COSMOGRAPHIA, OR A VIEW OF THE Terrestrial and Coelestial GLOBES, IN A Brief Explanation OF THE PRINCIPLES Of plain and solid GEOMETRY, Applied to Surveying and Gauging of CASK.
The Doctrine of the Primum Mobile.
With an Account of the Juilan & Gregorian Calendars, and the Computation of the Places of of the Sun, Moon, and Fixed Stars, from such Decimal Tables of their Middle Motion, as supposeth the whole Circle to be divided into an hundred Degrees or Parts.
To which is added an Introduction unto GEOGRAPHY.
By John Newton, D. D.
London, Printed for Thomas Passinger, at the Three Bibles on London-Bridge 1679.
TO THE Most Honourable HENRY SOMERSET, Lord Herbert, Baron of Chepstow, Raglan, and Gower, Earl and Marquess of Worcester, Lord President and Lord Lieutenant of Wales and the Marches, Lord Lieutenant of Gloucester, Hereford and Monmouth, and of the City and County of Bristol, Knight of the Most Noble Order of the Garter, and one of His Majestie's Most Honourable Privy Council.
HE that adventures upon any thing contrary to the General received practice, what ever his own courage and resolutions are, had need to be supported, not only by the most Wise and Honourable, but also [Page] the most Powerful Persons that are in a Nation or Kingdom; For let the Proposals be never so advantagious to the Publick, they shall not only be decried and neglected, but it is well, if the Promoter be not both abused and ruined: Yet I, notwithstanding all these discouragements, have not been silent, but in order to Childrens better Education, have long since published my thoughts, and have and do declare, that the multitude of Schools for the learning of the Latine and Greek Tongues, are destructive both to our youth and the Commonwealth; and if the Opinion of Sir Francis Bacon in his Advice to King Iames concerning Sutton's Hospital, be not sufficient to warrant my Assertion, I could heartily wish that no such Evidence could have been produced, as the late unhappy Wars, in the Bowels of this Kingdom hath afforded us; for what he saith there by way of Advice, we by woful Experience have found too true; that by reason of the multitude of Grammar Schools, more Scholars are dayly brought up, than all the Preferments in this Nation can provide for, and so they become uncapable of other Professions, and unprofitable in their own, and at last become, materia rerum novarum; whether this be an essential or an accidental Effect, [Page] I will not here dispute; the truth of it, I am sure, cannot be denied: but that is not all; by this means it comes to pass, that four of the seven Liberal Arts, are almost wholly neglected, as well in both Universities, as in all Inferiour Schools; and setting aside the City of London, there are but few Places in this Nation, where a man can put his Son, to be well instructed in Arithmetick, Geometry, Musick and Astronomy; and even that Famous City was without a Publick School for Mathematical Learning, till His present Majesty was pleased to lay the Foundation; nay so averse are men in the general to these Arts (which are the support of all Trade) that without a high hand, it will be almost impossible, to make this People wise for their own good: I come therefore to your Honour, humbly to beg your Countenance and Assistance, that the Stream of Learning may be a little diverted, in those Schools that are already erected, and to be instrumental for the erecting more, when they shall be wanting; that we may not be permitted still to begin at the wrong end; but that according to the practice of the Ancient Philosophers, Children may be instructed in Arithmetick, Geometry, Musick and Astronomy; before the Latine and Greek [Page] Grammars are thought on, these Arts in themselves, are much more easie to be learned, tend more to a general good, and will in a great measure facilitate the Learning of the Tongues, to as many as shall after this Foundation laid, be continued at School, and provided for in either Universities. Your Honour was instrumental to enlarge the Maintenance for God's Minister in the Place where I live, and perhaps it may please God to make you so, not only in making this Place in particular, but many other Places in this Land happy, by procuring Schools for these Sciences, and not only so, but by your Loyal and Prudent managing the several Trusts committed to you, you may do much for God's Glory, your Countries Good, and the continuance of your own Honour to all Future Generations, which is, and shall be the Prayer of,
TO THE READER.
MY Design in publishing these Introductions to Geometry and Astronomy, is so well known by all the Epistles, to my other Treatises of Grammar, Arithmetick, Rhetorick, and Logick, that I think it needless to tell thee here, that it is my Opinion, that all the Arts should be taught our Children in the English Tongue, before they begin to learn the Greek or Latin Grammar, by which means many thousands of Children would be fitted for all Trades, enabled to earn their own Livings, and made useful in the Commonwealth; and that before they attain to twelve years of age; and by consequence the swarming of Bees would be prevented, who being compelled to leave their Hives, for want of room, do spread themselves abroad, and instead of gathering of Honey, do [...] sting all that come in their way. We should not have such innumerable company of Gown-men to the loss and prejudice of themselves and the Commonwealth; and those we had would probably be more learned, and better regarded.
[Page] His Majesty being pleased to begin this Work, by His Bounty towards a Mathematical School in Christ's Church London; I am not now without hopes, to see the same effected in many other Places in this Kingdom; and to this purpose I have to my Introductions to the other Arts, added these also to Geometry and Astronomy; which I call by the name of Cosmographia; and this I have divided into four Parts; in the first I have briefly laid down the first Principles belonging to the three kinds of Magnitude or continued Quantity, Lines, Planes and Solids; which ought in some measure to be known, before we enter upon Astronomy, and this part I call an Introduction unto Geometry:
The second and third Parts treat of Astronomy; the first of which sheweth the Doctrine of the Primum Mobile, that is, the Declination, Right Ascension, and Oblique Ascensions of the Sun and Stars, and such other Problems, as do depend upon the Doctrine of Spherical Triangles.
The second Part of Astronomy, treateth of the motion of the Sun, Moon and Fixed Stars; in order whereunto, I have first given thee a brief account of the Civil Year, with the cause of the difference between our Julian and Gregorian Calendar, and of both from the true; for it must be acknowledged [Page] that both are erroneous, though ours be the worse of the two; yet not so bad, but that our Dissenting Brethren have I hope some better Arguments to justifie their Non-conformity▪ than what I see published in a little Book without any name to it, concerning two Easters in one Year; by the General Table, saith this learned man, who owneth the Feast of Easter was to be observed Anno 1674. upon the 19 day of April, so the Almanacks for that Year, as well as the General Table set before the Book of Common Prayer; but by the Rule in the said Book of Common Prayer given, the Feast of Easter should have been upon the twelfth of April, for Easter-Day must always be the first Sunday after the first Full Moon, which happeneth next after the one and twentieth day of March, and if the Full Moon happen upon a Sunday, Easter-Day is the Sunday after; Now in the Year 1674. the 19 of April being Friday was Full Moon, therefore by this Rule, Easter-Day should be the twelfth, and by the Table and the Common Almanacks April the tenth; but this learned man must know, that the mistake is in himself, and not in the Rule or Table set down in the Book of Common Prayer; for if he please to look into the Calendar, he will find that the Golden Number Three, (which was the Golden Number for that Year) is [Page] placed against the last day of March, and therefore according to the supposed motion of the Moon, that Day was New Moon; and then the Full Moon will fall upon the fourteenth day of April, and not upon the tenth, and so by consequence the Sunday following the first Full Moon after the 21 day of March was the nineteenth of April and not the twelfth. And thus the Rule and the Table in the Book of Common Prayer for finding the Feast of Easter are reconciled; and when Authority shall think sit, the Calendar may be corrected and all the moveable Feasts be observed upon the days and times at first appointed; but till that be, a greater difference than one Week will be found in the Feast of Easter between the Observation thereof according to the Moons true motion, and that upon which the Tables are grounded; for by the Fathers of the Nicene Council it was appointed, that the Feast of Easter should be observed upon the Sunday following the first Full Moon after the Vernal Equinox, which then indeed was the 21 of March; but now the tenth, and in the Year 1674. Wednesday the 11 of March was Full Moon, and therefore by this Rule, Easter-Day should have been upon March the fifteenth, whereas according to the Rules we go by, it was not till April the nineteenth.
[Page] The Tables of the Sun and Moons middle motions are neither made according to the usual Sexagenary Forms, nor according to the usual Degrees of a Circle and Decimal Parts, but according to a Circle divided into 100 Degrees and Parts, and this I thought good to do, to give the World a taste of the excellency of Decimal Numbers, which if a Canon of Sines and Tangents were fitted to it, would be found much better, as to the computing the Places of the Planets; but as to the Primum Mobile, by reason of the general dividing a Circle into 360 Degrees, I should think such a Canon with the Decimal Parts most convenient, and in some cases the common Sexagenary Canon may be very useful, and indeed should wish and shall endeavour to have all printed together, one Table of Logarithms will serve them all, and two such Canons, one for the Study and another for the Pocket, would be sufficient for all Mathematical Books in that kind; and then men may use them all or either of them as they shall have occasion, or as every one is perswaded in his own mind.
What I have done in this particular, as it was for mine own satisfaction, so I am apt to believe, that it will be pleasing to many others; and although I shall leave every one to abound in his own sense, yet I cannot think [Page] that Custom should be such a Tyrant, as to force us always to use the Sexagenary form, if so, I wonder that men did not always use the natural Canon; if no alteration may be admitted, what reason can be given for the use of Logarithms; and if that be found more ready than the natural, in things of this kind, where none but particular Students are concerned, I should think it reasonable, to reduce all things hereafter, into that form, which shall be found most ready and exact; now the Part Proportional in the Artificial Sines and Tangents in the three first Degrees cannot be well taken by the common difference, and the way of finding them otherwise will not be so easie in the Sexagenary Canon, as in either of the other, and this me thinks, should render that Canon which divides each Degree into 100 Parts more acceptable; but thus to retain the use of Sines, Degrees, and Decimal Parts, doth not to me seem convenient, and to reckon up, a Planets middle motion, by whole Circles will sometimes cause a Division of Degrees by 60, which hath some trouble in it also, but if a Circle be divided into 100 Degrees, this inconvenience is avoided, and were there no other reason to be given, this me thinks should make such a Canon to be desirable; but till I can find an opportunity of publishing such an one, I shall forbear to [Page] shew any further uses of it, and for what is wanting here in this subject, I therefore refer thee to Mr. Street's Astronomia Carolina, and the several Books written in English by Mr. Wing.
The fourth Part of this Treatise is an Introduction unto Geography, in which I have given general Directions, for the understanding how the habitable part of the World is divided in respect of Longitude and Latitude in respect of Climes and Parallels with such other Particulars as will be found useful unto such as shall be willing to understand History; in which three things are required; The time when, and this depends upon Astronomy; the place where, and this depends upon Geography; and the Person by whom any memorable Act was done, and this must be had from the Historical narration thereof; and he that reads History without some knowledge in Astronomy and Geography will find himself at a loss, and be able to give but a lame account of what he reads; but after the learning of these Arts of Grammar, (I mean so much thereof, as tends to the understanding of every ones Native Language) Arithmetick, Geometry and Astronomy; a Child may proceed profitably to Rhetorick and Logick, the reading of History, and the learning of the Tongues; and sure there is [Page] no studious and ingenious man, but will stand in need of some Recreation, and therefore if Musick in the Worship and Service of God be not Argument enough to allow that a place among the Arts, let that poor end of Delight and Pleasure be her Advocate; and although that all men have not Voyces, yet I can hardly believe, that he expects any Melodious Harmony in Heaven, that will not allow Instrumental Musick a place on Earth; and as for those that have Voyces, surely the time of learning Vocal Musick, must be in Youth, and I am perswaded that the Arts and Sciences to some good degree may be learned by Children before they be full twelve years old, and would our Grammer Masters leave off their horrible severity, and apply themselves to such ways of teaching Youth, as the World is not now unacquainted with, I am perswaded that it is no difficult matter, in four years time more to fit Children in some good measure for the University.
The great Obstruction in this Work, is the general Ignorance of Teachers, who being unacquainted with this Learning, cannot teach others what they know not themselves. I could propound a remedy for this, Sed Cynthius aurem vellit; Therefore I will forbear and leave what I have written, to be perused and censured as thou shall think fit.
Practical Geometry; OR, THE ART of SURVEYING.
CHAP. I.
Of the Definition and Division of Geometry.
GEometry is a Science explaining the kinds and properties of continued quantity or magnitude.
2. There are three Kinds or Species of Magnitude or continued Quantity, Lines, Superficies and Solids.
3. A Line is a Magnitude consisting only of length without either breadth or thickness.
4. In a Line two things are to be considered, the Terms or Limits, and the several Kinds.
5. The term or limit of a Line is a Point.
6. A Point is an indivisible Sign in Magnitude which cannot be comprehended by sense, but must be conceived by the Mind.
7. The kinds of Lines are two, Right and Oblique.
[Page 2] 8. A Right Line is that which lieth between his Points, without any going up or going down on either side. As the Line AB lieth streight and equally between the Points A and B. Fig. 1.
9. An Oblique Line is that which doth not lie equally between its Points, but goeth up and down sometimes on the one side and sometimes on the other. And this is either simple or various.
10. A simple Oblique Line, is that which is exactly Oblique, as the Arch of a Circle; of Various Oblique Lines there is but little use in Geometry.
11. Thus are Lines to be considered in themselves, they may be also considered as compared to one another, and that either in respect of their distances, or in respect of their meetings.
12. In respect of their distances, they may be either equally distant, or unequally.
13. Lines equally distant are two or more, which by an equal space are distant from one another, and these are called Parallels; and these though infinitely extended will never concur.
14. Lines unequally distant, are such as do more or less incline to one another, and these being extended will at last concur.
15. Concurring Lines are either perpendicular or not perpendicular.
16. A Perpendicular Line, is a Right Line falling directly upon another Right Line, not declining or inclining to one side more than another; as the Line AB in Fig. 1.
17. A Perpendicular Line is twofold, to wit, either falling exactly in the middle of another Line, or upon some other Point which is not the middle.
18. A line exactly Perpendicular, may be drawn in the same manner, as any Right Line [Page 3] may be divided into two equal Parts; the which may thus be done. If from the two Terms or Points of the Right Line given, there shall be described two Arches crossing one another above and below, a Line drawn through the Intersections of those Arches, shall be exactly Perpendicular, and also divide the Right Line given into her equal Parts. Fig. 1.
For Example; Let CD be the Right Line given, and let it be required, to bisect this Line, and to erect a Perpendicular in the middle thereof. 1. Then setting one of your Compasses in the Points C, draw the Arches E and F. 2. Setting one Foot of your Compasses in D, draw the Arches G and H, and from the Intersections of these Arches draw the Right Line KL, so shall the Right Line KL be Perpendicular to the Right-Line CD, and the Right Line CD also divided into two equal Parts, in the Point A.
19. A Line Perpendicular to any other Point than the middle is twofold: for it is either drawn from some Point given in the Line; or from some Point given without the Line.
20. From a Point given in the Line, at Perpendicular may thus be drawn. In Fig. 2. Let the given Line be CD, and let it be required to draw a Perpendicular Line to the Point C, your Compasses being opened to any reasonable distance, set one Foot in the Point C, and the other in any place on either side the Line CD, suppose at A, then describe the Arch ECF, this done draw the Line EA, and where that Line being extended shall cut the Arch ECF, a Right Line drawn from C to that Intersection shall be Perpendicular to the Point C in the Line CD, as was required.
21. From a Point given without the Line, a Perpendicular may be drawn in this manner. [Page 4] In Fig. 2. Let the given Line be CD, and let it be required to draw another Line Perpendicular thereunto, from the Point F without the Line. From the Point F draw a streight Line to some part of the Line CD at pleasure, as FE, which being bisected, the Point of Bisection will be A, if therefore at the distance of AF, you draw the Arch ECF, the Right Line CF shall be Perpendicular to the Line CD, as was required.
22. Hitherto concerning a Perpendicular Line. A Right Line not Perpendicular, is a Right Line falling indirectly upon another Right Line, inclining thereto on the one side more, and on the other less.
23. Lines unequally distant, and at last concurring, do by their meeting make an Angle.
24. An Angle therefore is nothing else, then the place, where two Lines do meet or touch one another, and the two Lines which constitute the Angle, are in Geometry called the sides of the Angle.
25. Every Angle is either Heterogeneous, or Homogeneous: that is called an Hetorogeneous Angle, which is made by the meeting of one Right Line, and another that is Oblique and Crooked; and that is called an Homogeneous Angle, which is made by the meeting of two Lines of the same kind, that is, of two Right Lines, or of two curved or Circular Lines.
26. An Homogeneous Angle made of two curved or Circular Lines, is to be considered in Geometry as in Spherical Triangles, but the other which is made of Right Lines, is in all the Parts of Geometry of more frequent use.
27. Right lined Angles are either Right or Oblique.
[Page 5] 28. A Right Angle is that whose legs or sides are Perpendicular to one another, making the comprehended space on both sides equal. Thus in Fig. 1. the Line AK is Perpendicular to the Line CD, and the Angles KAC and KAD, are right and equal to one another.
29. An Oblique Angle is that, whose sides are not Perpendicular to one another.
30. An Oblique Angle is either acute or obtuse.
31. An Acute Angle is that which is less than a Right.
32. An Obtuse Angle, is that which is greater than a Right. Thus in Fig. 1. The Angle BAC is an Acute Angle because less than the Right Angle CAK. And the Angle BAD is an Obtuse Angle being greater than the Right Angle DAK.
The Geometrical Propositions concerning Lines and Angles are very many, but these following we think sufficient for our present purpose.
Proposition I.
To divide a Right Line given into any Number of equal Parts.
Let it be required to divide the Right Line AB into five equal Parts. From the extream Points of the given Line A and B, let there be drawn two Parallel Lines, then from the Point A at any distance of the Compasses, set off as many equal Parts wanting one, as the given Line is to be divided into, which in our Example is four, and are noted thus, 1. 2. 3. 4. and from the Point B set off the like Parts in the Line BC, and let them be [Page 6] noted likewise thus, 1. 2. 3. 4. then shall the Parallel Lines, 14. 23. 32, and 41. divide the Right Line AB into 5 equal Parts, as was required.
Proposition II.
Two Right Lines being given, to find a Mean propertional between them.
Let the two Right Lines given be DB and CB, which let be made into one Line as CD, which being besected the Point of bisection is A, from which as from a Centre describe the Arch CED, and from the Point B erect the Perpendicular BE, so shall BE, be the Mean proportional required; for, BC. BE∷BE. BD.
Proposition III.
Three Right Lines being given, to find a fourth proportional.
Let the three given Lines be AB. BC. and AD. Fig. 5. to which a fourth proportional is required: draw AE at any Acute Angle, to the Line AD in the Point A; and make DE parallel to BC, so shall AE be the fourth proportional required; for, AB. BC∷AD. AE.
Proposition IV.
Vpon a Right Line given, to make a right-lined. Angle, equal to an Angle given.
Let it be required upon the Line CD in Fig. 6. [Page 7] to make an Angle, equal to the Angle DAE in Fig. 5. From the Point A as a Center, at any extent of the Compasses describe the Arch BG, between the sides of the Angle given, and with the same extent describe the Arch HL from the Point D, and then make HL equal to BG, then draw the Line DL, so shall the Angle CDL be equal to the Angle DAE given, as was required.
CHAP. II.
Of Figures in the general, more particularly of a Circle and the affections thereof.
HItherto we have spoken of the first kind of Magnitude, that is, of Lines, as they are considered of themselves, or amongst themselves.
2. The second kind of Magnitude is that which is made of Lines, that is, a Figure consisting of breadth as well as length, and this is otherwise called a Superficies.
3. And in a Superficies there are three things to be considered. 1. The Term or Limit. 2. The middle of the Term. 3. The Thing or Figure made by the Term or Limit.
4. The Term or Limit is that which comprehendeth and boundeth the Figure, it is commonly called the Perimeter or Circumference.
5. The Term of a Figure is either Simple or various.
6. A Simple Term is that which doth consist of a Simple Line, and is properly called a Circumference [Page 8] or Periphery: A Periphery therefore is the Term of a Circle or most Simple Figure.
7. A various Term is that which hath bending or crooked Lines, making Angles, and may therefore be called Angular.
8. The middle of Term is that which is the Center of the Figure; for every Figure, whether Triangular, Quadrangular, or Multangular, hath a Center as well as the Circular, differing in in this, that the Lines in a Circle drawn from the Center to the Circumference are all equal, but in other Figures they are not equal.
9. The Thing or Figure made by the Term or Limit, is all that Area or space which is included by the Term or Terms. And here it is to be observed, that the Term of a Figure is one thing, and the Figure it self another; for Example, A Periphery is the Term of a Circle, but the Circle it self is not properly the Periphery, but all that Area or space which is included by the Periphery, for a Periphery is nothing but a Line, but the Circle is that which is included by that Line.
10. As the Term of a Figure is either Simple or Various; so the Figure it self is either Simple and Round, or Various and Angular.
11. A Simple Figure is that which is contained by a Simple or Round Line, and is either a Circle or an Ellipsis.
12. A Circle therefore is such a Figure which is made by a Line so drawn into it self, as that it is every where equally distant from the middle or Center.
13. An Ellipsis is an oblong Circle.
14. In a Circle we are to consider the affections which are as it were the Parts or Sections [Page 9] thereof, as they are made by the various applications of Right Lines.
15. And Right Lines may be applied unto a Circle, either by drawing them within, or without the Circle.
16. Right Lines inscribed within a Circle, are either such as do cut the Circle into two equal or unequal Parts, as the Diameter and lesser Chords, or such as do cut the Diameter and lesser Chords into two equal or unequal Parts, as the Right and versed Sines.
17. A Diameter is a Right Line drawn through the Center from one side of the Circumference to the other, and divideth the Circle into two equal Parts, As in Fig. 7. The Right Line GD drawn through the Center B is the Diameter of the Circle GEDL dividing the same into the two equal Parts GED, and GLD: and this is also called the greatest Chord or Subtense.
18. A Chord or Subtense is a Right Line inscribed in a Circle, dividing the same into two equal or unequal Parts; if it divide the Circle into two equal Parts, it is the same with the Diameter, but if it divide the Circle into two unequal Parts it is less than the Diameter, and is the Chord or Subtense of an Arch less than a Semi-circle, and also of an Arch greater than a Semi-circle. As in the former Figure, the Right Line CAK divideth the Circle into two unequal Parts, and is the Chord or Subtense of the Arch CDK, less than a Semi-circle, and of the Arch CGK greater than a Semi-circle: and these are the Lines which divide the Circle into two equal or unequal Parts. And as they divide the Circle into two equal Parts, so do they also divide one another; The lesser Chords when they are divided by [Page 10] the Diameter into two equal Parts, those Parts are called Right Sines, and the two Parts of the Diameter made by the intersection of the Chords are called versed Sines.
19. Sines are right or versed.
20. Right Sines are made by being besected, by the Diameter, and are twofold, Sinus totus, the whole Sine or Radius, and this is the one half of the Diameter, as the Lines BE or BD, and all Lines drawn from the Center to the Circumference.
21. Sinus simpliter, or the lesser Sines, are the one half of any Chord less than the Diameter, as in the former Figure CA or AK, which are the equal Parts of the Chord CAK, are the Sines of the Arches CD. and DK less than a Quadrant, and also the Sines of CEG and KLG greater than a Quadrant.
22. Versed Sines are the Segments of the Diameter, made by the Chords intersecting it, at Right Angles, as AD is the versed Sine of CD or DG and the other Segment AG is the versed Sine of the Arch CEG or KLG.
23. The Right Lines drawn without the Circle are two, the one touching the Circle, and is called a Tangent, and the other cutting the Circle, and is called a Secant.
24. A Tangent is a Right Line touching the Circle, and drawn perpendicular to the Diameter, and extended to the Secant.
25. A Secant is a Right Line drawn from the Center through the Circumference, and extended to the Tangent. As in the former Figure, the Right Line DF is the Tangent of the Arch CD, and the Right Line BF is the Secant of the same Arch CD.
Proposition I.
The Arch of a Circle being given to describe the whole Periphery.
Let ABC be an Arch given, and let the Circumference of that Circle be required. Let there be three Points taken in the given Arch at pleasure, as A, B, C; open your Compasses to more than half the distance of A, B, and setting one Foot in A describe the Arch of a Circle, and the Compasses remaining at the same distance, setting one Foot in B, describe another Arch so as it may cut the former in two Points, suppose G, and H, and draw the Line HG towards that Part on which you suppose the Center of the Center of the Circle will fall.
In like manner, opening your Compasses to more than half your distance of B, C, describe two other Arches from the Points E and C, cutting each other in E and F, then draw the Line EF till it intersect the former Line HG, so shall the Point of Intersection be the Center of the Circumference or Circle required, as in Fig. may be seen.
Proposition II.
The Conjugate Diameters of an Ellipsis being given, to draw the Ellipsis.
Let the given Diameter in Fig. 24. be LB and ED, the greatest Diameter. LB being bisected in the Point of Bisection, erect the Perpendicular [Page 12] AD. which let be half of the lesser Diameter ED, then open your Compasses to the extent of AB, and setting one Foot in D, with the other make a mark at M and N in the Diameter BL, then cutting a thred to the length of BL, fasten the thred with your Compasses in the Points NM, and with your Pen in the inside of the thred describe the Arch BFKL, so shall you describe the one half of the Ellipsis required, and turning the Thred on the other side of the Compasses, you may with your Pen in the like manner describe the other half of the Ellipsis GBHL.
CHAP. III.
Of Triangles.
HItherto we have spoken of the most Simple Figure, a Circle. Come we now to those Figures that are Various or Angular.
2. And an Angular Figure is that which doth consist of three or more Angles.
3. An angular Figure consisting of three Angles, otherwise called a Triangle, is a Superficies or Figure comprehended by three Right Lines including three Angles.
4. A Triangle may be considered either in respect of its Sides, or of its Angles.
5. A Triangle in respect of its Sides, is either Isopleuron, Isosceles, or Scalenum.
6. An Isopleuron Triangle, is that which hath three equal sides. An Isoscecles hath two equal Sides. And a Scalenum hath all the three Sides unequal.
7. A Triangle in respect of its Angles is Right or Oblique.
[Page 13] 8. A Right angled Triangle is that which hath one Right Angle and two Acute.
9. An Oblique angled Triangle, is either Acute or Obtuse.
10. An Oblique acute angled Triangle, is that which hath all the three Angles Acute.
11. An Oblique obtuse angled Triangle, is that which hath one Angle Obtuse, and the other two Acute.
Proposition I.
Vpon a Right Line given to make an Isopleuron or an Equilateral Triangle.
In Fig. 8. let it be required to make an Equilateral Triangle upon the Right Line AB. Open your Compasses to the extent of the Line given, and setting one Foot of your Compasses in A, make an Arch of a Circle above or beneath the Line given, then setting one Foot of your Compasses in B, they being full opened to the same extent, with the other foot draw another Arch of a Circle crossing the former, and from the Intersection of those Arches draw the Lines AC and AB, so shall the Triangle ACB be Equilateral as was desired.
Proposition II.
Vpon a Right Line given to make an Isosceles Triangle, or a Triangle having two Sides equal.
In Fig. 8. let AB be the Right Line given, from the Points A and B as from two Centers, but at a lesser extent of the Compasses than AB; [Page 14] if you would have AB the greatest Side, at a greater extent; if you would have it to be the least Side, describe two Arches cutting one another, as at F, and from the Intersection draw the Lines AF, and FB, so shall the Triangle AFB have two equal Sides, as was required.
Proposition 3.
To make a Scalenum Triangle, or a Triangle, whose three Sides are unequal.
In Fig. 9. let the three unequal Sides be EFG make AB equal to one of the given Lines, suppose G, and from A as a Center, at the extent of E describe the Arch of a Circle; in like manner from B at the extent of F describe another Arch intersecting the former, then shall the Right Lines AC. CB and BA comprehend a Triangle, whose three sides shall be unequal, as was required.
CHAP. IV.
Of Quadrangular and Multangular Figures.
WE have spoken of Triangles or Figures consisting of three Angles, come we now to those that have more Angles than three, as the Quadrangle, Quinquangle, Sexangle, &c.
2. A Quadrangle is a Figure or Superficies, which is bounded with four Right Lines.
3. A Quadrangle is either a Parallelogram or a Trapezium.
4. A Parallelogram is a Quadrangle whose opposite [Page 15] Sides are parallel having equal distances from one another in all Places.
5. A Parallelogram is either Right angled or Oblique.
6. A Right angled Parallelogram, is a Quadrangle whose four Angles are all Right, and is either Square or Oblong.
7. A Square Parallelogram doth consist of four equal Lines. The Parts of a Square are, the Sides of which the Square is made, and the Diagonal or Line drawn from one opposite Angle to another through the middle of the Square.
8. An Oblong is a Right angled Parallelogram, having two longer and two shorter Sides.
9. An Oblique angled Parallelogram, is that whose Angles are all Oblique, and is either a Rhombus or a Rhomboides.
10. A Rhombus is an Oblique angled and equilateral Parallelogram.
11. A Rhomboides is an Oblique angled and inequilateral Parallelogram.
12. A Trapezium is a Quadrangular Figure whose Sides are not all parallel; it is either Right angled or Oblique.
13. A Right angled Trapezium hath two opposite Sides parallel, but unequal, and the Side between them perpendicular.
14. An Oblique angled Trapezium is a Quadrangle, but not a Parallelogram, having at least two Angles Oblique, and none of the Sides parallel.
15. Thus much concerning Quadrangles or four sided Figures. Figures consisting of more than four Angles are almost infinite, but are reducible unto two sorts, Ordinate and Regular, or Inordinate and Irregular.
[Page 16] 16. Ordinate and Regular Polygons are such, as are contained by equal Sides and Angles, as the Pentagon, Hexagon, and such like.
17. Inordinate or irregular Polygons, are such as are contained by unequal Sides and Angles. The construction of these Quadrangular and Multangular Figures is explained in the Propositions following.
Proposition. I.
Vpon a Right Line given to describe a Right angled Parallelogram, whether Square or Oblong.
In Fig. 10. let the given Line be AB, upon the Point A erect the Perpendic [...]lar AD equal to AB if you intend to make a Square, but longer or shorter, if you intend an oblong, and upon the Points D and B at the distance of AB and AD describe two Arches intersecting one another, and from the Intersection draw the Lines ED and EB, so shall the Right angled Figure AE be a Square, if AB and AD be equal, otherwise an Oblong, as was desired.
Proposition II.
To describe a Rhombus or Rhomboides.
In Fig. 11. To the Right Line AB draw the Line AD at any Acute Angle at pleasure, equal to AB if you intend a Rhombus, longer or shorter if you intend a Rhomboides, then upon your Compasses to the extent of AD and upon B as a Center describe an Arch; in like manner, at the extent of AB upon D as a Center describe another [Page 17] Arch intersecting the former, then draw the Lines ED and EB, so shall AE be the Rhombus or Rhomboides, as was required.
Proposition III.
Vpon a Right Line given to make a Regular Pentagon, or five sided Figure.
In Fig. 12. Let the given Line be AB, upon A and B as two Centers describe the Circles EBGH and CAGK, then open your Compasses to the extent of BC, and making G the Center, describe the Arch HAFK, then draw the Lines KFE and HFC: so shall AE and BC be two sides of the Pentagon desired, and opening your Compasses to the extent of AB, upon E and C as two Centers describe two Arches intersecting one another, and from the Point of Intersection draw the Lines ED and DC, so shall the Figure AB and DE be the Pentagon required.
Proposition IV.
To make a Regular Pentagon and Decagon in a given Circle.
In Fig. 13. upon the Diameter CAB describe the Circle CDBL, from the Center AErect the Perpendicular AD, and let the Semidiameter AC be bisected, the Point of Bisection is E, set the distance ED from E to G, and draw the Line GD, which is the side of a Pentagon, and AG the side of a Decagon inscribed in the same Circle.
Proposition V.
In a Circle given to describe a Regular Hexagon.
The side of a Hexagon is equal to the Radius of a Circle, the Radius of a Circle therefore being six times applied to the Circumference, will give you six Points, to which Lines being drawn from Point to Point, will constitute a Regular Hexagon, as was desired.
Proposition VI.
In a Circle given to describe a Regular Heptagon or Figure consisting of seven equal sides.
The side of a Heptagon is equal to half the side of a Triangle inscribed in a Circle, having therefore drawn an Hexagon in a Circle, the Chord Line subtending two sides of the Hexagon lying together, is the side of a Triangle inscribed in that Circle, and half that Chord applied seven times to the Circumference, will give seven Points, to which Lines being drawn from that Point, will constitute a Regular Heptagon, as in Fig. 14. is plainly shewed.
CHAP. V.
Of Solid Bodies.
HAving spoken of the two first kinds of Magnitude, Lines and Superficies, come we now to the third, a Body or Solid.
2. A Body or Solid is a Magnitude consisting of length, breadth and thickness.
3. A Solid is either regular or irregular.
4. That is called a regular Solid, whose Bases, Sides and Angles are equal and like.
5. And this either Simple or Compound.
6. A simple regular Solid, is that whith doth consist of one only kind of Superficies.
7. And this is either a Sphere or Globe, or a plain Body.
8. A Globe is a Solid included by one round and convex Superficies, in the middle whereof there is a Point, from whence all Lines drawn to the Circumference are equal.
9. A simple plain Solid, is that which doth consist of plain Superficies.
10. A plain Solid is either a Pyramid, a Prism, or a mixt Solid.
11. A Pyramid is a Solid, Figure or Body, contained by several Plains set upon one right lin'd Base, and meeting in one Point.
12. Of all the several sorts of Pyramids, there is but one that is Regular, to wit a Tetrahedron, or a Pyramid consisting of four regular or equilateral Triangles; the form whereof (as it may be cut in Pastboard) may be conceived by Figure 15.
[Page 20] 13. A Prism is a Solid contained by several Plains, of which those two which are opposite, are equal, like and parallel, and all others are Paralellogram.
14. A Prism is either a Pentahedron, a Hexahedron, or a Polyhedron.
15. A Pentahedron Prism, is a Solid comprehended of five Sides, and the Base a Triangle, as Fig. 16.
16. An Hexahedron Prism, is a Solid comprehended of six Sides, and the Base a Quadrangle, as Fig. 17.
17. An Hexahedron Prism, is distinguished into a Parallelipipedon and a Trapezium.
18. An Hexahedron Prism called a Trapezium is a Solid, whose opposites Plains or Sides, are neither opposite nor equal.
19. A Parallelipipedon is either right angled or oblique.
20. A right angled Parallelipipedon is an Hexahedron Prism, comprehended of right angled Plains or Sides; and it is either a Cube or an Oblong.
21. A Cube is a right angled Parallelipipedon comprehended of six equal Plains or Sides.
22. An Oblong Parallelipipedon, is an Hexahedron Prism, comprehended by unequal Plains or Sides.
23. An Oblique angled Parallelipipedon, is an Hexahedron Prism, comprehended of Oblique Sides.
24. A Polyhedron Prism, is a Solid comprehended by more than six Sides, and hath a multangled Base, as a Quincangle, Sexangle, &c.
25. A regular compound or mixt Solid, is such [Page 21] a Solid as hath its Vertex in the Center, and the several Sides exposed to view, and of this sort there are only three; the Octohedron, the Icosahedron, of both which the Base is a Triangle; and the Dodecahedron, whose Base is a Quincangle.
26. An Octohedron is a Solid Figure which is contained by eight equal and equilateral Triangles, as in Fig. 18.
27. An Icosahedron is a Solid, which is contained by twenty equal and equilateral Triangles, as Fig. 19.
28. A Dodecahedron is a Solid, which is contained by twelve equal Pentagons, equilateral and equiangled, as in Fig. 20.
29. A regular compound Solid, is such a Solid as is Comprehended both by plain and circular Superficies, and this is either a Cone or a Cylinder.
30. A Cone is a Pyramidical Body, whose Base is a Circle, or it may be called a round Pyramis, as Fig. 21.
31. A Cylinder is a round Column every where comprehended by equal Circles, as Fig. 22.
32. Irregular Solids are such, which come not within these defined varieties, as Ovals, Frustums of Cones, Pyramids, and such like.
And thus much concerning the description of the several sorts of continued Quantity, Lines, Plains and Solids; we will in the next place consider the wayes and means by which the Dimentions of them may be taken and determined, and first we will shew the measuring of Lines.
CHAP. VI.
Of the Measuring of Lines both Right and Circular.
EVery Magnitude must be measured by some known kind of Measure; as Lines by Lines, Superficies by Superficies, and Solids by Solids, as if I were to measure the breadth of a River, or height of a Turret, this must be done by a Right Line, which being applied to the breadth or height desired to be measured, shall shew the Perches, Feet or Inches, or by some other known measure the breadth or height desired: but if the quantity of some Field or Meadow, or any other Plain be desired, the number of square Perches must be enquired; and lastly, in measuring of Solids, we must use the Cube of the measure used, that we discover the number of those Cubes that are contained in the Body or Solid to be measured. First, therefore we will speak of the several kinds of measure, and the making of such Instruments, by which the quantity of any Magnitude may be known.
2. Now for the measuring of Lines and Superficies, the Measures in use with us, are Inches, Feet, Yards, Ells and Perches.
3. An Inch is three Barley Corns in length, and is either divided into halves and quarters, which is amongst Artificers most usual, or into ten equal Parts, which is in measuring the most useful way of Division.
4. A Foot containeth twelve Inches in length, and is commonly so divided; but as for such things as are to be measured by the Foot, it is far [Page 23] better for use, when divided into ten equal Parts, and each tenth into ten more.
5. A Yard containeth three Foot, and is commonly divided into halves and quarters, the which for the measuring of such things as are usually sold in Shops doth well enough, but in the measuring of any Superficies, it were much better to be divided into 10 or 100 equal Parts.
6. An Ell containeth three Foot nine Inches, aud is usually divided into halves and quarters, and needs not be otherwise divided, because we have no use for this Measure, but in Shop Commodities.
7. A Pole or Perch cotaineth five Yards and an half, and hath been commonly divided into Feet and half Feet. Forty Poles in length do make one Furlong, and eight Furlongs in length do make an English Mile, and for these kinds of of lengths, a Chain containing four Pole, divided by Links of a Foot long, or a Chain of fifty Foot, or what other length you please, is well enough, but in the measuring of Land, in which the number of square Perches is required; the Chain called Mr. Gunters, being four Pole in length divided into 100 Links, is not without just reason reputed the most useful.
8. The making of these several Measures is not difficult, a Foot may be made, by repeating an Inch upon a Ruler twelve times, a Yard is eight Foot, and so of the rest; the Subdivision of a Foot or Inch into halves and quarters, may be performed by the seventeenth of the first, and into ten or any other Parts by the first Proposition of the first Chapter, and all Scales of equal Parts, of what scantling you do desire. And this I [Page 24] think is as much as needs to be said concerning the dividing of such Instruments as are useful in the measuring Right Lines.
9. The next thing to be considered is the measuring of Circular Lines, or Perfect Circles.
10. And every Circle is supposed to be divided into 360 Parts called Degrees, every Degree into 60 Minutes, every Minute into 60 Seconds, and so forward this division of the Circle into 360 Parts is generally retained, but the Subdivision of those Parts, some would have be thus and 100, but as to our present purpose either may be used, most Instruments not exceeding the fourth part of a Degree.
11. Now then a Circle may be divided into 360 Parts in this manner, Having drawn a Diameter through the Center of the Circle dividing the Circle into two equal Parts, cross that Diameter with another at Right Angles through the Center of the Circle also, so shall the Circle be divided into four equal Parts or Quadrants, each Quadrant containing 90 Degrees, as in Fig. 7. GE. ED. DL and LG, are each of them 90 Degrees; and the Radius of a Circle being equal to the Chord of the sixth Part thereof, that is to the Chord of 60 Degrees, as in Fig. 14. if you set the Radius GB from L towards G, and also from G towards L, the Quadrant GL will be subdivided into three equal Parts, each Part containing 30 Degrees, GM. 30. MH 30 and HL 30, the like may be done in the other Quadrants also; so will the whole Circle be divided into twelve Parts, each Part containing 30 Degrees.
And because the side of a Pentagon inscribed in a Circle is equal to the Chord of 72 Degrees, or [Page 25] the first Part of 360, as in Fig. 13. therefore if you set the Chord of the first Part of the Circle given from G to L or L to G, in Fig. 7. you will have the Chord of 72 Degrees, and the difference between GP 72 and GH 60 is HP 12, which being bisected, will give the Arch of 6 Degrees, and the half of six will give three, and so the Circle will be divided into 120 Parts, each Part containing three Degrees, to which the Chord Line being divided into three Parts, the Arch by those equal Divisions may be also divided, and so the whole Circle will be divided into 360, as was desired.
12. A Circle being thus divided into 360 Parts, the Lines of Chords, Sines, Tangents and Secants, are so easily made (if what hath been said of them in the Second Chapter be but considered) that I think it needless to say any more concerning their Construction, but shall rather proceed unto their Use.
13. And the use of these Lines and other Lines of equal Parts we will now shew in circular and right lined Figures; and first in the measuring of a Circle and Circular Figures.
CHAP. VII.
Of the Measuring of a Circle.
THe squaring of a Circle, or the finding of a Square exactly equal to a Circle given, is that which many have endeavoured, but none as yet have attained: Yet Archimedes that Famous Mathematician hath sufficiently proved, That the Area of a Circle is equal to a Rectangle made of the Rodius and half the Circumference: Or thus, The Area of a Circle is equal to a Rectangle made of the Diameter and the fourth part of the Circumference. For Example, let the Diameter of a Circle be 14 and the Circumference 44; if you multiply half the Circumference 22 by 7 half the Diameter, the Product is 154; or if you multiply 11 the fourth part of the Circumference, by 14 the whole Diameter, the Product will still be 154. And hence the Superficies of any Circle may be found though not exactly, yet near enough for any use.
2. But Ludolphus Van Culen finds the Circumference of a Circle whose Diameter is 1.00 to be 3.14159 the half whereof 1.57095 being multiplied by half the Diameter 50, &c. the Product is 7.85395 which is the Area of that Circle, and from these given Numbers, the Area, Circumference and Diameter of any other Circle may be found by the Proportions in the Propositions following.
Proposition I.
The Diameter of a Circle being given to find the Circumference.
As 1. to 3.14159: so is the Diameter to the Circumference. Example. In Fig. 13. Let the Diameter IB be 13. 25. I say as 1. to 3. 14159. so IB. 13.25 to 41.626 the Circumference of that Circle.
Proposition II.
The Diameter of a Circle being given to find the Superficial Content.
As 1. to 78539; so is the Square of the Diameter given, to the Superficial Content required. Example, Let the Diameter given be as before IB 13.25 the Square thereof is 175.5625 therefore.
As 1. to 78539: so 175.5625 to 137.88 the Superficial Content of that Circle.
Proposition III.
The Circumference of a Circle being given, to find the Diameter.
This is but the Converse of the first Proposition: Therefore as 3.14159 is to 1: so is the Circumference to the Diameter; and making the Circumference an Unite, it is. 3. 14159. 1∷ 1. 318308, and so an Unite may be brought into the first place. Example, Let the given Circumference [Page 28] be 41. 626. I say,
As 1. to 318308: so 41.626 to 13. 25. the Diameter required.
Proposition IV.
The Circumference of a Circle being given to find the Superficial Content.
As the Square of the Circumference of a Circle given is to the Superficial Content of that Circle: so is the Square of the Circumference of another Circle given to the Superficial Content required. Example, As the Square of 3.14159 is to 7853938: so is 1. the Square of another Circle to 079578 the Superficial Content required, and so an Unite for the most easie working may be brought into the first place: Thus the given Circumference being 41. 626. I say,
As 1. to 0.79578: so is the Square of 41.626 to 137.88 the Superficial Content required.
Proposition V.
The Superficial Content of a Circle being given, to find the Diameter.
This is the Converse of the second Proposition, therefore as 78539 is to 1. so is the Superficial Content given, to the Square of the Diameter required. And to bring an Unite in the first place: I say.
As 7853978. 1∷1. 1. 27324, and therefore if the Superficial Content given be 137.88, to find the Diameter: I say,
[Page 29] As 1. to 1.27324: so 137.88 to 175.5625 whose Square Root is 13.25, the Diameter sought.
Proposition VI.
The Superficial Content of a Circle being given, to find the Circumference.
This is the Converse of the Fourth Proposition, and therefore as 079578 is to 1 : so is the Superficial Content given, to the Square of the Circumference required, and to bring an Unite in the first place: I say,
As 079578. 1 :: 1. 12.5664, and therefore if the Superficial Content given be 137.88, to find that Circumference: I say,
As 1. to 12.5664: so is the 137.88 to 1732.7 whose Square Root is 626 the Circumference.
Proposition VII.
The Diameter of a Circle being given to find the Side of the Square, which may be inscribed within the same Circle.
The Chord or Subtense of the Fourth Part of a Circle, whose Diameter is an Unite, is 7071067, and therefore, as 1. to 7071067: so is the Diameter of another Circle, to the Side required. Example, let the Diameter given be 13.25 to find the side of a Square which may be inscribed in that Circle: I say,
As 1. to 7071067: so is 13.25 to 9.3691 the side required.
Proposition VIII.
The Circumference of a Circle being given, to find the Side of the Square which may be inscribed in the same Circle.
As the Circumference of a Circle whose Diameter is an Unite, is to the side inscribed in that Circle; so is the Circumference of any other Circle, to the side of the Square that may be inscribed therein. Therefore an Unite being made the Circumference of a Circle.
As 3.14159 to 7071067: so 1. to 225078.
And therefore the Circumference of a Circle being as before 41.626, to find the side of the Square that may be inscribed: I say,
As 1. to 225078. so is 41.626 to 9.3691 the side inquired.
Proposition IX.
The Axis of a Sphere or Globe being given, to find the Superficial Content.
As the Square of the Diameter of a Circle, which is Unity, is to 3.14159 the Superficial Content, so is the Square of any other Axis given, to the Superficial Content required. Example, Let 13.25 be the Diameter given, to find the Content of such a Globe: I say,
As 1. to 3.14159: so is the Square of 13.25 to 551.54 the Superficial Content required.
Proposition X.
To find the Area of an Ellipsis.
As the Square of the Diameter of a Circle, is to the Superficial Content of that Circle; so is the Rectangle made of the Conjugate Diameters in an Ellipsis, to the Area of that Ellipsis; And the Diameter of a Circle being one, the Area is 7853975, therefore in Fig. 26. the Diameters AC8 and BD5 being given, the Area of the Ellipsis ABCD may thus be found.
As 1. to 7853975: so is the Rectangle AC in BD40 to 3.1415900, the Area of the Ellipsis required.
CHAP. VIII.
Of the Measuring of Plain Triangles.
HAving shewed the measuring of a Circle, and Ellipsis, we come now to Right lined Figures, as the Triangle, Quadrangle, and Multangled Figures, and first of the measuring of the plain Triangles.
2. And the measuring of Plain Triangles is either in the measuring of the Sides and Angles, or of their Area and Superficial Content.
3. Plain Triangles in respect of their Sides and Angles are to be measured by two sorts of Lines, the one is a Line of equal Parts, and by that the Sides must be measured, the other is a Line of Chords, the Construction whereof hath [Page 32] been shewed in the sixth Chapter, and by that the Angles must be measured, the Angles may indeed be measured by the Lines of Sines, Tangents or Secants, but the Line of Chords being not only sufficient, but most ready, it shall suffice to shew how any Angle may be protracted by a Line of Chords, or the Quantity of any Angle found, which is protracted.
4. And first to protract or lay down an Angle to the Quantity or Number of Degrees proposed, do thus, draw a Line at pleasure as AD in Figure 5, then open your Compasses to the Number of 60 Degrees in the Line of Chords, and setting one Foot in A, with the other describe the Arch BG, and from the Point A let it be required to make an Angle of 36 Degrees: open your Compasses to that extent in the Line of Chords, and setting one Foot in B, with the other make a mark at G, and draw the Line AG, so shall the Angle BAG contain 36 Degrees, as was required.
5. If the Quantity of an Angle were required, as suppose the Angle BAG, open your Compasses in the Line of Chords to the extent of 60 Degrees, and setting one Foot in A, with the other draw the Arch BG, then take in your Compasses the distance of BG, and apply that extent to the Line of Chords, and it will shew the Number of Degrees contained in that Angle, which in our Example is 36 Degrees.
6. In every Plain Triangle, the three Angles are equal to two right or 180 Degrees, therefore one Angle being given, the sum of the other two is also given, and two Angles being given, the third is given also.
[Page 33] 7. Plain Triangles are either Right Angled or Oblique.
8. In a Right Angled Plain Triangle, one of the Acute Angles is the Complement of the other to a Quadrant or 90 Degrees.
9. In Right Angled Plain Triangles, the Side subtending the Right Angle we call the Hypotenuse, and the other two Sides the Legs, thus in Fig. 5▪ AE is the Hypotenuse, and AD and ED are the Legs; these things premised, the several cases in Right Angled and Oblique Angled Plain Triangles may be resolved, by the Propositions following.
Proposition I.
In a Right Angled Plain Triangle, the Angles of one Leg being given, to find the Hypotenuse and the other Leg.
In the Right Angled Plain Triangle ADE in Fig. 5. Let the given Angles be DAE 36, and DEA 54, and let the given Leg be AD 476; to find the Hypotenuse AE, and the other Leg ED.
Draw a Line at pleasure, as AD, and by your Scale of equal Parts set from A to D 476 the Quantity of the Leg given, then erect a Perpendicular upon the Point D, and upon the Point A lay down your given Angle DAE 36 by the fourth hereof, and draw the Line AE till it cut the Perpendicular DE, then measure the Lines AE and DE upon your Scale of Equal Parts, so shall AE 588.3 be the Hypotenuse, and DE 345.8 the other Leg.
Proposition II.
The Hypotenuse and Oblique Angles given, to find the Legs.
Let the given Hypotenuse be 588, and one of the Angles 36 degrees, the other will then be 54 degrees, Draw a Line at pleasure, as AD, and upon the Point A by the fourth▪ hereof lay down one of the given Angles suppose the less, and draw the Line AC, and from your Scale of equal Parts, set off your Hypotenuse 588 from A to E, and from the Point E to the Line AD let fall the Perpendicular ED, then shall AD being measured upon the Scale be 476 for one Leg, and ED 345.8 the other.
Proposition III.
The Hypotenuse and one Leg given to find the Angles and the other Leg.
Let the given Hypotenuse be 588. and the given Leg 476. Draw a Line at pleasure as AD, upon which set the given Leg from A to D. 476, and upon the Point D, erect the Perpendicular DE, then open your Compasses in the Scale of Equal Parts to the Extent of your given Hypotenuse 588, and setting one Foot of that Extent in A, move the other till it touch the Perpendicular DE, then and there draw AE, so shall ED be 345.8 the Leg inquired, and the Angle DAE, will be found by the Line of Chords to be 36▪ whose Complement is the Angle DEA. 54.
Proposition IV.
The Legs given to find the Hypotenuse, and the Oblique Angles.
Let one of the given Legs be 476, and the other 345.8, Draw the Line AD to the extent of 476, and upon the Point D, erect the Perpendicular DE to the extent of 345.8, and draw the Line AE, so shall AE be the Hypotenuse 588, and the Angle DAE will by the Line of Chords be found to be 36 Degrees, and the Angle DEA 54, as before.
Hitherto we have spoken of Right angled plain Triangles: the Propositions following concern such as are Oblique.
Proposition V.
Two Angles in an Oblique angled plain Triangle, being given, with any one of the three Sides, to find the other two Sides.
In any Oblique angled plain Triangle, let one of the given Angles be 26.50 and the other 38. and let the given Side be 632, the Sum of the two given Angles being deducted from a Semi-circle, leaveth for the third Angle 115.50 Degrees, then draw the Line BC 632. and upon the Points B and C protract the given Angles, and draw the Lines BD and CD, which being measured upon your Scale of equal Parts BD will be fou [...]d to be 312.43, and BD 431.09,
Proposition VI.
Two Sides in an Oblique Angled Triangle being given, with an Angle opposite to one of them, to find the other Angles and the third Side, if it be known whether the Angle Opposite to the other Side given be Acute or Obtuse.
In an Oblique Angled Plain Triangle, let the given Angle be 38 Degrees, and let the Side adjacent to that Angle be 632, and the Side opposite 431. 1. upon the Line BC in Fig. 25. protract the given Angle 38 Degrees upon the Point C, and draw the Line DC, then open your Compasses to the Extent of the other Side given 431. 1. and setting one Foot in B, turn the other about till it touch the Line DC, which will be in two places, in the Points D and E; if therefore the Angle at B be Acute the third Side of the Triangle will he CE, according therefore to the Species of that Angle you must draw the Line BD or BE to compleat the Triangle, and then you may measure the other Angles, and the third Side as hath been shewed.
Proposition VII.
Two Sides of an Oblique Angled Plain Triangle being given, with the Angle comprehended by them to find the other Angles and the third Side.
Let one of the given Sides be 632, and the other 431.1, and let the Angle comprehended by them be Deg. 26.50, draw a Line at pleasure, [Page 37] as BC, and by help of your Scale of Equal Parts, set off one of your given Sides from B to C 632. then upon the Point B protract the given Angle 26. 50. and draw the Line BD, and from B to D, set off your other given Side 431. 1. and draw the Line DC, so have you constituted the Triangle BDC, in which you may measure the Angles and the third Side, as hath been shewed.
Proposition VIII.
The three Sides of an Oblique Angled Triangle being given, to find the Angles.
Let the length of one of the given Sides be 632, the length of another 431.1, and the length of the third 312.4, and Draw a Line at pleasure, as BC in Fig. 25, and by help of your Scale of Equal Parts, set off the greatest Side given 632 from B to C. then open your Compasses in the same Scale to the extent of either of the other Sides, and setting one Foot of your Compasses in B, with the other describe an occult Arch, then extend your Compasses in the same Scale to the length of the third Side, and setting one Foot in C with the other describe another Arch cutting the former, and from the Point of Intersection draw the Lines BD and DC. to constitute the Triangle BDC, whose Angles may be measured, as hath heen shewed.
And thus may all the Cases of Plain Triangles be resolved by Scale and Compass, he that desires to resolve them Arithmetically, by my Trigometria Britannica, or my little Geometrical, Trigonometry; only one Case of Right Angled Plain [Page 38] Triangles which I shall have occasion to use, in the finding of the Area of the Segment of a Circle I will here shew how, to resolve by Numbers.
Proposition IX.
In a Right Angled Plain Angle the Hypotenuse and one Leg being given to find the other Leg.
Take the Sums and difference of the Hypotenuse and Leg given, then multiply the Sum by the Difference, and of the Product extract the Square Root, which Square Root shall be the Leg inquired.
Example. In Fig. 5. Let the given Hypotenuse be AE 588.3, and the given Leg AD 476, and let DE be the Leg inquired. The Sum of AE and AD is 1064.3, and their Difference is 112.3, now then if you multiply 1064.3 by 112.3, the Product will be 119520.89, whose Square Root is the Leg DE. 345. 8.
Proposition X.
The Legs of a Right Angled Plain Triangle being gived, to find the Area or Superficial Content thereof.
Multiply one Leg by the other, half the Product shall be the Content. Example, In the Right angled plain Triangle ADE, let the given Legs be AD 476, and DE 345, and let the Area of that Triangle be required, if you multiply 476 by 345 the Product will be 164220, and the half thereof 82110 is the Area or Superficial Content required.
Proposition XI.
The Sides of an Oblique angled plain Triangle being given to find the Area or Superficial Content thereof.
Add the three Sides together, and from the half Sum subtract each Side, and note their Difference; then multiply the half Sum by the said Differences continually, the Square Root of the last Product, shall be the Content required.
Example. In Fig. 9. Let the Sides of the Triangle ABC be AB 20. AC 13, and BC 11 the Sum of these three Sides is 44, the half Sum is 22, from whence subtracting AB 20, the Difference is 2, from whence also if you substract AC 13, the Difference is 9, and lastly, if you subtract BC 11 from the half Sum 22, the Difference will be 11. And the half Sum 22 being multiplied by the first Difference 2, the Product is 44, and 44 being multiplied by the Second Difference 9, the Product is 396, and 396 being multiplied by the third Difference 11, the Product is 4356, whose Square Root 66, is the Content required.
Or thus, from the Angle C let fall the Perpendicular DC, so is the Oblique angled Triangle ABC, turned into two right, now then if you measure DC upon your Scale of Equal Parts, the length thereof will be found to be 6.6, by which if you multiply the Base AB 20, the Product will be 132.0, whose half 66, is the Area of the Triangle, as before.
Proposition XII.
The Sides of any Oblique angled Quadrangle being given, to find the Area or Superficial Content thereof.
Let the Sides of the Oblique angled Quadrangle ABED in Fig. 11. be given, draw the Diagonal AE, and also the Perpendiculars DC and BF, then measuring AE upon the same Scale by which the Quadrangular Figure was protracted, suppose you find the length to be 632, the length of DC 112, and the length of BF 136, if you multiply AE 632 by the Half of DC 56, the Product will be 35392 the Area of ACED. In like manner if you multiply AE 632, by the half of BF 68, the Product will be 42976 the Area of ACEB, and the Sum of these two Products is the Area of ABED as was required.
Or thus, take the Sum of DC 112, and BF 136; the which is 248, and multiply AE 632 by half that Sum, that is by 124, the Product will be 78368 the Area of the Quadrangular Figure ABED, as before.
Proposition XIII.
The Sides of a plain irregular multangled Figure being given, to find the Content.
In Fig. 26. Let the Sides of the multangled Figure. A. B. C. D. E. F. G. H. be given, and let the Area thereof be required, by Diagonals drawn from the opposite Angles reduce the Figure given, [Page 41] into Oblique angled plain Triangles, and those Oblique angled Triangles, into right by letting fall of Perpendiculars, then measure the Diagonals and Perpendiculars by the same Scale, by which the Figure it self was protracted, the Content of those Triangles being computed, as hath been shewed, shall be AF the Content required: thus by the Diagonals AG. BE and EC the multangled Figure propounded is converted into three Oblique angled quadrangular Figures, AFGH. AFEB and BEDC, and each of these are divided into four Right angled Triangles, whose several Contents may be thus computed. Let GA 94 be multiplied by half HL 27 more Half of KF 29, that is by 23, the Product will be 21, be the Area of AHGF. Secondly, OB is 11, and FN 13, their half Sum 12, by which if you multiply AE 132, the Product will be 1584 the Area of AFEB. Thirdly, let Bp be 18 m D 32, the half Sum is 25, by which if you multiply AEC 125 the Product will be 3125 the Area of BEDC, and the Sum of these Products is 6871 the Area of the whole irregular Figure. ABCDEFGH, as was required.
Proposition XIV.
The Number of Degrees in the Sector of a Circle being given, to find the Area thereof.
In Fig. 27. ADEG is the Sector of a Circle, in which the Arch DEG, is Degrees. 23.50, and by 1. Prop. of Archimed. de Dimensione Circuli, the length of half the Arch is equal to the Area of the Sector of the double Arch, there the length [Page 42] of DE or EG is equal to the Area of the Sector ADEG: and the length or circumference of the whole Circle whose Diameter is 1 according to Van Culen, is 3.14159265358979, therefore the length of one Centesme of a Degree, is. 0. 01745329259. Now then to find the length of any Number of Degrees and Decimal Parts, you must multiply the aforesaid length of one Centesme by the Degrees and Parts given, and the Product shall be the length of those Degrees and Parts required, and the Area of a Sector containing twice those Degrees and Parts. Example, the half of DEG 23.50 is DE or EG 11.75, by which if you multiply 0.01745329259, the Product will be 2050761879325, the length of the Arch DE, and the Area of the Sector ADEG.
Proposition XV.
The Number of Degrees in the Segment of a Circle being given, to find the Area of the Segment.
In Fig. 27. Let the Area of the Segment DEGK be required, in which let the Arch DEG be Degrees 23.50, then is the Area of the Sector ADEG 2050761879325 by the last aforegoing, from which if you deduct the Area of the Triangle ADG, the remainer will be the Area of the Segment DEGK. And the Area of the Triangle ADG may thus be found. DK is the Sine of DE 11.75, which being sought in Gellibrand's Decimal Canon is. 2036417511, and AK is the Sine of DH 78.25, or the Cosine of DE. 9790454724, which being multiplied by the Sine of DE, the Product will be 1993745344, or if you multiply AG [Page]
[Page] [Page 43] the Radius by half DF the Sine of the double Arch DEG, the Product will be 19937453445 as before, and this Product being deducted from the Area of the Sector ADEG 2050761879325, the remainer will be 57016434875 the Area of the Segment DEGL, as was desired.
Proposition XVI.
The Diameter of a Circle being cut into any Number of Equal Parts, to find the Area of any Segment made by the Chord Line drawn at Right Angles through any of those equal Parts of the Diameter.
In Fig. 28. The Radius AD is cut into five Equal Parts, and the Segment EDFL is made by the Chord Line ELF at Right Angles to AD in the fourth Equal Part, or at eight tenths thereof: now then to find the Area of this Segment we have given AE Radius, and AL 8, and therefore by the ninth hereof EL will be found to be 606000, the Sine of ED 36.87, by which if you multiply 0.0174532, the Product is the Area of the Sector AEDF 64350286, and the Area of the Triangle AEF is 48, which being deducted from the Area of the Sector, the Remainer 16350286 is the Area of the Sector EDFL, as was required. And in this manner was that Table of Segments made by the Chord Lines cutting the Radius into 100 Equal Parts.
Another way.
In Fig. 28. Let the Radius AD be cut into 10.100 or 1000 Equal Parts, and let the Area of [Page 44] the Segments made by the Chord Lines drawn at Right Angles through all those Parts be required: first find the Ordinates GK and M. PN. EL, the double of each Ordinate, will be the Chords of the several Arches, and the Sum of these Chords beginning with the least Ordinate, will orderly give you the Area of the several Segments made by those Chord Lines, but the Diameter must be be divided into 100000 Equal Parts, because of the unequal differences at the beginning of the Diameter: but taking the Area of the Circle to be 3. 1415926535, &c. as before, the Area of the Semicircle will be 1.57079632, from which if you deduct the Chord GH1999999, the Chord answering to 999 Parts of the Radius, the remainer is. 1.56879632 the Area of the Segment GDH. And in this manner by a continual deduction of the Chord Lines from the Area of the Segment of the Circle given, was made that Table shewing the Area of the Segments of a Circle to the thousandth part of the Radius.
And because a Table shewing the Area of the Segments of a Circle to the thousandth part of the Radius, whose whole Area is Unity, is yet more useful in Common Practice, therefore from this Table, was that Table also made by this Proportion.
As the Area of the Circle whose Diameter is. Unity, to wit 3.14149 is to the Area of any part of that Diameter, so is Unity the supposed Area of another Circle, to the like part of that Diameter. Example, the Area answering to 665 parts of the Radius of a Circlewhose Area is 3.14159 is 0.91354794 therefore,
As 3.14159265 is to 0.91354794: So is [...]. [Page]
[Page 45] to 290791, the Area required; and the Table being thus computed to the 1000 parts of the Radius, we have enlarged it by the difference to the 5000 parts of the Radius, and consequently to the ten thousandth part of the Diameter: The use of which Table shall be shewed when we come to the measuring of Solid Bodies.
CHAP. IX.
Of the Measuring of Heights and Distances.
HAving shewed in the former Chapter, how all plain Triangles may be measured, not only in respect of their Sides and Angles, but in respect of their Area, and the finding of the Area of all other plain Figures also, that which is next to be considered, is the practical use of those Instructions, in the measuring of Board, Glass, Wainscot, Pavement, and such like, as also the measuring or surveying of Land; and first we will shew the measuring of Heights and Distances.
2. And in the measuring of Heights and Distances, besides a Chain of 50 or 100 Links, each Link being a Foot, it is necessary to have a Quadrant of four or five Inches Radius, and the larger the Quadrant is, the more exactly may the Angles: be taken, though for ordinary Practice, four or five Inches Radius will be sufficient.
Let such a Quadrant therefore be divided in the Limb into 90 Equal Parts or Degrees, and numbred from the left hand to the right, at every tenth Degree, in this manner 10. 20. 30. 40. 50. 60. 70. 80. 90. and within the Limb of the Quadrant [Page 46] draw another Arch, which being divided by help of the Limb into two Equal Parts, in the Point of Interfection set the Figure 1. representing the Radius or Tangent of 45 Degrees, and from thence both ways the Tangents of 63.44 Deg. 71.57 Deg. 75.97 Deg.78.70 Deg. 80.54 Deg. that is, 2. 3. 4. 5 and 6 being set also, your Quadrant will be fitted for the taking of Heights several ways, as shall be explained in the Propositions following.
Proposition I.
To find the Height of a Tower, Tree, or other Object at one Station.
At any convenient distance from the Foot of the Object to be measured, as suppose at C in Fig. 30. and there looking through the Sights of your Quadrant till you espie the top of the Object at A, observe what Degrees in the Limb are cut by the Thread, those Degrees from the left Side or Edge of the Quadrant to the Right, is the Quantity of the Angle ACB, which suppose 35 Degrees; then is the Angle BAC 55 Degrees, being the Complement of the former to 90 Degrees. This done with your Chain or otherwise measure the distance from B the Foot of the Object, to your Station at C, which suppose to be 125 Foot. Then as hath been shewed in the 1. Prop. Chap. 8. draw a Line at pleasure as BC, and by your Scale of Equal Parts, set off the distance measured from B to C 125 Foot, and upon the Point C lay down your Angle taken by observation 35 Degrees, then erect a Perpendicular upon [Page 47] the Point B, and let it be extended till it cut the Hypothenusal Line AC, so shall AB measured on your Scale of Equal Parts, be 87.5 Foot for the Height of the Object above the Eye; to which the Height of the Eye from the Ground being added, their Sum is the Height required.
Another way.
Let AB represent a Tower whose Altitude you would take, go so far back from it, that looking through the Sights of your Quadrant, to the top of the Tower at A the Thread may cut just 45 Degrees in the Limb, then shall the distance from the Foot of the Tower, to your Station, be the Height of the Tower above the Eye.
Or if you remove your Station nearer and nearer to the Object, till your Thread hang over the Figures 2. 3. 4 or 5 in the Quadrant, the Height of the Tower at 2. will be twice as much as the distance from the Tower to the Station, at 3. it will be thrice as much, &c. As if removing my Station from C to D, the Thread should hang over 2 in the Quadrant, and the distance BD 62 Foot, then will 124 Foot be the Height of the Tower, above the Eye.
In like manner if you remove your Station backward till your Thread fall upon one of those Figures in the Quadrant; between 45 and 90 Degrees, the distance between the Foot of the Tower, and your Station will at 2. be twice as much as the Height, at 3. thrice as much, at 4. four times so much, and so of the rest.
[Page 48] A Third way by a Station at Random.
Take any Station at pleasure suppose at C, and looking through the Sights of your Quadrant, observe what Parts of the Quadrant the Thread falls upon, and then measure the distance between the Station, and the Foot of the Object, that distance being multiplied by the parts cut in the Quadrant, cutting off two Figures from the Product shall be the Height of the Object above the Eye?
Example, Suppose I standing at C, that the Thread hangs upon 36 Degrees, as also upon 72 in the Quadrant which is the Tangent of the said Arch, and let the measured distance be CB 125 Foot, which being multiplied by 72, the Product is 9000, from which cutting off his Figures because the Radius is supposed to be 100, the Height inquired will be 90 Foot, he that desires to perform this work with more exactness, must make use of the Table of Sines and Tangents Natural or Artificial, this we think sufficient for our present purpose.
Proposition II.
To find an inaccessible Height at two Stations.
Take any Station at pleasure as at D, and there looking through the Sights of your Quadrant to the top of the Object, observe what Degrees are cut by the Thread in the Limb, which admit to be 68 Degrees, then remove backward, till the Angle taken by the Quadrant, be but half so much [Page 49] as the former, that is 34 Degrees, then is the distance between your two Stations equal to the Hypothenusal Line at your first Station, viz. AD. if the distance between your two Stations were 326 foot, then draw a Line at pleasure as BD, upon the Point D protract, the Angle ADB 68 Degrees, according to your first Observation, and from your Line of equal parts set off the Hypothenusal 326 Foot from D to A, and from the Point A let fall the Perpendicular AB which being measured in your Scale of Equal Parts, shall be the Altitude of the Object inquired.
Or working by the Table of Sines and Tangents, the Proportion is.
As the Radius, is to the measured distance or Hypothenusal Line AD; so is the Sine of the Angle ADE, to the height AB inquired.
Another more General way, by any two Stations taken at pleasure.
Admit the first Station to be as before at D, and the Angle by observation to be 68 Degrees, and from thence at pleasure I remove to C, where observing aim I find the Angle at C to be 32 Degrees, and the distance between the Stations 150 Foot. Draw a Line at pleasure as BC, and upon Clay down your last observed Angle 32 Degrees, and by help of your Scale of Equal Parrs, set off your measured distance from C to D 150 Foot, then upon D lay down your Angle of 68 Degrees, according to your first Observation, and where the Lines AD and AC meet, let fall the Perpendicular AB, which being measured in your Scale of Equal Parts, shall be the height of the Object as before.
[Page 50] Or working by the Tables of Sines and Tangents, the Proportions.
1. As the Sine of DAC to the Distance DC. So the Sine of ACD, to the Side AD.
2. As the Radius, to the Side AD; so the Sine ADB, to the Perpendicular height AB inquired.
The taking of Distances is much after the same manner, but because there is required either some alteration in the sights of your Quadrant or some other kind of Instrument for the taking of Angles, we will particularly shew, how that may be also done several ways, in the next Chapter.
CHAP. X.
Of the taking of Distances.
FOr the taking of Distances some make use of a Semicircle, others of a whole Circle, with Ruler and Sights rather than a Quadrant, and although the matter is not much by which of these Instruments the Angles be taken, yet in all Cases the whole Circle is somewhat more ready, than either a Semicircle or Quadrant, the which with its Furniture is called the Theodolite.
2. A piece of Board or Brass then about twelve or fourteen Inches Diameter, being made Circular like a round Trencher, must be divided into four Quadrants, and each Quadrant divided into 90 Degrees, or the whole Circle into 360, and each Degree into as many other Equal Parts, as the largeness of the Degrees will well permit: let your Circle be numbred both ways to 360, that is from the right hand to the left, and from the left to the right.
[Page 51] 3. Upon the backside of the Circle there must be a Socket made fast, that it may be set upon a three legged Staff, to bear it up in the Field.
4. You must also have a Ruler with Sights fixed at each end, for making of Observation, either fixed upon the Center of your Circle, or loose, as you shall think best; your Instrument being thus made, any distance whether accessible or inaccessible may thus be taken.
5. When you are in the Field, and see any Church, Tower, or other Object, whose Distance from you, you desire to know, choose out some other Station in the same Field, from whence you may also see the Object, and measure the distance between your Stations; then setting your Ruler upon the Diameter of your Circle, set your Instrument so, as that by the Sights on your Ruler, you may look to the other Station, this done turn your Ruler to that Object whose distance you desire to know, and observe how many Degrees of the Circle are cut by the Ruler, as suppose 36 Degrees, as the Angle ACD in Fig. 30. Then removing your Instrument to D, lay the Ruler on the Diameter thereof, and then turn the whole Instrument about till through your Sights you can espy the mark set up at your first Station at C, and there fix your Instrument, and then upon the Centre of your Circle turn your Ruler till through the Sights you can espy the Object whose distance is inquired, suppose at A; and observe the Degrees in the Circle cut by the Ruler, which let be 112, which is the Angle ADC, and let the distance between your two Stations be DC 326 Foot; so have you two Angles and the side between them, in a plain Triangle given, by which [Page 52] to find the other sides, the which by protraction may be done as hath been shewed, in the fifth Proposition of Chapter 8. but by the Table of Sines and Tangents, the Proportion is.
As the Sine of DAC, is to DC; so is the Sine of ACD to the Side AD.
Or, as the Sine of DAC, is to the given Side DC.
So is the Sine of ADC to the Side AC.
6. There is another Instrument called the plain Table, which is nothing else, but a piece of Board, in the fashion and bigness of an ordinary sheet of paper, with a little frame, to fasten a sheet of paper upon it, which being also set upon a Staff, you may by help of your Ruler, take a distance therewith in this manner.
Having measured the distance between your two Stations at D and C, draw upon your paper a Line, on which having set off your distance place your Instrument at your first Station C, and laying your Ruler upon the Line so drawn thereon, turn your Instrument till through the Sights you can espy the Station at D, then laying your Ruler upon the Point C, turn the same about till through the Sights you can espy the Object at A, and there draw a Line by the side of your Ruler, and remove your Instrument to D, and laying your Ruler upon the Line DC, turn the Instrument about, till through the Sight you can espy the Mark at C, and then laying your Ruler upon the Point D, turn the same, till through the Sights you can espy the Object at A, and by the side of your Ruler draw a Line, which must be extended till it meet with the Line AC, so shall the Line AD being measured upon your Scale of Equal [Page 53] Parts, be the distance of the Object from D, and the Line AC shall be the distance thereof from C.
7. And in this manner may the distance of two, three or more Objects be taken, from any two Stations from whence the several Objects may be seen, and that either by the plain Table, or Theodolite.
CHAP. XI.
How to take the Plot of a Field at one Station, from whence the several Angles may be seen.
ALthough there are several Instruments by which the Plat of a Field may be taken, yet do I think it sufficient to shew the use of these two, the plain Table and Theodolite.
2. In the use of either of which the same chain which is used in taking of heights and distances, is not so proper. I rather commend that which is known by the Name of Gunter's Chain, which is four Pole divided into 100 Links; being as I conceive much better for the casting up the Content of a Piece of Ground, than any other Chain that I have yet heard of, whose easie use shall be explained in its proper place.
3. When you are therefore entered the Field with your Instrument, whether plain Table, or Theodolite, having chosen out your Station, let visible Marks be set up in all the Corners thereof, and then if you use the plain Table, make a mark upon your paper, representing your Station, and laying your Ruler to this Point, direct [Page 54] your Sights to the several Corners of the Field, where you have caused Marks to be set up, and draw Lines by the side of the Ruler upon the paper to the point representing your station, then measure the distance of every of these Marks from your Instrument, and by your Scale set those distances upon the Lines drawn upon the paper, making small marks at the end of every such distance, Lines drawn from Point to Point, shall give you upon your paper, the Plot of the Field, by which Plot so taken the content of the Field may easily be computed.
Example. Let Fig. 31. represent a Field whose Plot is required; your Table being placed with a sheet of paper thereupon, make a Mark about the middle of your Table, as at A. apply your Ruler from this Mark to B and draw the Line AB, then with your Chain measure the distance thereof which suppose to be 11 Chains 36 Links, then take 11 Chains 36 Links from your Scale, and set that distance from A to B, and at B make a mark.
Then directing the Sights to C, draw a Line by the side of your Ruler as before, and measure the distance AC, which suppose to be 7 Chains and 44 Links, this distance must be taken from your Scale, and set from A to C upon your paper.
And in this manner you must direct your Sights from Mark to Mark, until you have drawn the Lines and set down the distances, between all the Angles in the Field and your station, which being done, you must draw the Lines from one Point to another, till you conclude where you first began, so will those Lines BC. CD. DE. FG. and GB, give you the exact Figure of the Field.
4. To do this by the Theodolite, in stead of [Page 55] drawing Lines upon your paper in the Field, you must have a little Book, in which the Pages must be divided into five Columns, in the first Column whereof you must set several Letters to signifie the several Angles in the Field, from which Lines are to be drawn to your place of standing, in the second and third Columns the degrees and parts taken by your Instrument, and the fourth and fifth, to set down your distances Chains and Links, this being in readiness, and have placed your Instrument direct your Sights to the first mark at B, and observe how many Degrees are comprehended between the Diameter of your Instrument, and the Ruler, and set them in the second and third Columns of your Book against the Letter B, which stands for your first Mark, then measure the distance AB as before, and set that down, in the fourth and fifth Columns, and so proceed from Mark to Mark, until you have taken all the Angles and Distances in the Field, which suppose to be, as they are expressed in the following Table.
| Degr. | Part | Chains | Links | |
| B | 39 | 75 | 11 | 56 |
| C | 40 | 75 | 7 | 44 |
| D | 96 | 00 | 7 | 48 |
| E | 43 | 25 | 8 | 92 |
| F | 80 | 00 | 6 | 08 |
| G | 59 | 25 | 9 | 73 |
5. Having thus taken the Angles and Distances in the Field, to protract the same on Paper or [Page 56] Parchment, cannot be difficult; for if you draw a Line at pleasure as EB representing the Diameter of your Instrument about the middle thereof, as at A, mark a Mark, and opening your Compasses to 60 Degrees in your Line of Chords, upon A as a Center describe a Circle, then lay your Field book before you seeing that your first Observattion cut no Degrees, there are no Degrees to be marked out in the Circle, but the Degrees at C are 40.75 which being taken from your Line of Chords, you must set them from H to I, and draw the Line AI. the Degrees at D are 96 which must in like manner be set from I to K, and so the rest in order.
This done observe by your Field-book the length of every Line, as the Line AB at your first Observation was 11 Chains and 36 Links, which being by your Scale set from A will give the Point B in the Paper, the second distance being set upon AI will give the Point C, and so proceeding with the rest, you will have the Points BCDEF and G, by which draw the Lines BC. CD. DE; EF. FG and GB, and so at last you have the Figure of the Field upon your Paper, as was required.
And what is here done at one station, may be done at two or more, by measuring one or two distances from your first station, taking at every station, the Degrees and distances to as many Angles, as are visible at each station.
And as for taking the the Plot of a-Field by Intersection of Lines, he that doth but consider how the distances of several Objects may be taken at two stations, will be able to do the other also, and therefore I think it needless, to make any illustration by example.
CHAP. XII.
How to take the Plot of a Wood, Park or other Champion Plain, by going round the same, and making Observation at every Angle.
BY these Directions which have been already given, may the Plot of any Field or Fields be taken, when the Angles may be seen alone or more stations within the Field, which though it is the case of some Grounds, it is not the case of all; now where observation of the Angles cannot be observed within, they must be observed without, and although this may be done by the plain Table, yet as I judge it may be more conveniently done by the Theodolite, in these cases thereof I chiefly commend that Instrument, I know some use a Mariners Compass, but the working with a Needle is not only troublesom, but many times uncertain, yet if a Needle be joyned with the Theodolite the joynt Observations of the Angles may serve to confirm one another.
2. Suppose the Fig. 32. to be a large Wood whose Plot you desire to take; Having placed your Instrument at the Angle A, lay your Ruler on the Diameter thereof, turning the whole Instrument till through the Sights you espy the Angle at K, then fasten it there, and turn your Ruler upon the Center, till through the Sights you espy your second Mark at B, the Degrees cut by the Ruler do give the quantity of that Angle BAK, suppose 125 Degrees, and the Line AB 6 Chains, 45 Links, which you must note in your Field-book, as was shewed before.
[Page 58] 3. Then remove your Instrument to B, and laying your Ruler upon the Diameter thereof, turn it about, till through the Sights you can espy your third mark at C, and there fasten your Instrument, then turn the Ruler backward till through the Sights you see the Angle at A, the Degrees cut by the Ruler being 106.25 the quantity of the Angle ABC, and the Line BC containing 8 Chains and 30 Links, which note in your Field-book, as before.
4. Remove your Instrument unto C, and laying the Ruler on the Diameter thereof, turn the Instrument about till through the Sights you see the Angle at D, and fixing of it there, turn the Ruler upon the Center till you see your last station at B, and observe the Degrees cut thereby, which suppose to be 134 Degrees, and the Line CD 6 Chains 65 Links, which must be entered into your Field-book also, and because the Angle BCD is an inward Angle, note it with the Mark [...] for your better remembrance.
5. Remove your Instrument unto D, and Iaying the Ruler on the Diameter, turn the Instrument about, till through the Sights, you see the Angle at E, and there fixing your Instrument, turn your Ruler backward till you espy the Mark at C, where the Degrees cut are, suppose 68.0 and the Line DE 8 Chains and 23 Links.
6. Remove your Instrument unto E, and laying the Ruler on the Diameter, turn the Instrument about, till through the Sights you see the Angle at F, and there fix it, then turn the Ruler backward till you see the Angle at D, where the Degrees cut by the Ruler suppose to be 125 and the Line EF 7 Chains and 45 Links.
[Page 59] 7. Remove your Instrument unto F, and laying your Ruler upon the Diameter, turn the Instrument about, till through the Sights, you see the Angle at G, where fix the same, and turn the Ruler backward till you see the Angle at E, where the Degrees cut by the Ruler are 70, and the Line FG 4 Chains 15 Links, which must be set down with this [...] or the like Mark at the Angle.
8. Remove your Instrument unto G, and laying your Ruler upon the Diameter, turn the Instrument about, till through the Sights you see the Angle at H, where fix the same, and turn the Ruler backward till you see the Angle at F, where the Degrees cut by the Ruler are 65.25, and the Line GH 5 Chains 50 Links.
9. Remove your Instrument in like manner to H and K, and take thereby the Angles and Distances as before, and having thus made observation at every Angle in the Field, set them down in your Field-book, as was before directed, the which in our present Example will be as followeth.
| A | 151.00 | 6.45 |
| B | 106.25 | 8.30 |
| C [...] | 134.00 | 6.65 |
| D | 68.00 | 8.23 |
| E | 125.00 | 7.45 |
| F [...] | 70.25 | 4.15 |
| G | 65.25 | 5.50 |
| H | 130.00 | 6.50 |
| K | 140.00 | 11.00 |
The taking of the inward Angles BCD and EFG was more for Conformity sake than any [Page 60] necessity, you might have removed your Instrument from B to D, from E to G, the Length of the Lines BC. CD. EF and G, would have given by protraction the Plot of the Field without taking these Angles by observation; many other compendious ways of working there are, which I shall leave to the discretion of the Ingenious Practitioner.
10. The Angles and Sides of the Field being thus taken, to lay down the same upon Paper, Parchment, another Instrument called a Protractor is convenient, the which is so well known to Instrument-makers, that I shall not need here to describe it, the chief use is to lay down Angles, and is much more ready for that purpose than a Line of Chords, though in effect it be the same.
11. Having then this Instrument in a readiness draw upon your Paper or Parchment upon which you mean to lay down the Plot of that Field, a Line at pleasure as AB. Then place the Center of your Protractor upon the Point A, and because the Angle of your first observation at A was 115 Degrees 00 Parts, turn your Protractor about till the Line AK lie directly under the 115 Degree; and then at the beginning of your Protractor make a Mark, ând draw the Line AB, setting off 6 Chains 45 Links from A to B.
12. Then lay the Center of your Protractor upon the Point B, and here turn your Protractor about, till the line AB lie under 106 Degrees 25 Parts, and draw the Line BC, setting off the Distance 8 Chains, 30 Links from B to C.
13. Then lay the Center of your Protractor upon the Point C, and turn the same about till the Line BC lie under 134 Degrees, but remember [Page 61] to make it an inward Angle, as it is marked in your Field-Book, and there make a Mark, and draw the Line CD, setting off 6 Chains, 65 Links from C to D.
And thus must you do with the rest of the Sides and Angles, till you come to protract your last Angle at H, which being laid down according to the former Directions the Line HK will cut the Line AK making AK 11 Chains and HK 6 Chains, 50 Links. This work may be also performed by protracting your last observation first; for having drawn the Line AK, you may lay the Center of your Protractor upon the Point K, and the Diameter upon the Line AK; and because your Angle at K by observation was 140 Degrees, you must make a Mark by the Side of your Protractor at 140 Degrees; and draw the Line KH, setting off 6 Chains, 50 Links from K to H. And thus proceeding with the rest of the Lines and Angles, you shall find the Plot of your Field at last to close at A, as before it did at K.
CHAP. XIII.
The Plot of the Field being taken by any Instrument, how to compute the Content thereof in Acres, Roods, and Perches.
THe measuring of many sided plain Figures hath been already shewed in the 13 Proposition of the 8 Chapter, which being but well considered, to compute the Content of a Field cannot be difficult; It must be remembred indeed that 40 square Pearches do make an Acre.
[Page 62] 2. Now then if the Plot be taken by a four Pole Chain divided into 100 Links, as 16 square Poles are the tenth part of an Acre; so 10.000 square Links of such a Chain are equal to 16 square Pole, or Perches; and by consequence 100.000 square Links are equal to an Acre, or the square Pearches.
3. Having then converted your Plot into Triangles, you must cast up the Content of each Triangle as hath been shewed, and then add the several Contents into one Sum, and from the aggregate cut off five Figures towards the right hand; the remainer of the Figures towards the left hand are Acres, and the five Figures so cut off towards the right hand are parts of an Acre, which being multiplied by four, if you cut off five Figures from the Product, the Figures remaining towards the left hand are Roods, and the five Figures cut off are the parts of a Rood, which being multiplied by forty, if you cut off five Figures from the Product, the Figures remaining towards the left hand are Perches, and the Figures cut off are the Parts of a Pearch.
Example. Let 258.94726 be the Sum of several Triangles, or the Content of a Field ready cast up, the three Figures towards the left hand 258 are the Acres, and the other Figures towards the right hand 94726 are the Decimal Parts of an Acre, which being multiplied by 4, the Product is 3.78904, that is three Roods and 78904 Decimal Parts of a Rood, which being multiplied by 40, the product is 31.56160, that is 31 Perches and 56160 Decimal Parts of a Perch; and therefore in such a Field there are Acres 258, Roods 3, Pearches 31, and 56160 Decimal Parts of a Perch.
CHAP. XIV.
How to take the Plot of Mountainous and uneven Grounds, and how to find the Content.
