386 | 9. 964825 | 9. 622561 | 10. 377439 | 15
46 | 9. 587687 | 9. 964772 | 9. 622915 | 10. 377085 | 14
47 | 9. 587988 | 9. 964719 | 9. 623269 | 10. 376731 | 13
48 | 9. 588289 | 9. 964666 | 9. 623623 | 10. 376377 | 12
49 | 9. 588589 | 9. 964613 | 9. 623976 | 10. 376024 | 11
50 | 9. 588890 | 9. 964560 | 9. 624330 | 10. 375670 | 10
51 | 9. 589190 | 9. 964507 | 9. 624683 | 10. 375317 | 9
52 | 9. 589489 | 9. 964454 | 9. 625036 | 10. 374964 | 8
53 | 9. 589789 | 9. 964400 | 9. 625388 | 10. 374612 | 7
54 | 9. 590088 | 9. 964347 | 9. 625741 | 10. 374259 | 6
55 | 9. 590387 | 9. 964294 | 9. 626093 | 10. 373907 | 5
56 | 9. 590686 | 9. 964240 | 9. 626445 | 10. 373555 | 4
57 | 9. 590984 | 9. 964187 | 9. 626797 | 10. 373203 | 3
58 | 9. 591282 | 9. 964133 | 9. 627149 | 10. 372850 | 2
59 | 9. 591580 | 9. 964080 | 9. 627501 | 10. 372499 | 1
60 | 9. 591878 | 9. 964026 | 9. 627852 | 10. 372148 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 67.

[Page]

Degree 23.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 591878 | 9. 964026 | 9. 627852 | 10. 372148 | 60
1 | 9. 592175 | 9. 963972 | 9. 628203 | 10. 371797 | 59
2 | 9. 592473 | 9. 963919 | 9. 628554 | 10. 371446 | 58
3 | 9. 592770 | 9. 963865 | 9. 628905 | 10. 371095 | 57
4 | 9. 593067 | 9. 963811 | 9. 629255 | 10. 370744 | 56
5 | 9. 593363 | 9. 963757 | 9. 629606 | 10. 370394 | 55
6 | 9. 593659 | 9 963703 | 9. 629956 | 10. 370044 | 54
7 | 9. 593955 | 9. 963650 | 9. 630306 | 10. 369694 | 53
8 | 9. 594251 | 9. 963596 | 9. 630655 | 10. 369344 | 52
9 | 9. 594547 | 9. 963542 | 9. 631005 | 10. 368995 | 51
10 | 9. 594842 | 9. 963488 | 9. 631354 | 10. 368645 | 50
11 | 9. 595137 | 9. 963433 | 9. 631704 | 10. 368296 | 49
12 | 9. 595432 | 9. 963379 | 9. 632053 | 10. 367947 | 48
13 | 9. 595727 | 9. 963325 | 9. 632401 | 10. 367598 | 47
14 | 9. 596021 | 9. 963271 | 9. 632750 | 10. 367250 | 46
15 | 9. 596315 | 9. 963217 | 9. 633098 | 10. 366901 | 45
16 | 9. 596610 | 9. 963102 | 9. 633447 | 10. 366553 | 44
17 | 9. 596903 | 9. 963108 | 9. 633795 | 10. 366205 | 43
18 | 9. 597196 | 9. 963054 | 9. 634143 | 10. 365857 | 42
19 | 9. 597490 | 9. 962999 | 9. 634490 | 10. 365510 | 41
20 | 9. 597783 | 9. 962945 | 9. 634838 | 10. 365162 | 40
21 | 9 598075 | 9. 962892 | 9. 635185 | 10. 364815 | 39
22 | 9 598368 | 9. 962836 | 9. 635530 | 10. 364468 | 38
23 | 9. 598660 | 9. 962781 | 9. 635879 | 10. 364121 | 37
24 | 9. 598952 | 9 962726 | 9. 636226 | 10. 363774 | 36
25 | 9 599244 | 9. 962672 | 9. 636571 | 10. 363428 | 35
26 | 9. 599536 | 9. 962617 | 9. 636918 | 10. 363081 | 34
27 | 9. 599827 | 9. 962562 | 9. 637205 | 10. 362735 | 33
28 | 9. 600118 | 9. 962507 | 9. 637610 | 10. 362389 | 32
29 | 9. 600409 | 9. 962453 | 9. 637956 | 10. 362044 | 31
30 | 9. 600700 | 9. 962398 | 9. 638302 | 10. 361698 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 66.
Degree 23.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 600700 | 9. 967398 | 9. 638302 | 10. 361698 | 30
31 | 9. 600990 | 9. 962343 | 9. 638647 | 10. 361353 | 29
32 | 9. 601280 | 9. 962288 | 9. 638992 | 10. 361007 | 28
33 | 9. 601570 | 9. 962233 | 9. 639337 | 10. 360662 | 27
34 | 9. 601860 | 9 962178 | 9. 639682 | 10. 360318 | 26
35 | 9. 602149 | 9. 962122 | 9. 640027 | 10. 359973 | 25
36 | 9. 602439 | 9. 962067 | 9. 640371 | 10. 359629 | 24
37 | 9. 602728 | 9. 962012 | 9. 640716 | 10. 359284 | 23
38 | 9 603017 | 9. 961957 | 9. 641060 | 10. 358940 | 22
39 | 9. 603305 | 9. 961902 | 9. 641404 | 10. 358596 | 21
40 | 9. 603594 | 9. 961846 | 9. 641747 | 10. 358253 | 20
41 | 9. 603882 | 9. 961791 | 9. 642091 | 10. 357909 | 19
42 | 9. 604170 | 9. 961735 | 9. 642434 | 10. 357566 | 18
43 | 9. 604457 | 9. 961680 | 9. 642777 | 10. 357223 | 17
44 | 9. 604745 | 9. 961624 | 9. 643120 | 10. 356980 | 16
45 | 9. 605032 | 9. 961569 | 9. 643463 | 10. 356537 | 15
46 | 9. 605319 | 9. 961513 | 9. 643806 | 10. 356194 | 14
47 | 9. 605606 | 9. 961458 | 9. 644148 | 10. 355852 | 13
48 | 9. 605892 | 9. 961402 | 9. 644490 | 10. 355510 | 12
49 | 9. 606179 | 9. 961346 | 9. 644832 | 10. 355168 | 11
50 | 9. 606465 | 9. 961290 | 9. 645174 | 10. 354826 | 10
51 | 9. 606750 | 9. 961235 | 9. 645516 | 10. 354484 | 9
52 | 9. 607036 | 9. 961179 | 9. 645857 | 10. 354142 | 8
53 | 9. 607322 | 9. 961123 | 9. 646199 | 10. 353801 | 7
54 | 9. 607607 | 9. 961067 | 9. 646540 | 10. 353460 | 6
55 | 9. 607892 | 9. 961011 | 9. 646881 | 10. 353119 | 5
56 | 9. 608176 | 9. 960955 | 9. 647222 | 10. 352778 | 4
57 | 9. 608461 | 9. 960899 | 9. 647562 | 10. 352438 | 3
58 | 9. 608745 | 9. 960842 | 9. 647903 | 10. 352097 | 2
59 | 9. 609029 | 9. 960786 | 9. 648243 | 10. 351757 | 1
60 | 9. 609313 | 9. 960730 | 9. 648583 | 10. 351417 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 66.

[Page]

Degree 24.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 609313 | 9. 960730 | 9. 648583 | 10. 351417 | 60
1 | 9. 609597 | 9. 960674 | 9. 648923 | 10. 351077 | 59
2 | 9. 609880 | 9. 960617 | 9. 649263 | 10. 350737 | 58
3 | 9. 610163 | 9. 960561 | 9. 649602 | 10. 350398 | 57
4 | 9. 610446 | 9. 960505 | 9. 649942 | 10. 350058 | 56
5 | 9. 610729 | 9. 960448 | 9. 650281 | 10. 349319 | 55
6 | 9. 611012 | 9. 960392 | 9. 650620 | 10. 349380 | 54
7 | 9. 611294 | 9. 960335 | 9. 650959 | 10. 349041 | 53
8 | 9. 611576 | 9. 960279 | 9. 651297 | 10. 348703 | 52
9 | 9. 611858 | 9. 960222 | 9. 651636 | 10. 348364 | 51
10 | 9. 612140 | 9. 960165 | 9. 651974 | 10. 348026 | 50
11 | 9. 612421 | 9. 960109 | 9. 652312 | 10. 347688 | 49
12 | 9. 612702 | 9. 960052 | 9. 652650 | 10. 347350 | 48
13 | 9. 612983 | 9. 959995 | 9. 652988 | 10. 347012 | 47
14 | 9. 613264 | 9. 959938 | 9. 653326 | 10. 346674 | 46
15 | 9. 613545 | 9. 959881 | 9. 653663 | 10. 346337 | 45
16 | 9. 613825 | 9. 959824 | 9. 654000 | 10. 345999 | 44
17 | 9. 614105 | 9. 959768 | 9. 654337 | 10. 345662 | 43
18 | 9. 614385 | 9. 959710 | 9. 654674 | 10. 345325 | 42
19 | 9. 614665 | 9. 959653 | 9. 655011 | 10. 344989 | 41
20 | 9. 614944 | 9. 959596 | 9. 655348 | 10. 344652 | 40
21 | 9. 615223 | 9. 959539 | 9. 655684 | 10. 344316 | 39
22 | 9. 615502 | 9. 959482 | 9. 656020 | 10. 343980 | 38
23 | 9. 615781 | 9. 959425 | 9. 656356 | 10. 343643 | 37
24 | 9. 616060 | 9. 959367 | 9. 656692 | 10. 343308 | 36
25 | 9. 616338 | 9. 959310 | 9. 657028 | 10. 342972 | 35
26 | 9. 616616 | 9. 959253 | 9. 657363 | 10. 342636 | 34
27 | 9. 616894 | 9. 959195 | 9. 657699 | 10. 342301 | 33
28 | 9. 617172 | 9. 959138 | 9. 658034 | 10. 341966 | 32
29 | 9. 617450 | 9. 959080 | 9. 658369 | 10. 341531 | 31
30 | 9 617727 | 9. 959023 | 9. 658704 | 10. 341296 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 65.
Degree 24.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 617727 | 9. 959023 | 9. 658704 | 10. 341296 | 30
31 | 9. 618004 | 9. 958965 | 9. 659039 | 10. 340926 | 29
32 | 9. 618281 | 9. 958908 | 9659373 | 10. 340627 | 28
33 | 9. 618558 | 9. 958850 | 9. 659708 | 10. 340292 | 27
34 | 9. 618834 | 9. 958792 | 9. 660042 | 10. 339958 | 26
35 | 9. 619110 | 9. 958734 | 9. 660376 | 10. 339624 | 25
36 | 9. 619386 | 9. 958677 | 9. 660710 | 10. 339290 | 24
37 | 9. 619662 | 9. 958619 | 9. 661043 | 10. 338957 | 23
38 | 9. 619938 | 9. 958561 | 9. 661377 | 10. 338623 | 22
39 | 9. 620213 | 9. 958503 | 9. 661710 | 10. 338290 | 21
40 | 9. 620488 | 9. 958445 | 9. 662043 | 10. 337956 | 20
41 | 9. 620763 | 9. 958387 | 9. 662376 | 10. 337623 | 19
42 | 9. 621038 | 9. 958329 | 9. 662709 | 10. 337291 | 18
43 | 9. 621313 | 9. 958271 | 9. 663042 | 10. 336958 | 17
44 | 9. 621587 | 9. 958212 | 9. 663374 | 10. 336625 | 16
45 | 9. 621861 | 9. 958154 | 9. 663707 | 10. 336293 | 15
46 | 9. 622135 | 9. 958096 | 9. 664039 | 10. 335961 | 14
47 | 9. 622409 | 9. 958038 | 9. 664371 | 10. 335629 | 13
48 | 9. 622682 | 9. 957979 | 9. 664703 | 10. 335297 | 12
49 | 9. 622956 | 9. 957921 | 9. 665035 | 10. 334965 | 11
50 | 9. 623229 | 9. 957862 | 9. 665366 | 10. 334634 | 10
51 | 9. 623502 | 9. 957804 | 9. 665697 | 10. 334302 | 9
52 | 9. 623774 | 9. 957745 | 9. 666029 | 10. 333971 | 8
53 | 9. 624047 | 9. 957687 | 9. 666360 | 10. 333640 | 7
54 | 9. 624319 | 9. 957628 | 9. 666691 | 10. 333309 | 6
55 | 9. 624591 | 9. 957570 | 9. 667021 | 10. 332979 | 5
56 | 9. 624863 | 9. 957511 | 9. 667352 | 10. 332648 | 4
57 | 9. 625134 | 9. 957452 | 9. 667682 | 10. 332318 | 3
58 | 9. 625406 | 9. 957393 | 9. 668012 | 10. 331987 | 2
59 | 9. 625677 | 9. 957334 | 9. 668343 | 10. 331657 | 1
60 | 9. 625948 | 9. 957276 | 9. 668672 | 10. 331327 | 0
Co-sine | Sine | Co- [...]. | Tangent | M
Degree 65.

[Page]

Degree 25.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 625948 | 9. 957276 | 9. 668672 | 10. 331327 | 60
1 | 9. 626219 | 9. 957217 | 9. 669002 | 10. 330998 | 59
2 | 9. 626490 | 9. 957158 | 9. 669332 | 10. 330668 | 58
3 | 9. 626760 | 9 957099 | 9. 669661 | 10. 330339 | 57
4 | 9. 627030 | 9. 957040 | 9. 699990 | 10. 330009 | 56
5 | 9. 627300 | 9. 956981 | 9. 670320 | 10. 329680 | 55
6 | 9. 627570 | 9. 956922 | 9. 670649 | 10. 329351 | 54
7 | 9 627840 | 9. 956862 | 9. 670977 | 10. 329022 | 53
8 | 9. 628109 | 9. 956803 | 9. 671306 | 10. 328694 | 52
9 | 9. 628378 | 9. 956744 | 9. 671634 | 10. 328365 | 51
10 | 9. 628647 | 9. 956684 | 9. 671963 | 10. 328037 | 50
11 | 9. 628916 | 9 956625 | 9. 672291 | 10. 327709 | 49
12 | 9. 629184 | 9. 956565 | 9. 672619 | 10. 327381 | 48
13 | 9. 629453 | 9. 956208 | 9. 672947 | 10. 327053 | 47
14 | 9. 629721 | 9. 956446 | 9. 673274 | 10. 326725 | 46
15 | 9. 629989 | 9. 956387 | 9 673603 | 10. 326398 | 45
16 | 9. 630257 | 9. 956327 | 9 673929 | 10. 326070 | 44
17 | 9. 630524 | 9. 956267 | 9. 674256 | 10. 325743 | 43
18 | 9. 630792 | 9. 956208 | 9. 674584 | 10. 325416 | 42
19 | 9. 631059 | 9. 956148 | 9. 674910 | 10. 325089 | 41
20 | 9. 631326 | 6. 956088 | 9. 675237 | 10. 324763 | 40
21 | 9. 631592 | 9. 956029 | 9. 675564 | 10. 324436 | 39
22 | 9. 631859 | 9. 955969 | 9. 675890 | 10. 324110 | 38
23 | 9 632125 | 9. 955909 | 9. 676216 | 10. 323783 | 37
24 | 9. 632392 | 9. 955849 | 9. 676543 | 10. 323457 | 36
25 | 9. 632657 | 9. 955789 | 9. 676869 | 10. 323131 | 35
26 | 9. 632923 | 9. 955739 | 9. 677194 | 10. 322805 | 34
27 | 9. 633189 | 9. 955669 | 9. 677520 | 10. 322480 | 33
28 | 9 633454 | 9 955609 | 9. 677845 | 10. 322154 | 32
29 | 9. 633719 | 9. 955548 | 9. 678171 | 10. 321829 | 31
30 | 9. 633984 | 9. 955488 | 9. 678496 | 10. 321504 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 64.
Degree 25.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 633984 | 9. 955488 | 9. 678496 | 10. 321504 | 30
31 | 9. 634249 | 9. 955428 | 9. 678821 | 10. 321179 | 29
32 | 9. 634514 | 9. 955367 | 9. 679146 | 10. 320854 | 28
33 | 9. 634778 | 9 955307 | 9. 679471 | 10. 320529 | 27
34 | 9. 635042 | 9. 955246 | 9. 679795 | 10. 320205 | 26
35 | 9. 635306 | 9. 955186 | 9. 680120 | 10. 319880 | 25
36 | 9. 635570 | 9. 955125 | 9. 680444 | 10. 319556 | 24
37 | 9. 635833 | 9. 955065 | 9. 680768 | 10. 319232 | 23
38 | 9. 636097 | 9. 955004 | 9. 681092 | 10. 318908 | 22
39 | 9. 636360 | 9. 954944 | 9. 681416 | 10. 318584 | 21
40 | 9. 636623 | 9. 954883 | 9. 681740 | 10. 318260 | 20
41 | 9. 636886 | 9. 954823 | 9. 682063 | 10. 317937 | 19
42 | 9. 637148 | 9. 954762 | 9. 682386 | 10. 317613 | 18
43 | 9. 637411 | 9. 954701 | 9. 682710 | 10. 317290 | 17
44 | 9. 637673 | 9. 954640 | 9. 683033 | 10. 316967 | 16
45 | 9. 637935 | 9. 954579 | 9. 683356 | 10. 316644 | 15
46 | 9. 638197 | 9. 954518 | 9. 683678 | 10. 316321 | 14
47 | 9. 638458 | 9. 954457 | 9. 684001 | 10. 315999 | 13
48 | 9. 638720 | 9. 954396 | 9. 684324 | 10. 315676 | 12
49 | 9. 638981 | 9. 954335 | 9. 684646 | 10. 315354 | 11
50 | 9. 639242 | 9. 954274 | 9. 684968 | 10. 315032 | 10
51 | 9. 639503 | 9. 954213 | 9. 685290 | 10. 314710 | 9
52 | 9. 639764 | 9. 954152 | 9. 685612 | 10. 313388 | 8
53 | 9. 640024 | 9. 954090 | 9. 685934 | 10 314066 | 7
54 | 9. 640284 | 9. 954029 | 9. 686255 | 10. 313745 | 6
55 | 9. 640544 | 9. 954968 | 9. 686577 | 10. 313423 | 5
56 | 9. 640804 | 9. 953906 | 9. 686898 | 10. 313102 | 4
57 | 9. 641064 | 9. 953845 | 9. 687219 | 10 312781 | 3
58 | 9. 641323 | 9. 953783 | 9. 687540 | 10. 312460 | 2
59 | 9. 641583 | 9. 953722 | 9. 687861 | 10. [...] | 1
60 | 9. 641842 | 9. 953660 | 9. 688182 | 10 311818 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 64.

[Page]

Degree 26.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 641842 | 9. 953660 | 9. 688182 | 10. 311818 | 60
1 | 9. 642101 | 9. 953598 | 9. 688502 | 10. 311498 | 59
2 | 9. 642360 | 9. 953537 | 9. 688823 | 10. 311177 | 58
3 | 9. 642618 | 9. 953475 | 9. 689143 | 10. 310857 | 57
4 | 9. 642876 | 9. 953413 | 9. 689463 | 10. 310537 | 56
5 | 9. 643135 | 9. 953351 | 9. 689783 | 10. 310237 | 55
6 | 9. 643393 | 9. 953290 | 9. 690103 | 10. 309897 | 54
7 | 9. 643650 | 9. 953228 | 9. 690423 | 10. 309577 | 53
8 | 9. 643908 | 9. 953166 | 9. 690742 | 10. 309258 | 52
9 | 9. 644165 | 9. 953104 | 9. 691063 | 10. 308938 | 51
10 | 9. 644423 | 9. 953042 | 9. 691381 | 10. 308619 | 50
11 | 9. 644680 | 9. 952980 | 9. 691700 | 10. 308300 | 49
12 | 9. 644936 | 9. 952917 | 9. 692019 | 10. 307981 | 48
13 | 9. 645193 | 9. 952855 | 9. 692338 | 10. 307662 | 47
14 | 9. 645449 | 9. 952793 | 9. 692656 | 10. 307343 | 46
15 | 9. 645706 | 9. 952731 | 9. 692975 | 10. 307025 | 45
16 | 9. 645962 | 9. 952668 | 9. 693293 | 10. 306706 | 44
17 | 9. 646218 | 9. 952606 | 9. 693612 | 10. 306388 | 43
18 | 9. 646473 | 9. 952544 | 9. 693930 | 10. 306070 | 42
19 | 9. 646729 | 9. 952481 | 9. 694248 | 10. 305752 | 41
20 | 9. 646984 | 9. 952419 | 9. 694566 | 10. 305434 | 40
21 | 9. 647239 | 9. 952356 | 9. 694883 | 10. 305117 | 39
22 | 9. 647494 | 9. 952294 | 9. 695201 | 10. 304799 | 38
23 | 9. 647749 | 9. 952231 | 9. 695518 | 10. 304482 | 37
24 | 9. 648004 | 9. 952168 | 9. 695835 | 10. 304164 | 36
25 | 9. 648258 | 9. 952105 | 9. 696153 | 10. 303847 | 35
26 | 9. 648512 | 9. 952043 | 9 696470 | 10. 303530 | 34
27 | 9. 648766 | 9. 951980 | 9. 696786 | 10. 303213 | 33
28 | 9. 648020 | 9. 951917 | 9. 697103 | 10. 302897 | 32
29 | 9. 649274 | 9. 951854 | 9. 697420 | 10. 302580 | 31
30 | 9. 649527 | 9. 051791 | 9. 697738 | 10. 302264 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 63.
Degree 26.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 649527 | 9. 951791 | 9. 697738 | 10. 302264 | 30
31 | 9. 649781 | 9. 951728 | 9. 698052 | 10. 301947 | 29
32 | 9. 650034 | 9. 951665 | 9. 698369 | 10. 301631 | 28
33 | 9. 650287 | 9. 951602 | 9. 698685 | 10. 301315 | 27
34 | 9. 650519 | 9. 951539 | 9. 699001 | 10. 300999 | 26
35 | 9. 650798 | 9. 951476 | 9. 699316 | 10 300684 | 25
36 | 9. 651044 | 9. 951412 | 9 699632 | 10. 300368 | 24
37 | 9. 651296 | 9. 951349 | 9. 699947 | 10. 300052 | 23
38 | 9. 651648 | 9. 951286 | 9. 700263 | 10. 299737 | 22
39 | 9. 651800 | 9. 951222 | 9. 700578 | 10. 299422 | 21
40 | 9. 652052 | 9. 951159 | 9. 700893 | 10. 299107 | 20
41 | 9. 652303 | 9. 951095 | 9. 701208 | 10. 298792 | 19
42 | 9. 652555 | 9. 951032 | 9. 701522 | 10. 298477 | 18
43 | 9. 652806 | 9. 950968 | 9. 701837 | 10. 298163 | 17
44 | 9. 653057 | 9. 950905 | 9. 702152 | 10. 297848 | 16
45 | 9. [...] | 9. 950841 | 9. 702466 | 10. 297534 | 15
46 | 9. 653558 | 9. 950777 | 9. 702780 | 10. 297219 | 14
47 | 9. 653808 | 9. 950714 | 9. 703095 | 10. 296905 | 13
48 | 9. 654059 | 9. 950650 | 9. 703409 | 10. 296591 | 12
49 | 9. 654309 | 9. 950586 | 9. 703722 | 10. 296277 | 11
50 | 9. 654558 | 9. 950522 | 9. 704036 | 10. 295964 | 10
51 | 9. 654808 | 9. 950458 | 9. 704350 | 10. 295656 | 9
52 | 9. 655057 | 9. 950394 | 9. 704663 | 10. 295337 | 8
53 | 9. 655307 | 9. 950330 | 9. 704976 | 10. 295023 | 7
54 | 9. 655556 | 9. 950266 | 9. 705290 | 10. 294710 | 6
55 | 9. 655805 | 9. 950202 | 9. 705603 | 10. 294397 | 5
56 | 9. 656053 | 9. 950138 | 9. 705915 | 10. 294084 | 4
57 | 9. 656302 | 9. 950074 | 9. 706228 | 10. 293771 | 3
58 | 9. 656550 | 9. 950009 | 9. 706541 | 10. 293459 | 2
59 | 9. 656799 | 9. 949945 | 9. 706853 | 10. 293146 | 1
9. 656347 | 9. [...] | 9. 707166 | 10. 292834 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 63.

[Page]

Degree 27.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 657047 | 9. 949880 | 9. 707166 | 10. 292834 | 60
1 | 9. 657295 | 9. 949816 | 9. 707478 | 10. 292523 | 59
2 | 9. 657542 | 9. 949752 | 9. 707790 | 10. 292210 | 58
3 | 9. 657790 | 9. 949687 | 9. 708102 | 10. 291897 | 57
4 | 9. 658037 | 9. 949623 | 9. 708414 | 10. 291586 | 56
5 | 9. 658284 | 9. 949598 | 9. 708726 | 10. 291274 | 55
6 | 9. 658531 | 9. 949494 | 9. 709037 | 10. 290962 | 54
7 | 9. 658777 | 9. 949429 | 9. 709349 | 10. 290651 | 53
8 | 9. 659024 | 9. 949364 | 9. 709660 | 10. 290340 | 52
9 | 9. 659271 | 9. 949300 | 9. 709971 | 10. 290029 | 51
10 | 9. 659517 | 9. 949235 | 9. 710282 | 10. 289718 | 50
11 | 9. 659763 | 9. 949170 | 9. 710593 | 10. 289407 | 49
12 | 9. 660009 | 9. 949105 | 9. 710904 | 10. 289096 | 48
13 | 9. 660255 | 9. 949040 | 9. 711214 | 10. 288785 | 47
14 | 9. 660500 | 9. 948976 | 9. 711525 | 10. 288475 | 46
15 | 9. 660746 | 9. 948910 | 9. 711836 | 10. 288164 | 45
16 | 9. 660991 | 9. 948845 | 9. 712146 | 10. 287854 | 44
17 | 9. 661036 | 9. 948760 | 9. 712456 | 10. 287544 | 43
18 | 9. 661481 | 9. 948715 | 9. 712766 | 10. 287234 | 42
19 | 9. 661726 | 9. 948650 | 9. 713076 | 10. 286924 | 41
20 | 9. 661970 | 9. 948584 | 9. 713386 | 10. 286614 | 40
21 | 9. 662214 | 9. 948519 | 9. 713695 | 10. 286305 | 39
22 | 9. 662459 | 9. 948453 | 9. 714005 | 10. 285995 | 38
23 | 9. 662702 | 9. 948388 | 9. 714314 | 10. 285686 | 37
24 | 9. 662947 | 9. 948323 | 9. 714624 | 10. 285376 | 36
25 | 9. 663190 | 9. 948257 | 9. 714933 | 10. 285067 | 35
26 | 9. 663433 | 9. 948191 | 9. 715241 | 10. 284758 | 34
27 | 9. 663677 | 9. 948126 | 9. 715550 | 10. 284449 | 33
28 | 9. 663920 | 9. 948060 | 9. 715859 | 10. 284140 | 32
29 | 9. 664163 | 9. 947995 | 9. 716168 | 10. 283832 | 31
30 | 9. 664406 | 9. 947929 | 9. 716477 | 10. 283523 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 62.
Degree 27.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 664406 | 9. 947929 | 9. 716477 | 10. 283523 | 30
31 | 9. 664648 | 9. 947863 | 9. 716785 | 10. 283215 | 29
32 | 9. 664891 | 9. 947797 | 9. 717093 | 10. 282907 | 28
33 | 9. 665133 | 9. 947731 | 9. 717401 | 10. 282598 | 27
34 | 9. 665375 | 9. 947665 | 9. 717709 | 10. 282290 | 26
35 | 9. 665617 | 9. 947599 | 9. 718017 | 10. 281983 | 25
36 | 9. 665858 | 9. 947533 | 9. 718325 | 10. 281675 | 24
37 | 9. 666100 | 9. 947467 | 9. 718633 | 10. 281367 | 23
38 | 9. 666341 | 9. 947401 | 9. 718940 | 10. 281060 | 22
39 | 9. 666583 | 9. 947335 | 9. 719248 | 10. 280752 | 21
40 | 9. 666824 | 9. 947269 | 9. 719555 | 10. 280445 | 20
41 | 9. 667065 | 9. 947203 | 9. 719862 | 10. 280138 | 19
42 | 9. 667305 | 9. 947136 | 9. 720169 | 10. 279831 | 18
43 | 9. 667546 | 9. 947070 | 9. 720476 | 10. 279524 | 17 | 44 | 9. 667786 | 9. 947004 | 9. 720783 | 10. 279217 | 16
45 | 9. 668026 | 9. 946937 | 9. 721089 | 10. 278911 | 15
46 | 9. 668266 | 9. 946871 | 9. 721395 | 10. 278604 | 14
47 | 9. 668506 | 9. 946804 | 9. 721702 | 10. 278298 | 13
48 | 9. 668746 | 9. 946738 | 9. 722008 | 10. 277991 | 12
49 | 9. 668986 | 9. 946671 | 9. 722315 | 10. 277685 | 11
50 | 9. 669225 | 9. 946604 | 9. 722621 | 10. 277379 | 10
51 | 9. 669464 | 9. 946537 | 9. 722927 | 10. 277073 | 9
52 | 9. 669703 | 9. 946471 | 9. 723232 | 10. 276768 | 8
53 | 9. 669942 | 9. 946404 | 9. 723538 | 10. 276462 | 7
54 | 9. 670181 | 9. 946337 | 9. 723843 | 10. 276156 | 6
55 | 9. 670419 | 9. 946270 | 9. 724149 | 10. 275851 | 5
56 | 9. 670657 | 9. 946203 | 9. 724454 | 10. 275546 | 4
57 | 9. 670896 | 9. 946136 | 9. 724759 | 10. 275240 | 3
58 | 9. 671134 | 9. 946069 | 9. 725065 | 10. 274935 | 2
59 | 9. 671372 | 9. 946002 | 9. 725369 | 10. 274630 | 1
60 | 9. 671609 | 9. 945935 | 9. 725674 | 10. 274326 | 0
Co-sine. | Sine | Co-tang. | Tangent | M
Degree 62.

[Page]