VVHen you are to take the Plot of any Mountainous or uneven piece of Ground, such as is that in Figure 33, you must first place your Instrument at A, and direct your Sights to B, measuring the Line AB, observing the Angle GAB, as was shewed before, and so proceed from B to C, and because there is an ascent from C to D, you must measure the true length thereof with your Chain, and set that down in your Book, but your Plot must he drawn according to the length of the Horizontal Line, which must be taken by computing the Base of a right angled Plain Triangle, as hath been shewed before, and so proceed from Angle to Angle until you have gone round the Field, and having drawn the Figure thereof upon your Paper, reduce into Triangles and Trapezias, as ABC. CDE. ACEF and AFG. then from the Angles B. C. D. F and G; let fall the Perpendiculars, BK. CN. DL. FM. and GH. This done you must measure the Field again from Angle to Angle, setting down the Distance taken in a straight Line over Hill and Dale, and so likewise the several Perpendiculars, which will be much longer than the streight Lines measured on your Scale, and by these Lines thus measured with your Chain cast up the Content; which will be much more than the Horizontal Content of that Field according to the Plot, but if it should be otherwise plotted than by the Horizontal [Page 64] Lines, the Figure thereof could not be contained within its proper limits, but being laid down among other Grounds, would force some of them out of their places, and therefore such Fields as these must be shadowed off with Hills, if it be but to shew that the Content thereof is computed according to the true length of the Lines from Corner to Corner, and not according to their Distance measured by Scale in the Plot.
CHAP. XV.
How to reduce Statute Measure into Customary, and the contrary.
VVHereas an Acre of Ground by Statute Measure is to contain 160 square Perches, measured by the Pole or Perch of sixteen foot and a half: In many places of this Nation, the Pole or Perch doth by custom contain 18 foot, in some 20. 24. 28 Foot; it will be therefore required to give the Content of a Field according to such several quantities of the Pole or Perch.
2. To do this you must consider how many square Feet there is in a Pole according to these several Quantities.
- In 16.5 to the Pole, there are 272.25 sq. feet.
- In 18 to the Pole there are 324 square feet.
- In 20 to the Pole there are 400 square feet.
- In 24 to the Pole there are 576 square feet.
- In 28 to the Pole there are 784 square feet.
Now then if it were desired to reduce 7 Acres, 3 Roods, 27 Perches, according to Statute Measure, into Perches of 18 Foot to the Perch; first reduce [Page 65] your given quantity, 7 Acres. 3 Rods, 27 Poles into Perches, and they make 1267 Perches.
Then say, as 324. to 272. 25. so is 1267 to 1065. 6. that is 1065 Perches, and 6 tenths of a Perch. But to reduce customary Measure into statute measure, say as 272. 25. is to 324 so is 1267 Perches in customary measure, to 1507. 8 that is 1507 Perches and 8 tenths of a Perch in statute measure, the like may be done, with the customary measures of 20.24 and 28 or any other measure that shall be propounded.
CHAP. XVI.
Of the Measuring of solid Bodies.
HAving shewed how the content of all plains may be computed, we are now come to the measuring of solid Bodies, as Prisms, Pyramids and Spheres, the which shall be explained in the Propositions following.
Proposition. I.
The base of a Prism or Cylinder being given, to find the solid content.
The base of a Prism is either Triangular, as the Pentahedron; Quadrangular, as the Hexahedron, or Multangular, or the Polyhedron Prism, all which must be computed as hath been shewed, which done if you multiply the base given by the altitude, the product shall be the solid content required.
Example. In an Hexahedron Prism, whose base [Page 66] is quadrangular, one side of the Base being 65 foot and the other 43, the Superficies or Base will be 27. 95. Which being multiplyed by the Altitude, suppose 12. 5. the product. 359. 375. is the solid content required.
In like manner the Base of a Cylinder being 45. 6. and the altitude 15. 4. the content will be 702. 24.
And in this manner may Timber be measured whether round or squared, be the sides of the squared Timber equal or unequal.
Example. Let the Diameter of a round piece of Timber be 2. 75 foot. Then, As 1 it to 785397. so is the square of the Diameter 2. 75. to 5.9395 the Superficial content of that Circle.
Or if the circumference had been given 8. 64. then, As 1 is to 079578, so is the square of 8. 64. to 5.9404 the superficial content.
Now then if you multiply this Base 5. 94. by the length, suppose 21 foot, the content will be 124. 74.
If the side of a piece of Timber perfectly square be 1.15 this side being multiplyed by it self, the product will be 1.3225 the superficial content, or content of the Base, which being multiplyed by 21 the length, the content will be 27. 7745.
Or if a piece of Timber were in breadth 1. 15. in depth 1.5 the content of the Base would be 1.725 which being multiplied by 21 the length, the content will be. 36. 225.
Proposition. II.
The Base and Altitude of a Pyramid or Cone being given, to find the solid content.
[Page 67] Multiply the Altitude by a third part of the Base, or the whole Base by a third part of the Altitude, the Product shall be the solid content required.
Example. In a Pyramid having a Quadrangular Base as in Fig. 22. The side CF 17. CD 9. 5. the Product is the Base CDEF. 161. 5, which being multiplyed by 10.5 the third of the Altitude AB 31.5 the Product is 1695.75 the content. Or the third of the Base. viz. 53. & 3 being multiplied by the whole Altitude AB 31.5 the Product will be the content as before.
2. Example. In Fig. 21. Let there be given the Diameter of the Cone AB 3. 5. The Base will be 96. 25. whose Altitude let be CD 16.92 the third part thereof is 5.64 & 96.25 being multiplied by 5.64, the Product 542.85 is the solid content required.
Proposition. III.
The Axis of a Sphere being given, to find the solid content.
If you multiply the Cube of the Axis given by 523598 the solid content of a Sphere whose Axis is an unite, the Product shall be the solid content required.
Example. Let the Axis given be [...], the Cube thereof is 27, by which if you multiply. 523598, the Product 14.137166 is the solid content required.
Proposition. IV.
The Basis and Altitude of the Frustum of a Pyramid or Cone being given, to find the content.
If the aggregate of both the Bases of the Frustan and the mean proportional betwe [...]n them, shall be multiplied by the third part of the Altitude, the Product shall be the solid content of the Frustum.
Example. In Fig. 22. Let CDEF represent the greater Base of a Pyramid, whose superficial content let be 1. 92, and let the lesser Base be HGLKO. 85 the mean proportional between them is. 1. 2775 and the aggregate of these three numbers is. 4. 0475. Let the given Altitude be 15. the third part thereof is. 5 by which if you multiply 4.0475 the Product 20. 2375 is the content of the Frustum Pyramid.
And to find the content of the Frustum Cone. I say.
As. 1. ro 78539. so 20.23 to 15. 884397, the content of the Cone required.
But if the Bases of the Frustum Pyramid shall be square, you may find the content in this manner.
Multiply each Diameter by it self and by one another, and the aggregate of these Products, by the third part of the altitude, the last Product shall be the content of the Frustum Pyramid.
Example. Let the Diameter of the greater Base be 144, the Diameter of the lesser Base 108, and the altitude 60.
- The Square of 144 is
- 20736
- The Square of 108 is
- 11664
- The Product of 1444108 is
- 15552
- The Sum of these 3 Products is
- 47952
Which being multiplyed by 20 the third part of the Altitude, the Product 959040 is the content of the Frustum Pyramid.
And this content being multiplied by .785 39 the content of the Frustum Cone will be .753 .228.
Another way.
Find the content of the whole Pyramid of the greater and lesser Diameter, the lesser content deducted from the greater, the remain shall be the content of the Frustum. To find the content of the whole Pyramid, you must first find their several Altitudes in this manner.
As the difference between the Diameters,
Is to the lesser Diameter.
So is the Altitude given, to the Altitude cut off.
Example. The difference between the former Diameter. 144. and 108 is 36, the Altitude 60. now then As 36. 108∷60. 108. the altitude cut off.
Now then if you mnltiply the lesser Base 1 1664 by 60 the third part of 180 the Product 699840 is the content of that Pyramid.
And adding 60 to 180 the Altitude of the greater Pyramid is 240, the third part whereof is 80, by which if you multiply the greater Base before found, 70736, the Product is the content of the [Page 70] greater Pyramid. 1658880, from which if you deduct the lesser 699840 the remainer 959040 is the content of the Frustum Pyramid as before.
And upon these grounds may the content of Taper Timber, whether round or square, and of Brewers Tuns, whether Circular or Elliptical, be computed, as by the following Propositions shall be explained.
Proposition. V.
The breadth and depth of a Taper piece of Squared Timber, both ends being given together with the length, to find the content.
- At the Bottom be A. 5.75 and B 2.34
- At the Top. C. 2.16 and D. 1.83.
- And let the given length be 24 Foot.
According to the last Proposition, find the Area or Superficial content of the Tree at both ends thus.
| Multiply the breadth | 3.75 | 0.574031 |
| By the depth | 2.34 | 0.369215 |
| The Product | 8.7750 | 0.943246 |
| 2. Multiply the breadth | 2.16 | 0.334453 |
| By the depth | 1.82 | 0.262451 |
| The Product is | 3.9528 | 0.596904 |
| [Page 71] 3. Multiply the 1. Content. | 87750 | 0.943246 |
| by the second content. | 3.9528 | 0.596904 |
| And find the square root | 5.8986 | 1.540150 |
| 0.770075 |
The Sum of these 18.6264 being multiplyed by 8 one third of the length, the content will be found to be 149. 0112. Thus by the Table of Logarithms the mean proportional between the two Bases is easily found, and without extracting the square Root, may by natural Arithmetick be found thus.
A 4 2/2 CX A half C multiplyed by B: And C more half A multiplyed by D being added together and multiplyed by 30, the length shall give the content. Example.
| A. 3.75 | C 2.16 |
| 1/2 C. 1.08 | 1/2 A 1.875 |
| Sum 4.83 | Sum. 4.035 |
| B- 2.34 | D. 1.83 |
| 1932 | 12105 |
| 1449 | 32280 |
| 966 | 4035 |
| 11.3022 | 7.38405 |
| 11.30220 | |
| The sum of the Products | 18.68625 |
Being multiplyed by 8 the third of the length, the content will be. 149. 49000. The like may be done for any other.
Proposition VI.
The Diameters of a piece of Timber being given at the Top and and Bottom, together with the length, to find the content.
The Proposition may be resolved either by the Squares of the Diameters, or by the Areas of the Circles answering to the Diameters given, for which purpose I have here annexed not only a Table of the Squares of all numbers under a thousand, but a Table sharing the third part of the Areas of Circles in full measure, to any Diameter given under 3 foot.
And therefore putting S = The Sum of the Tabular numbers answering to the Diameters at each end.
X = The difference between these Diameters.
L = the length of the Timber, C = The content.
Then 1 ½ S = ½ - XX. + L. = C.
If you work by the Table of the squares of Numbers. you must multiply the less side of the Equation, by 0.26179 the third part of 0.78539 the Product being multiplyed by the length, will give the content.
But if you work by the Table of the third parts of the Areas of Circles in full measure, the tabular Numbers being multiplyed by the length will give the content. Only instead of the square of the difference of the Diameter, you must take half the Tabular number answering to that Difference, and you shall have the content as before. Example.
- Let the greatest Diameter by 2.75, and the less 1. 93.
- Their difference is 0.83
- The square of 2.75 is
- 7.5625
- The square of 1.93 is
- 3.7249.
- The Sum of the Squares
- 11.2874
- The half Sum
- 5.6437
- The Sum of them is
- 16.9311
- Half the square of 0.82 deduct.
- 0.3362
- The Difference is
- 16.5949
- Which being multiplyed by
- 26179
- 1493541
- 1161643
- 165949
- 995694
- 331898
- The Product will be.
- 4.344378871
- The Area of 2.75 is
- 1.979857
- The Area of 1.93 is
- 0.975176
- The Sum
- 2.955033
- The half Sum
- 1.477516
- The Sum of them
- 4.432549
- Half the Area of 0.82 deduct
- 0.088016
- The former Product
- 4.344533
- Which being multiplyed by
- 24
- 17378132
- 8689066
- The content is
- 104268792
But because that in measuring of round Timber the circumference is usually given and not the Diameter, I have added another Table by which the circumference being given, the Diameter may be found.
Example. Let the circumference of a piece of Timber be 8325220 looking this Number in the second column of that Table, I find the next less to be 8.168140 and thence proceeding in a streight Line, I find that in the seventh Column the Number given, and the Diameter answering thereunto to be 2. 65. and thus may any other Diameter be found not exceeding the three foot. The Proportion by which the Table was made, is thus. As 1. to 3.14159 so is the Diameter given, to the circumference required.
[Page 75] Or the Circumference being given, to find the Diameter, say: As. 1. to 0.3183, so is the Circumference given to the Diameter required.
And although by these two Tables all round Timber may be easily measured, yet it being more usual to take the Circumference of a Tree, then the Diameter, I have here added a third Table, shewing the third part of the Areas of Circles answering to any circumference under 10 foot, and that in Natural and Artificial numbers, the use of which Table shall be explained in the Proposition following.
Proposition. VII.
The Circumference of a piece of round Timber at both ends, with the length being given, to find the content.
The Circumference of a Circle being given, the Area thereof may be found as hath been shewed, in the 7 Chapter, Proposition 4. and by the first Proposition of this; and to find the third part of the Area, which is more convenient for our purpose I took a third part of the number given by which to find the whole, that is a third part of 07957747 that is 0.02652582 and having by the multiplying this number by the square of the Circumference computed three or four of the first numbers, the rest were found by the first and second differences.
The Artificial numbers were computed by adding the Logarithms of the Squares of the circumference, to 8.42966891 the Logarithm of 0. 02652582.
And by these Natural and Artificial numbers [Page 76] the content of round Timber may be found two ways
By the Natural numbers in the same manner as the content was computed, the Diameters being given, and by the Natural and Artificial numbers both, by finding a mean proportional between the two Areas at the top and bottom of the Tree, as by Example shall be explained.
Let the given Diamensions, or Circumferences be At the Bottom 9.95 Their difference is 6.20 At the Top 3.75
| Natural | Artificial. | |
| Answering to 9.95 | 2.626162 | 0.418931 |
| And to 3.75 | 0.373019 | 9.571731 |
- The Sum of the Logarith.
- 9.990662
- The half Sum or Logarith. 989300
- 9.995331
- The Sum of the Number is
- 3.988481
- The Sum of the Natural Numbers is
- 2.9 [...]9181
- The half Sum
- [...].499190
- The Sum of them
- 4.498771
- Half the number answer. to. 6. 20 is
- 0.509826
- The remainer is
- 3.988945
Which being multiplyed by the length 24, the content will be 95. 73468.
Mr. Darling in his Carpenters Rule made easie, doth propound a shorter way, but not so exact, which is by the Circumference given in the middle of the piece to find the side of the Square, namely [Page 77] by multiplying the Circumference given by 28209, or 2821. which side of the Square being computed in Inches, and lookt in his Table of Timber measure, doth give the content of the Tree not exceeding 31 foot in length, the which way of measuring may be as easily performed by this Table. Example.
The circumference at the top and bottom of the Tree being given 9.95 and 3.75 the Sum is 13.70 The half thereof is the mean circumfer. 6.85 Which sought in the Table, the Numbers are.
The Natural number is 1.244657, which being multiplyed by 3 the Product is 3.733971, which multiplyed by the length 24, the content is 89. 615304.
- The Artificial number is
- 0.095049
- The Logarithm of 24 is
- 1.380211
- The Absolute Number 29.871
- 1.475260
- Which multiplyed by 3, the Product is
- 89613
Proposition. VIII.
The Diameters of a Brewers Tun at top and bottom being given with the height thereof, to find the content.
In Fig. 29. Let the given Diameter.
At the top be AC 136 BD 128
At the bottom. KG 152 HF 144 Altit. 51 Inches.
The which by the 5 Proposition of this Chap. may thus be computed. AC 139 + ½ KG 76 = 212 × BD 128 the Product is 27136.
[Page 78] And KG 1524 ½ AC 68 = 220 × HF 144 the Product is 31680. the Sum of these 2 Products is 58816 which being multiplyed by onethird of 51, that is by 17, and that Product multiplyed by 26179 the third of 78539 will give the content.
- The Logarithm of 58816. is
- 54.76949
- The Logarithm of 17 is
- 1.230449
- The Product
- 1.999944
- The Logarithm of. 26179
- 9.417968
- The content is. 261765
- 5.417912
Thus the content of a Tun may be found in Inches, which being divided 282 the number of Inches in an Ale Gallon, the quotient will be the content in Gallons.
Or thus; divide the former. 26179 by 282 the quotient will be 00092836. by which the content may be found in Ale Gallons in this manner.
- The former Product
- 5.999944
- The Logarithm of 0.00092836
- 6.967719
- The content in Gallons 928.24
- 2.967663
Proposition. IX.
The Diameters of a close Cask, at head and bung with the length given, to find the content.
In the resolving of this Proposition, we are to consider the several forms of Casks, as will as the kind of the Liquor, with which it is filled, for one and the same Rule will not find the content in all Cask.
[Page 79] And a Coopers Cask is commonly taken, either for the middle Frustum of a Spheroid, the middle Frustum of a Parabolical Spindle, the middle Frustum of two Parabolick Conoids, or for the middle Frustum of two Cones abutting upon one common Base.
And the content of these several Casks may be found either by equating the Diameters, or by equating the Circles. for the one, a Table of Squares is necessary, and a Table shewing the third part of the Areas of a Circle to all Diameters. The making of the Table of Squares, every one knows, to be nothing else but the Product of a Number multiplyed, by it self, thus the Square of 3 is 9. the Square of 8 is 64 and so of the rest.
And the Area of a Circle to any given Diameter may be found, as hath been shewed, in Chap. 7 Proposition 2. But here the Area of a Circle in Inches, will not suffice, it will be more fit for use, if the third part of the Area be found in Ale and Wine Gallons both, the which may indeed be done by dividing the whole Area in Inches by 3 and the quotient by 282 to make the Table for Ale-measure, and by 231 to make the Table for Wine-measure; but yet these Tables (as I think) may be more readily made in this manner.
The Square of any Diameter in Inches, being divided by 3.81972 will give the Area of the Circle in Inches: And this Division being multiplyed by 282 will give you 1077.161 for a common Division, by which to find the Area in Ale-Gallons, or being multiplyed by 231 the Product, 882.355 will be a commou Division by which to find the Area in Wine-Gallons.
But because it is easier to multiply then divide: [Page 80] If you multiply the several Squares by 26178 the third part of 78539 the Product will give the Area in Inches, or if you divide. 26179 by 282 the quotient will be. 00092886 for a common Multiplicator, by which to find the Area in Ale-Gallons, or being divided by 231 the quotient will be 0011333 a common Multiplicator, by which to find the content in Wine-Gallons. An Example or two will be sufficient for illustration. Let the Diameter given be 32 Inches, the Square thereof 1024 being divided by 3.81970 the quotient is 268.083, and the same Square 1024 being multiplyed by 261799, the Product will be 268. 082.
Again if you divide 1024 by 1077.161 the quotient will be 9508, or being multiplied by 00092836, the Product will be 9508.
Lastly if you divide 1024 by 882.755, the quotient will be 1.1605, or being multiplied by 00113333 the Product is 1.1605,
And in this manner may the Tables be made for Wine and Beer-measure, but the second differences in these Numbers being equal, three or four Numbers in each Table being thus computed, the rest may be found by Addition only.
Thus the Squares of 1. 2. 3. and 4 Inches are. 1. 4. 9 and 16 by which if you multiply 00113333, the several Products will be third part of the Area, of the Circles answering to those Diameters in Wine-Gallons. Or 00092836 being multiplied by those Squares, the several Products, will be the third part of the Areas of the Circles answering to those Diameters in Ale-Gallons; the which with their first and second differences are as followeth.
| 1. | 00113333 | ||
| 2. | 00453332 | 33999 | 226666 |
| 3. | 01019997 | 566665 | 226666 |
| 4. | 01813328 | 796331 |
| 1. | 00092836 | ||
| 2. | 00371344 | 278508 | 185672 |
| 3. | 00835524 | 464180 | 185672 |
| 4. | 01485376 | 649852 |
And by the continual addition of the second differences to the first, and the first differences to the products before found, the Table may be continued as far as you please.
The construction of the Tables being thus shewed: We will now shew their use in finding the content of any Cask.
Let S = the Sum of the Tabular Numbers answering to the Diameters at the Head and Bung. D = their difference X = the difference of the Diameters themselves. L = the length of the Vessel, and C = the content thereof.
1. If a Cask be taken for the middle Frustum of a Spheroid, intercepted between two Planes parallel, cutting the Axis at right Angles: Then 1 ½ S + ½ D × L = C.
2. If a Cask be taken for the middle Frustum of a parabolical Spindle, intercepted between two planes parallel cutting the Axis at right Angles. Then 1 ½ S + ½ D × L = C.
[Page 82] 3. If a Cask be taken for the middle Frustum of two Parabolick Conoids, abutting upon one common Base, intercepted between two Planes parallel, cutting the Axis at right Angle: Then 1 ½ S: × L = C.
4. If a Cask be taken for the middle Frustum of two Cones, abutting upon one common bafe, intercepted between two Planes parallel cutting the Axis at Right Angles. Then 1 ½ S—⅓ XX. × L = C.
In all these four Equations, if you work by the Table of Squares of numbers, you must multiply the less side of the Equation by 262, if you would have the content in Cubical Inches; by 001133 if you would have the content in Wine-Gallons; and by 000928, if you would have the content in Ale-Gallons.
But if you work by the Tables of the third parts of the Areas Circle, the Tabular Numbers being multiplyed by the length only will give the content required, only in the fourth Equation instead of half the Square of the Difference of the Diameters, take half the Tabular Number answering to that difference, and you shall have the content required; as by the following Examples will better appear, then by many words.
Examples in Wine-measure by the Table of the Squares of Numbers.
The Diameter of a Vessel
At the Bung being 32 Inches.
At the Head 22 Inches.
The difference of the Diameters 10 Inches.
And the length of the Vessel 44 Inches.
| Spheroid. | Parabolick Spindle. |
| 1024 | 1024 |
| 484 | 484 |
| 1508 | 1508 |
| 754 | 754 |
| 270 | 540 |
| 2532 | 23160 |
| 2532 | 23160 |
| 7596 | 69480 |
| 7596 | 69480 |
| 7596 | 69480 |
| 28695156 | 262472280 |
| 44 | 44 |
| 114780624 | 104988912 |
| 114780624 | 104988912 |
| 126.2586864 | 115.4878032 |
| Parabolick Conoid | Cone. |
| 1024 | 1024 |
| 484 | 484 |
| 1508 | 1508 |
| 754 | 754 |
| 50 | |
| 2262 | 2212 |
| 2262 | 2212 |
| 6786 | 6636 |
| 6786 | 6636 |
| 6786 | 6636 |
| 25635246 | 25068596 |
| 44 | 44 |
| 102540984 | 200274384 |
| 102540984 | 100274384 |
| 112.79508241 | 110.30182224 |
[Page 85] This which hath been done by the Table of Squares may be more easily performed, by the Table of the third part of the Areas of Circles, ready reduced to Wine-Gallons.
| Spheroid | Parabolick Spindle. |
| 1.16053 | 1.16053 |
| 0.54853 | 0.54853 |
| 1.70906 | 1.70906 |
| 85453 | 85453 |
| 30600 | 61200 |
| 2.86959 | 2.624790 |
| 44 | 44 |
| 1147836 | 1049916 |
| 1147836 | 10499160 |
| 126.26196 | 115.490760 |
| Parabolick Conoid | Cone. |
| 1.16053 | 1.16053 |
| 0.54853 | 0.54853 |
| 1.70906 | 1.70906 |
| 85453 | 85453 |
| 56666 | |
| 2.56359 | 2.506924 |
| 44 | 44 |
| 1025436 | 10027696 |
| 1025436 | 10027696 |
| 112.79796 | 110.304656 |
[Page 87] Examples in Ale-measure by the Table of the Squares of Numbers.
| Spheroid. | Parabolick Spindle. |
| 1024 | 1024 |
| 484 | 484 |
| 1508 | 1508 |
| 754 | 754 |
| 270 | 540 |
| 2532 | 2316.0 |
| 00092836 | 00092836 |
| 22758 | 20844 |
| 5064 | 4632 |
| 20256 | 18528 |
| 7596 | 6948 |
| 15192 | 138960 |
| 235660752 | 2.150081760 |
| 44 | 44 |
| 948623008 | 860032704 |
| 940643008 | 860032704 |
| 103.22673088 | 94.60359744 |
| Parabolick Conoid | Cone. |
| 1024 | 1024 |
| 484 | 484 |
| 1508 | 1508 |
| 754 | 754 |
| 50 | |
| 2262 | 2212 |
| 20358 | 19909 |
| 4524 | 4424 |
| 18096 | 17696 |
| 6786 | 6636 |
| 13527 | 13272 |
| 2.09995032 | 2.05423232 |
| 44 | 44 |
| 8.39980128 | 821692928 |
| 839980128 | 821692928 |
| 92.39781408 | 90.38622208 |
| Spheroid. | Parabolick Spindle. |
| 0.95052 | 0.95052 |
| 0.44930 | 0.44930 |
| 1.39982 | 1.39982 |
| .69991 | 69991 |
| .25061 | 050122 |
| 2.35034 | 2.149852 |
| 34 | 44 |
| 940136 | 8599408 |
| 940136 | 8599408 |
| 103.41496 | 94.593488 |
| Parabolick Conoid. | Cone. |
| 0.95052 | 0.95052 |
| 0.44930 | 0.44930 |
| 1.39982 | 1.39982 |
| 69991 | .69991 |
| 46425 | |
| 209973 | 2.053305 |
| 44 | 44 |
| 839892 | 8213220 |
| 839892 | 8213220 |
| 90.345420 | 90.345420 |
And here for the Singularity of the Example, I will set the Dimensions of a Cask lately made in Herefordshire, for that excellent Liquor of Red streak Cyder, the like whereof either for the largeness of the Cask, or incomparable goodness of that kind of Drink, is not to be found in all England, nay and perhaps not in the World.
The length of the Cask is 104 Inches.
The Diameter at the Bung 92 Inches.
And the Diameter at the Head 74 Inches.
[Page 91] The Numbers in the Table of Ale-Gallons answering to these Dimensions are.
| Spheroid | Parabolick Spindle. |
| Bung. 92 7.859639 | 7.859639 |
| Head. 74 5.083699 | 5.083699 |
| 12.941338 | 12.941338 |
| 6.470669 | 6.470669 |
| 1.386770 | .277394 |
| 20.798777 | 19.689401 |
| 104 | 104 |
| 83195108 | 78.757604 |
| 20798777 | 19689401 |
| Con. 2163.072808 | 2047.697704 |
| Parabolick Conoid. | Cone. |
| 7.857639 | 7.857639 |
| 5.083699 | 5.083699 |
| 12.941338 | 12.941338 |
| 6.470669 | 6.470669 |
| 0.150394 | |
| 19.412007 | 19.261613 |
| 104 | 104 |
| 77648028 | 77046452 |
| 19412007 | 19261613 |
| 201. 8. 848728 | 2003.207752 |
And thus you have the content of this Cask by four several Ways of Gauging, but that which doth best agree with the true content, found by these that filled the same is the second way or that which takes a Cask to be the middle Frustum of a Parabolick Spindle, according to which the content is 2047 Gallons. That is allowing 64 Gallons to the Hogshead. 32 Hogsheads very near.
Proposition. X.
If a Cask be not full, to find the quantity of Liquor contained in it, the Axis being posited parallel to the Horizon.
To resolve this Proposition, there must be given the whole content of the Cask, the Diameter at the Bung, and the wet Portion thereof, then by help of the Table of Segments, whose Area is unity, and the Diameter divided into 10.000 equal parts, the content may thus be found.
As the whole Diameter, is to its wet Portion.
So is the Diameter in the Table. 10.000 to its like Portion, which being sought in the Table of Segments, gives you a Segment, by which if you multiply the whole content of the Cask, the Product is the content of the Liquor remaining in the Cask.
But in the Table of Segments in this Book, you have the Area, to the equal parts of one half of the Diameter only, when the Cask therefore is more then half full, you must make use of the dry part of the Diameter instead of the wet, so shall you find what quantity of Liquor is wanting to fill up the Cask, which being deducted from the whole content of the Cask; the remainer is the quantity of Liquor yet remaining, an Example in each will be sufficient, to explane the use of this Table.
1. Example, In a Wine Cask not half full, let the great Diameter be as before 32 Inches, the [Page 94] content 126.25 Gallons, and let the wet part of the Diameter be 12 Inches, First I say.
As the whole Diameter 32. is to the wet part 12. so is 10.000 to 3750, which being sought in the Table, I find, the Area of that Segment to be. 342518 which being multiplyed by the whole content of the Cask 126.25, the Product is 43.24289750 and therefore there is remaining in the Cask 43 & 1/4 ferè.
2. Example. In the same Cask let the wet part of the Diameter be 18 Inches. I say.
As 32.18 :: 10000.5625 whose Complement to 10000 is 4375 which being sought in the Table, I find the Area answering thereto to be 420630; now then I say.
As the whole Area of the Circle 1000000 is to the whole content of the Cask 126. 25.
So is the Area of the Segment sought. 420630, to the content 53.1044375 which is in this case the content of the Liquor that is wanting, this therefore being deducted from the content of the whole Cask, 136. 25. the part remaining in the Vessel is. 73. 1455625.
Thus may Casks be gauged in whole or in part, in which a Table of Squares is sometimes necessary, as being the Foundation, from whom the other Tables are deduced; such a Table therefore is here exhibited, for all Numbers under 1000, by help whereof the Square of any Number under 10.000 may easily be found in this manner.
The Rectangle made of the Sum and Difference of any two Numbers, is equal to the Difference of the Squares of these Numbers.
Example, Let the given Numbers be 36 and 85 [Page 95] their Sum is 121, their difference 49, by which if you multiply 121, the Product will be 5929. The Square of 36 is 1296, and the Square of 85 is 7225, the difference between which Squares is 5929 as before.
And hence the Square of any Number under 10.000 may thus be found, the Squares of all Numbers under 1000 being given.
Example. Let the Square of 5715 be required. The Square of 571 by the Table is 326041, therefore the Square of 5710 is 32604100: the Sum of 5710 and 5715 is 11425, and the difference 5, by which if you multiple 11425, the Product is 52125 which being added unto 32604100 the Sum 32656325 is the Square of 5715. The like may be done for any other.
| 1 | 1 | 3 |
| 2 | 4 | |
| 3 | 09 | 5 |
| 4 | 16 | 7 |
| 5 | 25 | 9 |
| 6 | 36 | 11 |
| 7 | 49 | 13 |
| 8 | 64 | 15 |
| 9 | 81 | 17 |
| 10 | 100 | 19 |
| 11 | 121 | 21 |
| 12 | 144 | 23 |
| 13 | 160 | 25 |
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| 25 | 625 | 49 |
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| 32 | 1024 | 63 |
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| 40 | 1600 | 81 |
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| 42 | 1764 | 85 |
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| 44 | 1936 | 89 |
| 45 | 2025 | 91 |
| 46 | 2116 | 93 |
| 47 | 2209 | 95 |
| 48 | 2304 | 97 |
| 49 | 2401 | 99 |
| 50 | 2500 | 101 |
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| 99 | 9801 | 199 |
| 100 | 10000 | 201 |
| [Page 100] 101 | 10201 | 203 |
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| 105 | 11025 | 211 |
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| 125 | 15625 | 251 |
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| 152 | 23104 | 305 |
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| 155 | 24025 | 311 |
| 156 | 24336 | 313 |
| 157 | 24649 | 315 |
| 158 | 24964 | 317 |
| 159 | 25281 | 319 |
| 160 | 25600 | 321 |
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| 162 | 26244 | 325 |
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| 164 | 26896 | 329 |
| 165 | 27225 | 331 |
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| 193 | 37249 | 387 |
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| 195 | 38025 | 391 |
| 196 | 38416 | 393 |
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| 198 | 39204 | 397 |
| 199 | 39601 | 399 |
| 200 | 40000 | 401 |
| 201 | 40401 | 403 |
| 202 | 40804 | 405 |
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| 204 | 41616 | 409 |
| 205 | 42025 | 411 |
| 206 | 42436 | 413 |
| 207 | 42849 | 415 |
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| 224 | 50176 | 449 |
| 225 | 50625 | 451 |
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| 230 | 52900 | 461 |
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| 260 | 67600 | 621 |
| 261 | 68121 | 523 |
| 262 | 68644 | 525 |
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| 264 | 69696 | 529 |
| 265 | 70225 | 531 |
| 266 | 70756 | 533 |
| 277 | 71289 | 535 |
| 267 | 71289 | 535 |
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| 278 | 77284 | 557 |
| 279 | 77841 | 559 |
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| 281 | 78961 | 563 |
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| 285 | 81225 | 571 |
| 286 | 81796 | 573 |
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| 295 | 87025 | 591 |
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| 297 | 88200 | 595 |
| 298 | 88804 | 597 |
| 299 | 89401 | 599 |
| 300 | 90000 | 601 |
| [Page 103] 301 | 090601 | 603 |
| 302 | 091204 | 605 |
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| 304 | 092416 | 609 |
| 305 | 093025 | 611 |
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| 310 | 096109 | 621 |
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| 317 | 100487 | 645 |
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| 319 | 101761 | 639 |
| 320 | 102400 | 641 |
| 321 | 103041 | 643 |
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| 324 | 104976 | 649 |
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| 327 | 106929 | 655 |
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| 350 | 122500 | 701 |
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| 400 | 160000 | 801 |
| 401 | 160801 | 803 |
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| 404 | 163216 | 809 |
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| 444 | 197136 | 889 |
| 445 | 198025 | 891 |
| 446 | 198916 | 893 |
| 447 | 199809 | 895 |
| 448 | 200704 | 897 |
| 449 | 201601 | 899 |
| 450 | 202500 | 901 |
| 451 | 203401 | 903 |
| 452 | 204304 | 905 |
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| 454 | 206116 | 909 |
| 455 | 207025 | 911 |
| 456 | 207936 | 913 |
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| 460 | 211600 | 921 |
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| 996 | 992016 | 1993 |
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| 998 | 996004 | 1997 |
| 999 | 998001 | 1999 |
| 1000 | 1000000 | 2001 |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 1 | 0.001133 | 001371 | 001631 | 001915 | 002221 | 002549 | 002901 | 003275 | 003671 | 004091 |
| 2 | 0.004533 | 004997 | 005485 | 005995 | 006527 | 007083 | 007661 | 008261 | 008885 | 009531 |
| 3 | 0.010199 | 010891 | 011605 | 012341 | 013101 | 013883 | 014687 | 015515 | 016365 | 017237 |
| 4 | 0.018133 | 019051 | 019991 | 020955 | 021941 | 022949 | 023981 | 025035 | 026111 | 027211 |
| 5 | 0.028333 | 029477 | 030645 | 031835 | 033047 | 034283 | 035541 | 036825 | 038125 | 039451 |
| 6 | 0.040799 | 042171 | 043565 | 044981 | 046421 | 047883 | 049367 | 050875 | 052405 | 053957 |
| 7 | 0.051533 | 057131 | 058751 | 060395 | 062061 | 063749 | 065461 | 067195 | 068951 | 070731 |
| 8 | 0.072533 | 074357 | 076205 | 078075 | 079967 | 681883 | 083821 | 085781 | 087765 | 089771 |
| 9 | 0.091799 | 093851 | 095925 | 098021 | 100141 | 102283 | 104447 | 106635 | 108845 | 111077 |
| 10 | 0.113333 | 115610 | 117911 | 120234 | 122580 | 124949 | 127340 | 129754 | 132191 | 134650 |
| 11 | 0.137132 | 139637 | 142164 | 144714 | 147287 | 149882 | 152500 | 155141 | 157804 | 160490 |
| 12 | 0.163199 | 165930 | 168684 | 171461 | 174266 | 177082 | 179927 | 182794 | 185684 | 188597 |
| 13 | 0.191532 | 194490 | 197471 | 200474 | 203500 | 206549 | 229620 | 212714 | 215831 | 218970 |
| 14 | 0.222132 | 225317 | 228524 | 231754 | 235007 | 238282 | 241589 | 244901 | 248244 | 251610 |
| 15 | 0.254999 | 258410 | 261844 | 265301 | 268780 | 272282 | 275807 | 279354 | 282924 | 286517 |
| 16 | 0.290132 | 293770 | 297431 | 301114 | 304820 | 308549 | 312300 | 316074 | 319871 | 323690 |
| 17 | 0.327532 | 331397 | 335284 | 339194 | 343126 | 374082 | 351060 | 355060 | 359084 | 363130 |
| 18 | 0.367198 | 371290 | 375404 | 379540 | 383700 | 387882 | 392086 | 396314 | 400564 | 404836 |
| 19 | 0.409132 | 413450 | 417790 | 422154 | 426540 | 430948 | 435380 | 439834 | 444310 | 448810 |
| 20 | 0.453332 | 457876 | 462443 | 467033 | 471646 | 476281 | 480939 | 485620 | 490323 | 495049 |
| 21 | 0.499798 | 504569 | 509363 | 514180 | 519019 | 523881 | 528766 | 533673 | 538603 | 543556 |
| 22 | 0.548531 | 553529 | 558550 | 563593 | 568659 | 573748 | 578859 | 583993 | 589150 | 594329 |
| 23 | 0.599531 | 604756 | 610003 | 615273 | 620566 | 625881 | 631219 | 696580 | 641963 | 647369 |
| 24 | 0.652798 | 658249 | 663723 | 669220 | 674739 | 680281 | 685845 | 691433 | 697043 | 702675 |
| 25 | 0.708331 | 714009 | 719709 | 725233 | 731179 | 736947 | 742739 | 748553 | 754389 | 760249 |
| 26 | 0.766131 | 772035 | 777963 | 783913 | 789885 | 795880 | 801898 | 807939 | 814002 | 820088 |
| 27 | 0.826197 | 832328 | 838482 | 844659 | 850858 | 857080 | 863325 | 869592 | 875882 | 882195 |
| 28 | 0.888530 | 894888 | 901269 | 907672 | 914098 | 920547 | 927018 | 933512 | 940029 | 946568 |
| 29 | 0.953130 | 959715 | 966322 | 972952 | 979605 | 986280 | 992978 | 999699 | 006442 | 013208 |
| 30 | 1.019997 | 026808 | 033642 | 040498 | 047378 | 054280 | 061204 | 068152 | 075122 | 082114 |
| 31 | 1.089130 | 096168 | 103228 | 110312 | 117418 | 124546 | 131698 | 138871 | 146068 | 153287 |
| 32 | 1.160529 | 167794 | 175081 | 182391 | 189724 | 197079 | 204457 | 211858 | 219281 | 226727 |
| 33 | 1.234196 | 241687 | 249201 | 256738 | 264297 | 271879 | 279484 | 287111 | 294761 | 302434 |
| 34 | 1.310129 | 317847 | 325588 | 333351 | 341137 | 348946 | 356777 | 364631 | 372507 | 380467 |
| [Page 116-117] 34 | 1.310129 | 317847 | 325588 | 333351 | 341137 | 348946 | 356777 | 364631 | 372507 | 380467 |
| 35 | 1.388329 | 396273 | 404241 | 412231 | 420243 | 428279 | 436337 | 444417 | 452521 | 460647 |
| 36 | 1.468795 | 476966 | 485160 | 493377 | 501616 | 509878 | 518163 | 526470 | 534800 | 543153 |
| 37 | 1.551528 | 559926 | 568347 | 576790 | 585256 | 593745 | 602256 | 610790 | 619347 | 627926 |
| 38 | 1.636528 | 645153 | 653800 | 662470 | 671163 | 679878 | 688616 | 697577 | 706160 | 714966 |
| 39 | 1.723794 | 732646 | 741520 | 750416 | 759336 | 768278 | 777242 | 786230 | 795240 | 804272 |
| 40 | 1.813328 | 822405 | 831506 | 840629 | 849775 | 858944 | 868135 | 877349 | 886586 | 895845 |
| 41 | 1.905127 | 914432 | 923759 | 933109 | 942482 | 951877 | 961295 | 970736 | 980199 | 989685 |
| 42 | 1.999194 | 008725 | 018279 | 027856 | 037455 | 047077 | 056721 | 066389 | 076079 | 085791 |
| 43 | 2.095527 | 105285 | 115065 | 124869 | 134695 | 144543 | 154414 | 164308 | 174225 | 184164 |
| 44 | 2.194126 | 204111 | 214118 | 224148 | 234201 | 244276 | 254374 | 264495 | 274638 | 284804 |
| 45 | 2.294993 | 305204 | 315438 | 325695 | 335974 | 346276 | 356601 | 366948 | 377318 | 387710 |
| 46 | 2.398126 | 408564 | 419024 | 429508 | 440014 | 450542 | 461094 | 471668 | 482264 | 492884 |
| 47 | 2.503525 | 514190 | 524877 | 535587 | 546310 | 557075 | 567853 | 578654 | 589477 | 600323 |
| 48 | 2.611192 | 622083 | 632997 | 643934 | 654893 | 665875 | 676880 | 697907 | 698957 | 710030 |
| 49 | 2.721125 | 732243 | 743383 | 754547 | 765733 | 776941 | 788173 | 799427 | 810703 | 822003 |
| 50 | 2.833325 | 844669 | 856036 | 867426 | 878839 | 890274 | 901732 | 913213 | 924716 | 936242 |
| 51 | 2.947791 | 959362 | 970956 | 982573 | 994212 | 005874 | 017559 | 029266 | 040996 | 052749 |
| 52 | 3.064524 | 076322 | 088142 | 099986 | 111852 | 123740 | 135652 | 147586 | 159542 | 171522 |
| 53 | 3.183523 | 195548 | 207595 | 219665 | 231758 | 243872 | 256011 | 268172 | 280355 | 292561 |
| 54 | 3.304790 | 317041 | 329315 | 341612 | 353931 | 366273 | 378638 | 391025 | 403435 | 415867 |
| 55 | 3.428323 | 440801 | 453301 | 465825 | 478371 | 490939 | 503531 | 516544 | 528781 | 541440 |
| 56 | 3.554122 | 566827 | 579554 | 592304 | 605077 | 617872 | 630690 | 643531 | 856394 | 669280 |
| 57 | 3.682189 | 695120 | 708074 | 721051 | 734050 | 747072 | 760116 | 773184 | 786274 | 799386 |
| 58 | 3.812522 | 825980 | 838860 | 852064 | 865289 | 878530 | 891809 | 905103 | 918420 | 931759 |
| 59 | 3.945121 | 958506 | 971913 | 985343 | 998796 | 012271 | 025769 | 039290 | 252833 | 066399 |
| 60 | 4 [...]079988 | 093599 | 107233 | 120889 | 134569 | 148271 | 161195 | 175743 | 189513 | 203305 |
| 61 | 4.217120 | 230958 | 244819 | 258702 | 272608 | 286537 | 300488 | 314462 | 328459 | 342478 |
| 62 | 4.356520 | 370585 | 384672 | 398782 | 412915 | 427070 | 441248 | 455448 | 469672 | 483918 |
| 63 | 4.498186 | 512778 | 526792 | 541128 | 555487 | 569869 | 584274 | 598701 | 613151 | 627624 |
| 64 | 4.642119 | 656637 | 671178 | 685741 | 700327 | 714936 | 729567 | 744221 | 758898 | 773597 |
| 65 | 4.788319 | 80 [...]06 [...] | 81 [...]831 | 832621 | 84773 [...] | 862269 | 877126 | 892007 | 906910 | 921836 |
| 66 | 4.936785 | 951756 | 966750 | 981767 | 996806 | 011868 | 026953 | 042060 | 057190 | 072341 |
| 67 | 5.087518 | 102716 | 117936 | 133180 | 148446 | 163734 | 179046 | 194380 | 209736 | 225115 |
| [Page 118-119] 67 | 5.087518 | 102716 | 117936 | 133180 | 148446 | 163734 | 179046 | 194380 | 209736 | 225115 |
| 68 | 5.240517 | 255942 | 271389 | 286859 | 302352 | 317567 | 333405 | 348966 | 364549 | 380155 |
| 69 | 5.395784 | 411435 | 427109 | 442805 | 458525 | 474267 | 490031 | 505819 | 521629 | 537461 |
| 70 | 5.553317 | 569194 | 585095 | 601018 | 616964 | 632933 | 648924 | 664938 | 680975 | 697034 |
| 71 | 5.713116 | 729221 | 745348 | 761498 | 777601 | 793868 | 810084 | 826324 | 842588 | 858874 |
| 72 | 5.875182 | 891514 | 907867 | 924244 | 940643 | 957065 | 973510 | 989977 | 006467 | 022980 |
| 73 | 6.039555 | 056073 | 072654 | 089257 | 105883 | 122531 | 139203 | 155897 | 172613 | 189353 |
| 74 | 6.206115 | 222899 | 239706 | 256536 | 273389 | 290264 | 307162 | 324083 | 341026 | 357992 |
| 75 | 6.374981 | 391992 | 409026 | 426083 | 443162 | 460264 | 477388 | 494536 | 511706 | 528898 |
| 76 | 6.546114 | 563352 | 580612 | 597895 | 615201 | 632530 | 649881 | 667255 | 684652 | 702071 |
| 77 | 6.719513 | 736978 | 754465 | 771975 | 789508 | 807063 | 824641 | 842241 | 859865 | 877511 |
| 78 | 6.895179 | 912871 | 930584 | 948321 | 966080 | 983862 | 001667 | 019494 | 037344 | 055117 |
| 79 | 7.073112 | 091030 | 108971 | 126934 | 144920 | 162928 | 180960 | 199012 | 217090 | 235190 |
| 80 | 7.253312 | 271456 | 289623 | 307813 | 326026 | 344261 | 362519 | 380800 | 399103 | 417429 |
| 81 | 7.455778 | 454149 | 472543 | 490959 | 509399 | 527861 | 546345 | 564853 | 583383 | 601935 |
| 82 | 7.620510 | 639108 | 657729 | 676372 | 695038 | 713727 | 732438 | 751172 | 769929 | 788708 |
| 83 | 7.807510 | 826334 | 845182 | 864052 | 882944 | 901860 | 920798 | 939758 | 958741 | 977747 |
| 84 | 7.996776 | 015827 | 034901 | 053998 | 073117 | 092259 | 111424 | 130611 | 149821 | 169053 |
| 85 | 8.188309 | 207587 | 226887 | 246211 | 265557 | 284925 | 304316 | 323730 | 343167 | 362626 |
| 86 | 8.382108 | 401613 | 421140 | 440690 | 460263 | 479858 | 499476 | 519116 | 538780 | 558466 |
| 87 | 8.578174 | 597906 | 617659 | 637436 | 657235 | 677057 | 696902 | 716769 | 736659 | 756572 |
| 88 | 8.776507 | 796465 | 816446 | 836449 | 856475 | 876523 | 896595 | 916689 | 936805 | 936944 |
| 89 | 8.977106 | 997291 | 017498 | 037728 | 057981 | 078256 | 098554 | 118875 | 139218 | 159584 |
| 90 | 9.179973 | 200384 | 220818 | 241274 | 261754 | 282256 | 302780 | 323327 | 343897 | 364490 |
| 91 | 9.385105 | 405743 | 426404 | 447087 | 467793 | 488522 | 509273 | 530047 | 550843 | 571663 |
| 92 | 9.592505 | 613369 | 634256 | 655166 | 676099 | 897054 | 718032 | 739033 | 760056 | 781102 |
| 93 | 9.802171 | 823262 | 844376 | 865512 | 886672 | 907854 | 929058 | 950286 | 971536 | 992808 |
| 94 | 10.014103 | 035421 | 056762 | 078125 | 099511 | 120920 | 142351 | 163805 | 185282 | 206781 |
| 95 | 10.228303 | 249847 | 271415 | 293005 | 314617 | 336252 | 337910 | 379591 | 401294 | 423020 |
| 96 | 10.444769 | 466540 | 488334 | 510151 | 531990 | 553852 | 575736 | 597644 | 619574 | 641526 |
| 97 | 10.663501 | 685499 | 707520 | 729563 | 751629 | 773718 | 795829 | 817963 | 840120 | 862299 |
| 98 | 10.884501 | 906725 | 928973 | 951243 | 973535 | 995850 | 018188 | 040549 | 062932 | 085338 |
| 99 | 11.107767 | 130218 | 152692 | 175189 | 197708 | 220250 | 242814 | 265402 | 288012 | 310644 |
| 100 | 11.333300 |
| [Page 120-121] | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 1 | 0.000928 | 001127 | 001336 | 001569 | 001819 | 002089 | 002376 | 002683 | 003008 | 003351 |
| 2 | 0.003713 | 004094 | 004493 | 004911 | 005347 | 005802 | 006275 | 006767 | 007278 | 007807 |
| 3 | 0.008355 | 008921 | 009506 | 010109 | 010731 | 011372 | 012031 | 012709 | 013405 | 014120 |
| 4 | 0.014855 | 015605 | 016376 | 017165 | 017973 | 018799 | 019644 | 020507 | 021389 | 022289 |
| 5 | 0.023209 | 024146 | 025102 | 026077 | 027070 | 028082 | 029113 | 030162 | 031230 | 032316 |
| 6 | 0.033420 | 034544 | 035686 | 036846 | 038025 | 039223 | 040439 | 041674 | 042927 | 044199 |
| 7 | 0.045489 | 046798 | 048126 | 049472 | 050836 | 052220 | 053322 | 055042 | 056481 | 057938 |
| 8 | 0.059415 | 060909 | 062422 | 063954 | 065505 | 067074 | 068661 | 070267 | 071892 | 073535 |
| 9 | 0.075197 | 076877 | 078576 | 080293 | 082029 | 083784 | 085557 | 087349 | 089159 | 090988 |
| 10 | 0.092836 | 094702 | 096586 | 098489 | 100411 | 102351 | 104310 | 106287 | 108283 | 110298 |
| 11 | 0.112331 | 114383 | 116453 | 118542 | 120649 | 122775 | 124920 | 127083 | 129264 | 131465 |
| 12 | 0.133683 | 135921 | 138177 | 140451 | 142744 | 145056 | 147386 | 149735 | 152102 | 154489 |
| 13 | 0.156892 | 159315 | 161757 | 164217 | 166696 | 169193 | 171709 | 174243 | 176796 | 179368 |
| 14 | 0.181958 | 184567 | 187194 | 189840 | 192504 | 195187 | 197889 | 200609 | 203347 | 206105 |
| 15 | 0.208881 | 211675 | 214488 | 217319 | 220169 | 223038 | 225925 | 228831 | 231755 | 234698 |
| 16 | 0.237660 | 240640 | 243638 | 246655 | 249691 | 252746 | 255818 | 258910 | 262020 | 265148 |
| 17 | 0.268296 | 271461 | 274646 | 277848 | 281070 | 284310 | 287568 | 290845 | 294141 | 297455 |
| 18 | 0.300788 | 304140 | 307509 | 310898 | 314305 | 317731 | 321175 | 324638 | 328119 | 331619 |
| 19 | 0.335137 | 338675 | 342230 | 345804 | 349397 | 353008 | 356638 | 360287 | 363954 | 367639 |
| 20 | 0.371344 | 375066 | 378808 | 382567 | 386346 | 390143 | 393958 | 397792 | 401645 | 405516 |
| 21 | 0.409406 | 413315 | 417242 | 421174 | 425151 | 429134 | 433135 | 437155 | 441193 | 445250 |
| 22 | 0.449326 | 463420 | 457532 | 461664 | 465813 | 469982 | 474169 | 478374 | 482598 | 486841 |
| 23 | 0.491102 | 495382 | 499680 | 503997 | 508338 | 512686 | 517059 | 521450 | 525860 | 530288 |
| 24 | 0.534735 | 539200 | 543684 | 548187 | 552708 | 557248 | 561806 | 566383 | 570978 | 575592 |
| 25 | 0.580225 | 584876 | 589545 | 594233 | 598940 | 603666 | 608410 | 613172 | 617953 | 622753 |
| 26 | 0.627570 | 632408 | 637263 | 642137 | 647029 | 651940 | 656870 | 661818 | 666785 | 671770 |
| 27 | 0.676774 | 681796 | 686837 | 691897 | 696975 | 702072 | 707187 | 712321 | 717473 | 722644 |
| 28 | 0.727834 | 733042 | 738269 | 743514 | 748778 | 754060 | 759361 | 764680 | 770018 | 775375 |
| 29 | 0.780750 | 786144 | 791556 | 796987 | 802437 | 807905 | 813391 | 818897 | 824420 | 829963 |
| 30 | 0.835524 | 841103 | 846701 | 852318 | 857953 | 863606 | 869279 | 874970 | 880679 | 886407 |
| 31 | 0.892133 | 897919 | 903702 | 909505 | 915325 | 921165 | 927023 | 932899 | 938794 | 944708 |
| 32 | 0.950640 | 956591 | 962560 | 968548 | 974555 | 980580 | 986623 | 992686 | 998766 | 004866 |
| 33 | 1.010984 | 017120 | 023275 | 029449 | 035641 | 041852 | 048081 | 054329 | 060595 | 066880 |
| 34 | 1.073184 | 079 [...]06 | 085846 | 092206 | 098584 | 104980 | 111395 | 117828 | 124281 | 130751 |
| [Page 122-123] 34 | 1.073184 | 079506 | 085846 | 092206 | 098584 | 104980 | 111395 | 117828 | 124281 | 130751 |
| 35 | 1.137241 | 143748 | 150275 | 156820 | 163383 | 169965 | 176566 | 183185 | 189823 | 196479 |
| 36 | 1.203154 | 209848 | 216560 | 223290 | 230039 | 236807 | 243593 | 250398 | 257222 | 264064 |
| 37 | 1.270924 | 277803 | 284701 | 291617 | 298852 | 305506 | 312478 | 319468 | 326477 | 333505 |
| 38 | 1.340551 | 347616 | 354700 | 361802 | 368922 | 376061 | 383219 | 390395 | 397590 | 404803 |
| 39 | 1.412035 | 419286 | 426555 | 433842 | 441148 | 448473 | 455817 | 463178 | 470549 | 477958 |
| 40 | 1.485376 | 492812 | 500266 | 507740 | 515232 | 522742 | 530271 | 537819 | 545385 | 552969 |
| 41 | 1.560573 | 568194 | 575835 | 583494 | 591171 | 598868 | 606582 | 614315 | 622067 | 629838 |
| 42 | 1.637627 | 645434 | 653260 | 661105 | 668968 | 676850 | 684750 | 692669 | 700606 | 708563 |
| 43 | 1.716537 | 724530 | 732542 | 740572 | 748621 | 756689 | 764775 | 772879 | 781002 | 789144 |
| 44 | 1.797304 | 805483 | 813681 | 821897 | 830131 | 838384 | 846656 | 854946 | 863255 | 871583 |
| 45 | 1.879929 | 888293 | 896676 | 905078 | 913498 | 921937 | 930394 | 938870 | 947365 | 955878 |
| 46 | 1.964409 | 972959 | 981528 | 990116 | 998721 | 007346 | 015989 | 024651 | 033331 | 042029 |
| 47 | 2.050747 | 059483 | 068237 | 077010 | 085802 | 094612 | 103440 | 112288 | 121154 | 130038 |
| 48 | 2.138941 | 147862 | 156803 | 165761 | 174739 | 183734 | 192749 | 201782 | 210833 | 219903 |
| 49 | 2.228992 | 238099 | 247225 | 256369 | 265532 | 274714 | 283914 | 293132 | 302369 | 311625 |
| 50 | 2.32090 | 330192 | 339504 | 348834 | 358182 | 267550 | 376935 | 386340 | 395762 | 405204 |
| 51 | 2.414664 | 424142 | 433640 | 443155 | 452689 | 462242 | 471814 | 481404 | 491012 | 508639 |
| 52 | 2.510285 | 519949 | 529632 | 539333 | 549053 | 558792 | 568549 | 578324 | 588119 | 597931 |
| 53 | 2.607763 | 617613 | 627481 | 637368 | 647274 | 657198 | 667141 | 677102 | 687082 | 697080 |
| 54 | 2.707097 | 717133 | 727187 | 737260 | 747351 | 757461 | 767589 | 777736 | 787902 | 798086 |
| 55 | 2.808289 | 818510 | 828750 | 839008 | 849285 | 859580 | 869894 | 880227 | 890578 | 900948 |
| 56 | 2.911336 | 921743 | 932169 | 942613 | 953076 | 963557 | 974056 | 984575 | 995112 | 005667 |
| 57 | 3.016241 | 026834 | 037445 | 048075 | 058723 | 069390 | 080075 | 090779 | 101502 | 112243 |
| 58 | 3.123003 | 133781 | 144578 | 155393 | 166227 | 177080 | 187951 | 198840 | 209748 | 220675 |
| 59 | 3.231621 | 242585 | 253567 | 264568 | 275588 | 286626 | 297683 | 308758 | 319852 | 330964 |
| 60 | 3.342096 | 353245 | 364413 | 375600 | 386805 | 398029 | 409272 | 420533 | 431812 | 443110 |
| 61 | 3.454427 | 465762 | 477116 | 488489 | 499880 | 511289 | 522717 | 534164 | 545629 | 557113 |
| 62 | 3.568615 | 580136 | 591676 | 603234 | 614811 | 626406 | 638020 | 649652 | 661303 | 672972 |
| 63 | 3.684660 | 696367 | 708092 | 719836 | 731598 | 743379 | 755179 | 766997 | 778838 | 790688 |
| 64 | 3.802562 | 814454 | 826365 | 838295 | 850243 | 862209 | 874194 | 886198 | 898220 | 910261 |
| 65 | 3.922321 | 934398 | 946495 | 958610 | 970744 | 982896 | 995067 | 007256 | 919464 | 031691 |
| 66 | 4.043936 | 056199 | 068481 | 080782 | 093102 | 105440 | 117796 | 130171 | 142565 | 154977 |
| 67 | 4.167408 | 179857 | 192325 | 204811 | 217316 | 229840 | 242382 | 254943 | 267522 | 280120 |
| [Page 124-125] 67 | 4.167408 | 179857 | 192325 | 204811 | 217316 | 229840 | 242382 | 254943 | 267522 | 280122 |
| 68 | 4.292736 | 305371 | 318025 | 330697 | 343387 | 356097 | 368825 | 381571 | 394336 | 407119 |
| 69 | 4.419921 | 432742 | 445581 | 458439 | 471315 | 484210 | 497124 | 510056 | 523007 | 535976 |
| 70 | 4.548964 | 561970 | 574995 | 588038 | 601100 | 614181 | 627280 | 670398 | 653534 | 666789 |
| 71 | 4.679862 | 693054 | 706265 | 819494 | 732724 | 746008 | 759293 | 772596 | 785918 | 799259 |
| 72 | 4.812618 | 825995 | 839392 | 852806 | 866240 | 879692 | 893162 | 906651 | 920159 | 933685 |
| 73 | 4.947230 | 960793 | 974375 | 987976 | 001595 | 015232 | 028888 | 042563 | 056257 | 069968 |
| 74 | 5.083699 | 097448 | 111215 | 125002 | 138806 | 152630 | 166471 | 180332 | 194211 | 208108 |
| 75 | 5.222025 | 235959 | 249912 | 263884 | 277875 | 291884 | 305911 | 319957 | 334022 | 348105 |
| 76 | 5.362207 | 376327 | 390466 | 404624 | 418800 | 432994 | 447208 | 461439 | 475690 | 489958 |
| 77 | 5.504246 | 518552 | 532877 | 547220 | 561581 | 575962 | 590361 | 604778 | 619214 | 633669 |
| 78 | 5.648142 | 662633 | 677144 | 691673 | 706220 | 720786 | 735370 | 479974 | 764595 | 779235 |
| 79 | 5.793894 | 808572 | 823268 | 837982 | 852715 | 867467 | 882237 | 897026 | 911833 | 926659 |
| 80 | 5.941504 | 956367 | 971248 | 986148 | 001067 | 016004 | 030960 | 045935 | 060928 | 075939 |
| 81 | 6.090969 | 106018 | 121085 | 136171 | 151276 | 166399 | 181540 | 196700 | 211879 | 227076 |
| 82 | 6.242292 | 257527 | 272779 | 288051 | 303341 | 318650 | 333977 | 349323 | 364687 | 380070 |
| 83 | 6.395472 | 410892 | 426330 | 441787 | 457263 | 472758 | 488270 | 503802 | 519352 | 534920 |
| 84 | 6.550508 | 566113 | 581738 | 597381 | 613042 | 628722 | 644421 | 660138 | 675873 | 691628 |
| 85 | 6.707401 | 723192 | 739002 | 754830 | 770678 | 786543 | 802427 | 818330 | 834252 | 850192 |
| 86 | 6.866150 | 882127 | 898123 | 914137 | 930170 | 946221 | 962291 | 978380 | 994487 | 010612 |
| 87 | 7.026756 | 042919 | 059100 | 075300 | 091519 | 107756 | 124011 | 140285 | 156578 | 172889 |
| 88 | 7.189219 | 205568 | 221935 | 239320 | 254724 | 271147 | 287588 | 304048 | 320527 | 337024 |
| 89 | 7.353539 | 370073 | 386626 | 403197 | 419787 | 436395 | 453022 | 469668 | 486332 | 503014 |
| 90 | 7.519716 | 536435 | 553174 | 569930 | 586706 | 603500 | 620313 | 637144 | 653993 | 670862 |