Degree 28.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 671609 | 9. 945935 | 9. 725674 | 10. 274326 | 60
1 | 9. 671847 | 9. 945868 | 9. 725979 | 10. 274021 | 59
2 | 9. 672084 | 9. 945800 | 9. 726284 | 10. 273816 | 58
3 | 9. 672321 | 9. 945733 | 9. 726588 | 10. 273412 | 57
4 | 9. 672558 | 9. 945666 | 9. 726892 | 10. 273107 | 56
5 | 9. 672795 | 9. 945598 | 9. 727197 | 10. 272803 | 55
6 | 9. 673032 | 9. 945531 | 9. 727501 | 10. 272499 | 54
7 | 9. 673268 | 9. 945463 | 9. 727805 | 10. 272195 | 53
8 | 9. 673505 | 9. 945396 | 9. 728109 | 10. 271891 | 52
9 | 9. 673741 | 9. 945328 | 9. 728412 | 10. 271587 | 51
10 | 9. 673977 | 9. 945261 | 9. 728716 | 10. 271284 | 50
11 | 9. 674213 | 9. 945193 | 9. 729020 | 10. 270980 | 49
12 | 9. 674448 | 9. 945125 | 9. 729323 | 10. 270677 | 48
13 | 9. 674684 | 9. 945058 | 9. 729626 | 10. 270374 | 47
14 | 9. 674919 | 9. 944990 | 9. 729929 | 10. 270070 | 46
15 | 9. 675154 | 9. 944922 | 9. 730232 | 10. 269767 | 45
16 | 9. 675389 | 9. 944854 | 9. 730535 | 10. 269464 | 44
17 | 9. 675623 | 9. 944786 | 9. 730838 | 10. 269162 | 43
18 | 9. 675859 | 9. 944718 | 9. 731141 | 10. 268859 | 42
19 | 9. 676094 | 9. 944650 | 9. 731443 | 10. 268559 | 41
20 | 9. 676328 | 9. 944582 | 9. 731746 | 10. 268254 | 40
21 | 9. 676562 | 9. 944514 | 9. 732048 | 10. 267952 | 39
22 | 9. 676796 | 9. 944446 | 9. 732351 | 10. 267649 | 38
23 | 9. 677030 | 9. 944377 | 9. 732653 | 10. 267347 | 37
24 | 9. 677264 | 9. 944309 | 9. 732955 | 10. 267045 | 36
25 | 9. 677497 | 9. 944241 | 9. 733257 | 10. 266743 | 35
26 | 9. 677731 | 9. 944172 | 9. 733558 | 10. 266441 | 34
27 | 9. 677964 | 9. 944104 | 9. 733860 | 10. 266140 | 33
28 | 9. 678197 | 9. 944016 | 9. 734162 | 10. 265838 | 32
29 | 9. 678430 | 9. 943967 | 9. 734463 | 10. 265537 | 31
30 | 9. 678663 | 9. 943898 | 9. 734764 | 10. 265236 | 30
Co-sine. | Sine | Co-tang. | Tangent | M
Degree 61.
Degree 28.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 678663 | 9. 943898 | 9734764 | 10. 265236 | 30
31 | 9. 678895 | 9. 943830 | 9. 735666 | 10. 264934 | 29
32 | 9. 679128 | 9. 943761 | 9. 735362 | 10. 264633 | [...]
33 | 9. 670360 | 9. 943692 | 9. 735668 | 10. 264332 | 27
34 | 9. 679592 | 9. 943624 | 9. 735968 | 10. 264031 | 26
35 | 9. 679824 | 9. 943555 | 9. 736269 | 10. 263731 | 25
36 | 9. 680056 | 9. 943486 | 9. 736570 | 10. 263430 | 24
37 | 9. 680288 | 9. 943417 | 9. 736870 | 10. 263130 | 23
38 | 9. 680519 | 9. 943348 | 9. 737171 | 10. 262829 | 22
39 | 9. 680750 | 9. 943279 | 9. 737471 | 10. 262529 | 21
40 | 9. 680982 | 9. 943210 | 9. 737771 | 10. 262229 | 20
41 | 9. 681213 | 9. 943141 | 9. 738071 | 10. 261929 | 19
42 | 9. 681443 | 9. 943072 | 9. 738371 | 10. 261629 | 18
43 | 9. 681674 | 9. 943003 | 9. 738671 | 10. 261329 | 17
44 | 9. 681904 | 9. 942933 | 9. 738971 | 10. 261029 | 16
45 | 9. 682135 | 9. 942864 | 9. 739271 | 10. 260729 | 15
46 | 9. 682365 | 9. 942795 | 9. 739570 | 10. 260430 | 14
47 | 9. 682595 | 9. 942725 | 9. 739870 | 10. 260130 | 13
48 | 9. 682825 | 9. 942656 | 9. 740169 | 10. 259831 | 12
49 | 9. 683055 | 9. 942587 | 9. 740468 | 10. 259532 | 11
50 | 9. 683284 | 9. 942517 | 9. 740767 | 10. 259233 | 10
51 | 9. 683514 | 9. 942448 | 9. 741066 | 10. 258934 | 9
52 | 9. 683743 | 9. 942378 | 9. 741365 | 10. 258635 | 8
53 | 9. 683972 | 9. 942308 | 9. 741664 | 10. 258336 | 7
54 | 9. 684201 | 9. 942239 | 9. 741962 | 10. 258038 | 6
55 | 9. 684430 | 9. 942169 | 9. 742261 | 10. 257739 | 5
56 | 9. 684658 | 9. 942099 | 9. 742559 | 10. 257441 | 4
57 | 9 684887 | 9. 942029 | 9. 742858 | 10. 257142 | 3
58 | 9. 685115 | 9. 941059 | 9. 743156 | 10. 256844 | 2
59 | 9. 685343 | 9. 941889 | 9 743454 | 10. 256546 | 1
60 | 9. 685571 | 9. 941819 | 9. 743751 | 10. 256248 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 61.

[Page]

Degree 29.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 685571 | 9. 941819 | 9743752 | 10. 256248 | 60
1 | 9. 685799 | 9. 941749 | 9. 744050 | 10. 255950 | 59
2 | 9. 686027 | 9. 941679 | 9. 744348 | 10. 255652 | 58
3 | 9. 686254 | 9. 941609 | 9. 744645 | 10. 255355 | 57
4 | 9. 686482 | 9. 941539 | 9. 744943 | 10. 255057 | 56
5 | 9. 686709 | 9. 941468 | 9. 745240 | 10. 254760 | 55
6 | 9. 686936 | 9. 941398 | 9. 745538 | 10. 254462 | 54
7 | 9. 687163 | 9. 941328 | 9. 745835 | 10. 254165 | 53
8 | 9. 687389 | 9. 941257 | 9. 746132 | 10. 253868 | 52
9 | 9. 687616 | 9. 941187 | 9. 746429 | 10. 253571 | 51
10 | 9. 687842 | 9. 941116 | 9. 746726 | 10. 253274 | 50
11 | 9. 688069 | 9. 941046 | 9. 747023 | 10. 252977 | 49
12 | 9. 688295 | 9. 940975 | 9. 747319 | 10. 252680 | 48
13 | 9. 688523 | 9. 940905 | 9. 747616 | 10. 252384 | 47
14 | 9. 688747 | 9. 940834 | 9. 747912 | 10. 252087 | 46
15 | 9. 688972 | 9. 940763 | 9. 748209 | 10. 251791 | 45
16 | 9. 689198 | 9. 940693 | 9. 748505 | 10. 251495 | 44
17 | 9. 689421 | 9. 940622 | 9. 748801 | 10. 251199 | 43
18 | 9. 689648 | 7. 940551 | 9. 749097 | 10. 250902 | 42
19 | 9. 689873 | 9. 940480 | 9. 749393 | 10. 250607 | 41
20 | 9. 690098 | 9. 940409 | 9 749689 | 10. 250311 | 40
21 | 9. 690323 | 9. 940338 | 9. 749985 | 10. 250015 | 39
22 | 9. 690548 | 9. 940267 | 9. 750281 | 10. 249719 | 38
23 | 9. 690772 | 9. 940196 | 9. 750576 | 10. 249424 | 37
24 | 9. 690996 | 9. 940125 | 9. 750872 | 10. 249128 | 36
25 | 9. 691220 | 9. 940053 | 9. 751167 | 10. 248833 | 35
26 | 9. 691444 | 9. 939982 | 9. 751462 | 10. 248538 | 34
27 | 9. 691668 | 9. 939911 | 9. 751757 | 10. 248243 | 33
28 | 9. 691892 | 9. 939840 | 9. 752052 | 10. 247948 | 32
29 | 9. 692115 | 9. 939768 | 9. 752347 | 10. 247653 | 31
30 | 9. 692339 | 9. 939697 | 9. 752642 | 10. 247358 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 60.
Degree 29.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 692339 | 9. 939697 | 9. 752642 | 10. 247358 | 30
31 | 9. 692562 | 9. 939625 | 9. 752937 | 10. 247063 | 29
32 | 9. 692785 | 9 939554 | 9. 753231 | 10. 246769 | 28
33 | 9. 693008 | 9. 939482 | 9. 753526 | 10. 246474 | 27
34 | 9. 693231 | 9. 939410 | 9. 753820 | 10. 246180 | 26
35 | 9. 693453 | 9. 939339 | 9. 754115 | 10. 245885 | 25
36 | 9. 693676 | 9. 939267 | 9. 754409 | 10. 245591 | 24
37 | 9. 693898 | 9. 939195 | 9. 754703 | 10. 245297 | 23
38 | 9. 694120 | 9. 939123 | 9. 754997 | 10. 245003 | 22
39 | 9. 694342 | 9. 939051 | 9. 755291 | 10. 244709 | 21
40 | 9. 694564 | 9. 938980 | 9. 755584 | 10. 244415 | 20
41 | 9. 694786 | 9. 938908 | 9. 755878 | 10. 244122 | 19
42 | 9. 695007 | 9. 938835 | 9. 756172 | 10. 243828 | 18
43 | 9. 695229 | 9. 938763 | 9 756465 | 10. 243535 | 17
44 | 9. 695450 | 9. 938691 | 9. 756759 | 10. 243241 | 16
45 | 9. 695671 | 9. 938619 | 9. 757052 | 10. 242948 | 15
46 | 9. 695892 | 9. 938547 | 9. 757345 | 10. 242655 | 14
47 | 9. 696113 | 9. 938475 | 9. 757638 | 10. 242362 | 13
48 | 9. 696334 | 9 938402 | 9. 757931 | 10. 242069 | 12
49 | 9. 696554 | 9. 938330 | 9. 758224 | 10. 241776 | 11
50 | 9. 696774 | 9. 938257 | 9. 758517 | 10. 241483 | 10
51 | 9. 696995 | 9. 938185 | 9. 758810 | 10. 241190 | 9
52 | 9. 697215 | 9. 938112 | 9. 759102 | 10. 240898 | 8
53 | 9. 697435 | 9. 938040 | 9. 759395 | 10. 240605 | 7
54 | 9. 697654 | 9. 937967 | 9. 759687 | 10. 240313 | 6
55 | 9. 697874 | 9. 937895 | 9. 759979 | 10. 240021 | 5
56 | 9. 698093 | 9. 937822 | 9. 760271 | 10. 239728 | 4
57 | 9. 698313 | 9. 937749 | 9. 760564 | 10. 239436 | 3
58 | 9. 698532 | 9. 937676 | 9. 760856 | 10. 239144 | 2
59 | 9. 698751 | 9. 937603 | 9. 761147 | 10. 238852 | 1
60 | 9. 698970 | 9. 937531 | 9. 761439 | 10. 238561 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 60.

[Page]

Degree 30.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 698970 | 9. 937531 | 9. 761439 | 10. 238561 | 60
1 | 9. 699189 | 9. 937458 | 9. 761731 | 10. 238269 | 59
2 | 9. 699407 | 9. 937385 | 9. 762023 | 10. 237977 | 58
3 | 9. 699626 | 9. 937312 | 9. 762314 | 10. 237686 | 57
4 | 9. 699844 | 9. 937238 | 9. 762606 | 10. 237394 | 56
5 | 9. 700062 | 9. 937165 | 9 762897 | 10. 237103 | 55
6 | 7. 700280 | 9 937092 | 9. 763188 | 10. 236812 | 54
7 | 9. 700498 | 9. 937019 | 9. 763479 | 10. 236521 | 53
8 | 9. 700716 | 9. 936945 | 9. 763770 | 10. 236230 | 52
9 | 9. 700933 | 9. 936872 | 9. 764061 | 10. 235939 | 51
10 | 9. 701151 | 9. 936799 | 9. 764352 | 10. 235648 | 50
11 | 9. 701568 | 9. 936725 | 9. 764643 | 10. 235357 | 49
12 | 9. 701585 | 9. 936652 | 9. 764933 | 10. 235067 | 48
13 | 9. 701802 | 9. 936578 | 9. 765224 | 10. 234776 | 47
14 | 9. 702019 | 9. 936505 | 9. 765514 | 10. 234486 | 46
15 | 9. 702236 | 9. 936431 | 9. 765805 | 10. 234195 | 45
16 | 9. 702452 | 9. 936357 | 9. 766095 | 10. 233905 | 44
17 | 9. 702669 | 9. 936284 | 9. 766385 | 10. 233615 | 43
18 | 9. 702885 | 9. 936210 | 9. 766675 | 10. 233325 | 42
19 | 9. 703101 | 9. 936136 | 9. 766965 | 10. 233035 | 41
20 | 9. 703317 | 9. 936062 | 9. 767255 | 10. 232745 | 40
21 | 9. 703533 | 9. 935988 | 9. 767545 | 10. 232455 | 39
22 | 9. 703748 | 9. 935914 | 9. 767834 | 10. 232166 | 38
23 | 9. 703964 | 9. 935840 | 9. 768124 | 10. 231876 | 37
24 | 9. 704179 | 9. 935766 | 9. 768413 | 10. 231587 | 36
25 | 9. 704395 | 9. 935692 | 9. 768703 | 10. 231297 | 35
26 | 9. 704610 | 9. 935618 | 9. 768992 | 10. 231008 | 34
27 | 9. 704820 | 9. 935543 | 9. 769281 | 10. 230719 | 33
28 | 9. 705040 | 9. 935469 | 9. 769570 | 10. 230430 | 32
29 | 9. 705254 | 9. 935395 | 9. 769859 | 10. 230141 | 31
30 | 9. 705469 | 9 935320 | 9. 770148 | 10. 229852 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 59.
Degree 30.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 705469 | 9. 935320 | 9. 770148 | 10. 229852 | 30
31 | 9. 705683 | 9. 935246 | 9. 770437 | 10. 229563 | 29
32 | 9. 705897 | 9. 935171 | 9. 770726 | 10. 229274 | 28
33 | 9. 706112 | 9. 935097 | 9. 771015 | 10. 228985 | 27
34 | 9. 706327 | 9. 935022 | 9. 771303 | 10. 228696 | 26
35 | 9. 706539 | 9. 934948 | 9. 771592 | 10. 228408 | 25
36 | 9. 706753 | 9. 934873 | 9. 771880 | 10. 228120 | 24
37 | 9. 706967 | 9. 934798 | 9. 772168 | 10. 227832 | 23
38 | 9. 707180 | 9. 934723 | 9. 772456 | 10. 227543 | 22
39 | 9. 707393 | 9. 934649 | 9. 772745 | 10. 227255 | 21
40 | 9. 707606 | 9. 934574 | 9. 773033 | 10. 226967 | 20
41 | 9. 707819 | 9. 934499 | 9. 773321 | 10. 226679 | 19
42 | 9. 708032 | 9. 934424 | 9. 773608 | 10. 226391 | 18
43 | 9. 708245 | 9. 934349 | 9. 773896 | 10. 226104 | 17
44 | 9. 708457 | 9. 934274 | 9. 774184 | 10. 225816 | 16
45 | 9. 708670 | 9. 934199 | 9. 774471 | 10. 225529 | 15
46 | 9. 708882 | 9. 934123 | 9. 774759 | 10. 225241 | 14
47 | 9. 709094 | 9. 934048 | 9. 775046 | 10. 224954 | 13
48 | 9. 709306 | 9. 933973 | 9. 775333 | 10. 224666 | 12
49 | 9. 709518 | 9. 933897 | 9. 775621 | 10. 224379 | 11
50 | 9. 709730 | 9. 933822 | 9. 775908 | 10. 224092 | 10
51 | 9. 709941 | 9. 933747 | 9. 776195 | 10. 223805 | 9
52 | 9. 710153 | 9. 933671 | 9. 776482 | 10. 223518 | 8
53 | 9. 710364 | 9. 933596 | 9. 776768 | 10. 223232 | 7
54 | 9. 710575 | 9. 933520 | 9. 777055 | 10. 222945 | 6
55 | 9. 710786 | 9. 933444 | 9. 777342 | 10. 222658 | 5
56 | 9. 710997 | 9. 933369 | 9. 777628 | 10. 222372 | 4
57 | 9. 711208 | 9. 933293 | 9. 777915 | 10. 222085 | 3
58 | 9. 711418 | 9. 933217 | 9. 778201 | 10. 221799 | 2
59 | 9. 711629 | 9. 933141 | 9. 778487 | 10. 221513 | 1
60 | 9. 711839 | 9. 933066 | 9. 778774 | 10. 221226 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 59.

[Page]

Degree 31.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 711839 | 9. 933066 | 9. 778774 | 10. 221226 | 60
1 | 9. 712049 | 9. 932990 | 9. 779060 | 10. 220940 | 59
2 | 9. 712259 | 9. 932914 | 9. 779346 | 10. 220654 | 58
3 | 9. 712469 | 9. 932838 | 9. 779632 | 10. 220368 | 57
4 | 9. 712679 | 9. 932761 | 9. 779918 | 10. 220082 | 56
5 | 9. 712889 | 9. 932685 | 9. 780203 | 10. 219796 | 55
6 | 9. 713098 | 9. 932609 | 9. 780489 | 10. 219511 | 54
7 | 9. 713308 | 9. 932533 | 9. 780775 | 10. 219225 | 53
8 | 9. 713517 | 9. 932457 | 9. 781060 | 10. 218940 | 52
9 | 9. 713726 | 9. 932380 | 9. 781346 | 10. 218654 | 51
10 | 9. 713935 | 9. 932304 | 9. 781631 | 10. 218369 | 50
11 | 9. 714144 | 9. 932227 | 9. 781916 | 10. 218084 | 49
12 | 9. 714352 | 9. 932151 | 9. 782202 | 10. 217799 | 48
13 | 9. 714561 | 9. 932074 | 9. 782480 | 10. 217514 | 47
14 | 9. 714769 | 9. 931998 | 9. 782771 | 10. 217229 | 46
15 | 9. 714977 | 9. 931921 | 9. 783056 | 10. 216944 | 45
16 | 9. 715186 | 9. 931845 | 9. 783341 | 10. 216659 | 44
17 | 9. 715394 | 9. 931768 | 9. 783626 | 10. 216374 | 43
18 | 9. 715601 | 9. 931691 | 9. 783910 | 10. 216090 | 42
19 | 9. 715809 | 9. 931614 | 9. 784195 | 10. 215805 | 41
20 | 9. 716017 | 9. 931537 | 9. 784479 | 10. 215520 | 40
21 | 9. 716224 | 9. 931460 | 9. 784764 | 10. 215236 | 39
22 | 9. 716431 | 9. 931383 | 9. 785048 | 10. 214952 | 38
23 | 9. 716639 | 9. 931306 | 9. 785332 | 10. 214668 | 37
24 | 9. 716846 | 9. 931229 | 9. 785616 | 10. 214384 | 36
25 | 9. 717053 | 9. 931152 | 9. 785900 | 10. 214099 | 35
26 | 9. 717259 | 9. 931075 | 9. 786184 | 10. 213816 | 34
27 | 9. 717466 | 9. 930998 | 9. 786468 | 10. 213532 | 33
28 | 9. 717672 | 9. 930920 | 9. 786752 | 10. 213248 | 32
29 | 9. 717879 | 9. 930843 | 9. 787036 | 10. 212964 | 31
30 | 9. 718085 | 9. 930766 | 9. 787319 | 10. 212681 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 58.
Degree 31.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 718085 | 9. 930766 | 9. 787319 | 10. 212681 | 30
31 | 9. 718291 | 9. 930688 | 9. 787603 | 10. 212397 | 29
32 | 9. 718497 | 9. 930611 | 9. 787886 | 10. 212114 | 28
33 | 9. 718703 | 9. 930533 | 9. 788170 | 10. 211830 | 27
34 | 9. 718909 | 9. 930456 | 9. 788453 | 10. 211547 | 26
35 | 9. 719114 | 9. 930378 | 9. 788736 | 10. 211264 | 25
36 | 9. 719320 | 9. 930300 | 9. 789019 | 10. 210981 | 24
37 | 9. 719525 | 9. 930223 | 9. 789302 | 10. 210698 | 23
38 | 9. 719730 | 9. 930145 | 9. 789585 | 10. 210415 | 22
39 | 9. 719935 | 9. 930067 | 9. 789868 | 10. 210132 | 21
40 | 9. 720140 | 9. 929989 | 9. 790151 | 10. 209849 | 20
41 | 9. 720345 | 9. 929911 | 9. 790433 | 10. 209566 | 19
42 | 9. 720549 | 9. 929833 | 9. 790716 | 10. 209284 | 18
43 | 9. 720754 | 9. 929755 | 9. 790999 | 10. 209001 | 17
44 | 9. 720958 | 9. 929677 | 9. 790281 | 10. 208719 | 16
45 | 9. 721162 | 9. 929599 | 9. 791563 | 10. 208436 | 15
46 | 9. 721366 | 9. 929521 | 9. 791846 | 10. 208154 | 14
47 | 9. 721570 | 9. 929442 | 9. 792128 | 10. 207872 | 13
48 | 9. 721774 | 9. 929364 | 9. 792410 | 10. 207590 | 12
49 | 9. 721978 | 9. 929286 | 9. 792692 | 10. 207308 | 11
50 | 9. 722181 | 9. 929207 | 9. 792974 | 10. 207024 | 10
51 | 9. 722385 | 9. 929129 | 9. 793256 | 10. 206744 | 9
52 | 9. 722588 | 9. 929050 | 9. 793538 | 10. 206462 | 8
53 | 9. 722791 | 9. 928972 | 9. 793819 | 10. 206180 | 7
54 | 9. 722994 | 9. 928893 | 9. 794101 | 10. 205899 | 6
55 | 9. 723197 | 9. 928814 | 9. 794383 | 10. 205617 | 5
56 | 9. 723400 | 9. 928736 | 9. 794664 | 10. 205336 | 4
57 | 9. 723603 | 9. 928657 | 9. 794945 | 10. 205054 | 3
58 | 9. 723805 | 9. 928578 | 9. 795227 | 10. 204773 | 2
59 | 9. 724007 | 9. 928499 | 9. 795508 | 10. 204492 | 1
60 | 9. 724210 | 9. 928420 | 9. 795789 | 10. 204211 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 58.

[Page]

Degree 32.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 724210 | 9. 928420 | 9. 795789 | 10. 204211 | 60
1 | 9. 724412 | 9. 928341 | 9. 796070 | 10. 203930 | 59
2 | 9. 724614 | 9. 928262 | 9. 796351 | 10. 203649 | 58
3 | 9. 724816 | 9. 928183 | 9. 796632 | 10. 203368 | 57
4 | 9. 725017 | 9. 928104 | 9. 796913 | 10. 203087 | 56
5 | 9. 725219 | 9. 928025 | 9. 797194 | 10. 202806 | 55
6 | 9. 725420 | 9. 927946 | 9. 797474 | 10. 202522 | 54
7 | 9. 725622 | 9. 921867 | 9. 797755 | 10. 202245 | 53
8 | 9. 725823 | 9. 927787 | 9. 798036 | 10. 201964 | 52
9 | 9. 726024 | 9. 927708 | 9. 798316 | 10. 201684 | 51
10 | 9. 726225 | 9. 927628 | 9. 798596 | 10. 201404 | 50
11 | 9. 726426 | 9. 927549 | 9. 798877 | 10. 201123 | 49
12 | 9. 726626 | 9. 927469 | 9. 799157 | 10. 200843 | 48
13 | 9. 726827 | 9. 927390 | 9. 799437 | 10. 200563 | 47
14 | 9. 727027 | 9. 927310 | 9. 799717 | 10. 200283 | 46
15 | 9. 727228 | 9. 927231 | 9. 799997 | 10. 200003 | 45
16 | 9. 727428 | 9. 927151 | 9. 800277 | 10. 199723 | 44
17 | 9. 727628 | 9. 927071 | 9. 800557 | 10. 199443 | 43
18 | 9. 727828 | 9. 926991 | 9. 800836 | 10. 199163 | 42
19 | 9. 728027 | 9. 926911 | 9. 801116 | 10. 198884 | 41
20 | 9. 728227 | 9. 926831 | 9. 801396 | 10. 198604 | 40
21 | 9. 728427 | 9. 926751 | 9. 801675 | 10. 198325 | 39
22 | 9. 728626 | 9. 926641 | 9. 801955 | 10. 198045 | 38
23 | 9. 728825 | 9. 926591 | 9. 802234 | 10. 197766 | 37
24 | 9. 729024 | 9. 926511 | 9. 802513 | 10. 197487 | 36
25 | 9. 729223 | 9. 926431 | 9. 802792 | 10. 197207 | 35
26 | 9. 729422 | 9. 926351 | 9. 803072 | 10. 196928 | 34
27 | 9. 729621 | 9. 926270 | 9. 803351 | 10. 196649 | 33
28 | 9. 729820 | 9. 926190 | 9. 803630 | 10. 196370 | 32
29 | 9. 730018 | 9. 926110 | 9. 803908 | 10. 196091 | 31
30 | 9. 730216 | 9. 926029 | 9. 804187 | 10. 195813 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 57.
Degree 32.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 730216 | 9. 926029 | 9. 804187 | 10. 195813 | 30
31 | 9. 730415 | 9. 925949 | 9. 804466 | 10. 195534 | 29
32 | 9. 730613 | 9. 925868 | 9. 804745 | 10. 195255 | 28
33 | 9. 730811 | 9. 925787 | 9. 805023 | 10. 194977 | 27
34 | 9. 731009 | 9. 925707 | 9. 805302 | 10. 194698 | 26
35 | 9. 731206 | 9. 9. 25626 | 9. 805580 | 10. 194420 | 25
36 | 9. 731404 | 9. 925545 | 9. 805859 | 10. 194141 | 24
37 | 9. 731601 | 9. 925464 | 9. 806137 | 10. 193863 | 23
38 | 9. 731799 | 9. 925384 | 9. 806415 | 10. 193585 | 22
39 | 9. 731996 | 9. 925303 | 9. 806693 | 10. 193309 | 21
40 | 9. 732193 | 9. 925222 | 9. 806971 | 10. 193028 | 20
41 | 9. 732390 | 9. 925141 | 9. 807249 | 10. 192751 | 19
42 | 9. 732587 | 9. 925060 | 9. 807527 | 10. 192433 | 18
43 | 9. 732784 | 9. 924978 | 9. 807805 | 10. 192195 | 17
44 | 9. 732980 | 9. 924897 | 9. 808083 | 10. 191917 | 16
45 | 9. 733177 | 9. 924816 | 9. 808361 | 10. 191639 | 15
46 | 9. 733373 | 9. 924735 | 9. 808638 | 10. 191362 | 14
47 | 9. 733569 | 9. 924653 | 9. 808916 | 10. 191084 | 13
48 | 9. 733765 | 9. 924572 | 9. 809193 | 10. 190807 | 12
49 | 9. 733961 | 9. 924491 | 9. 809471 | 10. 190529 | 11
50 | 9. 734157 | 9. 924409 | 9. 809748 | 10. 190252 | 10
51 | 9. 734353 | 9. 924328 | 9. 810025 | 10. 189975 | 9
52 | 9. 734548 | 9. 924246 | 9. 810302 | 10. 189697 | 8
53 | 9. 734744 | 9. 924164 | 9. 810580 | 10. 189420 | 7
54 | 9. 734939 | 9. 924083 | 9. 810857 | 10. 189143 | 6
55 | 9. 735134 | 9. 924001 | 9. 811134 | 10. 188866 | 5
56 | 9. 735330 | 9. 923919 | 9. 811410 | 10. 188589 | 4
57 | 9. 735525 | 9. 923837 | 9. 811687 | 10. 188313 | 3
58 | 9. 735719 | 9. 923755 | 9. 811964 | 10. 188036 | 2
59 | 9. 735914 | 9. 923673 | 9. 812241 | 10. 187759 | 1
60 | 9. 736109 | 9. 923591 | 9. 812517 | 10. 187483 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 57.

[Page]

Degree 33.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 736109 | 9. 923591 | 9. 812517 | 10. 187483 | 60
1 | 9. 736309 | 9. 923509 | 9. 812794 | 10. 187206 | 59
2 | 9. 736497 | 9. 923427 | 9. 813070 | 10. 186930 | 58
3 | 9. 736692 | 9. 923345 | 9. 813347 | 10. 186653 | 57
4 | 9. 736886 | 9. 923263 | 9. 813623 | 10. 186377 | 56
5 | 9. 737080 | 9. 923180 | 9. 813899 | 10. 186101 | 55
6 | 9. 737274 | 9. 923098 | 9. 814175 | 10. 185824 | 54
7 | 9. 737467 | 9. 923016 | 9. 814452 | 10. 185548 | 53
8 | 9. 737661 | 9. 922933 | 9. 814728 | 10. 185272 | 52
9 | 9. 737854 | 9. 922851 | 9. 815004 | 10. 184996 | 51
10 | 9. 738048 | 9. 922768 | 9 815279 | 10. 184720 | 50
11 | 9. 738241 | 9. 922686 | 9. 815555 | 10. 184445 | 49
12 | 9. 738434 | 9. 922603 | 9. 815831 | 10. 184169 | 48
13 | 9. 738627 | 9. 922520 | 9. 816107 | 10. 183893 | 47
14 | 9. 738820 | 9. 922438 | 9. 816382 | 10. 183617 | 46
15 | 9. 739013 | 9. 922355 | 9. 816658 | 10. 183342 | 45
16 | 9. 739205 | 9. 922272 | 9. 816933 | 10. 183066 | 44
17 | 9. 739398 | 9. 922189 | 9. 817209 | 10. 182791 | 43
18 | 9. 739590 | 9. 922106 | 9. 817484 | 10. 182516 | 42
19 | 9. 739783 | 9. 922023 | 9. 817759 | 10. 182240 | 41
20 | 9. 739975 | 9. 921940 | 9. 818035 | 10. 181965 | 40
21 | 9. 740167 | 9. 921857 | 9 818310 | 10. 181690 | 39
22 | 9. 740359 | 9. 921774 | 9. 818585 | 10. 181415 | 38
23 | 9. 740550 | 9. 921691 | 9. 818860 | 10. 181140 | 37
24 | 9. 740742 | 9. 921607 | 9. 819135 | 10. 180865 | 36
25 | 9. 740934 | 9. 921524 | 9. 819410 | 10. 180590 | 35
26 | 9. 741125 | 9. 921441 | 9. 819684 | 10. 180315 | 34
27 | 9. 741316 | 9. 921357 | 9. 819959 | 10. 180041 | 33
28 | 9. 741507 | 9. 921274 | 9. 820234 | 10. 179766 | 32
29 | 9. 741698 | 9. 921190 | 9. 820508 | 10. 179492 | 31
30 | 9. 741889 | 9. 921107 | 9. 820783 | 10. 179217 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 56.
Degree 33.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 741889 | 9. 921107 | 9. 820783 | 10. 179217 | 30
31 | 9. 742080 | 9. 921023 | 9. 821057 | 10. 178943 | 29
32 | 9. 742271 | 9. 920939 | 9. 821332 | 10. 178668 | 28
33 | 9. 742461 | 9. 920855 | 9. 821606 | 10. 178394 | 27
34 | 9. 742652 | 9. 920772 | 9. 821880 | 10. 178120 | 26
35 | 9. 742842 | 9. 920688 | 9. 822154 | 10. 177846 | 25
36 | 9. 743032 | 9. 920604 | 9. 822429 | 10. 177571 | 24
37 | 9. 743223 | 9. 920520 | 9. 822703 | 10. 177297 | 23
38 | 9. 743412 | 9. 920436 | 9. 822977 | 10. 177023 | 22
39 | 9. 743602 | 9. 920352 | 9. 823250 | 10. 176739 | 21
40 | 9. 743792 | 9. 920268 | 9. 823524 | 10. 176476 | 20
41 | 9. 743982 | 9. 920184 | 9. 823798 | 10. 176202 | 19
42 | 9. 744171 | 9. 920099 | 9. 824072 | 10. 175928 | 18
43 | 9. 744361 | 9. 920015 | 9. 824345 | 10. 175655 | 17
44 | 9. 744550 | 9. 919931 | 9. 824619 | 10. 175381 | 16
45 | 9. 744739 | 9. 919846 | 9. 824892 | 10. 175108 | 15
46 | 9. 744928 | 9. 919762 | 9. 825166 | 10. 174834 | 14
47 | 9. 745117 | 9. 919677 | 9. 825439 | 10. 174560 | 13
48 | 9. 745306 | 9. 919593 | 9. 825713 | 10. 174287 | 12
49 | 9. 745494 | 9. 919508 | 9. 825986 | 10. 174014 | 11
50 | 9. 745683 | 9. 919424 | 9. 826259 | 10. 173741 | 10
51 | 9. 745871 | 9. 919339 | 9. 826532 | 10. 173468 | 9
52 | 9. 746059 | 9. 919254 | 9. 826805 | 10. 173195 | 8
53 | 9. 746248 | 9. 919169 | 9. 827078 | 10. 172922 | 7
54 | 9. 746436 | 9. 919084 | 9. 827351 | 10. 172649 | 6
55 | 9. 746624 | 9. 918999 | 9. 827624 | 10. 172376 | 5
56 | 9. 746811 | 9. 918915 | 9. 827897 | 10. 172103 | 4
57 | 9. 746999 | 9. 918830 | 9. 828170 | 10. 171830 | 3
58 | 9. 747187 | 9. 918744 | 9. 828442 | 10. 171558 | 2
59 | 9. 747374 | 9. 918659 | 9. 828715 | 10. 171285 | 1
60 | 9. 747562 | 9. 918574 | 9. 828987 | 10. 171012 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 56.