| 91 | 7.687749 | 704654 | 721578 | 738521 | 755482 | 772462 | 789460 | 806477 | 823512 | 840566 |
| 92 | 7.857639 | 874730 | 891839 | 908968 | 926114 | 943280 | 960464 | 977666 | 994887 | 012127 |
| 93 | 8.029385 | 046662 | 063957 | 081271 | 098604 | 115955 | 133324 | 150713 | 168119 | 185545 |
| 94 | 8.202988 | 220451 | 237932 | 255432 | 272950 | 290486 | 308042 | 325616 | 343208 | 360819 |
| 95 | 8.378449 | 396097 | 413763 | 431449 | 449152 | 466875 | 484616 | 502375 | 520153 | 537950 |
| 96 | 8.555765 | 573599 | 591451 | 609322 | 627212 | 645120 | 663047 | 680992 | 698956 | 716938 |
| 97 | 8.734939 | 752958 | 770996 | 789053 | 807128 | 825222 | 843334 | 861465 | 879614 | 897782 |
| 98 | 8.915969 | 934174 | 952398 | 970640 | 988901 | 007180 | 025478 | 043795 | 062130 | 080484 |
| 99 | 9.098856 | 117247 | 135656 | 154084 | 172531 | 190996 | 209479 | 227981 | 246502 | 265042 |
| 100 | 9.283600 | 302176 | 320771 | 339385 | 358017 | 376668 | 395337 | 414025 | 432731 | 451456 |
| [Page 126-127] 100 | 9.283600 | 302176 | 302771 | 339385 | 358017 | 376668 | 395337 | 414025 | 432731 | 451456 |
| 101 | 9.470200 | 488962 | 507743 | 526524 | 545360 | 564196 | 583051 | 601925 | 620817 | 639728 |
| 102 | 9.658657 | 677605 | 696574 | 715556 | 734560 | 753582 | 772622 | 791682 | 810759 | 829856 |
| 103 | 9.848971 | 868104 | 887256 | 906427 | 925616 | 944824 | 964050 | 983295 | [...]02559 | 021841 |
| 104 | 10.041141 | 060460 | 079798 | 099154 | 118529 | 137923 | 157335 | 176765 | 196215 | 215682 |
| 105 | 10.235169 | 254673 | 274197 | 293739 | 323299 | 332878 | 352476 | 372092 | 391727 | 411381 |
| 106 | 10.431052 | 450743 | 470452 | 490180 | 509926 | 529691 | 549474 | 569276 | 589096 | 608936 |
| 107 | 10.628793 | 648669 | 668564 | 688477 | 708409 | 728360 | 748329 | 768316 | 788323 | 808347 |
| 108 | 10.828391 | 848452 | 868533 | 888632 | 908749 | 928886 | 949040 | 969213 | 989405 | 009616 |
| 109 | 11.029845 | 050092 | 070358 | 090643 | 110946 | 131268 | 151608 | 171967 | 192345 | 212741 |
| 110 | 11.233156 | 253589 | 274040 | 294511 | 315000 | 335507 | 356033 | 376578 | 397141 | 417723 |
| 111 | 11.438323 | 458942 | 479579 | 500235 | 520910 | 541603 | 562315 | 583045 | 603794 | 624561 |
| 112 | 11.645347 | 666152 | 686975 | 707817 | 728677 | 749556 | 770453 | 791369 | 812304 | 833257 |
| 113 | 11.854228 | 875219 | 896227 | 917255 | 938301 | 959365 | 980448 | 001550 | 022670 | 043809 |
| 114 | 12.064966 | 086142 | 107336 | 128549 | 149781 | 171031 | 192300 | 213587 | 234893 | 256218 |
| 115 | 12.277561 | 298922 | 320302 | 341701 | 363118 | 384554 | 406008 | 427481 | 448973 | 470483 |
| 116 | 12.492012 | 513559 | 535125 | 556709 | 578312 | 509934 | 621574 | 643232 | 664909 | 686605 |
| 117 | 12.708320 | 730052 | 751804 | 773574 | 795363 | 817170 | 838995 | 860840 | 882703 | 904584 |
| 118 | 12.926484 | 948403 | 970340 | 992296 | 014270 | 036263 | 058274 | 080304 | 102353 | 124420 |
| 119 | 13.146505 | 168610 | 190733 | 212874 | 235034 | 257212 | 279409 | 301625 | 323859 | 346112 |
| 120 | 13.368384 | 390673 | 412982 | 435309 | 457655 | 480019 | 502402 | 524803 | 547223 | 569661 |
| 121 | 13.592118 | 614594 | 637088 | 659601 | 682132 | 704682 | 727250 | 749837 | 772443 | 795067 |
| 122 | 13.817710 | 840371 | 863051 | 885749 | 908466 | 931202 | 953956 | 976729 | 999520 | 022320 |
| 123 | 14.045158 | 068005 | 090870 | 113754 | 136657 | 159578 | 182518 | 205476 | 228453 | 251449 |
| 124 | 14.274463 | 297495 | 320547 | 343616 | 366705 | 389812 | 412937 | 436081 | 459244 | 482425 |
| 125 | 14.505625 | 528843 | 552080 | 575335 | 598609 | 621902 | 645213 | 668542 | 691891 | 715257 |
| 126 | 14.738643 | 762047 | 785469 | 808910 | 832370 | 855848 | 879345 | 902860 | 926394 | 949947 |
| 127 | 14.973518 | 997108 | 020716 | 044343 | 067988 | 091652 | 115334 | 139035 | 162755 | 186493 |
| 128 | 15.210250 | 234025 | 257819 | 281631 | 305462 | 329312 | 353180 | 377067 | 400972 | 424896 |
| 129 | 15.448838 | 472799 | 496779 | 520777 | 544794 | 568829 | 592883 | 616955 | 641046 | 665155 |
| 130 | 15.689284 | 713430 | 737595 | 761779 | 785981 | 810202 | 834442 | 858700 | 882977 | 907272 |
| 131 | 15.931585 | 955918 | 982269 | 004638 | 029026 | 053433 | 077858 | 102302 | 126764 | 151245 |
| 132 | 16.175744 | 200262 | 224799 | 249354 | 273927 | 298520 | 323131 | 347760 | 372408 | 497074 |
| 133 | 16.421760 | 446463 | 471185 | 495926 | 502686 | 545464 | 570260 | 595075 | 619909 | 644761 |
| [Page 128-129] 134 | 16.669632 | 694521 | 719429 | 744355 | 769300 | 794264 | 819242 | 844247 | 869266 | 894304 |
| 135 | 16.919361 | 944436 | 969529 | 994641 | 019772 | 044921 | 070089 | 095275 | 120489 | 145704 |
| 136 | 17.170946 | 196207 | 221486 | 246784 | 272100 | 297435 | 322789 | 348161 | 373551 | 398961 |
| 137 | 17.424388 | 449835 | 475300 | 500783 | 526285 | 551806 | 577345 | 602903 | 628479 | 654074 |
| 138 | 17.679687 | 705319 | 730970 | 756639 | 782327 | 808033 | 833758 | 859501 | 885263 | 911044 |
| 139 | 17.936843 | 962661 | 988497 | 014352 | 040225 | 066117 | 092028 | 117957 | 143904 | 169871 |
| 140 | 18.195856 | 221859 | 247881 | 273921 | 299980 | 326058 | 325154 | 378269 | 404402 | 430554 |
| 141 | 18.456725 | 482914 | 509121 | 535347 | 561592 | 587856 | 614137 | 640438 | 666757 | 693094 |
| 142 | 18.719451 | 745825 | 772219 | 798630 | 825061 | 851510 | 877977 | 904463 | 930968 | 957491 |
| 143 | 18.984033 | 010059 | 037172 | 063770 | 090386 | 117021 | 143674 | 170346 | 197036 | 223745 |
| 144 | 19.250472 | 277219 | 303983 | 330766 | 357568 | 384388 | 411227 | 438085 | 464961 | 491855 |
| 145 | 19.518769 | 545700 | 527651 | 599619 | 626607 | 653613 | 680637 | 707680 | 734742 | 761822 |
| 146 | 19.788921 | 816039 | 843175 | 870329 | 897502 | 924694 | 951904 | 979133 | 006380 | 033646 |
| 147 | 20.060931 | 088234 | 115555 | 142896 | 170254 | 197632 | 225028 | 252442 | 279875 | 307327 |
| 148 | 20.334797 | 362286 | 389793 | 417319 | 444863 | 472426 | 500008 | 527608 | 555227 | 582864 |
| 149 | 20.610520 | 638194 | 665887 | 693599 | 721329 | 749078 | 776845 | 804631 | 832435 | 860258 |
| 150 | 20.888100 | 915960 | 943838 | 971735 | 999651 | 027586 | 055539 | 083510 | 111500 | 139509 |
| 151 | 21.167536 | 195582 | 223646 | 251729 | 279830 | 307950 | 336089 | 364246 | 392422 | 420616 |
| 152 | 21.448829 | 477060 | 505310 | 533579 | 561866 | 590172 | 618496 | 646839 | 675200 | 703580 |
| 153 | 21.731979 | 760396 | 788832 | 817286 | 845759 | 874250 | 902760 | 931288 | 959835 | 988401 |
| 154 | 22.016985 | 045588 | 074209 | 102849 | 131508 | 130185 | 188880 | 217595 | 246327 | 275079 |
| 155 | 22.303849 | 332637 | 361444 | 390270 | 419114 | 447976 | 476858 | 505758 | 534676 | 563613 |
| 156 | 22.592568 | 621543 | 650535 | 679547 | 708576 | 737625 | 766692 | 795777 | 824881 | 854004 |
| 157 | 22.883145 | 912305 | 941483 | 970680 | 999896 | 029130 | 058382 | 087654 | 116943 | 146252 |
| 158 | 23.175579 | 204924 | 234288 | 263671 | 293072 | 322492 | 351930 | 381387 | 410862 | 440356 |
| 159 | 23.469869 | 499400 | 528949 | 558515 | 588105 | 617710 | 647334 | 676976 | 706638 | 736317 |
| 160 | 23.766016 | 795732 | 825468 | 855222 | 884994 | 914785 | 944595 | 974423 | 004270 | 034135 |
| 161 | 24.064019 | 093922 | 123843 | 153782 | 183740 | 213717 | 243712 | 273726 | 303759 | 333810 |
| 162 | 24.363879 | 393967 | 424074 | 454199 | 484343 | 514506 | 544687 | 574886 | 605104 | 635341 |
| 163 | 24.665596 | 695870 | 726163 | 756474 | 786803 | 817151 | 847518 | 877903 | 908307 | 938729 |
| 164 | 24.969170 | 999630 | 030108 | 060604 | 091119 | 121653 | 152206 | 182776 | 213366 | 243974 |
| 165 | 25.274601 | 305246 | 335909 | 366592 | 397293 | 428012 | 458750 | 489507 | 520282 | 551075 |
| 166 | 25.581888 | 612718 | 643568 | 674436 | 705322 | 736228 | 767151 | 798093 | 829054 | 860034 |
| 167 | 25.89103 [...] | 922048 | 95308 | 984137 | 915209 | 046300 | 077409 | 108537 | 139683 | 170849 |
| [Page 130-131] 167 | 25.891032 | 922048 | 953083 | 984137 | 015209 | 046300 | 077409 | 108537 | 139683 | 170849 |
| 168 | 26.202032 | 233234 | 264455 | 295694 | 326952 | 358229 | 389524 | 420837 | 452169 | 483520 |
| 169 | 26.514689 | 546277 | 577684 | 609109 | 640552 | 672014 | 703495 | 734994 | 766512 | 798049 |
| 170 | 26.829604 | 861177 | 892769 | 924380 | 956009 | 987657 | 019323 | 051008 | 082712 | 114432 |
| 171 | 27.146174 | 177933 | 209711 | 241508 | 273322 | 305156 | 337008 | 368879 | 400768 | 432675 |
| 172 | 27.464602 | 496547 | 528510 | 560492 | 592493 | 624512 | 656549 | 688606 | 720681 | 752774 |
| 173 | 27.784886 | 817016 | 849166 | 881333 | 913520 | 945724 | 977948 | 010190 | 042450 | 074729 |
| 174 | 28.107027 | 139343 | 171678 | 204031 | 236403 | 268794 | 301213 | 333630 | 366076 | 398541 |
| 175 | 28.431025 | 463526 | 496047 | 528586 | 561142 | 593720 | 626314 | 658928 | 691559 | 724210 |
| 176 | 28.756879 | 789566 | 822273 | 854997 | 881740 | 920502 | 953283 | 986082 | 018899 | 051735 |
| 177 | 29.084590 | 117463 | 150355 | 183265 | 216194 | 249142 | 282108 | 315092 | 348096 | 381117 |
| 178 | 29.414158 | 447217 | 480294 | 513390 | 546505 | 579638 | 612790 | 645960 | 679149 | 712356 |
| 179 | 29.745582 | 778827 | 812090 | 845372 | 878672 | 919991 | 945328 | 978684 | 012059 | 045452 |
| 180 | 30.078864 | 112294 | 145743 | 179210 | 212696 | 246200 | 279723 | 313265 | 346825 | 380404 |
| 181 | 30.414001 | 447617 | 481252 | 514905 | 548577 | 582267 | 615975 | 649703 | 683449 | 717213 |
| 182 | 30.750996 | 784798 | 818618 | 852457 | 886314 | 920190 | 954084 | 987997 | 021929 | 055879 |
| 183 | 31.089848 | 123835 | 157841 | 192865 | 225908 | 252970 | 294050 | 328148 | 362265 | 396401 |
| 184 | 31.430556 | 464729 | 498920 | 533130 | 567359 | 601606 | 635872 | 670156 | 704459 | 738780 |
| 185 | 31.773121 | 807479 | 841856 | 876252 | 910666 | 945099 | 979551 | 014021 | 048509 | 083016 |
| 186 | 32.117542 | 152086 | 186649 | 221231 | 255831 | 290449 | 325508 | 359742 | 394416 | 429109 |
| 187 | 32.463820 | 498550 | 533299 | 568066 | 602892 | 637656 | 672479 | 707320 | 742180 | 777058 |
| 188 | 32.811955 | 846871 | 881805 | 916758 | 951729 | 986719 | 021728 | 056755 | 091800 | 126864 |
| 189 | 33.161947 | 197048 | 232168 | 267307 | 302464 | 337639 | 372833 | 408046 | 443277 | 478527 |
| 190 | 33.513796 | 549082 | 584388 | 619712 | 655055 | 690416 | 725796 | 761194 | 796611 | 832047 |
| 191 | 33.867501 | 902973 | 938464 | 973974 | 009503 | 045050 | 080615 | 116199 | 151802 | 187423 |
| 192 | 34.223063 | 258721 | 294398 | 330093 | 365807 | 401540 | 437291 | 473061 | 508849 | 544656 |
| 193 | 34.580481 | 616325 | 652188 | 688069 | 723968 | 759887 | 795824 | 831779 | 867753 | 903745 |
| 194 | 34.939756 | 975786 | 011834 | 047901 | 083986 | 120090 | 156213 | 192354 | 228514 | 264692 |
| 195 | 35.300889 | 337104 | 373338 | 409590 | 445861 | 482151 | 518459 | 554786 | 591131 | 627495 |
| 196 | 35.663877 | 700278 | 736698 | 773136 | 809593 | 846068 | 882562 | 919074 | 955605 | 992155 |
| 197 | 36.028723 | 065309 | 101915 | 138538 | 175181 | 211842 | 248521 | 285219 | 321936 | 358671 |
| 198 | 36.395425 | 432197 | 468988 | 505798 | 542626 | 579472 | 616337 | 653221 | 690124 | 727044 |
| 199 | 36.763984 | 800942 | 837918 | 874914 | 911927 | 948960 | 986010 | 023080 | 060168 | 097274 |
| 200 | 37.134400 |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 01 | 0.002617 | 003167 | 003769 | 004424 | 005131 | 005890 | 006702 | 007566 | 008482 | 009450 |
| 02 | 0.010471 | 011545 | 012671 | 013849 | 015079 | 016362 | 017697 | 019085 | 020525 | 022017 |
| 03 | 0.023561 | 025158 | 026808 | 028509 | 030264 | 032070 | 033929 | 035840 | 037803 | 039819 |
| 04 | 0.041887 | 044008 | 046181 | 048406 | 050684 | 053014 | 055396 | 057831 | 060318 | 062858 |
| 05 | 0.065449 | 068093 | 070790 | 073539 | 076340 | 079194 | 082100 | 085058 | 088069 | 091132 |
| 06 | 0.094247 | 097415 | 100635 | 103908 | 107232 | 110610 | 114039 | 117521 | 121055 | 124642 |
| 07 | 0.128281 | 131973 | 135716 | 139512 | 143361 | 147262 | 151215 | 155220 | 159278 | 163388 |
| 08 | 0.167551 | 171766 | 176033 | 180353 | 184725 | 189149 | 193626 | 198155 | 202737 | 207371 |
| 09 | 0.212057 | 216796 | 221586 | 226430 | 231325 | 236273 | 241274 | 246326 | 251432 | 256589 |
| 1.0 | 0.261799 | 267061 | 272376 | 277742 | 283162 | 288633 | 294157 | 299734 | 305362 | 311043 |
| 1.1 | 0.316777 | 322563 | 328401 | 334291 | 340234 | 346229 | 352277 | 358377 | 364529 | 370734 |
| 1.2 | 0.376991 | 383300 | 389662 | 396076 | 402542 | 409061 | 415632 | 422256 | 482932 | 435660 |
| 1.3 | 0.442440 | 449273 | 456159 | 463096 | 470086 | 477129 | 484224 | 491371 | 498570 | 505822 |
| 1.4 | 0.513126 | 520483 | 527892 | 535353 | 542867 | 550433 | 558051 | 565722 | 573445 | 581220 |
| 1.5 | 0.589048 | 596928 | 604861 | 612846 | 620883 | 628973 | 637114 | 645309 | 653555 | 661855 |
| 1.6 | 0.670206 | 678610 | 687066 | 695574 | 704135 | 712748 | 721414 | 730132 | 738902 | 747725 |
| 1.7 | 0.756600 | 765527 | 774507 | 783539 | 792623 | 801760 | 810949 | 820191 | 829485 | 838831 |
| 1.8 | 0.848230 | 857680 | 867184 | 876739 | 886348 | 896008 | 905721 | 915486 | 925303 | 935173 |
| 1.9 | 0.945097 | 955070 | 965097 | 975176 | 985308 | 995492 | 005728 | 016017 | 026358 | 036751 |
| 2.0 | 1.047197 | 057695 | 068246 | 078849 | 089504 | 100211 | 110971 | 121784 | 132648 | 143565 |
| 2.1 | 1.154535 | 165557 | 176631 | 187757 | 198936 | 210167 | 221451 | 232787 | 244175 | 255616 |
| 2.2 | 1.267109 | 278654 | 290252 | 301902 | 313604 | 325359 | 337166 | 349026 | 360937 | 372902 |
| 2.3 | 1.384918 | 396987 | 409109 | 421282 | 433508 | 445787 | 458117 | 470500 | 482936 | 495424 |
| 2.4 | 1.507964 | 520557 | 533201 | 545899 | 558648 | 571450 | 584305 | 597211 | 610170 | 623182 |
| 2.5 | 1.636246 | 649362 | 662530 | 675751 | 689024 | 702350 | 715728 | 729158 | 742641 | 756176 |
| 2.6 | 1.769763 | 783403 | 797095 | 810840 | 824637 | 838486 | 852387 | 866341 | 880347 | 894406 |
| 2.7 | 1.908517 | 922680 | 036896 | 951164 | 965485 | 979857 | 994283 | 008760 | 023290 | 037872 |
| 2.8 | 2.052507 | 067194 | 081933 | 096725 | 111569 | 126465 | 141414 | 156415 | 171468 | 186574 |
| 2.9 | 2.201732 | 216943 | 232206 | 247521 | 262889 | 278309 | 293781 | 309306 | 324883 | 340512 |
| 3.0 | 2.356194 |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 10 | 0.314159 | 345575 | 376991 | 408407 | 439822 | 471238 | 502654 | 534070 | 565486 | 596902 |
| 20 | 0.628318 | 659734 | 691150 | 722566 | 753982 | 785398 | 816814 | 848230 | 879645 | 911061 |
| 30 | 0.942477 | 974893 | 006309 | 037725 | 069148 | 100557 | 131973 | 162389 | 193805 | 225221 |
| 40 | 1.256637 | 288052 | 319468 | 350884 | 382300 | 413716 | 445132 | 476548 | 507964 | 539380 |
| 50 | 1.570796 | 602212 | 633628 | 665044 | 696460 | 727875 | 759291 | 790707 | 822123 | 853539 |
| 60 | 1.884955 | 916371 | 947787 | 979203 | 010619 | 042035 | 073451 | 104867 | 136283 | 167698 |
| 70 | 2.199114 | 230530 | 261946 | 293362 | 324778 | 356194 | 387610 | 419026 | 450442 | 481858 |
| 80 | 2.513274 | 544690 | 576105 | 607521 | 638937 | 670353 | 701769 | 733185 | 764601 | 796017 |
| 90 | 2.827433 | 858849 | 890265 | 921681 | 953097 | 984513 | 015928 | 047344 | 078760 | 110176 |
| 1.00 | 3.141592 | 173008 | 204424 | 235840 | 267256 | 298672 | 330088 | 361504 | 392920 | 424335 |
| 10 | 3.455751 | 487167 | 518583 | 549999 | 581415 | 613831 | 644247 | 675663 | 707079 | 738495 |
| 20 | 3.769911 | 801327 | 832743 | 864158 | 895574 | 926990 | 958406 | 989822 | 021238 | 052654 |
| 30 | 4.084070 | 115486 | 146902 | 178318 | 209734 | 241150 | 272566 | 303981 | 335397 | 366813 |
| 40 | 4.398229 | 429645 | 461061 | 492477 | 523893 | 555309 | 586725 | 618141 | 649557 | 680973 |
| 50 | 4.712388 | 743804 | 775220 | 806636 | 838052 | 869468 | 900884 | 932300 | 963716 | 995132 |
| 60 | 5.026548 | 057964 | 089380 | 120796 | 152211 | 183627 | 215043 | 246459 | 277875 | 309291 |
| 70 | 5.340707 | 372123 | 403539 | 434945 | 466371 | 497787 | 529203 | 560618 | 592034 | 623450 |
| 80 | 5.654866 | 686282 | 717698 | 749144 | 780530 | 811946 | 843362 | 874778 | 906194 | 937610 |
| 90 | 5.969026 | 000441 | 031857 | 063273 | 094689 | 126105 | 157521 | 188937 | 220353 | 251769 |
| 2.00 | 6.283105 | 314601 | 346017 | 377433 | 408849 | 440264 | 471689 | 503096 | 534512 | 565928 |
| 10 | 6.597344 | 628760 | 660176 | 691592 | 723008 | 754424 | 785840 | 817256 | 848671 | 880087 |
| 20 | 6.911503 | 942919 | 974336 | 005751 | 037167 | 068583 | 099999 | 131415 | 162831 | 194247 |
| 30 | 7.225663 | 257079 | 288494 | 319910 | 351326 | 382742 | 414158 | 445574 | 476990 | 508406 |
| 40 | 7.539822 | 571238 | 602654 | 634070 | 665486 | 696901 | 728317 | 759733 | 791149 | 822565 |
| 50 | 7.853981 | 885397 | 916813 | 948229 | 979645 | 011061 | 042477 | 073893 | 105309 | 136724 |
| 60 | 8.168140 | 199556 | 230972 | 262388 | 293804 | 325220 | 356636 | 388052 | 419468 | 450884 |
| 70 | 8.482300 | 513716 | 545132 | 576547 | 607963 | 639379 | 670795 | 702211 | 733627 | 765043 |
| 80 | 8.796458 | 827875 | 859291 | 890707 | 922123 | 953539 | 984954 | 016370 | 047786 | 079202 |
| 90 | 9.110618 | 142034 | 173450 | 204866 | 236282 | 267798 | 299114 | 330530 | 361946 | 393362 |
| 3.00 | 9.424777 |
| Natural | Artificial | |
| 0.01 | 0.000002 | 4.423668 |
| .02 | 0.000010 | 5.025728 |
| .03 | 0.000023 | 5.377911 |
| .04 | 0.000042 | 5.627788 |
| 0.05 | 0.000066 | 5.821608 |
| .06 | 0.000092 | 5.979971 |
| .07 | 0.000129 | 6.113864 |
| .08 | 0.000169 | 6.229848 |
| .09 | 0.000214 | 6.332153 |
| 0.10 | 0.000265 | 6.423668 |
| .11 | 0.000320 | 6.506454 |
| .12 | 0.000382 | 6.582031 |
| .13 | 0.000448 | 6.651555 |
| .14 | 0.000519 | 6.715924 |
| 0.15 | 0.000596 | 6.775851 |
| .16 | 0.000679 | 6.831908 |
| .17 | 0.000766 | 6.884566 |
| .18 | 0.000859 | 6.934213 |
| .19 | 0.000957 | 6.981176 |
| 0.20 | 0.001061 | 7.025728 |
| .21 | 0.001169 | 7.068107 |
| .22 | 0.001283 | 7.108514 |
| .23 | 0.001403 | 7.147124 |
| .24 | 0.001527 | 7.184091 |
| 0.25 | 0.001657 | 7.219548 |
| .26 | 0.001793 | 7.253615 |
| .27 | 0.001933 | 7.286396 |
| .28 | 0.002012 | 7.317984 |
| .29 | 0.002230 | 7.348464 |
| 0.30 | 0.002353 | 7.377911 |
| .31 | 0.002549 | 7.406392 |
| .32 | 0.002716 | 7.433968 |
| .3 [...] | 0.002888 | 7.460695 |
| .3 [...] | 0.003066 | 7.486626 |
| 0.35 | 0.003249 | 7.511804 |
| 0.36 | 0.003437 | 7.536273 |
| 0.37 | 0.003631 | 7.560071 |
| 0.38 | 0.003830 | 7.583236 |
| 0.39 | 0.004034 | 7.605798 |
| 0.40 | 0.004244 | 7.627788 |
| 0.41 | 0.004458 | 7.649236 |
| 0.42 | 0.004679 | 7.670167 |
| 0.43 | 0.004904 | 7.690605 |
| 0.44 | 0.005135 | 7.710574 |
| 0.45 | 0.005371 | 7.730093 |
| 0.46 | 0.005612 | 7.749184 |
| 0.47 | 0.005892 | 7.767864 |
| 0.48 | 0.006111 | 7.786151 |
| 0.49 | 0.006368 | 7.804060 |
| 0.50 | 0.006631 | 7.821608 |
| 0.51 | 0.006899 | 7.838809 |
| 0.52 | 0.007172 | 7.855675 |
| 0.53 | 0.007451 | 7.872220 |
| 0.54 | 0.007734 | 7.888456 |
| 0.55 | 0.008024 | 7.905394 |
| 0.56 | 0.008318 | 7.720044 |
| 0.57 | 0.008618 | 7.935418 |
| 0.58 | 0.008923 | 7.950524 |
| 0.59 | 0.009233 | 7.965372 |
| 0.60 | 0.009549 | 7.979971 |
| 0.61 | 0.009870 | 7.994328 |
| 0.62 | 0.010196 | 8.008452 |
| 0.63 | 0.010528 | 8.022358 |
| 0.64 | 0.010864 | 8.036028 |
| 0.65 | 0.011207 | 8.049495 |
| 0.66 | 0.01155 [...] | 8.062756 |
| 0.67 | 0.011907 | 8.075818 |
| 0.68 | 0.012265 | 8.088686 |
| 0.69 | 0.012628 | 8.101367 |
| 0.70 | 0.012997 | 8.113864 |
| [Page 137] 0.71 | 0.013371 | 8.126175 |
| 0.72 | 0.013750 | 8.138333 |
| 0.73 | 0.014135 | 8.150314 |
| 0.74 | 0.014525 | 8.162132 |
| 0.75 | 0.014920 | 8.173791 |
| 0.76 | 0.015321 | 8.185296 |
| 0.77 | 0.015727 | 8.190991 |
| 0.78 | 0.016138 | 8.207858 |
| 0.79 | 0.016554 | 8.218927 |
| 0.80 | 0.016976 | 8.229848 |
| 0.81 | 0.017403 | 8.240638 |
| 0.81 | 0.017835 | 8.251296 |
| 0.83 | 0.018273 | 7.261825 |
| 0.84 | 0.018716 | 8.272227 |
| 0.85 | 0.019164 | 8.282506 |
| 0.86 | 0.019618 | 8.292665 |
| 0.87 | 0.020077 | 8.302707 |
| 0.88 | 0.020541 | 8.312634 |
| 0.89 | 0.021011 | 8.312448 |
| 0.90 | 0.021489 | 8.332153 |
| 0.91 | 0.021999 | 8.341751 |
| 0.92 | 0.022451 | 8.351244 |
| 0.93 | 0.022942 | 8.360634 |
| 0.94 | 0.023438 | 8.369924 |
| 0.95 | 0.023939 | 8.379116 |
| 0.96 | 0.024446 | 8.388211 |
| 0.97 | 0.02495 [...] | 8.397212 |
| 0.98 | 0.025475 | 8.406121 |
| 0.99 | 0.025997 | 8.414939 |
| 1.00 | 0.026529 | 8.423668 |
| 1.01 | 0.027058 | 8.432311 |
| 1.02 | 0.027597 | 8.442369 |
| 1.03 | 0.028141 | 8.449343 |
| 1.04 | 0.028690 | 8.456735 |
| 1.05 | 0.029 [...]44 | 8.466047 |
| 1.06 | 0.029804 | 8.474280 |
| 1.07 | 0.030369 | 8.482437 |
| 1.08 | 0.030939 | 8.490516 |
| 1.09 | 0.031515 | 8.498521 |
| 1.10 | 0.032096 | 8.506454 |
| 1.11 | 0.032682 | 8.514314 |
| 1.12 | 0.033273 | 8.522104 |
| 1.13 | 0.033870 | 8.528825 |
| 1.14 | 0.034472 | 8.537478 |
| 1.15 | 0.035080 | 8.545064 |
| 1.16 | 0.035697 | 8.552584 |
| 1.17 | 0.036324 | 8.560040 |
| 1.18 | 0.036934 | 8.567432 |
| 1.19 | 0.037563 | 8.574762 |
| 1.20 | 0.038197 | 8.582031 |
| 1.21 | 0.038836 | 8.589239 |
| 1.22 | 0.039481 | 8.596388 |
| 1.23 | 0.040130 | 8.603471 |
| 1.24 | 0.040786 | 8.610512 |
| 1.25 | 0.041446 | 8.61 [...]488 |
| 1.26 | 0.042112 | 8.624410 |
| 1.27 | 0.042783 | 8.631276 |
| 1.28 | 0.043459 | 8.638088 |
| 1.29 | 0.044141 | 8.644848 |
| 1.30 | 0.044828 | 8.651555 |
| 1.31 | 0.045520 | 8.658211 |
| 1.31 | 0.046218 | 8.664816 |
| 1.33 | 0.046921 | 8.671372 |
| 1.34 | 0.047629 | 8.677878 |
| 1.35 | 0.048343 | 8.684336 |
| 1.36 | 0.049062 | 8.689746 |
| 1.37 | 0.049786 | 8.697110 |
| 1.38 | 0.050515 | 8.703427 |
| 1.39 | 0.051250 | 8.709698 |
| 1.40 | 0.051990 | 8.7 [...]5924 |
| [Page 138] 1.41 | 0.052735 | 8.722107 |
| 1.42 | 0.053453 | 8.728245 |
| 1.43 | 0.054242 | 8.734340 |
| 1.44 | 0.054007 | 8.740393 |
| 1.45 | 0.055770 | 8.746404 |
| 1.46 | 0.056542 | 8.752374 |
| 1.47 | 0.057319 | 8.758303 |
| 1.48 | 0.058102 | 8.764192 |
| 1.49 | 0.058889 | 8.770071 |
| 1.50 | 0.059687 | 8.775851 |
| 1.51 | 0.060481 | 8.781622 |
| 1.52 | 0.061285 | 8.793356 |
| 1.53 | 0.062094 | 8.793051 |
| 1.54 | 0.062908 | 8.798710 |
| 1.55 | 0.063728 | 8.804332 |
| 1.56 | 0.064556 | 8.809918 |
| 1.57 | 0.065383 | 8.815468 |
| 1.58 | 0.066219 | 8.820983 |
| 1.59 | 0.067059 | 8.826463 |
| 1.60 | 0.067906 | 8.831908 |
| 1.61 | 0.068754 | 8.837320 |
| 1.62 | 0.069614 | 8.842698 |
| 1.63 | 0.070476 | 8.848044 |
| 1.64 | 0.071343 | 8.853356 |
| 1.65 | 0.072216 | 8.858636 |
| 1.66 | 0.073091 | 8.863885 |
| 1.67 | 0.073944 | 8.869101 |
| 1.68 | 2.074533 | 8.874595 |
| 1.69 | 0.075760 | 8.879442 |
| 1.70 | 0.076659 | 8.884566 |
| 1.71 | 0.077564 | 8.889661 |
| 1.72 | 0.078440 | 8.894725 |
| 1.73 | 0.079355 | 8.899761 |
| 1.74 | 0.080309 | 8.904767 |
| 1.75 | 0.081235 | 8.909745 |
| 1.76 | 0.082166 | 8.914694 |
| 1.77 | 0.083102 | 8.919615 |
| 1.78 | 0.084044 | 8.924508 |
| 1.79 | 0.084991 | 8.929374 |
| 1.80 | 0.085943 | 8.934213 |
| 1.81 | 0.086901 | 8.939026 |
| 1.82 | 0.087864 | 8.943811 |
| 1.83 | 0.088832 | 8.948571 |
| 1.84 | 0.089805 | 8.953291 |
| 1.85 | 0.090784 | 8.958012 |
| 1.86 | 0.091768 | 8.962694 |
| 1.87 | 0.092758 | 8.967352 |
| 1.88 | 0.093752 | 8.971984 |
| 1.89 | 0.094418 | 8.976592 |
| 1.90 | 0.095748 | 8.981176 |
| 1.91 | 0.096768 | 8.985735 |
| 1.92 | 0.097784 | 8.990271 |
| 1.93 | 0.098806 | 8.994783 |
| 1.94 | 0.099832 | 8.999272 |
| 1.95 | 0.100864 | 9.003738 |
| 1.96 | 0.101901 | 9.008181 |
| 1.97 | 0.102944 | 9.012601 |
| 1.98 | 0.103981 | 9.016999 |
| 1.99 | 0.105044 | 9.021375 |
| 2.00 | 0.106103 | 9.025728 |
| 2.01 | 0.107166 | 9.030061 |
| 2.02 | 0.108235 | 9.034371 |
| 2.03 | 0.109310 | 9.038660 |
| 2.04 | 0.110389 | 9.042929 |
| 2.05 | 0.111441 | 9.047176 |
| 2.06 | 0.112564 | 9.051403 |
| 2.07 | 0.113660 | 9.055609 |
| 2.08 | 0.114427 | 9.059795 |
| 2.09 | 0.115867 | 9.063961 |
| 2.10 | 0.116978 | 9.068107 |
| [Page 139] 2.11 | 0.118095 | 9.072233 |
| 2.12 | 0.119210 | 9.076340 |
| 2.13 | 0.120345 | 9.080428 |
| 2.14 | 0.121477 | 9.084496 |
| 2.15 | 0.122614 | 9.088545 |
| 2.16 | 0.123758 | 9.092576 |
| 2.17 | 0.124907 | 9.096588 |
| 2.18 | 0.126061 | 9.100581 |
| 2.19 | 0.129220 | 9.104557 |
| 2.20 | 0.128384 | 9.108514 |
| 2.21 | 0.129554 | 9.112453 |
| 2.22 | 0.130729 | 9.116374 |
| 2.23 | 0.131910 | 9.120278 |
| 2.24 | 0.133095 | 9.124164 |
| 2.25 | 0.134286 | 9.128033 |
| 2.26 | 0.135483 | 9.131885 |
| 2.27 | 0.136684 | 9.135720 |
| 2.28 | 0.137891 | 9.139538 |
| 2.29 | 0.139104 | 9.143339 |
| 2.30 | 0.140321 | 9.147124 |
| 2.31 | 0.141544 | 9.150892 |
| 2.32 | 0.142772 | 9.154644 |
| 2.33 | 0.144006 | 9.158380 |
| 2.34 | 0.145244 | 9.162100 |
| 2.35 | 0.146488 | 9.165804 |
| 2.36 | 0.147738 | 9.169492 |
| 2.37 | 0.148992 | 9.173165 |
| 2.38 | 0.150252 | 9.176822 |
| 2.39 | 0.151518 | 9.180464 |
| 2.40 | 0.152788 | 9.184091 |
| 2.41 | 0.154064 | 9.187702 |
| 2.42 | 0.155345 | 9.191299 |
| 2.43 | 0.156632 | 9.194881 |
| 2.44 | 0.157924 | 9.198448 |
| 2.45 | 0.159221 | 9.202001 |
| 2.46 | 0.160523 | 9.205539 |
| 2.47 | 0.161831 | 9.209062 |
| 2.48 | 0.163144 | 9.212571 |
| 2.49 | 0.164462 | 9.216067 |
| 2.50 | 0.165786 | 9.219548 |
| 2.51 | 0.167115 | 9.222016 |
| 2.52 | 0.168449 | 9.226469 |
| 2.53 | 0.169789 | 9.229909 |
| 2.54 | 0.171133 | 9.233336 |
| 2.55 | 0.172484 | 9.236749 |
| 2.56 | 0.173839 | 9.240148 |
| 2.57 | 0.175200 | 9.243535 |
| 2.58 | 0.176566 | 9.246908 |
| 2.59 | 0.177937 | 9.250268 |
| 2.60 | 0.179314 | 9.253615 |
| 2.61 | 0.180685 | 9.256949 |
| 2.62 | 0.182083 | 9.260271 |
| 2.63 | 0.183476 | 9.263580 |
| 2.64 | 0.184874 | 9.266876 |
| 2.65 | 0.186277 | 9.270160 |
| 2.66 | 0.187686 | 9.273432 |
| 2.67 | 0.189099 | 9.276691 |
| 2.68 | 0.190519 | 9.279938 |
| 2.69 | 0.191943 | 9.283173 |
| 2.70 | 0.193373 | 9.286396 |
| 2.71 | 0.194808 | 9.289607 |
| 2.72 | 0.196248 | 9.292806 |
| 2.73 | 0.197694 | 9.295994 |
| 2.74 | 0.199145 | 9.299170 |
| 2.75 | 0.200601 | 9.302334 |
| 2.76 | 0.202630 | 9.305486 |
| 2.77 | 0.203529 | 9.308628 |
| 2.78 | 0.205002 | 9.311758 |
| 2.79 | 0.206479 | 9.314877 |
| 2.80 | 0.207962 | 9.317984 |
| [Page 140] | Natural | Artificial |
| 2.81 | 0.209450 | 9.321081 |
| 2.82 | 0.210943 | 9.324167 |
| 2.83 | 0.212442 | 9.327241 |
| 2.84 | 0.213946 | 9.330306 |
| 2.85 | 0.215455 | 9.333358 |
| 2.86 | 0.216970 | 9.336400 |
| 2.87 | 0.218490 | 9.339442 |
| 2.88 | 0.220015 | 9.342453 |
| 2.89 | 0.221546 | 9.344464 |
| 2.90 | 0.223082 | 9.348464 |
| 2.91 | 0.224623 | 9.351454 |
| 2.92 | 0.226169 | 9.354434 |
| 2.93 | 0.227721 | 9.357404 |
| 2.94 | 0.229278 | 9.360363 |
| 2.95 | 0.230840 | 9.363312 |
| 2.96 | 0.232408 | 9.366252 |
| 2.97 | 0.233981 | 9.369181 |
| 2.98 | 0.235559 | 9.372101 |
| 2.99 | 0.237143 | 9.375011 |
| 3.00 | 0.238732 | 9.377911 |
| 3.01 | 0.240326 | 9.380801 |
| 3.02 | 0.241926 | 9.383682 |
| 3.03 | 0.243530 | 9.386554 |
| 3.04 | 0.245141 | 9.389416 |
| 3.05 | 0.246756 | 6.392268 |
| 3.06 | 0.248377 | 9.395111 |
| 3.07 | 0.249003 | 9.397945 |
| 3.08 | 0.251634 | 9.400770 |
| 3.09 | 0.253271 | 9.403585 |
| 3.10 | 0.254713 | 9.406392 |
| 3.11 | 0.256627 | 9.409189 |
| 3.12 | 0.258212 | 9.411978 |
| 3.13 | 0.259870 | 9.414757 |
| 3.14 | 0.261534 | 9.417528 |
| 3.15 | 0.263202 | 9.420290 |
| 3.16 | 0.264876 | 9.423043 |
| 3.17 | 0.266555 | 9.425787 |
| 3.18 | 0.268239 | 9.428523 |
| 3.19 | 0.269926 | 9.431250 |
| 3.20 | 0.271624 | 9.433968 |
| 3.21 | 0.273324 | 9.436678 |
| 3.22 | 0.275030 | 9.439380 |
| 3.23 | 0.276741 | 9.442073 |
| 3.24 | 0.278457 | 9.444758 |
| 3.25 | 0.280179 | 9.447435 |
| 3.26 | 0.281905 | 9.450104 |
| 3.27 | 0.283604 | 9.452764 |
| 3.28 | 0.285375 | 9.455416 |
| 3.29 | 0.287114 | 9.458060 |
| 3.30 | 0.288866 | 9.460695 |
| 3.31 | 0.290619 | 9.463324 |
| 3.32 | 0.292378 | 9.465945 |
| 3.33 | 0.294142 | 9.468557 |
| 3.34 | 0.295911 | 9.471161 |
| 3.35 | 0.297686 | 9.473758 |
| 3.36 | 0.299465 | 9.476347 |
| 3.37 | 0.301251 | 9.478928 |
| 3.38 | 0.303041 | 9.481502 |
| 3.39 | 0.304504 | 9.484068 |
| 3.40 | 0.306638 | 9.486626 |
| 3.41 | 0.308444 | 9.489177 |
| 3.42 | 0.310256 | 9.491721 |
| 3.43 | 0.312073 | 9.494257 |
| 3.44 | 0.313895 | 9.496785 |
| 3.45 | 0.315726 | 9.499307 |
| 3.46 | 0.317556 | 9.501821 |
| 3.47 | 0.319394 | 9.504327 |
| 3.48 | 0.321238 | 9.506782 |
| 3.49 | 0.323087 | 9.509319 |
| 3.50 | 0.324941 | 9.511804 |
| [Page 141] 3.51 | 0.326800 | 9.514283 |
| 3.52 | 0.328665 | 9.516754 |
| 3.53 | 0.330502 | 9.519218 |
| 3.54 | 0.332411 | 9.521675 |
| 3.55 | 0.334291 | 9.524125 |
| 3.56 | 0.336177 | 9.526568 |
| 3.57 | 0.338068 | 9.528005 |
| 3.58 | 0.339965 | 9.531434 |
| 3.59 | 0.341867 | 9.533857 |
| 3.60 | 0.343774 | 9.536273 |
| 3.61 | 0.345687 | 9.538683 |
| 3.62 | 0.347604 | 9.541086 |
| 3.63 | 0.349528 | 9.543482 |
| 3.64 | 0.351456 | 9.545871 |
| 3.65 | 0.353390 | 9.547254 |
| 3.66 | 0.355329 | 9.550631 |
| 3.67 | 0.357273 | 9.553001 |
| 3.68 | 0.359223 | 9.555364 |
| 3.69 | 0.361178 | 9.557721 |
| 3.70 | 0.363138 | 9.560071 |
| 3.71 | 0.365104 | 9.562416 |
| 3.72 | 0.367074 | 9.564754 |
| 3.73 | 0.369051 | 9.567086 |
| 3.74 | 0.371032 | 9.569412 |
| 3.75 | 0.373019 | 9.571731 |
| 3.76 | 0.375011 | 9.574044 |
| 3.77 | 0.377008 | 9.576351 |
| 3.78 | 0.379011 | 9.578652 |
| 3.79 | 0.381019 | 9.580947 |
| 3.80 | 0.383032 | 9.583236 |
| 3.81 | 0.38505 [...] | 9.585518 |
| 3.82 | 0.387075 | 9.587795 |
| 3.83 | 0.389104 | 9.590066 |
| 3.8 [...] | 0.391139 | 9.592331 |
| 3.85 | 0.393179 | 9.594590 |
| 3.86 | 0.395224 | 9.596743 |
| 3.87 | 0.397274 | 9.599090 |
| 3.88 | 0.399330 | 9.601332 |
| 3.89 | 0.401391 | 9.603568 |
| 3.90 | 0.403457 | 9.605798 |
| 3.91 | 0.405529 | 9.608022 |
| 3.92 | 0.407606 | 9.610241 |
| 3.93 | 0.409688 | 9.612454 |
| 3.94 | 0.411776 | 9.614661 |
| 3.95 | 0.413869 | 9.616863 |
| 3.96 | 0.415967 | 9.619059 |
| 3.97 | 0.418070 | 9.621249 |
| 3.98 | 0.420179 | 9.623435 |
| 3.99 | 0.422293 | 9.625614 |
| 4.00 | 0.424413 | 9.627788 |
| 4.01 | 0.426504 | 9.629957 |
| 4.02 | 0.428667 | 9.632121 |
| 4.03 | 0.430803 | 9.634279 |
| 4.04 | 0.432943 | 9.636431 |
| 4.05 | 0.435089 | 9.638578 |
| 4.06 | 0.437241 | 9.640720 |
| 4.07 | 0.439397 | 9.642857 |
| 4.08 | 0.431559 | 9.644989 |
| 4.09 | 0.443726 | 9.647115 |
| 4.10 | 0.445899 | 9.649236 |
| 4.11 | 0.448076 | 9.651352 |
| 4.12 | 0.450259 | 9.653463 |
| 4.13 | 0.452448 | 9.655569 |
| 4.14 | 0.454642 | 9.657669 |
| 4.15 | 0.456840 | 9.659765 |
| 4.16 | 0.459045 | 9.661855 |
| 4.17 | 0.461254 | 9.663941 |
| 4.18 | 0.463469 | 9.666021 |
| 4.19 | 0.465690 | 9.668096 |
| 4.20 | 0.467915 | 9.670167 |
| [Page 142] 4.21 | 0.470146 | 9.672233 |
| 4.22 | 0.472382 | 9.674293 |
| 4.23 | 0.474623 | 9.676349 |
| 4.24 | 0.476870 | 9.678400 |
| 4.25 | 0.479122 | 9.680446 |
| 4.26 | 0.481380 | 9.682488 |
| 4.27 | 0.483642 | 9.684524 |
| 4.28 | 0.485910 | 9.686556 |
| 4.29 | 0.488183 | 9.688583 |
| 4.30 | 0.490462 | 9.690605 |
| 4.31 | 0.492746 | 9.692623 |
| 4.32 | 0.495032 | 9.694636 |
| 4.33 | 0.497330 | 9.696634 |
| 4.34 | 0.499629 | 9.698648 |
| 4.35 | 0.501934 | 9.700647 |
| 4.36 | 0.504245 | 9.702641 |
| 4.37 | 0.506560 | 9.704631 |
| 4.38 | 0.508882 | 9.706617 |
| 4.39 | 0.511208 | 9.708597 |
| 4.40 | 0.513539 | 9.710574 |
| 4.41 | 0.515876 | 9.712546 |
| 4.42 | 0.518219 | 9.714513 |
| 4.43 | 0.520566 | 9.716476 |
| 4.44 | 0.522919 | 9.718434 |
| 4.45 | 0.527277 | 9.720388 |
| 4.46 | 0.527641 | 9.722338 |
| 4.47 | 0.530009 | 9.724283 |
| 4.48 | 0.532383 | 9.726224 |
| 4.49 | 0.534429 | 9.728161 |
| 4.50 | 0.537147 | 9.730093 |
| 4.51 | 0.539537 | 9.732021 |
| 4.52 | 0.541933 | 9.733945 |
| 4.53 | 0.544333 | 9.735865 |
| 4.54 | 0.546739 | 9.737780 |
| 4.55 | 0.549150 | 9.739691 |
| 4.56 | 0.551567 | 9.741598 |
| 4.57 | 0.553989 | 9.743501 |
| 4.58 | 0.556416 | 9.745399 |
| 4.59 | 0.558848 | 9.747294 |
| 4.60 | 0.561286 | 9.749184 |
| 4.61 | 0.563729 | 9.751070 |
| 4.62 | 0.566177 | 9.752952 |
| 4.63 | 0.568631 | 9.754830 |
| 4.64 | 0.571090 | 9.756704 |
| 4.65 | 0.573554 | 9.758574 |
| 4.66 | 0.576024 | 9.760440 |
| 4.67 | 0.578499 | 9.762302 |
| 4.68 | 0.580979 | 9.764160 |
| 4.69 | 0.583464 | 9.766014 |
| 4.70 | 0.585955 | 9.767864 |
| 4.71 | 0.588451 | 9.769710 |
| 4.72 | 0.590952 | 9.771552 |
| 4.73 | 0.593459 | 9.773411 |
| 4.74 | 0.595971 | 9.775225 |
| 4.75 | 0.598488 | 9.777056 |
| 4.76 | 0.601011 | 9.778882 |
| 4.77 | 0.603539 | 9.780705 |
| 4.78 | 0.606072 | 9.782524 |
| 4.79 | 0.608611 | 9.784339 |
| 4.80 | 0.611154 | 9.786151 |
| 4.81 | 0.613704 | 9.787959 |
| 4.82 | 0.616258 | 9.789762 |
| 4.83 | 0.618818 | 9.791563 |
| 4.84 | 0.621383 | 9.793359 |
| 4.85 | 0.623953 | 9.795252 |
| 4.86 | 0.626529 | 9.796941 |
| 4.87 | 0.629110 | 9.798726 |
| 4.88 | 0.631696 | 9.800508 |
| 4.89 | 0.634288 | 9.802286 |
| 4.90 | 0.636885 | 9.804060 |
| [Page 143] 4.92 | 0.639487 | 9.805831 |
| 4.92 | 0.642094 | 9.807599 |
| 4.93 | 0.644707 | 9.809362 |
| 4.94 | 0.647325 | 9.811122 |
| 4.95 | 0.649948 | 9.812879 |
| 4.96 | 0.652577 | 9.814632 |
| 4.97 | 0.655211 | 9.815638 |
| 4.98 | 0.657851 | 9.818127 |
| 4.99 | 0.660495 | 9.819869 |
| 5.00 | 0.663145 | 9.821608 |
| 5.01 | 0.665800 | 9.823344 |
| 5.02 | 0.668461 | 9.825076 |
| 5.03 | 0.671127 | 9.826804 |
| 5.04 | 0.673798 | 9.828529 |
| 5.05 | 0.676474 | 9.830451 |
| 5.06 | 0.679156 | 9.831979 |
| 5.07 | 0.681843 | 9.833684 |
| 5.08 | 0.684536 | 9.835396 |
| 5.09 | 0.687267 | 9.837104 |
| 5.10 | 0.689936 | 9.838809 |
| 5.11 | 0.692644 | 9.840516 |
| 5.12 | 0.695358 | 9.842208 |
| 5.13 | 0.698077 | 9.843903 |
| 5.14 | 0.700801 | 9.845595 |
| 5.15 | 0.703531 | 9.847283 |
| 5.16 | 0.706265 | 9.848968 |
| 5.17 | 0.709006 | 9.850649 |
| 5.18 | 0.711751 | 9.852328 |
| 5.19 | 0.71450 [...] | 9.854003 |
| 5.20 | 0.717258 | 9.855675 |
| 5.21 | 0.720019 | 9.857344 |
| 5.22 | 0.722786 | 9:859009 |
| 5.23 | 0.725558 | 9.860672 |
| 5.24 | [...].728335 | 9.862331 |
| 5:25 | 0.731118 | 9.863987 |
| 5.26 | 0.733905 | 9.865640 |
| 5.27 | 0.736699 | 9.867290 |
| 5.28 | 0.739497 | 9.868936 |
| 5.29 | 0.742301 | 9.870580 |
| 5.30 | 0.745110 | 9.872220 |
| 5.31 | 0.747924 | 9.873857 |
| 5.32 | 0.750744 | 9.875492 |
| 5.33 | 0.753569 | 9.877223 |
| 5.34 | 0.756399 | 9.878751 |
| 5.35 | 0.759235 | 9.880376 |
| 5.36 | 0.762076 | 9.881998 |
| 5.37 | 0.764922 | 9.883617 |
| 5.38 | 0.767774 | 9.885233 |
| 5.39 | 0.770630 | 9.886846 |
| 5.40 | 0.773493 | 9.888456 |
| 5.41 | 0.776360 | 9.890063 |
| 5.42 | 0.779233 | 9.891667 |
| 5.43 | 0.782111 | 9.892268 |
| 5.44 | 0.784994 | 9.894866 |
| 5.45 | 0.78788 [...] | 9.896461 |
| 5.46 | 0.790777 | 9.898054 |
| 5.47 | 0.793676 | 9.899643 |
| 5.48 | 0.796581 | 9.900230 |
| 5.49 | 0.799490 | 9.902813 |
| 5.50 | 0.802406 | 9.905394 |
| 5.51 | 0.805326 | 9.905972 |
| 5.52 | 0.808252 | 9.907547 |
| 5.53 | 0.811283 | 9.909119 |
| 5.54 | 0.814129 | 9.910688 |
| 5.55 | 0.817061 | 9.912254 |
| 5.56 | 0.820008 | 9.913818 |
| 5.57 | 0.822994 | 9.915379 |
| 5.58 | 0.825918 | 9.916937 |
| 5.59 | 0.828881 | 9.918492 |
| 5.60 | 0.831849 | 9.920044 |
| [Page 144] 5.61 | 0.834823 | 9.921594 |
| 5.62 | 0.837802 | 9.923141 |
| 5.63 | 0.840786 | 9.924685 |
| 5.64 | 0.843775 | 9.926227 |
| 5.65 | 0.846770 | 9.927765 |
| 5.66 | 0.849770 | 9.929301 |
| 5.67 | 0.852776 | 9.930835 |
| 5.68 | 0.855786 | 9.932365 |
| 5.69 | 0.858802 | 9.933893 |
| 5.70 | 0.861824 | 9.935418 |
| 5.71 | 0.864850 | 9.936941 |
| 5.72 | 0.867882 | 9.938460 |
| 5.73 | 0.870919 | 9.939978 |
| 5.74 | 0.873963 | 9.941492 |
| 5.75 | 0.877010 | 9.943004 |
| 5.76 | 0.880063 | 9.944513 |
| 5.77 | 0.883121 | 9.946020 |
| 5.78 | 0.886185 | 9.947524 |
| 5.79 | 0.889254 | 9.949026 |
| 5.80 | 0.892328 | 9.950524 |
| 5.81 | 0.895408 | 9.952021 |
| 5.82 | 0.898493 | 9.953514 |
| 5.83 | 0.901586 | 9.955006 |
| 5.84 | 0.904679 | 9.956494 |
| 5.85 | 0.907779 | 9.957980 |
| 5.86 | 0.910886 | 9.959464 |
| 5.87 | 0.913997 | 9.960945 |
| 5.88 | 0.917114 | 9.962423 |
| 5.89 | 0.920236 | 9.963908 |
| 5.90 | 0.923363 | 9.965372 |
| 5.91 | 0.926496 | 9.966843 |
| 5.92 | 0.929967 | 9.968312 |
| 5.93 | 0.932777 | 9.969778 |
| 5.94 | 0.935926 | 9.971241 |
| 5.95 | 0.939080 | 9.972702 |
| 5.96 | 0.942239 | 9.974161 |
| 5.97 | 0.945404 | 9.975617 |
| 5.98 | 0.948574 | 9.977071 |
| 5.99 | 0.951749 | 9.978522 |
| 6.00 | 0.954929 | 9.979971 |
| 6.01 | 0.958115 | 9.981417 |
| 6.02 | 0.961306 | 9.982861 |
| 6.03 | 0.964502 | 9.984303 |
| 6.04 | 0.967704 | 9.985742 |
| 6.05 | 0.970911 | 9.987179 |
| 6.06 | 0.974123 | 9.988614 |
| 6.07 | 0.977341 | 9.990046 |
| 6.08 | 0.980564 | 9.991476 |
| 6.09 | 0.983792 | 9.992903 |
| 6.10 | 0.987025 | 9.994328 |
| 6.11 | 0.990264 | 9.995751 |
| 6.12 | 0.993508 | 9.997171 |
| 6.13 | 0.996758 | 9.998589 |
| 6.14 | 1.000012 | 0.000005 |
| 6.15 | 1.003272 | 0.001419 |
| 6.16 | 1.006538 | 0.002830 |
| 6.17 | 1.009808 | 0.004239 |
| 6.18 | 1.013084 | 0.005645 |
| 6.19 | 1.016366 | 0.007050 |
| 6.20 | 1.019652 | 0.008452 |
| 6.21 | 1.022944 | 0.009852 |
| 6.22 | 1.026241 | 0.011249 |
| 6.23 | 1.029544 | 0.012645 |
| 6.24 | 1.032851 | 0.014038 |
| 6.25 | 1.036164 | 0.015428 |
| 6.26 | 1.039450 | 0.016817 |
| 6.27 | 1.042807 | 0.018203 |
| 6.28 | 1.046136 | 0.019588 |
| 6.29 | 1.049470 | 0.020970 |
| 6.30 | 1.052809 | 0.022350 |
| [Page 145] 6.31 | 1.056154 | 0.023727 |
| 6.32 | 1.059505 | 0.025103 |
| 6.33 | 1.062860 | 0.026476 |
| 6.34 | 1.066331 | 0.027847 |
| 6.35 | 1.069587 | 0.029216 |
| 6.36 | 1.072958 | 0.030583 |
| 6.37 | 1.076335 | 0.031947 |
| 6.38 | 1.079717 | 0.033310 |
| 6.39 | 1.083105 | 0.034670 |
| 6.40 | 1.086497 | 0.036028 |
| 6.41 | 1.089899 | 0.037384 |
| 6.42 | 1.093298 | 0.038738 |
| 6.43 | 1.096707 | 0.040090 |
| 6.44 | 1.100121 | 0.041440 |
| 6.45 | 1.103540 | 0.042788 |
| 6.46 | 1.106965 | 0.044133 |
| 6.47 | 1.110394 | 0.045477 |
| 6.48 | 1.113829 | 0.046818 |
| 6.49 | 1.117270 | 0.048158 |
| 6.50 | 1.120716 | 0.049495 |
| 6.51 | 1.124167 | 0.050830 |
| 6.52 | 1.127623 | 0.052164 |
| 6.53 | 1.131084 | 0.053495 |
| 6.54 | 1.134551 | 0.054824 |
| 6.55 | 1.138024 | 0.056150 |
| 6.56 | 1.141501 | 0.057476 |
| 6.57 | 1.144984 | 0.058799 |
| 6.58 | 1.148472 | 0.060120 |
| 6.59 | 1.152966 | 0.061439 |
| 6.60 | 1.155464 | 0.062756 |
| 6.61 | 1.158968 | 0.064071 |
| 6.62 | 1.162478 | 0.065438 |
| 6.63 | 1.165992 | 0.066895 |
| 6.64 | 1.166951 | 0.068005 |
| 6.65 | 1.173038 | 0.069312 |
| 6.66 | 1.175688 | 0.070617 |
| 6.67 | 1.180104 | 0.071920 |
| 6.68 | 1.183645 | 0.073221 |
| 6.69 | 1.187192 | 0.074521 |
| 6.70 | 1.190744 | 0.075815 |
| 6.71 | 1.194301 | 0.077113 |
| 6.72 | 1.197863 | 0.078407 |
| 6.73 | 1.201431 | 0.079699 |
| 6.74 | 1.205004 | 0.080988 |
| 6.75 | 1.208582 | 0.082276 |
| 6.76 | 1.212166 | 0.083562 |
| 6.77 | 1.215755 | 0.084846 |
| 6.78 | 1.219349 | 0.086118 |
| 6.79 | 1.222949 | 0.087408 |
| 6.80 | 1.226554 | 0.088686 |
| 6.81 | 1.230164 | 0.089963 |
| 6.82 | 1.233779 | 0.091237 |
| 6.83 | 1.237073 | 0.092510 |
| 6.84 | 1.241026 | 0.093781 |
| 6.85 | 1.244657 | 0.095049 |
| 6.86 | 1.248294 | 0.096317 |
| 6.87 | 1.252936 | 0.097582 |
| 6.88 | 1.255587 | 0.098845 |
| 6.89 | 1.259236 | 0.100107 |
| 6.90 | 1.262894 | 0.101367 |
| 6.91 | 1.266557 | 0.102525 |
| 6.92 | 1.270226 | 0.103881 |
| 6.93 | 1.274900 | 0.105135 |
| 6.94 | 1.27757 | 0.106387 |
| 6.95 | 1.281263 | 0.107638 |
| 6.96 | 1.284953 | 0.108877 |
| 6.97 | 1.288648 | 0.110134 |
| 6.98 | 1.292348 | 0.111379 |
| 6.99 | 1.296054 | 0.112623 |
| 7.00 | 1.299765 | 0.113864 |
| [Page 146] 7.01 | 1.303481 | 0.115104 |
| 7.02 | 1.307203 | 0.116343 |
| 7.03 | 1.310930 | 0.117579 |
| 7.04 | 1.314662 | 0.118814 |
| 7.05 | 1.318399 | 0.120047 |
| 7.06 | 1.322142 | 0.121278 |
| 7.07 | 1.325890 | 0.122507 |
| 7.08 | 1.329644 | 0.123735 |
| 7.09 | 1.333402 | 0.124961 |
| 7.10 | 1.337166 | 0.126175 |
| 7.11 | 1.340936 | 0.127408 |
| 7.12 | 1.344710 | 0.128628 |
| 7.13 | 1.348490 | 0.129747 |
| 7.14 | 1.352275 | 0.131065 |
| 7.15 | 1.356066 | 0.132280 |
| 7.16 | 1.359862 | 0.133494 |
| 7.17 | 1.363667 | 0.134707 |
| 7.18 | 1.367469 | 0.135917 |
| 7.19 | 1.371281 | 0.137126 |
| 7.20 | 1.375098 | 0.138333 |
| 7.21 | 1.378921 | 0.139539 |
| 7.22 | 1.382748 | 0.140743 |
| 7.23 | 1.386581 | 0.141946 |
| 7.24 | 1.390419 | 0.143146 |
| 7.25 | 1.394263 | 0.144344 |
| 7.26 | 1.398112 | 0.145542 |
| 7.27 | 1.401966 | 0.146737 |
| 7.28 | 1.405826 | 0.147931 |
| 7.29 | 1.409691 | 0.149123 |
| 7.30 | 1.413561 | 0.150314 |
| 7.31 | 1.417436 | 0.151503 |
| 7.32 | 1.421317 | 0.152691 |
| 7.33 | 1.425203 | 0.153876 |
| 7.34 | 1.429094 | 0.155060 |
| 7.35 | 1.432991 | 0.156243 |
| 7.36 | 1.436893 | 0.157424 |
| 7.37 | 1.440800 | 0.158603 |
| 7.38 | 1.444713 | 0.159781 |
| 7.39 | 1.448630 | 0.160957 |
| 7.40 | 1.452554 | 0.162132 |
| 7.41 | 1.456482 | 0.163305 |
| 7.42 | 1.460416 | 0.164476 |
| 7.43 | 1.464355 | 0.165646 |
| 7.44 | 1.468299 | 0.166814 |
| 7.45 | 1.472249 | 0.167981 |
| 7.46 | 1.476204 | 0.169146 |
| 7.47 | 1.480164 | 0.170310 |
| 7.48 | 1.484130 | 0.171472 |
| 7.49 | 1.488101 | 0.172632 |
| 7.50 | 1.492077 | 0.173791 |
| 7.51 | 1.496059 | 0.174948 |
| 7.52 | 1.500045 | 0.176104 |
| 7.53 | 1.504038 | 0.177218 |
| 7.54 | 1.508035 | 0.178411 |
| 7.55 | 1.512038 | 0.179552 |
| 7.56 | 1.516046 | 0.180712 |
| 7.57 | 1.520059 | 0.181860 |
| 7.58 | 1.524078 | 0.183007 |
| 7.59 | 1.528102 | 0.184152 |
| 7.60 | 1.532131 | 0.185296 |
| 7.6 [...] | 1.536166 | 0.186438 |
| 7.62 | 1.540260 | 0.187578 |
| 7.63 | 1.544251 | 0.188717 |
| 7.64 | 1.548301 | 0.189855 |
| 7.65 | 1.552355 | 0.190991 |
| 7.66 | 1.556418 | 0.192126 |
| 7.67 | 1.560418 | 0.193259 |
| 7.68 | 1.564556 | 0.194391 |
| 7.69 | 1.568633 | 0.195521 |
| 7.70 | 1.572716 | 0.196650 |
| [Page 147] 7.71 | 1.576803 | 0.197777 |
| 7.72 | 1.580896 | 0.198903 |
| 7.73 | 1.584994 | 0.200027 |
| 7.74 | 1.589098 | 0.201150 |
| 7.75 | 1.593207 | 0.202272 |
| 7.78 | 1.597321 | 0.203402 |
| 7.77 | 1.601440 | 0.204500 |
| 7.78 | 1.605565 | 0.205628 |
| 7.79 | 1.609695 | 0.206744 |
| 7.80 | 1.613831 | 0.207858 |
| 7.81 | 1.617971 | 0.208970 |
| 7.82 | 1.622117 | 0.210082 |
| 7.83 | 1.626269 | 0.211192 |
| 7.84 | 1.630425 | 0.212311 |
| 7.85 | 1.634587 | 0.213408 |
| 7.86 | 1.638754 | 0.214514 |
| 7.87 | 1.642927 | 0.215618 |
| 7.88 | 1.647105 | 0.216721 |
| 7.89 | 1.651288 | 0.217822 |
| 7.90 | 1.655476 | 0.218923 |
| 7.91 | 1.659670 | 0.220021 |
| 7.92 | 1.663869 | 0.221119 |
| 7.93 | 1.668073 | 0.222215 |
| 7.94 | 1.672283 | 0.223309 |
| 7.95 | 1.676498 | 0.224403 |
| 7.96 | 1.680718 | 0.225495 |
| 7.97 | 1.684944 | 0.226585 |
| 7.98 | 1.689175 | 0.227674 |
| 7.99 | 1.693411 | 0.228762 |
| 8.00 | 1.697652 | 0.229848 |
| 8.01 | 1.701899 | 0.230933 |
| 8.02 | 1.706151 | 0.232017 |
| 8.03 | 1.710408 | 0.233100 |
| 8.04 | 1.714671 | 0.23418 [...] |
| 8.05 | 1.718939 | 0.235260 |
| 8.06 | 1.723212 | 0.236338 |
| 8.07 | 1.727491 | 0.237415 |
| 8.08 | 1.731775 | 0.238491 |
| 8.09 | 1.736064 | 0.239565 |
| 8.10 | 1.740359 | 0.240638 |
| 8.11 | 1.744659 | 0.241710 |
| 8.12 | 1.748964 | 0.242780 |
| 8.13 | 1.753274 | 0.243850 |
| 8.14 | 1.757590 | 0.244917 |
| 8.15 | 1.761911 | 0.245984 |
| 8.16 | 1.766237 | 0.247049 |
| 8.17 | 1.770569 | 0.248113 |
| 8.18 | 1.774906 | 0.249175 |
| 8.19 | 1.779248 | 0.250236 |
| 8.20 | 1.783596 | 0.251296 |
| 8.21 | 1.787949 | 0.252355 |
| 8.22 | 1.792307 | 0.253412 |
| 8.23 | 1.796670 | 0.254468 |
| 8.24 | 1.801039 | 0.255523 |
| 8.25 | 1.805413 | 0.256576 |
| 8.26 | 1.809793 | 0.257629 |
| 8.27 | 1.814177 | 0.258679 |
| 8.28 | 1.818568 | 0.259729 |
| 8.29 | 1.822963 | 0.260777 |
| 8.30 | 1.827363 | 0.261825 |
| 8.31 | 1.831769 | 0.262870 |
| 8.32 | 1.836181 | 0.263915 |
| 8.33 | 1.840597 | 0.264958 |
| 8.34 | 1.845019 | 0.266001 |
| 8.35 | 1.849480 | 0.267041 |
| 8.36 | 1.853879 | 0.268081 |
| 8.37 | 2.858316 | 0.269119 |
| 8.38 | 1.862760 | 0.270156 |
| 8.39 | 1.867208 | 0.271192 |
| 8.40 | 1.871662 | 0.272227 |
| [Page 148] 8.41 | 1.876121 | 0.273250 |
| 8.42 | 1.880585 | 0.274293 |
| 8.43 | 1.885054 | 0.275324 |
| 8.44 | 1.889529 | 0.276353 |
| 8.45 | 1.894010 | 0.277382 |
| 8.46 | 1.898495 | 0.278409 |
| 8.47 | 1.902886 | 0.279435 |
| 8.48 | 1.907482 | 0.280450 |
| 8.49 | 1.911983 | 0.281484 |
| 8.50 | 1.916490 | 0.282506 |
| 8.51 | 1.921002 | 0.283528 |
| 8.52 | 1.925520 | 0.283748 |
| 8.53 | 1.930042 | 0.285566 |
| 8.54 | 1.934570 | 0.286584 |
| 8.55 | 1.939104 | 0.287601 |
| 8.56 | 1.943642 | 0.288616 |
| 8.57 | 1.948186 | 0.289630 |
| 8.58 | 1.952735 | 0.290643 |
| 8.59 | 1.957290 | 0.291655 |
| 8.60 | 1.961849 | 0.292665 |
| 8.61 | 1.966414 | 0.293675 |
| 8.62 | 1.970985 | 0.294683 |
| 8.63 | 1.975561 | 0.295690 |
| 8.64 | 1.980142 | 0.296696 |
| 8.65 | 1.984728 | 0.297701 |
| 8.66 | 1.989320 | 0.298704 |
| 8.67 | 1.993916 | 0.299707 |
| 8.68 | 1.998529 | 0.300708 |
| 8.69 | 2.003126 | 0.301708 |
| 8.70 | 2.007406 | 0.302707 |
| 8.71 | 2.012357 | 0.303705 |
| 8.72 | 2.016981 | 0.304701 |
| 8.73 | 2.021609 | 0.305697 |
| 8.74 | 2.026243 | 0.306691 |
| 8.75 | 2.030883 | 0.307685 |
| 8.76 | 2.035528 | 0.308677 |
| 8.77 | 2.040177 | 0.309668 |
| 8.78 | 2.044833 | 0.310657 |
| 8.79 | 2.049493 | 0.311646 |
| 8.80 | 2.054159 | 0.312634 |
| 8.81 | 2.058830 | 0.313620 |
| 8.82 | 2.063507 | 0.314606 |
| 8.83 | 2.068189 | 0.315590 |
| 8.84 | 2.072876 | 0.316573 |
| 8.85 | 2.077568 | 5.317555 |
| 8.86 | 2.082266 | 0.318536 |
| 8.87 | 2.086969 | 0.319516 |
| 8.88 | 2.091677 | 0.320494 |
| 8.89 | 2.096391 | 0.321472 |
| 8.90 | 2.101110 | 0.322448 |
| 8.91 | 2.105834 | 0.323424 |
| 8.92 | 2.110564 | 0.324498 |
| 8.93 | 2.115299 | 0.325371 |
| 8.94 | 2.120039 | 0.326343 |
| 8.95 | 2.124784 | 0.327314 |
| 8.96 | 2.129535 | 0.328284 |
| 8.97 | 2.134291 | 0.329253 |
| 8.98 | 2.139053 | 0.330221 |
| 8.99 | 2.143819 | 0.331178 |
| 9.00 | 2.148591 | 0.332153 |
| 9.01 | 2.153368 | 2.333118 |
| 9.02 | 2.158151 | 0.334081 |
| 9.03 | 2.162939 | 0.335044 |
| 9.04 | 2.167732 | 0.336005 |
| 9.05 | 2.172531 | 0.337966 |
| 9.06 | 2.177335 | 0.338925 |
| 9.07 | 2.182144 | 0.338883 |
| 9.08 | 2.186958 | 0.339840 |
| 9.09 | 2.191778 | 0.340796 |
| 9.10 | 2.196603 | 0.341751 |
| [Page 149] 9.11 | 2.201433 | 0.342705 |
| 9.12 | 2.206269 | 0.343658 |
| 9.13 | 2.211110 | 0.344610 |
| 9.14 | 2.215956 | 0.345561 |
| 9.15 | 2.220808 | 0.346511 |
| 9.16 | 2.225665 | 0.347459 |
| 9.17 | 2.230527 | 9.348407 |
| 9.18 | 2.235394 | 0.349354 |
| 9.19 | 2.240267 | 0.350299 |
| 9.20 | 2.245145 | 0.351244 |
| 9.21 | 2.250029 | 0.352881 |
| 9.22 | 2.254917 | 0.353130 |
| 9.23 | 2.259811 | 0.354072 |
| 9.24 | 2.264711 | 0.355012 |
| 9.25 | 2.269615 | 0.355872 |
| 9.26 | 2.274525 | 0.356890 |
| 9.27 | 2.279440 | 0.357828 |
| 9.28 | 2.284361 | 0.358764 |
| 9.29 | 2.289287 | 0.359700 |
| 9.30 | 2.294218 | 0.360634 |
| 9.31 | 2.299154 | 0.361568 |
| 9.32 | 2.304096 | 0.362500 |
| 9.33 | 2.309043 | 0.363432 |
| 9.34 | 2.313996 | 0.364362 |
| 9.35 | 2.318953 | 0.365292 |
| 9.36 | 2.323916 | 0.366220 |
| 9.37 | 2.328885 | 0.367148 |
| 9.38 | 2.333858 | 0.368074 |
| 9.39 | 2.338847 | 0.369000 |
| 9.40 | 2.343821 | 0.369924 |
| 9.41 | 2.348144 | 0.370848 |
| 9.42 | 2.353806 | 0.371770 |
| 9.43 | 2.358806 | 0.372692 |
| 9.44 | 2.363478 | 0.373612 |
| 9.45 | 2.368489 | 0.374532 |
| 9.46 | 2.373838 | 0.375451 |
| 9.48 | 2.378893 | 0.376368 |
| 9.48 | 2.383886 | 0.377285 |
| 9.49 | 2.388918 | 0.378201 |
| 9.50 | 2.393955 | 0.37911 [...] |
| 9.51 | 2.398998 | 0.380029 |
| 9.52 | 2.404045 | 0.380942 |
| 9.53 | 2.409099 | 0.381854 |
| 9.54 | 2.414157 | 0.382765 |
| 9.55 | 2.419221 | 0.383675 |
| 9.56 | 2.424290 | 0.384584 |
| 9.57 | 2.429364 | 0.385492 |
| 9.58 | 2.434444 | 0.386399 |
| 9.59 | 2.439529 | 0.387306 |
| 9.60 | 2.444629 | 0.388211 |
| 9.61 | 2.449715 | 0.389115 |
| 9.62 | 2.454816 | 0.390019 |
| 9.63 | 2.459922 | 0.390921 |
| 9.64 | 2.465034 | 0.391822 |
| 9.65 | 2.470150 | 0.392723 |
| 9.66 | 2.475273 | 0.393623 |
| 9.67 | 2.480400 | 0.394521 |
| 9.68 | 2.485533 | 0.395419 |
| 9.69 | 2.490671 | 0.396316 |
| 9.70 | 2.495814 | 0.397212 |
| 9.71 | 2.500963 | 0.39810 [...] |
| 9.72 | 2.506117 | 0.399001 |
| 9.73 | 2.511243 | 0.399894 |
| 9.74 | 2.516441 | 0.400786 |
| 9.75 | 2.521611 | 0.4 [...]1678 |
| 9.76 | 2.526786 | 0.402568 |
| 9.77 | 2.531966 | 0.403458 |
| 9.78 | 2.537152 | 0.404346 |
| 9.79 | 2.542343 | 0.405234 |
| 9.80 | 2.547540 | 0.406121 |
| [Page 150] 9.81 | 2.552745 | 0.407006 |
| 9.82 | 2.557948 | 0.407891 |
| 9.83 | 2.563164 | 0.408775 |
| 9.84 | 2.568378 | 0.409659 |
| 9.85 | 2.573601 | 0.410541 |
| 9.86 | 2.578829 | 0.411422 |
| 9.87 | 2.584063 | 0.412303 |
| 9.88 | 2.589302 | 0.413182 |
| 9.89 | 2.594546 | 0.414061 |
| 9.90 | 2.599795 | 0.414939 |
| 9.91 | 2.605050 | 0.415816 |
| 9.92 | 2.610310 | 0.416692 |
| 9.93 | 2.615573 | 0.417567 |
| 9.94 | 2.620846 | 0.418441 |
| 9.95 | 2.626122 | 0.419315 |
| 9.96 | 2.631404 | 0.420187 |
| 9.97 | 2.636690 | 0.421059 |
| 9.98 | 2.641982 | 0.421929 |
| 9.99 | 2.647279 | 0.422789 |
| 10.00 | 2.652582 | 0.423668 |
[Page 151] A Table for the speedy finding of the Length or Circumference answering to any Arch in Degrees and Decimal Parts.
[Page 152] A Table for the speedy finding of the Length or Circumference answering to any Arch, in Degrees and Decimal Parts.
| 1 | 0.0174 | 5329 | 2519 |
| 2 | 0.0349 | 0658 | 5038 |
| 3 | 0.0523 | 5987 | 7557 |
| 4 | 0.0698 | 1317 | 0076 |
| 5 | 0.0872 | 6646 | 2595 |
| 6 | 0.1047 | 1975 | 5114 |
| 7 | 0.1221 | 7304 | 7633 |
| 8 | 0.1396 | 2634 | 0152 |
| 9 | 0.1570 | 7963 | 2671 |
| 10 | 0.1745 | 3292 | 5190 |
| 11 | 0.1919 | 8621 | 7709 |
| 12 | 0.2094 | 3951 | 0228 |
| 13 | 0.2268 | 9280 | 2747 |
| 14 | 0.2443 | 4609 | 5266 |
| 15 | 0.2617 | 9938 | 7785 |
| 16 | 0.2792 | 5268 | 0304 |
| 17 | 0.2967 | 0597 | 2823 |
| 18 | 0.3141 | 5926 | 5342 |
| 19 | 0.3316 | 1255 | 7861 |
| 20 | 0.3490 | 6585 | 0380 |
| 21 | 0.3665 | 1914 | 2899 |
| 22 | 0.3839 | 7245 | 5418 |
| 23 | 0.4014 | 2572 | 7937 |
| 24 | 0.4188 | 7902 | 0456 |
| 25 | 0.4363 | 3231 | 2975 |
| 26 | 0.4537 | 8560 | 5495 |
| 27 | 0.4712 | 3889 | 8013 |
| 28 | 0.4886 | 8219 | 0532 |
| 29 | 0.5061 | 4548 | 3051 |
| 30 | 0.5235 | 4877 | 5570 |
| 31 | 0.5410 | 5206 | 8089 |
| 32 | 0.5585 | 0536 | 0608 |
| 33 | 0.5759 | 5865 | 3127 |
| 34 | 0.5934 | 1194 | 5646 |
| 35 | 0.6108 | 6523 | 8165 |
| 36 | 0.6283 | 1853 | 0684 |
| 37 | 0.6457 | 7128 | 3203 |
| 38 | 0.6632 | 2511 | 5722 |
| 39 | 0.6806 | 7840 | 8241 |
| 40 | 0.6981 | 3170 | 0760 |
| 41 | 0.7155 | 8499 | 3279 |
| 42 | 0.7330 | 3828 | 5798 |
| 43 | 0.7504 | 9157 | 8317 |
| 44 | 0.7679 | 4487 | 0836 |
| 45 | 0.7853 | 9816 | 3355 |
| 46 | 0.8028 | 3145 | 5874 |
| 47 | 0.8203 | 0474 | 8393 |
| 48 | 0.8377 | 5804 | 0912 |
| 49 | 0.8552 | 1133 | 3431 |
| 50 | 0.8726 | 6462 | 4950 |
| [Page 153] 51 | 0.8901 | 1791 | 8469 |
| 52 | 0.9075 | 7121 | 0988 |
| 53 | 0.9250 | 2450 | 3507 |
| 54 | 0.9424 | 7779 | 6026 |
| 55 | 0.9599 | 3108 | 8545 |
| 56 | 0.9773 | 8438 | 1064 |
| 57 | 0.9948 | 3767 | 3583 |
| 58 | 1.0122 | 9096 | 6102 |
| 59 | 1.0297 | 4425 | 8621 |
| 60 | 1.0471 | 9755 | 1140 |
| 61 | 1.0646 | 5084 | 3659 |
| 62 | 1.0821 | 0413 | 6178 |
| 63 | 1.0995 | 5742 | 8697 |
| 64 | 1.1170 | 1072 | 1216 |
| 65 | 1.1344 | 6401 | 3735 |
| 66 | 1.1519 | 1730 | 6254 |
| 67 | 1.1693 | 7059 | 8773 |
| 68 | 1.1868 | 2389 | 1292 |
| 69 | 1.2042 | 7718 | 3811 |
| 70 | 1.2217 | 3047 | 6330 |
| 71 | 1.2391 | 8376 | 8849 |
| 72 | 1.2566 | 3706 | 1368 |
| 73 | 1.2740 | 9035 | 3887 |
| 74 | 1.2915 | 4364 | 6406 |
| 75 | 1.3089 | 9693 | 8925 |
| 76 | 1.3264 | 5023 | 1444 |
| 77 | 1.3439 | 0352 | 3963 |
| 78 | 1.3613 | 5681 | 6482 |
| 79 | 1.3788 | 1010 | 9001 |
| 80 | 1.3962 | 6340 | 1520 |
| 81 | 1.4137 | 1669 | 4039 |
| 82 | 1.4311 | 6998 | 6558 |
| 83 | 1.4486 | 2327 | 9057 |
| 84 | 1.4660 | 7657 | 1596 |
| 85 | 1.4835 | 2986 | 4115 |
| 86 | 1.5009 | 8315 | 6634 |
| 87 | 2.5184 | 3644 | 9153 |
| 88 | 1.5358 | 8974 | 1572 |
| 89 | 1.5533 | 4303 | 4191 |
| 90 | 1.5707 | 9632 | 6710 |
| 91 | 1.5882 | 4961 | 9229 |
| 92 | 1.6057 | 0291 | 1748 |
| 93 | 1.6231 | 5620 | 4267 |
| 94 | 1.6406 | 0949 | 6786 |
| 95 | 1.6580 | 6278 | 9305 |
| 96 | 1.6755 | 1608 | 1824 |
| 97 | 1.6929 | 6937 | 4343 |
| 98 | 1.7104 | 2266 | 6862 |
| 99 | 1.7278 | 7595 | 9381 |
| 100 | 1.7453 | 2925 | 1900 |
[Page 154] A Common Divisor for the speedy converting of the Table, shewing the Area of the Segments of a Circle whose Diameter is 2.0000 &c. into a Table shewing the Area of the Segment of any Circle whose Area is given.
| 1 | 0031 | 4159 | 2653 |
| 2 | 0062 | 8318 | 5306 |
| 3 | 0094 | 2477 | 7959 |
| 4 | 0125 | 6637 | 0612 |
| 5 | 0157 | 0796 | 3265 |
| 6 | 0188 | 4955 | 5918 |
| 7 | 0219 | 9114 | 8571 |
| 8 | 0251 | 3274 | 1224 |
| 9 | 0282 | 7433 | 3877 |
| 10 | 0314 | 1592 | 6530 |
| 11 | 0345 | 5751 | 9183 |
| 12 | 0376 | 9911 | 1836 |
| 13 | 0408 | 4070 | 4489 |
| 14 | 0439 | 8229 | 7142 |
| 15 | 0471 | 2388 | 9795 |
| 16 | 0502 | 6548 | 2448 |
| 17 | 0534 | 0707 | 5101 |
| 18 | 0565 | 4866 | 7754 |
| 19 | 0596 | 9026 | 0407 |
| 20 | 0628 | 3185 | 3060 |
| 21 | 0659 | 7344 | 5713 |
| 22 | 0691 | 1503 | 8366 |
| 23 | 0722 | 5663 | 1019 |
| 24 | 0753 | 9822 | 3672 |
| 25 | 0785 | 3981 | 6325 |
| 26 | 0816 | 8140 | 8978 |
| 27 | 0848 | 2300 | 1631 |
| 28 | 0889 | 6459 | 4284 |
| 29 | 0911 | 0618 | 6937 |
| 30 | 0942 | 4777 | 9590 |
| 21 | 0973 | 8937 | 2243 |
| 32 | 1005 | 3096 | 4896 |
| 33 | 1036 | 7255 | 7549 |
| 34 | 1068 | 1415 | 0202 |
| 35 | 1099 | 5574 | 2855 |
| 36 | 1130 | 9733 | 5508 |
| 37 | 1162 | 3892 | 8161 |
| 38 | 1193 | 8052 | 0814 |
| 39 | 1225 | 2211 | 3467 |
| 40 | 1256 | 6370 | 6120 |
| 41 | 1288 | 0529 | 8773 |
| 42 | 1319 | 4689 | 1426 |
| 43 | 1350 | 8848 | 4079 |
| 44 | 1382 | 3007 | 6732 |
| 45 | 1413 | 7166 | 9385 |
| 46 | 1445 | 1326 | 2038 |
| 47 | 1476 | 5485 | 4691 |
| 48 | 1507 | 9644 | 7344 |
| 49 | 1539 | 3803 | 9997 |
| 50 | 1570 | 7963 | 2650 |
| [Page 155] 51 | 1602 | 2122 | 5303 |
| 52 | 1633 | 6281 | 7956 |
| 53 | 1665 | 0441 | 0609 |
| 54 | 1696 | 4600 | 3262 |
| 55 | 1727 | 8759 | 5915 |
| 56 | 1759 | 2918 | 6568 |
| 57 | 1790 | 7078 | 1221 |
| 58 | 1822 | 1237 | 3874 |
| 59 | 1853 | 5396 | 6527 |
| 60 | 1884 | 9555 | 9180 |
| 61 | 1916 | 3715 | 1833 |
| 62 | 1947 | 7874 | 4486 |
| 63 | 1979 | 2033 | 7139 |
| 64 | 2010 | 6192 | 9792 |
| 65 | 2042 | 0352 | 2445 |
| 66 | 2073 | 4511 | 5098 |
| 67 | 2104 | 8670 | 7751 |
| 68 | 2136 | 2830 | 0404 |
| 69 | 2167 | 6989 | 3057 |
| 70 | 2199 | 1148 | 5710 |
| 71 | 2230 | 5307 | 8363 |
| 72 | 2261 | 9467 | 1016 |
| 53 | 2293 | 3626 | 3669 |
| 74 | 2324 | 7785 | 6322 |
| 75 | 2356 | 1944 | 8975 |
| 76 | 2387 | 6104 | 1628 |
| 77 | 2419 | 0263 | 4281 |
| 78 | 2450 | 4422 | 6934 |
| 79 | 2481 | 8581 | 9587 |
| 80 | 2513 | 2741 | 2240 |
| 81 | 2544 | 6900 | 4893 |
| 82 | 2576 | 1059 | 7546 |
| 83 | 2607 | 5219 | 0199 |
| 84 | 2638 | 9378 | 2852 |
| 85 | 2670 | 3537 | 5505 |
| 86 | 2701 | 7696 | 8158 |
| 87 | 2733 | 1856 | 0811 |
| 88 | 2764 | 6015 | 3464 |
| 89 | 2796 | 0174 | 6117 |
| 90 | 2827 | 4333 | 8770 |
| 91 | 2858 | 8493 | 1423 |
| 92 | 2890 | 2652 | 4076 |
| 93 | 2921 | 6811 | 6729 |
| 94 | 2953 | 0970 | 9382 |
| 95 | 2984 | 5130 | 2035 |
| 96 | 3015 | 9289 | 4688 |
| 97 | 3047 | 3448 | 7341 |
| 98 | 3078 | 7607 | 9994 |
| 99 | 3110 | 1767 | 2647 |
| 100 | 3141 | 5926 | 5300 |
[Page 156] A Table shewing the Ordinates, Arches and Areas of the Segments of a Circle, whose Diameter is 2000, &c. to every Hundredth Part of the Radius.