[Page]

Degree 34
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 747562 | 9. 918574 | 9. 828987 | 10. 171012 | 60
1 | 9. 747749 | 9. 918489 | 9. 829260 | 10. 170740 | 59
2 | 9. 747936 | 9. 918404 | 9. 829532 | 10. 170468 | 58
3 | 9. 748123 | 9. 918318 | 9. 829805 | 10. 170195 | 57
4 | 9. 748310 | 9. 918233 | 9. 830077 | 10. 169923 | 56
5 | 9. 748497 | 9. 918147 | 9. 830349 | 10. 169651 | 55
6 | 9. 748683 | 9. 918062 | 9. 830621 | 10. 169379 | 54
7 | 9. 748870 | 9. 917976 | 9. 830891 | 10. 166106 | 53
8 | 9. 749056 | 9. 917891 | 9. 831165 | 10. 168834 | 52
9 | 9. 749242 | 9. 917805 | 9. 831437 | 10. 168563 | 51
10 | 9. 749429 | 9. 917719 | 9. 831709 | 10. 168291 | 50
11 | 9. 749615 | 9. 917634 | 9. 831981 | 10. 168019 | 49
12 | 9. 749801 | 9. 917548 | 9. 832253 | 10. 167747 | 48
13 | 9. 749986 | 9. 917462 | 9. 832525 | 10. 167475 | 47
14 | 9. 750172 | 9. 917376 | 9. 832796 | 10. 167204 | 46
15 | 9. 750358 | 9. 917290 | 9. 833068 | 10. 166932 | 45
16 | 9. 750543 | 9. 917204 | 9. 833339 | 10. 166660 | 44
17 | 9. 750729 | 9. 917118 | 9. 833621 | 10. 166389 | 43
18 | 9. 750914 | 9. 917032 | 9. 833882 | 10. 166118 | 42
19 | 9. 751099 | 9. 916945 | 9. 834154 | 10. 165846 | 41
20 | 9. 751284 | 9. 916819 | 9. 834425 | 10. 165575 | 40
21 | 9. 751469 | 9. 916773 | 9. 834696 | 10. 165304 | 39
22 | 9. 751654 | 9. 916686 | 9. 834967 | 10. 165033 | 38
23 | 9. 751838 | 9. 916600 | 9. 835238 | 10. 164762 | 37
24 | 9. 752023 | 9. 916514 | 9. 835509 | 10. 164491 | 36
25 | 9. 752207 | 9. 916427 | 9. 835780 | 10. 164220 | 35
26 | 9. 752392 | 9. 916340 | 9. 836051 | 10. 163949 | 34
27 | 9. 752576 | 9. 916254 | 9. 836322 | 10. 163678 | 33
28 | 9. 752760 | 9. 916167 | 9. 836593 | 10. 163407 | 32
29 | 9. 752944 | 9. 916080 | 9. 836864 | 10. 163136 | 31
30 | 9. 753128 | 9. 915994 | 9. 837134 | 10. 162866 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 55.
Degree 34.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 753128 | 9. 915994 | 9. 837134 | 10. 162866 | 30
31 | 9. 753312 | 9. 915907 | 9. 837405 | 10. 162595 | 29
32 | 9. 753495 | 9. 915820 | 9. 837675 | 10. 162325 | 28
33 | 9. 753679 | 9. 915733 | 9. 837946 | 10. 162054 | 27
34 | 9. 753862 | 9. 915646 | 9. 838216 | 10. 161784 | 26
35 | 9. 754046 | 9. 915559 | 9. 838487 | 10. 161513 | 25
36 | 9. 754229 | 9. 915472 | 9. 838757 | 10. 161243 | 24
37 | 9. 754412 | 9. 915385 | 9. 839027 | 10. 160973 | 23
38 | 9. 754595 | 9. 915297 | 9. 839297 | 10. 160702 | 22
39 | 9. 754778 | 9. 915210 | 9. 839568 | 10. 160432 | 21
40 | 9. 754960 | 9. 915123 | 9. 839838 | 10. 160162 | 20
41 | 9. 755143 | 9. 915035 | 9. 840108 | 10. 159892 | 19
42 | 9. 755325 | 9. 914948 | 9. 840378 | 10. 159622 | 18
43 | 9. 755508 | 9. 914860 | 9. 840647 | 10. 159352 | 17
44 | 9. 755690 | 9. 914773 | 9. 840917 | 10. 159083 | 16
45 | 9. 755872 | 9. 914685 | 9. 841187 | 10. 158813 | 15
46 | 9. 756054 | 9. 914597 | 9. 841457 | 10. 158543 | 14
47 | 9. 756236 | 9. 914510 | 9. 841726 | 10. 158273 | 13
48 | 9. 756418 | 9. 914422 | 9. 841996 | 10. 158004 | 12
49 | 9. 756600 | 9. 914334 | 9. 842206 | 10. 157734 | 11
50 | 9. 756781 | 9. 914146 | 9. 842505 | 10. 157465 | 10
51 | 9. 756963 | 9. 914158 | 9. 842804 | 10. 157195 | 9
52 | 9. 757144 | 9. 914070 | 9. 843074 | 10. 156926 | 8
53 | 9. 757316 | 9. 913982 | 9. 843343 | 10. 156657 | 7
54 | 9. 757507 | 9. 913894 | 9. 843612 | 10. 156387 | 6
55 | 9. 757688 | 9. 913806 | 9. 843882 | 10. 156118 | 5
56 | 9. 757869 | 9. 913718 | 9. 844151 | 10. 155849 | 4
57 | 9. 758049 | 9. 913630 | 9. 844420 | 10. 155580 | 3
58 | 9. 758230 | 9. 913541 | 9. 844689 | 10. 155311 | 2
59 | 9. 758411 | 9. 913453 | 9. 844958 | 10. 155041 | 1
60 | 9. 758591 | 9. 913361 | 9. 845227 | 10. 154773 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 55.

[Page]

Degree 35.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 758591 | 9. 913364 | 9. 845227 | 10. 154774 | 60
1 | 9. 758772 | 9. 913276 | 9. 845496 | 10. 154504 | 59
2 | 9. 758952 | 9. 913187 | 9. 845764 | 10. 154235 | 58
3 | 9. 759132 | 9. 913099 | 9. 846033 | 10. 153967 | 57
4 | 9. 759312 | 9. 913010 | 9. 846302 | 10. 153698 | 56
5 | 9. 759492 | 9. 912921 | 9. 846570 | 10. 153429 | 55
6 | 9. 759672 | 9. 912833 | 9. 846839 | 10. 153161 | 54
7 | 9. 759851 | 9. 912744 | 9. 847107 | 10. 152892 | 53
8 | 9. 760031 | 9. 912655 | 9. 847376 | 10. 152624 | 52
9 | 9. 760210 | 9. 912566 | 9. 847644 | 10. 152356 | 51
10 | 9. 760390 | 9. 912477 | 9. 847913 | 10. 152087 | 50
11 | 9. 760569 | 9. 912388 | 9. 848181 | 10. 151819 | 49
12 | 9. 760748 | 9. 912299 | 9. 848449 | 10. 151551 | 48
13 | 9. 760927 | 9. 912210 | 9. 848717 | 10. 151283 | 47
14 | 9. 761106 | 9. 912121 | 9. 848985 | 10. 151015 | 46
15 | 9. 761285 | 9. 912031 | 9. 849254 | 10. 150746 | 45
16 | 9. 761464 | 9. 911942 | 9. 849522 | 10. 150478 | 44
17 | 9. 761642 | 9. 911853 | 9. 849789 | 10. 150214 | 43
18 | 9. 761821 | 9. 911763 | 9. 850057 | 10. 149943 | 42
19 | 9. 761999 | 9. 911674 | 9. 850325 | 10. 149675 | 41
20 | 9. 762177 | 9. 911584 | 9. 850593 | 10. 149407 | 40
21 | 9. 762356 | 9. 911495 | 9. 850861 | 10. 149139 | 39
22 | 9. 762534 | 9. 911405 | 9. 851128 | 10. 148872 | 38
23 | 9. 762712 | 9. 911315 | 9. 851396 | 10. 148604 | 37
24 | 9. 762889 | 9. 911226 | 9. 851664 | 10. 148336 | 36
25 | 9. 763067 | 9. 911136 | 9. 851931 | 10. 148069 | 35
26 | 9. 763245 | 9. 911046 | 9. 852199 | 10. 147801 | 34
27 | 9. 763422 | 9. 910956 | 9. 852466 | 10. 147534 | 33
28 | 9. 763599 | 9. 910866 | 9. 852731 | 10. 147267 | 32
29 | 9. 763777 | 9. 910776 | 9. 853001 | 10. 146999 | 31
30 | 9. 763954 | 9. 910686 | 9. 853268 | 10. 146732 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 54
Degree 35.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 763954 | 9. 910686 | 9. 853208 | 10. 146732 | 30
31 | 9. 764131 | 9. 910596 | 9. 853532 | 10. 146465 | 29
32 | 9. 764308 | 9. 910506 | 9. 853802 | 10. 146198 | 28
33 | 9. 764485 | 9. 910415 | 9. 854069 | 10. 145930 | 27
34 | 9. 764662 | 9. 910325 | 9. 854336 | 10. 145664 | 26
35 | 9. 764838 | 9. 910235 | 9. 854603 | 10. 145397 | 25
36 | 9. 765015 | 9. 910144 | 9. 854870 | 10. 145130 | 24
37 | 9. 765191 | 9. 910054 | 9. 855137 | 10. 144863 | 23
38 | 9. 765367 | 9. 909963 | 9. 855404 | 10. 144596 | 22
39 | 9. 765544 | 9. 909873 | 9. 855671 | 10. 144329 | 21
40 | 9. 765720 | 9. 909782 | 9. 855937 | 10. 144063 | 20
41 | 9. 765896 | 9. 909691 | 9. 856204 | 10. 143796 | 19
42 | 9. 766071 | 9. 909601 | 9. 856471 | 10. 143529 | 18
43 | 9. 766247 | 9. 909510 | 9. 856737 | 10. 143263 | 17
44 | 9. 766423 | 9. 909419 | 9. 857004 | 10. 142996 | 16
45 | 9. 766598 | 9. 909328 | 9. 857270 | 10. 142730 | 15
46 | 9. 766774 | 9. 909237 | 9. 857537 | 10. 142463 | 14
47 | 9. 766949 | 9. 909146 | 9. 857803 | 10. 142197 | 13
48 | 9. 767124 | 9. 909055 | 9. 858069 | 10. 141931 | 12
49 | 9. 767299 | 9. 908964 | 9. 858336 | 10. 141664 | 11
50 | 9. 767474 | 9. 908873 | 9. 858602 | 10. 141398 | 10
51 | 9. 767649 | 9. 908781 | 9. 858868 | 10. 141132 | 9
52 | 9. 767824 | 9. 908690 | 9. 859134 | 10. 140866 | 8
53 | 9. 767997 | 9. 908599 | 9859400 | 10. 140600 | 7
54 | 9. 768173 | 9. 908507 | 9. 859666 | 10. 140334 | 6
55 | 9. 768348 | 9. 908416 | 9. 859932 | 10. 140068 | 5
56 | 9. 768522 | 9. 908324 | 9. 860198 | 10. 139802 | 4
57 | 9. 768696 | 9. 908233 | 9. 860464 | 10. 139536 | 3
58 | 9. 768871 | 9. 908141 | 9860730 | 10. 139270 | 2
59 | 9. 769045 | 9. 908049 | 9. 860995 | 10. 139005 | 1
60 | 9. 769219 | 9. 907958 | 9. 861261 | 10. 138739 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 54.

[Page]

Degree 36.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 769219 | 9. 907958 | 9. 861261 | 10. 138739 | 60
1 | 9. 769392 | 9. 907866 | 9. 861527 | 10. 138473 | 59
2 | 9. 769566 | 9. 907774 | 9. 861792 | 10. 138208 | 58
3 | 9. 769740 | 9. 907682 | 9. 862058 | 10. 137942 | 57
4 | 9. 769913 | 9. 907590 | 9. 862323 | 10. 137677 | 56
5 | 9. 770087 | 9. 907498 | 9. 862589 | 10. 137411 | 55
6 | 9. 770260 | 9. 907406 | 9. 862854 | 10. 137146 | 54
7 | 9. 770433 | 9. 907314 | 9. 863119 | 10. 136880 | 53
8 | 9. 770606 | 9. 907221 | 9. 863385 | 10. 136615 | 52
9 | 9. 770779 | 9. 907129 | 9. 863650 | 10. 136350 | 51
10 | 9. 770952 | 9. 907037 | 9. 863915 | 10. 136085 | 50
11 | 9. 771125 | 9. 906945 | 9. 864180 | 10. 135820 | 49
12 | 9. 771298 | 9. 906852 | 9. 864445 | 10. 135554 | 48
13 | 9. 771470 | 9. 906760 | 9. 864710 | 10. 135289 | 47
14 | 9. 771643 | 9. 906667 | 9. 864975 | 10. 135024 | 46
15 | 9. 771815 | 9. 906574 | 9. 865240 | 10. 134759 | 45
16 | 9. 771987 | 9. 906482 | 9. 865505 | 10. 134495 | 44
17 | 9. 772159 | 9. 906389 | 9. 865770 | 10. 134230 | 43
18 | 9. 772331 | 9. 906296 | 9. 866035 | 10. 133965 | 42
19 | 9. 772503 | 9. 906203 | 9. 866300 | 10. 133700 | 41
20 | 9. 772675 | 9. 906111 | 9. 866564 | 10. 133436 | 40
21 | 9. 772847 | 9. 906018 | 9. 866829 | 10. 133171 | 39
22 | 9. 773018 | 9. 905925 | 9. 867094 | 10. 132906 | 38
23 | 9. 773190 | 9. 905832 | 9. 867358 | 10. 132642 | 37
24 | 9. 773361 | 9. 905738 | 9. 867623 | 10. 132377 | 36
25 | 9. 773533 | 9. 905645 | 9. 867887 | 10. 132113 | 35
26 | 9. 773704 | 9. 905552 | 9. 868152 | 10. 131848 | 34
27 | 9. 773875 | 9. 905459 | 9. 868416 | 10. 131584 | 33
28 | 9. 774046 | 9. 905365 | 9. 868680 | 10. 131320 | 32
29 | 9. 774217 | 9. 905272 | 9. 868945 | 10. 131055 | 31
30 | 9. 774388 | 9. 905179 | 9. 869209 | 10. 130791 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 53.
Degree 36.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 774388 | 9. 905179 | 9. 869209 | 10. 130791 | 30
31 | 9. 774558 | 9. 905085 | 9. 864773 | 10. 130527 | 29
32 | 9. 774729 | 9. 904992 | 9. 867337 | 10. 130263 | 28
33 | 9. 774899 | 9. 904898 | 9. 870001 | 10. 129999 | 27
34 | 9. 775070 | 9. 904804 | 9. 870265 | 10. 129735 | 26
35 | 9. 775240 | 9. 904711 | 9. 870529 | 10. 129471 | 25
36 | 9. 775410 | 9. 904617 | 9. 870793 | 10. 129207 | 24
37 | 9. 775580 | 9. 904523 | 9. 871057 | 10. 128943 | 23
38 | 9. 775750 | 9. 904429 | 9. 871321 | 10. 128679 | 22
39 | 9. 775920 | 9. 904335 | 9. 871585 | 10. 128415 | 21
40 | 9. 776090 | 9. 904241 | 9. 871849 | 10. 128151 | 20
41 | 9. 776259 | 9. 904147 | 9. 872112 | 10. 127888 | 19
42 | 9. 776429 | 9. 904053 | 9. 872376 | 10. 127624 | 18
43 | 9. 776598 | 9. 903959 | 9. 872640 | 10. 127360 | 17
44 | 9. 776768 | 9. 903864 | 9. 872903 | 10. 127097 | 16
45 | 9. 776937 | 9. 903770 | 9. 873167 | 10. 126833 | 15
46 | 9. 777106 | 9. 903676 | 9. 873430 | 10. 126570 | 14
47 | 9. 777275 | 9. 903581 | 9. 873694 | 10. 126306 | 13
48 | 9. 777444 | 9. 903486 | 9. 873957 | 10. 126043 | 12
49 | 9. 777613 | 9. 903392 | 9. 874220 | 10. 125780 | 11
50 | 9. 777781 | 9. 903298 | 9. 874484 | 10. 125516 | 10
51 | 9. 777950 | 9. 903203 | 9. 874747 | 10. 125253 | 9
52 | 9. 778119 | 9. 903108 | 9. 875010 | 10. 124990 | 8
53 | 9. 778287 | 9. 903013 | 9. 875273 | 10. 124727 | 7
54 | 9. 778455 | 9. 902919 | 9. 875536 | 10. 124464 | 6
55 | 9. 778623 | 9. 902824 | 9. 875799 | 10. 124201 | 5
56 | 9. 778792 | 9. 902729 | 9. 876063 | 10. 123937 | 4
57 | 9. 778960 | 9. 902634 | 9. 876326 | 10. 123674 | 3
58 | 9. 779129 | 9. 902539 | 9. 876589 | 10. 123411 | 2
59 | 9. 779295 | 9. 902444 | 9. 876851 | 10. 123149 | 1
60 | 9. 779463 | 9. 902349 | 9. 877114 | 10. 122886 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 53.

[Page]

Degree 37.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9 779463 | 9. 902349 | 9. 877114 | 10. 122885 | 60
1 | 9. 779631 | 9. 902253 | 9. 877377 | 10. 122623 | 59
2 | 9. 779798 | 9. 902158 | 9. 877640 | 10. 122360 | 58
3 | 9. 779965 | 9. 902063 | 9. 877903 | 10. 122097 | 57
4 | 9. 780133 | 9. 901967 | 9. 878165 | 10. 121834 | 56
5 | 9. 780300 | 9. 901872 | 9. 878428 | 10. 121572 | 55
6 | 9. 780467 | 9. 901776 | 9. 878691 | 10. 121309 | 54
7 | 9. 780634 | 9. 901681 | 9. 878953 | 10. 121047 | 53
8 | 9. 780801 | 9. 901585 | 9. 879216 | 10. 120784 | 52
9 | 9. 780968 | 9. 901488 | 9. 879478 | 10. 120522 | 51
10 | 9. 781134 | 9. 901391 | 9. 879741 | 10. 120259 | 50
11 | 9. 781301 | 9. 901298 | 9. 880003 | 10. 119997 | 49
12 | 9. 781467 | 9. 901202 | 9. 880265 | 10. 119734 | 48
13 | 9. 781634 | 9. 901106 | 9. 880528 | 10. 119472 | 47
14 | 9. 781800 | 9. 901010 | 9. 880790 | 10. 119210 | 46
15 | 9. 781966 | 9. 900914 | 9. 881052 | 10. 118948 | 45
16 | 9. 782132 | 9. 900828 | 9. 881314 | 10. 118686 | 44
17 | 9. 782298 | 9. 900722 | 9. 881576 | 10. 118424 | 43
18 | 9. 782464 | 9. 900626 | 9. 881839 | 10. 118161 | 42
19 | 9. 782690 | 9. 900529 | 9. 882101 | 10. 117899 | 41
20 | 9. 782796 | 9. 900433 | 9. 882363 | 10. 117637 | 40
21 | 9. 782961 | 9. 900337 | 9. 882625 | 10. 117375 | 39
22 | 9. 783127 | 9. 900240 | 9. 882886 | 10. 117114 | 38
23 | 9. 783292 | 9. 900144 | 9. 883148 | 10. 116852 | 37
24 | 9. 783457 | 9. 900047 | 9. 883410 | 10. 116590 | 36
25 | 9. 783623 | 9. 899951 | 9. 883672 | 10. 116328 | 35
26 | 9. 783788 | 9. 899854 | 9. 883934 | 10. 116066 | 34
27 | 9. 783953 | 9. 899757 | 9. 884195 | 10. 115805 | 33
28 | 9. 784118 | 9. 899660 | 9. 884457 | 10. 115543 | 32
29 | 9. 784282 | 9. 899563 | 9. 884719 | 10. 115281 | 31
30 | 9. 784447 | 9. 899467 | 9. 884980 | 10. 115020 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 52.
Degree 37.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 784447 | 9. 899467 | 9. 884980 | 10. 115025 | 30
31 | 9. 784616 | 9. 899370 | 9. 885242 | 10. 114758 | 29
32 | 9. 784776 | 9. 899273 | 9. 885503 | 10. 114497 | 28
33 | 9. 784941 | 9. 899175 | 9. 885765 | 10. 114235 | 27
34 | 9. 785105 | 9. 899078 | 9. 886026 | 10. 113974 | 26
35 | 9. 785269 | 9. 898981 | 9. 886288 | 10. 113712 | 25
36 | 9. 785433 | 9. 898884 | 9. 886549 | 10. 113451 | 24
37 | 9. 785591 | 9. 898787 | 9. 886810 | 10. 113190 | 23
38 | 9. 785761 | 9. 898689 | 9. 887072 | 10. 112928 | 22
39 | 9. 785925 | 9. 898592 | 9. 887333 | 10. 112667 | 21
40 | 9. 786088 | 9. 898494 | 9. 887594 | 10. 112406 | 20
41 | 9. 786252 | 9. 898397 | 9. 887855 | 10. 112145 | 19
42 | 9. 786416 | 9. 898299 | 9. 888116 | 10. 111884 | 18
43 | 9. 786579 | 9. 898201 | 9. 888377 | 10. 111623 | 17
44 | 9. 786742 | 9. 898104 | 9. 888638 | 10. 111362 | 16
45 | 9. 786909 | 9. 898006 | 9. 888899 | 10. 111101 | 15
46 | 9. 787069 | 9. 897908 | 9. 889160 | 10. 110840 | 14
47 | 9. 787232 | 9. 897810 | 9. 889421 | 10. 110579 | 13
48 | 9. 787395 | 9. 897112 | 9. 889682 | 10. 110318 | 12
49 | 9. 787557 | 9. 897614 | 9. 889943 | 10. 110057 | 11
50 | 9. 787720 | 9. 897516 | 9. 890204 | 10. 109796 | 10
51 | 9. 787883 | 9. 897418 | 9. 890465 | 10. 109535 | 9
52 | 9. 788045 | 9. 897320 | 9. 890725 | 10. 109275 | 8
53 | 9. 788208 | 9. 897222 | 9. 890986 | 10. 109014 | 7
54 | 9. 788370 | 9. 897123 | 9. 891248 | 10. 108753 | 6
55 | 9. 788532 | 9. 897025 | 9. 891507 | 10. 108493 | 5
56 | 9. 788694 | 9. 896926 | 9. 891768 | 10. 108232 | 4
57 | 9. 788856 | 9. 896828 | 9. 892028 | 10. 107972 | 3
58 | 9. 789018 | 9. 896729 | 9. 892289 | 10. 107711 | 2
59 | 9. 789180 | 9. 896631 | 9. 892549 | 10. 107451 | 1
60 | 9. 789342 | 9. 896532 | 9. 892810 | 10. 107190 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 52.

[Page]

Degree 38.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 789342 | 9. 896532 | 9. 892810 | 10. 107190 | 60
1 | 9. 789504 | 9. 896433 | 9. 893070 | 10. 106930 | 59
2 | 9. 789665 | 9. 896335 | [...] | 10. 106669 | 58
3 | 9. 789827 | 9. 896236 | 9. 893591 | 10. 106409 | 57
4 | 9. 789988 | 9. 896137 | 9. 893851 | 10. 106149 | 56
5 | 9. 790149 | 9. 896038 | 9. 894111 | 10. 105889 | 55
6 | 9. 790310 | 9. 895939 | 9. 894371 | 10. 105628 | 54
7 | 9. 790471 | 9. 895840 | 9. 894632 | 10. 105368 | 53
8 | 9. 790632 | 9. 895741 | 9. 894892 | 10. 105108 | 52
9 | 9. 790793 | 9. 895641 | 9. 895152 | 10. 104844 | 51
10 | 9. 790954 | 9. 895542 | 9. 895412 | 10. 104588 | 50
11 | 9. 791115 | 9. 895443 | 9. 895672 | 10. 104328 | 49
12 | 9. 791275 | 9. 895343 | 9. 895932 | 10. 104068 | 48
13 | 9. 791436 | 9. 895244 | 9. 896192 | 10. 103808 | 47
14 | 9. 791596 | 9. 895144 | 9. 896452 | 10. 103548 | 46
15 | 9. 791756 | 9. 895045 | 9. 896712 | 10. 103288 | 45
16 | 9. 791917 | 9. 894945 | 9. 896971 | 10. 103028 | 44
17 | 9. 792077 | 9. 894846 | 9. 897231 | 10. 102769 | 43
18 | 9. 792237 | 9. 894746 | 9. 897491 | 10. 102509 | 42
19 | 9. 792397 | 9. 894646 | 9. 897751 | 10. 102249 | 41
20 | 9. 792557 | 9. 894546 | 9. 898010 | 10. 101990 | 40
21 | 9. 792716 | 9. 894446 | 9. 898270 | 10. 101730 | 39
22 | 9. 792876 | 9. 894346 | 9. 898530 | 10. 101470 | 38
23 | 9. 793035 | 9. 894246 | 9. 898789 | 10. 101211 | 37
24 | 9. 793195 | 9. 894146 | 9. 899049 | 10. 100951 | 36
25 | 9. 793354 | 9. 894046 | 9 899308 | 10. 100692 | 35
26 | 9. 793513 | 9. 893946 | 9. 899568 | 10. 100432 | 34
27 | 9. 793673 | 9. 893845 | 9. 899827 | 10. 100173 | 33
28 | 9. 793832 | 9. 893745 | 9. 900086 | 10. 099913 | 32
29 | 9. 793991 | 9. 893645 | 9. 900346 | 10. 099654 | 31
30 | 9. 794149 | 9. 893544 | 9. 900605 | 10. 099395 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 51.
Degree 38.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 794149 | 9. 893544 | 9. 900605 | 10. 099395 | 30
31 | 9. 794308 | 9. 893444 | 9. 900864 | 10. 099135 | 29
32 | 9. 794467 | 9. 893343 | 9. 901124 | 10. 098876 | 28
33 | 9. 794626 | 9. 893243 | 9. 901383 | 10. 098617 | 27
34 | 9. 794784 | 9. 893142 | 9. 901642 | 10. 098358 | 26
35 | 9. 794942 | 9. 893041 | 9. 901901 | 10. 098099 | 25
36 | 9. 795101 | 9. 892940 | 9. 902160 | 10. 097839 | 24
37 | 9. 795259 | 9. 892839 | 9. 902419 | 10. 097580 | 23
38 | 9. 795417 | 9. 892738 | 9. 902678 | 10. 097321 | 22
39 | 9. 795575 | 9. 892637 | 9. 902937 | 10. 097062 | 21
40 | 9. 795733 | 9. 892536 | 9. 903196 | 10. 096803 | 20
41 | 9. 795891 | 9. 892435 | 9. 903455 | 10. 096544 | 19
42 | 9. 796049 | 9. 892334 | 9. 903714 | 10. 096285 | 18
43 | 9. 796206 | 9. 892233 | 9. 903973 | 10. 096027 | 17
44 | 9. 796364 | 9. 892132 | 9. 904232 | 10. 095768 | 16
45 | 9. 796521 | 9. 892030 | 9. 904491 | 10. 095509 | 15
46 | 9. 796678 | 9. 891929 | 9. 904750 | 10. 095250 | 14
47 | 9. 796836 | 9. 891827 | 9. 905008 | 10. 094991 | 13
48 | 9. 796993 | 9. 891726 | 9. 905267 | 10. 094733 | 12
49 | 9. 797150 | 9. 891624 | 9. 905526 | 10. 094474 | 11
50 | 9. 797307 | 9. 891522 | 9. 905784 | 10. 094215 | 10
51 | 9. 797464 | 9. 891421 | 9. 906043 | 10. 093957 | 9
52 | 9. 797621 | 9. 891319 | 9. 906302 | 10. 093698 | 8
53 | 9. 797777 | 9. 891217 | 9. 906560 | 10. 093440 | 7
54 | 9. 797934 | 9. 891115 | 9. 906819 | 10. 093181 | 6
55 | 9. 798091 | 9. 891013 | 9. 907077 | 10. 092923 | 5
56 | 9. 798247 | 9. 890911 | 9. 907336 | 10. 092664 | 4
57 | 9. 798403 | 9. 890809 | 9. 907594 | 10. 092406 | 3
58 | 9. 798560 | 9. 890707 | 9. 907852 | 10. 092147 | 2
59 | 9. 798716 | 9. 890605 | 9. 908111 | 10. 091889 | 1
60 | 9. 798872 | 9. 890503 | 9. 908369 | 10. 091631 | 0
Co-sine. | Sine | Co-tang. | Tangent | M
Degree 51.

[Page]

Degree 39.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 798872 | 9. 890503 | 9. 908369 | 10. 091631 | 60
1 | 9. 799028 | 9. 890400 | 9. 908627 | 10. 091373 | 59
2 | 9. 799184 | 9 890298 | 9. 908886 | 10. 091114 | 58
3 | 9. 799339 | 9. 890195 | 9. 909144 | 10. 090856 | 57
4 | 9. 799495 | 9. 890093 | 9. 909402 | 10. 090598 | 56
5 | 9. 799651 | 9. 889990 | 9. 909660 | 10. 090340 | 55
6 | 9. 799806 | 9. 889888 | 9. 909918 | 10. 090081 | 54
7 | 9. 799961 | 9. 889785 | 9. 910176 | 10. 089823 | 53
8 | 9. 800117 | 9. 889682 | 9. 910435 | 10. 089565 | 52
9 | 9. 800272 | 9. 889579 | 9. 910693 | 10. 089307 | 51
10 | 9. 800427 | 9. 889476 | 9. 910951 | 10. 089049 | 50
11 | 9. 800582 | 9. 889374 | 9. 911209 | 10. 088791 | 49
12 | 9. 800737 | 9. 889271 | 9. 911467 | 10. 088533 | 48
13 | 9. 800892 | 9. 889167 | 9. 911724 | 10. 088275 | 47
14 | 9. 801047 | 9. 889064 | 9. 911982 | 10. 088017 | 46
15 | 9. 801201 | 9. 888961 | 9. 912240 | 10. 087760 | 45
16 | 9. 801356 | 9. 888858 | 9. 912498 | 10. 087502 | 44
17 | 9. 801510 | 9. 888755 | 9. 912756 | 10. 087244 | 43
18 | 9. 801665 | 9. 888651 | 9. 913014 | 10. 086986 | 42
19 | 9. 801819 | 9. 888548 | 9. 913271 | 10 086729 | 41
20 | 9. 801973 | 9. 888444 | 9. 913529 | 10. 086471 | 40
21 | 9. 802127 | 9. 888341 | 9. 913787 | 10. 086213 | 39
22 | 9. 802282 | 9. 888237 | 9. 914044 | 10. 085956 | 38
23 | 9. 802435 | 9. 888133 | 9 914302 | 10. 085698 | 37
24 | 9. 802589 | 9. 888030 | 9. 914560 | [...] | 36
25 | 9. 802743 | 9. 887926 | 9. 914817 | 10. 085183 | 35
26 | 9. 802897 | 9. 887822 | 9. 915075 | 10. 084925 | 34
27 | 9. 803050 | 9. 887718 | 9. 915332 | 10. 084668 | 33
28 | 9. 803204 | 9. 887614 | 9. 915590 | 10. 084410 | 32
29 | 9. 803357 | 9. 887510 | 9. 915847 | 10. 084153 | 31
30 | 9. 803510 | 9. 887406 | 9. 916104 | 10. 083895 | 30
Co-sine. | Sine | Co-tang. | Tangent | M
Degree 50.
Degree 39.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 803510 | 9. 887406 | 9. 916104 | 10. 083895 | 30
31 | 9. 803664 | 9. 887302 | 9. 916362 | 10. 083638 | 29
32 | 9. 803817 | 9. 887198 | 9. 916619 | 10. 083381 | 28
33 | 9. 803970 | 9. 887093 | 9. 916876 | 10. 083123 | 27
34 | 9. 804123 | 9. 887989 | 9. 917134 | 10. 082866 | 26
35 | 9. 804276 | 9. 886884 | 9. 917391 | 10. 082609 | 25
36 | 9. 804428 | 9. 886780 | 9. 917648 | 10. 082352 | 24
37 | 9. 804581 | 9. 886675 | 9. 917905 | 10. 082094 | 23
38 | 9. 804734 | 9. 886571 | 9. 918162 | 10. 081837 | 22
39 | 9. 804886 | 9. 886466 | 9. 918420 | 10. 081580 | 21
40 | 9. 805038 | 9. 886361 | 9. 918677 | 10. 081323 | 20
41 | 9. 805191 | 9. 886257 | 9. 918934 | 10. 081066 | 19
42 | 9. 805343 | 9. 886152 | 9. 919191 | 10. 080809 | 18
43 | 9. 805495 | 9. 886047 | 9. 919448 | 10. 080552 | 17
44 | 9. 805647 | 9. 885942 | 9. 919705 | 10. 080295 | 16
45 | 9. 805799 | 9. 885837 | 9. 919962 | 10. 080038 | 15
46 | 9. 805951 | 9. 885732 | 9. 920219 | 10. 079781 | 14
47 | [...] | 9. 885627 | 9. 920476 | 10. 079524 | 13
48 | 9. 806254 | 9. 885521 | 9. 920733 | 10. 079267 | 12
49 | 9. 806406 | 9. 885416 | 9. 920990 | 10. 079010 | 11
50 | 9. 806557 | 9. 885311 | 9. 921247 | 10. 078753 | 10
51 | 9. 806709 | 9. 885205 | 9. 921503 | 10. 078496 | 9
52 | 9. 806860 | 9. 885100 | 9. 921760 | 10. 078240 | 8
53 | 9. 807011 | 9. 884994 | 9. 922017 | 10. 077983 | 7
54 | 9. 807162 | 9. 884889 | 9. 922274 | 10. 077726 | 6
55 | 9. 807314 | 9. 884783 | 9. 922530 | 10. 077469 | 5
56 | 9. 807464 | 9. 884677 | 9. 922787 | 10. 077213 | 4
57 | 9. 807615 | 9. 884572 | 9. 923044 | 10. 076956 | 3
58 | 9. 807766 | 9. 884466 | 9. 923300 | 10. 076699 | 2
59 | 9. 807917 | 9. 884360 | 9. 923557 | 10. 076443 | 1
60 | 9. 808067 | 9. 884254 | 9. 923813 | 10. 076186 | 0
Co-sine | Sine | Co-tang | Tangent | M
Degree 50