| Ordinates | Deg. & Dec. p. | Areas | |
| 100 | 10000000000 | 90.00000000 | 1.57079632 |
| 99 | 9999499971 | 89.42704196 | 1.55079682 |
| 98 | 9997999799 | 88.85400799 | 1.53079890 |
| 97 | 9995498987 | 88.28987110 | 1.51080538 |
| 96 | 99919967974 | 87.70756124 | 1.49081774 |
| 95 | 9987492177 | 87.13402020 | 1.47083808 |
| 94 | 9981983770 | 86.56018749 | 1.45086837 |
| 93 | 9975469913 | 85.98601581 | 1.43091081 |
| 92 | 9967948635 | 85.41143529 | 1.41096718 |
| 91 | 9959417653 | 84.83639513 | 1.39103966 |
| 90 | 9949874371 | 84.26083018 | 1.37113017 |
| 89 | 9939315871 | 83.68468641 | 1.35124084 |
| 88 | 9927738916 | 83.10789860 | 1.35137360 |
| 87 | 9915139938 | 82.53040793 | 1.31153053 |
| 86 | 9901515035 | 81.95215479 | 1.29171372 |
| 85 | 9886859966 | 81.37307468 | 1.27192518 |
| 84 | 9871170138 | 80.79310474 | 1.25216697 |
| 83 | 9854440623 | 80.21218180 | 1.23244118 |
| 82 | 9836666101 | 79.63024030 | 1.21274989 |
| 81 | 9817840903 | 79.04721672 | 1.19309522 |
| 80 | 9797958971 | 78.46304188 | 1.17347924 |
| 79 | 9777013859 | 77.87762112 | 1.15390361 |
| 78 | 9754998718 | 77.29096735 | 1.13437189 |
| 77 | 9731906288 | 76.70292903 | 1.11488481 |
| 76 | 9707728879 | 76.11243681 | 1.09544458 |
| 75 | 9682458365 | 75.52248845 | 1.07605462 |
| 74 | 9656086163 | 74.92996014 | 1.05671627 |
| 73 | 9628603221 | 74.33573392 | 1.03743102 |
| 72 | 9600000000 | 73.73979456 | 1.01820220 |
| 71 | 9570266454 | 73.14202474 | 0.99903143 |
| 70 | 9539392014 | 72.54239737 | 0.97992192 |
| 69 | 950 [...]365565 | 71.94076969 | 0.96087497 |
| 68 | 9474175425 | 71.33707564 | 0.94189323 |
| 67 | 9439809319 | 70.73122476 | 0.92297905 |
| [Page 158] 67 | 9439809319 | 70.73122476 | 0.92297905 |
| 66 | 9404254356 | 70.12312662 | 0.90413479 |
| 65 | 9367496997 | 69.51268522 | 0.88536283 |
| 64 | 9329523031 | 68.89980401 | 0.86666560 |
| 63 | 9290317540 | 68.28438326 | 0.84804557 |
| 62 | 9249864864 | 67.66631784 | 0.82950517 |
| 61 | 9208148564 | 67.04550117 | 0.81104695 |
| 60 | 9165151389 | 66.42182324 | 0.79267345 |
| 59 | 9120855222 | 65.79516567 | 0.77438721 |
| 58 | 9075241043 | 65.16541298 | 0.75619089 |
| 57 | 9028288874 | 64.53244020 | 0.73808713 |
| 56 | 8979977728 | 63.89612058 | 0.72007866 |
| 55 | 8930285549 | 63.25631645 | 0.70216884 |
| 54 | 8879189152 | 62.61289754 | 0.68435845 |
| 53 | 8826664149 | 61.96570387 | 0.66665234 |
| 52 | 8772684879 | 61.31459838 | 0.64905275 |
| 51 | 8717224755 | 60.65941181 | 0.63156249 |
| 50 | 8660254037 | 60.00000000 | 0.61418485 |
| 49 | 8601744009 | 59.33617061 | 0.59692260 |
| 48 | 8541662601 | 58.66774875 | 0.57977892 |
| 47 | 8479976415 | 57.99454553 | 0.56275702 |
| 46 | 8416650165 | 57.31636147 | 0.54586011 |
| 45 | 8351646544 | 56.63307065 | 0.52909299 |
| 44 | 8284926070 | 55.94420256 | 0.51245467 |
| 43 | 8216446926 | 55.24977433 | 0.49595300 |
| 42 | 8146264741 | 54.54945742 | 0.47959008 |
| 41 | 8074032449 | 53.84299205 | 0.46336957 |
| 40 | 8000000000 | 53.13010237 | 0.44725221 |
| 39 | 7924014134 | 52.41049708 | 0.43137885 |
| 38 | 7846018098 | 51.68386597 | 0.41560051 |
| 37 | 7765951325 | 50.94987748 | 0.39998818 |
| 36 | 7683749084 | 50.20810657 | 0.38453683 |
| 35 | 7599342076 | 49.45831012 | 0.36925312 |
| 34 | 7512655988 | 48.70012721 | 0.35414227 |
| [Page 159] 34 | 7512655988 | 48.70012721 | 0.35414227 |
| 33 | 7423610981 | 47.93293539 | 0.33920561 |
| 32 | 7332121111 | 47.15635717 | 0.32444946 |
| 31 | 7238093671 | 46.36989113 | 0.30987884 |
| 30 | 7141428428 | 45.57299618 | 0.29549884 |
| 29 | 7042016756 | 44.76508489 | 0.28131493 |
| 28 | 6939740629 | 48.94551977 | 0.26733268 |
| 27 | 6834471449 | 43.11360613 | 0.25355796 |
| 26 | 6726068688 | 42.26858452 | 0.23999689 |
| 25 | 6614378277 | 41.40962595 | 0.22665594 |
| 24 | 6499230723 | 40.53580228 | 0.21354168 |
| 23 | 6380438856 | 39.64611132 | 0.20066138 |
| 22 | 6257795138 | 38.73942400 | 0.18802248 |
| 21 | 6131068422 | 37.81448867 | 0.17563291 |
| 20 | 6000000000 | 36.86989765 | 0.16350111 |
| 19 | 5864298764 | 35.90406873 | 0.15163601 |
| 18 | 5723635208 | 34.91520640 | 0.14004722 |
| 17 | 5577633906 | 33.90125515 | 0.12874491 |
| 16 | 5425863986 | 32.85988059 | 0.11774053 |
| 15 | 5267826876 | 31.78833069 | 0.10704574 |
| 14 | 5102940328 | 30.68341722 | 0.09667379 |
| 13 | 4930517214 | 29.54136121 | 0.08663902 |
| 12 | 4749736834 | 28.35773666 | 0.07695728 |
| 11 | 4559605246 | 27.12675321 | 0.06764629 |
| 10 | 4358898943 | 25.84193282 | 0.05872590 |
| 09 | 4146082488 | 24.49464857 | 0.05021866 |
| 08 | 3919183588 | 23.07391815 | 0.04215095 |
| 07 | 3675595189 | 21.56518547 | 0.03455313 |
| 06 | 3411744421 | 19.94844363 | 0.02746204 |
| 05 | 3122498999 | 18.19487244 | 0.02092302 |
| 04 | 2800000000 | 16.260204 [...]1 | 0.01499411 |
| 03 | 2431049156 | 14.06986184 | 0.00975364 |
| 02 | 1989974874 | 11.47834097 | 0.00551730 |
| 01 | 1410673597 | 8.10961446 | 0.00188278 |
| [Page 160] 010 | 1410673597 | 8.10961446 | 0.00188278 |
| 009 | 1338618691 | 7.69281247 | 0.00160779 |
| 008 | 1262378707 | 7.25224680 | 0.00134761 |
| 007 | 1181143513 | 6.78328892 | 0.00110317 |
| 006 | 1093800713 | 6.279 [...]8064 | 0.00087554 |
| 005 | 0998749217 | 5.73196797 | 0.00066616 |
| 004 | 0893532316 | 5.12640010 | 0.00047674 |
| 003 | 0774015503 | 4.43922228 | 0.00030969 |
| 002 | 0632139225 | 3.62430750 | 0.00016860 |
| 001 | 0447101778 | 2.56255874 | 0.00005961 |
| 1.57079632 | |
| 999 | 199999 |
| 1.56879632 | |
| 998 | 199999 |
| 1.56679632 | |
| 199999 | |
| 997 | 1.56479633 |
| 199998 | |
| 996 | 1.56279634 |
| 199997 | |
| 995 | 1.56079636 |
| 199996 | |
| 994 | 1.55879639 |
| 199995 | |
| 993 | 1.55679644 |
| 199994 | |
| 992 | 1.55479649 |
| 199992 | |
| 991 | 1.55299657 |
| 199991 | |
| 990 | 1.55079666 |
| 199988 | |
| 989 | 1.54879677 |
| 199986 | |
| 988 | 1.54679690 |
| 199984 | |
| 987 | 1.54479706 |
| 199981 | |
| 986 | 1.54279724 |
| 199978 | |
| 985 | 1.54079745 |
| 199976 | |
| 984 | 1.53879769 |
| 199972 | |
| 983 | 1.53679796 |
| 983 | 1.53679796 |
| 199969 | |
| 982 | 1.53479827 |
| 199965 | |
| 981 | 1.53279862 |
| 199962 | |
| 980 | 1.53079899 |
| 199957 | |
| 979 | 1.52879941 |
| 199953 | |
| 978 | 1.52679988 |
| 199949 | |
| 977 | 1.52480039 |
| 199944 | |
| 976 | 1.52280095 |
| 199939 | |
| 975 | 1.52080156 |
| 199934 | |
| 974 | 1.51880222 |
| 199929 | |
| 973 | 1.51680293 |
| 199924 | |
| 972 | 1.51480369 |
| 199918 | |
| 971 | 1.51280451 |
| 199912 | |
| 970 | 1.51080539 |
| 199906 | |
| 969 | 1.50880633 |
| 199909 | |
| 968 | 1.50680733 |
| 199894 | |
| 967 | 1.50480839 |
| 199887 | |
| 966 | 1.50280952 |
| [Page 162] 966 | 1.50280952 |
| 199880 | |
| 965 | 1.50081072 |
| 199873 | |
| 964 | 1.49881199 |
| 199866 | |
| 963 | 1.49681333 |
| 199859 | |
| 962 | 1.49481474 |
| 199851 | |
| 961 | 1.49281623 |
| 199843 | |
| 960 | 1.49081774 |
| 199835 | |
| 959 | 1.48881938 |
| 199827 | |
| 958 | 1.48682110 |
| 199819 | |
| 957 | 1.48482291 |
| 199810 | |
| 956 | 1.48282480 |
| 199801 | |
| 955 | 1.48082678 |
| 199792 | |
| 954 | 1.47882885 |
| 199783 | |
| 953 | 1.47683102 |
| 199774 | |
| 952 | 1.47483328 |
| 199764 | |
| 951 | 1.47283563 |
| 199754 | |
| 950 | 1.47083808 |
| 199744 | |
| 949 | 1.46884063 |
| 949 | 1.46884063 |
| 199734 | |
| 948 | 1.46684328 |
| 199724 | |
| 947 | 1.46484604 |
| 199713 | |
| 946 | 1.46284890 |
| 199702 | |
| 945 | 1.46085187 |
| 199691 | |
| 944 | 1.45885496 |
| 199680 | |
| 943 | 1.45685815 |
| 199669 | |
| 942 | 1.45486146 |
| 199657 | |
| 941 | 1.45286489 |
| 199645 | |
| 940 | 1.45086837 |
| 199633 | |
| 939 | 1.44887204 |
| 199621 | |
| 938 | 1.44687583 |
| 199608 | |
| 937 | 1.44487975 |
| 199596 | |
| 936 | 1.44288379 |
| 199585 | |
| 935 | 1.44088794 |
| 199570 | |
| 934 | 1.43889224 |
| 199557 | |
| 933 | 1.43689667 |
| 199543 | |
| 932 | 1.43490124 |
| [Page 163] 932 | 1.43490124 |
| 199530 | |
| 931 | 1.43290594 |
| 199516 | |
| 930 | 1.43091078 |
| 199502 | |
| 929 | 1.42891578 |
| 199488 | |
| 928 | 1.42692090 |
| 199473 | |
| 927 | 1.42492617 |
| 199459 | |
| 926 | 1.42293158 |
| 199444 | |
| 925 | 1.42093714 |
| 199429 | |
| 924 | 1.41894305 |
| 199413 | |
| 923 | 1.41694892 |
| 199398 | |
| 922 | 1.41495494 |
| 199382 | |
| 921 | 1.41296112 |
| 199366 | |
| 920 | 1.41096746 |
| 199350 | |
| 919 | 1.40897396 |
| 199334 | |
| 918 | 1.40698062 |
| 199318 | |
| 917 | 1.40498744 |
| 199301 | |
| 916 | 1.40299443 |
| 199284 | |
| 915 | 1.40100159 |
| 915 | 1.40100159 |
| 199267 | |
| 914 | 1.39900892 |
| 199250 | |
| 913 | 1.39701642 |
| 199232 | |
| 912 | 1.39502410 |
| 199215 | |
| 911 | 1.39303195 |
| 199197 | |
| 910 | 1.39103998 |
| 199178 | |
| 909 | 1.38904820 |
| 199160 | |
| 908 | 1.38705660 |
| 199142 | |
| 907 | 1.38506518 |
| 199123 | |
| 906 | 1.38307395 |
| 199104 | |
| 905 | 1.38108291 |
| 199085 | |
| 904 | 1.37909206 |
| 199066 | |
| 903 | 1.37710140 |
| 199047 | |
| 902 | 1.37511093 |
| 199027 | |
| 901 | 1.37312066 |
| 199007 | |
| 900 | 1.37113059 |
| [Page 164] 900 | 1.37113017 |
| 198987 | |
| 899 | 1.36914030 |
| 198967 | |
| 898 | 1.36715063 |
| 198946 | |
| 897 | 1.36516117 |
| 198925 | |
| 896 | 1.36317192 |
| 198904 | |
| 895 | 1.36118288 |
| 198883 | |
| 894 | 1.35919405 |
| 198861 | |
| 893 | 1.35720544 |
| 198839 | |
| 892 | 1.35521705 |
| 198818 | |
| 891 | 1.35322887 |
| 198797 | |
| 890 | 1.35124090 |
| 198775 | |
| 889 | 1.34925315 |
| 198752 | |
| 888 | 1.34726563 |
| 198729 | |
| 887 | 1.34527834 |
| 198707 | |
| 886 | 1.34329127 |
| 198684 | |
| 885 | 1.34130443 |
| 198661 | |
| 884 | 1.33931782 |
| 198638 | |
| 88 [...] | 1.33733144 |
| 883 | 1.33733144 |
| 198619 | |
| 882 | 1.33534525 |
| 198590 | |
| 881 | 1.33335935 |
| 198566 | |
| 880 | 1.33137360 |
| 198541 | |
| 879 | 1.32938819 |
| 198517 | |
| 878 | 1.32740302 |
| 198499 | |
| 877 | 1.32541803 |
| 198480 | |
| 876 | 1.32343323 |
| 198449 | |
| 875 | 1.32144874 |
| 198418 | |
| 874 | 1.31946456 |
| 198393 | |
| 873 | 1.31748063 |
| 198367 | |
| 872 | 1.31549696 |
| 198341 | |
| 871 | 1.31351355 |
| 198315 | |
| 870 | 1.31153053 |
| 198289 | |
| 869 | 1.30954764 |
| 198262 | |
| 868 | 1.30756502 |
| 198235 | |
| 867 | 1.30558267 |
| 198209 | |
| [Page 165] | 198209 |
| 866 | 1.30360058 |
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| 865 | 1.30161876 |
| 198154 | |
| 864 | 1.29963722 |
| 198127 | |
| 863 | 1.29765595 |
| 198100 | |
| 862 | 1.29567495 |
| 198072 | |
| 861 | 1.29369423 |
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| 860 | 1.29171379 |
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| 859 | 1.28973357 |
| 197986 | |
| 858 | 1.28775371 |
| 197958 | |
| 857 | 1.2857741 [...] |
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| 856 | 1.28379484 |
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| 855 | 1.28181584 |
| 197871 | |
| 854 | 1.27983713 |
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| 853 | 1.27785872 |
| 197811 | |
| 852 | 1.27588061 |
| 197781 | |
| 851 | 1.27390280 |
| 197751 | |
| 850 | 1.27192529 |
| 850 | 1.27192518 |
| 197721 | |
| 849 | 1.26994797 |
| 197691 | |
| 848 | 1.26797106 |
| 197660 | |
| 847 | 1.26599446 |
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| 846 | 1.26401817 |
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| 845 | 1.26204219 |
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| 844 | 1.26006658 |
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| 843 | 1.25809124 |
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| 842 | 1.25611635 |
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| 841 | 1.25414178 |
| 197427 | |
| 840 | 1.25216751 |
| 197395 | |
| 839 | 1.25019356 |
| 197374 | |
| 838 | 1.24821982 |
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| 837 | 1.24624641 |
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| 836 | 1.24427333 |
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| 835 | 1.24230058 |
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| 834 | 1.24032817 |
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| 833 | 1.23835605 |
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| 831 | 1.23441293 |
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| 830 | 1.23244118 |
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| 829 | 1.23047046 |
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| 828 | 1.22850010 |
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| 827 | 1.22653009 |
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| 826 | 1.22456043 |
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| 825 | 1.22259113 |
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| 824 | 1.22062218 |
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| 823 | 1.21865357 |
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| 822 | 1.21668532 |
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| 821 | 1.21471745 |
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| 820 | 1.21274989 |
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| 819 | 1.21078275 |
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| 818 | 1.20881598 |
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| 817 | 1.20684958 |
| 817 | 1.20684954 |
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| 816 | 1.20488355 |
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| 815 | 1.20291790 |
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| 814 | 1.20 [...]95263 |
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| 813 | 1.19898774 |
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| 812 | 1.19702323 |
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| 811 | 1.19505910 |
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| 810 | 1.19309525 |
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| 809 | 1.19113254 |
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| 808 | 1.18916956 |
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| 807 | 1.18720698 |
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| 806 | 1.18524479 |
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| 805 | 1.18328291 |
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| 803 | 1.17936043 |
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| 784 | 1.14217966 |
| 784 | 1.14217966 |
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| 768 | 1.11099301 |
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| [Page 168] | 194518 |
| 767 | 1.10904783 |
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| 764 | 1.10321515 |
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| 760 | 1.09544514 |
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| 751 | 1.07799159 |
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| 750 | 1.07605485 |
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| 749 | 1.07411863 |
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| 747 | 1.07024775 |
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| 746 | 1.06831309 |
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| 745 | 1.06637895 |
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| 744 | 1.06444534 |
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| 742 | 1.06057971 |
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| 741 | 1.05864770 |
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| 740 | 1.05671623 |
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| 739 | 1.05478530 |
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| 738 | 1.05285491 |
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| 736 | 1.04899575 |
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| 735 | 1.04706699 |
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| [Page 169] | 192821 |
| 734 | 1.04513878 |
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| 733 | 1.04321112 |
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| 732 | 1.04128402 |
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| 731 | 1.03935747 |
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| 730 | 1.03743147 |
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| 729 | 1.03550604 |
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| 728 | 1.03358118 |
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| 727 | 1.03165688 |
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| 726 | 1.02973115 |
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| 725 | 1.02780999 |
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| 724 | 1.02588740 |
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| 723 | 1.02396527 |
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| 722 | 1.02204372 |
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| 721 | 1.02012286 |
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| 720 | 1.01820221 |
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| 719 | 1.01628251 |
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| 718 | 1.01436340 |
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| 717 | 1.01244487 |
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| 716 | 1.01052693 |
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| 715 | 1.00860959 |
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| 714 | 1.00669285 |
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| 713 | 1.00477670 |
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| 712 | 1.00286114 |
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| 711 | 1.00094609 |
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| 710 | 0.99903165 |
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| 707 | 0.99329226 |
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| 706 | 0.99138035 |
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| [Page 170] | 190818 |
| 700 | 0.97992192 |
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| 670 | 0.92297905 |
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| 666 | 0.91543272 |
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| 660 | 0.90413479 |
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| 644 | 0.87413542 |
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| 566 | 0.73087219 |
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| 550 | 0.70216834 |
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| 52815 | |
| 35 | 0.01228185 |
| 52079 | |
| 34 | 0.01176106 |
| 51331 | |
| [Page 190] 33 | 0.01124776 |
| 50572 | |
| 32 | 0.01074204 |
| 49801 | |
| 31 | 0.01024403 |
| 49016 | |
| 30 | 0.00975364 |
| 48217 | |
| 29 | 0.00927147 |
| 47405 | |
| 28 | 0.00879742 |
| 46578 | |
| 27 | 0.00833164 |
| 45734 | |
| 26 | 0.00787430 |
| 44874 | |
| 25 | 0.00742556 |
| 43997 | |
| 24 | 0.00698559 |
| 43102 | |
| 23 | 0.00655457 |
| 42185 | |
| 22 | 0.00613272 |
| 41244 | |
| 21 | 0.00572028 |
| 40273 | |
| 20 | 0.00531730 |
| 39291 | |
| 19 | 0.00492439 |
| 38297 | |
| 18 | 0.00454142 |
| 37248 | |
| 17 | 0.00416894 |
| 36176 | |
| 36176 | |
| 16 | 0.00380718 |
| 35071 | |
| 15 | 0.00345647 |
| 33929 | |
| 14 | 0.00311718 |
| 32746 | |
| 13 | 0.00278972 |
| 31517 | |
| 12 | 0.00247455 |
| 30236 | |
| 11 | 0.00217219 |
| 28897 | |
| 10 | 0.00188278 |
| 27442 | |
| 9 | 0.00160836 |
| 25959 | |
| 8 | 0.00134877 |
| 24434 | |
| 7 | 0.00110443 |
| 22749 | |
| 6 | 0.00087694 |
| 20925 | |
| 5 | 0.00066769 |
| 18922 | |
| 4 | 0.00047847 |
| 16675 | |
| 3 | 0.00031172 |
| 14061 | |
| 2 | 0.00017111 |
| 10792 | |
| 1 | 0.00006319 |
| 6319 | |
| 0 | 0.00000000 |
A TABLE SHEWING THE AREA OF THE SEGMENTS OF A CIRCLE WHOSE Whole Area is Unity, to the ten Thousandth part of the Diameter.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
| 0 | 000000 | 000004 | 000007 | 000011 | 000014 | 000018 | 000025 | 000032 | 000039 | 000046 | |
| 1 | 000053 | 000062 | 000071 | 000080 | 000089 | 000098 | 000108 | 000119 | 000130 | 000140 | |
| 2 | 000151 | 000163 | 000175 | 000187 | 000200 | 000212 | 000225 | 000238 | 000251 | 000265 | |
| 3 | 000278 | 000292 | 000307 | 000322 | 000336 | 000351 | 000366 | 000382 | 000397 | 000413 | |
| 4 | 000428 | 000444 | 000461 | 000478 | 000494 | 000511 | 000529 | 000546 | 000564 | 000581 | |
| 5 | 000599 | 000617 | 000636 | 000654 | 000673 | 000691 | 000710 | 000729 | 000748 | 000768 | 19 |
| 6 | 000787 | 000807 | 000827 | 000847 | 000867 | 000887 | 000908 | 000928 | 000949 | 000970 | 20 |
| 7 | 000991 | 001012 | 001034 | 001056 | 001077 | 001099 | 001121 | 001144 | 001166 | 001188 | 21 |
| 8 | 001211 | 001234 | 001257 | 001280 | 001304 | 001327 | 001350 | 001374 | 001398 | 001421 | 23 |
| 9 | 001445 | 001469 | 001494 | 001518 | 001542 | 001567 | 001592 | 001617 | 001642 | 001667 | 25 |
| 10 | 001692 | 001717 | 001743 | 001769 | 001794 | 001820 | 001846 | 001873 | 001899 | 001925 | 26 |
| 11 | 001952 | 001979 | 002005 | 002032 | 002059 | 002086 | 002113 | 002141 | 002168 | 002195 | 27 |
| 12 | 002223 | 002251 | 002279 | 002307 | 002335 | 002363 | 002392 | 002420 | 002449 | 002477 | 28 |
| 13 | 002506 | 002535 | 002564 | 002593 | 002623 | 002652 | 002681 | 002711 | 002741 | 002770 | 29 |
| 14 | 002700 | 002830 | 002860 | 002890 | 002921 | 002951 | 002982 | 003013 | 003043 | 003074 | 30 |
| 15 | 003105 | 003136 | 003167 | 003198 | 003229 | 003260 | 003291 | 003323 | 003355 | 003387 | 31 |
| 16 | 003419 | 003451 | 003483 | 003515 | 003548 | 003580 | 003612 | 003645 | 003678 | 003710 | 32 |
| 17 | 003743 | 003776 | 003809 | 003842 | 0038 [...]6 | 003909 | 003942 | 003976 | 004009 | 004043 | 33 |
| 18 | 004077 | 004111 | 004145 | 004179 | 004213 | 004247 | 004281 | 004316 | 004351 | 004385 | 34 |
| 19 | 004420 | 004455 | 004490 | 004525 | 004560 | 004595 | 004630 | 004665 | 004700 | 004735 | 35 |
| 20 | 004770 | 004806 | 004843 | 004879 | 004915 | 004952 | 004988 | 005024 | 005061 | 005097 | 36 |
| 21 | 005133 | 005170 | 005206 | 005243 | 005280 | 005317 | 005354 | 005391 | 005428 | 005465 | 37 |
| 22 | 005502 | 005539 | 005 [...]77 | 005615 | 005652 | 005690 | 005728 | 005766 | 005804 | 005842 | 38 |
| 23 | 005880 | 005918 | 005957 | 005995 | 006023 | 006072 | 006111 | 006150 | 006188 | 006227 | 39 |
| 24 | 006266 | 0063 [...]5 | 006344 | 006383 | 006423 | 006462 | 006501 | 006541 | 006581 | 006620 | 39 |
| 25 | 006660 | 006700 | 006739 | 006779 | 006819 | 006859 | 006899 | 006940 | 006980 | 007021 | 40 |
| 26 | 007061 | 007102 | 007142 | 007183 | 007223 | 007264 | 007305 | 007346 | 007387 | 007429 | 41 |
| 27 | 007470 | 007511 | 007553 | 007594 | 007635 | 007677 | 007719 | 007761 | 007802 | 007844 | 42 |
| 28 | 007886 | 007928 | 007970 | 008012 | 008055 | 008097 | 008140 | 008182 | 008225 | 008267 | 42 |
| 29 | 008310 | 008353 | 008396 | 008439 | 008482 | 008525 | 008568 | 008611 | 008654 | 008698 | 43 |
| 30 | 008471 | 008785 | 008828 | 008872 | 008916 | 008959 | 009203 | 009047 | 009091 | 009135 | 44 |
| 31 | 009179 | 009223 | 009267 | 009312 | 009356 | 009400 | 009445 | 009490 | 009535 | 009579 | 45 |
| 32 | 009624 | 009669 | 009714 | 009759 | 009804 | 009849 | 009894 | 009939 | 009985 | 010030 | 45 |
| 33 | 010075 | 010121 | 010167 | 010212 | 010258 | 010303 | 010349 | 010395 | 010441 | 010487 | 46 |
| [Page 194-195] 33 | 010075 | 010121 | 010167 | 010212 | 010258 | 010303 | 010349 | 010395 | 010441 | 010487 | 46 |
| 34 | 010533 | 010580 | 010626 | 010672 | 010719 | 010765 | 010812 | 010858 | 010905 | 010952 | 47 |
| 35 | 010999 | 011045 | 011093 | 011139 | 011186 | 011233 | 011281 | 011328 | 011375 | 011422 | 47 |
| 36 | 011469 | 011517 | 011565 | 011612 | 011660 | 011707 | 011755 | 011803 | 011851 | 011899 | 47 |
| 37 | 011947 | 011995 | 012043 | 012092 | 012140 | 012188 | 012237 | 012285 | 012334 | 012382 | 48 |
| 38 | 012431 | 012479 | 012528 | 012577 | 012626 | 012675 | 012724 | 012773 | 012823 | 012872 | 49 |
| 39 | 012921 | 012970 | 0130 [...]0 | 013069 | 013118 | 013168 | 013218 | 013267 | 013317 | 013367 | 50 |
| 40 | 013417 | 013467 | 013517 | 013567 | 013617 | 013667 | 013717 | 013767 | 013818 | 013868 | 50 |
| 41 | 013919 | 013969 | 014020 | 014071 | 014121 | 014172 | 014223 | 014274 | 014325 | 014375 | 51 |
| 42 | 014426 | 014478 | 014529 | 014580 | 014632 | 014683 | 014734 | 014786 | 014837 | 014889 | 51 |
| 43 | 014941 | 014992 | 015044 | 015096 | 015148 | 015199 | 015252 | 015304 | 015356 | 015408 | 52 |
| 44 | 015460 | 015512 | 015565 | 015617 | 015669 | 015721 | 015774 | 015827 | 015879 | 015932 | 52 |
| 45 | 015985 | 016038 | 016091 | 016144 | 016197 | 016249 | 016303 | 016356 | 0164 [...]9 | 016462 | 53 |
| 46 | 016515 | 016569 | 016622 | 016676 | 016729 | 016783 | 016837 | 016891 | 016944 | 016998 | 54 |
| 47 | 017052 | 017906 | 017160 | 017214 | 017268 | 017322 | 017376 | 017431 | 017485 | 017539 | 54 |
| 48 | 017593 | 017648 | 017703 | 017757 | 017812 | 017866 | 017921 | 017976 | 018031 | 018086 | 55 |
| 49 | 018141 | 018196 | 018251 | 018306 | 018361 | 018416 | 018471 | 018527 | 018582 | 018638 | 55 |
| 50 | 018693 | 018749 | 018804 | 018860 | 018916 | 018971 | 019027 | 019083 | 019139 | 019195 | 56 |
| 51 | 019251 | 019307 | 019363 | 019419 | 019475 | 019531 | 019588 | 019644 | 019701 | 019757 | 56 |
| 52 | 019813 | 019870 | 019927 | 019984 | 020040 | 020097 | 020154 | 020211 | 020268 | 020325 | 57 |
| 53 | 020381 | 020439 | 020496 | 020553 | 020610 | 020667 | 020725 | 020782 | 020840 | 020897 | 57 |
| 54 | 020954 | 021012 | 021070 | 021128 | 021185 | 021243 | 021301 | 021359 | 021416 | 021474 | 57 |
| 55 | 021532 | 021590 | 021649 | 021707 | 021765 | 021823 | 021882 | 021940 | 021999 | 022057 | 58 |
| 56 | 022115 | 022174 | 022233 | 022292 | 022350 | 022409 | 022468 | 022527 | 022586 | 022645 | 59 |
| 57 | 022703 | 022763 | 022822 | 022881 | 022949 | 022999 | 023058 | 023118 | 023177 | 023237 | 59 |
| 58 | 023296 | 023356 | 023415 | 023475 | 023534 | 023594 | 023654 | 023714 | 023774 | 023834 | 60 |
| 59 | 023894 | 023954 | 024014 | 024074 | 024134 | 024194 | 024254 | 024315 | 024375 | 024436 | 60 |
| 60 | 024496 | 024557 | 024617 | 024678 | 024738 | 024799 | 024860 | 024921 | 024981 | 025042 | 60 |
| 61 | 025103 | 025164 | 025225 | 025286 | 025347 | 025408 | 025470 | 025531 | 025692 | 025654 | 61 |
| 62 | 025715 | 025776 | 025838 | 025899 | 025961 | 026022 | 026084 | 026146 | 026208 | 026270 | 62 |
| 63 | 026331 | 026393 | 026455 | 026517 | 026579 | 026641 | 026703 | 026766 | 026828 | 026890 | 62 |
| 64 | 026952 | 027015 | 027077 | 027140 | 027202 | 027264 | 027327 | 027390 | 027453 | 027515 | 63 |
| 65 | 027578 | 027641 | 027704 | 027767 | 027830 | 027892 | 027956 | 028019 | 028082 | 028145 | 63 |
| 66 | 028208 | 028271 | 028335 | 028398 | 028461 | 028524 | 028588 | 028652 | 028715 | 028779 | 63 |
| [Page 196-197] 67 | 028842 | 028906 | 028970 | 029034 | 029097 | 029161 | 029225 | 029289 | 02935 [...] | 029417 | 64 |
| 68 | 029481 | 029545 | 029610 | 029674 | 029738 | 029802 | 029867 | 029931 | 029996 | 030060 | 65 |
| 69 | 030184 | 030189 | 030253 | 030318 | 030383 | 030447 | 030512 | 030577 | 030642 | 030707 | 65 |
| 70 | 030772 | 030837 | 030902 | 030967 | 031032 | 031097 | 031163 | 031228 | 031293 | 031359 | 65 |
| 71 | 031424 | 031489 | 031555 | 031620 | 031686 | 031751 | 031817 | 031883 | 031949 | 032014 | 66 |
| 72 | 032080 | 032146 | 032212 | 032278 | 032344 | 032409 | 032476 | 032542 | 032608 | 032674 | 66 |
| 73 | 032741 | 032807 | 032873 | 032939 | 033006 | 033072 | 033139 | 033205 | 033272 | 033338 | 66 |
| 74 | 033405 | 033472 | 033538 | 033605 | 033672 | 033738 | 033875 | 033879 | 033939 | 034006 | 67 |
| 75 | 034073 | 034140 | 034208 | 034275 | 034342 | 034409 | 034477 | 034544 | 034612 | 034679 | 67 |
| 76 | 034746 | 034814 | 034881 | 034949 | 035016 | 035084 | 035152 | 035219 | 035287 | 035355 | 68 |
| 77 | 035423 | 035491 | 035559 | 035627 | 035695 | 035763 | 035831 | 035899 | 035968 | 036036 | 68 |
| 78 | 036104 | 036172 | 036249 | 036309 | 036378 | 036446 | 036515 | 036583 | 036651 | 036720 | 69 |
| 79 | 036789 | 036858 | 036927 | 036995 | 037064 | 037133 | 037202 | 037271 | 037339 | 037408 | 69 |
| 80 | 037477 | 037546 | 037615 | 037684 | 037752 | 037822 | 037891 | 037961 | 038030 | 038099 | 69 |
| 81 | 038169 | 038239 | 038308 | 038378 | 038447 | 038517 | 038587 | 038657 | 038727 | 038797 | 70 |
| 82 | 038867 | 038937 | 039007 | 039077 | 039147 | 039217 | 039287 | 039357 | 039428 | 039498 | 70 |
| 83 | 039568 | 039638 | 039709 | 039779 | 039849 | 039919 | 039990 | 040061 | 040131 | 040202 | 71 |
| 84 | 040272 | 040343 | 040414 | 040485 | 040555 | 040636 | 040707 | 040778 | 040849 | 040920 | 71 |
| 85 | 040981 | 041052 | 041123 | 041194 | 041265 | 041336 | 041407 | 041479 | 041550 | 041621 | 71 |
| 86 | 041692 | 041764 | 041835 | 041907 | 041978 | 042050 | 042122 | 042193 | 042265 | 042336 | 72 |
| 87 | 042408 | 042480 | 042552 | 042624 | 042696 | 042768 | 042840 | 042912 | 042984 | 043056 | 72 |
| 88 | 043128 | 043200 | 043272 | 043345 | 043417 | 043489 | 043562 | 043634 | 043706 | 043779 | 73 |
| 89 | 043852 | 043924 | 043697 | 044069 | 044142 | 044214 | 044287 | 044360 | 044433 | 044505 | 73 |
| 90 | 044578 | 044651 | 044724 | 044797 | 044870 | 044943 | 045016 | 045089 | 045163 | 045236 | 73 |
| 91 | 045309 | 045382 | 045456 | 045529 | 045603 | 045676 | 045749 | 045823 | 045896 | 045969 | 74 |
| 92 | 046043 | 046117 | 046190 | 046264 | 046338 | 046411 | 046485 | 046559 | 046633 | 046707 | 74 |
| 93 | 046781 | 046855 | 046929 | 047003 | 047077 | 047151 | 047 [...]25 | 047299 | 047374 | 047448 | 74 |
| 94 | 047522 | 047596 | 047671 | 047745 | 047819 | 047894 | 047969 | 048043 | 048118 | 048192 | 75 |
| 95 | 048267 | 048342 | 048417 | 048491 | 048566 | 048641 | 048716 | 048791 | 048866 | 048941 | 75 |
| 96 | 049015 | 049091 | 049166 | 049241 | 049316 | 049391 | 049466 | 049542 | 049617 | 049692 | 75 |
| 97 | 049767 | 049843 | 049918 | 049994 | 050069 | 050144 | 050226 | 050296 | 050371 | 050447 | 75 |
| 98 | 050522 | 050598 | 050674 | 050750 | 050826 | 050901 | 050977 | 051053 | 051129 | 051205 | 76 |
| 99 | 051281 | 051358 | 051434 | 051510 | 051586 | 051662 | 051738 | 051815 | 051891 | 051968 | 76 |
| 100 | 052044 | 052120 | 052197 | 052273 | 052350 | 052426 | 052503 | 052579 | 052656 | 052733 | 76 |
| [Page 198-199] | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | D |
| 100 | 052044 | 052120 | 052197 | 052273 | 052350 | 052426 | 052503 | 052579 | 052656 | 052733 | 076 |
| 101 | 052810 | 052886 | 052963 | 053040 | 053117 | 053193 | 053271 | 053348 | 053425 | 053502 | 077 |
| 102 | 053579 | 053656 | 053733 | 053810 | 053887 | 053964 | 054041 | 054119 | 054196 | 054273 | 077 |
| 103 | 054351 | 054428 | 054506 | 054583 | 054661 | 054738 | 054816 | 054893 | 054973 | 055049 | 078 |
| 104 | 055127 | 055204 | 055282 | 055360 | 055438 | 055516 | 055594 | 055672 | 055750 | 055828 | 078 |
| 105 | 055906 | 055984 | 056062 | 056140 | 056218 | 056296 | 056374 | 056453 | 006531 | 056610 | 078 |
| 106 | 056688 | 056766 | 056845 | 056923 | 057002 | 057080 | 057159 | 057237 | 057316 | 057395 | 079 |
| 107 | 057474 | 057552 | 057631 | 057710 | 057789 | 057868 | 057946 | 058025 | 058104 | 058183 | 079 |
| 108 | 058262 | 058341 | 058420 | 058499 | 058578 | 058658 | 058737 | 058816 | 058895 | 058975 | 079 |
| 109 | 059054 | 059133 | 059213 | 059292 | 059372 | 059451 | 059531 | 059610 | 059690 | 059769 | 079 |
| 110 | 059849 | 059929 | 060009 | 060088 | 060168 | 060248 | 060328 | 060408 | 060488 | 060560 | 080 |
| 111 | 060648 | 060728 | 060808 | 060888 | 060968 | 061048 | 061128 | 061208 | 061289 | 061369 | 080 |
| 112 | 061449 | 061529 | 061610 | 061690 | 061771 | 061851 | 061932 | 062012 | 062093 | 062173 | 080 |
| 113 | 062254 | 062334 | 062415 | 062496 | 062576 | 062657 | 062738 | 062819 | 062900 | 062981 | 081 |
| 114 | 063062 | 063143 | 063224 | 063305 | 063386 | 063467 | 063548 | 063629 | 063710 | 063791 | 081 |
| 115 | 063873 | 063954 | 064035 | 064116 | 064198 | 064279 | 064360 | 064442 | 064523 | 064605 | 081 |
| 116 | 064686 | 064768 | 064849 | 064931 | 065013 | 065095 | 065176 | 065258 | 065340 | 065421 | 082 |
| 117 | 065503 | 065585 | 065667 | 065749 | 065831 | 065913 | 065995 | 066077 | 066159 | 066241 | 082 |
| 118 | 066323 | 066405 | 066488 | 066570 | 066652 | 066735 | 066817 | 066899 | 066981 | 067064 | 082 |
| 119 | 067146 | 067229 | 067311 | 067393 | 067476 | 067559 | 067642 | 067724 | 067807 | 067889 | 083 |
| 120 | 067972 | 068055 | 068138 | 068221 | 068304 | 068387 | 068469 | 068552 | 068635 | 068718 | 083 |
| 121 | 068801 | 068884 | 068967 | 069051 | 069134 | 069217 | 069300 | 069384 | 069467 | 069550 | 083 |
| 122 | 069633 | 069717 | 069800 | 069884 | 069967 | 070051 | 070134 | 070218 | 070301 | 070385 | 083 |
| 123 | 070468 | 070552 | 070635 | 070719 | 070803 | 070887 | 070970 | 071054 | 071138 | 071222 | 084 |
| 124 | 071306 | 071390 | 071474 | 071558 | 071642 | 071726 | 071810 | 071895 | 071979 | 072063 | 084 |
| 125 | 072148 | 072232 | 072316 | 072400 | 072484 | 072569 | 072653 | 072733 | 072822 | 072906 | 084 |
| 126 | 072991 | 073075 | 073160 | 073244 | 073329 | 073414 | 073498 | 073583 | 073668 | 073752 | 084 |
| 127 | 073837 | 073922 | 074007 | 074092 | 074177 | 074262 | 074347 | 074432 | 074517 | 074602 | 085 |
| 128 | 074687 | 074772 | 074857 | 074942 | 075027 | 075112 | 075197 | 075283 | 075368 | 075453 | 085 |
| 129 | 075539 | 075624 | 075709 | 075795 | 075880 | 075966 | 076051 | 076137 | 076222 | 076908 | 085 |
| 130 | 076394 | 076479 | 076565 | 076651 | 076736 | 076822 | 076908 | 076994 | 077079 | 077165 | 086 |
| 131 | 077251 | 077337 | 077423 | 077509 | 077595 | 077681 | 077767 | 077853 | 077939 | 078025 | 086 |
| 132 | 078112 | 078198 | 078184 | 078370 | 078457 | 078543 | 078629 | 078716 | 078802 | 078889 | 086 |
| 133 | 078975 | 079062 | 079148 | 079235 | 079321 | 079408 | 079494 | 079581 | 079668 | 079754 | 086 |
| [Page 200-201] 133 | 078975 | 079062 | 079148 | 079235 | 079321 | 079408 | 079494 | 079581 | 079668 | 079754 | 086 |
| 134 | 079841 | 079928 | 080015 | 080101 | 080188 | 080275 | 080362 | 080449 | 080536 | 080623 | 087 |
| 135 | 080710 | 080797 | 080885 | 08g972 | 081059 | 081147 | 081234 | 081321 | 081408 | 081495 | 087 |
| 136 | 081582 | 081669 | 081756 | 081841 | 081931 | 032018 | 082106 | 082193 | 082281 | 082368 | 087 |
| 137 | 082456 | 082543 | 082631 | 082718 | 082806 | 082894 | 082981 | 083069 | 083157 | 083245 | 087 |
| 138 | 083333 | 083420 | 083508 | 083596 | 083684 | 083772 | 083860 | 083948 | 084036 | 084124 | 088 |
| 139 | 084212 | 084300 | 084388 | 084477 | 084565 | 084653 | 084741 | 084830 | 084918 | 085006 | 088 |
| 140 | 085095 | 085183 | 085271 | 085359 | 085448 | 085536 | 085625 | 085714 | 085802 | 085891 | 088 |
| 141 | 085979 | 086068 | 086157 | 086246 | 086334 | 086423 | 086512 | 086601 | 086689 | 086778 | 089 |
| 142 | 086867 | 086956 | 087045 | 087134 | 087223 | 087312 | 087401 | 087490 | 087579 | 087668 | 089 |
| 143 | 087757 | 087846 | 087935 | 088025 | 088114 | 088203 | 088292 | 088383 | 088471 | 088561 | 089 |
| 144 | 088650 | 088740 | 088829 | 088920 | 089009 | 089099 | 089188 | 089278 | 089366 | 089456 | 090 |
| 145 | 089545 | 089635 | 089725 | 089814 | 089904 | 089994 | 090084 | 090174 | 090264 | 090354 | 090 |
| 146 | 090444 | 090533 | 090623 | 090713 | 090803 | 090893 | 090983 | 091073 | 091163 | 091253 | 90 |
| 147 | 091344 | 091434 | 091524 | 091614 | 091704 | 091795 | 091885 | 091975 | 092066 | 092156 | 90 |
| 148 | 092247 | 092337 | 092427 | 092518 | 092608 | 092699 | 092790 | 092880 | 092971 | 093061 | 90 |
| 149 | 093152 | 093243 | 093333 | 093424 | 0 [...]3515 | 093606 | 093696 | 093787 | 093878 | 093969 | 91 |
| 150 | 094060 | 094151 | 094242 | 094333 | 094424 | 094515 | 094606 | 094697 | 094788 | 094879 | 91 |
| 151 | 094971 | 095062 | 095153 | 095244 | 095335 | 095427 | 095518 | 095609 | 095701 | 095792 | 91 |
| 152 | 095884 | 095975 | 096067 | 096158 | 096249 | 096341 | 096433 | 096524 | 096616 | 096707 | 92 |
| 153 | 096799 | 096891 | 096982 | 097074 | 097166 | 097258 | 097349 | 097441 | 097533 | 097625 | 92 |
| 154 | 097717 | 097809 | 097901 | 097993 | 098085 | 098177 | 098269 | 098361 | 098453 | 098545 | 92 |
| 155 | 098637 | 098729 | 098822 | 098914 | 099006 | 099098 | 099191 | 099283 | 099375 | 099468 | 92 |
| 156 | 099569 | 099653 | 099745 | 099838 | 099830 | 100023 | 100115 | 100208 | 100300 | 100393 | 93 |
| 157 | 100486 | 100578 | 100671 | 100763 | 100856 | 100949 | 101042 | 101135 | 101227 | 101320 | 93 |
| 158 | 101413 | 101506 | 101599 | 101692 | 101785 | 101878 | 101971 | 102064 | 102157 | 101250 | 93 |
| 159 | 102343 | 102436 | 102529 | 102622 | 102715 | 102809 | 102902 | 102995 | 103088 | 103182 | 93 |
| 160 | 103275 | 103368 | 103462 | 103555 | 103649 | 103742 | 103836 | 103929 | 104023 | 104116 | 93 |
| 161 | 104210 | 104304 | 104397 | 104491 | 104584 | 104678 | 104772 | 104866 | 104959 | 105053 | 94 |
| 162 | 105147 | 105241 | 105335 | 105429 | 105523 | 105617 | 105711 | 105805 | 105899 | 105991 | 94 |
| 193 | 106087 | 106181 | 106275 | 106369 | 106463 | 106557 | 106651 | 106745 | 106840 | 106934 | 94 |
| 164 | 107028 | 107122 | 107217 | 107311 | 107406 | 107500 | 107594 | 107689 | 107783 | 107878 | 94 |
| 165 | 107972 | 108067 | 108161 | 108256 | 108350 | 108445 | 108540 | 108634 | 108727 | 108824 | 95 |
| 166 | 108919 | 109013 | 109108 | 109203 | 109298 | 109393 | 109488 | 109583 | 109678 | 109773 | 95 |
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| 168 | 110818 | 110913 | 111009 | 111104 | 111295 | 111390 | 111485 | 111485 | 111581 | 111676 | 95 |
| 169 | 111772 | 111867 | 111963 | 112058 | 112154 | 112249 | 112344 | 112440 | 112535 | 112631 | 96 |
| 170 | 112726 | 112822 | 112918 | 113013 | 113109 | 113205 | 113301 | 113397 | 113492 | 113588 | 96 |
| 171 | 113684 | 113780 | 113876 | 113972 | 114068 | 114164 | 114260 | 114356 | 114452 | 114548 | 96 |
| 172 | 114644 | 114740 | 114836 | 114932 | 115028 | 115125 | 115221 | 115318 | 115414 | 115511 | 96 |
| 173 | 115607 | 115703 | 115799 | 115896 | 115992 | 116088 | 116184 | 116281 | 116377 | 116473 | 97 |
| 174 | 116570 | 116666 | 116763 | 116860 | 116956 | 117053 | 117150 | 117246 | 117343 | 117439 | 97 |
| 175 | 117536 | 117633 | 117730 | 117827 | 117924 | 118021 | 118118 | 118215 | 118312 | 118409 | 97 |
| 176 | 118506 | 118603 | 118700 | 118797 | 118894 | 118991 | 119088 | 119185 | 119282 | 119379 | 97 |
| 177 | 119476 | 119574 | 119671 | 119768 | 119866 | 119963 | 120060 | 120158 | 120255 | 120352 | 97 |
| 178 | 120449 | 120547 | 120644 | 120742 | 120839 | 120936 | 121034 | 121132 | 121229 | 121327 | 98 |
| 179 | 121424 | 121562 | 121620 | 121717 | 121815 | 121913 | 122010 | 122108 | 122206 | 122304 | 98 |
| 180 | 122402 | 122500 | 122597 | 122696 | 122794 | 122892 | 122990 | 123088 | 123186 | 123284 | 98 |
| 181 | 123382 | 123480 | 123578 | 123676 | 123774 | 123872 | 123970 | 124068 | 124167 | 124265 | 98 |
| 182 | 124363 | 124461 | 124560 | 124658 | 124756 | 124855 | 124953 | 125051 | 125150 | 125248 | 98 |
| 183 | 125347 | 125445 | 125544 | 125642 | 125741 | 125840 | 125938 | 126037 | 126136 | 126234 | 98 |
| 184 | 126333 | 126432 | 126530 | 126629 | 126728 | 126827 | 126925 | 127024 | 127123 | 127221 | 99 |
| 185 | 127320 | 127419 | 127518 | 127617 | 127716 | 127815 | 127914 | 128013 | 128112 | 128211 | 99 |
| 186 | 128310 | 128409 | 128508 | 128607 | 128706 | 128805 | 128905 | 129004 | 129103 | 129202 | 99 |
| 187 | 129302 | 129401 | 129500 | 129600 | 129699 | 129799 | 129897 | 139997 | 130096 | 130196 | 99 |
| 188 | 130296 | 130395 | 130495 | 130594 | 130694 | 130794 | 130893 | 130993 | 131093 | 131192 | 100 |
| 189 | 131292 | 131392 | 131491 | 131591 | 131691 | 131791 | 131890 | 131990 | 132090 | 132190 | 100 |
| 190 | 132290 | 132390 | 132490 | 132590 | 132690 | 132790 | 132890 | 132990 | 133090 | 133190 | 100 |
| 191 | 133290 | 133390 | 133490 | 133590 | 133690 | 133790 | 133890 | 133990 | 134090 | 134191 | 100 |
| 192 | 134292 | 134392 | 134492 | 134593 | 134693 | 134794 | 134894 | 134994 | 135095 | 135195 | 100 |
| 193 | 135296 | 135396 | 135497 | 135597 | 135697 | 135798 | 135899 | 135999 | 136100 | 136201 | 101 |
| 194 | 136302 | 136402 | 136503 | 136604 | 136704 | 136805 | 136906 | 137007 | 137108 | 137209 | 101 |
| 195 | 137310 | 137410 | 137511 | 137612 | 137713 | 137814 | 137915 | 138016 | 138117 | 138218 | 101 |
| 196 | 138320 | 138421 | 138522 | 138623 | 138724 | 138825 | 138926 | 139027 | 139129 | 139230 | 101 |
| 197 | 139331 | 139432 | 139534 | 139635 | 139737 | 139838 | 139939 | 140041 | 140142 | 140244 | 101 |
| 198 | 140345 | 140447 | 140548 | 140650 | 140751 | 140853 | 140954 | 141055 | 141157 | 141259 | 101 |
| 199 | 141361 | 141462 | 141564 | 141665 | 141767 | 141869 | 141970 | 142072 | 142174 | 142276 | 102 |
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| 201 | 143398 | 142500 | 143602 | 143704 | 143806 | 143909 | 144011 | 144113 | 144215 | 144317 | 10 [...] |
| 202 | 144419 | 144521 | 144624 | 144726 | 144828 | 144931 | 145033 | 145135 | 145238 | 145340 | 102 |
| 203 | 145443 | 145545 | 145648 | 145750 | 145853 | 145955 | 146058 | 146160 | 146263 | 146365 | 102 |
| 204 | 146468 | 146571 | 146673 | 146776 | 146878 | 146981 | 147084 | 147187 | 147289 | 147392 | 102 |
| 205 | 147495 | 147598 | 147701 | 147803 | 147906 | 148009 | 148112 | 148215 | 148318 | 141421 | 103 |
| 206 | 148524 | 148627 | 148730 | 148833 | 148936 | 149039 | 149142 | 149245 | 149348 | 149451 | 103 |
| 207 | 149555 | 149658 | 149761 | 149864 | 149967 | 150071 | 150174 | 150277 | 150387 | 150484 | 103 |
| 208 | 150588 | 150691 | 150794 | 150898 | 151001 | 151105 | 151208 | 151312 | 151415 | 151519 | 103 |
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| 210 | 152659 | 152762 | 152866 | 152970 | 153074 | 153178 | 153281 | 153385 | 153489 | 153592 | 103 |
| 211 | 153696 | 153800 | 153903 | 154009 | 154111 | 154215 | 154319 | 154423 | 154523 | 154632 | 103 |
| 212 | 154736 | 154840 | 154944 | 155048 | 155152 | 155257 | 155361 | 155465 | 155569 | 155674 | 104 |
| 213 | 155778 | 155882 | 155987 | 156092 | 156195 | 156300 | 156404 | 156508 | 156613 | 156717 | 104 |
| 214 | 156822 | 156926 | 157030 | 157135 | 157239 | 157344 | 157448 | 157553 | 157657 | 157762 | 104 |
| 215 | 157867 | 157971 | 158076 | 158180 | 158285 | 158390 | 158494 | 158599 | 158704 | 158809 | 105 |
| 216 | 158914 | 159018 | 159123 | 159228 | 159333 | 159438 | 159453 | 159648 | 159753 | 159858 | 105 |
| 217 | 159963 | 160068 | 160173 | 160278 | 160383 | 160488 | 160593 | 160698 | 160803 | 160908 | 105 |
| 218 | 161013 | 161118 | 161223 | 161329 | 161434 | 161539 | 161644 | 161749 | 161855 | 161959 | 105 |
| 219 | 162065 | 162170 | 162276 | 162387 | 162487 | 162592 | 162697 | 162803 | 162908 | 162014 | 105 |
| 220 | 163119 | 163225 | 163330 | 163436 | 163542 | 163648 | 163753 | 163859 | 163964 | 164070 | 106 |
| 221 | 164176 | 164281 | 164387 | 164493 | 164598 | 164704 | 164810 | 164916 | 165021 | 165127 | 106 |
| 222 | 165233 | 165339 | 165445 | 165550 | 165656 | 165762 | 165868 | 165974 | 166080 | 166186 | 106 |
| 223 | 166292 | 166398 | 166504 | 166610 | 166716 | 166822 | 166928 | 167034 | 167141 | 167247 | 106 |
| 224 | 167353 | 167459 | 167565 | 167672 | 167778 | 167884 | 167990 | 168096 | 168203 | 168309 | 106 |
| 225 | 168415 | 168521 | 168521 | 168734 | 168841 | 168947 | 169053 | 169160 | 169266 | 169373 | 107 |
| 226 | 169479 | 169586 | 169692 | 169799 | 169905 | 170012 | 170119 | 170225 | 120332 | 170438 | 107 |
| 227 | 170545 | 170652 | 170758 | 170865 | 170972 | 171079 | 171185 | 171292 | 171399 | 171506 | 107 |
| 228 | 171613 | 171719 | 171826 | 171933 | 172040 | 172147 | 172254 | 172361 | 172468 | 172575 | 107 |
| 229 | 172682 | 172789 | 172896 | 173003 | 173110 | 173217 | 173324 | 173431 | 173538 | 173645 | 107 |
| 230 | 173753 | 173860 | 173967 | 174074 | 174181 | 174289 | 174396 | 174503 | 174610 | 174718 | 107 |
| 231 | 174825 | 174932 | 175040 | 175147 | 175255 | 175362 | 175469 | 175577 | 175684 | 175792 | 107 |
| 232 | 175899 | 176007 | 176114 | 176222 | 176329 | 176437 | 176554 | 176652 | 176759 | 176867 | 108 |
| 233 | 176975 | 177082 | 177190 | 177298 | 177405 | 177513 | 177621 | 177729 | 177836 | 177944 | 108 |
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| 235 | 179131 | 179239 | 179347 | 179455 | 179563 | 179672 | 179780 | 179888 | 179996 | 180104 | 108 |
| 236 | 180212 | 180320 | 180428 | 180536 | 180644 | 180752 | 180860 | 180969 | 181077 | 181085 | 108 |
| 237 | 181294 | 181402 | 181510 | 181618 | 181727 | 181835 | 181943 | 182052 | 182160 | 182269 | 108 |
| 238 | 182377 | 182486 | 182594 | 182703 | 182811 | 182920 | 183028 | 183137 | 183245 | 183354 | 108 |
| 239 | 183462 | 183571 | 183680 | 183788 | 183897 | 184006 | 184114 | 184223 | 184332 | 184440 | 109 |
| 240 | 184549 | 184658 | 184767 | 184875 | 184984 | 185093 | 185202 | 185311 | 185419 | 185528 | 109 |
| 241 | 185637 | 185746 | 185855 | 185964 | 186073 | 186182 | 186291 | 186400 | 186509 | 186618 | 109 |
| 242 | 186728 | 186837 | 186946 | 187055 | 187164 | 187273 | 187382 | 187491 | 187600 | 187709 | 109 |
| 243 | 187819 | 187928 | 188037 | 188146 | 188256 | 188365 | 188474 | 188584 | 188693 | 188802 | 109 |
| 244 | 188911 | 189021 | 189130 | 189240 | 189349 | 189458 | 189568 | 189677 | 189787 | 189896 | 109 |
| 245 | 199006 | 190116 | 190225 | 190335 | 190444 | 190554 | 190664 | 190773 | 190883 | 190992 | 110 |
| 246 | 191102 | 191212 | 191322 | 191431 | 191541 | 191651 | 191761 | 191871 | 191980 | 192090 | 110 |
| 247 | 192200 | 192310 | 192420 | 192529 | 192639 | 192749 | 192859 | 192969 | 193079 | 193189 | 110 |
| 248 | 193298 | 193408 | 193518 | 193628 | 193738 | 193848 | 193958 | 194068 | 194178 | 194288 | 110 |
| 249 | 194399 | 194509 | 194619 | 194729 | 194839 | 194949 | 195059 | 195169 | 195279 | 195389 | 110 |
| 250 | 195500 | 195611 | 195721 | 195831 | 195942 | 196052 | 196163 | 196273 | 196384 | 196494 | 110 |
| 251 | 196605 | 196715 | 196826 | 196936 | 197047 | 197157 | 197268 | 197378 | 197489 | 197599 | 110 |
| 252 | 197710 | 197821 | 197931 | 198041 | 198152 | 198262 | 198373 | 198484 | 198594 | 198705 | 110 |
| 253 | 198815 | 198926 | 199037 | 199148 | 199258 | 199369 | 199480 | 199591 | 199701 | 199812 | 111 |
| 254 | 199923 | 200034 | 200145 | 200256 | 200367 | 200478 | 200 [...]88 | 200699 | 200811 | 200921 | 111 |
| 255 | 201032 | 201144 | 201255 | 201366 | 201477 | 201587 | 201699 | 201810 | 201921 | 202032 | 111 |
| 256 | 202143 | 202254 | 202365 | 202477 | 202588 | 202699 | 202810 | 202921 | 203033 | 203144 | 111 |
| 257 | 203255 | 203366 | 203478 | 203589 | 203700 | 203811 | 203922 | 204033 | 204145 | 204256 | 111 |
| 258 | 204368 | 204480 | 204591 | 204703 | 204814 | 204926 | 205037 | 205149 | 205261 | 205372 | 111 |
| 259 | 205483 | 205595 | 205707 | 205818 | 205930 | 206041 | 206153 | 206264 | 206376 | 206488 | 112 |
| 260 | 206599 | 206711 | 206823 | 206935 | 207047 | 207158 | 207270 | 207382 | 207494 | 207606 | 112 |
| 261 | 207717 | 207829 | 207941 | 208053 | 208165 | 208277 | 208389 | 208501 | 208613 | 208725 | 112 |
| 262 | 208836 | 208948 | 209060 | 209172 | 209284 | 209396 | 209508 | 209620 | 209733 | 209845 | 112 |
| 263 | 209957 | 210069 | 210181 | 210293 | 210405 | 210517 | 210630 | 210742 | 210854 | 210966 | 112 |
| 264 | 211078 | 211191 | 211303 | 211415 | 211528 | 211640 | 211752 | 211865 | 211977 | 212089 | 112 |
| 265 | 212202 | 212314 | 212427 | 212539 | 212651 | 212763 | 212876 | 212989 | 213101 | 213214 | 112 |
| 266 | 213326 | 213439 | 213552 | 213665 | 213778 | 222891 | 214003 | 214115 | 214227 | 214340 | 113 |
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| 267 | 214452 | 214565 | 214677 | 214780 | 214893 | 214015 | 215128 | 215241 | 215354 | 215467 | 113 |
| 268 | 215579 | 215692 | 215805 | 215918 | 216031 | 216143 | 216256 | 216369 | 216482 | 216595 | 113 |
| 269 | 216708 | 216821 | 216934 | 217047 | 217160 | 217272 | 217386 | 217499 | 217612 | 217725 | 113 |
| 270 | 217838 | 217951 | 218064 | 218177 | 218290 | 218403 | 218517 | 218630 | 218743 | 218856 | 113 |
| 271 | 218969 | 219082 | 219195 | 219309 | 219422 | 219535 | 219648 | 219762 | 219875 | 219988 | 113 |
| 272 | 220101 | 220215 | 220328 | 220442 | 220555 | 220668 | 220782 | 220895 | 221009 | 221122 | 113 |
| 273 | 221235 | 221349 | 221462 | 221576 | 221689 | 221803 | 221916 | 222030 | 222143 | 222257 | 113 |
| 274 | 222371 | 222484 | 222598 | 222711 | 222825 | 222538 | 223052 | 223166 | 223280 | 223393 | 114 |
| 275 | 223507 | 223621 | 223734 | 223848 | 223962 | 224075 | 224189 | 224303 | 224417 | 224531 | 114 |
| 276 | 224644 | 224754 | 224871 | 224986 | 225100 | 225214 | 225328 | 225442 | 225555 | 225670 | 114 |
| 277 | 225784 | 225897 | 226011 | 226125 | 226239 | 226353 | 226467 | 226581 | 226695 | 226809 | 114 |
| 278 | 226923 | 227038 | 227152 | 227266 | 227380 | 227494 | 227608 | 227723 | 227837 | 227951 | 114 |
| 279 | 228065 | 228179 | 228294 | 228408 | 228522 | 228636 | 228751 | 228865 | 228979 | 229094 | 114 |
| 280 | 229208 | 229322 | 229437 | 229551 | 229666 | 229780 | 229894 | 230009 | 230123 | 230238 | 114 |
| 281 | 230352 | 230467 | 230581 | 230695 | 230810 | 230924 | 231039 | 231153 | 231268 | 231382 | 114 |
| 282 | 231497 | 231612 | 231726 | 231841 | 231955 | 232070 | 232185 | 232299 | 232414 | 232529 | 115 |
| 283 | 232644 | 232758 | 232873 | 232989 | 233103 | 233218 | 233332 | 233447 | 233562 | 233677 | 115 |
| 284 | 233792 | 233906 | 234021 | 234136 | 234251 | 234366 | 234480 | 234595 | 234710 | 234825 | 115 |
| 285 | 234946 | 235055 | 235170 | 235285 | 235400 | 235515 | 235630 | 235745 | 235860 | 235975 | 115 |
| 286 | 236091 | 236206 | 236321 | 236436 | 236551 | 236666 | 236781 | 236896 | 237012 | 237127 | 115 |
| 287 | 237242 | 237357 | 237472 | 237588 | 237703 | 237818 | 237933 | 238049 | 238164 | 238279 | 115 |
| 288 | 238395 | 238510 | 238625 | 238740 | 238856 | 238971 | 239086 | 339202 | 239317 | 239433 | 115 |
| 289 | 239548 | 239664 | 239779 | 239895 | 230010 | 240126 | 240241 | 240357 | 240472 | 240588 | 115 |
| 290 | 240703 | 240819 | 240934 | 241050 | 241165 | 241281 | 241397 | 241522 | 241628 | 251753 | 116 |
| 291 | 241859 | 241975 | 242090 | 242206 | 242322 | 242438 | 242553 | 242669 | 242785 | 242908 | 116 |
| 292 | 243016 | 243132 | 243248 | 243364 | 243480 | 243595 | 243711 | 243827 | 243943 | 243059 | 116 |
| 293 | 244175 | 244290 | 244496 | 244522 | 244638 | 244754 | 244870 | 244986 | 245102 | 245218 | 116 |
| 294 | 245334 | 245450 | 245566 | 245682 | 245748 | 245915 | 246031 | 246147 | 246263 | 246379 | 116 |
| 295 | 246495 | 246611 | 246727 | 246843 | 246959 | 247075 | 247192 | 247308 | 247424 | 247541 | 116 |
| 296 | 247657 | 247773 | 247889 | 248006 | 248122 | 248238 | 248354 | 248471 | 248587 | 248704 | 116 |
| 297 | 248820 | 248936 | 249053 | 249169 | 249285 | 249402 | 249518 | 249634 | 249751 | 249867 | 116 |
| 298 | 249984 | 250101 | 250217 | 250334 | 250450 | 250567 | 250683 | 250799 | 250916 | 251033 | 117 |
| 299 | 251149 | 251266 | 251382 | 251499 | 251615 | 251732 | 251849 | 251965 | 252082 | 252198 | 117 |
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| 301 | 253483 | 253599 | 253716 | 253833 | 253950 | 254067 | 254184 | 254301 | 254418 | 254535 | 117 |
| 302 | 254652 | 254769 | 254886 | 255003 | 255119 | 255237 | 255353 | 255471 | 255588 | 255701 | 117 |
| 303 | 255822 | 255939 | 256056 | 256173 | 256 [...]89 | 256407 | 256524 | 256641 | 256758 | 256875 | 117 |
| 304 | 256992 | 257109 | 257226 | 257344 | 257461 | 257573 | 257695 | 257812 | 257930 | 258047 | 117 |
| 305 | 258164 | 258281 | 258398 | 258516 | 258633 | 258750 | 258867 | 258985 | 259102 | 259220 | 117 |
| 306 | 259337 | 259454 | 259752 | 159689 | 259807 | 259924 | 259041 | 260159 | 260276 | 260394 | 117 |
| 307 | 260511 | 260269 | 250746 | 260864 | 260981 | 261099 | 261216 | 261334 | 261451 | 261569 | 117 |
| 308 | 261686 | 261803 | 251921 | 262039 | 262156 | 262274 | 262392 | 262509 | 262627 | 262744 | 117 |
| 309 | 262862 | 262980 | 253097 | 263215 | 263332 | 263540 | 263568 | 263686 | 263804 | 263922 | 118 |
| 310 | 264040 | 264157 | 264275 | 264393 | 264511 | 264269 | 264746 | 264864 | 264982 | 265100 | 118 |
| 311 | 265218 | 265336 | 265454 | 265572 | 265690 | 265808 | 265925 | 266043 | 266161 | 266279 | 118 |
| 312 | 266397 | 266515 | 266633 | 166751 | 266869 | 266987 | 267105 | 267223 | 267341 | 267459 | 118 |
| 313 | 267578 | 267696 | 267814 | 267932 | 268050 | 268168 | 268286 | 268404 | 268522 | 268640 | 118 |
| 314 | 268759 | 268877 | 268995 | 269113 | 269231 | 269349 | 269468 | 269586 | 269704 | 269823 | 118 |
| 315 | 269941 | 270059 | 270177 | 270295 | 270413 | 270532 | 270650 | 270769 | 270887 | 271006 | |
| 316 | 271124 | 271242 | 271361 | 271479 | 271598 | 271716 | 271835 | 271953 | 272072 | 272190 | |
| 317 | 272 [...]09 | 272427 | 272546 | 272664 | 272783 | 272901 | 273020 | 273138 | 273257 | 273375 | |
| 318 | 273494 | 273612 | 273731 | 273849 | [...]73968 | 274087 | 274205 | 274324 | 274443 | 274561 | |
| 319 | 274680 | 274799 | 274917 | 275036 | 275155 | 275274 | 275392 | 275511 | 275630 | 275749 | |
| 320 | 275868 | 275986 | 276105 | 276224 | 276343 | 276462 | 276580 | 276699 | 276818 | 276937 | |
| 321 | 277856 | 277175 | 277294 | 277413 | 277532 | 277651 | 277769 | 277888 | 278007 | 278126 | |
| 322 | 278245 | 278364 | 278483 | 278602 | 278721 | 278840 | 278959 | 279078 | 279197 | 279316 | |
| 323 | 279436 | 279555 | 279674 | 279793 | 27991 [...] | 280031 | 280150 | 280269 | 280389 | 280508 | |
| 324 | 280627 | 280746 | 280865 | 280985 | 281104 | 281223 | 281342 | 281461 | 281581 | 281700 | |
| 325 | 281819 | 281938 | 282058 | 282177 | 282296 | 282416 | 282535 | 282654 | 282773 | 282893 | |
| 326 | 283012 | 283131 | 283251 | 283370 | 283490 | 283609 | 283729 | 283848 | 283968 | 284087 | |
| 327 | 284207 | 284326 | 284446 | 288565 | 284685 | 284804 | 284924 | 285043 | 285163 | 285282 | |
| 328 | 285402 | 285521 | 285641 | 285760 | 28 [...]880 | 286000 | 286119 | 286239 | 286358 | 286478 | |
| 329 | 286598 | 286717 | 286837 | 286956 | 287076 | 287196 | 287315 | 287435 | 287555 | 287675 | |
| 330 | 287795 | 287915 | 288034 | 288154 | 288273 | 288393 | 288513 | 288633 | 288752 | 288872 | |
| 331 | 288992 | 289112 | 289232 | 289352 | 28947 [...] | 289592 | 289711 | 289831 | 289951 | 290071 | |
| 332 | 280191 | 290311 | 290431 | 290551 | 290671 | 290791 | 290911 | 291031 | 291751 | 291271 | |
| 333 | 291 [...]91 | 291511 | 291631 | 291751 | 291871 | 291991 | 292111 | 292231 | 292351 | 292471 | |
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| 334 | 292592 | 292712 | 292983 | 292952 | 293072 | 293192 | 293312 | 293432 | 293552 | 293672 | |
| 335 | 293793 | 293913 | 294033 | 294153 | 294273 | 294394 | 294514 | 294634 | 294754 | 294875 | |
| 336 | 294995 | 295115 | 295236 | 295356 | 295476 | 295197 | 295717 | 295837 | 295957 | 296078 | |
| 337 | 296198 | 296318 | 296779 | 296559 | 296680 | 296 [...]00 | 296021 | 297041 | 297162 | 297282 | |
| 338 | 297403 | 297523 | 297643 | 297764 | 297884 | 298005 | 298125 | 298246 | 298366 | 298487 | |
| 339 | 298608 | 298728 | 298849 | 298969 | 299 [...]90 | 299210 | 299331 | 299401 | 299572 | 299692 | |
| 340 | 299813 | 299934 | 300054 | 300175 | 300296 | 300417 | 300537 | 300658 | 300779 | 300899 | |
| 341 | 301020 | 301141 | 301281 | 301382 | 301503 | 301624 | 301744 | 301865 | 301986 | 302107 | |
| 342 | 302228 | 302348 | 302469 | 302590 | 302711 | 302832 | 302952 | 303073 | 303194 | 303315 | |
| 343 | 303436 | 303557 | 303678 | 303799 | 303920 | 304041 | 304161 | 304282 | 304043 | 304524 | |
| 344 | 304645 | 304766 | 304887 | 305008 | 305129 | 305250 | 305371 | 305492 | 305613 | 305734 | |
| 345 | 305856 | 305977 | 306098 | 306219 | 306340 | 306461 | 306582 | 306703 | 306824 | 306945 | |
| 346 | 307066 | 307187 | 307308 | 307430 | 307551 | 307672 | 307192 | 307914 | 308036 | 308157 | |
| 347 | 308278 | 308399 | 308520 | 308642 | 308763 | 308884 | 309005 | 309127 | 309248 | 309369 | |
| 348 | 309491 | 309612 | 309733 | 309854 | 309976 | 310097 | 310219 | 310340 | 310461 | 310583 | |
| 349 | 310704 | 310825 | 310947 | 311068 | 311 [...]90 | 311311 | 311432 | 311554 | 311675 | 311797 | |
| 350 | 311918 | 312040 | 312167 | 312283 | 312404 | 312526 | 312647 | 312769 | 312890 | 313012 | |
| 351 | 313133 | 313255 | 313376 | 313498 | 313619 | 313741 | 313863 | 313984 | 314106 | 314227 | |
| 352 | 314349 | 314471 | 314592 | 314714 | 314835 | 314957 | 315079 | 315200 | 315322 | 315444 | |
| 353 | 315566 | 315687 | 315890 | 315931 | 316052 | 316174 | 316296 | 316418 | 316539 | 316661 | |
| 354 | 316783 | 316905 | 312027 | 317148 | 317270 | 317392 | 317514 | 317636 | 317757 | 317879 | |
| 355 | 318001 | 318123 | 318245 | 318367 | 318489 | 318611 | 318732 | 318054 | 318976 | 319098 | |
| 356 | 319220 | 319342 | 319464 | 319586 | 319708 | 319830 | 319952 | 320074 | 320196 | 320318 | |
| 357 | 320440 | 320562 | 320684 | 320806 | 320928 | 321050 | 321172 | 321294 | 321416 | 321538 | |
| 358 | 321660 | 321782 | 321904 | 322026 | 322148 | 322271 | 322393 | 322515 | 322637 | 322759 | |
| 359 | 322881 | 323003 | 323125 | 323248 | 323370 | 323492 | 323614 | 323736 | 323859 | 323981 | |
| 360 | 324103 | 324225 | 324348 | 324470 | 324592 | 324715 | 324837 | 324959 | 325081 | 325204 | |
| 361 | 325326 | 325448 | 325571 | 325693 | 325815 | 325938 | 326060 | 326182 | 326305 | 326427 | |
| 362 | 326550 | 326672 | 326795 | 326917 | 327040 | 327162 | 327284 | 327406 | 327529 | 327651 | |
| 363 | 327773 | 327896 | 328018 | 328141 | 328263 | 328386 | 328509 | 328631 | 328754 | 328876 | |
| 364 | 328999 | 329125 | 329244 | 329366 | 329489 | 329611 | 329734 | 329856 | 329979 | 330101 | |
| 365 | 330224 | 330346 | 330469 | 330592 | 330714 | 330837 | 330960 | 331082 | 331205 | 331327 | |
| 366 | 331450 | 331572 | 331695 | 331818 | 331941 | 332064 | 332186 | 332309 | 332432 | 332555 | |
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| 368 | 333905 | 334028 | 334151 | 334273 | 334396 | 334519 | 334642 | 334765 | 334888 | 335011 | |
| 369 | 335134 | 335256 | 335379 | 335502 | 335625 | 335748 | 335871 | 335994 | 336117 | 336240 | |
| 370 | 336313 | 336486 | 336609 | 336732 | 336855 | 336978 | 337101 | 337224 | 337347 | 337470 | |
| 371 | 337593 | 337716 | 337839 | 337962 | 338085 | 338208 | 338331 | 338451 | 338577 | 338700 | |
| 372 | 338823 | 338946 | 339069 | 339192 | 339315 | 339438 | 339561 | 339684 | 339807 | 339930 | |
| 373 | 340054 | 340177 | 340300 | 340423 | 340546 | 340670 | 340793 | 340196 | 341039 | 341162 | |
| 374 | 341286 | 341409 | 341532 | 341655 | 341779 | 341902 | 342025 | 342148 | 342271 | 342394 | |
| 375 | 342518 | 342641 | 342764 | 342888 | 343011 | 343135 | 343258 | 343381 | 343505 | 343628 | |
| 376 | 343752 | 343875 | 343998 | 344121 | 344245 | 344368 | 344491 | 344615 | 344738 | 344862 | |
| 377 | 344985 | 345108 | 345233 | 345355 | 345479 | 345602 | 345726 | 345849 | 345973 | 346096 | |
| 378 | 346220 | 346343 | 346467 | 346590 | 346714 | 346837 | 347961 | 347082 | 347208 | 347331 | |
| 379 | 347455 | 347578 | 347702 | 347825 | 347949 | 348073 | 348916 | 348320 | 348443 | 348567 | |
| 380 | 348691 | 348814 | 348938 | 349061 | 349185 | 349309 | 349432 | 349556 | 349679 | 349803 | |
| 381 | 3499 [...]7 | 350050 | 350174 | 350298 | 350421 | 350545 | 350669 | 350792 | 350916 | 351040 | |
| 382 | 351164 | 351287 | 351411 | 851535 | 351659 | 351783 | 351806 | 352030 | 352154 | 352277 | |
| 383 | 352401 | 352525 | 352649 | 352772 | 352896 | 353020 | 353144 | 353268 | 353392 | 353516 | |
| 384 | 353640 | 353763 | 353887 | 354011 | 354135 | 354259 | 354382 | 354506 | 354030 | 354754 | |
| 385 | 354878 | 355002 | 355126 | 355250 | 355374 | 355498 | 355622 | 355746 | 355870 | 355994 | |
| 386 | 356118 | 356242 | 356366 | 356490 | 356614 | 356738 | 356862 | 356986 | 357110 | 357234 | |
| 387 | 357358 | 357482 | 357606 | 357730 | 357854 | 357978 | 358102 | 358226 | 358350 | 358474 | |
| 388 | 358598 | 358722 | 358846 | 358970 | 359094 | 359219 | 359343 | 359467 | 359591 | 359715 | |
| 389 | 359840 | 359964 | 360088 | 360212 | 360336 | 360460 | 360584 | 360708 | 360832 | 360956 | |
| 390 | 361081 | 361208 | [...]61329 | 361453 | 361578 | 361702 | 361826 | 361951 | 362075 | 362 [...]99 | |
| 391 | 362324 | 362448 | 362572 | 362698 | 362821 | 362945 | 363069 | 363194 | 363318 | 363442 | |
| 392 | 363567 | 363191 | 363815 | 363939 | 364064 | 364188 | 364312 | 364437 | 364561 | 364686 | |
| 393 | 364810 | 364934 | 365059 | 365183 | 36530 [...] | 365432 | 365556 | 315681 | 365805 | 365930 | |
| 394 | 366054 | 366178 | 366303 | 396427 | 366552 | 366676 | 366808 | 366925 | 36704 [...] | 367173 | |
| 395 | 367298 | 367422 | 367547 | 367672 | 367796 | 367921 | 368046 | 368170 | 368295 | 368419 | |
| 396 | 368 [...]44 | 368668 | 368793 | 369917 | 369042 | 369166 | 369290 | 369415 | 369539 | 369663 | |
| 397 | 369788 | 369912 | 370037 | 370162 | 370287 | 370412 | 370537 | 370661 | 370786 | 370911 | |
| 398 | 371036 | 371160 | 371285 | 371409 | 371534 | 371659 | 371783 | 376908 | 372033 | 372157 | |
| 399 | 372282 | 372407 | 37253 [...] | 372656 | 372181 | 372906 | 373030 | 373155 | 373280 | 373405 | |
| [Page 216-217] 400 | 373530 | 373654 | 373779 | 373904 | 37 [...]029 | 374154 | 374278 | 374403 | 374528 | 374655 | |
| 401 | 334778 | 374902 | 375027 | 375152 | 375277 | 375402 | 375526 | 375651 | 375776 | 375901 | |
| 402 | 376026 | 376150 | 376275 | 376400 | 376525 | 376650 | 376775 | 376900 | 377025 | 377150 | |
| 403 | 377275 | 377399 | 377524 | 377649 | 377774 | 377899 | 378024 | 378149 | 378274 | 378399 | |
| 404 | 378524 | 378649 | 378774 | 378899 | 379024 | 379149 | 379274 | 379399 | 379524 | 379649 | |
| 405 | 379774 | 379899 | 380024 | 380149 | 380274 | 380399 | 380524 | 380649 | 380774 | 380899 | |
| 406 | 381024 | 381149 | 381274 | 381399 | 381524 | 381649 | 381774 | 381899 | 382024 | 382149 | |
| 407 | 382275 | 382400 | 382525 | 38265 [...] | 382775 | 382901 | 383026 | 383151 | 383276 | 383401 | |
| 408 | 383526 | 385651 | 383776 | 383902 | 384027 | 384152 | 384277 | 384402 | 384528 | 384653 | |
| 409 | 384778 | 384903 | 385028 | 385154 | 385279 | 385404 | 385529 | 385654 | 385779 | 385904 | |
| 410 | 386029 | 386155 | 386280 | 386406 | 386531 | 386657 | 386782 | 386907 | 387032 | 387157 | |
| 411 | 387283 | 387408 | 387533 | 387658 | 387784 | 387909 | 388034 | 388160 | 388285 | 388421 | |
| 412 | 388536 | 388661 | 388787 | 388912 | 389037 | 389163 | 389288 | 389413 | 389539 | 389664 | |
| 413 | 389790 | 389915 | 390040 | 390166 | 390291 | 390417 | 390542 | 390667 | 390793 | 390918 | |
| 414 | 391044 | 391169 | 391294 | 391420 | 391545 | 391671 | 391796 | 391922 | 391047 | 392173 | |
| 415 | 392298 | 392424 | 392549 | 392675 | 392800 | 392926 | 393051 | 393177 | 393302 | 393428 | |
| 416 | 393553 | 393679 | 393804 | 393930 | 394055 | 394181 | 394306 | 394432 | 394557 | 394683 | |
| 417 | 394809 | 394934 | 395060 | 395185 | 395311 | 395437 | 397562 | 395688 | 395813 | 395939 | |
| 418 | 396065 | 396190 | 396316 | 396441 | 396567 | 396693 | 396818 | 396944 | 397069 | 397195 | |
| 419 | 3973 [...]1 | 397446 | 397572 | 397607 | 397823 | 397949 | 398074 | 398200 | 398326 | 398451 | |
| 420 | 398577 | 3987 [...]3 | 398828 | 398954 | 349080 | 399206 | 399331 | 399457 | 399583 | 399708 | |
| 421 | 399834 | 399960 | 400085 | 400211 | 400337 | 400463 | 400588 | 400714 | 400840 | 400966 | |
| 422 | 401092 | 401217 | 401343 | 401469 | 401595 | 401721 | 401846 | 401972 | 402098 | 40 [...]223 | |
| 423 | 402349 | 402475 | 402601 | 402727 | 402853 | 402979 | 403104 | 403230 | 403356 | 403482 | |
| 424 | 403608 | 403733 | 403859 | 403985 | 404111 | 404237 | 404362 | 404488 | 404614 | 404740 | |
| 425 | 404866 | 404992 | 405118 | 405244 | 405370 | 405496 | 405621 | 405747 | 405873 | 405999 | |
| 426 | 406125 | 406251 | 406377 | 406503 | 406629 | 406755 | 406881 | 407007 | 407133 | 407259 | |
| 427 | 407385 | 407511 | 407637 | 407763 | 407889 | 408015 | 408141 | 408267 | 408393 | 408519 | |
| 428 | 408645 | 408771 | 408897 | 409023 | 409149 | 409275 | 409401 | 409527 | 409653 | 409779 | |
| 429 | 409405 | 410031 | 410157 | 410283 | 401409 | 410535 | 410661 | 410787 | 410913 | 411039 | |
| 430 | 411165 | 411291 | 411417 | 411543 | 411669 | 411795 | 411921 | 412047 | 412173 | 412299 | |
| 431 | 412426 | 412552 | 412678 | 412804 | 412930 | 413056 | 413182 | 413308 | 413434 | 413560 | |
| 432 | 413687 | 413813 | 413939 | 414065 | 414191 | 414318 | 414444 | 414570 | 414696 | 414822 | |
| 433 | 414948 | 415074 | 415207 | 415327 | 415453 | 415579 | 415705 | 415831 | 415958 | 410084 | |
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| 435 | 417473 | 417599 | 417725 | 417851 | 417977 | 418104 | 418230 | 418356 | 418482 | 418609 | |
| 436 | 418735 | 418861 | 418988 | 419114 | 419240 | 419367 | 419493 | 419619 | 419745 | 419872 | |
| 437 | 419998 | 420124 | 420251 | 420377 | 420503 | 420630 | 420756 | 420882 | 421008 | 421135 | |
| 438 | 421261 | 421387 | 421514 | 421640 | 421767 | 421895 | 422019 | 422146 | 422272 | 422399 | |
| 439 | 422525 | 422651 | 422778 | 422904 | 423030 | 423157 | 423283 | 423409 | 4235 [...]6 | 423662 | |
| 440 | 423789 | 423915 | 424041 | 424168 | 424294 | 424421 | 424547 | 424674 | 424800 | 424927 | |
| 441 | 425053 | 425179 | 425306 | 425432 | 425559 | 425685 | 425812 | 425938 | 425065 | 426191 | |
| 442 | 426318 | 426444 | 426570 | 426697 | 426823 | 426950 | 427076 | 427203 | 427329 | 427456 | |
| 443 | 427582 | 427709 | 427853 | 427962 | 428088 | 428215 | 428341 | 428468 | 428594 | 428721 | |
| 444 | 428247 | 428974 | 429100 | 429227 | 429353 | 429480 | 429607 | 429733 | 429860 | 429986 | |
| 445 | 430113 | 430239 | 430366 | 430492 | 430619 | 430746 | 430872 | 430999 | 431125 | 431252 | |
| 446 | 431378 | 431505 | 431631 | 431758 | 431884 | 432011 | 432138 | 432264 | 432391 | 432517 | |
| 447 | 432644 | 432771 | 432897 | 433024 | 433050 | 433277 | 433404 | 433530 | 433657 | 433784 | |
| 448 | 433911 | 434037 | 434164 | 434290 | 434411 | 43 [...]544 | 434670 | 434797 | 434924 | 435050 | |
| 449 | 435177 | 435304 | 435430 | 435557 | 435683 | 435810 | 435937 | 436063 | 436190 | 436317 | |
| 450 | 436444 | 436570 | 436697 | 436824 | 436950 | 437077 | 437204 | 437330 | 437457 | 437584 | |
| 451 | 437717 | 437837 | 437964 | 438091 | 438217 | 438344 | 438471 | 438598 | 438724 | 438851 | |
| 452 | 438978 | 439105 | 439231 | 439358 | 439485 | 439612 | 439738 | 439865 | 439990 | 440119 | |
| 453 | 440246 | 440372 | 440499 | 440626 | 440752 | 440879 | 441006 | 441133 | 441259 | 441386 | |
| 454 | 441513 | 441640 | 441767 | 441893 | 442020 | 442147 | 442274 | 442400 | 442527 | 442654 | |
| 455 | 442781 | 442907 | 443034 | 443161 | 443288 | 443415 | 443541 | 443667 | 443793 | 443929 | |
| 456 | 444045 | 444172 | 444300 | 444428 | 444556 | 444684 | 444810 | 444937 | 445064 | 445191 | |
| 457 | 445318 | 445444 | 445571 | 445698 | 445825 | 445952 | 446079 | 446206 | 446332 | 446459 | |
| 458 | 446586 | 446713 | 446840 | 446967 | 447094 | 447221 | 447347 | 447474 | 447601 | 447728 | |
| 459 | 447854 | 447982 | 448089 | 448216 | 448343 | 448490 | 448616 | 448743 | 448870 | 448997 | |
| 460 | 449124 | 449251 | 449378 | 449505 | 449632 | 449759 | 449886 | 450013 | 450140 | 450267 | |
| 461 | 450394 | 450520 | 450647 | 450774 | 450801 | 451028 | 451155 | 451282 | 451409 | 451536 | |
| 462 | 451663 | 451790 | 451917 | 452044 | 452171 | 452298 | 452425 | 452552 | 452679 | 452806 | |
| 463 | 452933 | 453060 | 453187 | 453314 | 453441 | 453568 | 453695 | 453822 | 453949 | 454076 | |
| 464 | 454203 | 454330 | 454457 | 454584 | 454711 | 454838 | 454965 | 455092 | 455219 | 455346 | |
| 465 | 455473 | 455600 | 455727 | 455854 | 455981 | 456108 | 456235 | 456362 | 456489 | 456616 | |
| 466 | 456743 | 456870 | 456997 | 457124 | 457251 | 457 [...]78 | 457505 | 457632 | 457759 | 457886 | |
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| 468 | 459284 | 459411 | 459538 | 459665 | 459792 | 459919 | 460046 | 460173 | 460300 | 460427 | |
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| 489 | 485992 | 486122 | 486249 | 486376 | 486504 | 486631 | 486758 | 486886 | 487013 | 487140 | |
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COSMOGRAPHIA, THE Second Part. OR, THE DOCTRINE OF THE PRIMUM MOBILE.
AN INTRODUCTION TO Astronomy.
The First Part.
Of the Primum Mobile.
CHAP. I.
Of the General Subject of Astronomy.
AStronomy, is a Science concerning the Measure and Motion of the Spheres and Stars.
2. Astronomy hath two parts, the first is Absolute, and the other Comparative.
3. The Absolute part of Astronomy is that which treateth of the Measure and Motion of the Orbs and Stars absolutely without respect to any distinction of Time.
[Page 226] 4. The Comparative part of Astronomy is that, which treateth [...] the Motion of the Stars, in reference to some certain distinction of Time.
5. The Absolute part of Astronomy treateth of the Primum Mobile, or Diurnal Motion of all the Celestial Orbs or Spheres.
6. The Primum Mobile, or Diurnal Motion of the Heavens, is that Motion, by which the several Spheres are moved round the World in a Day [...] from East towards West, and [...]o forward [...] from West towards East, and so continually returning to the same point from whence they began their Motion.
7. This first and common Motion of the Heavens, will be best understood, by help of an Instrument called a Globe, which is an Artificial representation of the Heavens, or the Earth and Waters under that Form and Figure of Roundness which they are supposed to have.
8. This Representation or Description of the Visible World is by Circles, great and small, some of which are expressed upon, and others are framed without the Globe.
9. The Circles without the Globe are chiefly two; the Meridian and the Horizon, the one of Brass, and the other of Wood▪ And these two Circles are variable or mutable; for although there is but one Horizon and one Meridian in respect of the whole World, or in respect of the whole Heaven and Earth, yet in respect of the particular parts of Heaven, or rather in respect of the diverse Provinces, Countries and Cities on the Earth, there are diverse both Horizons and Meridians.
[Page 227] 10. The Meridian then is a great Circle without the Globe, dividing the Globe, and consequently the Day and Night into two equal parts, from the North and South ends whereof a strong Wyre of Brass or Iron is drawn or supposed to be drawn through the Center of the Globe representing the Axis of the Earth, by means whereof the whole Globe turneth round within the said Circle, so that any part may be brought directly under this Brass Meridian at pleasure.
11. This Brass Meridian is divided into 4 equal parts or Quadrants, and each of them are subdivided into 90 Degrees, that is 360 for the whole Circle. The reason why this Circle is not divided in 360 Degrees throughout, but still stoping at 90, beginneth again with 10. 20. 30 &c. is, for that the use of this Meridian, in reference to its Division in Degrees, requireth no more than that Number.
12. The Horizon is a great Circle without the Globe, which divides the upper part of Heaven from the lower, so that the one half is always above that Circle, and the other under it.
13. The Poles of this Circle are two, the one directly over our Heads, and is called the Zenith; the other is under feet, and is called the Nadir.
14. The Horizon is either Rational or Sensible.
15. The Rational Horizon is that, which divideth the Heavens and the Earth into two equal parts, which though it cannot be perceived and distinguished by the eye, yet may be conceived i [...] our minds, in which respect all the Stars may be conceived to rise and set as in our view.
[Page 228] 16. The Visible Horizon is that Circle which the eye doth make at its farthest extent of sight, when the body in any particular place doth turn it self round. Of these two Circles there needeth no more to be said at present, only we may observe, that it was ingeniously devised by those, who first thought upon it, to set one Meridian and one Horizon without the Globe, to avoid the confusion, if not the impossibility, of drawing a several Meridian and a several Horizon for every place, which must have been done if this or the like device had not been thought upon.
17. Besides these two great Circles without the Globe, there are 4 other great Circles drawn upon the Globe it self besides the Meridian. 1. The AEquator or Equinoctial Circle. 2. The Zodiack. 3. The AEquinoctial Colure. 4. Solstitial Colure. And these four Circles are imm [...] table, that is, in whatsoever part of the World you are, these Circles have no variation, as the other two have.
18. The AEquator is a great Circle drawn upon the Globe, in the middle between the two Poles▪ and plainly dividing the Globe into two equal parts.
19. The AEquator is the measure of the Motion of the Primum Mobile, for 15 Degrees of this Circle do always arise in an hours time; the which doth clearly shew, that the whole Heavens are turned round by equal intervals in the space of one day or 24 hours.
20. In this Circle the Declinations of the Stars are computed from the mid-Heaven towards the North or South.
[Page 229] 21. This Circle gives denomination to the AEquinox, for the Sun doth twice in a Year and no more cross this Circle, to wit, when he enters the first points of Aries, and Libra, and then he maketh the Days and the Nights equal: His entrance into Aries is in March, and is called the Vernal Equinox; and his entrance into Libra, is in September, and is called the Autumnal Equinox.
22. And from one certain point in this Circle, the Longitude of Places upon the Earth are reckoned; and the Latitude of Places are reckoned from this Circle towards the North, or the South Poles.
23. The Zodiack is a great Circle drawn upon the Globe, cutting the AEquinoctial Points at Oblique Angles: for although it divides the whole World into two equal parts, in reference to its own Poles; yet in reference to the Poles of the World, it hath an Oblique Motion.
24. The Poles of this Circle are as far distant from the Poles of the World, as the greatest Obliquity thereof is from the Equinoctial, that is 23 Degrees, and 31 Minutes or thereabouts.
25. This Circle doth differ from all other Circles upon the Globe in this: other Circles (to speak properly) have Longitude assigned them, but no Latitude; but this hath both. Whereas other Circles are in reference to their Longitude or Rotundity only divided into 360 Degrees, this Circle in respect of its Latitude is supposed to be divided into 16 Degrees in Latitude.
26. The Zodiack then in respect of Longitude is commonly divided into 360 Degrees as other [Page 230] Circles are: but more peculiarly in respect of its self it is divided into 12 Parts called Signs, and each Sign into 30 Degrees, and 12 times 30 do make 360.
27. The 12 Signs into which the Zodiack is divided, have these Names and Characters. Aries ♈. Taurus ♉. Gemini ♊. Cancer ♋. Leo ♌. Virgo ♍. Libra ♎. Scorpio ♏. Sagittarius ♐. Capricornus ♑. Aquarius ♒. and Pisces ♓.
28. These two Circles of the Equator and Zodiack are crossed by two other great Circles, which are called Colures: They are drawn through the Poles of the World, and cut one another as well as the Equator at Right Angles. One of them passeth through the Intersections of the Equinoctial points, and is called the Equinoctial Colure. The other passeth through the points of the greatest distance of the Zodiack, from the Equator, and is called the Solstitial Colure.
29. The other great Circles described upon the Globe are the Meridians: Where we must not think much to hear of the Meridians again. That of Brass without the Globe is to serve all turns, and the Globe is framed to apply it self thereto. The Meridians upon the Globe, will easily be perceived to be of a new and another use.
30. The Meridians upon the Globe are either the great or the less: Not that the great are any greater than the less, for they have all one and the same center, and equally pass through the Poles of the Earth; But those which are called less, are of less use than that, which is called the great.
[Page 231] 31. The great is otherwise called the fixt and first Meridian, to which the less are second, and respectively moveable. The great Meridian is as it were the Landmark of the whole Sphere, from whence the Longitude of the Earth, or any part thereof is accounted. And it is the only Circle which passing through the Poles is graduated or divided into Degrees, not the whole Circle but the half, because the Longitude is to be reckoned round about the Earth.
32. The lesser Meridians are those black lines; which you see to pass through the Poles and succeeding the great at 10 and 10 Degrees, as in most Globes; or at 15 and 15 Degrees difference, as in some. Every place never so little more East or West than another, hath properly a several Meridian, yet because of the huge distance of the Earth from the Heavens, there is no sensible difference between the Meridians of places that are less than one Degree of Longitude asunder, and therefore the Geographers as well as the Astronomers allow a new Meridian to every Degree of the Equator; which would be 180 in all: but except the Globes were made of an extream and an unusual Diameter, so many would stand too thick for the Description. Therefore most commonly they put down but 18, that is, at 10 Degrees distance from one another; the special use of the lesser Meridians being to make a quicker dispatch, in the account of the Longitudes. Others set down but 12, at 15 Degrees difference; aiming at this, That the Meridians might be distant from one another a full part of time, or an hour: for seeing that the Sun is carried 15 Degrees of the Equinoctial every hour, the Meridians set at that [Page 232] distance must make an hours difference in the rising or setting of the Sun in those places which differ 15 Degrees in Longitude.
And to this purpose also upon the North end of the Globe, without the Brass Meridian, there is a small Circle of Brass set, and divided into two equal parts, and each of them into twelve, that is, twenty four all; to shew the hour of the Day and Night, in any place where the Day and Night exceed not 24 hours; for which purpose it hath a little Brass Pin turning about upon the Pole, and pointing to the several hours, which is therefore the Index Horarius, or Hour Index.
33. Having described the great Circles framed without and drawn upon the Globe, we will now describe the lesser Circles also; And these lesser Circles are called Parallels, that is, such as are in all places equally distant from the Equator; and these Circles how little soever, are supposed to be divided into 360 Degrees: but these Degrees are not so large as in the great Circles, but do proportionably decrease according to the Radius by which they are drawn.
34. These lesser Circles are either the Tropicks or the Polar Circles.
35. The Tropicks are two small Circles drawn upon the Globe, one beyond the Equator towards the North Pole, and the other towards the South, Shewing the way which the Sun makes in his Diurnal Motion, when he is at his greatest distance from the Equator either North or South. These Circles are called Tropicks [...], that is, from the Suns returning: for the Sun coming to these Circles, he is at his greatest distance from the Equator, and in the same Moment of time [Page 233] sloping as it were his course, he returns nearer and nearer to the Equator again.
36. These Tropical Circles do shew the point of Heaven in which the Sun doth make either the longest Day, or the Shortest Day in the Year, according as he is in the Northern or the Southern Tropick: And are drawn at 23 Degrees and a half distant from the Equator.
37. The Polar Circles are two lesser Circles drawn upon the Globe at the Radius of 23 Degrees and a half distant from the Poles of the World, shewing thereby the Poles of the Zodiack, which is so many Degrees distant from the Equator on both sides thereof.
38. These Polar Circles are 66 Degrees and a half distant from the Equator, and 43 Degrees distant from his nearest Tropick. They are called the Arctick and Antarctick Circles.
39. The Arctick Circle is that which is described about the Arctick Pole, and passeth almost through the middle of the Head of the greater Bear. It is called the Arctick Circle [...] from the two conspicuous Stars towards the North; called the greater and the lesser Bear.
40. The Antarctick Circle is that which is described about the Antarctick or South Pole. It is so called [...] that is, from being opposite to the greater and lesser Bear.
Having thus described the Globe or Astronomical Instrument by which the Frame of the World is represented to our view, I will proceed to shew the use for which it is intended.
CHAP. II.
Of the Distinctions and Affections of Spherical Lines or Arches.
THE uses of the Globe as to practice, are either such as concern the Heavens or the Earth, in either of which, if we should descend unto particulars, the uses would be more in number, than a short Treatise will contain: Seeing therefore that all Problems which concern the Globe, may be best and most accurately resolved by the Doctrine of Spherical Triangles, we will contract these uses of the Globe (which otherwise might prove infinite) to such Problems as come within the compass of the 28 Cases of Right and Oblique angled Spherical Triangles.
2. And that the nature of Spherical Triangles may be the better understood, and by which of the 28 Cases the particular Problems may be best resolved, I will set down some General Definitions and Affections, which do belong to such Lines or Arches of which the Triangle must be framed, with the Parts and Affections of those Triangles, and how the things given and required in them, may be represented and resolved upon and by the Globe, as also how they may be represented and resolved by the Projection of the Sphere, and by the Canon of Triangles.
3. A Spherical Triangle then is a Figure consisting of three Arches of the greatest Circles upon the Superficies of a Sphere or Globe, every one being less than a semicircle.
[Page 235] 4. A great Circle is that which divideth the Sphere or Globe into two equal parts, and thus the Horizon, Equator, Zodiack and Meridians before described are all of them great Circles: And of these Circles or any other, there must be three Arches to make a Triangle, and every one of these Arches severally must be less than a semicircle: To make this plain.
In Fig. 1. The streight Line HAR doth represent the Horizon, PR the height of the Pole above the Horizon, PMS a Meridian, and these three Arches by their intersecting one another do visibly constitute four Spherical Triangles. 1. PMR. 2. PMH. 3. SHM. 4. SMR. And every Arch is less than a semicircle, as in the Triangle PMR, the Arch PR is less than the Semicircle PRS, the Arch MR is less than the Semicircle AMR, and the Arch PM is less than the Semicircle PMS, the like may be shewed in the other Triangles.
5. Spherical or circular Lines are Parallel or Angular.
6. Parallel Arches or Circles, are such as are drawn upon the same Center within, without, or equal to another Arch or Circle. Thus in Fig. 1. The Arches ♋ M ♋ and ♑ O ♑ are though lesser Circles, parallel to the Equinoctial AE A Q and do in that Scheme represent the Tropicks of Cancer and Capricorn. The manner of describing them or any other Parallel Circle is thus, set off their distance from the great Circle, to which you are to draw a parallel with your Compasses, by help of your Line of Chords, which in this Example is 23 Degrees and a half from AE to ♋, then draw the Line A ♋, and upon the point ♋ erect a Perpendicular, [Page 236] where that Perpendicular shall cut the Axis PAS extended, is the Center of that Parallel.
7. A Spherical Angle, is that which is conteined by two Arches of the greatest Circles upon the Superficies of the Globe intersecting one another: Angles made by the Intersection of two little Circles, or of a little Circle with a great, we take no notice of in the Doctrine of Spherical Triangles.
8. A Spherical Angle is either Right or Oblique.
9. A Spherical Right Angle is that which is conteined, by two Arches of the greatest Circles in the Superficies of the Sphere cutting one another at Right Angles, that is, the one being right or perpendicular to the other: thus the Brass Meridian cutteth the Horizon at right Angles; and thus the Meridians drawn upon the Globe, as well as the Brass Meridian, do all of them cut the Equator at Right Angles.
10. An Oblique Spherical Angle, is that which is conteined by two Arches of the greatest Circles in the Superficies of the Sphere, not being right or perpendicular to one another.
11. An Oblique Spherical Angle is Obtuse, or Acute.
12. An Obtuse Spherical Angle, is that which is greater than a Right Angle. An Acute is that which is less than a Right Angle.
13. If two of the greatest Circles of the Sphere shall pass through one anothers Poles, those two great Circles shall cut one another at Right Angles: Thus the Brazen Meridian doth intersect the Equinoctial and Horizon.
[Page 237] 14. If two of the greatest Circles of the Sphere shall intersect one another, and pass through each others Poles, they shall intersect one another at unequal or Oblique Angles, the Angle upon the one side of the intersection being Obtuse, or more than a Right, and the Angle upon the other side of the intersection being Acute or less than a Right. Thus in Fig. 1. The Arch PM doth intersect the Meridian and Horizon, but not in the Poles of either, therefore the Angle HPM upon one side of the intersection of that Arch with the Meridian, is more than a Right Angle; And the Angle MPR upon the other side of the Intersection is less. And so likewise the Angle PMH upon the one side of the intersection of the Arch PM with the Horizon HR, is greater than a right Angle; and the Angle RMP upon the other side of the Intersection is less than a Right.
15. A Spherical Angle is measured by the Arch of a great Circle described from the Angular point between the sides of the Angle, those sides being continued unto Quadrants. Thus the Arch of the Equator TQ in Fig. 1. is the measure of the Angle MPR, or TPQ, the sides PT and PQ being Quadrants.
And the measure thereof in the Projection may thus be found: lay a Ruler from P to T, and it will cut the Primitive Circle in V; and the Arch VQ being taken in your Compasses and applyed to your Line of Chords, will give the Quantity of the Angle propounded.
16. The Complement of a Spherical Arch or Angle, is so much as it wanteth of a Quadrant, if the Arch or Angle given be less than a Quadrant; or so much as it wanteth of a Semicircle, [Page 238] if it be more than a Quadrant.
17. An Arch of a great Circle cutting the Arch of another great Circle, shall intersect one another at Right Angles, or make two Angles; which being taken together, shall be equal unto two Right. Thus in Fig. 1. The Axis PAS or Equinoctial Colure doth cut the Equator AE A Q at Right Angles; but the Meridian PMS doth cut the Horizon HMR at Oblique Angles, making the Angle PMR less than a Right, and the Angle SMR more than a Right, and both together equal to a Semicircle.
18. From these general Definitions proper to Spherical Lines or Arches, the general Affections of these Arches may easily be discerned; I mean the various Positions of the Globe of the Earth, in respect of all and singular the Inhabitants thereof.
19. And the whole Body of the Sphere or Globe, in respect of the Horizon, is looked upon by the Earths Inhabitants, either in a Parallel, a Right, or an Oblique Sphere.
20. A Parallel Sphere is, when one of the Poles of the World is elevated above the Horizon to the Zenith, the other depressed as low as the Nadir, and the Equinoctial Line joyned with the Horizon. They which there inhabite (if any such be) see not the Sun or other Star rising or setting, or higher or lower in their diurnal revolution. And seeing that the Sun traverseth the whole Zodiack in a Year, and that half the Zodiack, is above the Horizon and half under it, it cometh to pass, that the Sun setteth not with them, for the space of six Months, nor giveth them any Light for the space of other six Months, and so [Page 239] maketh but one Day and Night of the whole Year.
21. A Right Sphear is, when both the Poles of the World do lie in the Horizon, and the Equinoctial Circle is at his greatest distance from it, passing through the Zenith of the place. And in this position of the Sphere, all the Coelestiall Bodies, Sun, Moon, and other Planets, and fixed Stars, by the daily turning about of the Heaven, do directly ascend above, and also directly descend below the Horizon, because the Motions which they make in their Daily motion do cut the Horizon Perpendicularly, and as it were at Right Angles. In this Position of the Sphere, all the Stars may be observed to rise and set in an equal space of time, and to continue as long above the Horizon, as they do under it, the Day and Night to those Inhabitants, being always of an equal length.
22. An Oblique Sphere is, when the Axis of the World (being neither Direct nor Parallel to the Horizon) is inclined obliquely towards both sides of the Horizon, as in Fig. 1. Whence it cometh to pass; that so much as one of the Poles is elevated above the Horizon, upon the one side; so much is the other depressed under the Horizon, upon the other side.
And in this Position of Sphere, the Days are sometimes longer than the Nights, sometimes shorter, and sometimes of equal length. When the Sun is in either of the Equinoctial Points, the Days and Nights are equal; but when he declineth from the Equator towards the elevated Pole, the Days are observed to encrease; and when he declineth from the Equator towards the opposite Pole, or the Pole [Page 240] depressed, the Days do decrease▪ as is manifest in Fig. 1. For when the Sun riseth at M, the Line M ♋ above the Horizon is the Semidiurnal Arch of the longest day. When he riseth at C, the Arch C ♑ above the Horizon, is the Semidiurnal Arch of the shortest Day: And when he riseth at A, the Days and Nights are of equal Length, the Semidiurnal Arch AAE, being equal to the Seminocturnal Arch AQ.
CHAP. III.
Of the kind and parts of Spherical Triangles; and how to project the same upon the Plane of the Meridian.
HAving shewed what a Spherical Triangle is, and of what Circles it is composed, with the general Affections of such Lines: I will now shew how many several sorts of Triangles there are, of what Circular parts they do consist, and such Affections proper to them as will render the so [...]ition of them more clear and certain.
2. Spherical Triangles are either Right or Oblique.
3. A Right Angled Spherical Triangle, is that which hath one or more Right Angles.
4. A Spherical Triangle which hath three Right Angles, hath always his three sides Quadrants. As in Fig. 1. The Spherical Triangle AZR, the Angles ZRA, RAZ and AZR are right Angles, and the three sides AZ, ZR and AR are Quadrants also.
[Page 241] 5. A Triangle that hath two right Angles, hath the sides opposite to those Angles Quadrants, and the third side is the measure of the third Angle. As in Fig. 1. The sides of the Spherical Triangle TPQ, namely TP and PQ are Quadrants, and the Angles opposite to these sides, to wit, PTQ and TQP are Quadrants also, and the third Angle TQ is the measure of the third Angle TPQ. But the Right Angled Triangle which hath one Right and two Acute Angles, is that which cometh most commonly to be resolved.