[Page]

Degree 40.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 808067 | 9. 884254 | 9. 923813 | 10. 076180 | 60
1 | 9. 808218 | 9. 884148 | 9. 924070 | 10. 075930 | 59
2 | 9. 808368 | 9. 884042 | 9. 924327 | 10. 075673 | 58
3 | 9. 808519 | 9. 883936 | 9. 924583 | 10. 075417 | 57
4 | 9. 808669 | 9. 883829 | 9. 924839 | 10. 075160 | 56
5 | 9. 808819 | 9. 883723 | 9. 925096 | 10. 074904 | 55
6 | 9. 808969 | 9. 883617 | 9. 925352 | 10. 074647 | 54
7 | 9. 809119 | 9. 883510 | 9. 925609 | 10. 074391 | 53
8 | 9. 809269 | 9. 883404 | 9. 925865 | 10. 074135 | 52
9 | 9. 809419 | 9. 883297 | 9. 926121 | 10. 073878 | 51
10 | 9. 809569 | 9. 883191 | 9. 926378 | 10. 073622 | 50
11 | 9. 809718 | 9. 883084 | 9. 926634 | 10. 073366 | 49
12 | 9. 809868 | 9. 882977 | 9. 926890 | 10. 073110 | 48
13 | 9. 810017 | 9. 882871 | 9. 927147 | 10. 072853 | 47
14 | 9. 810166 | 9. 882764 | 9. 927403 | 10. 072597 | 46
15 | 9. 810316 | 9. 882657 | 9. 927659 | 10. 072341 | 45
16 | 9. 810465 | 9. 882550 | 9. 927915 | 10. 072085 | 44
17 | 9. 810614 | 9. 882443 | 9. 928171 | 10. 071829 | 43
18 | 9. 810763 | 9. 882336 | 9. 928427 | 10. 071573 | 42
19 | 9. 810912 | 9. 882228 | 9. 928683 | 10. 071317 | 41
20 | 9. 810061 | 9. 882121 | 9. 928940 | 10. 071060 | 40
21 | 9. 811210 | 9. 882014 | 9. 929196 | 10. 070804 | 39
22 | 9. 811358 | 9. 881907 | 9. 929452 | 10. 070548 | 38
23 | 9. 811506 | 9. 881799 | 9. 929708 | 10. 070292 | 37
24 | 9. 811655 | 9. 881692 | 9. 929964 | 10. 070036 | 36
25 | 9. 811804 | 9. 881584 | 9. 930219 | 10. 069781 | 35
26 | 9. 811952 | 9. 881477 | 9. 930475 | 10. 069525 | 34
27 | 9. 812100 | 9. 881369 | 9. 930731 | 10. 069269 | 33
28 | 9. 812248 | 9. 881261 | 9. 930987 | 10. 069013 | 32
29 | 9. 812396 | 9. 881153 | 9. 931243 | 10. 068757 | 31
30 | 9. 812544 | 9. 881045 | 9. 931499 | 10. 068501 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 55.
Degree 40.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 812544 | 9. 881045 | 9. 931499 | 10. 068501 | 30
31 | 9. 812692 | 9. 880937 | 9. 931755 | 10. 068245 | 29
32 | 9. 812840 | 9. 880829 | 9. 932010 | 10. 067989 | 28
33 | 9. 812988 | 9. 880721 | 9. 932266 | 10. 067734 | 27
34 | 9. 813135 | 9. 880613 | 9. 932522 | 10. 067478 | 26
35 | 9. 813283 | 9. 880505 | 9. 932778 | 10. 067222 | 25
36 | 9. 813430 | 9. 880397 | 9. 933033 | 10. 066967 | 24
37 | 9. 813578 | 9. 880289 | 9. 933289 | 10. 066711 | 23
38 | 9. 813725 | 9. 880180 | 9. 933545 | 10. 066455 | 22
39 | 9. 813872 | 9. 880072 | 9. 933800 | 10. 066200 | 21
40 | 9. 814019 | 9. 879963 | 9. 934056 | 10. 065944 | 20
41 | 9. 814166 | 9. 879855 | 9. 934311 | 10. 065688 | 19
42 | 9. 814313 | 9. 879746 | 9. 934567 | 10. 065433 | 18
43 | 9. 814460 | 9. 879637 | 9. 934822 | 10. 065177 | 17
44 | 9. 814607 | 9. 879529 | 9. 935078 | 10. 064922 | 16
45 | 9. 814753 | 9. 879420 | 9. 935333 | 10. 064666 | 15
46 | 9. 814900 | 9. 879311 | 9. 935589 | 10. 064411 | 14
47 | 9. 815046 | 9. 879202 | 9. 935844 | 10. 064156 | 13
48 | 9. 815193 | 9. 879093 | 9. 936100 | 10. 063900 | 12
49 | 9. 815339 | 9. 878984 | 9. 936355 | 10. 063645 | 11
50 | 9. 815485 | 9. 878875 | 9. 936610 | 10. 063389 | 10
51 | 9. 815631 | 9. 878766 | 9. 936866 | 10. 063134 | 9
52 | 9. 815777 | 9. 878656 | 9. 937121 | 10. 062879 | 8
53 | 9. 815923 | 9. 878547 | 9. 937376 | 10. 062623 | 7
54 | 9. 816069 | 9. 878438 | 9. 937632 | 10. 062368 | 6
55 | 9. 816215 | 9. 878328 | 9. 937887 | 10. 062113 | 5
56 | 9. 816361 | 9. 878219 | 9. 938142 | 10. 061858 | 4
57 | 9. 816506 | 9. 878109 | 9. 938397 | 10. 061602 | 3
58 | 9. 816652 | 9. 877999 | 9. 938653 | 10. 061347 | 2
59 | 9. 816797 | 9. 877890 | 9. 938908 | 10. 061092 | 1
60 | 9. 816943 | 9. 877780 | 9. 939163 | 10. 060837 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 49.

[Page]

Degree 41.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 816943 | 9. 877780 | 9. 939163 | 10. 060837 | 60
1 | 9. 817088 | 9. 877670 | 9. 939418 | 10. 060582 | 59
2 | 9. 817233 | 9. 877560 | 9. 939673 | 10. 060327 | 58
3 | 9. 817378 | 9. 877450 | [...] | 10. 060072 | 57
4 | 9. 817523 | 9. 877340 | 9. 940183 | 10. 059816 | 56
5 | 9. 817668 | 9. 877230 | 9. 940438 | 10. 059562 | 55
6 | 9. 817813 | 9. 877120 | 9. 940693 | 10. 059307 | 54
7 | 9. 817958 | 9. 877009 | 9. 940948 | 10. 059052 | 53
8 | 9. 818103 | 9. 876899 | 9. 941203 | 10. 058797 | 52
9 | 9. 818247 | 9. 876789 | 9. 941458 | 10. 058542 | 51
10 | 9. 818392 | 9. 876678 | 9 941713 | 10. 058287 | 50
11 | 9. 818536 | 9. 876568 | 9. 941968 | 10. 058032 | 49
12 | 9. 818681 | 9. 876457 | 9. 942223 | 10. 057777 | 48
13 | 9. 818825 | 9. 876347 | 9. 942478 | 10. 057522 | 47
14 | 9. 818969 | 9. 876236 | 9. 942733 | 10. 057267 | 46
15 | 9. 818113 | 9. 876125 | 9. 942988 | 10. 057012 | 45
16 | 9. 819257 | 9. 876014 | 9. 943243 | 10. 056757 | 44
17 | 9. 819401 | 9. 875904 | 9. 943498 | 10. 056502 | 43
18 | 9. 819545 | 9. 875793 | 9. 943752 | 10. 056248 | 42
19 | 9. 819689 | 9. 875682 | 9. 944007 | 10. 055993 | 41
20 | 9. 819832 | 9. 875571 | 9. 944262 | 10. 055728 | 40
21 | 9. 819976 | 9. 875459 | 9. 944517 | 10. 055483 | 39
22 | 9. 820119 | 9. 875348 | 9. 944771 | 10. 055229 | 38
23 | 9. 820263 | 9. 875237 | 9. 945026 | 10. 054974 | 37
24 | 9. 820406 | 9. 875125 | 9. 945281 | 10. 054719 | 36
25 | 9. 820549 | 9. 875014 | 9. 945535 | 10. 054464 | 35
26 | 9. 820693 | 9. 874903 | 9. 945790 | 10. 054210 | 34
27 | 9. 820836 | 9. 874791 | 9. 946045 | 10. 053955 | 33
28 | 9. 820979 | 9. 874679 | 9. 946299 | 10. 053701 | 32
29 | 9. 821122 | 9. 874568 | 9. 946554 | 10. 053446 | 31
30 | 9. 821264 | 9. 874456 | 9. 946808 | 10. 053192 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 48.
Degree 41.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 821264 | 9. 874456 | 9. 946808 | 10. 053192 | 30
31 | 9. 821407 | 9. 874344 | 9 947063 | 10. 052937 | 29
32 | 9. 821550 | 9. 874232 | 9. 947317 | 10. 052682 | 28
33 | 9. 821692 | 9. 874120 | 9. 947572 | 10. 052128 | 27
34 | 9. 821835 | 9 874008 | 9. 947826 | 10. 052173 | 26
35 | 9. 821977 | 9. 873896 | 9. 948081 | 10. 051919 | 25
36 | 9. 822120 | 9. 873784 | 9. 948335 | 10. 051664 | 24
37 | 9. 822262 | 9. 873672 | 9. 948590 | 10. 051410 | 23
38 | 9 822404 | 9 873560 | 9. 948844 | 10. 051156 | 22
39 | 9. 822546 | 9. 873447 | 9. 949099 | 10. 050901 | 21
40 | 9. 822688 | 9. 873335 | 9. 949353 | 10. 050647 | 20
41 | 9. 822830 | 9 873223 | 9. 949607 | 10. 050393 | 19
42 | 9. 822972 | 9. 873110 | 9 949862 | 10. [...] | 18
43 | 9. 823114 | 9. 872998 | 9. 950116 | 10. 049884 | 17
44 | 9. 823255 | 9. 872885 | 9 950370 | 10. 049630 | 16
45 | 9. 823397 | 9. 872772 | 9. 950625 | 10 049375 | 15
46 | 9. 823538 | 9. 872659 | 9. 950879 | 10. 049121 | 14
47 | 9. 823680 | 9. 872546 | 9. 951133 | 10. 048867 | 13
48 | 9. 823821 | 9. 872434 | 9. 951388 | 10. 048612 | 12
49 | 9. 823962 | 9. 872321 | 9. 951642 | 10. 048358 | 11
50 | 9. 824104 | 9. 872208 | 9. 951896 | 10. 048104 | 10
51 | 9. 824245 | 9. 872094 | 9. 952150 | 10. 047850 | 9
52 | 9. 824386 | 9. 871981 | 9. 952404 | 10. 047575 | 8
53 | 9. 824527 | 9. 871868 | 9. 952659 | 10. 047341 | 7
54 | 9. 824667 | 9. 871755 | 9. 952913 | 10. 047087 | 6
55 | 9. 824808 | 9. 871641 | 9. 953167 | 10. 046833 | 5
56 | 9. 824949 | 9. 871528 | 9. 953421 | 10. 046579 | 4
57 | 9. 825090 | 9. 871414 | 9. 953675 | 10. 046325 | 3
58 | 9. 825230 | 9. 871301 | 9. 953929 | 10. 046071 | 2
59 | 9. 825370 | 9. 871187 | 9. 954183 | 10. 045817 | 1
60 | 9. 825511 | 9. 871073 | 9. 954437 | 10. 045563 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 48.

[Page]

Degree 42.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 825511 | 9. 871073 | 9. 954437 | 10. 045562 | 60
1 | 9. 825651 | 9. 870960 | 9. 954691 | 10. 045308 | 59
2 | 9. 825791 | 9. 870846 | 9. 954945 | 10. 045054 | 58
3 | 9. 825931 | 9. 870732 | 9. 955199 | 10. 044800 | 57
4 | 9. 826071 | 9. 870618 | 9. 955453 | 10. 044546 | 56
5 | 9. 826211 | 9. 870504 | 9. 955707 | 10. 044292 | 55
6 | 9. 826351 | 9. 870390 | 9. 955961 | 10. 043038 | 54
7 | 9. 826491 | 9 870275 | 9. 956215 | 10. 043784 | 53
8 | 9. 826631 | 9. 870161 | 9. 956469 | 10. 043531 | 52
9 | 9. 826770 | 9. 870047 | 9. 956723 | 10. 043276 | 51
10 | 9. 826910 | 9. 869933 | 9. 956977 | 10. 043023 | 50
11 | 9. 827049 | 9. 869818 | 9. 957231 | 10. 042769 | 49
12 | 9. 827189 | 9. 869704 | 9. 957485 | 10. 042515 | 48
13 | 9. 827328 | 9. 869589 | 9. 957739 | 10. 042261 | 47
14 | 9. 827467 | 9. 869474 | 9. 957993 | 10. 042007 | 46
15 | 9. 827606 | 9. 869360 | 9. 958246 | 10. 041753 | 45
16 | 9. 827745 | 9. 869245 | 9 958500 | 10. 041500 | 44
17 | 9. 827884 | 9. 869130 | 9. 958754 | 10. 041246 | 43
18 | 9. 828023 | 9. 869015 | 9. 959008 | 10. 040992 | 42
19 | 9. 828162 | 9. 868900 | 9. 959262 | 10. 040738 | 41
20 | 9. 828301 | 9. 868785 | 9. 959515 | 10. 040485 | 40
21 | 9. 828439 | 9. 868670 | 9. 959769 | 10. 040231 | 39
22 | 9. 828578 | 9. 868555 | 9. 960023 | 10. 039977 | 38
23 | 9. 828716 | 9. 868439 | 9. 960277 | 10. 039723 | 37
24 | 9. 828855 | 9. 868324 | 9. 960530 | 10. 039469 | 36
25 | 9. 828993 | 9. 868209 | 9. 960784 | 10. 039216 | 35
26 | 9. 829131 | 9. 868093 | 9. 961038 | 10. 038962 | 34
27 | 9. 829269 | 9. 867978 | 9. 961291 | 10. 038608 | 33
28 | 9. 829407 | 9. 867862 | 9. 961545 | 10. 038451 | 32
29 | 9. 829545 | 9. 867747 | 9. 961799 | 10. 038201 | 31
30 | 9. 829683 | 9. 867631 | 9. 962052 | 10. 037947 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 47.
Degree 42.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 829683 | 9. 867631 | 9. 962052 | 10. 037947 | 30
31 | 9. 829821 | 9. 867515 | 9. 962306 | 10. 037694 | 29
32 | 9. 829959 | 9. 867399 | 9. 962560 | 10. 037440 | 28
33 | 9. 830096 | 9. 867283 | 9. 962813 | 10. 037187 | 27
34 | 9. 830234 | 9. 867167 | 9. 963067 | 10. 036933 | 26
35 | 9. 830372 | 9. 867051 | 9. 963320 | 10. 036680 | 25
36 | 9 830509 | 9. 866935 | 9. 963574 | 10. 036426 | 24
37 | 9. 830646 | 9. 866819 | 9. 963827 | 10. 036173 | 23
38 | 7. 830784 | 9. 866703 | 9. 964081 | 10. 035919 | 22
39 | 9. 830921 | 9. 866586 | 9. 964335 | 10. 035665 | 21
40 | 9. 831058 | 9. 866470 | 9. 964588 | 10. 035412 | 20
41 | 9. 831195 | 9. 866353 | 9. 964842 | 10. 035158 | 19
42 | 9. 831332 | 9. 866237 | 9. 965095 | 10. 034905 | 18
43 | 9. 831469 | 9. 866120 | 9. 965348 | 10. 034652 | 17
44 | 9. 831606 | 9. 866004 | 9. 965602 | 10. 034398 | 16
45 | 9. 831742 | 9. 865887 | 9. 965855 | 10. 034144 | 15
46 | 9. 831879 | 9. 865770 | 9. 966109 | 10. 033891 | 14
47 | 9. 832015 | 9. 865653 | 9 966362 | 10. 033638 | 13
48 | 9. 832152 | 9. 865536 | 9. 966616 | 10. 033384 | 12
49 | 9. 832288 | 9. 865419 | 9. 966869 | 10. 033131 | 11
50 | 9. 832425 | 9. 865302 | 9. 967122 | 10. 032878 | 10
51 | 9. 832561 | 9. 865185 | 9. 967376 | 10. 032624 | 9
52 | 9. 832697 | 9. 865068 | 9. 967629 | 10. 032371 | 8
53 | 9. 832833 | 9. 864950 | 9. 967883 | 10. 032117 | 7
54 | 9. 832969 | 9. 864833 | 9. 968136 | 10. 031864 | 6
55 | 9. 833105 | 9. 864716 | 9. 968389 | 10. 031611 | 5
56 | 9. 833241 | 9. 864598 | 9. 968643 | 10. 031357 | 4
57 | 9. 833376 | 9. 864480 | 9. 968896 | 10. 031104 | 3
58 | 9. 833512 | 9. 864363 | 9. 969149 | 10. 030851 | 2
59 | 9. 833648 | 9. 864245 | 9 969403 | 10. 030597 | 1
60 | 9. 833783 | 9. 864127 | 9. 969656 | 10. 030344 | 0
Co-sine. | Sine | Co-tang. | Tangent | M
Degree 47.