6. The Legs of a right Angled Spherical Triangle are of the same Affection with their opposite Angles; as in the Triangle ZAR Fig. 1. The side ZA is a Quadrant, and the Angle at A is right, because Z is the Pole of the Arch AR and ZA is perpendicular thereunto. And in the Triangle RAAE the side RZAE being more then a Quadrant the Angle RAAE is more then a Quadrant also, being more then the Right Angle RAZ. And in the right Angled Spherical Triangle APR the side PR being less then a Quadrant, the Angle PAR is less then a Quadrant also, being less then the right Angle RAZ.
7. An Oblique angled Spherical Triangle is either acute or obtuse.
8. An Acute angled Spherical Triangle hath all his Angles Acute, and each Side less then a Quadrant; As in the Triangles, ZFP. Fig. 2. The Angles at Z and P are acute, as appeareth by inspection; and the Angle at F is acute also because the Measure thereof CD = EM is less then a Quadrant.
[Page 242] 9. An Oblique Angled Spherical Triangle hath all his Angles either acute or obtuse: viz. Acute and mixt.
10. The Sides of a Spherical Triangle may be turned into Angles, and the Angles into Sides; The Complement of the greatest Side or greatest Angle to a Semicircle being taken in each conversion. For Example. If it were required to turn the Angles of the Oblique Angled Spherical Triangle ZFP into sides in Fig. 3. EAE is the measure of the Angle at P, and AD in the Triangle ADC equal thereunto, AC is the Complement of FZP to a Semicircle, and KM the the Measure of the Angle at F is equal to DC, and so the Sides of the Spherical Triangle ADC are equal to the Angles of the Spherical Triangle FZP, making the side AC equal to the Complement of the Angle Z to a Semicircle.
11. In Right Angled Spherical Triangles the Sides intending the Right Angle we call the Legs; The Side subtending it the Hypotenuse.
12. In every Spherical Triangle besides the Area or space contained, there are six parts. viz. Three Sides and three Angles and of these six there must be always three given to find the rest, but in right Angled Spherical Triangles there are but five of the six parts parts which come into question, because one of the Angles being right is allways known, and so any two of the other five being given, the three remaining parts whether Sides or Angles, may be found. But before I come to the solution of these Triangles whether right or oblique, I will first shew how they may be represented upon the Globe, and projected upon the plane of of the Meridian.
[Page 243] 13. A right Angled Spherical Triangle may be represented upon the Globe in this manner: Elevate one of the Poles of the Globe above the Horizon, to the quantity of one of the given Legs, so shall the distance between the AEquinox and the Zenith be equal thereunto, and at the Zenith fasten the Quadrant of altitude, so shall there be delineated upon the Globe the right Angled Spherical Triangle AEZB as may be seen in Fig. 1. In which the outward Circle HZR doth represent the Brass Meridian, AEAQ the Equator, and ZC the Quadrant of altitude.
14. An Oblique Angled Spherical Triangle may be represented upon the Globe in this manner. Number one of the given sides from one of the Poles to the Zenith; and there fasten the Quadrant of Altitude, upon which number another side, the third upon the great Meridian, from the Pole towards the Equinoctial, then turn the Globe till the Side numbred upon the Quadrant of Altitude, and the Side numbred upon the great Meridian shall intersect one another; so shall there be delineated upon the Globe the Oblique Angled Spherical Triangle ZFP in Fig. 3. In which ZP is numbred upon the Brass Meridian from S the Pole of the World to Z the Zenith, ZF the Azimuth Circle represents the Quadrant of Altitude, and PF the great Meridian upon the Globe intersecting the Quadrant of Altitude at F.
15. A Right or Oblique Angled Spherical Triangle being thus delineated upon the Globe, there needs no further instructions, as to the measure of the sides, all that is wanting, is the laying down the Angles comprehended by those sides, and the finding out the measure of these Angles [Page 244] being so laid down. And that this may be the better understood, I will first shew; how the several Circles upon the Globe before described, may be projected upon the Plane of the Meridian, and the several useful Triangles that are described by such Projection with such Astronomical Propositions as are conteined and resolvable by these Triangles.
16. The Circles in the first Figure are the Meridian, AEquator, Horizon, AEquinoctial Colure, and the Tropicks. The Brass Meridian without the Globe, is a perfect Circle described by taking 60 Degrees from your Line of Chords, as the Circle HZRN in Fig. 1. Within which all the other are projected. The Horizon, AEquator, AEquinoctial Colure, East and West Azimuths are all streight Lines. Thus the Diameter HAR represents the Horizon, AEAQ the Equator, PAS the Equinoctial Colure and ZAN the East and West Azimuths, in the drawing of these there is no difficulty, PMS is a Meridian, and ZCN an Azimuth Circle, for the drawing of which there are three points given and the Centers of the Meridians do always fall in the Equinoctial extended if need be, the Centers of the Azimuth Circles do fall in the Horizon extended if need be, and for the drawing of these Circles there needs no further direction, supposing the middle point given to be in the AEquator or Horizon, but yet the Centers of these Circles may be readily found, by the Lines of Tangents or Secants, for the Tangent of the Complement of AT set from A to D, or the Secant of the Complement set from A to D will give the Center of the Meridian PTS. The other two Circles in the 1. Fig. [Page 245] are the Tropicks whose Centers are thus found; each Tropick is Deg. 23 ½ from the Equinoctial, which distance being set upon the Meridian from AE to ♋ and AE to ♑, if you draw a Line from A to ♋ and another perpendicular thereunto from ♋ it will cut the Axis SAP extended in the Center of that Tropick, by which extent of the compasses the other Tropick may be drawn also. Or thus the Co-tangent of AE ♋ set from ♋ to the Axis extended will give the Center as before, and thus may all other Parallels be described.
17. In the second and third figures, the two extream points given in the Meridians are not eqnidistant from the third, for the drawing of which Circles, if the common way of bringing three points into a Circle be not liked; you may do thus, from the given point at F and the Center A draw the Diameter TAS, and cross the same at Right Angles with the Diameter BAG, a Ruler laid from G to F will cut the primitive Circle in L, make EL = BL a Ruler laid from G to E will cut the Diameter SAT in V the Center of the Circle BDG. Which Circle doth cut the Diameter HAR in the Pole of ZF, and the Diameter AEAQ in D in the Pole of PFX, and a Ruler laid from Z to C will cut the Primitive Circle in Y, and making Y O equal to Y a Ruler laid from Z to O will cut the Diameter HAR, extended in the Center of the Circle ZF.
18. Having drawn the Circle ZFI, in Fig. 13. The Circle PEX, or any other passing through the point F, may easily be described. Draw AEQ at right Angles to PX, a Ruler laid from G unto (e) will cut the Primitive Circle in (m) make mn = Bn, a Ruler laid from G to n [Page 246] shall cut the Diameter TFS in p make Fq = Fp so shall FQ be the Radius, and the Center of the Circle PFX as was desired.
19. The preceeding directions are sufficient for the projecting of several Circles of the Globe before described upon the Plane of the Meridian, and the parts of those Circles so described may thus be measured. In Fig. 1. HZ = CZ = AZ 90 Degrees. Whence it followeth, that the Quadrant CZ is divided into Degrees from its Pole M, by the Degrees of the Quadrant HZ, that is a Ruler laid from M to any part of the Quadrant HZ will cut as many Degrees in CZ as it doth in the Quadrant HZ, and thus the Arch CF = HK the Arch CB = HL, and the Arch BF = LK.
20. That which is next to be considered is the projecting or laying down the Angles of a Triangle, and the measuring of them being projected, and the Angles of a Triangle are either such as are conteined between two right Lines as the Angle A in the Triangle PAR; or such as are conteined by a streight and a Circular Line, as the Angle PMR. Fig. 1. Or such as are conteined by two circular Lines, as the Angles FZP or ZFP in Fig. 3. The projecting or measuring the first sort of these Angles, needs no direction.
21. To project an Angle conteined by a streight and a circular line as the Angle AEBZ in Fig. 1. Do thus, lay a Ruler from N to C, and it will cut the Primitive Circle in K make ZX = HK, a Ruler laid from N to X will cut the Diameter HAR in the point M the Pole of the Circle ZCN, a Ruler laid from M to B the Angular point propounded, will cut the primitive Circle in I, make [Page 247] NY = HL a Ruler laid from N to Y will cut the Circle ZCN in W, a Ruler laid from B to W will cut the Primitive Circle in A, make AQ equal to the Angle propounded, and draw the Diameter BAQ, then is the Angle AEBZ or NBQ = NQ as was required.
22. If the Angle had been projected and the measure required, a Ruler laid from M to B would give L and making NY = HL a Ruler laid from M to Y would give W, from B to W would give A, and AQ would be the measure of the Angle propounded.
23. To project an Angle conteined by two circular lines, one of them being an Arch of the Primitive Circle, as the Angle AEZB, Fig. 1. Do thus, set off the quantity of the Angle given from H to G, a Ruler laid from Z to G will cut the Diameter HAR in the point C, so may you draw the Circle ZCN and the Angle HZC will be equal to the Arch HG = HC as was required.
24. If the Angle had been projected and the measure required, a Ruler laid from Z to C would cut the Primitive Circle in G and HG would be the measure of the Angle propounded.
25. To project an Angle conteined by two oblique Arches of a Circle, as the Angle ZFP in Fig. 3. You must first find the Pole of one of the two Circles conteining the Angle propounded, suppose ZBI, a Ruler laid from C the Pole thereof to F, the Angular point propounded, will cut the Primitive Circle in a make ab equal to the Angle propounded, a Ruler laid from F to b will cut the Diameter AEAQ in D the Pole of the Circle PEX, a Ruler laid from G to e will cut the Primitive Circle in m, make mn = Bm [Page 248] ler laid from G to n will cut the Diameter TAS in p, make Aq = Ap so shall Fp be the Radius and the Center of the Circle PFX and the Angle ZFP = ab, as was propounded.
26. If the Angle had been projected and the measure required; through the point F draw the Diameter TFS and the Diameter BAG at right Angles thereunto, a Ruler laid from G to F will cut the Primitive Circle in K, and making KE = BK a Line drawn from G to E will cut the Diameter TAS in the Center of the Circle GDB cutting the Diameter HAR in C the Pole of the Circle ZBI, and the Diameter AEAQ in D, the Pole of the Circle PEX and a Ruler laid from F to C and D will cut the Primitive Circle in a and b the measure of the Angle required.
Or a Ruler laid from F to K and M will cut the Primitive Circle in Deg. the measure of the Angle propounded as before.
Or thus a Ruler laid from C and D to F will cut the Primitive Circle in ae and h set 90 Degrees from e and h to f and l a Ruler laid from C to f will cut ZBI in M and a Ruler laid from D to l will cut PEX in K. This done a Ruler laid from F to K and M will cut the Primitive Circle in g and d the measure of the Angle as before.
And in Fig. 2. The quantity of the Angle ZEP may thus be found. A Ruler laid from C the Pole of the Circle ZFI to F the angular point will cut the Primitive Circle in a, set off a Quadrant from a to b, a Ruler laid from C to b will cut the Circle ZFI in the point M. In like manner a Ruler laid from D the Pole of the Circle PEX, will cut the Primitive Circle in D, set off a Quadrant from A to h, a Ruler laid from D to P will [Page 249] cut the Circle PFX in K: Lastly a Ruler laid from F to K, and M will cut the Primitive Circle in NS the measure of the Angle KFM or ZFP, as was propounded.
27. Having shewed how a right or oblique Angled Spherical Triangle may be projected upon the Plane of the Meridian, as well as delineated upon the Globe, we will now consider the several Triangles usually represented upon the Globe, with the several Astronomical and Geographical Problems conteined in them, and resolved by them.
28. The Spherical Triangles usually represented upon the Globe are eight, whereof there are five Right angled Triangles, have their Denomination from their Hypotenusas.
The first is called the Ecliptical Triangle, whose Hypotenusa is an Arch of the Ecliptick, the Legs thereof are Arches of the AEquator and Meridian, this is represented upon the Globe, by the Triangle ADF, in Fig. 1. In which the five Circular parts, besides the Right Angle are;
1. The Hypotenuse or Arch of the Ecliptick AF.
2. The Leg or Arch of the AEquator, AD.
3. The Leg or Arch of the Meridian DF.
4. The Oblique Angle of the Equator with the Ecliptick and the Suns greatest Declination DAF.
5. The Oblique Angle of the Ecliptick and Meridian, or the Angle of the Suns position AFD.
The two next I call Meridional, because the Hypotenusas in them both, are Arches of a Meridian. One of these is noted with the Letters [Page 250] MPR in Fig. 1. In which the five Circular parts are;
1. The Hypotenusa or Arch of a Meridian PM.
2. The Leg or Arch of the Horizon MR, the Suns Azimuth North.
3. The Leg or Arch of the Brass Meridian, representing the height of the Pole PR.
4. The Oblique Angle of the Meridian upon the Globe; with the Brass Meridian, or Angle of the Hour from Midnight. P.
5. The Oblique Angle of the Suns Meridian with the Horizon, or the Complement of the Suns Angle of Position PMR.
The other Right Angled Meridional Triangle is noted with the Letters AEG in Fig. 1. In which the 5 Circular parts are.
1. The Hypotenusa or present Declination of the Sun, AE.
2. The Leg or Suns Amplitude at the hour of six, AG.
3. The other Leg or Suns height at the same time EG.
4. The Angle of the Meridian with the Horizon, or Angle of the Poles elevation, EAG.
5. The Angle of the Meridian with the Azimuth, or the Angle of the Suns position, AEG.
The fourth Right Angled Spherical Triangle, I call an Azimuth Triangle, because the Hypotenusa doth cut the Horizon in the East and West Azimuths, as is represented by the Triangle ADV. in Fig. 1. In which the 5 Circular parts are,
[Page 251] 1. The Hypotenusa, or Arch of the Sun or Stars Altitude AV.
2. The Leg or Declination of the Sun or Star, DV.
3. The other Leg, or Right Ascension of the Sun or Star, AD.
4. The Oblique Angle or Angle of the Poles elevation, DAV.
5. The other Oblique Angle or Angle of the Sun or Stars Position, DVA.
The fifth and last Right Angled Spherical Triangle, that I shall mention, I call an Horizontal Triangle, because the Hypotenusa thereof is an Arch of the Horizon, and is represented by the Triangle AMT in Fig. 1. In which the 5 Circular parts are;
1. The Hypotenusa and Arch of the Horizon, or Amplitude of the Sun at his rising or setting, AM.
2. The Leg conteining the Sun or Stars Declination TM.
3. The other Leg or Ascensional difference AT, that is, the difference between DT the Right Ascension and DA the Oblique Angle.
4. The Oblique Angle of the Horizon and Equator, or height of the Equator TAM.
5. The other Oblique Angle, or Angle of the Horizon and Meridian AMT.
The Oblique Angled Spherical Triangles usually represented upon the Globe are three. The first I call the Complemental Triangle, because the sides thereof are all Complements, and this is represented by the Triangle FZP in Fig. 1. Whose Circular parts are;
1. The Complement of the Poles elevation ZP.
[Page 252] 2. The Complement of the Suns Declination, FP.
3. The Complement of the Suns Altitude or Almicantar FZ.
4. The Suns Azimuth or Distance from the North FZP.
5. The hour of the day or distance of the Sun from Noon ZPF.
6. The Angle of the Suns Position ZFP.
The second Oblique Angled Spherical Triangle, I call a Geographical or Nautical Triangle, because it serveth to resolve those Problems, which concern Geographie and Navigation, and this is also represented by the Triangle FZP in Fig. 1. Whose parts are.
1. The Complement of Latitude as before ZP.
2. The distance between the two places at Z and F or side FZ.
3. The Complement of the Latitude of the place at F or side FP.
4. The difference of Longitude between the two places at Z and F or the Angle FBZ.
5. The point of the compass leading from Z to F or Angle FZP.
6. The point of the Compass leading from F to Z, or Angle ZFP.
The third Oblique Angled Spherical Triangle is called a Polar Triangle, because one side thereof is the distance between the Poles of the World, and the Poles of the Zodiack. This Triangle is represented upon the Coelestial Globe, by the Triangle FSP in Fig. 4. In which the Circular parts are;
1. The distance between the Pole of the [Page 253] World, and the Pole of the Ecliptick, or the Arch SP.
2. The Complement of the Stars Declination, FP.
3. The Complement of the Stars North Latitude, from the Ecliptick or the Arch FS.
4. The Angle of the Stars Right Ascension FPS.
5. The Complement of the Stars Longitude FSP.
6. The Angle of the Stars Position SFP.
29. And thus at length I have performed, what was proposed in the 15 of this Chapter, that is, I have shewed how the several Circles of the Globe, may be projected upon the Plane of the Meridian, the several useful Triangles that are described by such projection, with such Astronomical Propositions as are contained and resolveable by those Triangles; And although the most accurate way of resolution is by the Doctrine of Trigonometry and the Canon of Lines and Tangents, yet it is not impertinent to do the same upon the Globe it self, which as to the sides is easie, but to measure or lay down the Angles is sometimes a little labourious.
In the Right Angled Spherical Triangle AEBZ in Fig. 1. The measure of the Angle AEZB is reckoned in the Horizon from H to C but to lay down or measure the Angle AEBZ the readiest way is to describe the Triangle again, making AEZ = AEB and AEB = AEZ, so will the Angle AEBZ stand where the Angle AEZB is, and may be measured in the Horizon as the other was.
[Page 254] And so in the Oblique Angled Spherical Triangle FZP in Fig. 1. The Angles at Z and P are easily measured or laid down upon the Globe, but to perform the same with the Angle ZFP, you may represent it at the Pole or Zenith and find the measure in the Equator or Horizon.
30. And now having, as I hope, sufficiently prepared the young Student for the first part of Astronomy, the Doctrine of the Primum Mobile, by shewing how the Heavens and the Earth are represented upon the Globe, or may be projected in Plane, I will now proceed to such Astronomical Propositions as are generally useful, and may be sufficient for an Introduction to this noble Science: to go through the several Triangles before propounded, will be very tedious, I will therefore shew the several Problems in one Right Angled and one Oblique Angled Spherical Triangle and the Canons by which they are to be resolved, and leave the rest for the Practice of my Reader. To this purpose I will next acquaint you with my Lord Nepiers Catholick Proposition for the solution of all Right and Oblique Angled Spherical Triangles.
CHAP. IV.
Of the solution of Spherical Triangles.
IN Spherical Triangles there are 28 Varieties or Cases, 16 in Rectangular, and 12 in Oblique, whereof all the Rectangular and ten of the Oblique may be resolved by the two Axioms following.
1. Axiom. In all Right Angled Spherical Triangles having the same Acute Angle at the Base, the Sines of the Hypotenusas are proportional to the Sines of their Perpendicular.
2. Axiom. In all right Angled Spherical Triangles, the Sines of the Bases and the Tangents of the Perpendicular are proportional.
That all the Cases of a Right Angled Spherical Triangle may be resolved by these two Axioms, the several parts of the Spherical Triangle proposed, that so the Angles may be turned into sides, the Hypotenusa, into Bases and Perpendiculars and the contrary. By which means the proportions as to the parts of the Triangle given, are sometimes changed into Co-sines instead of Sines, and into Cotangents instead of Tangents. Which the Lord Nepier observing; those parts of the Right Angled Spherical Triangle, which in conversion do change their proportion, he noteth by their Complements. viz. The Hypotenuse and the two Acute Angles: But the sides or Legs are not so noted, [Page 256] as in the Right Angled Spherical Triangle MPR in Fig. 1. And these five he calleth the Circular parts of the Triangle, amongst which the Right Angle is not reckoned.
2. Now if you reckon five Circulat parts in a Triangle, one of them must needs be in the middle, and of the other four, two are adjacent to that middle part, the other two are disjunct, and which soever of the five you call the middle part, for every one of them may by supposition be made so; those two Circular parts which are on each side of the middle are called extreams adjunct, and the other two remaining parts, are called extream disjunct, as in the Triangle MPR if you make the Leg PR the middle part, then the other Leg MR and the Angle Comp. P. Are the extreams conjunct, the Hyp. Comp. MP and Comp. M, are the extreams disjunct, and so of the rest, as in the following Table.
| Mid. Part | Exctr. conj. | Extr. disj [...] |
| Leg. MR | Comp. M | |
| Leg PR | ||
| Comp. P | Comp. MP | |
| Leg. PR | Comp. MP | |
| Leg MR | ||
| Comp. M | Comp. P | |
| Leg. MR | Comp. P | |
| Comp. M | ||
| Comp. MP | Leg. PR | |
| Comp. M | Leg. PR | |
| Comp. MP | ||
| Comp. P. | Leg. MR | |
| Comp. MP | Leg. MR | |
| Comp. P | ||
| Leg. PR | Comp. M |
3. These things premised, the Ld. Nepier as a consectory from the two preceeding Axioms hath composed this Catholick and Universal Proposition.
The Rectangle made of the Sine of the middle part and Radius is equal to the Rectangle made of the Tangents of the Extremes conjunct or the cosines of the Extremes disjunct.
Therefore if the middle part be sought, the Radius must be in the first place; if either of the extremes, the other extreme must be in the first place.
Only note that if the middle part, or either of the extremes propounded be noted with its [Page 258] Comp. in the circular parts of the Triangle, instead of the Sine or Tangent you must use the Cosine or Cotangent of such circular part or parts.
That these directions may be the better conceived, we have in the Table following set down the circular parts of a Triangle under their respective Titles, whether they be taken for the middle part, or for the extremes, conjunct or disjunct, and unto these parts, we have prefixed the Sine or Cosine, the Tangent or Cotangent, as it ought to be by the former Rule.
| Mid. Par. | Extr. Conj. | Ext. Disj. |
| Tang. MR | Sine M | |
| Sine PR | ||
| Tang. P. | Sine MP | |
| Tang. PR | Sine MP | |
| Sine MR | ||
| Cotang. M. | Sine P | |
| Tang. MR | Sine P | |
| Cosine M | ||
| Cot. MP | Cos. PR | |
| Cotang. M. | Cos. PR | |
| Cos. MP | ||
| Cotang. P | Cos. MR | |
| Cot. MP | Cos. MR | |
| Cosine P | ||
| Tang. PR | Sine M |
- [Page 259]1. Sine PR x Rad.= t MR x ct P.
- 2. Sine PR x Rad.= s M x s MP.
- 3. Sine MR x Rad.= t PR x ct M.
- 4. Sine MR x Rad.= s MP x s P.
- 5. Cos. M x Rad.= t MR x ct MP.
- 6. Cos. M x Rad.= s P x cs PR.
- 7 Cos. MP x Rad.= ct M x ct P.
- 8. Cos. MP x Rad.= cos. PR x cs MR.
- 9. Cos. P x Rad.= ct MP x t PR.
- 10. Cos. P x Rad.= cos. MR x s M.
By these 10 Rectangles may the 16 Cases of a Right angled Spherical Triangle be resolved, and some of them twice over; for although there are but 16 varieties in all Right angled Spherical Triangles, yet 30 Astronomical Problems may be resolved by one Triangle, as by the following Examples shall more clearly appear.
Of Right angled Spherical Triangles.
CASE 1.
The Legs given, to find the Angles.
IN the Right angled Spherical Triangle MPR. The given Legs are MR and RP. The Angles at M and P are required.
By the first of the 10 equal Rectangles s PR x Rad.=t MR x ct P. in which P is fought, therefore putting MR in the first place. The proportion is. t MR. x Rad.∷s PR. ct P.
And by the third equal Rectangle. t PR. Rad.∷s MR. ct M.
CASE 2.
The Legs given, to find the Hypotenuse.
In the Right angled Spherical Triangle MPR. The given Legs are MR and PR. The Hypotenuse MP is required.
By the eighth of the 10 Rectangles cos. MP x Rad.=cos. PR x cos. MR in which MP the middle part is sought, therefore Radius must be put in the first place, and then the proportion is.
Rad: cos. PR∷ cos. M. R. cos. MP.
CASE 3.
A Leg with an Angle opposite thereunto being given, to find the other Leg.
In the Right angled Spherical Triangle MPR, let there be given. The Leg MR. The Angle P. The Leg PR inquired.
By the first of the 10 Rectangles. Rad. tMR ∷cot. P. Sine PR. or The Leg PR and the Angle M given, to [...]ind MR.
By the 3 of the 10 Rectangles. Rad. tPR∷ct M. Sine MR.
CASE 4.
A Leg with an Angle conterminate therewith being given, to find the other Leg.
In the Right angled Spherical Triangle, MPR, The given Leg is MR, with the Angle M. The Leg PR is required.
By the 3 Rectangle. cot. M. Rad∷Sine MR. tPR.
The given Leg RP, and Angle P. The Leg MR is required.
By the 1. Rectangle. ctP. Rad∷sine RP. tang. MR.
CASE 5.
A Leg and an Angle conterminate therewith being given, to find the Hypotenuse.
In the Right angled Spherical Triangle MPR, let there be given,
- The Leg
- MR and the Angle M
- PR and the Angle P
- to find MP.
By the 5. Rectangle, t MR. Rad∷cos. M. ct MP.
By the 9. Rectangle. t PR. Rad.∷cos. P. ct MP.
CASE 6.
The Hypotenuse and a Leg given, to find the contained Angle.
In the Right angled Spherical Triangle MPR, let there be given,
- The Hypotenuse MP,
- and Leg
- MR. PR.
- To find
- M.
By the 5. Rectangle, Rad. ct MP∷t MR. cos. M.
By the 9. Rectangle, Rad. ct MR∷t PR. cos. P.
CASE 7.
The Hypotenuse and one Angle given, to find the other Angle.
In the Right angled Spherical Triangle MPR, let there be given,
- [Page 263]The Hypotenuse MP
- & Angle
- M P.
- To find the Angle
- P. M.
By the 7. Rectangle, cot. M. Rad∷cos. MP. cot. P.
By the 7. Rectangle cot. P. Rad∷cos. MP. cot. M.
CASE 8.
The Oblique A [...]gles given, to find the Hypotenuse.
In the Right angled Spherical Triangle MPR, let there be given The Angles at P and M, To find the Hypotenuse PM.
By the 7. Rectangle. Rad. ct P∷cot. M. cos. MP.
CASE 9.
The Hypotenuse and an Angle given, to find the Leg conterminate with the given Angle.
In the Right angled Spherical Triangle MPR, let there be given,
- The Hypotenuse PM
- Angle
- P. M.
- To find
- PR. MR.
By the 9. Rectangle, ct PM. Rad∷cos. P. t PR.
By the 5. Rectangle, ct PM. Rad∷cos. M. tMR.
CASE 10.
The Hypotenuse and an Angle given, to find the Leg opposite to the given Angle.
In the Right angled Spherical Triangle MPR, let there be given,
- The Hypotenuse PM
- and the Angle
- M. P.
- To find
- PR. MR.
By the 2. Rectangle, Rad. s MP∷s M. Sine PR.
By the 4. Rectangle, Rad. s MP∷s P. Sine MR▪
CASE 11.
A Leg and an Angle opposite thereunto being given, to find the Hypotenuse.
In the Right angled Spherical Triangle MPR, let there be given,
- The Leg
- PR. MR.
- and the Angle
- M P
- to find the Hypotenuse PM.
By the 2. Rectangle, s M. Rad∷s PR. s MP.
By the 4. Rectangle, s P. Rad∷s MR. s PM.
CASE 12.
The Hypotenuse and a Leg given, to find the Angle opposite to the given Leg.
In the Right angled Spherical Triangle PMR, [Page 265] the Hypotenuse MP and the Leg MR are given, the Angle at P is required.
By the fourth Rectangle Sine MP to, Rad∷s MR. s P.
The Hypotenuse MP and Leg PR given, the Angle M is required.
By the second Rectangle. sMP. Rad∷s PR. s M.
CASE 13.
The Angle and Leg conterminate with it being given, to find the other Angle.
In the Right angled Spherical Triangle PMR, let there be given,
- The Angle
- M P
- and the Leg
- MR PR
- to find the Angle
- P. M.
By the tenth Rectangle, Rad. cs MR∷s M. cs P.
By the sixth Rectangle, Rad. s P∷cs PR. cs M.
CASE 14.
An Angle and a Leg opposite thereunto being given, to find the other Angle.
In the Right angled Spherical Triangle MPR, let there be given,
- The Angle
- P M
- and the Leg
- MR PR
- to find the Angle
- M. P.
By the 10. Rectangle, cs MR. Rad∷cs P. csM.
By the 6. Rectangle, cs PR. Rad∷cs M. sP.
CASE 15.
The Oblique Angles given, to find a Leg.
In the Right angled Spherical Triangle MPR, let there be given, the Angles at M and P, to find the Legs MR and PR.
By the 10. Rectangle, sM. Rad∷cs P. cs MR.
By the 6. Rectangle, s P. Rad∷cs M. cs PR.
CASE 16.
The Hypotenuse and one Leg given, to find the other Leg.
In the Right angled Spherical Triangle MPR, let there be given,
- The Hypotenuse MP and the Leg
- PR MR
- to find the Leg
- MR. PR.
- By the 8. Rectangle,
- csPR. Rad∷csMP. csMR.
- csMR. Rad∷csMP. csPR.
Thus I have given you the Proportions by which the 16 Cases of a Right angled Spherical Triangle may be resolved, In which there are contained 30 Astronomical Problems. Two in every Case except the Second and the Eighth. In both which Cases there are but two Problems. And thus I have done with Right angled Spherical Triangles.
4. In Oblique angled Spherical Triangles [Page]
[Page] [Page 267] there are twelve Cases, ten whereof may be resolved by the Catholick Proposition; If the Spherical Triangle propounded be first converted into two right, by letting fall of a Perpendicular, sometimes within, sometimes without the Triangle.
5. If the Angles at the Base be both acute or both obtuse, the Perpendicular shall fall within the Triangle; but if one of the Angles of the Base be acute and the other obtuse, the Perpendicular shall fall without the Triangle.
6. However the Perpendicular falleth, it must be always opposite to a known Angle, for your better direction, take this General Rule. ‘From the end of a Side given, being adjacent to an Angle given, let fall the Perpendicular.’
As in the Triangle FPS in Fig. 4. If there were given the Side F S and the Angle at S, the Perpendicular by this Rule must fall from P upon the Side S P extended, if need require.
But if there were given the Side P S and the Angle at S, the Perpendicular must fall from F upon the Side F S.
7. To divide an Oblique angled Spherical Triangle into two Right, by letting fall a Perpendicular upon the Globe it self, is not necessary, because all the Cases may be resolved without it, but in projection it is convenient to inform the fancy: and seeing the reason by which it is done in projection doth depend upon the nature of the Globe, I will here shew it both ways, first upon the Globe, and then by projection.
[Page 268] An Oblique angled Spherical Triangle may be divided into two Right, by letting fall a Perpendicular upon the Globe it self, in this manner. In the Oblique angled Spherical Triangle FPS in Fig. 4. let it be required to let fall a Perpendicular from P upon the Side FS. Suppose the Point P to stand in the Zenith, where the Arch FS shall cut the Zodiack, which in this Figure is at K, make a mark, and from this Point of Intersection of the Circle upon which the Perpendicular is to fall with the Zodiack, reckon 90 Degrees, which suppose to be at P; a thin Plate of Brass with a Nut at one end thereof, whereby to fasten it to the Meridian, as you do the Quadrant of Altitude, being graduated as that is, but of a larger extent (for that a Quadrant in this case will not suffice) being fastned at P and turned about till it cut the Point L in the Zodiack, will describe upon the Globe the Arch of a great Circle PEL, intersecting the Side F S at Right Angles in the Point E, because the Point L in the Zodiack is the Pole of the Circle SFK, now all great Circles which passing through the Point L, shall intersect the Circle SKG, shall intersect it at Right Angles; by the 13. of the 2. Chapter.
9. And hence to divide an Oblique angled Spherical Triangle into two Right by projection is easie, as in the Triangle FPS, the Pole of the Circle SFK is L, therefore the Circle BLP shall cut the Arch FS at Right Angles in the Point E. And because the Point M is the Pole of the Circle BFP, therefore the Circle GMS shall cut the Circle BFP at Right Angles in the Point D, the Side F P being extended. [Page 269] Come we now to the several Cases which after this preparation may be resolved, by the Catholick Proposition.
CASE 1.
Two Sides with an Angle opposite to one of them being given, to find the Angle opposite to the other.
In the Oblique angled Spherical Triangle F P S, in Fig. 4. the Sides and Angles given and required will admit of six Varieties; all which may be resolved by the Catholick Proposition, at two operations, but those two may be reduced to one, as by the following Analogies to every Variety will plainly appear.
| Given | Required | |
| FP | Rad. s PS∷s PSF. s PE | |
| 1. PS | PFS | s. PF. Rad∷s PE. s PFS |
| PSF | s PF. s PS∷s PSF. s PFS | |
| FP | Rad. s FP∷s F. s PE | |
| 2. PS | PSF | s PS. Rad∷s PE. s PSF |
| PFS | s. PS. s FP∷s PFS. s PSF | |
| PS | Rad. s SF∷s F. sDS | |
| 3. FS | FPS | s PS. Rad∷s DS. s SPD |
| PFS | s. PS. s SF∷s PFS. s PSF | |
| PS | Rad. s PS∷s SPD. s DS | |
| 4. FS | PFS | s FS. Rad∷s DS. s SF |
| FPS | s FS. s PS∷s SPF. s SF | |
| FS | Rad. s FS∷s S. s FC | |
| 5. FP | FPS | s. FP. Rad∷s FC. s FPC |
| FSP | s. FP. s FS∷s PSF. s FPS | |
| [Page 270] FS | Rad. s FP∷s FPC. s FC | |
| 6. FP | FSP | s FS. Rad∷s FC. s S |
| FPS | s. FS. s FP∷FPS. s PSF. |
CASE 2.
Two Sides with an Angle appo [...]ite to one of them being given, to find the contained Angle.
In this Case there are six Varieties, all which may be resolved by the Catholick Proposition, according to the Table following.
| Given | Required | |
| FP | 1: cot PSF. Rad∷cs PS. ct EPS | |
| 1. PS | FPS | 2. ct PS. Rad∷cs EPS. t EP |
| PSF | 3. Rad. t EP∷ct FP. cs FPE | |
| EPS+EPF=FPS | ct PS. cs EPS∷ct FP. ct FPE | |
| FP | 1. cot PFS. Rad∷cs PF .ct EPF | |
| 2. PS | FPS | 2. ct PF. Rad∷cs EPF. t EP |
| PFS | 3. Rad. t EP∷cot PS. cs EPS | |
| EPS+EPF=FPS | cot PF. cs EPF∷ct PS ct EPS | |
| PS | 1. cot PFS. Rad∷cs FS. ct FSD | |
| 3. FS | PSF | 2. ct FS. cs FSD∷ Rad. t DS |
| PFS | 3. Rad. t DS∷ ct PS. cs PSD | |
| FSD-PSD=PSF | ct FS. cs FSD∷ct PS. cs PSD | |
| PS | 1. cot FPS. Rad∷cs PS. ct PSD | |
| 4. FS | PSF | 2. ct PS. cs PSD∷Rad. t DS |
| FPS | 3. Rad. t DS∷ct FS. cs FSD | |
| FSD-PSD=PSF | ct PS. cs PSD∷ct FS. cs FSD | |
| FS | 1. cot FSP. Rad∷cs FS. ct SFC | |
| 5. FP | PFS | 2. ct FS. cs SFC∷Rad. t FC |
| FSP | 3. Rad. t FC∷ct FP. cs PFC | |
| SFC-PFC=PFS | ct FS. cs SFC∷ct FP. cs PFC | |
| [Page 271] FS | 1. cot FPS. Rad∷cs PF. ct PFC | |
| 6. FF | PFS | 2. cot FP. cs PFC∷Rad. t FC |
| FPS | 3. Rad. t FC∷ct FS. cs SFC | |
| SFC-PFC=PFS | ct FP. cs PFC∷ct FS. cs SFC. |
CASE 3.
Two Sides and an Angle opposite to one of them being given, to find the third side.
The Varieties in this Case, with their resolution by the Catholick Proposition, are as followeth.
| Given | Required | |
| FP | 1. ct PS∷cs PSF. t ES | |
| 1. PS | FS | 2. cs ES. cs PS∷Rad. cs EP |
| PSF | 3. Rad. cs EP∷cs FP. cs FE | |
| ES+FE=FS | cs ES. cs PS∷cs FP. cs FE | |
| FP | 1. cot FP. Rad∷cos PFS. t FE | |
| 2. PS | FS | 2. cos FE. cos FP∷Rad. cos EP |
| PFS | 3. Rad. cos EP∷cos PS. cos SE | |
| SE+FE=FS | cos FE. cos FP∷cos PS. cos SE | |
| PS | 1. cot FS. Rad∷cos PFS. t FD | |
| 3. FS | FP | 2. cos FD. cos FS∷Rad. cs SD |
| PFS | 3. Rad. cos SD∷cos PS. cs PD | |
| FD-PD=FP | cos FD. cos FS∷cs PS. cs PD | |
| PS | 1. cot PS. Rad∷cos FPS. t PD | |
| 4. FS | FP | 2. cos PD. cos PS∷Rad. cos SD |
| FPS | 3. Rad. cos SD∷cos FS. cs FD | |
| FD-PD=FP | cos PD. cos PS∷cos FS. cs FD | |
| [Page 272] FS | 1. cot FS. Rad∷cos FSP. t SC | |
| 5. FP | PS | 2. cos SC. cos FS∷Rad. cos FC |
| FSP | 3. Rad. cos FC∷cos FP. cos PC | |
| SP-PC=PS | cos SC. cos FS. cos FP. cos PC | |
| FS | 1. cot FP. Rod∷cos FPS. t PC | |
| 6. FP | PS | 2. cos PC. cos FP∷Rad. cos FC |
| FPS | 3. Rad. cos FC∷cos FS. cos SC | |
| SC-PC=PS | cos PC. cos FP∷cos FS. cos SC |
CASE 4.
Two Angles with a Side opposite to one of them being given, to find the Side opposite unto the other.
The Varieties in this Case, with their Resolution by the Catholick Proposition, are as followeth.
| Given | Required | |
| PFS | Rad. s. PS∷s DPS. s SD | |
| 1. FPS | FS | S. FP. Rad∷s SD. s FS |
| PS | ||
| s. PFS. s PS∷s FPS. s FS | ||
| PFS | Rad. s FS∷s PFS. s. SD | |
| 2. FPS | PS | s. FPS. Rad∷s SD. s PS |
| FS | ||
| s. FPS. s FS∷s PFS. s PS | ||
| FPS | Rad. s FP∷s FPS. s FC | |
| 3. PSF | FS | s. PSF. Rad∷s FC. s FS |
| FP | ||
| s. PSF. s FP∷s FPS. s FS | ||
| [Page 273] FPS | Rad. s FS∷s PSF. s FC | |
| 4. PSF | FP | s. FPS. Rad∷s FC. s FP |
| FS | s. FPS s FS∷s PSF. s FP | |
| PSF | Rad. s PS∷s PSF. s PE | |
| 5. SFP | FP | s. SFP. Rad∷s PE. s FP |
| PS | s. SFP. s PS∷s PSF. s FP | |
| PSF | Rad. s FP∷s PFS. s PE | |
| 6. SFP | PS | s. PSF. Rad∷s PE. s PS |
| FP | s. PSF. s FP∷s PFS. s PS |
CASE 5.
Two Angles and a side opposite to one of them being given, to find the Side between them.
The Varieties and Proportions, are as followeth.
| Given | Required | |
| PFS | 1. ct PS. Rad∷cs DPS. PD | |
| 1. FPS | FP | 2. ct DPS. s PD∷Rad. t DS |
| PS | 3. Rad. t DS∷ct PFS. s FD | |
| FD-PD=FP | ct DPS. s PD∷ct PFS. s FD | |
| PFS | 1. ct DFS. Rad∷cs PFS. t FD | |
| 2. FPS | FP | 2. cot PFS. s FD∷Rad. t DS |
| FS | 3. Rad. t DS∷ct FPS. s PD | |
| FD-PD=FP | ||
| FPS | 1. cot FP. Rad∷cs FPC. t PC | |
| 3. PSF | PS | 2. cot FPC. s PC∷Rad. t FC |
| FP | 3. Rad. t FC∷ct PSF. s SC | |
| SC-PC=PS | cot FPC. s PC∷ct PSF. CS | |
| [Page 274] FPS | 1. cot FS. Rad∷cs PSF. t SC | |
| 4. PSF | PS | 2. cot PSF. s SC∷Rad. t FC |
| FS | 3. Rad. t FC∷cot FPS. s PC | |
| SC-PC=PS | cot PSF. s SC∷cot FPS. s PC | |
| PSF | 1. cot PS. Rad∷cs PSF. t SE | |
| 5. SFP | FS | 2. cot PSF. s SE∷Rad. t PE |
| PS | 3. Rad. t PE∷cot SFP. s FE | |
| FE+SE=FS | cot PSF. s SE∷cot SFP. s FE | |
| PSF | 1. cot FP. Rad∷cs SFP. t FE | |
| 6. SFP | FS | 2. cot SFP. s FE∷Rad. t PE |
| FP | 3. Rad. t PE∷cos PSF. s SE | |
| FE+SE=FS | cot. SFP. s FE∷cs PSF. s SE |
CASE 6.
Two Angles and a Side opposite to one of them being given, to find the third Angle.
The Varieties and Proportions are as followeth.
| Given | Required | |
| PFS | 1. ct DPS. Rad∷cs PS. ct PSD | |
| 1. FPS | PSF | 2. s PSD. cs DPS∷Rad. cs DS |
| PS | 3. cs DS. Rad∷cs DFS. s FSD | |
| FSD-PSD=PSF | cs DPS. s PSD∷cs DFS. s FSD | |
| PFS | 1. ct PFS. Rad∷cs FS. ct FSD | |
| 2. FPS | PSF | 2. s FSD. cs PFS∷Rad. cs DS |
| FS | 3. cs PDS. Rad∷cs DPS. cs PSD | |
| FSD-PSD=PSF | cs PFS. s FSD∷cs DPS. cs PSD | |
| [Page 275] FPS | 1. ct FPC. Rad∷cs FP. ct PFC | |
| 3. PSF | PFS | 2. s PFC. cs FPG∷Rad. cs FC |
| FP | 3. cs FC. Rad∷cs PSF. s FC | |
| SFC-PFC=PFS | cs FPC. s PFC∷cs PSF. s SFC | |
| FPS | 1. cot PSF. Rad∷cos FS. ct SFC | |
| 4. PSF | PFS | 2. s SFC. cs PSF∷Rad. cs FC |
| FS | 3. cs FC. Rad∷cs CPF. s PFC | |
| SFC-PFC=PFS | cs PSF. s SFC∷cs CPF. s PFC | |
| PSF | 1. cot PSF. Rad∷cs PS. ct SPE | |
| 5. SFP | FPS | 2. s SPE. cs PSF∷Rad. cs PE |
| PS | 3. cs PE. Rad∷cs SFP. s FPE | |
| FPE+SPE=FPS | cs PSF. s SPE∷cs SFP. s FPE | |
| PSF | 1. cot SFP. Rad∷cs FP. ct FPE | |
| 6. SFP | FPS | 2. s FPE. cs SFP∷Rad. cs PE |
| FP | 3. cos PE Rad∷cs PSF. s SPE | |
| FPE+SPE=FPS | cs SFP. s FPE∷cs PSF. s SPE |
CASE 7.
Two Sides and their contained Angle being given, to find either of the other Angles.
The Varieties and Proportions are as followeth.
| Given | Required | |
| FS | 1. ct FP. Rad∷cs PFS. t FE | |
| 1. FP | FSP | 2. ct PFS. s FE∷Rad. t PE |
| PFS | 3. t PE. Rad.∷s ES. ct PSF | |
| FS-FE=ES | s EF. ct PFS∷s ES. ct PSF | |
| [Page 276] FS | 1. cot FS. Rad::cs PFS. t DF | |
| 2. FP | FPS | 2. cot PFS. s DF::Rad. t DS |
| PFS | 3. t DS. Rad::s PD. ct SPD | |
| FD - FP=PD | s DF. ct PFS::s PD. ct SPD | |
| FP | 1. cot FP. Rad::cos FPC. t PC | |
| 3. PS | PSF | 2. cot FPC. s PC::Rad. t FC |
| FPS | 3. t FC. Rad::s CS. cot FSP | |
| PS+PC=CS | s PC. ct FPC::s CS. ct FSP | |
| FP | 1. cot PS. Rad::cos SPD. t PD | |
| 4. PS | SFP | 2. cot SPD. s PD::Rad. t DS |
| FPS | 3. t DS Rad::s FD. cot SFP | |
| FP+PD=FD | s PD. ct SPD::s FD. cot SFP | |
| PS | 1. cot PS. Rad:: cs PSF. t SE | |
| 5. FS | SFP | 2. cot PSF. s SE::Rad. t PE |
| PSF | 3. t PE. Rad::s FE. cot SFP | |
| FS-SE=FE | s SE. ct PSF::s FE. ct SFP | |
| PS | 1. cot FS. Rad::cs PSF. t SC | |
| 6. FS | FPS | 2. cot PSF. s SC::Rad. t FC |
| PSF | 3. t FC. Rad::s PC. cot FPC | |
| SC-PS=PC | s SC. cot PSF::s PC. ct FPC |
CASE 8.
Two Sides and their contained Angle being given, to find the third Side.
The Varieties and Proportions are as followeth.
| Given | Required | |
| FS | 1. ct FP. Rad::cs PFS. t FE | |
| 1. FP | PS | 2. cs FE. cs FP::Rad. cos PE |
| PFS | 3. Rad. cs PE :: cs ES. cs PS | |
| FS-FE=ES | cs FE. cs FP::cs ES. cs PS | |
| FP | 1. ct PS. Rad::cs SPD. t PD | |
| 2. SP | FS | 2. cs PD. cs PS::Rad. cos DS |
| FPS | 3. Rad. cos DS::cs FD. cs FS | |
| FP+PD=FD | cs PD. cs PS::cs FD. cs FS | |
| PS | 1. ct PS. Rad::cs PSF. t. ES | |
| 3. FS | FP | 2. cs ES. cs PS::Rad. cos PE |
| PSF | 3. Rad. cos PE::cos FE. cos FP | |
| FS-ES=FE | cs ES. cs PS::cos FE. cs FP |
CASE 9.
Two Angles and their contained Side being given, to find one of the other Sides.
| Given | Required | |
| PFS | 1. ct PFS. Rad::cs FP ct FPE | |
| 1. FPS | PS | 2. ct FP. cs FPE::Rad. t PE |
| FP | 3. t PE. Rad::cs EPS. ct PS | |
| FPS-FPE=EPS | cs FPE. ct FP::cs EPS. ct PS | |
| PFS | 1. cot FPC. Rad::cs FP. t PFC | |
| 2. FPS | FS | 2. cot FP. cs PFC::Rad. t FC |
| FP | 3. t FC. Rad::cs SFC. ct SF | |
| SFP+PFC=SFC | ct FP. cs PFC::cs SFC. ct SF | |
| [Page 278] FPS | 1. ct SPD. Rad::cs PS. ct PSD | |
| 3. PSF | SF | 2. ct PS. cos PSD::Rad. t DS |
| PS | 3. t DS. Rad::cs FSD. ct SF | |
| PSF+PSD=FSD | cs PSD. ct PS::cs FSD. ct SF | |
| FPS | 1. ct PSF. Rad::cs PS. ct SPE | |
| 4. PSF | FP | 2. ct PS. cs SPE::Rad. t PE |
| PS | 3. t PE. Rad::cs PPE. ct FP | |
| FPS-EPS=FPE | cs SPE. ct PS::cs FPE. ct FP | |
| PSF | 1. ct PSF. Rad::cs SF. ct SFC | |
| 5. SFP | FP | 2. ct SF. cs SFC::Rad. t FC |
| SF | 3. t FC. Rad::cs CFP. ct FP | |
| SFC-SFP=CFP | cs SFC. ct SF::cs CFP. ct FP | |
| PSF | 1. ct SFP. Rad::cs FS. ct FSD | |
| 6. SFP | PS | 2. ct FS. cs FSD::Rad. t SD |
| SF | 3. t SD. Rad::cos PSD. ct PS | |
| FSD-FSP=PSD | cs FSD. ct FS::cs PSD. ct PS |
CASE 10.
Two Angles and the Side between them being given, to find the third Angle.
The Varieties and Proportions are as followeth.
| Given | Required | |
| SFP | 1. ct SFP. Rad::cs FP. ct FPE | |
| 1. FPS | PSF | 2. s FPE. cs F::Rad. cs PE |
| FP | 3. Rad. cs PE::s EPS. cs PST | |
| FPS-FPE=EPS | s FPE. cs PFS::s SPE. cs PSF | |
| [Page 279] FPS | 1. ct SPD. Rad::cs PS. ct PSD | |
| 2. PSF | SFP | 2. s PSD. cs SPD::Rad. cs DS |
| PS | 3. Rad. cs DS::s FSD. cs SFP | |
| PSF+PSD=FSD | s PSD. cs SPD::s FSD. cs SFP | |
| PSF | 1. ct PSF. Rad::cs SF. ct SFC | |
| 3. SFP | FPS | 2. s SFC. cs PSF::Rad. cs FC |
| SF | 3. Rad. cs FC::s PFC. cs FPS | |
| SFC-SFP=PFC | s SFC. cs PSF::s PFC. cs FPS |
CASE 11.
The three Sides being given, to find an Angle.
This Case may be resolved by the Catholick Proposition also, according to the direction of the Lord Nepier, as I have shewed at large in the Second Book of my Trigonometria Britannica, Chap. 2. but may as I conceive be more conveniently solved, by this Proposition following.
As the Rectangle of the Square of the Sides containing the Angle inquired;
Is to the Square of Radius: So is the Rectangle of the Square of the difference of each containing Side, and the half sum of the three Sides given.
To the Square of the Sine of half the Angle inquired.
In this Case there are three Varieties, as in the Triangle FZP. Fig. 3.
| Given | Required | |
| ZP | s ZP x s PF. Rad. q. | |
| 1. PF | ZPF | s ½ Z-ZP x s ½ Z-PF. Q |
| FZ | s ½ ZPF | |
| ZP | s PF x s PZ. Rad. q. | |
| 2. PF | PFZ | s ½ Z-PF x s ½ Z-FZ. Q |
| FZ | s ½ PFZ | |
| ZP | s ZP x s FZ Rad. q. | |
| 3. PF | FZP | s. ½ Z-ZP x ½ Z-ZF. Q |
| FZ | s ½ FZP |
CASE 12.
The three Angles given, to find a Side.
This is the Converse of the last, and to be resolved after the same manner, if so be we convert the Angles into Sides, by the tenth of the third Chapter: for so the Sides of the Triangle ACD will be equal to the Angles of the Triangle FZP n Fig. 3.
- AD=AEE the measure of the Angle ZPF.
- DC=KM the measure of the Angle ZFP.
- AC=HB the Complement of FZP to a Semicircle.
- [Page 281]DAC = QR = ZP.
- ACD = rM = Hf = Zoe = ZF.
- ADM = sK = AEl = Ph = PF.
And thus the Sides of the Triangle ZPF are equal to the Angles of the Triangle ACD. The Complement of the greatest Side PF to a Semicircle being taken for the greatest Angle ADC.
And in this Case therefore, as in the preceding, there are three Varieties which make up sixty Problems in every Oblique angled Spherical Triangle; which actually to resolve in so many Triangles, as have been mentioned, would be both tedious, and to little purpose; I will therefore select some few, that are of most general use in the Doctrine of the Sphere, and leave the rest to thine own practice.
CHAP. V.
Of such Spherical Problems as are of most General Use in the Doctrine of the Primum Mobile or Diurnal Motion of the Sun and Stars.
PROBLEM 1.
The greatest Declination of the Sun being given, to find the Declination of any Point of the Ecliptick.
THe Declination of the Sun or other Star, is his or their distance from the Equator, and as they decline from thence either Northward or Southward; so is their Declination reckoned North or South.
2. The Sun's greatest Declination, which in this and many other Problems is supposed to be given, with the Distance of the Tropicks, Elevation of the Equator, and Latitude of the Place, may thus be found.
Take with a Quadrant, the Sun's greatest and least Meridian Altitudes, on the longest and shortest days of the year, which suppose at London to be as followeth. [Page 283]
- [Page 283]The Sun's
greatest Meridian H♋. 61.9916 least Altitude H♑. 14.9416 - Their difference is the distance of the Tropicks ♋. ♑. 47. 050
- Half that Difference, is the Sun's greatest Declination AE ♋. 23. 525
- Which deduct from the Sun's greatest Altitude, the remainer is the height of the Equator HAE. 38. 467
- The Complement is the height of the Pole AEZ or PR. 51. 533
Now then in the Right angled Spherical Triangle ADF in Fig. 1. there being given.
1. The Angle of the Sun's greatest Declination DAF. 23. 525.
2. The Sun's supposed distance from ♈ to ♎ AF. 60 deg.
The Sun's present Declination DF may be found, by the 10 Case of Right angled Spherical Triangles.
As the Radius
- Is to the Sine of DAF. 23. 525.
- 9.60113517
- So is the Sine of AF 60.
- 9.93753063
- To the Sine of DF. 20. 22.
- 9.53866580
PROBLEM 2.
The Sun's groatest Declination, with his Distance from the next AEquinoctial Point being given, to find his Right Ascension.
In the Right angled Spherical Triangle ADF in Fig. 1. Having the Angle of the Sun's greatest Declination DAF. 23. 525. And his supposed distance from ♈ or ♎, the Hypotenusa AF. 60. The Right Ascension of the Sun, or Arch of the AEquator, AD may be found, by the ninth Case of Right angled Spherical Triangles,
- As the Cotang. of the Hypot. AF. 60.
- 9.76143937
- Is to the Radius
- 10.00000000
- So is the Cosine of DAF. 23. 525.
- 9▪96231533
- To the Tang. of AD. 57. 80.
- 10.20087596
PROBLEM 3.
To find the Declination of a Planet or Fixed Star with Latitude.
In the Oblique angled Spherical Triangle FPS in Fig. 4. we have given, 1. PS = AE ♋ the greatest Declination of the Ecliptick, 2. The Side FS the Complement of the Stars Latitude from the Ecliptick at K. 3. The Angle PSF the Complement of the Stars Longitude. To find PF the Complement of Declination. By the eighth Case of Oblique angled Spherical Triangles, the Proportions are.
- As the Cot. of PS. 23. 525.
- 10.3611802
- Is to the Radius.
- 10.0000000
- So is the Cos. of PSF. 20 deg.
- 9.9729858
- To the Tang. of SE. 22. 25.
- 9.6118056
- FS. 86 deg. - ES. 22. 25. = FE.
- 63. 75.
- As the Cos. of ES. 22. 25. Comp. Arith.
- 0.0336046
- To the Cosine of PS. 23. 525.
- 9.9623154
- So the Cos. FE. 63. 75.
- 9.6457058
- To the Cos. PF. 64. 01.
- 9.6416258
Whose Complement, is FT. 25. 99. the Declination sought.
PROBLEM 4.
To find the Right Ascension of a Planet, or other Star with Latitude.
The Declination being found by the last Problem, we have in the Oblique angled Spherical Triangle PFS in Fig. 4. All the Sides with the Angle FSP 20 deg. or the Complement of the Stars Longitude. Hence to find FPS by the first Case of Oblique angled Spherical Triangles, I say
- As the Sine of PF. 64. 01. Comp. Arith.
- 0.0463059
- Is to the Sine of FSP. 20.
- 9.5340516
- So is the Sine of FS. 86.
- 9.9984407
- To the Sine of FPS. 22. 28.
- 9.5787982
Whose Complement 67. 72. is the Right Asc. of a Star II. 10. North Lat. 4.
PROBLEM 5.
The Poles Elevation, Sun's greatest Declination and Meridian Altitude being given, to find his true place in the Zodiack.
If the Meridian Altitude of the Sun be less than the height of the AEquator, deduct the Meridian Altitude from the height of the AEquator, the Remainer is the Sun's Declination towards the South Pole: but if the Meridian Altitude of the Sun be more than the height of the AEquator, deduct the height of the AEquator from the Meridian Altitude, what remaineth, is the Sun's Declination towards the North Pole, in these Northern Parts of the World: the contrary is to be observed in the Southern Parts.
Then in the Right angled Spherical Triangle ADF in Fig. 1. we have given the Angle FAD the Sun's greatest Declination.
The Leg DF the Sun's present Declination, To find AF the Sun's distance from the next Equinoctial Point.
Therefore by the Case of Right angled Spherical Triangles.
- As the Sine of FAD. 23. 525. Comp. Ar.
- 0.3988648
- Is to the Sine of DF. 23. 5.
- 9.5945468
- So is the Radius.
- 10.0009000
- To the Sine of AF. 80. 04.
- 9.9934116
PROBLEM 6.
The Poles Elevation and Sun's Declination being given, to find his Amplitude.
The Amplitude of the Sun's rising or setting is an Arch of the Horizon intercepted betwixt the AEquator and the place of the Sun's rising or setting; and it is either Northward or Southward, the Northward Amplitude is when he riseth or setteth on this Side of the AEquator towards the North Pole; and the Southern when he riseth or setteth on that Side of the AEquator which is towards the South Pole: That we may then find the Sun's Amplitude or Distance from the East or West Point, at the time of his rising or setting. In the Right angled Spherical Triangle ATM, in Fig. 2. let there be given the Angle TAM. 38. 47. the Complement of the Poles Elevation; and TM. 23. 15. the Sun's present Declination: To find AM the Sun's Amplitude.
By the eleventh Case of Right angled Spherical Triangles.
- As the Sine of MAT. 38. 47. Comp. Ar.
- 0.2061365
- Is to the Radim.
- 10.0000000
- So is the Sine of MT. 23. 15.
- 9.5945468
- To the Sine of AM. 39. 19.
- 9.8006833
PROBLEM 7.
To find the Ascensional Difference.
The Ascensional Difference is nothing else, but the Difference between the Ascension of any Point of the Ecliptick in a Right Sphere, and the Ascension of the same Point in an Oblique Sphere; As in Fig. 1. AT is the Ascensional difference between DA the Sun's Ascension in a Right Sphere, and DT the Sun's Ascension in an Oblique Sphere. Now then in the Right angled Spherical Triangle AMT, we have given. The Angle MAT. 38. 47. the Complement of the Poles Elevation. And MT. 23. 15. To find AT the Ascensional difference.
- As Rad.
- To the Cot. of MAT. 38. 47. Com. Ar.
- 10.0999136
- So is Tang. MT. 23. 55.
- 9.6310051
- To the Sine of AT. 32. 56.
- 9.7309187
PROBLEM 8.
Having the Right Ascension and Ascensional Difference, to find the Oblique Ascension and Descension.
In Fig. 1. DT represents the Right Ascension, AT the Ascensional Difference. DA the Oblique Ascension which is found by deducting the Ascensional Difference AT. from the Right Ascension DT. according to the Direction following.
- [Page 289]If the Declination be
- N.
- North
- Subt.
- Add
- The Ascentional Difference from the Right, and it giveth the Oblique Ascension.
- The Ascensional Difference to the Right, and it giveth the Oblique Descension.
- South
- Add
- Subt.
- The Ascensional Difference to the Right, and it giveth the Oblique Ascension.
- The Ascensional Difference from the Right, and it giveth the Oblique Descension.
- Right Ascension of ♊. 0 deg.
- 57.80
- Ascensional Difference
- 27.62
- Oblique Ascension ♊. 0 deg.
- 30.18
- Oblique Descension ♊. 0 deg.
- 85.42
PROBLEM 9.
To find the time of the Sun's rising and setting, with the length of the Day and Night.
The Ascensional Difference of the Sun being added to the Semidiurnal Arch in a Right Sphere, that is, to 90 Degrees in the Northern Signs, or substracted from it in the Southern, their Sum or Difference will be the Semidiurnal Arch, which [Page 290] doubled is the Right Arch, which bisected is the time of the Sun rising, and the Day Arch bisected is the time of his setting.
As when the Sun is in 0 deg. ♊. his Ascensional Difference is 27. 62. which being added to 90 degrees, because the Declination is North, the Sum will be 117.62 the Semidiurnal Arch.
The double whereof is 235.22 the Diurnal Arch, which being converted into time makes 15 hours 41 minutes: for the length of the Day, whose Complement to 24; is 8 hours 19 minutes the length of the Night; the half whereof is 4 hours 9 minutes 30 Seconds the time of the Sun's rising.
PROBLEM 10.
The Poles Elevation and the Sun's Declination being given, to find his Altitude at any time assigned.
In this Problem there are three Varieties. 1. When the Sun is in the AEquator., that is, in the beginning of ♈ and ♎ in which case supposing the Sun to be at B, 60 degrees or four hours distant from the Meridian, then in the Right angled Spherical Triangle BZ AE, in Fig. 1. we have given, AE Z, 51. 53. the Poles Elevation, and B AE 60 degrees, to find BZ.
Therefore by the 2 Case of Right angled Spherical Triangles.
- As the Radius
- To the Cosine of AE Z. 51. 53.
- 9.7938635
- So is the Cosine of B. AE. 60.
- 9.6989700
- To the Cosine of B Z. 71. 88.
- 9.4928335
Whose Complement BC. 18. 12. is the ☉ Altitude required.
The second Variety is when the Sun is in the Northern Signs, that is, in ♈. ♉. ♊. ♋. ♌. ♍. in which Case supposing the Sun to be at F in Fig. II Then in the Oblique angled Spherical Triangle FZP, we have given. 1. PZ 38. 47 the Complement of the Poles Elevation. 2. FP. 67. 97 the Complement of Declination. 3. ZPF. 45 the Distance of the ☉ from the Meridian, To find FZ.
Therefore by the eighth Case of Oblique angled Spherical Triangles.
- As the Cotang. of ZP. 38. 47.
- 10.0997059
- Is to the Radius.
- 10.0000000
- So is the Cosine of ZPF. 45.
- 9.8494850
- To the Tang. of SP. 29. 33.
- 9.7497791
- Then from FP.
- 67.97
- Deduct SP.
- 29.33
- There rests FS.
- 38.64
- As the Cosine of SP. 29. 33. Comp. Ar.
- 0.0595768
- To the Cosine of PZ. 38. 47.
- 9.8937251
- So is the Cosine of FS. 38. 64.
- 9.8926982
- To the Cosine of FZ. 45. 45.
- 9.8460001
Whose Complement FC. 44. 55 is the ☉ Altitude required.
The third Variety is when the Sun is in the Southern Signs as in ♎. ♏. ♐. ♑. ♒. ♓. And in this Case supposing the ☉ to be ♐ 10 degrees, and his Declination South Db 22. 03. and his Distance from the Meridian 45 as before, then in the Oblique angled Spherical Triangle Z bP in Fig. 1. we have given Z P. 38. 47. The Side bP 112. 03. and the Angle ZPb 45. To find Zb.
Therefore by the 8 Case of Oblique angled Spherical Triangles.
- As the Cotang. of ZP. 38. 47.
- 10.0997059
- Is to the Radius.
- 10.0000000
- So is the Cosine of ZPb. 45.
- 9.8494850
- To the Tang. of SP. 29. 33.
- 9.7497791
- Then from bP.
- 112.03
- Deduct SP.
- 29.33
- There rests bS.
- 82.70
- As the Cosine of P S. 29. 33. Comp. Ar.
- 0.0595768
- To the Cosine of ZP. 38. 47.
- 9.8937251
- So the Cosine of bS. 82. 70
- 9.1040246
- To the Cosine of Zb. 83. 45.
- 9.0573265
Whose Complement 6.55 is the ☉ Altitude required.
PROBLEM 11.
Having the Altitude of the Sun, his Distance from the Meridian, and Declination, to find his Azimuth.
The Azimuth of the Sun is an Arch of the Horizon intercepted between the Meridian and the Vertical Line passing by the Sun, being understood by the Angle HZC in Fig. 1. or Arch HC. And in all the Varieties of the last Problem, may be found, by the first Case of Oblique angled Spherical Triangles.
Thus in the Triangle ZBP.
- As the Sine of BZ. 71. 88. Comp. Ar.
- 0.022090 [...]
- Is to the Sine of BPZ. 60.
- 9.9375306
- So is the Sine of BP. 90.
- 10.0000000
- To the Sine of BZP. 65. 67.
- 9.9596209
In the Triangle ZFP. I say.
s. ZF. s. ZPF▪ :: s. FP. s. FZP.
In the Triangle ZbP. I say.
Sine Zb. Sine ZPb :: Sine bP. Sine bZP.
PROBLEM 12.
The Poles Elevation, with the Sun's Altitude and Declination given, to find his Azimuth.
In the Oblique angled Spherical Triangle FZP in Fig. 1. let there be given.
1. FP. 67. 97 the Complement of the ☉ Declination.
2. ZP. 38. 47 the Complement of the Poles Elevation.
3. FZ. 45. 46 the Complement of the ☉ Altitude.
And let the Angle FZP the ☉ Azimuth be required.
By the 11 Case of Oblique Angled Spherical Triangles.
As the Sine ZP x Sine FZ, Is to the Square of Radius.
So is the Sine 1/2 Z of the Sides ZP x 1/2 Z cr—ZF.
To the Square of the Sine of half the Angle FZP.
The Sum of the three Sides is 151.89
The half Sum is 75.945 from which deduct PZ 38. 47. The difference is 37.475 And the Difference between 75.945 and FZ is 30. 495.
- Sine of PZ. 38. 47. Comp. Ar.
- 0.2061365
- Sine of FZ. 45. 45. Comp. Ar.
- 0.1471308
- s. 1/2 Z cr—PZ. 37. 475.
- 9.7842000
- s. 1/2 Z cr—FZ. 30. 495.
- 9.7054045
- Square of the Sine of 1/2 FZP.
- 19.8428618
- Sine of 57. 94.
- 9.9214309
The double whereof is 115.88 the ☉ Azimuth from the North. And the Complement 64.12, is the ☉ Azimuth from the South.
PROBLEM 13.
To find the Point of the Ecliptick Culminating, and its Altitude.
Before we can know what Sign and Degree of the Ecliptick is in the Medium Coeli; we must find the Right Ascension thereof, to do which, we must add the Sun's Right Ascension to the time afternoon, being reduced into Degrees and Minutes of the AEquator, the Sum is the Right Ascension of the Medium Coeli.
Example. Let the time given, be March the 20. 1674. at one of the Clock in the Afternoon.
At which time the Sun's place is in ♈. 10 deg. 23 Centesms.
To find the Right Ascension thereof, in the Right angled Spherical T [...]iangle ADF in Fig. 1. we have given; The Angle of the Sun's greatest Declination DAF 23. 525 and the Sun's distance from the next Equinoctial Point AF 10. 23.
Therefore by the ninth Case of Right angled Spherical Triangles.
- As the ct. AF. 10. 23.
- 10.7435974
- Is to Radius.
- 10.0000000
- So is cs DAF 22. 525.
- 19.9623154
- To t AD 9. 39.
- 9.2187180
To which adding the Equinoctial Degrees answering to one hour, viz. 15. the Sum is 24.39 the Right Ascension of the Mid Heaven. Hence to find the Point culminating; in the Right angled Spherical Triangle ADF in Fig. 1. we have given AD 24. 39 and DAF 23. 525 to find AF.
Therefore by the fifth Case of Right Angled Spherical Triangles.
- As t AD 24. 39.
- 10.6564908
- Is to Radius.
- 10.0000000
- So is cs DAF 23. 525.
- 9.9623154
- To ct. AF 26. 31.
- 10.3058246
Therefore the Point culminating is ♈ 26. 31.
To find the Altitude thereof above the Horizon we have given in the same Triangle DAF 23. 525. and AF 26. 31. to find DF.
Therefore by the tenth Case of Right angled Spherical Triangles.
- As Radius.
- 10.0000000
- Is to s AF—26, 31.
- 9.6466268
- So is s DAF 23. 525.
- 9.6011352
- To the s DF 10. 19.
- 9.2477628
[Page 297] Which is the North Declination of the Point of the Ecliptick culminating, and being added to the height of the AEquator at London 38. 47 the Sum is 48.66 the Altitude of the Mid Heaven as was required.
PROBLEM 14.
Having the greatest obliquity of the Ecliptick together with the Distance of the Point given from the Equinoctial, to find the Meridian Angle, or Intersection of the Meridian with the Ecliptick.
Having drawn the Primitive Circle HZRN in Fig. 5. representing the Meridian, and the two Diameters HAR, and ZAN, set off the height of the Pole from R to P. 51. 53, and from N to S, and draw the Diameters PAS for the Axis of the World, and AE AQ for the AEquator; this done, the Right Ascension of the Mid Heaven being given, as in the last Problem 24.39 with the Point culminating. ♈. 26.31, and the Declination thereof 10.19, if you set 10 deg. 19 Centesmes from AE to F and e to X, you may draw the Diameters FAX and cAd at Right Angles thereunto, and because the Imum Coeli is directly opposite to the Point culminating, that is, in ♎ 26.31, if you set 26.31 from X to b, a Ruler laid from c to b will cut the Diameter FX in G, and then making Xh Z Xb you have the three Points b G h, by which to draw that Circle, which will cut the AEquator AE AQ in ♎, and so you have the three Points X ♎ F by which to describe the Arch of the Ecliptick ♈ F ♎ X.
[Page 298] And in the Right angled Spherical Triangle ♈ AEF we have given. The Angle AE ♈ F. 23.525 the Sun's greatest Declination, and ♈ F. 26. 31. the Point culminating, to find the Angle ♈ F AE.
Therefore by the seventh Case of Right angled Spherical Triangles.
- As the ct AE ♈ F. 23. 525.
- 10.3611802
- Is to the Radius.
- 10.0000000
- So is the cs ♈ F. 26. 31.
- 9.9525062
- To the cot. ♈ FAE. 68. 60.
- 9.5913260
Which is the Angle of the Ecliptick with the Meridian.
PROBLEM 15.
To find the Angle Orient, or Altitude of the Nonagesime Degree of the Ecliptick.
In Fig. 5. the Pole of the Ecliptick ♈ F ♎ X is at m, and so you have the three Points Z m N to draw the Vertical Circle Z k N cutting the Ecliptick at Right Angles in the Point a: And then in the Right angled Spherical Triangle F a Z, we have given; FZ 41. 34 the Complement of FH the Altitude of the Mid Heaven; And the Angle a FZ 68. 68 the Angle of the Ecliptick with the Meridian. To find Z a.
Therefore by the tenth Case of Right angled Spherical Triangles.
[Page 299] As the Radius.
- To the Sine of FZ. 41. 34.
- 9.819889 [...]
- So is the Sine of Z F a. 68. 68.
- 9.9691128
- To the Sine of Z a. 37. 97.
- 9.7891027
Whose Complement is ak the Measure of the Angle agk 52. 03 the Angle of the Ecliptick with the Horizon, or Altitude of the Nonagesime Degree.
PROBLEM 16.
To find the place of Nonagosime Degree of the Ecliptick.
In Fig. 5. F represents the Point of the Ecliptick in the Mid Heaven, which according to Problem 14 is ♈. 26.31 which being known, in the Triangle FZa, we have also given, FZ 41. 34 and the Angle ZFa. 68. 68. To find Fa.
Therefore by the ninth Case of Right angled Spherical Triangles.
- As the cot. of FZ. 41. 34.
- 10.0556361
- Is to the Radius.
- 10.0000000
- So is the cos. of ZFa. 6. 8. 6. 8.
- 9.5605957
- To the tang. of Fa. 17. 73.
- 9.5049596
Which being added to ♈ F 26. 31 the sum is ♈ a. 44. 04 the place of the Nonagesime Degree of the Ecliptick at a.
PROBLEM 17.
The Mid Heaven being given, to find the Points of the Ecliptick Ascending and Descending.
Having found by the last Problem, the place of Nonagesime Degree of the Ecliptick at a to be in ♉. 14.04, if you add 90 Degrees or three Signs thereto, the Ascendant at g will be in ♌ 14. 04, and the Point descending by adding of six Signs will be in ♒ 14. 02. But these with the Cusps of the other Houses of Heaven may be otherwise found in this manner.
To the Right Ascension of the Medium Coeli or the tenth House, add 30, it giveth the Ascension of the eleventh House, to which adding 90 Degrees more, it giveth the Ascension of the twelfth House, &c. According to which direction, the Ascensions of the six Houses towards the Orient, are here set down in the following Table.
| 10. | 24.39 |
| 11. | 54.39 |
| 12. | 84.39 |
| 1. | 114.39 |
| 2. | 144.39 |
| 3. | 174.39 |
Now because the Circles of Position must according to these Directions cut the AEquator at 30 and 30 Degrees above the Horizon, if in Fig. 5. you set 30 Degrees from AE to n, and n to r. A Ruler laid from P to n and r, shall cut the AEquator at B and K, and then you may describe the Circles of Position HBR and HKR, make AT = AK and AV= AB, and so you may describe the Circles HTR and HVR, and where these Circles do cut the [Page 301] Arch of the Ecliptick ♈ F ♎ there are the Cusps of the Coelestial Houses.
Thus a Ruler laid from m. the Pole of the Ecliptick to the Intersections ct s. t. g. [...]. will cut the Primitive Circle in [...]. and the Arches [...] = Fs. [...] = Ft. [...] = Fg. [...] = [...]. and [...] = [...] being added to [...] B will give you the Cusps of the 11. 12. 1. 2 and 3 Houses, the other six are the same Degrees and Parts in the Opposite Signs.
Thus a Figure in Heaven may be erected by Projection, the Arithmetical Computation now followeth; In which the height of the Pole above each Circle of Position is required, the which in the Projection is easily found; as the Pole of the Circle of Position HBR is at the Point D. and so you have the three Points S, D, P, to describe that Circle by, which will cut the Circle HBR at Right Angles in the Point C. and the Arch PC is the height of the Pole above that Circle of Position, and may be measured by the Directions given in the nineteenth of the third Chapter.
In like manner the height of the Pole above the Circle of Position HKR, will be the Arch PE.
To compute the same Arithmetically in the Right angled Spherical Triangle HAEB in Fig. 5. we have given AEH. 38. 47 the height of the Equator. AEB 30. the difference of Ascension between the 10 and 11 Houses, to find HBAE the Angle of that Equator with the Circle of Position.
Therefore by the first Case of Right angled Spherical Triangles.
- As the Tang. of H AE, 38. 47.
- 9.90000652
- Is to the Radius.
- 10.00000000
- So is the Sine of AE B. 30
- 9.69897000
- To the Cotang. of AE B H. 57. 81626.
- 9.79888348
Whose Measure in the Scheme is EC, and the Complement thereof is CP. 32. 18374 the height of the Pole required.
Therefore the height of the Pole above the Circle of Position HKR. In the Triangle HAEK, we have given, H AE as before, and AE K. 60 to find HKAE. Therefore.
- As the Tang. of H AE 38. 47.
- 9.90008652
- Is to the Radius.
- 10.00000000
- So is the Sine of AE K 60.
- 9.93753063
- To the Cotang. of HK AE 42. 53308.
- 10.03744411
Whose Measure in the Scheme is GL, and the Complement thereof is PL 47. 46692. the height of the Pole required.
The height of the Pole above HDR is the same with HBR, and the height of the Pole above HTR is the same with HKR.
Having found the Ascensions of the several Houses together with the Elevation of the Pole above their Circles of Position, in the Oblique angled Spherical Triangle ♈ BS, we have given.
1. The Angle ♈ BS the Complement of HBAE.
2. The Angle B ♈ S. 23. The Sun's greatest Declination.
3. Their included Side ♈ B. 54. 39 the Ascension of the eleventh House. To find ♈ S the [Page 303] Point of the Ecliptick, which is resolvable by the ninth Case of Right angled Spherical Triangles.
But in my Trigonometria Britannica, Problem. 5. for the resolving of Oblique angled Spherical Triangles, I have shewed how this Case as to our present purpose may be resolved, by these Proportions following.
- 1. s 1/2 Z Ang. s 1/2 X Ang :: t 1/2 ♈ B. t 1/2 X Cru.
- 2. cs 1/2 Z Ang. cs 1/2 X Ang :: t 1/2 ♈ B. t 1/2 Z Cru.
- 1/2 Z Cru + 1/2 X Cru=♈ S the Arch of the Ecliptick desired.
For the Cusp of the Eleventh House.
- T B Arch ♈ B. 4439 the half whereof is 27. 195.
- ♈ B S. 122. 18374.
- B ♈ S. 23. 525.
- Z 145.70874—1/2 Z 72. 85437.
- X. 198.65874—1/2 X. 49. 32937.
- s 1/2 Z. 72. 85437. Comp. Arith.
- 0.01977589
- s 1/2 X. 49. 32937.
- 9.88000800
- t 1/2 ♈ B. 27. 195.
- 9.71081089
- t 1/2 X Cru. 22. 192.
- 9.61059478
2. Operation.
- cs. 1/2 Z. 72. 85437. Comp. Arith.
- 0.53012277
- ss 1/2 X. 49. 32937.
- 9.81395860
- t 1/2 ♈ B. 27. 195.
- 9.71081089
- t 1/2 Z Cru. 48. 611.
- 10.05489226
1. Arch. 22. 192. Their Sum is 70.803 the Point of the Ecliptick.
- cs. ½ Z. 82. 51916. Comp. Arish.
- 0.88517901
- cs ½ X. 59. 00416.
- 9.71164750
- t. ½ ♈ A. 57. 195.
- 10.19072348
- t▪ ½ Z Cru: 78. 397.
- 10.68754999
1. Arch—53. 296. Their Sum 121.693 is the Point of the Ecliptick for the Ascendant.
For the Cusp of the Second House.
In the Oblique angled Spherical Triangle ♈ T [...]. we have given,
- 1. ♈ T. 144. 39.
- The half whereof is 72. 195.
- 2. ♈ T [...]. 122. 18374
- To find ♈ [...]. The Angles are the same with those of the Twelfth House. Therefore.
- 3. T [...] y. 23. 525
- s. ½ Z. 80. 49596. Comp. Arith.
- 0.00601663
- s ½ X. 56. 97096.
- 9.92351651
- Their Sum
- 9.92953314
- t ½ ♈ T. 72. 195.
- 10.49327695
- t ½ X Cru. 69. 306.
- 10.42281009
- cs ½ Z 80. 49596. Comp. Arith.
- 0.78170174
- cs ½ X 56. 97096.
- 9.73628614
- Their Sum
- 10.51798788
- t ½ ♈ T. 72. 195.
- 10.49327695
- t ½ Z Cru. 84. 34.
- 11.01126483
1. Arch. 69. 306. Their Sum is 53.740 is the Point of the Ecliptick for the Second House.
For the Cusp of the Third House.
In the Oblique angled Spherical Triangle ♈ [...], we have,
1. ♈ [...]. 174. 39. The half whereof is 87. 195. The Angls ♈ [...] and [...] ♈ [...] are the same with those of the Eleventh House.
- s ½ Z. 72. 85437. Comp. Arith.
- 0.01977580
- s ½ [...] 49. 32937.
- 6.88000800
- Their Sum
- 9.89978389
- t ½ [...]. 87. 195.
- 11.30984054
For the Eleventh House.
For the Cusp of the Twelfth House.
In the Oblique angled Spherical Triangle ♈ KF, we have given.
- 1. [...] K. 84. 39.
- The half whereof is. 42. 195.
- 2. ♈ Kt. 137.46692
- 3. K ♈ t. 23.525
- To find ♈ t.
- Z. 160.99192
- ½ Z. 80.49596
- X. 113.94192
- ½ X. 56.97096
- s ½ Z. 80. 49596. Comp. Arith.
- 0.00601663
- s ½ X. 56. 97096.
- 9.92351651
- t ½ ♈ K. 42. 195.
- 9.95740882
- t ½ X Cru. 37. 625.
- 9.88694196
- cs. ½ Z. 80. 49596. Comp. Arith.
- 0.78170174
- cs ½ X. 56. 97096.
- 9.73628614
- t ½ ♈ K. 42. 195.
- 9.95740882
- t ½ Z Cru. 71. 496.
- 10.47539670
1. Arch. 37. 625. Their Sum 113.6691 is the Point of the Ecliptick for the Twelfth House.
For the Cusp of the Ascendant.
In the Oblique angled Spherical Triangle ♈ AG we have,
- 1. ♈ A. 114. 39.
- The half whereof is 57. 195.
- 2. ♈ AZ. 141. 5333.
- The Complement of HAAE 38. 46667.
- 3. A [...] y. 23. 525.
- Z. 165.05833
- ½ Z. 82.51916
- X. 118.00833
- ½ X. 59.00416
- s. ½ Z. 82. 51916. Comp. Arith.
- 0.0037162 [...]
- s. ½ X. 59. 00416.
- 9.93313477
- t ½ ♈ A. 57. 195.
- 10.19072348
- t ½ X. 53. 296.
- 10.12757454
- t ½ X Cru. 86. 468.
- 11.20962043
- cs ½ Z. 72. 85437. Comp. Arith.
- 0.53012277
- cs ½ X. 49. 32937.
- 9.81395860
- Their Sum
- 10.34408137
- t ½ ♈ [...]. 87. 195.
- 11.39984054
- t ½ X Cru. 88. 729.
- 11.65392191
1. Arch. 86. 468. Their Sum 175.197 is the Point of the Ecliptick for the Third House.
And thus we have not only erected a Figure for the Time given, but composed a Table for the general erecting of a Figure in that Eatitude; for by adding together the first and second Numbers in each Proportion for the first, second and third Houses there is composed two Numbers for each House, to each of which the Artificial Tangent of half the Ascension of each House being added, their Aggregates are the Tangents of two Arches, which being added together, do give the distance of the Cusp of the House, from the first Point of Aries, as in the preceding Operations hath been shewed.
Only note, That if the Ascension of any House be more than a Semicircle, you must take the Tangent of half the Complement to a whole Circle. And to find the Cusp of the House, you must also take the Complement of the Sum of the Arches added together.
The Numbers according to the former Operations which do constitute a Table of Houses for the Latitude of London. 51. 53 are as followeth.
| 11 and 3 Houses | Ascendant | 12 and 2 Houses | |
| 1. Oper. | 9.89978389 | 9.93685106 | 9.92953314 |
| 2. Oper. | 10.34408137 | 10.59682651 | 10.51798788 |
The Six Oriental Houses, by the preceding Operations.
- 10 House ♈ 26.311
- 11 House ♊ 10.803
- 12 House ♋ 23.691
- Ascendant ♌ 11.693
- 2 House ♏ 3.740
- 3 House ♏ 25.197
- 4 House ♎ 26.311
- 5 House ♐ 10.803
- 9 House ♑ 23.691
- 7 House ♒ 11.693
- 8 House ♓ 3.740
- 9 House ♓ 25.197
A Figure of the Twelve Coelestial Houses.
AN INTRODUCTION TO Astronomy.
The Second Book.
CHAP. I.
Of the Year Civil and Astronomical.
HAving shewed the Motion of the Primum Mobile, or Doctrine of the Sphere, which I call the Absolute Part of Astronomy; I come now unto the Comparative, that is, to shew the Motion of the Stars in reference to some certain Distinction of Time.
2. And the Distinction of Time is to be considered either according to Nature, or according to Institution.
[Page 312] 3. The Distinction of Time according to Nature, is that space of Time, in which the Planets do finish their Periodical Revolutions from one certain Point in the Zodiack, to the same again, and this in reference to the Sun is called a Year, in reference to the Moon a Month.
4. The Sun doth pass through the Zodiack in 365 Days, 5 Hours, and 49 Minutes. And the Moon doth finish her course in the Zodiack, and return into Conjunction with the Sun, in 29 Days, 12 hours, 44 Minutes, and 4 Seconds. And from the Motion of these two Planets, the Civil Year in every Nation doth receive its Institution.
5. Twelve Moons or Moneths is the measure of the Common Year, in Turkey in every Moneth they have 29 or 30 Days, in the whole Year 354 Days, and in every third Year 355 Days.
6. The Persians and Egyptians do also account 12 Moneths to their Year; but their moneths are proportioned to the Time of the Suns continuance in every of the Twelve Signs; in their Year therefore which is Solar, there are always 365 Days, that is eleven Days more than the Lunar Year.
7. And the Iulian Year which is the Account of all Christendom, doth differ from the other in this; that by reason of the Sun's Excess in Motion above 365 Days, which is 5 Hours, 49 Minutes, it hath a Day intercalated once in 4 Years, and by this intercalation, it is more agreeable to the Motion of the Sun, than the former, and yet there is a considerable difference between them, which hath occasioned the Church of Rome to make some further amendment of the Solar [Page 313] Year, but hath not brought it to that exactness, which might be wished.
8. This intercalation of one Day once in 4 Years, doth occasion the Sunday Letter still to alter till 28 Years be gone about; The Days of the Week which use to be signed by the seven first Letters in the Alphabet, do not fall alike in every Common Year, but because the Year consisteth of 52 Weeks and one Day, Sunday this Year will fall out upon the next Year's Monday, and so forward for seven years, but every fourth year consisting of 52 weeks and two days, doth occasion the Sunday Letter to alter, till four times seven years, that is till 28 years be gone about. This Revolution is called the Cycle of the Sun, taking its name from the Sunday Letter, of which it sheweth all the Changes that it can have by reason of the Bissextile or Leap-year. To find which of the 28 the present is, add nine to the year of our Lord, (because this Circle was so far gone about, at the time of Christs Birth) and divide the whole by 28, what remaineth is the present year, if nothing remain the Cycle is out, and that you must call the last year of the Cycle, or 28.
9. This Intercalation of one day in four years, doth occasion the Letter F to be twice repeated in February, in which Moneth the day is added, that is, the Letter F is set to the 24 and 25 days of that Moneth, and in such a year S. Matthias day is to be observed upon the 25 day, and the next Sunday doth change or alter his Letter, from which leaping or changing, such a year is called Leap-year, aud the number of days in each Moneth is well expressed by these old Verses.
[Page 314] Thirty days hath September, April, June and November.
February hath 28 alone, All the rest have thirty and one.
That this year is somewhat too long, is acknowledged by the most skilful Astronomers, as for the number of days in a year the Emperours Mathematicians were in the right, for it is certain, that no year can consist of more than 365 days, but for the odd hours it is as certain that they cannot be fewer than five, nor yet so many as six; so then the doubt is upon the minutes, 60 whereof do make an hour, a small matter one would think, but how great in the consequence we shall see. The Emperours year being more than 10 minutes greater than the Suns, will in 134 years rise to one whole day, and by this means the Vernal or Spring Equinox, which in Iulius Caesar's time was upon the 24 of March, is now in our time upon the 10 of March, 13 days backward, and somewhat more, and so if it be let alone will go back to the first of March, and first of February, and by degrees more and more backward still.
10. To reform this difference, some of the late Roman Bishops have earnestly endeavoured. And the thing was brought to that perfection it now standeth, by Gregory the Thirteenth, in the year 1582. His Mathematicians, whereof Lilius was the Chief, advised him thus: That considering there had been an Agitation in the [Page 315] Council of Nice somewhat concerned in this matter upon the motion of that Question, about the Celebration of Easter. And that the Fathers of the Assembly, after due deliberation with the Astronomers of that time, had fixed the Vernal Equinox at the 21 of March, and considering also that since that time a difference of ten whole days had past over in the Calendar, that is, that the Vernal Equinox, which began upon the 21 of March, had prevented so much, as to begin in Gregorie's days at the 10 of the same, they advised, that 10 days should be cut off from the Calendar, which was done, and the 10 days taken out of October in the year 1582. as being the moneth of that year in which that Pope was born; so that when they came to the fifth of the moneth they reckoned the 15, and so the Equinox was come up to its place again, and happened upon the 21 of March, as at the Council of Nice.
But that Lilius should bring back the beginning of the year to the time of the Nicene Council and no further, is to be marvelled at, he should have brought it back to the Emperours own time, where the mistake was first entered, and instead of 10, cut off 13 days; however this is the reason why these two Calendars differ the space of 10 days from one another. And thus I have given you an account of the year as it now stands with us in England, and with the rest of the Christian World in respect of the Sun, some other particulars there are between us and them which do depend upon the motion of the Moon, as well as of the Sun, and for the better underderstanding of them, I will also give you a brief [Page 316] account of her revolution. But first I will shew you, how the day of the moneth in any year propounded in one Couutry, may be reduced to its correspondent time in another.
11. Taking therefore the length of the year, to be in several Nations as hath been before declared, if we would find what day of the moneth in one Conntry is correspondent to the day of that moneth given in another, there must be some beginning to every one of these Accounts, and that beginning must be referred to some one, as to the common measure of the rest.
12. The most natural beginning of All Accounts, is the time of the Worlds Creation, but they who could not attain to the Worlds Beginning, have reckoned from their own, as the Romans from the building of Rome, the Greeks from their Olympicks, the Assyrians from Nabonassar, and all Christians from the Birth of Christ: the beginning of which and all other the most notable Epochaes, we have ascertained to their correspondent times in the Julian Period, which Scaliger contrived by the continual Multiplication of those Circles, all in former time of good use, and two of them do yet remain; the Circles yet in use are those of the Sun and Moon, the one, to wit, the Sun, is a Circle of 28 years, and the Circle of the Moon is 19, as shall be shewed hereafter. The third Circle which now serves for no other use than the constituting of the Julian Period, is the Roman Indiction, or a Circle of 15 years; if you multiply 28 the Circle of the Sun, by 19 the Circle of the Moon, the Product is 532, which being multiplied by 15, the Circle of the Roman Indiction, the Product is 7980, the [Page 317] Number of years in the Julian Period: whose admirable condition is to distinguish every year within the whole Circle by a several certain Character, the year of the Sun, Moon, and Indiction being never the same again until the revolution of 7980 years be gone about, the beginning of this Period was 764 Julian years before the most reputed time of the Worlds Creation; which being premised, we will now by Example shew you how to reduce the years of Forreigners to our Julian years, and the contrary.
1. Example.
I desire to know at what time in the Turkish Account, the fifth of Iune in the year of our Lord 1640. doth fall.
The Julian years complete are 1648, and are thus turned into days, by the Table of days in Julian years.
- 1000 Julian years give days
- 365250
- 600 Julian years give days▪
- 219150
- 40 Julian years give days
- 14610
- 8 Years give days
- 2922
- May complete
- 151
- Days
- 5
- The Sum is
- 602088
Now because the Turkish Account began Iuly 16. Anno Christi. 622. you must convert these years into days also.
- 600 Julian years give days
- 219150
- 20 Years give days
- 7305
- 1 Year giveth days
- 365
- Iune complete
- 181
- Days
- 15
- The Sum is
- 227016
- Which being substracted from
- 602088
- There resteth days
- 375072
- 900 Turkish years give days
- 318930
- There resteth
- 56142
- 150 Turkish years give days
- 53155
- There resteth
- 02987
- 8 Turkish years give days
- 2835
- There resteth
- 152
- Giumadi. 4.
- 148
- There resteth
- 4
Therefore the fifth of Iune 1649. in our English Account doth fall in the year 1058. of Mahomet, or the Turkish Hegira, the fourth day of the moneth Giumadi. 11
2. Example.
I desire to know upon what day of our Julian year the 17 day of the moneth in the 1069 year complete of the Persian Account from Ieshagile doth fall.
- The beginning of this Epocha is from the Epocha of Christ in complete days
- 230639
- 1000 Persian years give
- 365000
- 60 Years give
- 21900
- 9 Years give
- 3285
- Chortal complete
- 90
- Days complete
- 16
- The Sum
- 620930
- 1000 Julian years Substracted
- 365250
- There rests
- 255680
- 700 Julian years
- 255675
- There rests
- 5
Therefore it falls out in the Julian year from Christ 1700. the fifth day of Ianuary.
He that understands this may by the like method convert the years of other Epochas, into our Julian years and the contrary.
The Anticipation of the Gregorian Calendar is more easily obtained, for if you enter the Table with the years of Christ complete, you have the days to be added to the time in the Julian Account, to make it answer to the Gregorian, which will be but ten days difference till the year 1700. and then the difference will be a day more, until the year 1800. and so forward three days difference more in every 400 years to come, unless our year shall be reformed as well as theirs.
CHAP. II.
Of the Cycle of the Moon, what it is, how placed in the Calendar, and to what purpose.
THat the Civil Year in use with us and all Christians, doth consist of 365 days, and every fourth year of 366, hath been already shewed, with the return of the Sunday Letter in 28 years. In which time the Moon doth finish her course in the Zodiack no less than twelve times, which twelve Moons, or 354 days, do fall short of the Sun's year, eleven days in every common year, and twelve in the Bissextile or Leap-year.
And by Observation of Meton an Athenian, it was found out about 432 years before Christ, that the Moon in nineteen years did return to be in Conjunction with the Sun on the self same day, and this Circle of nineteen years is called the Cycle of the Moon, which being written in the Calendar against the day in every Moneth, in which the Moon did change, in Letters of Gold, was also called the Golden Number, or from the excellent use thereof, which was at first, only to find the New Moons in every Moneth for ever, but amongst Christians it serveth for another purpose also, even the finding of the time when the Feast of Easter is to be observed. The New Moons by this Number are thus found. In the first year of the Circle, or when the Golden Number is 1, where the Number 1 was set in the Calendar in any Moneth, that day is New Moon, in [Page 319] the second Year where you find the golden Number 2, in the third Year where you find the golden Number 3, and so forward till the whole Circle be expired; then you must begin with one again, and run through the whole Circle as before.
2. And the reason why the Calendar begins with the golden Number 3, not 1, is this. The Christians in Alexandria had used this Circle of the Moon two Years before the Nicene Council. And in the first of these Years the new Moon next to the Vernal Equinox was upon the 27th Day of the Egyptian month Phamenoth answering to the 23d of our March, against that Day therefore they placed the golden Number 1. And because there are 29 Days and a half from one new Moon to another, they made the distance between the new Moons to be interchangeably 29 and 30 Days, and so they placed the same golden Number against the 26 Day of Phurmuthi the Month following, and against the 26 Day of the Month Pachon and so forward, and upon this ground by the like progression was the golden Number set in the Roman Calendar; and so the golden Number 1 by their example was set against March 23. April 21. Iune 19. Iuly 19. August 17. September 16. October 15. November 14. December. 13. But then because in the following Year the golden Number was 2. reckoning 30 Days from the 13th of December, the golden Number 2 was set to Innuary 12. February 10. March 12. April 10. May 10. Iune 8. Iuly 8. August 6. September 5. October 4. November 3. December. 2. From whence reckoning 13 Days as before, the golden Number 3 [Page 320] comes in course for the third Year to be set against the first of Ianuary.
But that you may know how the golden Number comes to be distributed in the Calendar according to the form in which it now is, you must consider that in 19 Solar Years there are not only 228 Lunar Months or 12 times 19 Lunar Months but 235 for the 11 Days which the common Solar Year doth exceed the Lunar, do in 19 Years arise to 209 Days, out of which there may be appointed 7 Months, 6 whereof will contain 30 Days apiece, and one Month 29 days; and these 7 Months are called Embolismical Months, because by a kind of injection or interposition they are reckoned in some of the 19 Years. And those Years in which they are reckoned are called Embolismical Years, to distinguish them from the common Years which always contain 354 Days, whereas 6 of these Embolismical Years do each of them contain 384 Days, and the seventh Embolismical Year in which the Month of 29 Days is reckoned, doth contain 383 Days.
3. The Embolismical Years in the Cycle of the Moon are properly these Seven. 3, 6, 9, 11, 14, 17, 19. because in the third Year 11 Days being thrice reckoned do amount to 33 Days, that is one Month of 30 Days and 3 Days over. Again in the sixth Year the 11 Days which the Solar exceed the Lunar, being thrice numbred, do amount to 33 Days. which with the 3 Days formerly reserved do make 36 Days, that is one Month of 30 Days and 6 Days over. Again in the Ninth Year there are also 33 Days, to which the 6 Days reserved being added, there will arise one Month more and 9 Days over. But in [Page 321] the Eleventh Year twice 11 Days being added to the 9 Days reserved, do make 31 Days, that is, one Month of 30 days and one day over, which being added to the supernumerary days in the fourteenth Year do make another Month of 30 Days and 4 Days over, and these being added to the supernumerary Days in the sevententh Year do make another Month of 30 and 7 Days over, and these 7 Days being added to the 22 supernumerary Days in the Ninteenth Year of the Moons Cycle do make another Month of 29 Days.
4. But because there are 6939 Days and 18 Hours in 19 Solar Years, that is, 4 Days 18 Hours more then in the common and Embolismical Lunar Years, in which the excess between the Lunar and the Solar Year is supposed to be no more then 11 Days in each Year, whereas in every fourth Year the excess is one Day more, that is, 12 Days, that is, in 16 Years 4 Days, and in the remaining 3 Years three fourths of a day more. And that the new Moons after 19 Lunar Years or 235 Lunations do not return to the same days again, but want almost 5 days, it is evident that the civil Lunations do not agree with the Astronomical and that there must be yet some kind of intercalation used.
5. Now therefore in distributing the golden Number throughout the Calendar. If the new Moons should interchangeably consist of 30 and 29 days, and so but 228 Lunations in 19 Years; we might proceed in the same order in which we have begun, and by which as hath been shewed the third Year of the Golden Number falls upon the Calends of Ianuary. But for as much as there are first six Lunations of 30 days apiece and [Page 322] one of 29 days to be interposed, therefore there must be 6 times 2 Lunations together consisting of 30 days and once three Lunations of 29 days. And that respect may be also had to the Bissextile days, although they are not exprest in the Calendar, that Lunation which doth contain the Bissertile day, if it should have been 29 days, it must be 30, if it should have consisted of 30 days it must consist of 31.
6. And because it was thought convenient, as hath been shewed, to begin with the third Year of the Cycle of the Moon, because the Golden Number 3 is set to the Calends of Ianuary, therefore in this Cycle the Embolismical Years are, 2, 5, 8, 11, 13, 16, 19. But yet that it may appear, that these Years are in effect the same, as if we had begun with the first Year of the Golden Number, save only that the eighth Year instead of the ninth is to be accounted Embolismical, I have added the Table follwing, in which it is apparent that the former Embolismical years do agree with these last mentioned.
7. But as I said before, it was thought more convenient to begin the account from the number 3 set to the Calends of Ianuary, because by so reckoning 30 and 29 days to each Lunation interchangeably, the same Number 3 falls upon Ianuary 31. March 1, and 31. April 29. May 29. Iune 27. Iuly 27. August 25. September 14. October 23. November 22. December 21. As if the Lunar years were compleated upon the 20 of December there remain just 11 Days, which the Solar years doth exceed the Lunar.
8. And by ranking on and accounting 4 for the Golden Number of the next year, you will [Page 323] find it set on Ianuary 20, February 18, March 20, April 18, May 18, Iune 16, Iuly 16, August 14, September 13, Octob. 12, Novemb. 11, Decemb. 10.
| Cycle of the Moon. | Cycle of the Moon. | Embolismical Years. | Number of Days. |
| 1 | 3 | 354 | |
| 2 | 4 | 354 | |
| 3 | 5 | Embol. | 384 |
| 4 | 6 | 354 | |
| 5 | 7 | 354 | |
| 6 | 8 | Embol. | 384 |
| 7 | 9 | 354 | |
| 8 | 10 | 354 | |
| 9 | 11 | Embol. | 384 |
| 10 | 12 | 354 | |
| 11 | 13 | Embol. | 384 |
| 12 | 14 | 354 | |
| 13 | 15 | 354 | |
| 14 | 16 | Embol. | 384 |
| 15 | 17 | 354 | |
| 16 | 18 | 354 | |
| 17 | 19 | Embol. | 384 |
| 18 | 1 | 354 | |
| 19 | 2 | Embol. | 384 |
9. But in going on, and taking 5 for the Golden Number in the third year, we must remember that that is an Embolismical Year, and therefore that somewhere there must be 2 Months together of 30 days. And for this reason the Golden Number 5, is set to Ianuary 9, February 7, March 9, April 7, May 7, Iune 5, Iuly 5, August 3, September 2, as also upon the second day of October, and not upon the first, that so there may be 2 Lunations together of 30, and the same Number 5 is also set to the thirty first of October, to make the Lunation to consist of 29 days, and to the thirtieth of November instead of the twenty ninth, that so a Lunation of [Page 324] 30 may again succeed as it ought.
10. In like manner in the sixth Year, having gone through the fourth and fifth as common years, you may see the Golden Number 8 set to the fifth of April, which should have been upon the fourth, and in the ninth Year the Golden Number 11 is set to the second of February which should have been upon the first.
And there is a particular reason, for which these numbers are otherwise placed from the eighth of March to the fifth of April, namely, that all the paschal Lunations may consist of 29 days: For thus from the eighth of March to the sixth of April, to both which days the Golden Number is 16, there are but 29 days. And from the ninth of March to the seventh of April, to both which days the Golden Number is 5, there are also 29 days, and so of the rest till you come to the fifth of April, which is the last Paschal Lunation, as the eighth of March is the first, but at any other time of the Year, the length of the Month in the Embolismical Year, may be fixed as you please.
12. And in this manner in the 17 years, in which the lunations of the whole Circle are finished, and in which the Golden Number is 19, the Month of Iuly is taken at pleasure, to the thirtieth day whereof is set the Golden Number 19, which should have been upon the thirty first, and the same Number being notwithstanding placed upon the twenty eighth of August, that by the two Lunations of 29 days together, it might be understood, that the seventh Embolismical Month consisting of 29 days is there inserted, instead of a Month of 30 days. In which place the Embolismical or leaping Year of the Moon may [Page 325] plainly be observed for that year is one day less than the rest, which the Moon doth as it were pass over. The which one day is again added to the 29 days of the last Month, that we may by that means come, as in other Years, to the Golden Number, which sheweth the New Moon in Ianuary following. And for this reason the Epact then doth not consist of 11 but of 12 days. And thus you see the reason, for which the Golden Numbers are thus set in the Calendar as here you see. In which we may also observe, that every following Number is made by adding 8 to the Number preceding, and every preceding Number is also made by adding 11 to the Number next following, and casting away 19 when the addition shall exceed it.
For Example, if you add 8 to the Golden Number 3 set against the first of Ianuary, it maketh 11, to which add 8 more and it maketh 19, to which adding 8 it maketh 27, from which substracting 19 the remainer is 8, to which again adding 8, the sum is 19, to which adding 8 the sum is 24, from which deducting 19 the remainer is 5, and so of the rest. In like manner receding backward, to the 5 add 11 they make 16, to the 16 add 11 they make 27, from which deducting 19 the remainer is 8, to which 11 being added the same is 19, to which 11 being added the sum is 30, from which deducting 19 the remainer is 11, to which 11 being added the sum is 22, from which deducting 19 the remainer is 3. And by this we may see that every following number will be in use 8 years after the preceding, and every preceding Number will be in use 11 years after the following, that is, the same will return to be in use after [Page 326] 8 Years and 11, and the other after 11 Years and 8, or once in 19 years.
CHAP. III.
Of the Vse of the Golden Number in finding the Feast of Easter.
THe Cycle of the Moon or Golden Number is a circle of 19 years, as hath been said already, which being distributed in the Calendar as hath been shewn in the last Chapter, doth shew the day of the New Moon for ever; though not exactly: But the use for which it was chiefly intended, was to find the Paschal New Moons, that is, those new Moons on which the Feast of Easter and other moveable Feasts depend. To this purpose we must remember,
1. That the vernal Equinox is supposed to be fixed to the twenty first day of March.
2. That the fourteenth day of the Moon on which the Feast of Easter doth depend, can never happen before the Equinox; though it may fall upon it or upon the day following.
3. That the Feast of Easter is never observed upon the fourteenth day of the Moon, but upon the Sunday following; so that if the fourteenth day of the Moon be Sunday, the Sunday following is Easter day.
4. That the Feast of Easter may fall upon the fifteenth day of the Moon, or upon any other day unto the twenty first, inclusively.
5. That the Paschal Sunday is discovered by [Page 327] the proper and Dominical Letter for every Year The which may be found as hath been already declared, or by the proper Table for that purpose. Hence it followeth,
1. That the New Moon immediately preceding the Feast of Easter, cannot be before the eighth day of March, for if you suppose it to be upon March 6, the Moon will be 14 days old March 19, which is before the Equinox, contrary to the second Rule before given, and upon the seventh day of March there is no Golden Number fixed; and therefore the Golden Number 16, which standeth against March 8, is the first by which the Paschal New Moon may be discovered.
2. It followeth hence, That the last Paschal New Moon cannot happen beyond the fifth day of April, because all the 19 Golden Numbers are expressed from the eighth of March to that day. And if a New Moon should happen upon the sixth of April, there would be two Paschal New Moons that year, one upon the eighth of March and another upon the sixth of April, the same Golden Number 16 being proper to them both, but this is absurd because Easter cannot be observed twice in one year.
3. It followeth hence, That the Feast of Easter can never happen before the twenty second day of March, nor after the twenty fifth day of April: For if the first New Moon be upon the eighth of March, and that the Feast of Easter must be upon the Sunday following the fourteenth day of the Moon; it is plain that the fourteenth day of the Moon must be March 21 at the soonest: So that supposing the next day to be Sunday, Easter cannot [Page 228] not be before March the twenty second. And because the fourteenth day of the last Moon falleth upon the eighteenth day of April, if that day be Saturday, and the Dominical Letter D, Easter shall be upon the nineteenth day, but if it be Sunday, Easter cannot be till the twenty fifth.
4. It followeth hence, That although there are but 19 days, on which the fourteenth day of the Moon can happen, as there are but 19 Golden Numbers, yet there are 35 days from the twenty second of March to the twenty fifth of April, on which the Feast of Easter may happen, because there is no day within those Limits, but may be the Sunday following the fourteenth day of the Moon. And although the Feast of Easter can never happen upon March 22, but when the fourteenth day of the Moon is upon the twenty first, and the Sunday Letter D, nor upon the twenty fifth of April, but when the fourteenth day of the Moon is upon April 18, and the Dominical Letter C. Yet Easter may fall upon March 23, not only when the fourteenth day of the Moon is upon the twenty second day which is Saturday, but also if it fall upon the twenty first which is Friday. In like manner Easter may fall upon April 24, not only when the fourteenth day of the Moon is upon the eighteenth day which is Monday, but also if it happen upon the seventeenth being Sunday. And for the same reason it may fall oftner upon other days that are further distant from the said twenty second of March and twenty fifth of April.
5. It followeth hence, That the Feast of Easter may be easily found in any Year propounded: For the Golden Number in any Year being given, if [Page 329] you look the same between the eighth of March and fifth of April both inclusively, and reckon 14 days from that day, which answereth to the Golden Number given, where your account doth end is the fourteenth day of the Moon: Then consider which is the Dominical Letter for that Year, and that which followeth next after the fourteenth day of the Moon is Easter day. Example, In the year 1674 the Golden Number is 3, and the Sunday Letter D, which being sought in the Calendar between the aforesaid limits, the fourteenth day of the Moon is upon April the thirteenth, and the D next following is April 19. And therefore Easter day that Year is April 19. Otherwise thus.
6. Thus the Feast of Easter may be found in the Calendar, and from thence a brief Table shewing the same, may be extracted in this manner. Write in one Column the several Golden Numbers in the Calendar from the eighth of March to the fifth of April, in the same order observing the same distance. In the second Column set the Dominical Letters in number 35 so disposed, as that no Dominical Letter may stand against the Golden Number 16, but setting the Letter D against the Golden Number 5, write the rest in [Page 330] this order. E, F, G, A, B, &c. and when you come to the Golden Number 8, set the Letter C, and there continue the Letters till you come to C again, because when the Golden Number is 16, which in the Calendar is set to the eighth day of March, is new Moon, and the fourteenth day of that Moon doth fall upon the twenty first, to which the Dominical Letter is C, upon which the Feast of Easter cannot happen; and therefore in the third Column containing the day in which the Feast of Easter is to be observed, is also void. But in the next place immediately following, to wit, against the letter D is set March 22, because if the fourteenth day of the Moon shall fall upon the twenty first of March being Saturday, the next day being Sunday, shall be the Feast of Easter.
To the Letters following, E, F, G, A, B, &c. are set 23, 24, 25, and so orderly to the last of March, and so forward till you come to the twenty fifth of April, by which Table thus made, the Feast of Easter may be found until the Calendar shall be reformed.
For having found the Golden Number in the first Column, the Dominical Letter for the Year next after it, doth shew the Feast of Easter, as in the former Example, the Golden Number is 3 and the Dominicall Letter D, therefore Easter day is upon April 19. The other moveable Feasts are thus found.
Advent Sunday is always the nearest Sunday to St. Andrews, whether before or after.
[Page 331] Septuagesima Sunday is Nine Weeks before Easter.
Sexagesima Sunday is Eight Weeks before Easter.
Qainquagesima Sunday is Seven Weeks before Easter.
Quadragesima Sunday is Six Weeks before Easter.
Rogation Sunday is five Weeks after Easter.
Ascension day is Forty Days after Easter.
Whitsunday is Seven Weeks after Easter.
Trinity Sunday is Eight Weeks after Easter.
| G. N. | D. L. | Easter. |
| XVI | ||
| V | D | 22 March |
| E | 23 | |
| XIII | F | 24 |
| II | G | 25 |
| A | 26 | |
| X | B | 27 |
| C | 28 | |
| XVIII | D | 29 |
| VII | E | 30 |
| F | 31 | |
| XV | G | 1 April |
| IV | A | 2 |
| B | 3 | |
| XII | C | 4 |
| I | D | 5 |
| E | 6 | |
| IX | F | 7 |
| G | 8 | |
| XVII | A | 9 |
| VI | B | 10 |
| C | 11 | |
| XIV | D | 12 |
| III | E | 13 |
| F | 14 | |
| XI | G | 15 |
| A | 16 | |
| XIX | B | 17 |
| VIII | C | 18 |
| D | 19 | |
| E | 20 | |
| F | 21 | |
| G | 22 | |
| A | 23 | |
| B | 24 | |
| C | 25 |
CHAP. IV.
Of the Reformation of the Calendar by Pope Gregory the Thirteenth; and substituting a Cycle of Epacts in the room of the Golden Number.
HItherto we have spoken of the Calendar which is in use with us, we will now shew you for what reasons it is alter'd in the Church of Rome, and how the Feast of Easter is by them observed.
The Year by the appointment of Iulius Caesar consisting of 365 days 6 hours, whereas the Sun doth finish his course in the Zodiack, in 365 days 5 hours 49 minutes or thereabouts, it cometh to pass that in 134 Years or less, there is a whole day in the Calendar more than there ought; in 268 years 2 days more; in 4002 years 3 days: and so since Iulius Caesar's time the vernal Equinox hath gone backward 13 or 14 days, namely from the 24 of March to the tenth. Now because the Equinox was at the time of the Nicene Council upon the twenty first of March, when the time for the observing of Easter was first universally established, they thought it sufficient to bring the Equinox back to that time, by cutting off 10 days in the Calendar as hath been declared, and to prevent any anticipation for the time to come, have appointed, that the Leap-year shall be thrice omitted in every 400 Years to come, and for memory sake, appointed the first omission to be accounted from the Year 1600, not from 1582, in which the reformation was made, because it was not only near the time, in which the emendation was begun, but also because the Equinox has not fully made an anticipation of 10 days from the [Page 333] place thereof, at the time of the Nicene Council, which was March 21.
The Years then 1700, 1800, 1900, which should have been Bissextile Years, are to he accounted common years, but the Year 2000 must be a Bissextile: In like manner the Years 2100, 2200, 2300, shall be common years, and the Year 2400 Bissextile, and so forward.
2. Again, because it was supposed that the Cycle of the Moon, or Golden Number was so fixed, that the new and full Moons would in every 19 years return to the same days again; whereas their not returning the same hours, but making an anticipation of one hour 27 minutes or thereabouts, it must needs be that in 17 Cycles or little more than 300 Years, there would be an anticipation of a whole day. And hence it is evident that in 1300 Years since the Nicene Council, the New and Full Moons do happen more than 4 days sooner than the Cycle of the Moon or Golden Number doth demonstrate: Whence also it comes to pass, that the fourteenth day of the Moon by the Cycle is in truth the eighteenth day, and so the Feast of Easter should be observed not from the fifteenth day of the Moon to the twenty first, but from the nineteenth to the twenty fifth.
3. That the Moon therefore being once brought into order, might not make any anticipation for the time to come, it is appointed that a Cycle of 30 Epacts should be placed in the Calendar instead of the Golden Number, answering to every day in the Year; to shew the New Moons in these days, not only for 300 Years or thereabouts, but that there might be new Epacts without [Page 334] altering the Calendar, to perform the same thing upon other days as need shall require.
4. For the better understanding whereof, to the Calendar in use with us, we have annexed the Gregorian Calendar also: In the first Column whereof you have 30 numbers from 1 to 30, save only that in the place of 30 you have this Asterisk *, But they begin with the Calends of Ianuary, and we continued and repeated after a Retrograde order in this manner, *29, 28, 27, &c. and that for this cause especially, that the number being given which sheweth the New Moons in every Month for one Year, you might by numbring 11 upwards exclusively find the number which will shew the New Moons the Year following, to wit, the Number which falleth in the eleventh place.
5. And these Numbers are called Epacts, because they do in order shew those 11 days, which are yearly to be added to the Lunar Year consisting of 354 days, that it may be in conformity with the Solar Year consisting of 365 days. To this purpose, as hath been said concerning the Golden Number, these Epacts being repeated 12 times, and ending upon the twentieth day of December, the same Numbers must be added to the 11 remaining days, which were added to the first 11 days in the Month of Ianuary.
6. And because 12 times 30 do make 360, whereas from the first of Ianuary to the twentieth of December inclusively, there are but 354 days, you must know that to gain the other six days, the numbers 25 and 24 are in every other Month both placed against one day, namely, to February 5, April 5, Iune 3, August 1, September 29, and [Page 335] November 27. But why these two Numbers are chosen rather then any other, and why in these 6 Months the number 25 is sometimes writ to XVI, sometimes to XXV in a common character, and why the number 19 is set to the last day of December in a common Character, shall be declared hereafter.
7. Here only note that this Asterisk * is set instead of the Epact 30, because the Epact shewing the Number of days which do remain after the Lunation in the Month of December, it may sometimes fall out that 2 Lunations may so end, that the one may require 30 for the Epact, and the other 0, which would, if both were written, cause some inconveniences, and therefore this * Asterisk is there set, that it might indifferently serve to both. And the Epact 29 is therefore set to the second day of Ianuary, because after the compleat Lunation in the second of December there are 29 days, and for the like reason the Epact 28 is set against the third of Ianuary, because after the compleat Lunation in the third of December there are then 28 days over, and so the rest in order till you come to the thirtieth of Ianuary, where you find the Epact 1. because after the compleat Lunation on the thirtieth day there is only one day over.
8. And besides the shewing of the New Moons in every Month, which is and may be done by the Golden Number, the Epacts have this advantage, that they may be perpetual and keep the same place in the Calendar in all future ages, which can hardly be effected with the Golden Number, for in little more then 700 years, the New Moons do make an anticipation of one day, and then it [Page 336] will be necessary to set the Golden Number one degree backward, and so the Golden Number which at the time of the Nicene Council was set to the first of Ianuary, should in 300 years be set to the last of December, and so of the rest, but the Epacts being once fixed shall not need any such retraction or commutation. For as often as the New Moons do change their day either by Anticipation or by Suppression of the Bissextile year, you shall not need to do any more than to take another rank of 19 Epacts, insteed of those which were before in use. For instance, the Epacts which are and have been in use in the Church of Rome since the year of reformation 1582, and will continue till the year 1700, are these 10 following 1. 12. 23. 4. 15. 26. 7. 18. 29. 10. 21. 2. 13. 24. 5. 16. 27. 8. 19. And from the year 1700 the Epacts which will be in use are these. * 11. 22. 3. 14. 25. 6. 17. 28. 9. 20. 1. 12. 23. 4. 15. 26. 7. 18. and shall continue not only to the year 1800, but from thence until the year 1900 also; and although in the year 1800 the Bissextile is to be suppressed, yet is there a compensation for that Suppression, by the Moons Anticipation. To make this a little more plain, the motion of the Moon, which doth occasion the change of the Epact, must be more fully considered.
CHAP. V.
Of the Moons mean Motion, and how the Anticipation of the New Moons may be discovered by the Epacts.
THe Moon according to her middle motion doth finish her course in the Zodiack in 29 days, 12 hours 44 minutes, three seconds or thereabout, and therefore a common Lunar year doth consist of 354 days, 8 hours, 48 minutes, 38 seconds and some few thirds, but an Embolismical year doth consist of 383 days, 21 hours, 32 minutes, 41 seconds and somewhat more; and therefore in 19 years it doth exceed the motion of the Sun 1 hour, 27 minutes, 33 seconds feré.
2. Hence it cometh to pass, that although the New Moons do after 19 years return to the same days; yet is there an Anticipation of 1 hour, 27 minutes, 33 seconds. And in twice 19 years, that is, in 38 years, there is an Anticipation of 2 hours, 55 minutes, 6 seconds, and after 312 years and a half, there is an Anticipation of one whole day and some few Minutes. And therefore after 312 years no new Moon can happen upon the same day it did 19 years before, but a day sooner. Hence it comes to pass that in the Julian Calendar, in which no regard is had to this Anticipation, the New Moons found out by the Golden Number must needs be erroneous, and from the time of the Nicene Council 4 days after the New Moons by a regular Computation.
3. And hence it follows also, that if the Golden Number, after 312 were upon due consideration [Page 338] removed a day forwarder or nearer the beginning of the Months, they would shew the New Moons for 312 years to come. And being again removed after those years, a day more would by the like reason do the same again. But it was thought more convenient so to dispose 30 Epacts, that they keeping their constant places, 19 of them should perform the work of the Golden Number, until by this means there should be an Anticipation of one day. And when such an Anticipation should happen, those 19 Epacts being let alone, other 19 should be used, which do belong to the preceding day, without making any alteration in the Calendar.
4. And if this Anticipation would do the whole work, nothing were more plain, then to make that commutation of the 19 Epact once in 312 years: but because the detraction of the Bissextile days doth variously interpose and cause the 19 Epacts sometimes to be changed into these that do precede, sometimes into these that follow, sometimes into neither, but to continue still the same; therefore some Tables are to be made, by which we may know, when the commutation was to be made and into what Epacts.
4. First therefore there was made a Table called Tabula Epactarum Expansa, in this manner.
First on the top were placed the 19 Golden Numbers in order, beginning with the Number 3, which in the old Calendar is placed against the Calends of Ianuary, and under every one of these Golden Numbers there are placed 30 Epacts all constituted from the lowest number in the first rank in which the Epact is 1, and in that first rank the Golden Number is 3, the rest from [Page 339] thence towards the right Hand are made by the constant addition of it, and the casting away of 30, as often as they shall exceed that number, only when you come to the 27, the Epact under the Golden Number 19, there must be added 12 instead of 11, that so the Epact following may be 9 not 8, for the Reasons already given in this Discourse concerning the Golden Number and Embolismical years. And this rank being thus made, the other Epacts are disposed in their natural order ascending upwards, and the number once again resumed after the Epact 30 or rather this Asterisk * set in the place thereof: only observe that under the Golden Number 12. 13. 14. 15. 16. 17. 18. 19. in the place of XX there is yet 25 in the common Character. And to the Epacts under the Golden Number 19, 12 must still be added to make that Epact under the Golden Number 1. As was said before concerning the lowest Rank.
5. And on the left hand of these Epacts before those under the Golden Number 3. are set 30 Letters of the Alphabet, 19 in a small Character, and 11 in a great, in which some are passed by, for no other reason save only this, that their similitude with some of the small Letters, should not occasion any mistake in their use, which shall be shewed in its place.
6. Besides this Table there was another Table made which is called Tabula AEquationis Epactarum, in which there is a series of years, in which the Moon, by reason of her mentioned anticipation doth need AEquation, and in which the number of Epacts signed with the letters of the Alphabet, are to be changed; being otherwise AEquated [Page 340] where it needeth, by the suppression of the Bissextile days.
7. But it supposeth, that it was convenient to suppress the Bissextiles once only in 100 years; and the Moon to be aequated, or as far as concerns her self, the rank of Epacts to be changed, once only in 300 years, and the 12 years and a half more, to be referred till after the years 2400, they do amount unto 100 years, and then an aequation to be made: but then it must be made by reason of the interposing this hundred not in the three hundredth but the hundredth year. Moreover this aequation is to be made as in referece to the Moon only, because as the suppression of the Bissextiles intervene, the order of changing the ranks of Epacts is varied, as shall be shewed hereafter.
8. Again this Table supposeth, that seeing the New Moon at the time of the Nicene Council was upon the Calends of Ianuary, the golden Number 3 being there placed, that it would have been the same if the Epact * had been set to the same Calends, that is if the Epacts had been then in use. And therefore at that time the highest or last rank of Epacts was to be used, whose Index is P, and then after 300 years, the lowest or first rank should succeed, whose Index is a, (for the letters return in a Circle) and after 300 years more, the following rank whose Index is b and so forward; but that it is conceived, that the New Moon in the Calends of Ianuary, is more agreeable to the year of Christ 500, than the time of the Nicene Councel; and therefore as if the rank of Epacts under the letter l were sutable to the year 500, it seemed [Page 341] good to make use of that rank under the letter a in the year of Christ 800, and those under the letter b, in the year 1100, and those under the letter e in the year 1400.
9. Which being granted, because in the year 1582, ten days were cut off from the Calendar, we must run backward, or in an inverted order count 10 series, designed, suppose, by the letters b. a. P. N. M. H. G. F. E. D. so that from the year 1582 the series of Epacts whose literal Index is D, is to be used, and this is that rank of Epacts which is now used in the Church of Rome.
10. And therefore as if this Table had its beginning from that year; the first number in the second column is 1582, and then in order under it. 1600. 1700. 1800. 1900. 2000. &c. And in the third Column every fourth hundred year is marked for a Bissextile, that is, 1600. 2400. 2800, &c. and in the fourth Column to every three hundreth▪ Year is set this Character C, to shew in what year the Moon by her Anticipation of one day, doth need aequation; but in the year 1800 the double character is set CC, to signify that then another hundred years are gotten by the 12 years and a half reserved, besides and above the other 300 years; and this character is also set to the years 4300. 6800, and for the same reason.
But in the first Column, or on the left hand of these years are placed the Letters or Indices of those ranks of Epacts in the former Table, which are to be used in those years and when the Letters are charged. Thus against the year 1600 the Letter D is continued, to shew that from that [Page 342] year, to the year 1700 the rank of Epacts is still to be used, which do belong to that Letter. And for as much as the Letter C is set to the year 1700, it sheweth that that rank of Epacts is then to be used, which do belong thereto, and so of the rest.
11. The reason why these Letters in the first Column are sometimes changed in 100 years, sometimes in 200, sometimes not in less then 300 Years, and that they are sometimes taken forward, sometimes backward, according to the order of the Alphabet, is because the suppression of the Bissextiles do intervene with the lunar aequation: for if the Bissextile were only to be suppressed, in these 300 or sometimes 400 years, in which the Moon needeth aequation, the rank of Epacts in that case would need no commutation, but would continue the same for ever; and the golden Number would have been sufficient, if the suppression of the Bissextile, and anticipation of the Moon, did by a perpetual compensation cause the new Moons still to return to the same days: but because the Bissextile is ofttimes suppressed, when the Moon hath no aequation, the Moon hath sometimes an aequation when the Bissextile is not suppressed, sometimes also both are to be done and sometimes neither; all which varieties may yet be reduced to these three Rules.
1. As often as the Bissextile is suppressed without any aequation of the Moon, then the letter which served to that time shall be changed to the next below it contrary to the order of the Alphabet. And the new Moons shall be removed [Page 343] one day towards the end of the Year.
2. As often as the Moon needeth aequation, without suppression of the Bissextile, then the Letter which was in use to that time shall be changed to the next above it according to the order of the Alphabet, that the New Moons may again return one day towards the beginning of the year.
3. As often as there is a Suppression and an aequation both, or when there is neither, the Letter is not changed at all but that which served for the former Centenary, shall also continue in the succeeding; because the compensation so made, the New Moons do neither go forward nor backward, but happen in the compass of the same days.
1. And this is enough to shew for what reason the letters are so placed in the Table, as there you see them: for in the year 1600 the Bissextile being neither suppressed, nor the Moon aequated, the letter D used in the former Centenary or in the latter part thereof from the year 1582, is still the same.
In the year 1700, because there is a suppression, but no aequation, the commutation is made to the Letter C descending.
In the Year 1800, because there is both a suppression and an aequation, the same letter C doth still continue.
In the Year 2400, because there is an aequation and no suppression, there is an ascension to the Letter A.
And thus you see not only the construction of this Table, but how it may be continued to any other Year, as long as the World shall last.
[Page 344] 12. And by these two Tables we may easily know which rank of the 30 Epacts doth belong to, or is proper for any particular age: for as in our age, that is, from the Year 1600 to the Year 1700 exclusively, that series is proper whose Index is D. Namely, 23, 4, 15, 26, &c. so in the two Ages following, that is, from the Year 1700 to the Year 1900 exclusively, that series is proper whose Index is C, namely these, 22, 3, 14, 25. and in the three ages following thence, that is from the Year 1900 to the Year 2100 exclusively, that series is proper whose Index is B, namely these, 21, 2, 13, 24, &c. And so for any other.
Hence also it may be known, which of the 19 doth belong to any particular Year, for which no more is necessary, than only to know the Golden Number for the year given, which being sought in the head of the Table, and the Index of that Age in the side, the common Angle, or meeting of these two, will shew you the Epact desired: As in the year 1674 the Golden Number is 3 and the Index D; therefore in the common Angle I find 23 for the Epact that year, and sheweth the New Moons in every Month thereof.
And here it will not be unseasonable to give the reason, for which the Epact 25 not XXV is written under the Golden Numbers 12, 13, 14, 15, 16, 17, 18, 19. namely, because the ranks of Epacts, which under these greater Numbers hath this Epact 25, hath also XXIV, it would follow that in these Ages in which any of these Ranks were in use, the New Moon in 19 years will happen twice upon the same days; in those six Months in which the Epacts XXV and XXIV are set to the same day: Whereas the New Moons do not happen [Page 245] on the same day till 19 years be gone about. To avoid this inconvenience, the Epact 25 not XXV is set under these great numbers, and the Epact 25 is in the Calendar, in these Months set with the Epact XXVI, but in the other Months with the Epact XXV.
14. Hence it cometh to pass, 1. That in these Years the Epacts 25 and XXIV do never meet on the same day. 2. That there is no danger that the Epacts 25 and XXVI should in these 6 Months cause the same inconvenience, seeing that the Epacts 25 and XXVI are never both found in the same Rank. 3. That the Epact 25 may in other Months without inconvenience be set to the same day with the Epact XXVI, because in these there is no danger of their meeting with the Epact XXIV on the same days. 4. That there is no fear that the Epacts XXV and XXIV being set on the same days, should in future Ages cause the same inconvenience, because the Epacts XXV and XXIV are not found together in any of the other Ranks. But that either one or both of them are wanting. Besides, when one of these Epacts is in use, the other is not, and that only which is in use is proper to the day. As in this our Age until the Year 1700 the Epacts in use are those in the rank whose Index is D. In which these two XXIV and XXV are not both found. And in the two following Ages, because the rank of Epacts in use is that whose Index is C, in which there is the Epact XXV, not XXIV, the New Moons are shewed by the Epact XXV not by XXIV. But because in three following Ages, the rank of Epacts in use is that whose Index is B, in which 25 and XXIV are both found, the New [Page 346] Moons are shewed by the Epact XXIV when the golden Number is 6. And by the Epact 25 when the golden Number is 17, and not by the Epact XXV.
15. And if it be asked why the Epact 19 in the common Character is set with the Epact XX against the last day of December; know that for the reasons before declared, the last Embolismical Month within the space of 19 years, ought to be but 29 days and not 30, as the rest are; and therefore when the Epact 19 doth concur with the golden Number 19, the last Month or last Lunation beginning the second of December, shall end upon the 30 and not upon the 31 of that Month, and the New Moon should be supposed to happen upon the 31 under the same Epact 19, that 12 being added to 19 and not 11, you may have one for the Epact of the year following, which may be found upon the 30 of Ianuary, as if the Lunation of 30 days had been accomplished the Day before.
CHAP. VI.
How to find the Dominical Letter and Feast of Easter according to the Gregorian account.
HAving shewed for what reason, and in what manner the Epacts are substituted in the place of the golden Number, and how the New Moons may be by them found in the Calendar for ever; I shall now shew you how to find the Feast of Easter and the other moveable [Page 347] Feasts according to the Gregorian or new account; and to this purpose I must first shew you how to find the Dominical Letter, for that the Cycle of 28 years will not serve the turn, because of the suppression of the Bissextile once in a hundred years, but doth require 7 Cycles of 28 years apeice. The first whereof begins with CB, and endeth in D. The second begins with DC, and endeth in E. The third begins with ED, and endeth in F &c. The first of these Cycles began to be in use 1582, in which year the dominical Letter according to the Julian account was G, but upon the fifteenth day of October, that Year was changed to C: for the fifth of October being Friday and then called the fifteenth, the Letter A became Friday, B Saturday, and C Sunday, the remaining part of the year, in which the Cycle of the Sun was 23, and the second after the Bissextile or leap Year, and so making C, which answereth to the fifteenth year of that Circle, to be 23, the Circle will end at D; and consequently CB, which in the old account doth belong to the 21 year of the Circle, hath ever since been called the first, and so shall continue until the year 1700, in which the Bissextile being suppressed, the next Cycle will begin with DC as hath been said already. Under the first rank or order of Dominical Letters are written the years 1582 and 1600, under the second 1700, under the third 1800, under the fourth 1900 and 2000, under the fifth 2100, under the sixth 2200 and under the seventh 2300 and 2400. And again under the first Order, 2500, under the second 2600, under the third 2700 and 2800, and so forward as far as you [Page 348] please, always observing the same order, that the 100 Bissextile years may still be joyned with the not Bissextile immediately preceding.
1. And hence it appears, that the seven orders of Dominical Letters, are so many Tables, successively serving all future Generations. For as the first Order serveth from the year 1582 and 1600 to the year 1700 exclusively, and the second Order from thence to the year 1800 exclusively, so shall all the rest in like manner which here are set down, and to be set down at pleasure. And hence the Dominical Letter or Letters may be found for any year propounded, as if it were required to find the dominical Letter for the year 1674, because the year given is contained in the centenary 1600. I find the Cycle of the Sun by the Rule already given to be 3. In the first order against the number 3, I find G for the Sunday Letter of that year, in like manner because the year 1750 is contained under the Centenary 1700, the Cycle of the Sun being 27, I find in the second rank the Letter D answering to that Number, and that is the Dominical Letter for that year, and so of the rest.
3. Again for as much as the fifth Order is the same with that Table, which serves for the old account, therefore that order will serve the turn for ever where that Calendar is in use, and so this last will be of perpetual use to both the Calendars.
4. Now then to find the time in which the Feast of Easter is to be observed, there is but little to be added to that which hath been already said concerning the Julian Calendar. For the Paschal [Page 349] Limits are the same in both, the difference is only in the Epacts, which here are used instead of the golden Number.
5. For the terms of the Paschal New Moons are always the eighth of March and the fifth of April: but whereas there are 11 days within these Limits to which no golden Number is affixed, there is now one day to which an Epact is not appointed, because there is no day within those Limits, on which in process of time a New Moon may not happen. And the reason for which the two Epacts XXV and XXIV are both set to the fifth of April, is first general, which was shewed before, namly that by doing the same in 5 other Months, the 12 time 30 Epacts might be contracted to the Limits of the lunar Year which consists of 354 days: but there is a particular reason also for it, that the Antients having appointed that all the Paschal lunations should consist of 29 days, it was necessary that some two of the Epacts should be set to one of these days in which the Paschal lunation might happen, the Epacts being 30 in number. And it was thought convenient to choose the last day, to which the Epact XXV belonging, the Epact XXIV should also be set; and hence by imitation it comes to pass, that these and not other Epacts are set to that day in other Months, in which two Epacts are to be set to the same days.
6. The use of these Epacts in finding the Feast of Easter, is the same with that which hath been shewed concerning the golden Numbers. For the Epact and the Sunday Letter for that year propounded being given, the Feast of Easter may be found in the Calendar after the same [Page 350] manner. Thus in the year 1674, the Epact is 23 and the Sunday Letter G, and therefore reckoning fourteen days from the eighth of March to which the Epact is set, the Sunday following is March 25, which is the day on which the Feast of Easter is observed.
7. And hence as hath been shewed in the third Chapter concerning the Julian Calendar, a brief table may be made to shew the feast of Easter and the other moveable Feasts for ever, in which there is no other difference, save only that the Epacts as they are in this new Calendar, are to be used as the golden Numbers are, which stand in the old Calendar. And a Table having the golden Numbers of the old Calendar set in one Column, and the Epacts as they are in the new Calendar set in another, will indifferently shew the movable Feasts in both accounts, as in the Year 1674, the golden Number is 3 and the Sunday Letter according to the Julian account is D, according to the Gregorian G, and the Epact 23, and therefore according to this Table our Easter is April 19, and the other, to wit, the Gregorian, is March 25. The like may be done for any other year past or to come.
CHAP. VII.
How to reduce Sexagenary numbers into Decimal, and the Contrary.
EVery Circle hath antiently, and is yet generally supposed to be divided into 300 degrees, each degree into 60 Minutes, each Minute into 60 Seconds, and so forward as far as need shall require. But this partition is somewhat troublesom in Addition and Subtraction, much more in Multiplication and Division; and the Tables hitherto contrived to ease that manner of computation, do scarce sufficiently perform the work, for which they are intended. And although the Canon published by the learned H. Gellibrand, in which the Division of the Circle into 360 degrees is retained, but every degree is divided into 100 parts, is much better than the old Sexagenary Canon, yet some are of opinion, that if the Antients had divided the whole circle into 100 or 1000 parts, it would have proved much better then either; only they think Custome such a Tyrant, that the alteration of it now will not be perhaps so advantagious; leaving them therefore to injoy their own opinions, they will not I hope be offended if others be of another mind: for their sakes therefore, that do rather like the Decimal way of calculation▪ Having made a Canon of artificial Signs and Tangents for the degrees and parts of a Circle divided into 100 parts, I shall here also shew you, how to reduce sexagenary Numbers into Decimal, and the contrary, [Page 352] as well in time as motion.
2. The parts of a Circle consisting of 360 degrees, may be reduced into the parts of a circle divided into 100 degrees or parts, by the rule of Three in this manner.
As 360 is to 100, so is any other Number of degrees, in the one, to the correspondent degrees and parts in the other.
But if the sexagenary degrees have Minutes and Seconds joyned with them, you must reduce the whole Circle as well as the parts propounded into the least Denomination, and so proceed according to the rule given.
Example. Let it be required to convert 125 degrees of the Sexagenary Circle, into their correspondent parts in the Decimal. I say, as 360 is to 100, so is 125 to 34, 722222, &c. that is, 34 degrees and 722222 Parts.
2. Example. Let the Decimal of 238 degrees 47 Minutes be required. In a whole Circle there are 21600 Minutes, and in 238 degrees, there are 14280 Minutes, to which 47 being added the sum is 14327. Now then I say if 21600 give 100, what shall 14327. The Answ. is 66, 3287 &c. In like manner if it were required to convert the Hours and Minutes of a Day into decimal Parts, say thus, if 24 Hours give 100, what shall any other number of Hours give. Thus if the Decimal of 18 hours were required, the answer would be 75, and the Decimal answering to 16 Hours 30 Minutes is 68, 75.
But if it be required to convert the Decimal Parts of a Circle into its correspondent Parts in Sexagenary. The proportion is; as 100 is [Page 353] to the Decimal given, so is 360 to the Sexagenary degrees and parts required.
Example. Let the Decimal given be 349 722222, if you multiply this Number given by 360, the Product will be 1249999992, that is cutting off 7 Figures, 124 degrees and 9999992 parts of a degree. If Minutes be required, multiply the Decimal parts by 60, and from the product cut off as many Figures, as were in the Decimal parts given, the rest shall be the Minutes desired.
But to avoid this trouble, I have here exhibited two Tables, the one for converting sexagenary degrees and Minutes into Decimals, and the contrary. The other for converting Hours and Minutes into Decimals, and the contrary. The use of which Tables I will explain by example.
Let it be required to convert 258 degrees 34′. 47″, into the parts of a Circle decimally divided.
The Table for this purpose doth consist of two Leaves, the first Leaf is divided into 21 Columns, of which the 1. 3. 5. 7. 9. 11. 13. 15. 17. 19 doth contain the degrees in a sexagenary Circle, the 2. 4. 6. 8. 10. 12. 14. 16. 18 and 20 doth contain the degrees of a Circle Decimally divided, answering to the former, and the last Column doth contain the Decimal parts, to be annexed to the Decimal degrees. Thus the Decimal degrees answering to 26 Sexagenary are 7, and the parts in the last Column are 22222222 and therefore the degrees and parts answering to 26 Sexagenary degrees are 7. 22222222.
[Page 354] In like manner the Decimal of 62 degrees, 17. 22222222. And the Decimal of 258 degrees, 34′. 47″, is thus found.
- The Decimal of 258 degrees is
- 71.66666666
- The Decimal of 34 Minutes is
- .15747040
- The Decimal of 47 seconds is
- .00362652
- Their Sum
- 71.82776358
is the Decimal of 258 degrees, 34′. 47″ as was required.
In like mauner the Decimal of any Hours and Minutes may be found by the Table for that purpose.
Example. Let the Decimal of 7 Hours 28′ be required.
- The Decimal answering to 7h. is
- 29.16666667
- The Decimal of 28 Minutes is
- 1.94444444
- The Sum
- 31.11111111
is the Decimal Sought.
To find the degrees and Minutes in a sexagenary Circle, answering to the degrees and parts of a Circle Decimally divided, is but the contrary work.
As if it were required to find the Degrees and minutes answering to this decimal 71. 02776359, the Degrees or Integers being sought in the 2. 4. 6 or 8 Columns &c. of the first Leaf of that Table, right against 71. I find 256 and in the last Column these parts 11111111, which being less than the Decimal given, I proceed [Page 355] till I come to 6666667, which being the nearest to my number given, I find against these parts under 71. Degrees 258, so then 258 are the degrees answering to the Decimal given and,
- To find the Minutes and Seconds from
- 71.82776359
- I Substract the number in the Table
- 71.66666667
- The remainer is
- 16109692
- which being Sought in the next Leaf under the title Minutes, the next leaf is
- 11747640
- And the Minutes 34, and this number being Subtracted the remainer is
- 00362652
Which is the Decimal of 47 seconds, and so the degrees and Minutes answering to the Decimal given are 258 degrees 34′ and 47″, the like may be done for any other.
CHAP. VIII.
Of the difference of Meridiens.
HAving in the first part shewed how the places of the Planets in the Zodiack may be found by observation, and how to reduce the time of an observation made in one Country, to the correspondent time in another, as to the day of the Month, by considering the several [Page 356] measures of the year in several Nations, there is yet onething wanting, which is, by an observation made of a Planets place in one Country to find when the Planet is in that place in reference to another; as suppose the ☉ by observation was found at Vraniburg to be in ♈. 3d. 13′. 14″. March the fourteenth 1583 at what time was the Sun in the same place at London? To resolve this and the like questions, the Longitude of places from some certain Meridian must be known; to which purpose I have here exhibited a Table shewing the difference of Meridians in Hours and Minutes, of most of the eminent places in England from the City of London, and of some places beyond the Seas also. The use whereof is either to reduce the time given under the Meridian of London to some other Meridian, or the time given in some other Meridian to the Meridian of London.
1. If it be required to reduce the time given under the Meridian of London to some other Meridian, seek the place desired in the Catalogue, and the difference of time there found, either add to or subtract from the times given at London, according as the Titles of Addition or Subtraction shew, so will the time be reduced to the Meridian of the other place as was required. Example. The same place at London was in the first Point of ♉, 6 Hours P. M. and it is required to reduce the same to the Meridian of Vraniburg I therefore seek in Vraniburg in the Catalogue of places, against which I find 50′ with the Letter A annexed, therefore I conclude, that the Sun was that day at Vraniburg in the first point of ♉, 6 Hours 50′. P. M.
[Page 357] 2. If the time given be under some other Meridian, and it be required to reduce the same to the Meridian of London, you must seek the place given in the Catalogue, and the difference of time there found, contrary to the Title is to be added or subtracted from the time there given.
Example. Suppose the place of the Sun had been at Vraniburg, at 6 Hours 50′. P. M. and I would reduce the same to the Meridian of London; against Vraniburg as before I find 50′ A. therefore contrary to the Title I Subtract 50′ and the remainder 6 Hours is the time of the Suns place in the Meridian of London.
CHAP. IX.
Of the Theory of the Sun's or Earth's Motion.
IN the first part of this Treatise we have spoken of the primary Motion of the Planets and Stars, as they are wheeled about in their diurnal motion from East to West, but here we are to shew their own proper motions in their several Orbs from West to East, which we call their second motions.
1. And these Orbs are supposed to be Elliptical, as the ingenious Repler, by the help of Tycho's accurate observations, hath demonstrated in the Motions of Mars and Mercury, and may therefore be conceived to be the Figure in which the rest do move.
2. Here then we are to consider what an Ellipsis is, how it may be drawn, and by what Method [Page 358] the motions of the Planets according to that Figure may be computed.
3. What an Ellipsis is Apollonius Pergaeus in Conicis, Claudius Mydorgius and others have well defined and explained, but here I think it sufficient to tell the Reader, that it is a long Circle, or a circular Line drawn within or without a long Square; or a circular Line drawn between two Circles of different Diameters.
4. The usual and Mechanical way of drawing this Ellipsis is thus; first draw a line to that length which you would have the greatest Diameter to be, as the Line AP in Figure 8, and from the middle of this Line at X, set off with your compasses the Equal distance XM and XH.
5. Then take a piece of thred of the same length with the Diameter AP and fasten one end thereof in the point M and the other in the point H, and with your Pen extend the thred thus fastened to the point A, and from thence towards P keeping the thread stiff upon your Pen, draw a line from A by B to P, the line so drawn shall be half an Ellipsis, and in like manner you may draw the other half from P by D to A. In which because the whole thred is equal to the Diameter AP. therefore the two Lines made by thred in drawing of the Ellipsis, must in every point of the said Ellipsis be also equal to the same Diameter AP. They that desire a demonstration thereof geometrically, may consult Apollonius Pergaeus, Claudius Mydorgius or others, in their treatises of Conical Sections, this is sufficient for our present purpose, and from the equality of these two Lines with the Diameter, a brief Method of calculation of the [Page 359] Planets place in an Ellipsis, is thus Demonstrated by Dr. Ward now Bishop of Salisbury.
6. In this Ellipsis H denotes the place of the Suns Center, to which the true motion of the Planet is referred, M the other Focus whereunto the equal or middle motion is numbred, A the Aphelion where the Planet is farthest distant from the Sun and slowest in motion, P the Perihelion where the Planet is nearest the Sun and slowest in motion. In the points A and P the Line of the mean and true motion do convene, and therefore in either of these places the Planet is from P in aequality, but in all other points the mean and true motion differ, and in D and C is the greatest elliptick AEquation.
8. Now suppose the Planet in B, the line of the middle motion according to this Figure is MB, the line of the true motion HB. The mean Anomaly AMB. The Eliptick aequation or Prosthaphaeresis MBH, which in this Example subtracted from AMB, the remainer AHB is the true Anomaly. And here note that in the right lined Triangle MBH, the side MH is always the same, being the distance of the Foci, the other two sides MB and HB are together equal to AP. Now then if you continue the side MB till BE be equal to BH and draw the line HE, in the right lined Triangle MEH, we have given ME=AD and MH with the Angle EMH, to find the Angles MEH and MHE which in this case are equal, because EB=BH by Contraction, and therefore the double of BEH or BHE=MBH, which is the Angle required.
And that which yet remaineth to be done, is [Page 360] the finding the place of the Aphelion, the true Excentricity or distance of the umbilique points, and the stating of the Planets middle motion.
CHAP. X.
Of the finding of the Suns Apogeon, quantity of Excentricity aend middle motion.
THe place of the Suns Apogaeon and quantity of Excentricity may from the observations of our countrey man Mr. Edward Wright be obtained in this manner, in the years 1596, and 1497, the Suns entrance into ♈ and ♎ and into the midst of ♉. ♌. ♍. and ♒ were as in the Table following expressed.
| 1596 | 1597 | ||
| D. H. M. | D. H. M. | ||
| Ianuary. | 25. 00.07 | 24. 05.54 | ♒. 15 |
| March. | 9. 18.43 | 10. 00.37 | ♈. 0 |
| April. | 24. 21.47 | 25. 03.54 | ♉. 15 |
| Iuly. | 28. 01.43 | 28. 09.56 | ♌. 15 |
| September. | 12. 13.48 | 12. 19.15 | ♎. 0 |
| October. | 27. 15.23 | 27. 21.50 | ♍. 15 |
And hence the Suns continuance in the Northern Semicircle from ♈ to ♎ in the year 1596 being Leap year, was thus found.
- d. h.
- From the 1. of Ianuary to ☉ Entrance ♎.
- 256. 13. 48.
- From the 1. of Iun to ☉ Entrance ♈
- 69. 18.43
- Their difference.
- 186. 19.05
- In the year 1597 from the 1 of Ianuary to the time of the ☉ Entrance into ♎.
- 255. 19.15
- To the ☉ entrance into ♈.
- 69. 09.37
- Their difference is
- 186. 18.38
And the difference of the Suns continuance in these Arks in the year 1596 and 1597 is 27′. and therefore the mean time of his continuance in those Arks is days 186. hours 18. minutes 51. seconds 30. And by consequence his continuance in the Southern Semicircle that is from ♎ to ♈ is 178 days. 11 hours, 8 minutes and 30 seconds.
- In like manner in the year 1596 between his entrance into ♉ 15. and ♍ 15, there are days
- 185. 17.36
- And in the year 1597 there are days
- 185. 17.56
And to find the middle motion answering to days 186. hours 18. Minutes 51. seconds 30 I say.
As 365 days, 6 hours, the length of the Julian, year is to 360, the degrees in a Circle.
So is 186 days, 18 hours, 51′. 30″ to 184 degrees. 03′. 56″.
In like manner the mean motion answering [Page 362] to 185 days, 17h. 46′ is 183 degrees, 02′.09.
- Apparent motion from ♈ to ♎
- 180. 00.00
- Middle motion
- 184. 03.56
- Their Sum
- 364. 03.56
- Half Sum is the Arch. SME
- 182. 01.58
In 1596 from 15 ♒ to 15 ♌ there are days 185, hours 01, minutes 36. In 1597. days 135. hours 4. 02′.
And the mean motion answering thereunto is. 182d. 30′. 36″.
- Apparent motion from
- 15 ♉ to 15 ♍. 180.
- Middle motion
- 185. 17. 56. 181. 04.53
- Half Sum is
- 183. 32. 26
From 15 ♒ to 15 ♌ Days. 185. 04h. 02′
- Apparent motion
- 180.
- Middle motion
- 182. 30. 36
- Half Sum
- 181. 15. 18
Now then in Fig. from PGC. 181. 32. 26 deduct NKD 180, the Remainer is DC+NP. 1. 32. 26. Therefore DC or NP. 46. 13, whose Sine is HA.
And from XPG. 181. 15. 18 deduct TNK 180, the Remainer is KG+TX 1. 15. 18. Therefore KG or TX 37. 39, whose Sine is HR.
- As HA 46′.13″
- 5.12851105
- To Rad. So HR 37′.39″
- 15.03948202
- To Tang. HAR. 39d.10′.04″
- 9.91097097
- GAM.
- 45
- Apogaeon
- 95. 49. 56.
- As the Sine HAR. 39. 10.04
- 9.80043756
- To Rad. So HR. 37.39
- 15.03948202
- To RA. 1733.99
- 5.23904446
In the Triangle [...] we have given [...]. and [...].
- As [...]. 37.39
- 5.03948202
- To Rad. So [...]. 46. 13.
- 15.12851105
- To Tang. R [...]. 50. 49.56
- 10.08902903
- PAS.
- 45.
Apogaeon 95 deg. 49′. 56″. as before.
- As the Sine of R [...]. 50. 49. 56
- 9.88945938
- Is to [...]. 46′.13″
- 5.12851105
- So is Radius. To RA.1734.01
- 5.23905167
And this agreeth with the excentricity, used [Page 364] by Mr. Street in his Astron. Carolina, Pag. 23. But Mr. Wing as well by observation in former ages, as our own, in his Astron. Instaur. Pag. 39. doth find it to be 1788 or 1791. The work by both observations as followeth.
2. And first in the time of Ptolemy, Anno Christi 139 by comparing many observations together, he sets down for the measure nearest truth, the interval between the vernal Equinox and the Tropick of Cancer to be days 93. hours 23. and minutes 03. And from the Vernal to the Autumnal Equinox, days 186. hours 13. and minutes 5.
- D.
- The apparent motion from ♈ to ♎
- 90. 36.00
- Middle motion for 93d. 23h. 3′. is
- 92. 36.42
- The half Sum is GP
- 91. 18.21
- Apparent motion from ♈ to ♎
- 180. 00.00
- Middle motion for 186d. 13h. 5′. is
- 183. 52.03
- The half Sum is GEK
- 181. 56.02
- The half of GEK is GE.
- 90. 58.01
- And GP less GE is
- 00. 20.20
- Whose Sum is AC 59146.
Again from GEK 181. 56. 02. deduct the Semicircle FED 180. the remainer is the summ DK and FG. 1. 56. 2. and therefore DK=FG. 58′. 01″. whose sign is BC. 168755. L is the place of the Aphelion, and AB the Excentricity.
Now then in the Triangle ABC. in the Fig. 6 we have given the two sides AC and BC. To find the Angle BAC and the Hypotenuse AB. [Page 365] For which the proportions are.
- As the side AC. 59146
- 4.77192538
- Is to the Radius.
- 10.00000000
- So is the side BC▪ 168755
- 5.22725665
- To Tang. BAC. 70. 41. 10.
- 10.45533127
- As the Sine of BAC. 70. 41. 10.
- 9.97484352
- Is to the side AB. 168755.
- 5.22725665
- So is the Radius.
- 10.00000000
- To the Hypot. AB. 1788. 10.
- 5.25241313
Therefore the Aphelion at that time was in II 10. 41. 10. And the excentricity. 1788.
3. Again Anno Christi 1652 the Suns place by observation was found to be as followeth.
| April. 24. | hours. 10. | ♉. 15 |
| October. 27. | hours. 7. 10′ | ♍. 15 |
| Ianuary. 24. | hours. 11.20′ | ♒. 15 |
| Iuly. 27. | hours. 16.30, | ♌. 15 |
Hence it appeareth that the Sun is running through one Semicircle of the Ecliptick, that is from ♉ 15 to ♍ 15. 185 days 21 hours and 10′. And through the other Semicircle from ♒ 15 to ♌ 15, days 184. hours 5. therefore the Suns mean motion, according to the practice in the last example, from ♉ 15 to ♍ 15 is 181. 30. 26. and from ♒ 15 to ♌ 15. 181. 16. 30.
Now then in Fig. 7. if we subtract the semicircle of the Orb KMH. 180. from WPV 181. 36. 26. the remainer is the sum of KW and HV [Page 366] 1. 36. 26. the Sine of half thereof 48′. 13″ is equal to AC. 140252.
Again the mean motion of the Sun in his Orb from ♒ 15 to ♉ 15 is the Arch SKP. 181. 16. 30. whose excess above the Semicircle being bisected is 38. 15. whose Sine CB. 111345. now then in the Triangle ABC to find the Angle BAC, the proportion is.
- As the side AC. 140252
- 5.14690906
- Is to the Radius.
- 10.00000000
- So is the Side CB 111345
- 5.04667072
- To Tang. BAC. 38. 36. 21,
- 9.89966166
Which being deducted out of the Angle. 69 A ♌. 45 it leaveth the Angle 69 AL 6. 33. 39. the place of the ☉ Aphelion sought, and this is the quantity which we retain.
- As the Sum of BAC. 38. 26. 21
- 9.79356702
- Is to the Radius.
- 10.00000000
- So is the side BC 111345
- 5.04667072
- To the Hypot. AB. 179103
- 5.25310370
- So then Anno Christi. 1652. Aphel.
- 96. 33.39
- Anno Christi. 139. the Aphelion
- 70. 41.10
- Their difference is
- 25. 52.29
And the difference of time is 1513 Julian years.
Hence to find the motion of the Aphelion for 2. years, say I, if 1513 years give 25. 52.29, what shall one year give, and the answer is 00d. 01′ [Page 367] 01″. 33‴. 56iv. 44v. that is in Decimal numbers. 0. 00475. 04447. 0555.
And the motion for. 1651 years. 7. 84298. 4208862, which being deducted from the place of the Aphelion Anno Christi. 1652—26. 82245. 3703703. The remainer, viz. 18. 97946. 9494841 is the place thereof in the beginning of the Christian AEra, which being reduced is, 68 deg. 19. min. 33. sec. 56. thirds.
4. The Earths middle motion, Aphelion and Excentricity being thus found, we will now shew how the same may be stated to any particular time desired, and this must be done by help of the Sun or Earths place taken by observation. In the 178 year then from the death of Alexander, Mechir the 27 at 11 hours P. M. Hipparcus found in the Meridian of Alexand. that the Sun entered ♈ 0. the which Vernal Equinox happened in the Meridian of London according to Mr. Wings computation at 9 hours 14′, and the Suns Aphelion then may thus be found.
The motion of the Aphelion for one year, was before found to be. 0. 00475. 04447. 0555. therefore the motion thereof for one day is 0. 00001. 501491722. The Christian AEra began in the 4713 year compleat of the Julian Period, in which there are days 1721423. The AEra Alexandri began November the twelfth, in the year 4390 of the Julian Period, in which there are 1603397 days. And from the death of Alexander to the 27 of Mechir 178, there are days 64781, therefore from the beginning of the Julian Period, to the 178 year of the AEra Alexandri, there are days 1668178 which being deducted from the days in the Christian AEra, [Page 368] 1721423, the remainer is 53245, the number of days between the 178 year after the death of Alexander, Mechir 27, and the beginning of the Christian AEra.
Or thus. From the AEra Alexandri to the AEra Christi there are 323 Julian years, and 51 days, that is 118026 days. And from the AEra Alexandri to the time of the observation, there are 64781 days, which being deducted from the former, the remainer is 53245 as before. Now then if you multiply the motion of the Aphelion for one day, viz. 0. 00001. 3014917 by 53245, the product is 0. 69297. 9255665, which being deducted from the place of the Aphelion in the beginning of the Christian AEra, before found. 18. 97946. 9494841. the remainer 18. 28649. 0239176 is the place of the Aphelion at the time of the observation, that is in Sexagenary numbers. deg. 65. 49′. 53″.
5. The place of the Aphelion at the time of the observation being thus found to be deg. 65. 49′. 53″. The Suns mean Longitude at that time, may be thus computed.
In Fig. 8. In the Triangle EMH we have given the side ME 200000, the side MH 3576, the double excentricity before found, and the Angle EMH 114. 10′. 07″. the complement of the Aphelion to a Semicircle, to find the Angle MEH, for which the proportion is,
As the Summ of the sides, is to the difference of the sides, so is the Tangent of the half Summ of the opposite Angles, to the Tangent of half their difference.
- The side ME. 200000.
- The side MH 3576.
- Z. Of the sides. 203576. Co. ar.
- 4.69127343
- X. Of the sides. 196424.
- 5.29321855
- Tang. ½ Z Angles. 32′. 54′. 56.
- 9.91111512
- Tang. ½ X Angles. 31. 59. 21.
- Angle MEH. 0. 55. 35.
- 9.79560710
The double whereof is the Angle MBH 1. 51. 10. which being Subtracted from 360 the remainer 358. 08. 50. is the estimate middle motion of the Sun, from which subtracting the Aphelion before found, 65. 49. 53. the remainer 292. 18. 57. is the mean Anomaly by which the absolute AEquation may be found according to the former operation.
- Z. ME+MH. 203576. Co. ar.
- 4.69127343
- X. ME-MH. 196424
- 5.29321855
- Tang. ½ Anom. 56. 09. 28.
- 10.17359517
- Tang. ½ X. 55. 12. 18.
- 10.15808715
- Differ. 00. 57. 10.
Doubled 1. 54. 20, which added to the middle motion before found gives the ☉ true place ♈. 00. 3′. 10″, which exceeds the observation 3′. 10″. therefore I deduct the same from the middle motion before found, and the remainer 358. 05. 50. is the middle motion at the time of the observation of Hipparchus, to which if you add the middle motion of the Sun for 53245 days, or for 323 AEgyptian years 131 days, 280. 46. 08′ the Summ, rejecting the whole Circles, is 278. 51. 48 the Suns mean Longitude in the beginning of the Christian AEra.
6. But one observation is not sufficient, whereby [Page 370] to state the middle motion for any desired Epocha, we will therefore examine the same by another observation made by Albategnius at Aracta in the year of Christ 882, March: 15. hours 22. 21. but in the Meridian of London at 18 hours. 58′.
The motion of the Aphelion for 881 years, 74 days is 3. 806068653737, which being added to the place thereof in the beginning of the Christian AEra, the place at the time of the observation will be found to be 22. 785538148578, that is reduced, Deg. 82. 01′. 40″. And hence the AEquation according to the former operations is Deg. 2. 01′. 16″ which being deducted from a whole Circle, the remainer 357d. 58′. 44″ is the estimate middle motion at that time, from which deducting the Aphelion deg. 82. 01. 40. the remainer 275. 57. 04 is the mean anomaly, and the AEquation answering thereto is deg. 2. 02′. 18″ which being added to the middle motion before found, gives the ☉ place ♈. 00. 01′. 02″ which exceeds the observation 01′. 02″. therefore deduct the same from the middle motion before found, the remainer 357. 57′. 22″ is the middle motion of the ☉ at the time of the observation, from which deducting the middle motion for 881 years, 74 days, 18 hours, 58 minutes, viz. 80d. 06′. 10″. the remainer 277 deg. 51′. 12″. is the ☉ mean Longitude in the beginning of the Christian AEra.
- By the first observation it is deg.
- 278. 51′. 48″
- By the second
- 277. 51. 12
- Their difference is
- 1. 00. 36
[Page 371] He that desires the same to this or any other Epocha, to more exactness, must take the pains to compare the Collection thereof from sundry Observations, with one another, this is sufficient to shew how it is to be found. Here therefore I will only add the measures set down by some of our own Nation, and leave it to the Readers choice to make use of that which pleaseth him best.
- Vincent Wing is
- 9. 8d. 00′. 31″
- Tho. Street is
- 9. 7. 55. 56
- Iohn Flamsted is
- 9. 7. 54. 39
- By our first Computation
- 9. 8. 51. 48
- By our second
- 9. 7. 51. 12
In the Ensuing Tables of the ☉ mean Longitude, we have made use of that measure given by Mr. Flamsted, a little pains will fit the Tables to any other measure.
CHAP. XI.
Of the quantity of the Tropical and Sydereal Year.
THe year Natural or Tropical (so called from the Greek word [...], (which signifies to turn) because the year doth still turn or return into it self) is that part of time in which the ☉ doth finish his course in the Zodiack [Page 372] by coming to the same point from whence it began.
2. That we may determine the true quantity thereof, we must first find the time of the ☉ Ingress into the AEquinoctial Points, about which there is no small difference amongst Astronomers, and therefore an absolute exactness is not to be expected, it is well that we are arrived so near the Truth as we are. Leaving it therefore to the scrutiny of after Ages, to make and compare sundry Observations of the ☉ entrance into the AEquinoctial Points, it shall suffice to shew here how the quantity of the Tropical year may be determined, from these following observations.
3. Albategnius, Anno Christi 882 observed the ☉ entrance into the Autumnal AEquinox at Aracta in Syria to be Sept. 19. 1 hour 15′ in the Morning. But according to Mr. Wings correction in his Astron. Instaur. Page 44, it was at 1 hour 43′ in the Morning, and therefore according to the ☉ middle motion, the mean time of this Autumnal AEquinox was Sept. 16. 12h. 14′. 25″. that is at London at 8h. 54′. 25″.
4. Again by sundry observations made in the year 1650. the second from Bissextile as that of Albategnius was, the true time of the ☉ ingress into ♎ was found to be Sept. 12. 14h. 40′. and therefore his ingress according to his middle motion was Sept. 10. 13h. 02.
5. Now the interval of these two observations is the time of 768 years, in which space by subtracting the lesser from the greater, I find an anticipation of 5 days, 9 hours, 52′. 25″. which divided by 768 giveth in the quotient [Page 373] 10′. 55″. 39 which being subtracted for 365 days, 6 hours, the quantity of the Julian year, the true quantity of the Tropical year will be 365 days, 5 hours, 49′. 04″. 21‴. Others from other observations have found it somewhat less, our worthy countryman Mr. Edward Wright takes it to be 365d. 5 hours. 48′.
Mr. Iohn Flamsted, 5h. 29′. Mr. Tho. Street 5h. 49′. 01″. taking therefore the Tropical year to consist of 365 days, 5 hours, 49 Minutes, the Suns mean motion for one day is 0 deg. 59′. 8″. 19‴. 43iv. 47v. 21vi. 29vii. 23viii. or in decimal Numbers, the whole Circle being divided into 100 degrees, the ☉ daily motion is 0. 27379. 08048. 11873.
6. The Sydereal or Starry year is found from the Solar by adding the Annual Motion of the eighth Orb or praecession of the AEquinoctial Points thereunto, that praecession being first converted into time.
7. Now the motion of the fixed Stars is found to be about 50″. in a years time, as Mr. Wing hath collected from the several observations of Timocharis, Hipparchus, Tycho and others; and to shew the manner of this Collection, I will mention onely two, one in the time of Timocharis, and another in the time of Tycho.
8. Timocharis then as Ptolemy hath it in his Almagist, sets down the Virgins Spike more northwardly than the AEquinoctial, 1 deg. 24′. the time of this observation is supposed to be about 291 years before Christ, the Latitude 1 deg. 59′ South, and therefore the place of the Star was in ♍. 21d. 59′. And by the observation of Tycho 1601 current, it was in ♎ [Page 374] 18. 16′. and therefore the motion in one year 50″, which being divided by 365 days, 6 hours, the quotient is the motion thereof in a days time. 00′. 8‴. 12iv. 48v. 47vi. 18vii. 30viii. 13ix. and in decimal Numbers, the motion for a year is 00385. 80246. 91358. The motion for a day. 00001. 05626. 95938.
9. Now the time in which the Sun moveth 50″, is 20′. 17″. 28‴, therefore the length of the sydereal year is 365 days, 6 hours, 9′. 17″. 28‴. And the Suns mean motion for a day 59′. 8″. 19‴. 43iv. 47v. 21vi. 29vii. 23viii. converted into time is 00. 03′. 56″. 33‴. 18iv. 55v. 9vi. 23vii. 57viii. which being added to the AEquinoctial day, 24 hours, giveth the mean solar day, 24 hours. 3.′ 56″. 33‴. 18iv. 55. 9. 23. 57.
10. And the daily motion of the fixed Stars, being converted into time is 32iv. 51v. 15vi. 9vii. 14viii. 24ix. and therefore the AEquinoctial day being 24 hours, the sydereal day is 24 hours, 00′. 00″. 00‴. 32iv. 51. 15. 9. 14 24.
11. Hence to find the praecession of the AEquinoctial Points, or Longitude of any fixed Star, you must add or subtract the motion thereof, from the time of the observation, to the time given, to or from the place given by observation, and you have your desire.
Example. The place of the first Star in Aries found by Tycho in the year 1601 current, was in ♈. 27d. 37′. 00. and I would know the place thereof in the beginning of the Christian AEra.
- The motion of the fixed Stars for 1600 years,
- 22d. 13′. 20″
- Which being deducted from the place found by observ.
- 27. 37. 00
- The remainer.
- 5. 231. 40
is the place thereof in the beginning of the Christian AEra.
12. Having thus found the ☉ middle motion, the motion of the Aphelion and fixed Stars, with their places, in the beginning of the Christian AEra; we will now set down the numbers here exhibited AEra Christi. Mr. Wing from the like observations, takes the ☉ motion to be as followeth.
- The ☉ mean Longitude
- 9. 8. 00. 31
- Place of Aphelion
- 2. 8. 20. 03
- The Anomaly
- 06. 29. 40. 28
- The ☉ mean Longitude
- 77. 22460. 86419
- Place of the Aphelion
- 18. 98171. 29629
- The Anomaly
- 58. 24289. 56790
- The ☉ mean Longitude
- 99. 93364. 37563. 34
- The Aphelion
- 00. 00475. 04447. 05
- The ☉ mean Anomaly
- 99. 92889. 33116. 29
- The ☉ mean Longitude
- 00. 27379. 08048. 11
- The Aphelion
- 00. 00001. 30149. 17
- The mean Anomaly
- 00. 27377. 77898. 94
And according to these measures are the Tables made shewing the ☉ mean Longitude and Anomaly, for Years, Months, Days and Hours.
CHAP. XII.
The Suns mean motions otherwise stated.
SOme there are in our present age, that will not allow the Aphelion to have any motion, or alteration, but what proceeds from the motion of the fixed Stars, the which as hath been shewed, do move 50 seconds in a year, and hence the place of the first Star in Aries, in the beginning of the Christian AEra was found to be ♈. 5. 23d. 40.
Now then, if from the place of the Aphelion Anno Christi. 1652 as was shewed in the tenth Chapter, deg. 96. 33′. 39. we deduct the motion of the fixed Stars for that time. 28. 19. 12. the remainer 68. 14. 27 is the constant place of the Aphelion; but Mr. Street in his Astronomia Carolina Page 23, makes the constant place of the Aphelion to be 68d. 20. 00, and the ☉ excentricity 1732.
And from the observation of Tycho 1590 March the eleventh. in the Meridian of Vraniburg, but reduced to the Meridian of London. March the tenth, hour 23. 2′. He [Page 376]
[Page] [Page 377] determines the Earths mean Anomaly thus.
- The place of the Sun observed
- ♈. 0. 33. 19
- The praecession of AEquinox
- 0. 27. 27. 22
- The Earths Sydereal Longitude
- 5. 03. 05. 57
- The place of the Aphelion Subtract
- 8. 08. 20. 00
- The Earths true Anomaly
- 8. 24. 45. 57
- AEquation Subtract
- 1. 58. 47
- The remainer is the Estimate M. Anom.
- 8. 22. 47. 10
- AEquation answering thereto add.
- 1. 58. 27
- The Earths true Anomaly
- 8. 24. 45. 37
- The place of the Aphelion
- 8. 08. 20. 00
- Praecession of the AEquinox
- 0. 27. 27. 22
- Place of the Sun
- ♈. 00. 32. 59
- But the place by observation
- ♈. 00. 33. 19
- The difference is
- 001. 001. 20
- Which being added to the mean Anom.
- 8. 22. 47. 10
- The mean Anomaly is
- 8. 22. 47. 30
- The absolute AEquation
- 1. 58. 27
- The true Anomaly
- 8. 24. 45. 57
- Agreeing with observation.
And so the mean Anomaly AEra Christi is 6. 23. 19. 56. But Mr. Flamsted according to whose measure the ensuing Tables are composed, takes the mean Anomaly AEra Christi. to be 6. 24. 07. 091. The place of the Aphelion to be 8, 08. 23. 50. And so the Praecession of the AEquinox and Aphelion in the beginning of [Page 378] the Christian AEra. 8, 13. 47. 30. in decimal Numbers.
- The Suns mean Anomaly
- 56. 69976. 85185
- The Suns Apogaeon and Praec. AEq.
- 20. 49768. 51851
- The ☉ mean Longitude
- 99. 93364. 37563. 34
- The Praecession of AEquin.
- 00385. 80246. 91
- The ☉ mean Anomal.
- 99. 92978. 57316. 43
- The ☉ mean Longitude
- 00. 27379. 08048. 11
- The Praecession of AEqui.
- 00. 00001. 05699. 30
- The ☉ mean Anom.
- 00. 27378. 02348. 81
CHAP. XIII.
How to Calculate the Suns true place by either of the Tables of middle motion.
VVRite out the Epocha next before the given time, and severally under that set the motions belonging to the years, months and days compleat, to the hours, scruples, current every one under his like (only remember that in the Bissextile years after the end of Frebruary the days must be increased by an unite) then adding all together, the sum shall be the ☉ mean motion for the time given.
[Page 379] Example.
Let the given time be Anno Christi 1672. February 23. hours 11. 34′. 54″. by the Tables of the ☉ mean Longitude and Anomaly, the numbers are as followeth.
| M. Longitud. | M. Anomal. | ||
| The Epocha | 1660 | 80. 67440. | 53.79815 |
| Years | 11 | 99. 81766. | 99.76526 |
| Ianuary | 08. 48751. | 08.48711 | |
| Day. | 23 | 06. 29718. | 06.29688 |
| Hours | 11 | 00. 12548. | 00.12548 |
| 34 | 00. 00646. | 00.00646 | |
| 54 | 00. 00017. | 00.00017 | |
| 95. 40886. | 68.47951 |
By the Tables of the Suns mean Anomaly and praecession of the AEquinox, the numbers are these.
| Anomaly. | Praece. AEqui. | ||
| The Epocha | 1660 | 53. 76721. | 26.90200 |
| Years | 11 | 99. 77520. | 00.04243 |
| Ianuary | 8. 48718. | 00032 | |
| Days | 23 | 6. 29694. | 00024 |
| Hours | 11 | 0. 12548. | 26.94499 |
| 34′ | .00646. | 68.45882 | |
| 54″ | .00035. | 95.40381 | |
| ☉ mean Anomaly | 68.45882 |
[Page 380] There is no great difference between the ☉ mean Longitude and Anomaly found by the Tables of mean Longitude and Anomaly, and that found by the Tables of mean Anomaly and Precession of the AEquinox. The method of finding the Elliptical AEquation is the same in both, we will instance in the latter only, in which the ☉ mean Anomaly is Degrees 68. 45882. And the precession of AEquin. deg. 26. 94499.
But because there is no Canon of Sines and Tangents as yet published, suitable to this division of the Circle into an 100 deg. or parts: We must first convert the ☉ mean Anomaly, and prec. of of the AEquin, given, into the degrees and parts of the common Circle: And this may be done either into degrees and decimal parts of a degree, or into deg. and minutes: if it were required to be done into degrees and minutes, the Table here exhibited for that purpose will serve the turn, but if it be required to be done into degrees and decimal parts, I judge the following method to be more convenient.
Multiply the degrees and parts given by 36, the Product, if you cut off one figure more towards the right hand than there are parts in the number given, shall be the degrees and parts of the common Circle.
| Anomaly. 68. 45882 | Praec. AEquinox. 26. 94499 | |
| 36 | 36 | |
| 41075292 | 16166994 | |
| 20537646 | 8983497 | |
| Anom. 246. 451752 | Prae. AEq. 97. 001964 |
[Page 38] And if you multiply the parts of these Products, you will convert them into minutes.
Otherwise thus. Multiply the degrees and parts given by 6 continually, the second Product, if you cut off one figure more towards the right hand than are parts in the number given, shall be the degrees and parts of the common Circle. The third Product of the parts only shall give minutes, the fourth seconds, and so forward as far as you please. Example.
- ☉ Mean Anom. 68. 45882
- Praec. AEq. 26. 94499
- 6
- 6
- 41075292
- 16166994
- 246.451752
- 97.001964
- 6
- 6
- 27.10512
- 0.11784
- 6
- 6
- 6.3072
- 7.0704
And thus the mean Anom. is deg. 246. 451742 or 27′. 06. The Prec. AEq. 97. 001964. or 00′. 07″.
Hence to find the Elliptical AEquation in degrees and decimal parts: In Fig. 8. we have given in the right lined plain Triangle EMH, the sides ME, and MH, and the Angle EMH, 66. 451742. the excess of the mean Anomaly above a Semicircle, to find the Angle MEH.