[Page]

Degree 43.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 833783 | 9. 864127 | 9. 969656 | 10. 030344 | 60
1 | 9. 833919 | 9. 864010 | 9. 969909 | 10. 030091 | 59
2 | 9. 834054 | 9. 863892 | 9. 970162 | 10. 029838 | 58
3 | 9. 834189 | 9. 863774 | 9. 970416 | 10. 029584 | 57
4 | 9. 834324 | 9. 863656 | 9 970669 | 10. 029331 | 56
5 | 9. 834460 | 9. 863537 | 9. 970922 | 10. 029078 | 55
6 | 9 834595 | 9 863419 | 9.971175 | 10. 028827 | 54
7 | 9. 834730 | 9. 863301 | 9. 971428 | 10. 028571 | 53
8 | 9. 834865 | 9. 863183 | 9. 971682 | 10. 028318 | 52
9 | 9. 834999 | 9. 863064 | 9. 971935 | 10. 028065 | 51
10 | 9. 835134 | 9. 862946 | 9. 972188 | 10. 027812 | 50
11 | 9. 835269 | 9. 862827 | 9. 972441 | 10. 027559 | 49
12 | 9. 835503 | 9. 862709 | 9. 972694 | 10. 027306 | 48
13 | 9. 835538 | 9. 862590 | 9. 972948 | 10. 027052 | 47
14 | 9. 835672 | 9. 862471 | 9. 973201 | 10. 026799 | 46
15 | 9. 835806 | 9. 862353 | 9. 973454 | 10. 026546 | 45
16 | 9. 835941 | 9. 862234 | 9. 973707 | 10. 026293 | 44
17 | 9. 836075 | 9. 862115 | 9. 973960 | 10. 026040 | 43
18 | 9. 836209 | 9. 861996 | 9. 974213 | 10. 025787 | 42
19 | 9. 836343 | 9. 861877 | 9. 974466 | 10. 025533 | 41
20 | 9. 836477 | 9. 861757 | 9. 974719 | 10. 025280 | 40
21 | 9. 836611 | 9. 861638 | 9 974973 | 10. 025027 | 39
22 | 9. 836745 | 9 861519 | 9 975229 | 10. 024774 | 38
23 | 9. 836878 | 9 861399 | 9 975479 | 10. 024521 | 37
24 | 9. 837012 | 9. 861280 | 9. 975732 | 10. 024268 | 36
25 | 9. 837146 | 9. 861161 | 9. 975985 | 10. 024015 | 35
26 | 9. 837279 | 9. 861041 | 9. 976238 | 10. 023762 | 34
27 | 9. 837412 | 9. 860921 | 9. 976491 | 10. 023509 | 33
28 | 9 837546 | 9. 860802 | 9. 976744 | 10. 023256 | 32
29 | 9. 837679 | 9. 860682 | 9. 976997 | 10. 023003 | 31
30 | 9 837812 | 9. 860562 | 9. 977250 | 10 022750 | 30
Co-sine. | Sine | Co-tang. | Tangent | M
Degree 46.
Degree 43.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 837812 | 9. 860562 | 9. 977250 | 10. 022750 | 30
31 | 9. 837945 | 9. 860442 | 9. 977503 | 10. 022497 | 29
32 | 9. 838078 | 9. 860322 | 9. 977756 | 10. 022244 | 28
33 | 9. 838211 | 9. 860202 | 9. 978009 | 10. 021991 | 27
34 | 9. 838344 | 9. 860082 | 9. 978262 | 10. 021738 | 26
35 | 9. 838477 | 9. 859962 | 9. 978515 | 10. 021485 | 25
36 | 9. 838009 | 9. 859842 | 9. 978768 | 10. 021232 | 24
37 | 9. 838742 | 9. 859721 | 9. 979021 | 10. 020979 | 23
38 | 9. 838875 | 9. 859601 | 9. 979274 | 10. 020726 | 22
39 | 9. 839007 | 9. 859480 | 9. 979527 | 10. 020473 | 21
40 | 9. 839140 | 9. 859360 | 9. 979780 | 10. 020220 | 20
41 | 9. 839272 | 9. 859239 | 9. 980033 | 10. 019967 | 19
42 | 9. 839484 | 9. 859118 | 9. 980285 | 10. 019714 | 18
43 | 9. 839536 | 9. 858998 | 9. 980538 | 10. 019461 | 17
44 | 9. 839668 | 9. 858877 | 9. 980791 | 10. 019209 | 16
45 | 9. 839800 | 9. 858756 | 9. 981044 | 10. 018956 | 15
46 | 9. 839932 | 9. 858639 | 9. 981297 | 10. 018703 | 14
47 | 9. 840064 | 9. 858514 | 9. 981550 | 10. 018450 | 13
48 | 9. 840196 | 9. 858398 | 9. 981803 | 10. 018197 | 12
49 | 9. 840428 | 9. 858272 | 9 982056 | 10. 017944 | 11
50 | 9. 840459 | 9. 858150 | 9. 982309 | 10 017691 | 10
51 | 9. 840591 | 9. 858029 | 9. 982562 | 10. 017438 | 9
52 | 9. 840722 | 9. 857908 | 9. 982814 | 10. 017185 | 8
53 | 9. 840854 | 9. 857786 | 9 983067 | 10. 016933 | 7
54 | 9. 840985 | 9. 857665 | 9. 983320 | 10. 016683 | 6
55 | 9. 841116 | 9. 857543 | 9. 983573 | 10. 016427 | 5
56 | 9. 841247 | 9. 857421 | 9. 983826 | 10. 016174 | 4
57 | 9. 841378 | 9. 857300 | 9. 984079 | 10. 015921 | 3
58 | 9. 841509 | 9. 857178 | 9. 984331 | 10. 015668 | 2
59 | 9. 841640 | 9. 857056 | 9. 984584 | 10. 015416 | 1
60 | 9. 841771 | 9. 856934 | 9. 984837 | 10. 015163 | 0
Co-sine | Sine | Co-tang. | Tangent | M
Degree 46.

[Page]

Degree 44.
M | Sine | Co-sine | Tangent | Co-tang.
0 | 9. 841771 | 9. 856934 | 9. 984837 | 10. 015162 | 60
1 | 9. 841902 | 9. 856812 | 9. 985090 | 10. 014910 | 59
2 | 9. 842033 | 9. 856690 | 9. 985343 | 10. 014657 | 58
3 | 9. 842163 | 9. 856568 | 9. 985596 | 10. 014404 | 57
4 | 9. 842294 | 9. 856445 | 9. 985848 | 10. 014151 | 56
5 | 9. 842424 | 9. 856323 | 9. 986101 | 10. 013899 | 55
6 | 9. 842555 | 9 856201 | 9. 986354 | 10. 013646 | 54
7 | 9. 842685 | 9. 856078 | 9. 986607 | 10. 013393 | 53
8 | 9. 842815 | 9. 855956 | 9. 986859 | 10. 013140 | 52
9 | 9. 842945 | 9. 855833 | 9. 987112 | 10. 012888 | 51
10 | 9. 843076 | 9. 855710 | 9. 987365 | 10. 012635 | 50
11 | 9. 843206 | 9. 855588 | 9. 987618 | 10. 012382 | 49
12 | 9. 843336 | 9. 855465 | 9. 987871 | 10. 012129 | 48
13 | 9. 843465 | 9. 855342 | 9. 988123 | 10. 011877 | 47
14 | 9. 843595 | 9. 855219 | 9. 988376 | 10. 011624 | 46
15 | 9. 843725 | 9. 855096 | 9. 988629 | 10. 011371 | 45
16 | 9. 843855 | 9. 854973 | 9. 988882 | 10. 011118 | 44
17 | 9. 843934 | 9. 854850 | 9. [...] | 10. 010866 | 43
18 | 9. 844114 | 9. 854727 | 9. 989387 | 10. 010613 | 42
19 | 9. 844243 | 9. 854603 | 9. 989640 | 10. 010360 | 41
20 | 9. 844372 | 9. 854480 | 9. 989893 | 10. 010107 | 40
21 | 9. 844502 | 9. 854356 | 9. 990145 | 10. 009855 | 39
22 | 9. 844631 | 9. 854233 | 9. 990398 | 10. 009602 | 38
23 | 9 844760 | 9 854109 | 9. 990651 | 10. 009349 | 37
24 | 9. 844889 | 9. 853986 | 9. 990903 | 10. 009096 | 36
25 | 9. 845018 | 9. 853862 | 9. 991156 | 10. 008844 | 35
26 | 9. 845147 | 9. 853738 | 9. 991409 | 10. 008591 | 34
27 | 9. 845276 | 9. 853614 | 9. 991662 | 10. 008338 | 33
28 | 9. 845404 | 9. 853490 | 9. 991914 | 10. 008086 | 32
29 | 9. 845533 | 9. 853366 | 9. 992167 | 10. 007833 | 31
30 | 9. 845662 | 9 853242 | 9. 992420 | 10. 007580 | 30
Co-sine | Sine | Co-tang. | Tangent | M
Degree 45.
Degree 44.
M | Sine | Co-sine | Tangent | Co-tang.
30 | 9. 845662 | 9. 853242 | 9. 992420 | 10. 007580 | 30
31 | 9. 845790 | 9. 853118 | 9. 992672 | 10. 007328 | 29
32 | 9. 845919 | 9. 852994 | 9. 992925 | 10. 007075 | 28
33 | 9. 846047 | 9. 852869 | 9. 993178 | 10. 006822 | 27
34 | 9. 846175 | 9. 852745 | 9. 993430 | 10. 006569 | 26
35 | 9. 846304 | 9. 852620 | 9. 993683 | 10. 006317 | 25
36 | 9. 846432 | 9. 852496 | 9. 993936 | 10. 006064 | 24
37 | 9. 846560 | 9. 852371 | 9. 994189 | 10. 005811 | 23
38 | 9. 846688 | 9. 852246 | 9. 994441 | 10. 005559 | 22
39 | 9. 846816 | 9. 852122 | 9. 994694 | 10. 005306 | 21
40 | 9. 846944 | 9. 851997 | 9. 994947 | 10. 005053 | 20
41 | 9. 847071 | 9. 851872 | 9. 995199 | 10. 004801 | 19
42 | 9. 847199 | 9. 851747 | 9. 995452 | 10. 004548 | 18
43 | 9. 847327 | 9. 851622 | 9. 995701 | 10. 004295 | 17
44 | 9. 847454 | 9. 851497 | 9. 995957 | 10. 004043 | 16
45 | 9. 847582 | 9. 851372 | 9. 996210 | 10. 003790 | 15
46 | 9. 847709 | 9. 851246 | 9. 996463 | 10. 003537 | 14
47 | 9. 847836 | 9. 851121 | 9. 996715 | 10. 003285 | 13
48 | 9. 847964 | 9. 850996 | 9. 996968 | 10. 003032 | 12
49 | 9. 848091 | 9. 850870 | 9. 997220 | 10. 002779 | 11
50 | 9. 848218 | 9. 850745 | 9. 997473 | 10. 002527 | 10
51 | 9. 848345 | 9. 850619 | 9. 997726 | 10. 002274 | 9
52 | 9. 848472 | 9. 850493 | 9. 997979 | 10. 002021 | 8
53 | 9. 848599 | 9. 850367 | 9. 998231 | 10. 001769 | 7
54 | 9. 848726 | 9. 850242 | 9. 998484 | 10. 001516 | 6
55 | 9. 848852 | 9. 850116 | 9. 998737 | 10. 001263 | 5
56 | 9. 848979 | 9. 849990 | 9. 998989 | 10. 001011 | 4
57 | 9. 849106 | 9. 849864 | 9. 999242 | 10. 001758 | 3
58 | 9. 849232 | 9. 849737 | 9. 999495 | 10. 000505 | 2
59 | 9. 849359 | 9. 849611 | 9. 999747 | 10. 000253 | 1
60 | 9. 849485 | 9. 849485 | 10. 000000 | 10. 000000 | 0
Co-sine | Sine | Co-tang | Tangent | M
Degree 45.

FINIS.

Some Books sold by W. Freeman at the Artichoke next St. Dunstan's Church in Fleet-street.

[Page]

REports in the Court of King's-Bench from the 12th. to the 30th. Year of the Reign of our late Sovereign Lord King Charles II. in 3 Vol. Taken by J. Keble of Grey's-Inn, Esquire; with new and usefull Tables to all the 3 Vol.

Sheppard's President of Presidents. 8o.

Zenophon's History of the Affairs of Greece, inseven Books, being a Continuation of the Peloponnesian War, from the time where Thucydides ends, to the Battel at Mantinea; to which is prefixed an Abstract of Thucydides, and a brief Account of the Land and Naval Forces of the ancient Greeks. Translated from the Greek by John Newman.

The Institution and Life of Cyrus the Great, written by the famous Philosopher and General, Zenophon of Athens. Translated by F. Digby and J. Norris.

Holy Devotions with Directions to Pray: Also a Brief Exposition on the Lord's Prayer, Creed, Ten Commandments, seven Penitential Psalms, and the seven Psalms of Thanksgiving, by the Right Reverend Father in God Lancelot Andrews late Bishop of Winchester.

Daily Exercise for a Christian, or a Manual of Private Devotions, as well for every day in the Week, as upon Particular Occasions, by J. Bridal, Senior, Esq late of the Rolls.

A New Years Gift composed of Prayers and Meditations with Devotions for several Occasions; the whole six parts compleat.

A Catalogue of Books.

[Page]

The Old Religion: A Treatise wherein is laid down the true State of the Difference betwixt the Reformed and Roman Church. Serving for the Vindication of Our Innocence, for the setling of Wavering Minds, for a Preservation against Popish Insinuations. By the Reverend Father in God Jos. Hall late Lord Bishop of Excester, and afterwards of Norwich.

The Manners of the Israelites in 3 parts. 1. Of the Patriarchs. 2. Of the Israelites, after their coming out of Egypt, untill the Captivity of Babylon. 3. Of the Jews, after their Return from the Captivity, untill the Preaching of the Gospel, in 12o.

The Means to preserve Peace in Marriage; being an ingenious Treatise written (Originally in French) by the Authour of the Rules of Civility.

The Penitent Pardoned, or a Discourse of the Nature of Sin and Efficacy of Repentance, under the Parable of the Prodigal Son, by Dr. Goodman.

An infallible way to Contentment in the midst of Publick and Personal Calamities, together with the Christians Courage and Encouragement against evil Tidings and the fear of Death.

Argalus and Parthenia, by F. Quarles.

Nanis History of Venice, Fol.

History of the Government of Venice, 8o.

Policy of the Venetians, 12o.

The Mistaken Beauty, a Comedy.

The Dutchess of Malphey.

The Empress of Morocco, a Farce.

The Court of the Gentiles in 4 parts compleat, by Theophilus Gale.

Information about this book

Title statement

Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.

Taylor, John, mathematician.

Edition statement

1687

Publication

Publisher

Text Creation Partnership,

Place of publication

Ann Arbor, MI ; Oxford (UK) :

Date

2003-01 (EEBO-TCP Phase 1).

ID [DLPS]

A64224

ID [STC]

Wing T534

ID [STC]

ESTC R23734

ID [EEBO-CITATION]

07887917

ID [OCLC]

ocm 07887917

ID [VID]

40282

Availability

This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal. The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission.

Series

Early English books online.

Notes

(EEBO-TCP ; phase 1, no. A64224)

Transcribed from: (Early English Books Online ; image set 40282)

Images scanned from microfilm: (Early English books, 1641-1700 ; 1215:4)

Source

— Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor., Taylor, John, mathematician..

Extent

[15], 507 p., 8 leaves of plates : ill., port.

Printed by J.H. for W. Freeman,. London :: 1687.. (The logarithm tables have separate t.ps.) (Reproduction of original in the Cambridge University Library.)

Creation

Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford.

Editorial practices

EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO.

EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org).

The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source.

Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data.

Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so.

Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor.

The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines.

Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements).

Keying and markup guidelines are available at the Text Creation Partnership web site.