- The side ME
- 200000
- The side MH
- 3468
| Zcru. | 203468 | Co. ar. | 4.69150389 |
| Xcru. | 196532 | 5.29343327 | |
| t frac12 Zangle. | 56.774129 | 10.18374097 | |
| t frac12 Xangle. | 55.857087 | 10.16867813 |
MEH. 0. 917042 the double whereof is the Angle MBH. 1. 834084 or Elliptick AEquation sought, which being added to the mean Anomaly and praecession of the AEquinox, because the Anomaly is more than a Semicircle, the same is the Suns true place.
- The ☉ mean Anomaly
- 246.451742
- The Praecession of the AEquinox
- 97.001964
- Elliptick AEquation
- 1.834084
- The Suns true place.
- 345.287790
But because the Elliptick AEquation thus found doth not so exactly agree to observation as is desired, Bullialdus in Chap. 3. of his Book entituled Astronomiae Philolaicae fundamenta clarius explicata, Printed at Paris, 1657. shews how to correct the same by an Angle applied to the Focus of middle motion, subtended by the part of the ordinate line, intercepted between the Ellipsis and the Circle circumscribing it. This Mr. Street maketh use of in his Astronomia Carolina, and this I thought not amiss to add here.
In Fig. 9. let ABCPDF be supposed an Ellipsis, and the Circle AGPK described upon the extremes of the transverse Diameter, and the Ordinates KN and OB extended to G and M in [Page]
[Page] [Page 383] the Periphery of the Circle: then by the 21 of the first of Apollonius.
XN. GX∷OB tang. OEB. OM tang. OEM.
And the Angle OEM-OEB=BEM=ETY, the variation to be deducted from the Elliptick AEquation ETH, the Remainer is the absolute AEquation YTS in the first Quadrant.
In the second and third Quadrants, the variation or difference between the mean and corrected Anomaly, must be added to the Elliptick AEquation, to find the true and absolute AEquation.
For XN. XG. QV. tang. QEV. the comp. m. Anom. QR. t. QER. and the Angle VER=ECO is the variation, and ECO+ECH=OCH is the absolute AEquation sought in the second Quadrant.
Again, XN. SG∷a D, tang. a ED. a b, tang. aEB. And aEB—aED=DEf the variation=EFO and EfO+EfH=OfH the absolute AEquation sought in the third Quadrant.
Lastly, in the fourth Quadrant of mean Anomaly it is.
XN. XG∷ch. tang. eEH. eg. tang. eFg. and hEg is the variation: And EFH— [...]= [...] the absolute AEquation sought in the fourth Quadrant.
And to find XN the conjugate Semi-diameter, in the right angled Triangle ENX, we have given, EN=AX and EX the semi-distance of the umbilick points. And Mr. Briggs in Chap. 19. of his Arithm. Logar. hath shewed, that the half Sum of the Logarith. of the sum and difference of the Hypotenuse, and the given leg. shall be the Logarith. of the other leg.
| [Page 384] Now then EN=AX. 100000 | ||
| The Leg EX. | 1734 | |
| Their Sum | 101734 | 5.00745001 |
| Their difference | 98266 | 4.99240328 |
| The Z of the Logarithms, | 9.99985329 | |
| frac12; Z. Logarith. XN. 99983 | 4.99992664 |
Now then in the former Example the mean Anomaly is 246 deg. 451741. and the excess above a semicircle is the ang. aED. 66. 451742. Therefore.
- As XN. 99983
- 4.99992664
- Is to XG. 100000
- 5.00000000
- So is the tang. aED 66.451742
- 10.36069857
- To the tang. aEB 66.455296
- 10.36077193
aEB—aED=DEf .003544 the variation, which being added to the Elliptick AEquation before found, the absolute AEquation is 1. 837628. and therefore the ☉ true place 345. 291334. that is X. 15. 17. 28.
CHAP. XIV.
To find the place of the fixed Stars.
THe annual motion of the fixed Stars is, as hath been shewed, 50 Seconds, hence to find their places at any time assigned, we have exhibited a Table of the Longitude and Latitude of some of the most fixed Stars, from the Catalogue [Page]
[Page] [Page 385] of noble Tycho for the year of our Lord 1600 compleat. Now then the motion of the fixed Stars according to our Tables being computed, for the difference of time between 1600 and the time propounded, and subtracted from the place in the Table, when the time given is before 1600, or added to it, when the time given is after; the Summ or difference shall be the place desired. The Latitude and Magnitude are still the same.
Example. Let the given time be 1500, the difference of time is 100 years, and the motion of the fixed Stars for 100 years is 0. 38580.
- The place of the 1 * in ♈, 1600
- 7.67129
- Motion for 100 years subtract
- 0.38580
- Place required in the year 1500
- 7.28549
2. Example.
- Let the time given be
- 1674.
- The place of the first Star in ♈ 1600 was
- 7.67129
- Motion for 60 years is
- 0.23148
- Motion for 14 years is
- 0.05401
- Place required in the year 1674 compl.
- 7.95678
CHAP. XV.
Of the Theory of the Moon, and the finding the place of her Apogaeon, quantity of excentricity and middle motion.
THe Moon is a secondary Planet, moving about the Earth, as the Earth and other [Page 386] Planets do about the Sun, and so not only the Earth but the whole System of the Moon, is also carried about the Sun in a year. And hence, according to Hipparchus, there arises a twofold, but according to Tycho a three-fold Inequality in the Moons Motion. The first is Periodical and is to be obtained after the same manner, as was the excentrick AEquation of the Sun or Earth: in order whereunto, we will first shew how the place of her Apogaeon and excentricity may be found.
At Bononia in Italy, whose Longitude is 13 degrees Eastward from the Meridian of London, Ricciolus and others observed the apparent times of the middle of three lunar Eclipses to be as followeth.
- The first 1642. April the 4. at 14 hours and 4 Minutes.
- The second 1642, September 27 at 16 hours and 46 minutes.
- The third 1643. September 17 at 7 hours and 31 Minutes.
The equal times reduced to the Meridian of London, with the places of the Sun in these three observations, according to Mr. Street in the 25 Page of his Astronomia Carolina, are thus.
- Anno Mens. D. h.
- d.
- 1642. April 4. 13. 37.
- ♈. 25. 6. 54
- 1642. Septemb. 27. 15. 57
- ♎. 14. 50. 09
- 1643. Sehtemb. 17. 6. 46
- ♎ 4. 20. 20
Hence the place of the Moon in the first observation [Page 387] is in ♎ 25. 6′. 54. in the second ♈ 14. 50. 9. in the third ♓ 4. 20. 20. Now then in Fig. 10. let the Circle BHDGFE denote the Moons AEquant T the Center of the Earth, the Semidiameters TD, TE and TF the apparent places of the Moon, in the first, second and third observations, C the Center of the Excentrick, CD, CE and CF the Lines of middle motion.
- From the first observation to the second there are
- 176d. 2h. 20′
- The true motion of the Moon is deg.
- 169. 43. 15″
- The motion of the Apogaeon subtract
- 19. 37. 07
- The motion of the true Anomaly is the arch DE
- 150. 06. 08
- The motion of the mean Anomaly DCE
- 140. 42. 28
- From the first observation to the third, there are
- 530d. 17h. 9.
- The true motion of the Moon is degrees
- 159. 13. 26
- The motion of the Apogaeon subtract
- 159. 07. 32
- The motion of the true Anomaly is the Arch DF
- 100. 05. 54
- The motion of the mean Anomaly DCF
- 93. 46. 45
- And deducting the Arch DGF from the Arch DFE, the remainer is the Arch FE
- 50. 00. 14
- And deducting the Angle DCF from the Angle DCE, the remainer is the Angle FCE
- 46. 55. 43
Suppose 10.00000000 the Logarithm of DC, continue FC to H, and with the other right Lines compleat the Diagram.
1. In the Triangle DCH we have given the Angle DCH 86. 13. 15. the complement of DCF 93. 46. 45 to a Semicircle. The Angle DHC 50. 02. 57. The half of the Arch DF and the side CD 1000000. To find CH.
- As the Sine of DHC 50. 02. 57
- 9.88456640
- To the Side DC, so the Sine of HDC 43. 43. 48.
- 19.83964197
- To the Side CA
- 9.95507557
2 In the Triangle HCE we have given CH as before, the Angle CHE 25. 00. 07. The half of the Arch FE, the Angle HCE 133. 04. 17 the complement of FCE, and by consequence the Angle CEH 21. 55. 36 To find the Side CE.
- As the Sine of CEH 21. 55. 36
- 9.57219707
- To the Side CH
- 19.95507557
- So is the Sine of CHE 25. 00. 07
- 9.62597986
- To the Sine CE
- 19.58105543
- 10.00885836
3. In the Triangle DCE, we have given DC. CE and the Angle DCE 140. 42. 28. whose complement 39. 17. 32 is the Summ of the Angles, to find the Angle CED and DE,
- As the greater Side CE
- 10.00885836
- Is to the lesser Side DC
- 10.00000000
- So is the Radius
- 10.00000000
- To the tang. of 44. 24. 54
- 19.99114164
- Which subtracted from 45. 2
- the remainer is the half.
- Difference of the acute angles 35. 16.
- To the tang. of the com. 35. 16
- 8.01109962
- Is to the tang. of the frac12; Z. 19. 38. 46
- 9.55265735
- To the tang. of frac12; X. 00. 12. 35
- 7.56375697
- Their Sum 19. 51. 21. is the angle—CDE.
- Their difference 19. 26. 11. is the angle CED.
- As the Sine of CED. 19. 26. 11.
- 9.52216126
- Is to the Sine of DCE. 140. 42. 28.
- 9.80159290
- So is the Side EC.
- 10.00000000
- To the Side DE.
- 10.27943164
4. In the Isosceles Triangle DTE we have given the Side DE, the angle DTE 150. 06. 08 whose complement 29. 53. 52 is the Summ of the other two angles, the half whereof is the angle TDE 14. 56. 56 which being subtracted from the angle CDE. 19. 51. 21 the remainer is the angle CDT. 4. 54. 25.
- As the Sine of DTE 150. 06. 08 Co. ar.
- 0.30237482
- Is to the Sine of DET. 14. 56. 56
- 9.41154778
- So is the Side DE
- 10.27943164
- To the Side DT
- 9.99335424
5. In the Triangle CDT we have given DC. DT and the angle CDT, to find CTD and CT.
- As the Side DT
- 9.99335424
- Is to the Side DC
- 10.00000000
- So is the Rad.
- 10.00000000
- To the tang. of 26. 18
- 10.00664576
- Deduct 45.
- As the Radius.
- Is to the Sine of the remainer 0. 26. 18.
- 7.88368672
- So is the tang. of the frac12; Z angle 87. 32. 57
- 11.36854996
- To the tang. frac12; X angle 10. 08. 04
- 9.25223668
- Their Summ 97. 41. 01
- is the angle CTD
- As the Sine of CTD. 97. 41. 01. Co. ar.
- 0.00391693
- Is to the Side DC
- 10.00000000
- So is the Sine of CDT 4. 54. 25
- 8.93215746
- To the Side CT
- 8.93607439
- s. d.
- The place of the Moon in the first Observation
- 6. 25. 06. 54
- The true Anomaly CTD sub.
- 3. 07. 41. 01
- The place of the Apogaeon
- 3. 17. 25. 53
- ☽ place in the first Observation
- 6. 25. 06. 54
- The AEquation CDT Add.
- 04. 54. 25
- The ☽ mean Longitude
- 7. 00. 01. 19
- From which subtract the place of the Apogeon
- 3. 27. 25. 53
- There rests the mean Anomaly BCD
- 3. 12. 35. 26
And for the excentricity in such parts, as the Radius of the AEquant is 100000 the Proportion is.
- DT
- 9.99335424
- CT
- 8.93607439
- 100000
- 5.00000000
- 8764
- 3.94272015
And this is the Method for finding the place of the Moons Apogaeon and excentricity. And from these and many other Eclipses as well Solar as Lunar, Mr. Street limits the place of the ☽ Apogaeon to be at the time of the first observation 21′. 04″ more, and the mean Anomaly 20. 41″ less, and the excentricity 8765 such parts as the Radius of the AEquant is 100000.
And by comparing sundry observations both antient and modern, he collects the middle motion of the Moon, from her Apogaeon, to be in the space of four Julian years or 146 days, 53 revolutions, 0 Signes, 7 degrees, 56 minutes, 45 Seconds. And the Apogaeon from the AEquinox 5 Signes, 12 degrees, 46 minutes. And hence the daily motion of her mean Anomaly will be found to be 13d. 03′. 53″. 57‴. 09iv. 58v. 46vi. Of her Apogaeon 0. 06. 41. 04. 03. 25. 33.
And according to these Measures, if you deduct the motion of the ☽ mean Anomaly for 1641 years April
- 4. hours 13. 37′, viz.
- 8. 22. 02. 00.
- from
- 3. 121. 35. 26
- The remainer is
- 6. 201. 33. 26
- from which abating 20′. 41″ the ☽ mean Anom. AEra Chr. 6. 20. 12. 45.
- In like manner the motion of her Apogaeon for the same time is
- 6. 05. 311. 57
- which being deducted from
- 3. 17. 25. 57
- The remainer is
- 9. 11. 55. 56
- To which if you add
- 21.04
- The Sum
- 91. 121. 15200
- is the place of the ☽ Apogaeon in the beginning of the Christian AEra.
CHAP. XVI.
Of the finding of the place and motion of the Moons Nodes.
ANno Christi 1652, March 28, hour. 22. 16′, the Sun and Moon being in conjunction, Mr. Street in Page 33, computes the ☽ true place in the Meridian of London to be in ♈. 19. 14. 18 with latitude North 46′. 15″.
And Anno Christi 1654 August 1. hour. 21. 19′. 30″ was the middle of a Solar Eclipse at London. at which time the Moons true place was found to be in ♌ 18. 58′. 12″ with North Latitude 32′. 01″.
- 1654 August 1. 21. 19′. 30″ ☽ place ♌ 18. 58. 12
- 1652 March 28. 22. 16. 00 ☽ place ♈ 19. 14. 18
From the first observation to the second there are 27 years, 4 months, 5 days, 23 hours 03′. 30″.
- Mean motion of the Nodes in that time, deg.
- 45. 19. 41
- The true motion of the ☽
- 119. 43. 54
- Their Summ is in Fig. 11. The angle DPB
- 165. 03. 35
Therefore in the oblique angled Spherical Triangle DPB we have given BP. 89. 13. 45 the complement of the Moons Latitude in the first Observation 2. PD 89. 27. 50 the complement of the Moons Latitude in the second observation, and the angle DPB 165. 03. 35, whose complement to a Semicircle is DPF 14. 56. 25. The angle PBD is required.
1. Proportion.
- As the Cotangent of PD 89. 27. 50
- 9.97114485
- Is to the Radius
- 10.00000000
- So is the Cosine of DPF 14. 56. 25
- 9.98506483
- To the tang. of PF 89. 26. 42
- 12.01191998
- BP 89. 13. 45
Their Z is FPB 178. 40. 27. whose complement Is the Arch FG 1. 19. 33.
2. Proportion.
- As the Sine of FP 89. 26. 42. Co. ar.
- 0.00002037
- Is to the Cotang. of DPF 14. 56. 25
- 10.57376158
- So is the Sine of FG 1. 19. 33
- 8.36418419
- To the Cotang. of FGD 85. 02. 56
- 8.93796614
- FGD=PBD inquired.
[Page 395] And in the right angled Spherical Triangle BA☊ right angled at A we have given AB 046′. 15″ the Latitude in the first observation, and the Angle AB☊=PBD 85. 02. 56. to find A☊ the Longitude of the Moon from the ascending Node.
- As the Cot. of AB☊ 85. 02. 56
- 8.93796614
- Is to the Radius
- 10.00000000
- So is the Sine of AB 0.46′. 15″
- 8.12882290
- To the tang. of A☊ 8. 49. 17
- 9.19085676
2. To find the Angle A☊B.
- As the tang. of AB 0. 46. 15
- 8.12886212
- Is to the Radius
- 10.00000000
- So is the Sine of A☊ 8. 49. 17
- 9.18569718
- To the Cotang. of A☊B 5. 0. 41
- 11.05682506
- The angle of the ☽ orbite with the Ecliptick
- The first observed place of the ☽
- ♈. 19. 14. 18
- A☊ Subtract
- 8. 49. 17
- There rests the true place of the ☊
- ♈. 10. 25. 01
The retrograde motion whereof in 4 Julian years or 2461 days, is by other observations found to be Sign 2. deg. 17. 22′. 06″. and therefore the daily motion deg. 0. 03′. 10″. 38‴. 11iv. 35v.
And the motion thereof for 1651 years, March 28. h. 22. 16′, viz. Sign 8. deg. 18. 26′. 58″ being added to the place of the Node before found Sig. 0. 10. 25: 01. Their Sum is the place thereof in the beginning of the Christian AEra Sign 8. deg. 28. 51′. 59″.
[Page 396] But the Rudolphin Tables as they are corrected by Mr. Horron and reduced to the Meridian of London, do differ a little from these measures, for according to these Tables, the Moons mean motions are.
- The Moons mean Longitude is
- Sign. 04. deg. 02. 25. 55
- The Moons Apogaeon
- Sign. 09. deg. 13. 46. 59
- The Moons mean Anomaly
- Sign. 06. deg. 18. 38. 56
- The Moons Node Retrograde
- Sign. 08. deg. 28. 33. 16
And according to these measures, the Moons mean motions in decimal Numbers are.
- The Moons mean Longitude, deg.
- 34. 00887.345677
- The Moons Apogaeon, deg.
- 78. 82862.654320
- The Moons mean Anomaly, deg.
- 55. 18024.691357
- The Moons Node Retrograde, deg.
- 74. 69845.679010
- The Moons mean Longitude, deg.
- 35. 94001. 44893. 1
- The Moons Apogeaon, deg.
- 11. 29551. 126365
- The Moons mean Anomaly, deg.
- 24. 64450. 322566
- The Moons Node Retrograde, deg.
- 05. 36900. 781604
- The Moons mean Longitude, deg
- 03. 66010. 962873
- The Moons Apogaeon, deg.
- 00. 03094. 660620
- The Moons mean Anomaly, deg.
- 03. 62916. 302253
- The Moons Node Retrograde, deg.
- 00. 01470. 961045
And according to these measures are the Tables made shewing the Moons mean Longitude, Apogaeon, Anomaly, and Node retrograde for Years, Months, Days and Hours.
And hence to compute the Moons true place in her Orbit, I shall make use of the Method, which Mr. Horron in his Posthumas works lastly published by Mr. Flamsted, in which from the Rudolphin Tables he sets down these Dimensions.
- The Moons mean Semidiameter deg.
- 00. 15′. 30″
- Her mean distance in Semid. of the Earth Deg.
- 11. 47. 22
- The half whereof deg. 5. 53. 41. he adds 45 the whole is
- deg. 50. 53. 41
- Whose Artificial cotangent is
- 9.91000022
- And the double thereof makes this standing Numb.
- 9.82000044
| Greatest 6685. 44 | ||
| The Moons | Mean 5523. 69 | Excentricity |
| Least 4361. 94 | ||
| And her greatest variation 00. 36′. 27″. | ||
These things premised his directions for computing the Moons place, are as followeth.
CHAP. XVII.
How to Calculate the Moons true place in her Orbit.
TO the given time find the true place of the Sun, or his Longitude from the Vernal AEquinox, as hath been already shewed.
2. From the Tables of the Moons mean motions, write out the Epocha next before the given time, and severally under that set the motions, belonging to the years, months and days compleat, and to the hours and scruples current, every one under his like (only remember that in the Bissextile years, after the end of February, the days must be increased by one Unite) then adding them all together, the Summ shall be [Page 399] the Moons mean motions for the time given: But in her Node Retragrade you must leave out the Radix or first number, and the Summ of the remainer being deducted from the Radix, shall be the mean place of her Node required.
3. Deduct the Moons Apogaeon from the ☉ true place, the rest is the annual Augment, the tangent of whose Complement 180 or 360, being added to the artificial Number given 9. 82000044. the Summ shall be the tangent of an Arch, which being deducted from the said Complement, giveth the Apogaeon AEquation to be added to the mean Apogaeon, in the first and third quadrants of the annual Augment, and Subtracted in the second and fourth, their Summ or difference is the true Apogaeon.
4. The true Apogaeon being Deducted from the ☽ mean Longitude gives the Moons mean Anomaly.
5. Double the annual Augment, and to the Cosine thereof add the Logarithm of 1161. 75. the difference between the Moons mean and extream Excentricity, viz. 3. 06511268, the Summ shall be the Logarithm of a number which being added to the mean Excentricity, if the double annual Augment be in the first or fourth quadrants; or Subtracted from it, if in the second or third quadrants; the Summ or difference shall be the Moons true Excentricity.
6. The Moons true Excentricity being taken for a natural Sine, the Arch answering thereto shall be the ☽ greatest Physical AEquation.
7. To the half of the Moons greatest Physical AEquation add 45 deg. the cotagent of the Summ is the artificial Logarithm of the Excentrick. [Page 400] To the double whereof if you add the tang. of half the mean Anomaly, the Summ shall be the tangent of an Arch, which being added to half the mean Anomaly, shall give the Excentrick Anomaly.
8. To the Logarithm of the Excentrick, add the tangent of half the Excentrick Anomaly, the Summ shall be the tangent of an Arch, whose double shall be the Coequated Anomaly, and the difference between this and the mean Anomaly is the terrestrial Equation, which being added to the Moons mean Longitude, if the mean Anomaly be in the first Semicircle, or Subtracted from it, if in the latter, the Summ or difference shall be the place of the Moon first Equated.
9. From the place of the Moon first Equated, Deduct the true place of the Sun, and double the remainer, and to the Sine of the double add the Sine of the greatest variation 0. 36. 27, viz. 8. 02541571, the Summ shall be the Sine of the true variation, at that time, which being added to the Moons place first Equated, when her single distance from the Sun is in the first or third quadrants, or Subtracted when in the second or fourth, the Summ or difference shall be the Moons true place in her Orbit.
Example.
Let the given time be Anno Christi 1672. Feb. 23. h. 11. 34′. 54″ at which time the Suns true place is in ♓ 15. 29133 and the Moons middle motions are as followeth.
| ☽ Longitude | ☽ Apogaeon | ☊ Retrograde | ||
| 1660 | 13. 36650. | 41. 78372. | 55.85177 | |
| 11. | 02. 66032. | 24. 31246. | 59.08943 | |
| Ianuary. | 13. 46339. | 00.95934 | .45599 | |
| D. 23 | 84. 18252. | .71177 | .33832 | |
| H. 11 | 1. 67755. | .01418 | .00674 | |
| 34′ | .08641. | .00072 | .00054 | |
| 54 | .00228. | .00012 | .00001 | |
| ☽ Longitude | 15. 43897. | 67.78229 | 59.89082 | |
| 95.96094 |
- These Numbers reduced to the Degrees and Parts of the common Circle are for the ☽ mean Longitude.
- 55.580292
- The ☽ Apogaeon.
- 244.015956
- The ☉ true place is
- 345.29133
- The ☽ Apogaeon subtract.
- 244.01595
- The Annual Augment.
- 101.27538
- The Complement whereof is
- 78.72462
- The Tang. of deg. 78. 72462
- 10.70033391
- The standing Number.
- 9.82000044
- The Tang. of deg. 73. 20288
- 10.52033435
- Their difference. 5. 52174 the Apogaeon Equation
- Mean Apogaeon 244. 01595
- Their difference 238. 49421 is the true Apogaeon.
[Page 402] Secondly.
- The ☽ mean Longitude.
- 55.58029
- The true Apogaeon subtract.
- 238.49421
- Rests the ☽ mean Anom. correct.
- 177.08608
Or thus.
- The ☽ mean Anomaly in the Tables for the time propounded, will be found to be 67. 78221, which converted into the deg. and parts of the common Circle is
- 171.56434
- To which the Apogaeon Equation being added
- 5.52174
- Their Sum is the mean Anom. correct.
- 177.08608
And hence it appears that working by the mean Anomaly instead of the mean Longitude, the true Apogaeon Equation must be added to the mean Anomaly, in the second and fourth Quadrants of the ☽ Annual Augment, and subtracted from it in the first and third.
Thirdly.
- The Annual Augment. 101. 27538 being doubled is deg. 202. 55076, the Cosine of whose excess above 180, that is the Cosine of 22. 54076 is
- 9.96545577
- The Logarithm of 1161. 75
- 3.06511268
- The Logarithm of 1072. 92
- 3.03056845
- The ☽ mean Excentr. 5523. 69
- Their difference 4450. 77 is the ☽ true Excentricity. [Page 403] Which taken as a natural Sine, the Arch answering thereunto Deg. 2. 55094 is the ☽ greatest Physical Equation.
Fourthly.
- To the half of the Physical Equation. deg. 01. 27547 add 45 degrees, the Sum is deg. 46. 27547, the Cotangent whereof; viz. 9. 98080957 is the Logarithm of the Excentrick, the double of which Logarithm is
- 9.96161914
- Tangent frac12 Anomaly corrected 88. 54304
- 11.59455229
- Tang. of deg. 88. 40849
- 11.55620143
- Their Sum deg. 176. 95153 is the excentrick Anomaly.
Fifthly.
- The Logarithm of the Excentrick is
- 9.98080957
- Tang. frac12 excent. Anom. 88. 475765
- 11.57505878
- Tangent of deg. 88. 407268
- 11.55586835
- The double whereof 176. 814536
- is the coequated Anomaly.
- M. Anomaly correct.
- 177.086080
- Their difference 0. 271544
- is the Equation sought to
- be subst. from ☽ mean Long.
- 55.580292
- The Remainer 55. 308748
- is the ☽ place first Equated.
[Page 404] Sixthly,
- From the place of the ☽ first Equated.
- 55.308748
- Deduct the true place of the Sun
- 345.291330
- The Remainer is the Distance of the ☽ à ☉
- 70.017418
- The double whereof is 140. 034836. The Sine of whose Complement to a Semi-circle, 39. 965164 is
- 9.80775260
- The Sine of the greatest variation
- 8.02541571
- The Sine of the true var. 0. 390206
- 7.83316831
- The ☽ place first Equa. 55. 308748
- The ☽ place in Orbit 55. 698954 that is in Sexagenary Numbers. 8. 25. 41. 54.
CHAP. XVIII.
To compute the true Latitude of the Moon, and to reduce her place, from her Orbit to the Ecliptick.
THe greatest Obliquity of the Moon's Orb with the Ecliptick or Angle A ☊ B Fig. 11. is by many Observations confirmed to be 5 Degrees just, at the time of the Conjunction or Opposition of the Sun and Moon, but in her Quarters deg. 5. 18′. Now then then find her Latitude at all times, the said Mr. Horrox refers us to pag. 87. in the Rudolphin Tables, to find from thence the Equation of the Nodes, and Inclination limitis menstrui, in this manner.
1. From the mean place of the Node, deduct [Page 405] the ☉ true place, the Remainer is the distance of the ☉ from the ☊. with which entring the said Table, he finds the Equation of the Node and Inclination limitis menstrui, which being added to or subtracted from the Nodes mean place according to the title, the Sum or difference is the true place of the Node, which being deducted from the place of the Moon in her Orb, the Remainer shall be the Augment of Latitude or Distance of the Moon from the Node, or Leg A ☊.
2. With the Augment of Latitude, enter the Table of the Moon's Latitude, and take thence her Simple and Latitude and Increase answering to it. Then say, as the whole excess of Latitude 18′, or in Decimals 30. is to the Inclination of the Monethly limit: So is the increase of Latitude to the Part Proportional; which being added to the simple Latitude, will give you the true Latitude of the Moon.
3. With the same Augment of Latitude, enter the Table of Reduction, and take thence the Reduction and Inclination answering thereto: Then say again, as 18′. 00″. or 0. 30. is to the Inclination of the Monethly limit: So is the increase of Reduction, to the Part Proportional; which being added to the simple Reduction, shall give the true, to be added to, or subtracted from the place of the Moon in the Ecliptick.
- Example. By the former Chapter, we found the mean motion of the Node to be 95. 96094, which reduced to the Degrees and Parts of the common Circle is
- 345.459384
- And the Suns true place to be
- 345.291334
- Their difference is the distance ☉ à ☊
- . 168050
[Page 406] with which entring the Table, Entituled Tabula AEquationis Nodorum Lunae. I find the Node to need no Equation, and the Inclination limitis menstrui to be deg. 00. 30.
- The place of the ☽ in her Orbit
- 55.698954
- The Nodes true place, subtract.
- 345.459384
- The Augment of Latitude
- 70.239570
- 2. With this Augment of Latitude I enter the Table shewing the Moons simple Latitude, and thereby find her simple Latitude to be Degrees. 04. 70476. North; And the increase
- 00.28234
- And therefore the Moons true Latitude is deg.
- 4.98610
- 3. With the same Augment of Latitude, I enter the Table of Reduction, and thereby find the Reduction to be
- 00.06955
- And the increase of Reduction to be deg.
- 00.00855
- And therefore the whole Reduction to be sub.
- 00.07810
- From the ☽ place in her Orbit
- 55.69895
- The ☽ true place in the Ecliptick
- 55.62085
- That is in Sexagenary Numbers.
- 8. 25. 37′. 15″.
CHAP. XIX.
To find the Mean Conjunction and Opposition of the Sun and Moon.
TO this purpose we have here exhibited a Table shewing the Moons mean motion from the Sun, the construction whereof is this: By the Tables of the Moons mean motions, her mean
- Longitude AEra Christi is
- 34.0088734567
- The ☉ mean Anomaly.
- 56.6997085185
- Praecession of the AEquinox.
- 20.4976851851
- Their Sum is the ☉ mean longit. AEra a Christi.
- 77.4973937036
- Which being deducted from the ☽ mean longitude, the remainer is the Moons mean
- 56.8114797531
- distance from the Sun, in the beginning of the Christian AEra.
In like manner the Moons mean distance from the Sun in a year or a day is thus found.
- ☉ Anomaly for a year.
- 99.9297857316
- Praecession of the AEquinox.
- 0038580246
- Their Sum subtract.
- 99.9336437562
- From the ☽ mean Longitude.
- 35.9400144893
- Moons distance from the ☉.
- 36.0063707331
[Page 408] Moons distance from the Sun in a days time.
- ☉ mean Anomaly.
- 27378.02348
- Praecession of the AEquinox.
- 1.05699
- Their Sum subtract.
- 27379.08047
- From the ☽ mean Longitude.
- 03. 66010.96287
- ☽ Daily motion from the ☉.
- 03. 38631.88240
And according to these measures are the Tables made, shewing the Moons mean motion from the Sun, by which the mean conjunction of the ☽ and Moon may be thus computed.
To the given year and Month gather the middle motions of the Moon from the Sun, and take the complement thereof to a whole Circle, from which subtracting continually the nearest lesser middle motions, the day, hour, and minute enfuing thereto is the mean time of the Conjunction.
Example, Anno Christi 1676. I would know the time of the mean Conjunction or New Moon in October.
- Epocha 1660
- 32.697283
- Years Compl. 15.
- 50.254463
- Septemb. Compl.
- 24.465038
- 1. day for Leap-year.
- 03.386318
- Their Sum is the Moons motion from the ☉.
- 10.803102
- Complement to a whole Circle.
- 89.196898
- Days 26 Subtract.
- 88.044289
- Hours 8. substract.
- 1.152609
- 1.128772
- [Page 409] Minutes 10 Subtract.
- 0.023837
- 0.023516
- The Remainer giveth 8″.
- .00321
Therefore the mean Conjunction in October, 1676. was the 26 day, 10 min. 8 seconds after 8 at night.
And to find the mean opposition. To the complement of the middle motion, add a semicircle, and then subtract the nearest lesser middle motions as before, the day, hour, and minute ensuing thereto, shall be the mean opposition required.
Example, Anno Christi, 1676. I desire to know the mean opposition in November.
- Epocha 1660
- 32.697283
- Years Compl. 15
- 50.254463
- October Compl.
- 29.440922
- 1 day for Leap-year.
- 03.386318
- The ☽ mean motion from the ☉
- 15.778986
- Complement to a whole Circle.
- 84.221014
- To which add a Semicircle.
- 50.
- The Sum is
- 34.221014
- Day 10 subtract.
- 33.863188
- Hours 2.
- .357826
- .282193
- Minutes 32.
- .075633
- .075251
- The Remainer giveth 9 seconds.
- .000382
[Page 410] Therefore the Full Moon or mean Opposition of the Sun and Moon was November the 10th, Hours 2, 32′ 09″. The like may be done for any other.
And here I should proceed to shew the manner of finding the true Conjunction or Opposition of the Sun and Moon, but there being no decimal Canon yet extant, suitable to the Tables of middle motions here exhibited, I chuse rather to refer my Reader to Mr. Street's Astronomia Carolina, for instructions in that particular, and what else shall be found wanting in this Subject.
AN INTRODUCTION TO Geography, OR, The Fourth Part of COSMOGRAPHY.
CHAP. I.
Of the Nature and Division of Geography.
GEOGRAPHY is a Science concerning the measure and distinction of the Earthly Globe, as it is a Spherical Body composed of Earth and Water, for that both these do together make but one Globe.
[Page 412] 2. And hence the parts of Geography are two, the one concerns the Earthy part, and the other the Water.
3. The Earthy part of this Globe is commonly divided into Continents and Islands.
4. A Continent is a great quantity of Land not separated by any Sea from the rest of the World, as the whole Continent of Europe, Asia, and Africa, or the Continents of France, Spain, and Germany.
5. An Island is a part of Earth environed round about with some Sea or other; as the Isle of Britain with the Ocean, the Isle of Sicily with the Mediterranean, and therefore in Latine it is called Insula, because it is scituate in Salo, in the Sea.
6. Both these are subdivided into Peninsula, Isthmus, Promontorium.
7. Peninsula, quasi pene insula, is a tract of land which being almost encompassed round by water, is joyned to the main land by some little part of Earth.
8. Isthmus is that narrow neck of Land which joyneth the Peninsula to the Continent.
9. Promontorium is a high mountain which shooteth it self into the Sea, the outmost end whereof is called a Cape or Foreland, as the Cape of Good Hope in Africk.
10. The Watry part of this Globe may be also distinguished by diverse Names, as Seas, Rivers, Ponds, Lakes, and such like.
11. And this Terrestrial Globe may be measured either in whole, or in any particular part.
12. The measure of this Earthly Globe in whole, is either in respect of its circumference, [...]o its bulk and thickness.
[Page 413] 13. For the measuring of the Earths circumference, it is supposed to be compassed with a great Circle, and this Circle in imitation of Astronomers, is divided into 360 degrees or parts, and each degree is supposed to be equal to 15 common German miles, or 60 miles with us in England, and hence the circumference of the Earth is found, by multiplying 360 by 15, to be 5400 German miles, or multiplying 360 by 60, the circumference is 21600 English miles.
14. The circumference of the Earth being thus obtained, the Diameter may be found by the common proportion between the Circumference and the Diameter of a Circle, the which according to Archimedes is as 22 to 7, or according to Van Culen as 1 to 3. 14159. and to bring an Unite in the first place.
As the circumference 3. 14159. is to 1 the Diameter, so is 1 the circumference to 318308 the Diameter, which being multiplied by 5400, the Earths Diameter will be found to be 1718 German miles and 8632 parts, but being multiplied by 21600, the Diameter will be 6875 English miles, and parts 4528.
15. The measure of the Earth being thus found in respect of its whole circumference and Diameter, that which is next to be considered, is the distinction of it into convenient spaces.
16. And this is either Primary or Secondary.
17. The Primary distinction of the Earthly Globe into convenient spaces, is by Circles considered absolutely in themselves, dividing the Globe into several parts without any reference to one another.
[Page 416] Dutch Geographer inclines much to the bringing back the great Meridian to the Fortunate Islands, more particularly to the Peak a Mountain so called from the sharpness in the top, in the Isle Teneriff, which is believed to be the highest Mountain in the World; therefore the same Iohnson in his greatest Globe of the year 1616, hath drawn the great Meridian in that place, and it were to be wished, that this might be made the common and unchangeable practice.
25. The Horizon is a great Circle, designing so great a Part of the Earth, as a quick sight can discern in an open field; it is twofold Rational and Sensible.
26. The Rational Horizon is that which is supposed to pass through the Center of the Earth, and is represented by the wooden Circle in the Frame, as well of the Celestial, as the Terrestrial Globe, this Rational Horizon belongeth more to Astronomy than Geography.
27. The Sensible Horizon is that before defined, the use of it is to discern the divers risings and settings of the Stars, in divers places of the Earth, and why the days are sometimes longer, and sometimes shorter.
28. The great but less principal Circle upon the Terrestrial Globe is the Zodiack, in which the Sun doth always move. This Circle is described upon Globes and Maps for ornament sake, and to discover under what part of the Zodiack the several Nations lie.
29. The lesser Circles are those which do not divide the Terrestial Globe into two equal, but into two unequal Parts, and these by a general name are called Parallels, or Circles aequidistant [Page 417] from the Equinoctial; of which as many may be drawn, as there can Meridians, namely 180 if but to each degree, but they are usually drawn to every ten Degrees in each Quadrant from the AEquator to the Poles.
30. These Parallels are not of the same Magnitude, but are less and less as they are nearer and nearer to each Pole: and their use is to distinguish the Zones, Climates and Latitudes of all Countries, with the length of the Day and Night▪ in any Part of the World.
31. Again, a Parallel is either named or unnamed.
32. An unnamed Parallel is that which is drawn with small black Circular Lines.
33. A named Parallel is that which is drawn upon the Globe with a more full ruddy and circular Line: such as are the Tropicks of Cancer and Capricorn, with the Arctick and Antarctick Circles, of which having spoken before in the general description of the Globe, there is no need of adding more concerning them now.
CHAP. II.
Of the Distinction or Dimension of the Earthly Globe by Zones and Climates.
HAving shewed the primary distinction of the Globe into convenient spaces by Circles considered absolutely in themselves, we come now to consider the secondary Dimension or distinction of convenient spaces in the Globe, by the same Circles compared with one another, [Page 418] and by the spaces contained between those Circles.
2. This secundary Dimension or Distinction of the terrestial Globe into Parts, is either a Zone or a Clime.
3. A Zone is a space of the Terrestial Globe included either between two of the lesser nominated Circles, or between one and either Pole. They are in Number five, one over hot, two over cold, and two temperate.
4. The over hot or Torrid Zone, is between the two Tropicks, continually scorched with the presence of the Sun.
5. The two over cold or Frigid Zones, are scituated between the two polar Circles and the very Poles, continually wanting the neighbour hood of the Sun.
6. The two temperate Zones, are one of them between the Tropick of Cancer and the Arctick Circles and the other between the Tropick of Capricorn and the Antarctick Circle, enjoyning an indifferency between Heat and Cold; so that the parts next the Torrid Zone are the hotter, and the parts next the Frigid Zone are the Colder.
7. The Inhabitants of these Zones, in respect of the diversity of their noon Shadows are divided into three kinds, Amphiscii, Heteroscii and Periscii. Those that inhabit between the two Tropicks are called Amphiscii, because that their noon Shadows are diversly cast, sometimes towards the South as when the Sun is more Northward than their vertical point, and sometimes towards the North, as when the Sun declines Southward from the Zenith.
[Page 419] Those that live between the Tropick of Cancer and the Arctick Circle or between the Tropick of Capricorn and the Antarctick Circle are [...]alled Heteroscii, because the Shadows at noon are cast one only way, and that either North or South. They that inhabit Northward of the Tropick of Cancer have their Shadows always towards the North, and they that inhabit Southward of the Tropick of Capricorn, have their noon Shadows always towards the South.
Those that inhabit between the Poles and the Arctick or Antarctick Circles are called Periscii, because that their Gnomons do cast their Shadows circulary, and the reason hereof is, for that the Sun is carried round about above their Horizon in his whole diurnal revolution.
8. The next secundary Dimension or distinction of the earthy Globe into convenient parts or spaces, is by Climes.
9. And a Clime or Climate is a space of Earth conteined between three Paralells, the middlemo [...] whereof divideth it into two equal parts, serving for the setting out the length and shortness of the days in every Country.
10. These Climates and the Parallels by which they are conteined are none of them of equal quantity, for the first Clime as also the Parallel beginning at the AEquator is larger than the second, and the second is likewise greater than the third.
11. The Antients reckoned but seven Climates at the first, to which Number there were afterward added two more, so that in the first of these Numbers were comprehended fourteen parallels, but in the latter eighteen.
[Page 420] 12. Ptolemy accounted the Paralells 38 each way from the Equator, that is 38 towards the North, and as many towards the South, 24 of which he reckoned by the difference of one quarter of an hour, 4 by the difference of half an hour, 4 by an whole hours difference, and 6 by a Months difference, but now the parallels being reckoned by the difference of a quarter of an hour, the Climates are 24 in Number till you come to the Latitude of 66 degrees 31 Minutes, to which are afterwards added 6 Climates more unto the Pole it self, where the Artificial day is 6 Months in length.
13. The distances of all both Climates and Parallels, together with their Latitudes from the AEquator, and difference of the quantity of the longest days, are here fully exprest in the Table following.
| Inhabitants belonging to the several Climes | Climes | Paralells | Length of the Day | Poles Elevation | Bea of the Clime |
| 0 | 12.0 | 0.0 | |||
| 0 | 4.18 | ||||
| 1 | 12.15 | 4.18 | |||
| 2 | 12.30 | 8.34 | |||
| 1 | 8.25 | ||||
| Amphiscii | 3 | 12.45 | 12.43 | ||
| 4 | 13.0 | 16.43 | |||
| 2 | 7.50 | ||||
| 5 | 13.15 | 20.33 | |||
| 6 | 13.30 | 23.10 | |||
| 3 | 7.3 | ||||
| 7 | 13.45 | 27.36 | |||
| 8 | 14.0 | 30.47 | |||
| 4 | 6.9 | ||||
| 9 | 14.15 | 33.45 | |||
| 10 | 14.30 | 36.30 | |||
| 5 | 5.17 | ||||
| 11 | 14.45 | 39.02 | |||
| 12 | 15.0 | 41.22 | |||
| 6 | 4.30 | ||||
| 13 | 15.15 | 43.32 | |||
| 14 | 15.30 | 45.29 | |||
| 7 | 3.48 | ||||
| 15 | 15.45 | 47.20 | |||
| 16 | 16.0 | 49.21 | |||
| 8 | 3.13 | ||||
| 17 | 16.15 | 50.13 | |||
| 18 | 16.30 | 51.58 | |||
| 9 | 2.44 | ||||
| 19 | 15.45 | 53.17 |
| Climes | Paralells | Length of the Days | Poles Elevation | Breadth of the Clime | |
| 20 | 17.00 | 54.29 | |||
| 10 | 2.17 | ||||
| Heteroscii | 21 | 17.15 | 55.34 | ||
| 22 | 17.30 | 56.37 | |||
| 11 | 2.0 | ||||
| 23 | 17.45 | 57.34 | |||
| 24 | 18.00 | 58.26 | |||
| 12 | 1.40 | ||||
| 25 | 18.15 | 59.14 | |||
| 26 | 18.30 | 59.59 | |||
| 13 | 1.26 | ||||
| 27 | 18.45 | 60.40 | |||
| 28 | 19.00 | 61.18 | |||
| 14 | 1.13 | ||||
| 29 | 19.15 | 61.53 | |||
| 30 | 19.30 | 62.25 | |||
| 15 | 1.0 | ||||
| 31 | 19.45 | 62.54 | |||
| 32 | 20.00 | 63.22 | |||
| 16 | 0.52 | ||||
| 33 | 20.15 | 63.46 | |||
| 34 | 20.30 | 64.06 | |||
| 17 | 0.44 | ||||
| 35 | 20.45 | 64.30 | |||
| 36 | 21.00 | 64.49 | |||
| 18 | 0.36 | ||||
| 37 | 21.15 | 65.06 | |||
| 38 | 21.30 | 65.21 | |||
| 19 | 0.29 | ||||
| 39 | 21.45 | 65.35 | |||
| 40 | 22.00 | 65.47 | |||
| 20 | 0.22 | ||||
| 41 | 22.15 | 65.57 | |||
| 42 | 22.30 | 66.00 | |||
| 21 | 0.17 | ||||
| 43 | 22.45 | 66.14 |
| [Page 423] | Clime | Paralells | Length of the Day | Poles Elevation | Breadth of the Clime |
| 44 | 23.00 | 66.20 | |||
| 22 | 0.11 | ||||
| 45 | 23.15 | 66.25 | |||
| 46 | 23.30 | 66.28 | |||
| 23 | 0.5 | ||||
| 47 | 23.45 | 66.30 | |||
| 24 | 48 | 24.00 | 66.31 | 0.0 |
| Periscii | Here the Climates begin to be accounted by Months, from 66. 31 where the day is 24 hours long; unto the Pole it self, where it is 6 Months in length. | 1 | 67.15 |
| 2 | 69.30 | ||
| 3 | 73.20 | ||
| 4 | 78.20 | ||
| 5 | 84.0 | ||
| 6 | 90.0 |
[Page 424] 14. Hitherto we have considered the inhabitants of the Earth in respect of the several Zones and Climes into which the whole Globe is divided; there is yet another distinction behind into which the inhabitants of the Earth are divided in respect of their site and position in reference to one another, and thus the inhabitants of the Earth are divided into the Perioeci, Antoec [...] and Antipodes.
15. The Perioeci are such as dwell in the same Parallel on the same side of the AEquator, how far distant soever they be East and West, the season of the year and the length of the days being to both alike, only the midnight of the one is the moon to the other.
16. The Antoeci are such as dwell under the same Meridian and in the same Latitude, or Parallel distance from the AEquator, the one Northward and the other Southward, the days in both places being of the same length, but differ in the Seasons of the year, for when it is Summer in the one it is Winter in the other.
17. The Antipodes are such as dwell Feet to Feet, so as a right Line drawn from the one unto the other, passeth from North to South through the Center of the World. These are distant 180 degrees or half the compass of the Earth, they differ in all things, as Seasons of the year, length of days, rising and setting of the Sun and such like. A matter reckoned so ridiculous and impossible in former times, that Boniface Arch-Bishop of Mentz seeing a Treatise concerning these Antipodes written by Virgilius Bishop of Salisburg, and not knowing what damnable Doctrine might be couched under that [Page 425] strange Name, made complaint first to the Duke of Bohemia, and after to Pope Zachary Anno 745 by whom the poor Bishop (unfortunate only in being learned in such a time of Ignorance) was condemned of Heresie, but God hath blest this latter age of the World with more understanding, whereby we clearly see those things, which either were unknown, or but blindly guessed at by the Antients.
18. The second part of the Terrestial Globe is the Water which is commonly divided into these parts, or distinguished by these Names, Oceanus, Mare, Fretum, Sinus, Lacus and Flumen.
19. And first Oceanus or the Ocean is that general Collection of all Waters, which encompasseth the Earth on every side.
20. Mare the Sea, is a part of the main Ocean, to which we cannot come but through some Fretum or Strait, as Mare Mediterraneum. And it takes it name first either from the adjacent Shore, as Mare Adriaticum, from the City of Adria; or secondly from the first discoverer, as Mare Magellanicum, from Magellanus who first found it, or thirdly from some remarkable accident, as Mare Icarium from the drowning of Icarus the Son of Daedalus.
21. Fretum, a Strait is a part of the Ocean penned within some narrow Bounds, and opening a way into some Sea, or out of some Sea into the Ocean, as the Strait of Hellespont, Gibralter, &c.
22. Lacus, a Lake is a great body or collection of Water, which hath no visible Intercourse with the Sea, or influx into it; as the Lake of [Page 426] Thrasymene in Italy, and Lacus Asphaltites, or the dead Sea in the Land of C [...]naan.
23. Flumen or Fluvius is a water-course continually running, (whereby it differs frum Stagnum a standing Pool) issuing from some Spring or Lake, and emptying it self into some part of the Sea, or some other great River, the mouth or outlet of which is called Ostium.
And thus we have gon over those particulars both of Earth and Water, which appertain to this Science of Geography in the general; We will now proceed to a more particular Consideration of the several parts into which the Terrestial Globe is commonly divided.
CHAP. III.
Of Europe.
THe Terrestial Globe is divided into two parts, known or unknown.
2. The unknown or the parts of the World not fully discovered, are distinguished into North and South, the unknown parts of the World towards the North, are those which lie between the North part of Europe or America and the North Pole; and the unknown parts of the World toward the South, are those which ly between the South part of America and the South Pole.
3. The known parts of the World were antiently these three, Europe, Asia and Africk, to which in latter ages a fourth hath been added which is called America.
[Page 427] 4. Europe is bounded on the North with the Northern Ocean, and on the South with the Mediterranean Sea, on the East with the River Tanais, and on the West with the Western Ocean, and is contained between the Tropick of Cancer, and the Pole Arctick, or 44 degrees as most do say, taking its beginning Southward from Sicily where the Pole is elevated 36 degrees, and is thence continued to 80 degrees of North Latitude, and so the whole Latitude of Europe is in English miles 2640, but some allow to Europe 45 degrees of Latitude, that is in English miles 2700.
5. The Longitude of Europe is reckoned from the furthest part of Spain and the Atlantick Ocean, to the River Tanais, which some reckon to be 60 Degrees, to one of which Degrees passing through the middle of Europe, they allow fifteen German miles almost, or sixty English, and so the Longitude in German miles is 900, in English 3600.
6. Europe though the least of all the four Quarters of the World, is yet of most renown amongst us: First, because of the temperature of the Air, and fertility of the Soil: Secondly, from the study of Arts, both ingenuous and mechanical: Thirdly, of the Roman and Greek Monarchies: Fourthly, from the purity and sincerity of the Christian Faith: Fifthly, because we dwell in it, and so give it the first place.
7. Europe may be considered as it stands divided into the Continent and the Islands: the Continent lying all together, containeth these Countries. 1. Spain. 2. France. 3. Germany. 4. Italy, and the Alpes. 5. Belgium. 6. Denmark▪ [Page 428] 7. Swethland. 8. Russia. 9. Poland. 10. Hungary. 11. Sclavonia, 12. Dacia, and 13. Greece. Of each of which I will give some short account; as also of the chief Islands as they are dispersed, in the Greek, AEgaean, Cretan and Io [...]ian Seas, with those in the Adriatick, Mediterranean, and in the British and Northern Ocean.
8. Amongst these I give Spain the first place, as being the most Western Part of all the Continent of Europe environed on all sides with the Sea, except towards France; from which it is separated by the Pyrenaean Hills: but more particularly, it is bounded upon the North with the Cantabrian, on the West with the Atlantick Ocean, on the South with the Straits of Gibralter, on the East with the Mediterranean, and on the North East with the said Pyren [...]ean Hills. The Figure of it is compared by Strabo to an Oxes hide spread upon the Ground; the Neck whereof being that Isthmus which unites it to France.
9. The greatest length hereof, it reckoned at 800 miles, the breadth where it is broadest at 500, the whole Circumforence 2480 Italian miles: but Mariana measuring the compass of it by the bendings of the Pyrenaean Hills, and the creeks and windings of the Sea, makes the full circuit of it to be 2816 miles of Italian measure.
10. It is situate in the more Southerly Part of the Northern temperate Zone, and almost in the midst of the fourth and sixth Climates; the longest day being 15 hours and a quarter in length in the most Northern Parts hereof: but in the extream South near to Gibralter not above [Page 429] fourteen, which Situation of this Country, rendreth the Air here very clear and calm, seldom obscured with mists and vapours, and not so much subject to Diseases as the more Northern Regions are.
11. This Continent is subdivided into the Kingdoms of Navarr. 2. Biscay. 3. Guipusco [...]. 4. Lean and Oviodo. 5. Gallicia. 6. Corduba. 7. Granada. 8. Murcia. 9. Toledo. 10. Castile. 11. Portugal. 12. Valentia. 13. Catalonia. 14. Majorca. And 15. Aragon; but all of them are now united in the Monarchy of Spain.
12. France according to the present dimensions of it, is bounded on the East with a Branch of the Alpes which divide Dauphine and Piemont, as also with the Countries of Savoy, Switzerland, and some Parts of Germany and the Netherlands. On the West with the Aquitanick Ocean, and a Branch of the Pyrenaean Mountains which divide it from Spain. On the North with the English Ocean, and some Parts of Belgium, and on the South with the rest of the Pyrenaean Mountains, and the Mideterranean.
13. The Figure of it is almost square, each side of the Quadrature being reckoned 600 miles in length, but they that go more exactly to work upon it, make the length thereof to be 660 Italian miles, the breadth 570, the whole Circumference 2040. It is seated in the Northern temperate Zone, between the middle Parallel of the first Clime, where the longest day is 15 hours, and the middle Parallel of the eighth Clime, where the longest day is 16 hours and a half.
14. The Principal Provinces in this flourishing [Page 430] Country, are. 1. France specially so called. 2. Champagne. 3. Picardy. 4. Normandy. 5. Bretagne. 6. The Estates of Angiou. 7. La Beausio. 8. Nivernois. 9. The Dukedom of Bourbon. 10. Berry. 11. Poictou. 12. Limosin. 13. Piregort. 14. Quercu. 15. Aquitain. 16. Languedoc. 17. Provence. 18. Daulphine. 19. La Bresse. 20. Lionnois. 21. The Dutchy. 22. The County of Burgundy. 23. The Islands in the Aquitanick and Gallick Ocean: Those of most note are these six. 1. Oleron. 2. Re [...]. 3. Iarsey. 4. Gernsey. 5. Sarke. 6. Aldernay on the shores of Normandy, of which the four last are under the Kings of England.
15. Italy once the Empress of the greatest part of the then known World, is compassed with the Adriatick, Ionian and Tyrrhenian Seas, except it be towards France and Germany, from which it is parted by the Alpes; so that it is in a manner, a Peninsula, or a Demi-Island. But more particularly it hath on the East the lower part of the Adriatick, and the Ionian Sea, by which it is divided from Greece; on the West the River Varus, and some part of the Alpes, by which it is parted from France, on the North in some part the Alpes which divide it from Germany; and on the other, part of the Adriatick, which divides it from Dalmaria; and on the South the Tyrrhenian and Tuscan Seas, by which it is separated from the main Land of Africa.
16. It containeth in length from Augusta Praetoria, now called Aost, at the foot of the Alpes, unto Otranto in the most Eastern Point of the Kingdom of Naples 1020 miles; in breadth from [Page 431] the River Varo, which parts it from Provence, to the mouth of the River Arsia in Friuli, where it is broadest, 410 miles; about Otranto, where it is narrowest not above 25 miles; and in the middle parts from the mouth of Peseara in the Adriatick or upper Sea to the mouth of Tiber in the Tuscan or lower Sea, 126 miles. The whole compass by Sea reckoning in the windings and turnings of the shore, comes to 3038 miles; which added to the 410 which it hath by Land, make up in all 3448 miles: but if the Coasts on each side be reckoned by a straight Line, then as Castaldo computes it, it comes to no more than 2550 miles.
17. The whole Country lieth under the first and sixth Climates of the Northern temperate Zone, which it wholly taketh up: so that the longest day in the most Northern Parts is 15 hours and three first parts of an hour; the longest in the Southern Parts, falling short a full hour of that length.
18. Italy as it stands now is divided into the Kingdoms of Naples, Sicily and Sardinia. 2. The Land or Patrimony of the Church. 3. The great Dukedom of Tuscany. 4. The Commonwealths of Venice, Genoa and Luca. 5. The Estates of Lombardy, that is the Dukedoms of 1. M [...]llain. 2. Mantua. 3. Modena. 4. Parma. 5. Montferrat, and the Principality of Piedmont.
19. To the Peninsula of Italy belong the Alpes, aridge of Hills, wherewith as with a strong and defensible Rampart Italy is assured against France and Germany. They are said to be five days Journey high, covered continually with Snow, [Page 432] from the whiteness whereof they took this name, it doth contain the Dukedom of Savoy; the Seigniory of Geneva; the Country of Wallisland, Switzerland and the Grisons.
20. Belgium, or the Netherlands, is bounded on the East with Westphalin, Gulick, Cleve, and the Land of Triers, Provinces of the higher Germany; on the West with the main Ocean, which divides it from Britain; on the North with the River Ems, which parts it from East-Friezeland; on the South with Picardie and Campagne two French Provinces; upon the South-East with the Dukedom of Lorrain.
21. It is in compass 1000 Italian or 280 German miles, and is situated in the Northern temperate Zone, under the seventh, eighth and ninth Climates: the longest day in the midst of the seventh Climate where it doth begin, being 16 hours, iu the beginning of the ninth Climate increased to 16 hours 3 quarters, or near 17 hours.
22. It containeth those Provinces which in these later Ages were possessed by the House of Burgundy, that is the Lordship of West-Friezeland, given to the Earls of Holland by Charles the Bald; the Earldom of Zutphen united unto that of Gelder by Earl Otho of Nassau, and finally the Estate of Groening, Over-Yssel, and some part of Vtrecht, by Charles the Fifth. As it stands now divided between the Spaniards and the States it containeth the Provinces of 1. Flanders. 2. Artois. 3. Hainault. 4. The Bishoprick of Cambray. 5. Namur. 6. Luxemburg. 7. Limbourg. 8. Luyckland, or the Bishoprick of Leige. 9. Brabant. 10. Marquisate. 11. Meohlin. The rest of the Netherlands [Page 433] which have now for sometime withdrawn their obedience from the Kings of Spain, are 1. Holland. 2. Zeland. 3. West-Friezeland. 4. Vtrecht. 5. Over-Yssel. 6. Gelderland. 7. Zutphen. 8. Groening.
23. Germany is bounded on the East with Prussia, Poland, and Hungary; on the West with France, Switzerland and Belgium; on the North with the Baltick Seas, the Ocean, and some part of Denmark; on the South with the Alps which part it from Italy.
24. The length from East to West, that is from the Vistula or Weissel to the Rhine, is estimated at 840 Italian miles, the breadth from North to South, that is from the Ocean to the Town of Brixen in Tyrol, 740 of the same miles. So that the Figure of it being near a Square, it may take up 3160 miles in compass, or thereabouts. Situate in the Northern temperate Zone, between the middle Parallels of the sixth and tenth Climates; the longest day in the most Southern Parts being 15 hours and an half, and in the most Northern 17 hours and a quarter.
25. The Principal Parts of this great Continent, are 1. Cleveland. 2. The Estates of the three spiritual Electors, Colen, [...]ntz, and Triers. 3. The Palatinate of the Rhine. 4. Alsatia. 5. Lorrain. 6. Suevia or Schwaben. 7. Bavaria. 8. Austria and its Appendices. 9. The Confederation of Waderaw. 10. Franconia. 11. Wirtenberg. 12. Baden. 13. The Palatinate of Northgoia, or the Upper Palatinate. 14. Bohemia and the Incorporate Provinces. 15. Pomerania. 16. Mecklenburg. 17. The Marquisate of Brandenburg. [Page 434] 18. Saxony, and the Members of it. 19. The Dukedom of Brunswick and Lunenburg. 20. The Lantgravedom of Hassia. 21. Westphalen. 22. East-Friezeland.
26. Denmark or Danemark, reckoning in the Additions of the Dukedom of Holstein, and the great Continent of Norway, with the Isles thereof, now all united and incorporated into one Estate is bounded on the East with the Baltick Sea and some part of Sweden; on the West with the main Western Ocean; on the North-East with a part of Sweden; full North with the main frozen Seas; and on the South with Germany, from which it is divided on the South-West by the River Albis, and on the South-East by the Trave; a little Isthmus or neck of Land uniting it to the Continent.
27. It lieth partly in the Northern temperate Zone, and partly within the Arctick Circle; extending from the middle Parallel of the tenth Clime, or 55 degree of Latitude where it joyneth with Germany, as far as the 71 degree where it hath no other bound but the frozen Ocean; by which account the longest day in the most Southern Parts is 17 hours and a quarter, but in the Parts extreamly North, they have no Night for two whole Moneths, three Weeks, one Day, and about seven hours; as on the other side no day for the like quantity of time, when the Sun is most remote from them, in the other Tropick.
28. The whole Body of the Estate consisteth chiefly of three Members. viz. 1. The Dukedom [Page 435] of Holstein; containing Waggerland, Dilmarsh, Starmaria, and Holstein, especially so called. 2. The Kingdom of Denmark; comprehending both Iuitlands, part of Scandia, and the Hemodes, or Baltick Islands. 3. The Kingdom of Norway consisting of Norway it self, and the Islands of the Northern Ocean.
29. Swethland is bounded on the East with Muscovy, on the West with the Doferine Hills, which divide it from Norway; on the North with the great frozen Ocean spoken of before; on the South with Denmark, Liefland, and the Baltick Sea.
30. It is situate under the same Parallels and Degrees with Norway, that is, from the first Parallel of the 12 Clime, where the Pole is elevated 58 degrees 26 minutes, as far as to the 71 degree of Latitude, by which account the longest day in the Southern Point is but 18 hours, whereas on the farthest North of all the Countrey, they have no Night for almost three whole Moneths together.
31. The whole Kingdom is divided into two Parts, the one lying on the East, the other on the West of the Bay or Gulf of Bodener, being a large and spacious branch of the Baltick Sea, extending from the most Southerly Point of Gothland, as far as to Lapland on the North. According to which Division we have the Provinces of 1. Gothland. 2. Sweden lying on the West side of the Gulph. 3. Lapland shutting it up upon the North. 4. Bodia or Bodden. 5. Finland on the East side thereof. 6. The Sweedish Islands, where it mingleth with the rest of the Baltick Seas.
32. Russia is bounded on the East by Tartary, [Page 436] on the West with Livonia and Finland, from which it is divided by great mountains and the River Poln, on the North by the frozen Ocean, and some part of Lapland, and on the South by Lituania a Province of the Kingdom of Poland, and the Crim Tartar inhabiting on the Banks of Palus Maeotis, and the Euxine Sea. It standeth partly in Europe and partly in Asia, the River Tanais or Don running through it, the common boundary of those great and noted parts of the world.
33. It is scituate North within the Artick Circle so far, that the longest day in Summer will be full six months, whereas the longest day in the southern parts is but 16 hours and an half.
34. It is divided into the Provinces of 1. Moscovy specially so called. 2. Snol [...]usio, 3. Masaisky, 4. Plesco, 5. Novagrod the great, 6. Corelia, 7. Blarmia, 8. Petzora, 9. Condora, 10 Obdora, 11. Iugria, 12. Severia, 13. Permia, 14. Rozan, 15. Wiathka, 16. Casau, 17. Astracan, 18. Novogordia inferiour, 10. The Morduits or Mordua, 20. Worotime, 21. Tuba, 22. Wolodomir, 23. Duina, 24. the Russian Islands.
35. Poland is bounded on the East with Russia, and the Crim-Tartar, from whom it is parted by the River Borysthenes; on the West with Germany, on the North with the Baltick Sea and some part of Russia, on the South with the Carpathian Mountains, which divide it from Hungary, Transilvania, and Moldavia. It is of figure round in compass 2600 miles, scituate under the 8 and 12 Climates, so that the longest day in the southern parts is but 16 hours, and about 18 [Page 437] hours in the parts most North.
36. The several Provinces of which this Kingdom doth consist, are 1. Livonia, 2. Samogitia, 3. Lituania, 4. Volkinia, 5. Podolia, 6. Russia nigra, 7. Massovia, 8. Podlassia, 9. Prussia, 10. Pomerellia, 11. Poland specially so called.
37. Hungary is bounded on the East with Transilvania and Walachia, on the West with Sterria, Austria and Moravia, on the North with the Carpathian mountains which divide it from Poland, and on the South with Sclavonia, and some part of Dacia: it extendeth in length from Presburg along the Danow to the borders of Transilvania, for the space of 300 English miles, and 190 of the same miles in breadth.
38. It lieth in the Northern temperate Zone, betwixt the middle parallels of the 7 and 9 Climates, so that the longest Summers day in the Southern parts is but 15 hours and an half, and not above 16 hours in the parts most North.
40. This Country is commonly divided into the upper Hungary and the lower, the upper lying on the North of the River Danow, the lower lying on the South of that River, comprehending all Pannonia inferior and part of Superior, and is now possessed by the King of Hungary and the Great Turk, who is Lord of the most part by Arms and Conquest.
04. Sclavonia is bounded on the East with Servia, Macedonia and Epirus, from which it is parted by the River Drinus, and a line drawn from thence unto the Adriatick, on the West with Carniola in Germany, and Istria in the Seigniory [Page 438] of Venice, from which last it is divided by the River Arsia; on the North with Hungary, on the South with the Adriatick Sea.
41. It is scituate in the Northern temperate Zone, between the middle Parallels of the sixth and seventh Climates, so that the longest day in Summer is about 15 hours and an half.
42. This Country as it came at last to be divided, between the Kings of Hungary and the State of Venice; is distinguished into 1. Windischland, 2. Croatia, 3. Bosnia, 4. Dalmatia, 5. Liburnia or Cantado di Zara, and 6. The Sclavonian Islands.
43. Dacia is bounded on the East with the Euxine Sea and some part of Thrace; on the West with Hungary and Sclavonia; on the North with Podolia, and some other members of the Realm of Poland, on the South with the rest of Thrace and Macedonia.
44. It lieth on both sides of the Danow fronting all along the upper and the lower Hungary, and some part of Sclavonia; extended from the 7 Climate to the 10; so that the longest Summers day in the most northern parts thereof, is near 17 hours, and in the most southern 15 hours 3 quarters.
45. The several Provinces comprehended under the name of Dacia, are 1. Transilvania, 2. Moldavia, 3. Walachia, 7. Rascia, 5. Servia, 6. Bulgaria, the first four in old Dacia, on the North side of the Danow; the two last in new Dacia, on the South thereof.
45. Greece in the present Latitude and extent thereof, is bounded on the East with the [Page 439] Propontick, Hellespont, and AEgean Seas, on the West with the Adriatick; on the North with Mount Haemus which parteth it from Bulgaria, Servia and some part of Illyricum; and on the South with the Sea-Ionian; so that it is in a manner a Peninsula or Demi-Island, environed on three sides by the Sea, on the fourth only united to the rest of Europe.
46. It is scituate in the northern temperate Zone, under the fifth and sixth Climates, the longest day being 15 hours.
47. In this Country formerly so famous for learning and government, the several Provinces are 1. Peloponnesus, 2. Achaia, 3. Epirus, 4. Albania, 5. Macedon, 6. Thrace, 7. The Islands of the Propontick; 8. AEgean, and 9. The Ionian Seas, and 10. finally the Isle of Crete.
And thus I have given you a brief description of those Countries which are comprehended in the Continent of Europe; the Islands in this part of the world are many; I will mention only some few. These two in the British and Northern Ocean, known by the names of Great Britain and Ireland are the most famous, to which may be added Greenland. In the Mediterranaen Sea you have the Islands of Sicilia, Sardinia, Corsica and Crete, which is now called Candia the greater and the less: As for the other Islands belonging to this part of the world, the Reader may expect a more particular description from them who have or shall write more largely of this subject: This we deem sufficient for our present purpose. Let this then suffice for the description of the first part of the World called Europe.
CHAP. IV.
Of Asia.
ASia is bound on the West with the Mediteranean and AEgaean Seas, the Hellespont, Propontis, Thracian Bosphorus and the Euxine Sea, the Palus Maeotis, the Rivers Tanais and Duina, a Line being drawn from the first of the two said Rivers unto the other, by all which it is parted from Europe; on the North it hath the main Scythick Ocean; but on the East the Indian Ocean, and Mare del Eur by which it is separated from America; on the South the Mediterranean, or that part of it, which is called the Carpathian Sea, washing the shoars of Anatolia, and the main Southern Ocean, passing along the Indian, Persian and Arabian Coasts: and finally on the south-west, the red Sea or Bay of Arabia, by which it is parted from Affrick. Environed on all sides with the Sea, or some Sea like Rivers, except a narrow Isthmus in the south-west, which joyns it to Africk, and the space of ground (whatsoever it be) between Duina and Tanais, on the North-west which unites it to Europe.
2. It is situated East and West, from the 52 to the 169 degree of Longitude; and North and South from the 82 degree of Latitude to the very AEquator; some of the Islands only lying on the South of that Circle: so that the longest summers day in the southern parts, is but twelve hours, but in the most northern parts hereof almost four whole Months together.
[Page 441] 3. This Country hath heretofore been had in special honour; 1. For the creation of Man, who had his first making in this part of the World. 2. Because in this part of it stood the Garden of Eden, which he had for the first place of his habitation. 3. Because here flourished the four first great Monarchies of the Assyrians, Babylonians, Medes and Persians. 4. Because it was the Scene of almost all the memorable Actions which are recorded by the penmen of the Scriptures. 5. Because our Saviour Christ was borne here, and here wrought his most divine Miracles, and accomplished the great work of our Redemption. 6. And finally, because from hence all Nations of the World had their first beginning, on the dispersion which was made by the Sons of Noah after their vain attempt at Babel.
4. This part of the World for the better understanding of the Greek and the Roman Stories and the estate of the Assyrian, Babylonian and the Persian Monarchies, to which the holy Scriptures do so much relate, we shall consider as divided into the Regions of 1. Anatolia or Asia minor. 2. Cyprus. 3. Syria. 4. Arabia. 5. Chaldea. 6. Assyria. 7. Mesopotamia. 8. Turcomania. 9. Media. 10. Persia. 11. Tartaria. 12. China. 13. India. and 14. the Oriental Islands.
Anatolia or Asia minor.
Anatolia or Asia minor, is bounded on the East with the River Euphrates, by which it is parted from the greater Asia; on the West with the Thracian Bosphorus, Propontis, Hellespont, and [Page 442] the AEgean Sea, by which it is parted from Europe; on the North with Pontus Euxinus, called also the black Sea, and Mare Maggiore, and on the South by the Rhodian, Lydian and Pamphilian Seas, several parts of the Mediterranean. So that it is a Demi-Island or Peninsula environed on all sides with water, excepting a small Isthmus or Neck of Land extending from the head of Euphrates to the Euxine Sea, by which it is joyned to the rest of Asia.
It reacheth from the 51 to the 72 degree of Longitude, and from the 36 to the 45 degree of Latitude, and lyeth almost in the same position with Italy, extending from the middle Parallel of the fourth Clime, to the middle Parallel of the sixth, so that the longest summers day in the Southern Parts, is about 14 hours and a half; and one hour longer in those parts which lie most towards the North.
The Provinces into which it was divided before the Roman Conquest were 1. Bithynia. 2. Pontus. 3. Paphlagonia. 4. Galatia. 5. Cappadocia. 6. Armenia Major & Minor. 7. Phrygia minor. 8. Phrygia major. 9. Mysia the greater and the less. 10. Asia specially so called, comprehending AEolis and Ionia. 11. Lydia. 12. Caria. 13. Lycia. 14. Lycaonia. 15. Pisidia. 16. Pamphylia. 17. Isauria. 18. Cilicia. 19. The Province of the Asian Isles, whereof the most principal are 1. Tenedos. 2. Chios. 3. Samos. 4. Choos. 5. Icaria. 6. Lesbos. 7. Patmos. 8. Claros. 9 Carpathos. 10. Rhodes.
Cyprus.
Cyprus is situated in the Syrian and Cilician Seas, extended in length from East to West 200 miles, in breadth 60 the whole compass reckoned 550, distant about 60 miles from the rocky Shores of Cilicia in Asia minor, and about one hundred from the main Land of Syria.
It is situated under the fourth Climate, so that the longest day in Summer is no more than 14 hours and a half.
Divided by Ptolemy into the 4 provinces of 1. Paphia. 2. Amathasia. 3. Lepathia. 4. Salamine.
Syria.
Syria is bounded on the East with the River Euphrates by which it is parted from Mesopotamia; on the West with the Mediterranean Sea; on the North with Cilicia and Armenia minor, parted from the last by mount Taurus; and on the South with Palestine, and some parts of Arabia. The length hereof from Mount Taurus to the Edge of Arabia, is said to be 525 Miles; the breadth from the Mediterranean to the River Euphrates 470 Miles, drawing somewhat near unto a Square.
The whole Country was antiently divided into these six parts. 1. Phoenicia. 2. Palestine. 3. Syria specially so called. 4. Comagena. 5. Palmyrene. and Caelosyria, or Syria Cava.
Arabia.
Arabia hath on the East Chaldaea and the Bay or Gulf of Persia; on the West Palestine, some part of Egypt, and the whole course of the red Sea, on the North the River Euphrates with some parts of Syria and Palestine, and on the South the main southern Ocean. It is in circuit about 4000 Miles, but of so unequal and heteregeneous Composition, that no general Character can be given of it, and therefore we must look upon it as it stands divided into Arabia Deserta, 2. Arabia Petraea. 3. Arabia Felix and 4. The Arabick Islands.
Chaldea.
Chaldea is bounded on the East with Susiana a Province of Persia; on the West with Arabia deserta; on the North with Mesopotamia; and on the South with the Persian Bay and the rest of Deserta.
Assyria.
Assyria is bounded on the East with Media, from which it is parted by the Mountain called Coathras; on the West with Mesopotamia, from which it is divided by the River Tygris; on the South with Susiana; and on the North with some part of Turcomania; it was antiently divided into six parts. 1. Arraphachitis. 2. Adiabene. 3. Calacine. 4. Aobelites. 5. Apolloniates.
Mesopotamia.
Mesopotamia is bounded on the East with the River Tygris by which it is parted from Assyria; on the West with Euphrates which divides it from Comagena a Province of Syria; on the North with Mount Taurus; by which it is separated from Armenia major; and on the South with Chaldea and Arabia deserta from which last it is parted by the bendings of Euphrates also. It was antiently divided into, 1. Anthemasia. 2. Chalcitis. 3. Caulanitis. 4. Acchabene. 5. Ancorabitis and 6. Ingine.
Turcomania.
Turcomania is bounded on the East with Media and the Caspian Sea; on the West with the Euxine Sea, Cappadocia and Armenia minor; on the North with Tartary, and on the South with Mesopotamia and Assyria. A Countrey which consisteth of four Provinces. 1. Armenia major or Turcomania properly and specially so called. 2. Colchis. 3. Iberia. 4. Albania.
Media.
Media is bounded on the East with Parthia, and some part of Otyrcania, Provinces of the Persian Empire; on the West with Armenia major, and some part of Assyria; on the North with the Caspian Sea and those parts of Armenia major, which now pass in the account of Iberia, Georgia; and on the South with Persia. It is now [Page 446] divided into two Provinces. 1. Atropatia. 2. Media major.
Persia.
Persia is bounded on the East with India; on the West with Media, Assyria, and Chaldea; on the North with Tartary, on the South with the main Ocean.
It is divided into the particular Provinces of 1. Susiana. 2. Persis. 3. Ormur. 4. Carmania. 5. Gedrosia. 6. Drangiana. 7. Arachosia. 8. Paropamisus. 9. Aria. 10. Parthia. 11. Hyrcania. 12. Margiana and 13. Bactria.
Tartaria.
Tartaria is bounded on the East with China, the Oriental Ocean, and the Straits of Anian, by which it is parted from America, on the West with Russia and Podolia, a Province of the Realm of Poland; on the North with the main Scythick or frozen Ocean; and on the South with part of China, from which it is separated by a mighty Wall, some part of India, the River Oxus parting it from Bactria and Margiana, two Persian Provinces; the Caspian Sea which separates it from Media and Hyrcania; the Caucasian Mountains interposing between it and Turcomania; and the Euxine Sea which divideth it from Anatolia and Thrace.
It reacheth from the 50 degree of Longitude to the 195 which is 145 degrees from West to East; and from the 40 degree of Northern Latitude, unto the 80, which is within 10 degrees [Page 447] of the Pole it self, By which accompt it lieth from the beginning of the sixth Clime, where the longest day in Summer is 15 hours, till they cease measuring the Climates, the longest day in the most Northen parts hereof being full six Months, and in the winter half of the Year, the night as long.
It is now divided into these five parts. 1. Tartaria Precopensis. 2. Asiatica. 3. Antiqua. 4. Zagathay. 5. Cathay.
China.
China is bounded on the North with Altay and the Eastern Tartars, from which it is separated by a continued Chain of Hills, part of those of Ararat, and where that chain is broken off or interrupted, with a great wall extended 400 Leagues in length; on the South partly with Cauchin China a Province of India, partly with the Ocean; on the East with the oriental Ocean, and on the West with part of India and Cathay.
It reacheth from the 130 to the 160 degree of Longitude, and from the Tropick of Cancer to the 53 degree of Latitude; so that it lieth under all the Climes from the third to the ninth inclusively. The longest summers day in the southern parts being 13 hours and 40 Minutes increased in the most northern parts to 16 hours and 3 quarters.
It containeth no fewer than 15 Provinces. 1. Canton. 2. Foquien. 3. Olam. 4. Sisnam 5. Tolenchia. 6. Causay. 7. Minchian. 8. Ochian. 9. Honan. 10. Pagnia. 11. Taitan. 12. Quinchen. 13. Chagnian 14. Susnan. 15. Cunisay. [Page 448] Besides the provinces of Suehuen, the Island of Chorea and the Island of Cheaxan.
India.
India is bounded on the East with the Oriental Ocean and some part of China; on the West with the Persian Empire; on the North with some Branches of Mount Taurus, which divide it from Tartary; on the South with the Indian Ocean.
Extended from 106 to 159 degrees of Longitude, and from the AEquator to the 44th degree of Northern Latitude, by which account it lieth from the beginning of the first to the end of the sixth Clime, the longest Summers day in the southern Parts being 12 hours onely, and in the parts most North 15 hours and a half.
The whole Country is divided into two main parts, India intra Gangem, and India extra Gangem.
The Oriental Islands.
The Oriental Islands are 1. Iapan. 2. The Philippine and Isles adjoyning. 3. The Islands of Bantam. 4. The Moluccoes. 5. Those called Sinda or the Celebes. 6. Iava. 7. Borneo. 8. Sumatra. 9. Ceilan. and 10. others of less note.
CHAP. V.
Of Africk.
AFrick is bounded on the East by the Red Sea, and Bay of Arabia, by which it is parted from Asia; on the West by the main Atlantick Oceans interposing between it and America; on the North by the Mediterranean Sea, which divides it from Europe and Anatolia; and on the South with the AEthiopick Ocean, separating it from Terra Australis incognita or the southern continent, parted from all the rest of the World except Asia only, to which it is joyned by a narrow Isthmus not above 60 miles in length.
It is situate for the most part under the Torrid Zones, the AEquator crossing it almost in the midst. It is now commonly divided into these seven parts. 1. AEgypt. 2. Barbary or the Roman Africk. 3. Numidia. 4. Lybia. 5. Terra Nigritarum. 6. AEthiopia superior. and 7. AEthiopia rinferior.
AEgypt.
AEgypt is bounded on the East with Idumea, and the Bay of Arabia; on the West with Barbary, Numidia, and part of Lybia; on the North with the Mediterranean Sea; on the South with AEthiopia superior, or the Abyssyn Emperor; it is situate under the second and fifth Climates, so that the longest day in Summer is but thirteen hours and a half.
Barbary.
Barbary is bounded on the East with Cyrenaica; on the West with the Atlantick Ocean; on the North with the Mediterranean Sea, the Straits of Gibralter and some part of the Atlantick also; on the South with Mount Atlas, by which it is separated from Lybia inferior or the Desarts of Lybia.
It is situated under the third and fourth Climates: so that the longest Summers day in the parts most South, amounteth to 13 hours and 3 quarters, and in the most northern parts it is 14 hours and a quarter. This country is now reduced to the Kingdoms of 1. Tunis. 2. Tremesch or Algiers. 3. Fesse and 4. Morocco.
Numidia.
Numidia is bounded on the East with Egypt, on the West with the Atlantick Ocean; on the North with Mount Atlas, which parteth it from Barbary and Cyrene; on the South with Lybia Deserta.
Lybia.
Lybia is either Interior or Deserta, Libia interior is bounded on the North with Mount Atlas by which it is parted from Barbary and Cyrenaica; on the East with Lybia Marmarica interposed between it and Egypt, and part of AEthiopia superior, or the Habassine Empire; on the South with AEthiopia inferior, and the Land of the Negroes; [Page 451] and on the West with the main Atlantick Ocean.
Lybia deserta is bounded on the North with Numidia or Biledulgerid; on the South with the Land of the Negroes; and on the West with Gulata another Province of the Negroes interposed between it and the Atlantick.
Terra Nigritarum.
Terra Nigritarum or the Land of the Negroes is bounded on the East with AEthiopia Superior; on the West with the Atlantick Ocean; on the North with Lybia deserta and on the South with the Ethiopick Ocean, and part of AEthiopia Inferior.
AEthiopia Superior.
AEthiopia Superior is bounded on the East with the Red Sea and the Sinus Barbaricus; on the West with Lybia Interior, the Realm of Nubia in the Land of the Negroes and part of the Kingdoms of Congo in the other AEthiopia; on the North with Egypt and Lybia Marmarica, and on the South with the Mountains of the Moon, by which it is parted from the main Body of AEthiopia Inferior.
It is situate on both sides of the AEquinoctial, extending from the South Parallel of seven degrees, where it meeteth with some part of the other AEthiopia to the Northern end of the Isle of Meroz situated under the fifth Parallel on the North of that Circle.
AEthiopia Inferior.
AEthiopia inferior is bounded on the East with the Red Sea; on the West with the Ethiopick Ocean; on the North with Terra Nigritarum, and the higher AEthiopia; and on the South where it endeth, is a point of a Conus, with the main Ocean parting it from the Southern undiscovered Continent. This in Ptolemyes time went under the name of Terra incognita.
CHAP. IV.
Of America.
AMerica the fourth and last part of the World is bounded on the East with the Atlantick Ocean and the Vergivian Seas, by which it is parted from Europe and Africa; on the West with the Pacifick Ocean, which divides it from Asia; on the South with some part of Terra Australis incognita, from which it is separated by a long, but narrow Strait, called the Straits of Magellan; the North bounds of it hither to not so well discovered, as that we can certainly affirm it to be Island or Continent.
It is called by some and that most aptly, The new World; New for the late discovery, and World for the vast greatness of it. The whole is naturally divided into two great Peninsules, whereof that towards the North is called Mexicana. That towards the South hath the name of Peruana: the Isthmus which joyneth these two [Page 453] together is very long, but narrow in some places not above 120 miles from Sea to Sea, in many not above seventeen.
The Northern Peninsula called Mexicana, may be most properly divided into the Continent and Islands: The Continent again into the several Provinces of 1. Estotiland, 2. Nova Francia, 3. Virginia, 4. Florida, 5. California, 6. Nova Gallicia, 7. Nova Hispania, 8. Guntimala. The Southern Peninsula called Peruana, taking in some part of the Isthmus, hath on the Continent the Provinces of 1. Castella Aurea, 2. Nova Granada, 3. Peru, 4. Chile, 5. Paraguay, 6. Brasil, 7. Guiana, and 8. Paria. The Islands which belong to both, are dispersed either in the Southern Ocean called Mare del Zur, where there is not any one of Note but those called Los Ladrones and the Islands of Solomon. Or in the Northern Ocean called Mare del Noords, reduced unto the Caribes, Porto-Rico, Hispaniola, Cuba and Iamaica. And thus much concerning the real and known parts of the Terrestrial Globe.
CHAP. XV.
Of the Description of the Terrestrial Globe by Maps Vniversal and Particular.
HItherto we have spoken of the true and real Terrestrial Globe, and of the measure thereof by Circles, Zones, and Climates, as it is usually represented by a Sphere or Globe; which must be confessed to be the nearest and the most▪ commensurable to nature: Yet it may also be [Page 454] described upon a plain, in whole or in part many several ways: But those which are most useful and artificial are these two, by Parallelogram and by Planisphere.
2. The description thereof by Parallelogram is thus, the Parallelogram is divided in the midst by a line drawn from North to South, passing by the Azores or Canaries for the great Meridian. Cross to this and at eight Angles, another line is drawn from East to West for the AEquator; then two parallels to each to comprehend the figure, in the squares whereof there are set down four parts of the world rather than the whole: And this way of description though not exact or near to the natural, hath yet been followed by such as ought still to be accounted excellent, and is the form of our plain Charts, and in places near the AEquinoctial may be used without committing any great error; because the Meridians about the AEquinoctial are equi-distant, but as they draw up towards the Pole, they do upon the Globe come nearer and nearer together, to shew that their distance is proportionably diminished till it come to a concurrence, and answerably the Parallels as they are deeper in latitude, so they grow less and less with the Sphere; so that at 60 degrees, the Equinoctial is double to the parallel of Latitude, and so proportionably of the rest.
3. Hence it followeth, that if the picture of the earth be drawn upon a Parallelogram, so that the Meridians be equally distant throughout, and the Parallels equally extended, the Parellel of 60 degrees shall be as great as the line of the AEquator it self is, and he that coasteth about the world [Page 455] in the latitude of 60 degrees, shall have as far to go by this Map, as he that doth it in the AEquator, though the way be but half as long. For the longitude of the Earth in the AEquator it self, is 21600; but in the Parallel of 60 but 10800 miles. So two Cities under the same parallel of 60, shall be of equal Longitude to other two under the Line, and yet the first two shall be but 50, the other two an hundred miles distant. So two Ships departing from the AEquator at 60 miles distance, and coming up to the Parallel of 60, shall be thirty miles nearer, and yet each of them keep the same Meridians and sail by this Card upon the very points of the Compass at which they set forth. This was complained of by Martin Cortez and others, and the learned Mercator considering well of it, caused the degrees of the Parallel to increase by a proportion towards the Pole. The Mathematical Generation whereof, Mr. Wright in the second Chapter of his Correction of Errors in Navigation, hath sought by the inscription of a Planisphere into a Concave Cylinder. And this description of the Earth upon a Parallelogram, may indeed be so ordered by Art, as to give a true account of the scituation and distance of the parts, but cannot be fitted to represent the figure of the whole.
4. The description therefore of the whole by Planisphere is much better, because it represents the face of the Earth upon a plain, in its own proper Spherical Figure as upon the Globe it self. This description cannot well be contrived upon so few as one Circle or more than two.
Suppose then the Globe to be divided into [Page 456] two equal parts or Hemispheres, which cannot be done but by a great Circle: And therefore it must be done by the AEquator or Meridian (for the Colure is all one with the Meridian) the Horizon cannot fix, and the Zodiack hath nothing to do here.
5. Suppose then the Globe to be flatted upon the plain of the AEquator, and you have the first way of projection dividing the Globe into the North and South Hemispheres.
In this projection the Pole is the Centre, the AEquator is the Circumference divided into 360 degrees of Longitude, the Paralles are whole Circles, the Meridians are streight lines, the Parallels are Parallels indeed, and the Meridians equi-distantly concur, and therefore all the degrees are equal. After this way of projection, Ptolemy describes that part of the habitable world which was discovered to his time.
6. Suppose the Globe to be flatted upon the plain of the Meridian, and you have the other way of projection; the AEquator here is a streight line, the great Meridian a whole Circle, in this Section the Meridians do not equi-distantly concur, the Parallels are not Parallels indeed, and therefore the degrees are all unequal.
However, this latter way is that which is now most and indeed altogether in use.
7. Particular Maps are but limbs of the Globe, and therefore though they are drawn asunder, yet are they still to be done with that proportion, as a remembring eye may suddenly acknowledge, and joyn them to the whole Body.
[Page 457] The Projection is most commonly upon a Parallelogram, in which the Latitude is to be expressed by Paralles from North to South, and the Longtitude by Meridians from West to East at 10 or 15 degrees distance, as you please, and may be drawn either by circle or right Lines; but if they be right Lines, the Meridians are not to be drawn parallel, but inclining and concurring, to shew the nature of the whole, whereof they are such parts. For the Graduation; the degrees of Longitude are most commonly divided upon the North and South sides of the Parallelogram; the degrees of Latitude upon the East and West sides, or otherwise upon the most Eastern or Western Meridian of the Map, within the square. But it hath seemed good to some in these particular descriptions to make no graduation or projection at all; but to put the matter off to a scale of Miles, and leave the rest to be believed.
The difference of Miles in several Countries is great, but it will be enough to know that the Italian and English, are reckoned for all one, and four of these do make a German Mile; two a French League. The Swedish or Danish Mile consisteth of 5 Miles English and somewhat more. Sixty common English and Italian Miles answer to a degree of a great Circle.
Now as the Miles of several Countries do very much differ, so those of the same do not very much agree: and therefore the scales are commonly written upon with Magna, Mediocria and Parva, to shew the difference. In some Maps you shall find the Miles thus hiddenly set down, and the meaning is, that you should measure the [Page 458] Milliaria magna upon the lowermost Line, the Parva upon the uppermost, and the Mediocria upon the middlemost.
Scala Milliarium.
The use of the Scale is for the measuring the distances of places in the Map, by setting one foot of your Compasses in the little circle representing one place, and the other foot in the like little circle representing another, the Compasses kept at that distance being applied to the Scale, will shew the number of great or middle Miles according as the inhabitants of those places are known to reckon.
Soli Deo Gloria.
| Epochae. | Years of the Julian Period. | Months |
| The Julian Period | 1 | Ian. 1 |
| Creation of the World | 765 | Ian. 1 |
| AEra of the Olympiades | 3938 | Iuly 8 |
| The building of Rome | 4961 | Ap. 21 |
| Epochae of Nabonasser | 3667 | Feb. 26 |
| The beginning of Metons Cyrcle. | 4281 | Iune 26 |
| The beginning of the periods of Calippus | 4384 | Iune 28 |
| The Death of Alexander the great | 4390 | No. 12 |
| AEra of the Caldees | 4403 | Oct. 15 |
| The AEra of Dionysrus | 4429 | Mar. 25 |
| The beginning of the Christian AEra falls in the 4713 year of the Julian Period. | Years of Christ | Month |
| The Dioclesian AEra | 284 | Aug. 29 |
| The Turkish AEra or Hegyra | 622 | Iuly 16 |
| The Persian AEra from Iesdagird | 632 | Iune 16 |
| The AEra from the Persian Sultan | 1079 | Mar. 14 |
| Julian Accompt | AEgypt and Persian Accompt | |||||||||||||||||
| 1 | 0 | 0 | 0 | 365 | 2 | 5 | 0 | 1 | 0 | 0 | 0 | 365 | 0 | 0 | 0 | |||
| 2 | 0 | 0 | 0 | 730 | 5 | 0 | 0 | 2 | 0 | 0 | 0 | 730 | 0 | 0 | 0 | |||
| 3 | 0 | 0 | 0 | 1095 | 7 | 5 | 0 | 3 | 0 | 0 | 0 | 1095 | 0 | 0 | 0 | |||
| 4 | 0 | 0 | 0 | 1461 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 1460 | 0 | 0 | 0 | |||
| 5 | 0 | 0 | 0 | 1826 | 2 | 5 | 0 | 5 | 0 | 0 | 0 | 1825 | 0 | 0 | 0 | |||
| 6 | 0 | 0 | 0 | 2191 | 5 | 0 | 0 | 6 | 0 | 0 | 0 | 2190 | 0 | 0 | 0 | |||
| 7 | 0 | 0 | 0 | 2556 | 7 | 5 | 0 | 7 | 0 | 0 | 0 | 2555 | 0 | 0 | 0 | |||
| 8 | 0 | 0 | 0 | 2922 | 0 | 0 | 0 | 8 | 0 | 0 | 0 | 2920 | 0 | 0 | 0 | |||
| 9 | 0 | 0 | 0 | 3287 | 2 | 5 | 0 | 9 | 0 | 0 | 0 | 3285 | 0 | 0 | 0 | |||
| 10 | 0 | 0 | 0 | 3652 | 5 | 0 | 0 | 10 | 0 | 0 | 0 | 3650 | 0 | 0 | 0 | |||
| Days in Julian Months | Days in AEgyptian Months | Days in Persian Months | |
| Comon | Bissex | Thoth 30 | Pharvadin 30 |
| Ianuary 31 | 30 | Paophi 60 | Aripehast 60 |
| February 59 | 60 | Athyr 90 | Chortat 90 |
| March 90 | 91 | Chaeae 120 | Tirma 120 |
| April 120 | 121 | Tybi 150 | Mertat 150 |
| May 151 | 152 | Michir 180 | Sachriur 180 |
| Iune 181 | 182 | Phamenoth 210 | Macherma 210 |
| Iuly 212 | 213 | Pharmuthi; 240 | Apenina Wahak 245 |
| August 243 | 244 | Pachon 270 | |
| September 273 | 274 | Payny 300 | Aderma 275 |
| October 304 | 305 | Ephephi 330 | Dima 305 |
| November 334 | 335 | Mesori 330 | Pechmam 335 |
| December 365 | 366 | Epagomena 365 | Aphander 365 |
| Days in Turkish or Arabical Years | Days in Turkish Months | ||||
| 1 | 354 | Muharran 30 | |||
| 2 | 709 | Sapher 59 | |||
| 3 | .1063 | Rabie 1. 89 | |||
| 4 | .1417 | Rabie 2. 118 | |||
| 5 | .1772 | Giumadi 1. 148 | |||
| 6 | .2126 | Giumadi 2. 177 | |||
| 7 | .2480 | Regeb 207 | |||
| 8 | .2835 | Sahahen 236 | |||
| 9 | .3189 | Ramaddan 266 | |||
| 10 | .3543 | Scheval 295 | |||
| 11 | .3898 | Dulkadati 325 | |||
| 12 | .4252 | Dulhajati Dsilhittsche true 354 | |||
| 13 | .4607 | ||||
| 14 | .4961 | ||||
| 15 | .5315 | In anno Abundanti 355 | |||
| 16 | .5670 | ||||
| 17 | .6024 | ||||
| 18 | .6378 | ||||
| 19 | .6733 | ||||
| 20 | .7087 | ||||
| 21 | 7442 | ||||
| 22 | 7796 | ||||
| 23 | 8150 | ||||
| 24 | 8505 | ||||
| 25 | 8859 | ||||
| 26 | 9213 | ||||
| 27 | 9568 | ||||
| 28 | 9922 | ||||
| 29 | 10276 | ||||
| 30 | 0 | 10631 | 0 | ||
| 60 | 0 | 21262 | 0 | ||
| 90 | 0 | 31893 | 0 | ||
| 120 | 0 | 42524 | 0 | ||
| 150 | 0 | 53155 | 0 | ||
| 180 | 0 | 63786 | 0 | ||
| 210 | 0 | 74417 | 0 | ||
| 240 | 0 | 05048 | 0 | ||
| 270 | 0 | 95679 | 0 | ||
| 300 | 0 | 106310 | 0 | ||
| Ianuary | February | March | |||||||
| 1 | 3 | A | Circumcis. | D | Purificat | 3 | D | ||
| 2 | B | 11 | E | E | |||||
| 3 | 11 | C | 19 | F | 11 | F | |||
| 4 | D | 8 | G | G | |||||
| 5 | 19 | E | A | 19 | A | ||||
| 6 | 8 | F | Epiphany | 16 | B | 8 | B | ||
| 7 | G | 5 | C | C | |||||
| 8 | 16 | A | D | 16 | D | ||||
| 9 | 5 | B | 13 | E | 5 | E | |||
| 10 | C | 2 | F | F | |||||
| 11 | 13 | D | G | 13 | G | ||||
| 12 | 2 | E | 10 | A | 2 | A | |||
| 13 | F | B | B | ||||||
| 14 | 10 | G | 18 | C | 10 | C | |||
| 15 | A | 7 | D | D | |||||
| 16 | 18 | B | E | 18 | E | ||||
| 17 | 7 | C | 15 | F | 7 | F | |||
| 18 | D | 4 | G | G | |||||
| 19 | 15 | E | A | 15 | A | ||||
| 20 | 4 | F | 12 | B | 4 | B | |||
| 21 | G | 1 | C | C | |||||
| 22 | 12 | A | D | 12 | D | ||||
| 23 | 1 | B | 9 | E | 1 | E | |||
| 24 | C | F | F | ||||||
| 25 | 9 | D | Conv. S. Paul | 17 | G | S. Matthias | 9 | G | Anunc. |
| 26 | E | 6 | A | A | |||||
| 27 | 17 | F | B | 17 | B | ||||
| 28 | 6 | G | 14 | C | 6 | C | |||
| 29 | A | D | |||||||
| 30 | 14 | B | 14 | E | |||||
| 31 | 3 | C | 3 | F | |||||
| April | May | Iune | |||||||
| 1 | G | 11 | B | Phil. & Jac. | E | ||||
| 2 | 11 | A | C | 19 | F | ||||
| 3 | B | 19 | D | 8 | G | ||||
| 4 | 19 | C | 8 | E | 16 | A | |||
| 5 | 8 | D | F | 5 | B | ||||
| 6 | 16 | E | 16 | G | C | ||||
| 7 | 5 | F | 5 | A | 13 | D | |||
| 8 | G | B | 2 | E | |||||
| 9 | 13 | A | 14 | C | F | ||||
| 10 | 2 | B | 2 | D | 10 | G | |||
| 11 | C | E | A | S. Barnaby | |||||
| 12 | 10 | D | 10 | F | 18 | B | |||
| 13 | E | G | 7 | C | |||||
| 14 | 18 | F | 18 | A | D | ||||
| 15 | 7 | G | 7 | B | 15 | E | |||
| 16 | A | C | 4 | F | |||||
| 17 | 15 | B | 15 | D | G | ||||
| 18 | 4 | C | 4 | E | 12 | A | |||
| 19 | D | F | 1 | B | |||||
| 20 | 12 | E | 12 | G | C | ||||
| 21 | 1 | F | 1 | A | 9 | D | |||
| 22 | G | B | E | ||||||
| 23 | 9 | A | 9 | C | 17 | F | |||
| 24 | B | D | 6 | G | S. John B. | ||||
| 25 | 17 | C | Mark Evang. | 17 | E | A | |||
| 26 | 6 | D | 6 | F | 14 | B | |||
| 27 | E | G | 3 | C | |||||
| 28 | 14 | F | 14 | A | D | ||||
| 29 | 5 | G | 3 | B | 11 | E | Pet. Ap. | ||
| 30 | A | C | F | ||||||
| 31 | 11 | D | |||||||
| Iuly | August | September | |||||||
| 1 | 19 | G | 8 | C | 16 | F | |||
| 2 | 8 | A | 16 | D | 5 | G | |||
| 3 | B | 5 | E | A | |||||
| 4 | 16 | C | F | 13 | B | ||||
| 5 | 5 | D | 13 | G | 2 | C | |||
| 6 | E | 2 | A | D | |||||
| 7 | 13 | F | B | 10 | E | ||||
| 8 | 2 | G | 10 | C | F | ||||
| 9 | A | D | 18 | G | |||||
| 10 | 10 | B | 18 | E | 7 | A | |||
| 11 | C | 7 | F | B | |||||
| 12 | 18 | D | G | 15 | C | ||||
| 13 | 7 | E | 15 | A | 4 | D | |||
| 14 | F | 4 | B | E | |||||
| 15 | 15 | G | C | 12 | F | ||||
| 16 | 4 | A | 12 | D | 1 | G | |||
| 17 | B | 1 | E | A | |||||
| 18 | 12 | C | F | 9 | B | ||||
| 19 | 1 | D | 9 | G | C | ||||
| 20 | E | Margaret | A | 17 | D | ||||
| 21 | 9 | F | 17 | B | 6 | E | S. Matth | ||
| 22 | G | 6 | C | F | |||||
| 23 | 17 | A | D | 14 | G | ||||
| 24 | 6 | B | 14 | E | Barthol. | 3 | A | ||
| 25 | C | 3 | F | B | |||||
| 26 | 14 | D | 11 | G | 11 | C | |||
| 27 | 3 | E | 19 | A | 19 | D | |||
| 28 | F | B | 8 | E | |||||
| 29 | 11 | G | 8 | C | F | S. Mich. | |||
| 30 | 9 | A | D | G | |||||
| 31 | B | E | |||||||
| October | November | December | |||||||
| 1 | 16 | A | D | All Saints | 13 | F | |||
| 2 | 5 | B | 13 | E | All Souls | 2 | G | ||
| 3 | 13 | C | 2 | F | A | ||||
| 4 | 2 | D | G | 10 | B | ||||
| 5 | E | 10 | A | P. Conspir. | C | ||||
| 6 | 10 | F | B | 18 | D | ||||
| 7 | G | 18 | C | 7 | E | ||||
| 8 | 18 | A | 7 | D | F | ||||
| 9 | 7 | B | E | 15 | G | ||||
| 10 | C | 15 | F | 4 | A | ||||
| 11 | 15 | D | 4 | G | B | ||||
| 12 | 4 | E | A | 12 | C | ||||
| 13 | F | 12 | B | 1 | D | ||||
| 14 | 12 | G | 1 | C | E | ||||
| 15 | 13 | A | D | 9 | F | ||||
| 16 | B | 9 | E | G | |||||
| 17 | 9 | C | F | 17 | A | ||||
| 18 | D | Luke Evang. | 17 | G | 6 | B | |||
| 19 | 17 | E | 6 | A | C | ||||
| 20 | 6 | F | B | 14 | D | ||||
| 21 | G | 14 | C | 3 | E | S. Thomas | |||
| 22 | 14 | A | 3 | D | F | ||||
| 23 | 3 | B | E | 11 | G | ||||
| 24 | C | 11 | F | 19 | A | ||||
| 25 | 11 | D | 19 | G | B | Chri. Nat. | |||
| 26 | 19 | E | A | 8 | C | S. Steph. | |||
| 27 | F | 8 | B | D | S. John | ||||
| 28 | 8 | G | Sim. & Jude | C | 16 | E | Innocents | ||
| 29 | A | 16 | D | 5 | F | ||||
| 30 | 16 | B | 5 | E | S. Andrew | G | |||
| 31 | 5 | C | 13 | A | Sylvester | ||||
| Ianuary | February | March | ||||
| 1 | * | A | XXIX | D | * | D |
| 2 | XXIX | B | XXVIII | E | XXIX | E |
| 3 | XXVIII | C | XXVII | F | XXVIII | F |
| 4 | XXVII | D | 25. XXVI | G | XXVII | G |
| 5 | XXVI | E | XXV. XXIV | A | XXVI | A |
| 6 | 25. XXV | F | XXIII | B | 25. XXV | B |
| 7 | XXIV | G | XXII | C | XXIV | C |
| 8 | XXIII | A | XXI | D | XXIII | D |
| 9 | XXII | B | XX | E | XXII | E |
| 10 | XXI | C | XIX | F | XXI | F |
| 11 | XX | D | XVIII | G | XX | G |
| 12 | XIX | E | XVII | A | XIX | A |
| 13 | XVIII | F | XVI | B | XVIII | B |
| 14 | XVII | G | XV | C | XVII | C |
| 15 | XVI | A | XIV | D | XVI | D |
| 16 | XV | B | XIII | E | XV | E |
| 17 | XIV | C | XII | F | XIV | F |
| 18 | XIII | D | XI | G | XIII | G |
| 19 | XII | E | X | A | XII | A |
| 20 | XI | F | IX | B | XI | B |
| 21 | X | G | VIII | C | X | C |
| 22 | IX | A | VII | D | IX | D |
| 23 | VIII | B | VI | E | VIII | E |
| 24 | VII | C | V | F | VII | F |
| 25 | VI | D | IV | G | VI | G |
| 26 | V | E | III | A | V | A |
| 27 | IV | F | II | B | IV | B |
| 28 | III | G | I | C | III | C |
| 29 | II | A | II | D | ||
| 30 | I | B | I | E | ||
| 31 | * | C | * | F | ||
| April | May | Iune | ||||
| 1 | XXIX | G | XXVIII | B | XXVII | E |
| 2 | XXVIII | A | XXVII | C | 25. XXVI | F |
| 3 | XXVII | B | XXVI | D | XXV. XXIV | G |
| 4 | 25. XXVI | C | 25. XXV | E | XXIII | A |
| 5 | XXV. XXIV | D | XXIV | F | XXII | B |
| 6 | XXIII | E | XXIII | G | XXI | C |
| 7 | XXII | F | XXII | A | XX | D |
| 8 | XXI | G | XXI | B | XIX | E |
| 9 | XX | A | XX | C | XVIII | F |
| 10 | XIX | B | XIX | D | XVII | G |
| 11 | XVIII | C | XVIII | E | XVI | A |
| 12 | XVII | D | XVII | F | XV | B |
| 13 | XVI | E | XVI | G | XIV | C |
| 14 | XV | F | XV | A | XIII | D |
| 15 | XIV | G | XIV | B | XII | E |
| 16 | XIII | A | XIII | C | XI | F |
| 17 | XII | B | XII | D | X | G |
| 18 | XI | C | XI | E | IX | A |
| 19 | X | D | X | F | VIII | B |
| 20 | IX | E | IX | G | VII | C |
| 21 | VIII | F | VIII | A | VI | D |
| 22 | VII | G | VII | B | V | E |
| 23 | VI | A | VI | C | IV | F |
| 24 | V | B | V | D | III | G |
| 25 | IV | C | IV | E | II | A |
| 26 | III | D | III | F | I | B |
| 27 | II | E | II | G | * | C |
| 28 | I | F | I | A | XXIX | D |
| 29 | * | G | * | B | XXVIII | E |
| 30 | XXIX | A | XXIX | C | XXVII | F |
| 31 | XXVIII | D | ||||
| Iuly | August | September | ||||
| 1 | XXVI | G | XXV. XXIV | C | XXIII | F |
| 2 | 25. XXV | A | XXIII | D | XXII | G |
| 3 | XXIV | B | XXII | E | XXI | A |
| 4 | XXIII | C | XXI | F | XX | B |
| 5 | XXII | D | XX | G | XIX | C |
| 6 | XXI | E | XIX | A | XVIII | D |
| 7 | XX | F | XVIII | B | XVII | E |
| 8 | XIX | G | XVII | C | XVI | F |
| 9 | XVIII | A | XVI | D | XV | G |
| 10 | XVII | B | XV | E | XIV | A |
| 11 | XVI | C | XIV | F | XIII | B |
| 12 | XV | D | XIII | G | XII | C |
| 13 | XIV | E | XII | A | XI | D |
| 14 | XIII | F | XI | B | X | E |
| 15 | XII | G | X | C | IX | F |
| 16 | XI | A | IX | D | VIII | G |
| 17 | X | B | VIII | E | VII | A |
| 18 | IX | C | VII | F | VI | B |
| 19 | VIII | D | VI | G | V | C |
| 20 | VII | E | V | A | IV | D |
| 21 | VI | F | IV | B | III | E |
| 22 | V | G | III | C | II | F |
| 23 | IV | A | II | D | I | G |
| 24 | III | B | I | E | * | A |
| 25 | II | C | * | F | XXIX | B |
| 26 | I | D | XXIX | G | XXVIII | C |
| 27 | * | E | XXVIII | A | XXVII | D |
| 28 | XXIX | F | XXVII | B | 25. XXVI | E |
| 29 | XXVIII | G | XXVI | C | XXV. XXIV | F |
| 30 | XXVII | A | 25. XXV | D | XXIII | G |
| 31 | 25. XXVI | B | XXIV | E | ||
| October | November | December | ||||
| 1 | XXII | A | XXI | D | XX | F |
| 2 | XXI | B | XX | E | XIX | G |
| 3 | XX | C | XIX | F | XVIII | A |
| 4 | XIX | D | XVIII | G | XVII | B |
| 5 | XVIII | E | XVII | A | XVI | C |
| 6 | XVII | F | XVI | B | XV | D |
| 7 | XVI | G | XV | C | XIV | E |
| 8 | XV | A | XIV | D | XIII | F |
| 9 | XIV | B | XIII | E | XII | G |
| 10 | XIII | C | XII | F | XI | A |
| 11 | XII | D | XI | G | X | B |
| 12 | XI | E | X | A | IX | C |
| 13 | X | F | IX | B | VIII | D |
| 14 | IX | G | VIII | C | VII | E |
| 15 | VIII | A | VII | D | VI | F |
| 16 | VII | B | VI | E | V | G |
| 17 | VI | C | V | F | IV | A |
| 18 | V | D | IV | G | III | B |
| 19 | IV | E | III | A | II | C |
| 20 | III | F | II | B | I | D |
| 21 | II | G | I | C | * | E |
| 22 | I | A | * | D | XXIX | F |
| 23 | * | B | XXIX | E | XXVIII | G |
| 24 | XXIX | C | XXVIII | F | XXVII | A |
| 25 | XXVIII | D | XXVII | G | XXVI | B |
| 26 | XXVII | E | 25. XXVI | A | 25. XXV | C |
| 27 | XXVI | F | XXV. XXIV | B | XXIV | D |
| 28 | 25. XXV | G | XXIII | C | XXIII | E |
| 29 | XXIV | A | XXII | D | XXII | F |
| 30 | XXIII | B | XXI | E | XXI | G |
| 31 | XXII | C | XX | A |
| 1672 | 1 | GF | CB |
| 1673 | 2 | E | A |
| 1674 | 3 | D | G |
| 1675 | 4 | C | F |
| 1676 | 5 | BA | ED |
| 1677 | 6 | G | C |
| 1678 | 7 | F | B |
| 1679 | 8 | E | A |
| 1680 | 9 | DC | GF |
| 1681 | 10 | B | E |
| 1682 | 11 | A | D |
| 1683 | 12 | G | C |
| 1684 | 13 | FE | BA |
| 1685 | 14 | D | G |
| 1686 | 15 | C | F |
| 1687 | 16 | B | E |
| 1688 | 17 | AG | DC |
| 1689 | 18 | F | B |
| 1690 | 19 | E | A |
| 1691 | 20 | D | G |
| 1692 | 21 | CB | FE |
| 1693 | 22 | A | D |
| 1694 | 23 | G | C |
| 1695 | 24 | F | B |
| 1696 | 25 | ED | AG |
| 1697 | 26 | C | F |
| 1698 | 27 | B | E |
| 1699 | 28 | A | D |
| Year | G | Julian | Gregor. |
| N | Epact | Epact | |
| 1672 | 1 | 11 | 1 |
| 1673 | 2 | 22 | 12 |
| 1674 | 3 | 3 | 23 |
| 1675 | 4 | 14 | 4 |
| 1676 | 5 | 25 | 15 |
| 1677 | 6 | 6 | 26 |
| 1678 | 7 | 17 | 7 |
| 1679 | 8 | 28 | 18 |
| 1680 | 9 | 9 | 29 |
| 1681 | 10 | 20 | 10 |
| 1682 | 11 | 1 | 21 |
| 1683 | 12 | 12 | 2 |
| 1684 | 13 | 23 | 13 |
| 1685 | 14 | 4 | 24 |
| 1686 | 15 | 15 | 5 |
| 1687 | 16 | 26 | 16 |
| 1688 | 17 | 7 | 17 |
| 1689 | 18 | 18 | 8 |
| 1690 | 19 | 29 | 19 |
- From 5 October 1582 D. 10
- From 24 Feb. 1700 D. 11
- From 24 Feb. 1800 D. 12
- From 24 Feb. 1900 D. 13
- From 24 Feb. 2100 D. 14
- From 24 Feb. 2200 D. 15
- From 24 Feb. 2320 D. 16
| III | IV | V | VI | VII | VIII | ||
| 1 | P | * | XI | XXII | III | XIV | XXV |
| 2 | N | XXIX | X | XXI | II | XIII | XXIV |
| 3 | M | XXVIII | IX | XX | I | XII | XXIII |
| 4 | H | XXVII | VIII | XIX | * | XI | XXII |
| 5 | G | XXVI | VII | XVIII | XXIX | X | XXI |
| 6 | F | XXV | VI | XVII | XXVIII | IX | XX |
| 7 | E | XXIV | V | XVI | XXVII | VIII | XIX |
| 8 | D | XXIII | IV | XV | XXVI | VII | XVIII |
| 9 | C | XXII | III | XIV | XXV | VI | XVII |
| 10 | B | XXI | II | XIII | XXIV | V | XVI |
| 11 | A | XX | I | XII | XXIII | IV | XV |
| 12 | u | XIX | * | XI | XXII | III | XIV |
| 13 | t | XVIII | XXIX | X | XXI | II | XIII |
| 14 | s | XVII | XXVIII | IX | XX | I | XII |
| 15 | r | XVI | XXVII | VIII | XIX | * | XI |
| 16 | q | XV | XXVI | VII | XVIII | XXIX | X |
| 17 | p | XIV | XXV | VI | XVII | XXVIII | IX |
| 18 | n | XIII | XXIV | V | XVI | XXVII | VIII |
| 19 | m | XII | XXIII | IV | XV | XXVI | VII |
| 20 | l | XI | XXII | III | XIV | XXV | VI |
| 21 | k | X | XXI | II | XIII | XXIV | V |
| 22 | i | IX | XX | I | XII | XXIII | IV |
| 23 | h | VIII | XIX | * | XI | XXII | III |
| 24 | g | VII | XVIII | XXIX | X | XXI | II |
| 25 | f | VI | XVII | XXVIII | IX | XX | I |
| 26 | e | V | XVI | XXVII | VIII | XIX | * |
| 27 | d | IV | XV | XXVI | VII | XVIII | XXIX |
| 28 | c | III | XIV | XXV | VI | XVII | XXVIII |
| 29 | b | II | XIII | XXIV | V | XVI | XXVII |
| 30 | a | I | XII | XXIII | IV | XV | XXVI |
| IX | X | XI | XII | XIII | XIV | XV |
| VI | XVII | XXVIII | IX | XX | I | XII |
| V | XVI | XXVII | VIII | XIX | * | XI |
| IV | XV | XXVI | VII | XVIII | XXIX | |
| III | XIV | XXV | VI | XVII | XXVIII | IX |
| II | XIII | XXIV | V | XVI | XXVII | VIII |
| I | XII | XXIII | IV | XV | XXVI | VII |
| * | XI | XXII | III | XIV | 25 | VI |
| XXIX | X | XXI | II | XIII | XXIV | V |
| XXVIII | IX | XX | I | XII | XXIII | IV |
| XXVII | VIII | XIX | * | XI | XXII | III |
| XXVI | VII | XVIII | XXIX | X | XXI | II |
| XXV | VI | XVII | XXVIII | IX | XX | I |
| XXIV | V | XVI | XXVII | VIII | XIX | * |
| XXIII | IV | XV | XXVI | VII | XVIII | XXIX |
| XXII | III | XIV | XXV | VI | XVII | XXVIII |
| XXI | II | XIII | XXIV | V | XVI | XXVI |
| XX | I | XII | XXIII | IV | XV | XXVII |
| XIX | * | XI | XXII | III | XIV | 25 |
| XVIII | XXIX | X | XXI | II | XIII | XXIV |
| XVII | XXVIII | IX | XX | I | XII | XXIII |
| XVI | XXVII | VIII | XIX | * | XI | XXII |
| XV | XXVI | VII | XVIII | XXIX | X | XXI |
| XIV | XV | VI | XVII | XXVIII | IX | XX |
| XIII | XXIV | V | XVI | XXVII | VIII | XIX |
| XII | XXIII | IV | XV | XXVI | VII | XVIII |
| XI | XXII | III | XIV | 25 | VI | XVII |
| X | XXI | II | XIII | XXIV | V | XVI |
| IX | XX | I | XII | XXIII | IV | XV |
| VIII | XIX | * | XI | XXII | III | XIV |
| VII | XVIII | XIX | X | XXI | II | XIII |
| XVI | XVII | XVIII | XIX | I | II | |
| P | XXIII | IV | XV | XXVI | VIII | XIX |
| N | XXII | III | XIV | 25 | VII | XVIII |
| M | XXI | II | XIII | XXIV | VI | XVII |
| H | XX | I | XII | XXIII | V | XVI |
| G | XIX | * | XI | XXII | IV | XV |
| F | XVIII | XXIX | X | XXI | III | XIV |
| E | XVII | XXVIII | IX | XX | II | XIII |
| D | XVI | XXVII | VIII | XIX | I | XII |
| C | XV | XXVI | VII | XVIII | * | XI |
| B | XIV | 25 | VI | XVII | XXIX | X |
| A | XIII | XXIV | V | XVI | XXVIII | IX |
| u | XII | XXIII | IV | XV | XXVII | VIII |
| t | XI | XXII | III | XIV | XXVI | VII |
| t | X | XXI | II | XIII | 25 | VI |
| r | IX | XX | I | XII | XXIV | V |
| q | VIII | XIX | * | XI | XXIII | IV |
| p | VII | XVIII | XXIX | X | XXII | III |
| n | VI | XVII | XXVIII | IX | XXI | II |
| m | V | XVI | XXVII | VIII | XX | I |
| l | IV | XV | XXVI | VII | XIX | * |
| k | III | XIV | 25 | VI | XVIII | XXIX |
| i | II | XXIII | XXIV | V | XVII | XXVIII |
| h | I | XII | XXIII | IV | XVI | XXVII |
| g | * | XI | XXII | III | XV | XXVI |
| f | XXIX | X | XXI | II | XIV | 25 |
| e | XXVIII | IX | XX | I | XIII | XXIV |
| d | XXVII | VIII | XIX | * | XII | XXIII |
| c | XXVI | VII | XVIII | XXIX | XI | XXII |
| b | 25 | VI | XVII | XXVIII | X | XXI |
| a | XXIV | V | XVI | XXVII | IX | XX |
| N | I | ||
| P | 320 | ||
| P | 580 | Biss. | |
| a | 800 | Biss. | C |
| b | 1100 | Biss. | C |
| c | 1400 | Biss. | C |
| D | 1484 | ||
| D | 1600 | Biss. | |
| C | 1700 | ||
| C | 1800 | CC | |
| B | 1900 | ||
| B | 2000 | Biss. | |
| B | 2100 | C | |
| A | 2200 | ||
| u | 2300 | ||
| A | 2409 | Biss. | C |
| u | 2500 | ||
| t | 2600 | ||
| t | 2700 | C | |
| t | 2800 | Biss. | |
| s | 2900 | ||
| s | 3000 | C | |
| r | 3100 | ||
| r | 3200 | Biss. | |
| r | 3300 | C | |
| q | 3400 | ||
| p | 3500 |
| q | 3600 | Biss. | C |
| p | 3700 | ||
| n | 3800 | ||
| n | 3900 | ||
| n | 4000 | Biss. | C |
| m | 4100 | ||
| l | 4200 | ||
| l | 4300 | CC | |
| l | 4400 | Biss. | |
| k | 4500 | ||
| k | 4600 | C | |
| i | 4700 | ||
| i | 4800 | Biss. | |
| i | 4900 | C | |
| h | 5000 | ||
| g | 5100 | ||
| h | 5200 | Biss. | C |
| g | 5300 | ||
| f | 5400 | ||
| f | 5500 | C | |
| f | 5600 | Biss. | |
| e | 5700 | ||
| e | 5800 | C | |
| d | 5900 | ||
| d | 6000 | Biss. | |
| d | 6100 | C | |
| c | 6200 | ||
| b | 6300 | ||
| c | 6400 | Biss. | C |
| b | 6500 |
| Cy. ☉ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | C B | D C | E D | F E | G F | A G | B A |
| 2 | A | B | C | D | E | F | G |
| 3 | G | A | B | C | D | E | F |
| 4 | F | G | A | B | C | D | E |
| 5 | E D | F E | G F | A G | B A | C B | D C |
| 6 | C | D | E | F | G | A | B |
| 7 | B | C | D | E | F | G | A |
| 8 | A | B | C | D | E | F | G |
| 9 | G F | A G | B A | C B | D C | E D | F E |
| 10 | E | F | G | A | B | C | D |
| 11 | D | E | F | G | A | B | C |
| 12 | C | D | E | F | G | A | B |
| 13 | B A | C B | D C | E D | F E | G F | A G |
| 14 | G | A | B | C | D | E | F |
| 15 | F | G | A | B | C | D | E |
| 16 | E | F | G | A | B | C | D |
| 17 | D C | E D | F E | G F | A G | B A | C B |
| 18 | B | C | D | E | F | G | A |
| 19 | A | B | C | D | E | F | G |
| 20 | G | A | B | C | D | E | F |
| 21 | F E | G F | A G | B A | C B | D C | E D |
| 22 | D | E | F | G | A | B | C |
| 23 | C | D | E | F | G | A | B |
| 24 | B | C | D | E | F | G | A |
| 25 | A G | B A | C B | D C | E D | F E | G F |
| 26 | F | G | A | B | C | D | E |
| 27 | E | F | G | A | B | C | D |
| 28 | D | E | F | G | A | B | C |
| Anni | 1582 | 1900 | 2300 | ||||
| 1600 | 1700 | 1800 | 2000 | 2100 | 2200 | 2400 | |
| 2700 | 3100 | ||||||
| Chr. | 2500 | 2600 | 2900 | 3000 | |||
| 2800 | 3200 | 3300 |
| LXX | Ash. | East. | Asci. | Pent. | Corp. Christi. | Adv. | |||
| 16 | XXIII | Ian. | Feb. | Mar. | Apr. | May. | May. | Nov. | |
| 5 | XXII | d | 18 | 4 | 22 | 30 | 10 | 21 | 29 |
| XXI | e | 19 | 5 | 23 | Ma. 1 | 11 | 22 | 30 | |
| 13 | XX | f | 20 | 6 | 24 | 2 | 12 | 23 | De. 1 |
| 2 | XIX | g | 21 | 7 | 25 | 3 | 13 | 24 | 2 |
| XVIII | a | 22 | 8 | 26 | 4 | 14 | 25 | 3 | |
| 10 | XVII | b | 23 | 9 | 27 | 5 | 15 | 26 | No. 27 |
| XVI | c | 24 | 10 | 28 | 6 | 16 | 27 | 28 | |
| 18 | XV | d | 25 | 11 | 29 | 7 | 17 | 28 | 29 |
| 7 | XIV | e | 26 | 12 | 30 | 8 | 18 | 29 | 30 |
| XIII | f | 27 | 13 | 31 | 9 | 19 | 30 | Dec. 1 | |
| 15 | XII | g | 28 | 14 | Ap. 1 | 10 | 20 | 31 | 2 |
| 4 | XI | a | 29 | 15 | 2 | 11 | 21 | Iun. 1 | 3 |
| X | b | 30 | 16 | 3 | 12 | 22 | 2 | No. 27 | |
| 12 | IX | c | 31 | 17 | 4 | 13 | 23 | 3 | 28 |
| 1 | VIII | d | Feb. 1 | 18 | 5 | 14 | 24 | 4 | 29 |
| VII | e | 2 | 19 | 6 | 15 | 25 | 5 | 30 | |
| VI | f | 3 | 20 | 7 | 16 | 26 | 6 | Dec. 1 | |
| 9 | V | g | 4 | 21 | 8 | 17 | 27 | 7 | 2 |
| 17 | IV | a | 5 | 22 | 9 | 18 | 28 | 8 | 3 |
| 6 | III | b | 6 | 23 | 10 | 19 | 29 | 9 | No. 27 |
| II | c | 7 | 24 | 11 | 20 | 30 | 10 | 28 | |
| 14 | I | d | 8 | 25 | 12 | 21 | 31 | 11 | 29 |
| 3 | * | e | 9 | 26 | 13 | 22 | Iun. 1 | 12 | 30 |
| XXIX | f | 10 | 27 | 14 | 23 | 2 | 13 | Dec. 1 | |
| 11 | XXVIII | g | 11 | 28 | 15 | 24 | 3 | 14 | 2 |
| XXVII | a | 12 | Ma. 1 | 16 | 25 | 4 | 15 | 3 | |
| 19 | 25. XXVI | b | 13 | 2 | 17 | 26 | 5 | 16 | No. 27 |
| 8 | XXV. XXIV | c | 14 | 3 | 18 | 27 | 6 | 17 | 28 |
| d | 15 | 4 | 19 | 28 | 7 | 18 | 29 | ||
| e | 16 | 5 | 20 | 29 | 8 | 19 | 30 | ||
| f | 17 | 6 | 21 | 30 | 9 | 20 | Dec. 1 | ||
| g | 18 | 7 | 22 | 31 | 10 | 21 | 2 | ||
| a | 19 | 8 | 23 | Iun. 1 | 11 | 22 | 3 | ||
| b | 20 | 9 | 24 | 2 | 12 | 23 | No. 27 | ||
| c | 21 | 10 | 25 | 3 | 13 | 24 | 28 |
| 1 | 00 | 37 | 10 | 73 | 20 | 109 | 30 | 145 | 40 | 181 | 50 |
| 2 | 38 | 74 | 110 | 146 | 182 | ||||||
| 3 | 39 | 75 | 111 | 147 | 183 | ||||||
| 4 | 01 | 40 | 11 | 76 | 21 | 112 | 31 | 148 | 41 | 184 | 51 |
| 5 | 41 | 77 | 113 | 149 | 185 | ||||||
| 6 | 42 | 78 | 114 | 150 | 186 | ||||||
| 7 | 43 | 79 | 115 | 151 | 187 | ||||||
| 8 | 02 | 44 | 12 | 80 | 22 | 116 | 32 | 152 | 42 | 188 | 52 |
| 9 | 45 | 81 | 117 | 153 | 189 | ||||||
| 10 | 46 | 82 | 118 | 154 | 190 | ||||||
| 11 | 03 | 47 | 13 | 83 | 23 | 119 | 33 | 155 | 43 | 191 | 53 |
| 12 | 48 | 84 | 120 | 156 | 192 | ||||||
| 13 | 49 | 85 | 121 | 157 | 193 | ||||||
| 14 | 50 | 86 | 122 | 158 | 194 | ||||||
| 15 | 04 | 51 | 14 | 87 | 24 | 123 | 35 | 159 | 44 | 195 | 54 |
| 16 | 52 | 88 | 124 | 160 | 196 | ||||||
| 17 | 53 | 89 | 125 | 161 | 197 | ||||||
| 18 | 05 | 54 | 15 | 90 | 25 | 126 | 35 | 162 | 45 | 198 | 55 |
| 19 | 55 | 91 | 127 | 163 | 199 | ||||||
| 20 | 56 | 92 | 128 | 164 | 200 | ||||||
| 21 | 57 | 93 | 129 | 165 | 201 | ||||||
| 22 | 06 | 58 | 16 | 94 | 26 | 130 | 36 | 166 | 46 | 202 | 56 |
| 23 | 59 | 95 | 131 | 167 | 203 | ||||||
| 24 | 60 | 96 | 132 | 168 | 204 | ||||||
| 25 | 61 | 97 | 133 | 169 | 205 | ||||||
| 26 | 07 | 62 | 17 | 98 | 27 | 134 | 37 | 170 | 47 | 206 | 57 |
| 27 | 63 | 99 | 135 | 171 | 207 | ||||||
| 28 | 64 | 100 | 136 | 172 | 208 | ||||||
| 29 | 08 | 65 | 18 | 101 | 28 | 137 | 38 | 173 | 48 | 209 | 58 |
| 30 | 66 | 102 | 138 | 174 | 210 | ||||||
| 31 | 67 | 103 | 139 | 175 | 211 | ||||||
| 32 | 68 | 104 | 140 | 176 | 212 | ||||||
| 33 | 09 | 69 | 19 | 105 | 29 | 141 | 39 | 177 | 49 | 213 | 59 |
| 34 | 70 | 106 | 142 | 178 | 214 | ||||||
| 35 | 71 | 107 | 143 | 179 | 215 | ||||||
| 36 | 10 | 72 | 20 | 108 | 30 | 144 | 40 | 180 | 50 | 216 | 60 |
| [Page 477] 217 | 60 | 253 | 70 | 289 | 80 | 325 | 90 | 277777778 | |||
| 218 | 254 | 290 | 326 | 555555555 | |||||||
| 219 | 255 | 291 | 327 | 833333333 | |||||||
| 220 | 61 | 256 | 71 | 292 | 81 | 328 | 91 | 111111111 | |||
| 221 | 257 | 293 | 329 | 388888889 | |||||||
| 222 | 258 | 294 | 330 | 666666667 | |||||||
| 223 | 259 | 295 | 331 | 944444444 | |||||||
| 224 | 62 | 260 | 72 | 296 | 82 | 332 | 92 | 222222222 | |||
| 225 | 261 | 297 | 333 | 500000000 | |||||||
| 226 | 262 | 298 | 334 | 777777778 | |||||||
| 227 | 63 | 263 | 73 | 299 | 83 | 335 | 93 | 055555555 | |||
| 228 | 264 | 300 | 336 | 333333333 | |||||||
| 229 | 265 | 301 | 337 | 511111111 | |||||||
| 230 | 266 | 302 | 338 | 888888889 | |||||||
| 231 | 64 | 267 | 74 | 303 | 84 | 339 | 94 | 166666667 | |||
| 232 | 268 | 304 | 340 | 444444444 | |||||||
| 233 | 269 | 305 | 341 | 722222222 | |||||||
| 234 | 65 | 270 | 75 | 306 | 85 | 342 | 95 | 000000000 | |||
| 235 | 271 | 307 | 343 | 277777778 | |||||||
| 236 | 272 | 308 | 344 | 555555555 | |||||||
| 237 | 273 | 309 | 345 | 833333333 | |||||||
| 238 | 66 | 274 | 76 | 310 | 86 | 346 | 96 | 111111111 | |||
| 239 | 275 | 311 | 347 | 388888889 | |||||||
| 240 | 276 | 312 | 348 | 666666667 | |||||||
| 241 | 277 | 313 | 349 | 944444444 | |||||||
| 242 | 67 | 278 | 77 | 314 | 87 | 350 | 97 | 222222222 | |||
| 243 | 279 | 315 | 351 | 500000000 | |||||||
| 244 | 280 | 316 | 352 | 777777778 | |||||||
| 245 | 68 | 281 | 78 | 317 | 88 | 353 | 98 | 055555555 | |||
| 246 | 282 | 318 | 354 | 333333333 | |||||||
| 247 | 283 | 319 | 355 | 611111111 | |||||||
| 248 | 284 | 320 | 356 | 888888889 | |||||||
| 249 | 69 | 285 | 79 | 321 | 89 | 357 | 99 | 166666667 | |||
| 250 | 286 | 322 | 358 | 444444444 | |||||||
| 251 | 287 | 323 | 359 | 722222222 | |||||||
| 252 | 70 | 288 | 80 | 324 | 90 | 360 | 100 | 000000000 |
| Minutes | Seconds | Thirds | |
| 1 | 00462962 | 00007716 | 00000128 |
| 2 | 00925925 | 15432 | 257 |
| 3 | 01388889 | 23148 | 385 |
| 4 | 01851851 | 30864 | 515 |
| 5 | 02314814 | 00038580 | 00000643 |
| 6 | 02777778 | 46296 | 771 |
| 7 | 03240740 | 54012 | 900 |
| 8 | 03703703 | 61728 | 1028 |
| 9 | 04166667 | 69444 | 1157 |
| 10 | 04629629 | 00077160 | 00001286 |
| 11 | 05092592 | 084876 | 1414 |
| 12 | 05555555 | 092592 | 1543 |
| 13 | 06018518 | 100308 | 1671 |
| 14 | 06481480 | 108024 | 1800 |
| 15 | 06944444 | 00115740 | 1929 |
| 16 | 07409407 | 123450 | 2057 |
| 17 | 07870370 | 131172 | 2186 |
| 18 | 08333333 | 138889 | 2314 |
| 19 | 08796296 | 146604 | 2443 |
| 20 | 09259259 | 00154320 | 2572 |
| 21 | 00722222 | 162036 | 2700 |
| 22 | 10185185 | 169752 | 2829 |
| 23 | 10648148 | 177468 | 2957 |
| 24 | 11111111 | 185184 | 3086 |
| 25 | 11574074 | 00192900 | 3215 |
| 26 | 12037037 | 200616 | 3343 |
| 27 | 12500000 | 208332 | 3472 |
| 28 | 12962962 | 216048 | 3600 |
| 29 | 13425926 | 223764 | 3729 |
| 30 | 13888889 | 00231481 | 00003858 |
| [Page 479] 31 | 14351852 | 00239670 | 00003986 |
| 32 | 14814814 | 246913 | 4115 |
| 33 | 15277777 | 254629 | 4243 |
| 34 | 15747040 | 262345 | 4372 |
| 35 | 16203703 | 270061 | 4581 |
| 36 | 16666666 | 00277777 | 00004629 |
| 37 | 17129629 | 285493 | 4758 |
| 38 | 17592592 | 293209 | 4886 |
| 39 | 18055555 | 300925 | 5015 |
| 40 | 18518518 | 308640 | 5144 |
| 41 | 18981481 | 00316356 | 00005272 |
| 42 | 19444444 | 324072 | 5401 |
| 43 | 19907407 | 331788 | 5529 |
| 44 | 20370370 | 339504 | 5658 |
| 45 | 20833333 | 347220 | 5787 |
| 46 | 21296296 | 00354936 | 00005915 |
| 47 | 21759259 | 362652 | 6044 |
| 48 | 22222222 | 370 [...]70 | 6172 |
| 49 | 22685185 | 378084 | 6301 |
| 50 | 23148148 | 385802 | 6430 |
| 51 | 23611111 | 00393518 | 00006558 |
| 52 | 24074074 | 401234 | 6687 |
| 53 | 24537037 | 408950 | 6815 |
| 54 | 25000000 | 416666 | 6944 |
| 55 | 25462963 | 424382 | 7073 |
| 56 | 25925926 | 00432098 | 00007201 |
| 57 | 26388888 | 439814 | 7330 |
| 58 | 26851852 | 447530 | 7458 |
| 59 | 27314814 | 455256 | 7587 |
| 60 | 27777777 | 00462962 | 00007716 |
| Hours. | |
| 1 | 04.16666667 |
| 2 | 08.33333333 |
| 3 | 12.5 |
| 4 | 16.16666667 |
| 5 | 20.83333333 |
| 6 | 25.0 |
| 7 | 29.16666667 |
| 8 | 33.33333333 |
| 9 | 37.5 |
| 10 | 41.66666667 |
| 11 | 45.83333333 |
| 12 | 50. |
| 13 | 54.16666667 |
| 14 | 58.33333333 |
| 15 | 62.5 |
| 16 | 66.66606667 |
| 17 | 70.83333333 |
| 18 | 75.00 |
| 19 | 79.16660667 |
| 20 | 83.33333333 |
| 21 | 87.5 |
| 22 | 91.66666667 |
| 23 | 95.83333333 |
| 24 | 100.00000000 |
| Minutes | |
| 1 | 0.06944444 |
| 2 | 0.13888888 |
| 3 | 0.20833333 |
| 4 | 0.27777777 |
| 5 | 0.34722222 |
| 6 | 0.41666666 |
| 7 | 0.48611111 |
| 8 | 0.55555555 |
| 9 | 0.625 |
| 10 | 0.69444444 |
| 11 | 0.76388888 |
| 12 | 0.83333333 |
| 13 | 0.90277777 |
| 14 | 0.97222222 |
| 15 | 1.04166666 |
| 16 | 1.11111111 |
| 17 | 1.18055555 |
| 18 | 1.25 |
| 19 | 1.31944444 |
| 20 | 1.38888888 |
| 21 | 1.45833333 |
| 22 | 1.52777777 |
| 23 | 1.59722222 |
| 24 | 1.66666666 |
| 25 | 1.73611111 |
| 26 | 1.80555555 |
| 27 | 1.875 |
| 28 | 1.94444444 |
| 29 | 2.01388888 |
| 30 | 2.08333333 |
| Seconds | Minutes | Seconds | |
| .00115740 | 31 | 2.15277777 | .03587963 |
| .00231481 | 32 | 2.22222222 | .03703704 |
| .00347222 | 33 | 2.29166666 | .03819444 |
| .00462962 | 34 | 2.36111111 | .03935185 |
| .00578703 | 35 | 2.43055555 | .04050926 |
| .00694444 | 36 | 2.5 | .04166666 |
| .00810184 | 37 | 2.56944444 | .04282407 |
| .00925925 | 38 | 2.63888888 | .04398148 |
| .01041660 | 39 | 2.70833333 | .04513888 |
| .01157405 | 40 | 2.77777777 | .04629629 |
| .01273148 | 41 | 2.84722222 | .04745370 |
| .01388888 | 42 | 2.91666666 | .04861111 |
| .01504630 | 43 | 2.98611111 | .0497685 [...] |
| .01620371 | 44 | 3.05555555 | .05092592 |
| .01736111 | 45 | 3.125 | .05208333 |
| .01851853 | 46 | 3.19444444 | .05324074 |
| .01967593 | 47 | 3.26388888 | .05439814 |
| .02083333 | 48 | 3.33333333 | .05555555 |
| .02199074 | 49 | 3.40277777 | .05671296 |
| .02314810 | 50 | 3.47222222 | .05787037 |
| .02430555 | 51 | 3.54166666 | .05902777 |
| .02546295 | 52 | 3.61111111 | .06018518 |
| .02662037 | 53 | 3.68055555 | .06134 [...]59 |
| .02777777 | 54 | 3.75 | .0625 |
| .02893518 | 55 | 3.81944444 | .06365741 |
| .03009259 | 56 | 3.88888888 | .06481481 |
| .03125000 | 57 | 3.95833333 | .06597222 |
| .03240741 | 58 | 4.02777777 | .06712963 |
| .03356482 | 59 | 4.09722222 | .06828704 |
| .03472222 | 60 | 4.16666666 | .06944444 |
| Names of Citties | Differ. Merid. | Hight Pole |
| St. Albons | 0 1 s | 55.55 |
| Barwick | 0 6 s | 55.49 |
| Bedford | 0 2 s | 52.18 |
| Bristol | 0 11 s | 51.32 |
| Boston | 0 0 | 53.2 |
| Cambridge | 0 1 a | 52.17 |
| Canterbury | 0 5 a | 51.27 |
| Carlile | 0 10 s | 54.57 |
| Chester | 0 11 s | 53.20 |
| Coventry | 0 4 s | 52.30 |
| Carmarthen | 0 17 s | 52.2 |
| Chichester | 0 3 s | 50.56 |
| Colchester | 0 5 a | 52.4 |
| Darby | 0 5 s | 53.6 |
| Dublin in Ireland | 0 26 s | 53.11 |
| Duresm [...] | 0 5 s | 54.45 |
| Dartmouth | 0 15 s | 50.32 |
| Eely | 0 1 a | 52.20 |
| Grantha [...] | 0 2 s | 52.58 |
| Glocester | 0 9 s | 52.00 |
| Halefax | 0 6 s | 52.49 |
| Hartford | 0 1 s | 52.50 |
| Hereford | 0 11 s | 52.14 |
| Huntington | 0 1 s | 52.19 |
| Hull | 0 1 s | 53.58 |
| Lancaster | 0 11 s | 54.08 |
| Leicester | 0 4 s | 52.40 |
| [Page 483] Lincoln | 0 1 s | 53.12 |
| Middle of the Isle of Man | 0 17 s | 54.22 |
| Nottingham | 0 4 s | 53.03 |
| Newark | 0 3 s | 53.02 |
| Newcastle | 0 6 s | 54.58 |
| N. Luffingham | 0 3 s | 52.41 |
| Norwich | 0 4 a | 52.44 |
| Northampton | 0 4 s | 52.18 |
| Oxford | 0 5 s | 51.54 |
| Okenham | 0 3 s | 52.44 |
| Peterborough | 0 2 s | 52.35 |
| Richmond | 0 6 s | 54.26 |
| Rochester | 0 3 a | 51.28 |
| Ross | 0 10 s | 52.07 |
| St. Michaels Mount in Cornwal | 0 23 s | 50.38 |
| Stafford | 0 8 s | 52.55 |
| Stamford | 0 2 s | 52.41 |
| Shrewsbury | 0 11 s | 52.48 |
| Tredah in Ireland | 0 27 s | 53.28 |
| uppingham in Rutland | 0 3 s | 52.40 |
| Warwick | 0 6 s | 52.25 |
| Winchester | 0 5 s | 50.10 |
| Waterford in Ireland | 0 27 s | 52.22 |
| Worcester | 0 9 s | 52.20 |
| Yarmouth in Suffolk | 0 6 a | 52.45 |
| York | 0 4 s | 54.00 |
| London | 0 00 | 51.32 |
| ☉ Mean Longitude | ☉ Mean Anomaly | |
| 1 | 99.9336437563 | 99.9288933116 |
| 2 | 99.8672875126 | 99.8577866232 |
| 3 | 99.8009312690 | 99.7866799348 |
| 4 | 99.7345750253 | 99.7155732465 |
| 5 | 99.6682187816 | 99.6444665581 |
| 6 | 99.6018625380 | 99.5733598697 |
| 7 | 99.5355062943 | 99.5022531814 |
| 8 | 99.4691500506 | 99.4211464930 |
| 9 | 99.4027938070 | 99.3600398046 |
| 10 | 99.3364375633 | 99.2889331162 |
| 100 | 93.3643756334 | 92.8893311628 |
| 1000 | 33.6437563341 | 28.8933116289 |
| Year. | ☉ Mean Anomaly | Praecession AEquinox. |
| 1 | 99.9297857316 | 00.0038580246 |
| 2 | 99.8595714632 | 00.0077160493 |
| 3 | 99.7893571949 | 00.0115740740 |
| 4 | 99.7191429265 | 00.0154320987 |
| 5 | 99.6489286582 | 00.0192901234 |
| 6 | 99.5787143898 | 00.0231481481 |
| 7 | 66.5085001114 | 00.0270061728 |
| 8 | 99.2978573164 | 00.0308641975 |
| 9 | 99.3680715847 | 00.0347222221 |
| 10 | 99.2978573164 | 00.0385802469 |
| 100 | 92.9785731642 | 00.3858024691 |
| 1000 | 99.7857316427 | 03.8580246913 |
| ☉ Mean Longitude | ☉ Mean Anomaly | |
| 1 | 99.9336437563 | 99.9288933116 |
| 2 | 99.8672875126 | 99.8577866232 |
| 3 | 99.8009312689 | 99.7866799348 |
| B 4 | 00.008365830 | 99.9892901234 |
| 5 | 99.9420095864 | 99.9181834350 |
| 6 | 99.875633427 | 99.8470767466 |
| 7 | 99.8092970990 | 99.7759700583 |
| B 8 | 00.0167316602 | 99.9785802468 |
| 9 | 99.9503754165 | 99.9074735584 |
| 10 | 99.8840191728 | 99.8363668700 |
| 11 | 99.8176629291 | 99.7652591816 |
| B 12 | 00.0250974903 | 99.9678703702 |
| 13 | 99.9587412466 | 99.8967636818 |
| 14 | 99.8923850029 | 99.8256569934 |
| 15 | 99.8260287592 | 99.7545503050 |
| B 16 | 00.0334633205 | 99.9571604936 |
| 17 | 99.9671070768 | 99.8860548052 |
| 18 | 99.9007508331 | 99.8149481168 |
| 19 | 99.8343945894 | 99.7438414284 |
| B 20 | 00.0418291506 | 99.9164506171 |
| 40 | 00.0836583012 | 99.8929012342 |
| 60 | 00.1254874518 | 99.8393518513 |
| 80 | 00.1673166024 | 99.7858024684 |
| 100 | 00.2091457530 | 99. 73225308 [...]5 |
| 200 | 00.4182015060 | 99.4645061710 |
| 300 | 00.6274372590 | 99.1967592565 |
| 400 | 00.8365830120 | 98.9290123420 |
| 500 | 01.0457287650 | 98.6612654275 |
| 600 | 01.2548745180 | 98.3935185130 |
| 700 | 01. 4640 [...]02710 | 98.1257715985 |
| AEra | ☉ mean Longitude | ☉ mean Anomaly |
| Chr. | 77. 22400.86419 | 58. 24289.56790 |
| 1600 | 80. 54891.97529 | 53. 95880.62961 |
| 1620 | 80. 59074.89035 | 53. 90525.69132 |
| 1640 | 80. 63257.80541 | 53. 85170.75303 |
| 1660 | 80. 67440.72047 | 53. 79815.81474 |
| 1680 | 80. 71623.63553 | 53. 74460.87645 |
| 1700 | 80. 75806.55059 | 53. 69105.93816 |
| 1720 | 80. 79989.46665 | 53. 63750.99987 |
| 1740 | 80. 84172.38171 | 53. 58396.06158 |
| 1760 | 80. 88265.29677 | 53. 53041.12329 |
| ☉ mean Lon. in Mon. | ☉ mean Ano. in Mo. | |
| Ianu. | 08. 48751.49488 | 08. 48711.14867 |
| Febr. | 16. 15365.74832 | 16. 15288.96037 |
| Mar. | 24. 64117.24320 | 24. 64000.10904 |
| April | 32. 85489.65760 | 32. 85333.47872 |
| May | 41. 34241.15248 | 41. 34044.62739 |
| Iune | 49. 55613.56688 | 49. 55377.99708 |
| Iuly | 58. 04365.06176 | 58. 04089.14575 |
| Aug. | 66. 53116.55664 | 66. 52800.29442 |
| Sept. | 74. 74488.97104 | 74. 74133.66410 |
| Octo. | 83. 23240.46592 | 85. 22844.81277 |
| Nov. | 91. 44612.88032 | 91. 44178.18245 |
| Dec. | 99. 93364.37563 | 99. 92889.33116 |
In Anno [...]issentili; post Februarium adde unum diem & unius dies motum.
| ☉ mean Longitude | ☉ mean Anomaly | |
| 1 | 0.2737908048 | 0.2737777898 |
| 2 | 0.5475816096 | 0.5475555796 |
| 3 | 0.8213724144 | 0.8213333694 |
| 4 | 1.0951632192 | 1.0951111592 |
| 5 | 1.3 [...]89540240 | 1.3688889490 |
| 6 | 1.6427448288 | 1.6426667388 |
| 7 | 1.9165356336 | 1.9164445286 |
| 8 | 2.1903264384 | 2.1902223184 |
| 9 | 2.4641172432 | 2.4640001082 |
| 10 | 2.7379080480 | 2.7377778980 |
| 11 | 3.0116988528 | 3.0115556878 |
| 12 | 3.2854896576 | 3.2853334776 |
| 13 | 3.5592804624 | 3.5591112674 |
| 14 | 3.8330712672 | 3.8328890572 |
| 15 | 4.1068620720 | 4.1066668470 |
| 16 | 4.3806428768 | 4.3804446368 |
| 17 | 4.6544436816 | 4.6542224266 |
| 18 | 4.9282344864 | 4.9280002164 |
| 19 | 5.2020252912 | 5.2077780062 |
| 20 | 5.4758160960 | 5.4755557960 |
| 21 | 5.7496069008 | 5.7493335858 |
| 22 | 6.0233977056 | 6.0231113756 |
| 23 | 6.2971885104 | 6.2968891654 |
| 24 | 6.5709793152 | 6.5706669552 |
| 25 | 6.8447701200 | 6.8444447450 |
| 26 | 7.1185609248 | 7.1182225348 |
| 27 | 7.3923517296 | 7.3920003246 |
| 28 | 7.6661425344 | 7.6957781144 |
| 29 | 7.9399333392 | 7.9395559042 |
| 30 | 8.2137241440 | 8.2133336940 |
| 31 | 8.4875149488 | 8.4871114838 |
| ☉ Mean Longitude | ☉ Mean Anomaly. | |
| 1 | 0.0114079502 | 0.0114074079 |
| 2 | 0.0228159004 | 0.0228148158 |
| 3 | 0.0342238506 | 0.0342222237 |
| 4 | 0.0456318008 | 0.0456296316 |
| 5 | 0.0570397510 | 0.0570370395 |
| 6 | 0.0684477012 | 0.0684444474 |
| 7 | 0.0798556514 | 0.0798518553 |
| 8 | 0.0912636016 | 0.0912592632 |
| 9 | 0.1026715518 | 0.1026666711 |
| 10 | 0.1140795020 | 0.1140740790 |
| 11 | 0.1254874522 | 0.1254814869 |
| 12 | 0.1368954024 | 0.1368888948 |
| 13 | 0.1483033526 | 0.1482963027 |
| 14 | 0.1597113028 | 0.1597037106 |
| 15 | 0.1711192530 | 0.1711111185 |
| 16 | 0.1825272032 | 0.1825185264 |
| 17 | 0.1939351534 | 0.1939259343 |
| 18 | 0.2053431036 | 0.2053333422 |
| 19 | 0.2167510538 | 0.2167407501 |
| 20 | 0.2281590040 | 0.2281481580 |
| 21 | 0.2395669542 | 0.2395555659 |
| 22 | 0.2509749044 | 0.2509629738 |
| 23 | 0.2623828546 | 0.2623703817 |
| 24 | 0.2737777048 | 0.2737777896 |
| AEra | ☉ Anomaly. | Praecess. AEquinox |
| Chr. | 56. 69976.85185 | 20. 49768.51851 |
| 1600 | 53. 87323.10751 | 26. 67052.46907 |
| 1620 | 53. 83789.15687 | 26. 74768.51845 |
| 1640 | 53. 80255.20623 | 26. 82484.56783 |
| 1660 | 53. 76721.25559 | 26. 90200.61721 |
| 1680 | 53. 73187.30495 | 26. 97916.66659 |
| 1700 | 53.69653▪35431 | 27. 05632.71597 |
| 1720 | 53. 66119.40367 | 27. 13348.76535 |
| 1740 | 53. 65585.45303 | 27. 21064.81473 |
| 1760 | 53. 59051.50230 | 27. 28780.86411 |
| ☉ Anomaly in Months | Praecess. AEquinox in Months | |
| Ianu | 08. 48718.72813 | 0. 00032.76678 |
| Febr. | 16. 15303.38579 | 0. 00062.36258 |
| Mar. | 24. 64022.11392 | 0. 00095.12937 |
| April | 32. 85362.81857 | 0. 00126.83916 |
| May | 41. 34081.54670 | 0. 00159.60594 |
| Iune | 49. 55422.25134 | 0. 00191.31573 |
| Iuly | 58. 04140.97947 | 0. 00224.08251 |
| Aug. | 66. 52859.70760 | 0. 00256.84929 |
| Sept. | 74. 74200.41225 | 0. 00288.55908 |
| Octo. | 83. 22919.14038 | 0. 00321.32587 |
| Nov. | 91. 44259.84502 | 0. 00353.03566 |
| Dec. | 99. 92978.57315 | 0. 00385.80244 |
| ☉ mean Anomaly | Praecess. AEquinox | |
| 1 | 99.9297857316 | 00.0038580246 |
| 2 | 99.8595714612 | 00.0077160493 |
| 3 | 99.7893571949 | 00.0115740740 |
| B 4 | 99.9929231686 | 00.0154320987 |
| 5 | 99.9227089002 | 00.0192901233 |
| 6 | 99.8524946318 | 00.0231481479 |
| 7 | 99.7822803634 | 00.0270061725 |
| B 8 | 99.9858463372 | 00.0308641974 |
| 9 | 99.9156320688 | 00.0347222220 |
| 10 | 99.8454178004 | 00.0385802466 |
| 11 | 99.7752035321 | 00.0424382714 |
| B 12 | 99.9787695058 | 00.0462962961 |
| 13 | 99.9085552374 | 00.0501543207 |
| 14 | 99.8383409690 | 00.0540123453 |
| 15 | 99.7681266066 | 00.0578703699 |
| B 16 | 99.9716926744 | 00.0617283948 |
| 17 | 99.9014784060 | 00.0655864194 |
| 18 | 99.8312647376 | 00.0694444440 |
| 19 | 99.7610498692 | 00.0733024686 |
| B 20 | 99.9646158434 | 00.0771604938 |
| 40 | 99.9292306868 | 00.1543209876 |
| 60 | 99.8938465302 | 00.2314814814 |
| 80 | 99.8584623736 | 00.3086419752 |
| 100 | 99.8270782170 | 00.3858024690 |
| 200 | 99.6461564340 | 00.7716049380 |
| 300 | 99.4692346510 | 01.1574074070 |
| 400 | 99.2923128680 | 01.5432098760 |
| 500 | 99.1153910850 | 01.9290123450 |
| 600 | 98.9384693020 | 02.3148148140 |
| 700 | 98.7615475190 | 02.7006172830 |
| D | ☉ Anomaly | Praecess. AEquinox |
| 1 | 0.2737802348 | 0.0000105699 |
| 2 | 0.5475604697 | 0.0000211398 |
| 3 | 0.8213407046 | 0.0000317097 |
| 4 | 1.0951209395 | 0.0000422797 |
| 5 | 1.3689011744 | 0.0000528496 |
| 6 | 1.6426814092 | 0.0000634195 |
| 7 | 1.9164616441 | 0.0000739894 |
| 8 | 2.1902418790 | 0.0000845593 |
| 9 | 2.4640221139 | 0.0000951292 |
| 10 | 2.7378023488 | 0.0001056993 |
| 11 | 3.0115825836 | 0.0001162692 |
| 12 | 3.2853628184 | 0.0001268391 |
| 13 | 3.5591430532 | 0.0001374090 |
| 14 | 3.8329232880 | 0.0001479789 |
| 15 | 4.1067035228 | 0.0001585488 |
| 16 | 4.3804837576 | 0.0001691187 |
| 17 | 4.6542639924 | 0.0001796886 |
| 18 | 4.9280442272 | 0.0001902585 |
| 19 | 4.2018244620 | 0.0002008284 |
| 20 | 5.4756046976 | 0.0002113986 |
| 21 | 5.7493849324 | 0.0002219685 |
| 22 | 6.0231651672 | 0.0002325384 |
| 23 | 6.2969454020 | 0.0002431083 |
| 24 | 6.5707256368 | 0.0002536782 |
| 25 | 6.8445058716 | 0.0002642481 |
| 26 | 7.1182861064 | 0.0002748180 |
| 27 | 7.3920663412 | 0.0002853879 |
| 28 | 7.6658455766 | 0.0002959580 |
| 29 | 7.9396258115 | 0.0003065279 |
| 30 | 8.2134070464 | 0.0003170979 |
| 31 | 8.4871872813 | 0.0003276678 |
| D | ☉ mean Anomaly | Praecess. AEquinox |
| 1 | 0.0114075097 | 0.0000004404 |
| 2 | 0.0228150195 | 08808 |
| 3 | 0.0342225293 | 13212 |
| 4 | 0.0456300391 | 17616 |
| 5 | 0.0570375489 | 22020 |
| 6 | 0.0684450587 | 0.0000026424 |
| 7 | 0.0798525684 | 30828 |
| 8 | 0.0912600782 | 35232 |
| 9 | 0.1026675881 | 39636 |
| 10 | 0.1140750978 | 44041 |
| 11 | 0.1254826075 | 0.0000048445 |
| 12 | 0.1368901174 | 0.0000052849 |
| 13 | 0.1482976271 | 57253 |
| 14 | 0.1597051368 | 61657 |
| 15 | 0.1711126465 | 66061 |
| 16 | 0.1825201562 | 70465 |
| 17 | 0.1939276659 | 74869 |
| 18 | 0.2053351761 | 0.0000079272 |
| 19 | 0.2167426858 | 83677 |
| 20 | 0.2281501955 | 88081 |
| 21 | 0.2395577052 | 92485 |
| 22 | 0.2509652149 | 96889 |
| 23 | 0.2623727246 | 101293 |
| 24 | 0.2737802348 | 0.0000105698 |
THE TABLES OF THE MOONS MEAN MOTIONS.
| AEra | ☽ Mean Longitude | ☽ Apogaeon |
| Chr. | 34.0088734567 | 78.8286265432 |
| 1600 | 02.0644290122 | 63.5892746911 |
| 1620 | 39.1651134566 | 89.6540895059 |
| 1640 | 76.2658079010 | 15.7189033207 |
| 1660 | 13.3665023454 | 41.7837191355 |
| 1680 | 50.4671967898 | 67.6485339503 |
| 1700 | 87.5675912342 | 93.9133487651 |
| 1620 | 29.6685801230 | 19.9781635799 |
| 1740 | 61.7692801230 | 46.0429783947 |
| 1760 | 98.8699745674 | 72.1077932095 |
| ☽ Mean Long. in Mon. | ☽ Apogaeon in Mont. | |
| Ianu. | 13.4633984897 | 00.9593447922 |
| Febr. | 15.9464670933 | 01.8258497658 |
| Mar. | 29.4098665830 | 02.7851945580 |
| April | 39.2131554440 | 03.7135927440 |
| May. | 52.6765539337 | 04.6729375362 |
| Iune | 62.4798427947 | 05.6013357222 |
| Iuly | 75.9432412844 | 06.5606805144 |
| Aug. | 89.4066397741 | 07.5200253066 |
| Sept. | 99.2099286451 | 08.4484234926 |
| Octo. | 12.6733271348 | 09.4077682848 |
| Nov. | 22.4766159958 | 10.3361664708 |
| Dec. | 35.9400144893 | 11.2955112636 |
| AEra | ☽ Mean Anomaly | ☽ Node Retrograde |
| Chr. | 55.1802469135 | 74.6984567901 |
| 1600 | 38.4751543211 | 78.2198302468 |
| 1620 | 49.5110239507 | 70.7638117283 |
| 1640 | 60.5469035803 | 63.3077932098 |
| 1660 | 71.5827832099 | 55.8517746913 |
| 1680 | 82.6186628395 | 48.3957561728 |
| 1700 | 93.6545424691 | 40.9397376543 |
| 1720 | 04.6904220987 | 33.4837191358 |
| 1740 | 15.7263017283 | 26.0277006173 |
| 1760 | 26.7621813579 | 18.5716820988 |
| ☽ Mean Ano. in Mon. | Node Ret. in Mont. | |
| Ianu. | 12.5040536975 | 00.4559979224 |
| Febr. | 14.7206183275 | 00.8678670136 |
| Mar. | 27.2246720250 | 01.3238649360 |
| April | 35.4995627000 | 01.7651532480 |
| May. | 48.0036163975 | 02.2211511704 |
| Iune | 56.8785070725 | 02.6624394824 |
| Iuly | 69.3825607700 | 03.1184374048 |
| Aug. | 81.8866144675 | 03.5744353272 |
| Sept. | 90.7615051425 | 04.0157236392 |
| Octo. | 03.2655588400 | 04.4717215616 |
| Nov. | 12.1404495150 | 04.9130098736 |
| Dec. | 24.6445032256 | 05.3690078260 |
| ☽ Mean Longitude | ☽ Apogaeon | |
| 1 | 35.9400144893 | 11.2955112636 |
| 2 | 71.8800289786 | 22.5910225272 |
| 3 | 07.8200434679 | 33.8865337908 |
| B 4 | 47.4201388888 | 45.2129629629 |
| 5 | 83.3601533781 | 56.5084742265 |
| 6 | 19.3001678674 | 67.8039854901 |
| 7 | 55.2401823567 | 79.0994967537 |
| B 8 | 94.8402777777 | 90.4259259258 |
| 9 | 30.7802922670 | 01 [...]7214371894 |
| 10 | 66.7203067563 | 13.0169484530 |
| 11 | 02.6603212456 | 24.3124597166 |
| B 12 | 42.2604166666 | 35.6388888888 |
| 13 | 78. [...]004311559 | 46.9344001524 |
| 14 | 14.1404456652 | 58.2299114 [...]60 |
| 15 | 50.0804601545 | 69. [...]2 [...]4226706 |
| B 16 | 89.6805555555 | 80.8518518518 |
| 17 | 25.6205700448 | 91.1473631154 |
| 18 | 61.5605845341 | 02.4428743790 |
| 19 | 97.5005990234 | 13.7383856426 |
| B 20 | 37.1006944404 | 26.0648 [...]48148 |
| 40 | 74.2013888888 | 52.1296296296 |
| 60 | 11.3020833333 | 78.1944444444 |
| 80 | 48.4027777777 | 04.2592592592 |
| 100 | 85.5034722222 | 30.3240740740 |
| 200 | 71.0069444444 | 60.648 [...]81 [...]81 |
| 300 | 56.5104166666 | 90.9722222222 |
| 400 | 42.0138888888 | 21.2962962962 |
| 500 | 27.517361111 [...] | 51.6003703700 |
| 600 | 13.0208333333 | 91.9444444442 |
| 700 | 98.5243055555 | 12.2685185182 |
| ☽ Mean Anomaly | ☽ Nodes Retrograde | |
| 1 | 24.6445032256 | 05.3690078260 |
| 2 | 49.2890064512 | 10.7380156520 |
| 3 | 73.9335096768 | 16.1070234780 |
| B 4 | 02.2071759259 | 21.4912037037 |
| 5 | 26.8516791515 | 26.7602115297 |
| 6 | 5 [...].4951823771 | 32.1292193557 |
| 7 | 76.1396856027 | 37.4982271817 |
| B 8 | 04.4143518518 | 42.9824074074 |
| 9 | 29.0588550774 | 48.3514152334 |
| 10 | 53.7033583030 | 53.7204230594 |
| 11 | 78.3478615286 | 59.0894308854 |
| B 12 | 06.6215277777 | 64.4736111111 |
| 13 | 37.2660310033 | 69.8426189371 |
| 14 | 55.9105342289 | 74.2116267631 |
| 15 | 80.5550374545 | 79.5806345891 |
| B 16 | 08.8287037037 | 85.9648148148 |
| 17 | 33.4732069293 | 91.3338226408 |
| 18 | 58.1177101549 | 96.7028304668 |
| 19 | 82. [...]622133805 | 02.0718382928 |
| B 20 | 11.0 [...]58796 [...]97 | 07.4560185185 |
| 40 | 22.0717592594 | 14.9120370370 |
| 60 | 33.1076388891 | 22.3680555555 |
| 80 | 44.1435185188 | 29.8040740740 |
| 100 | 55.1793981487 | 37.2800925925 |
| 200 | 10.3587062074 | 74.5601851850 |
| 300 | 65.5381944461 | 11.8402777775 |
| 400 | 20.7175925948 | 49.1203703700 |
| 500 | 75.8969907435 | 86.4004629629 |
| 600 | 31.0763888922 | 23.6805555555 |
| 700 | 86.2557870409 | 60.9606481480 |
| Days | ☽ Mean Longitude | ☽ Apogaeon |
| 1 | 03.6601096287 | 00.0309466062 |
| 2 | 07.3202192574 | 00.0618932124 |
| 3 | 10.9803288861 | 00.0928398186 |
| 4 | 14.6404385148 | 00.1237864248 |
| 5 | 18.3005481435 | 00.1547330310 |
| 6 | 21.9606577722 | 00.1856796372 |
| 7 | 25.6207674009 | 00.2166262434 |
| 8 | 29.2808770296 | 00.2475728496 |
| 9 | 32.9409866583 | 00.2785194558 |
| 10 | 36.6010962870 | 00.3094660620 |
| 11 | 40.2612059157 | 00.3404126682 |
| 12 | 43.9213155444 | 00.3713592744 |
| 13 | 47.5814251731 | 00.4023058806 |
| 14 | 51.2415348018 | 00.4332524868 |
| 15 | 54.9016444305 | 00.4641990930 |
| 16 | 58.5617540592 | 00.4951456992 |
| 17 | 62.2218636879 | 00.5260923054 |
| 18 | 65.8819733166 | 00.5570389116 |
| 19 | 69.5420829453 | 00.5879855178 |
| 20 | 73.2021925740 | 00.6189321240 |
| 21 | 76.8623022037 | 00.6498787302 |
| 22 | 80.5224118314 | 00.6808233364 |
| 23 | 84.1825214601 | 00.7117719426 |
| 24 | 87.8426310898 | 00.7427185488 |
| 25 | 91.5027407175 | 00.7736651550 |
| 26 | 95.1628503462 | 00.8046117612 |
| 27 | 98.8229599749 | 00.8355583674 |
| 28 | 02.4830696036 | 00.8665049736 |
| 29 | 06.1431792323 | 00.8974515798 |
| 30 | 09.8032888610 | 00.9283981860 |
| 31 | 13.4633984897 | 00.9593447922 |
| Days | ☽ Mean Anomaly | ☽ Node Retrograde |
| 1 | 03.6291630225 | 00.0147096104 |
| 2 | 07.2583260450 | 00.0294192208 |
| 3 | 10.8874890675 | 00.0441288312 |
| 4 | 14.5166520900 | 00.0588384416 |
| 5 | 18.1458151125 | 00.0735480520 |
| 6 | 21.7749781350 | 00.0882576624 |
| 7 | 25.4041411575 | 00.1029672728 |
| 8 | 29.0333041800 | 00.1176768832 |
| 9 | 32.6624672025 | 00.1323864936 |
| 10 | 36.2916302250 | 00.1470961040 |
| 11 | 39.9207932475 | 00.1618057144 |
| 12 | 43.5499562700 | 00.1765153248 |
| 13 | 47.1791192925 | 00.1912249352 |
| 14 | 50.8082823150 | 00.20 [...]9345456 |
| 15 | 54.4374453375 | 00.2206441560 |
| 16 | 58.0666083600 | 00.2353537664 |
| 17 | 61.6957713825 | 00.2500633768 |
| 18 | 65.3249344050 | 00.2647729872 |
| 19 | 68.9540974275 | 00.2794825976 |
| 20 | 72.5832604500 | 00.2941922080 |
| 21 | 76.2124234725 | 00.3089018184 |
| 22 | 79.8415864950 | 00.3236114288 |
| 23 | 83.4707495175 | 00.3383210392 |
| 24 | 87.0999125400 | 00.3530306496 |
| 25 | 90.7290755625 | 00.3677402600 |
| 26 | 94.3582385850 | 00.3824498704 |
| 27 | 97.9874016075 | 00.3971594808 |
| 28 | 01.6165646300 | 00.4118690912 |
| 29 | 05.2457276525 | 00.4265787016 |
| 30 | 08.8748906750 | 00.4412883120 |
| 31 | 12.5040536975 | 00.4559979224 |
| Hours | ☽ Mean Longitude | ☽ Apogaeon |
| 1 | 00.1525045678 | 00.0012894419 |
| 2 | 00.3050091357 | 00.0025788838 |
| 3 | 00.4575137035 | 00.0038683257 |
| 4 | 00.6100182713 | 00.0041577676 |
| 5 | 00. [...]625228391 | 00.0064172095 |
| 6 | 00.9150274071 | 00.0077366515 |
| 7 | 01.0675319749 | 00.0090260934 |
| 8 | 01.2200365427 | 00.0103155353 |
| 9 | 01.3725411105 | 00.0116049772 |
| 10 | 01.5250456786 | 00.0128044192 |
| 11 | 01.6775502464 | 00.0141838611 |
| 12 | 01.8300548143 | 00.0154733031 |
| 13 | 01.9825593821 | 00.0167627450 |
| 14 | 02.1350639499 | 00.0180521869 |
| 15 | 02.2875685177 | 00.0193416288 |
| 16 | 02.4400730855 | 00.0206310707 |
| 17 | 02.5925776533 | 00.0219205126 |
| 18 | 02.7450822211 | 00.0232099545 |
| 19 | 02.8975867891 | 00.0244993964 |
| 20 | 03.0500 [...]13560 | 00.0257888384 |
| 21 | 03.2025959250 | 00.0270782803 |
| 22 | 03.3551004928 | 00.0283677222 |
| 23 | 03.5076050607 | 00.0296571642 |
| 24 | 03.6601096285 | 00.0309466061 |
| Hours | ☽ Mean Anomaly | ☽ Node Retrograde |
| 1 | 00.1512151259 | 00.0006129004 |
| 2 | 00.3024302518 | 00.0012258008 |
| 3 | 00.4536453778 | 00.0018387013 |
| 4 | 00.6048605037 | 00.0024516017 |
| 5 | 00.7560756296 | 00.0030645021 |
| 6 | 00.9072907556 | 00.0036774026 |
| 7 | 01.0585058815 | 00.0042903030 |
| 8 | 01.2097210074 | 00.0049032034 |
| 9 | 01.3609361333 | 00.0055161038 |
| 10 | 01.5121512593 | 00.0061290043 |
| 11 | 01.6633663852 | 00.0067419047 |
| 12 | 01.8145815112 | 00.0073548052 |
| 13 | 01.9657066371 | 00.0079677056 |
| 14 | 01.1170117630 | 00.0085806060 |
| 15 | 02.2682068889 | 00.0091935064 |
| 16 | 02.4194420148 | 00.009 [...]064068 |
| 17 | 02.5706571407 | 00.0 [...]04193072 |
| 18 | 02.7218722666 | 00.0110722076 |
| 19 | 02.8730873926 | 00.0116451081 |
| 20 | 03.9243025 [...]85 | 00.0122580085 |
| 21 | 03.1755176445 | 00.0128709090 |
| 22 | 03.3267327704 | 00.0134838004 |
| 23 | 03.4779478964 | 00.0140967099 |
| 24 | 03.6291630223 | 00.0147096103 |
| M. | ☽ M. Long. | ☽ Apog. | ☽ M. Au. | [...] Retrog. |
| 1 | .0025414 | .0000214 | .0025202 | .0000102 |
| 2 | .0050828 | .0000429 | .0050405 | .0000204 |
| 3 | .0076242 | .0000643 | .0075607 | .0000306 |
| 4 | .0101656 | .0000859 | .0100810 | .0000408 |
| 5 | .0127070 | .0001074 | .0126012 | .0000510 |
| 6 | .0152484 | .0001288 | .0151214 | .0000612 |
| 7 | .0177898 | .0001502 | .0176416 | .0000714 |
| 8 | .0203312 | .0001716 | .0201618 | .0000816 |
| 9 | .0228726 | .0001930 | .0226820 | .0000918 |
| 10 | .0254141 | .0002149 | .0252025 | .0001021 |
| 11 | .0279555 | .0002363 | .0277227 | .0001123 |
| 12 | .0304969 | .0002577 | .0302429 | .0001225 |
| 13 | .0330383 | .0002791 | .0327631 | .0001327 |
| 14 | .0355797 | .0003004 | .0352833 | .0001429 |
| 15 | .0381211 | .0003218 | .0378035 | .0001531 |
| 16 | .0406624 | .0003432 | .0403237 | .0001633 |
| 17 | .0432038 | .0003646 | .0428439 | .0001735 |
| 18 | .0457452 | .0003860 | .0453641 | .0001837 |
| 19 | .0482867 | .0004079 | .0478843 | .0001939 |
| 20 | .0508284 | .0004298 | .0504045 | .0002041 |
| 21 | .0533696 | .0004512 | .0529247 | .0002143 |
| 22 | .0559110 | .0004726 | .0554449 | .0002245 |
| 23 | .0584524 | .0004940 | .0579651 | .0002347 |
| 24 | .0609938 | .0005154 | .0604853 | .0002442 |
| 25 | .0635352 | .0005368 | .0630055 | .0002544 |
| 26 | .0660766 | .0005582 | .0655257 | .0002642 |
| 27 | .0686180 | .0005795 | .0680459 | .0002744 |
| 28 | .0711594 | .0006008 | .0705661 | .0002846 |
| 29 | .0737008 | .0006222 | .0730863 | .0092948 |
| 30 | .0762422 | .0006437 | .0756075 | .0003064 |
| ☽ M. Long. | ☽ Apog. | ☽ M. Au. | ☊ Retrog. | |
| 1 | 0000423 | 0000003 | 0000420 | 0000002 |
| 2 | 0000847 | 0000007 | 0000840 | 0000003 |
| 3 | 0001270 | 0000010 | 0001260 | 0000005 |
| 4 | 0001693 | 0000013 | 0001680 | 0000006 |
| 5 | 0002116 | 0000016 | 0002100 | 0000009 |
| 6 | 0002539 | 0000019 | 0002520 | 0000010 |
| 7 | 0002969 | 0000022 | 0002940 | 0000012 |
| 8 | 0003392 | 0000025 | 0003360 | 0000013 |
| 9 | 0003815 | 0000028 | 0003780 | 0000015 |
| 10 | 0004275 | 0000035 | 0004200 | 0000017 |
| 11 | 0004658 | 0000038 | 0004620 | 0000019 |
| 12 | 0005078 | 0000041 | 0005040 | 0000020 |
| 13 | 0005504 | 0000044 | 0005460 | 0000022 |
| 14 | 0005930 | 0000047 | 0005880 | 0000023 |
| 15 | 0006357 | 0000050 | 0006300 | 0000025 |
| 16 | 0006784 | 0000053 | 0006720 | 0000027 |
| 17 | 0007207 | 0000056 | 0007140 | 0000028 |
| 18 | 0007630 | 0000059 | 0007560 | 0000029 |
| 19 | 0008050 | 0000062 | 0007980 | 0000031 |
| 20 | 0008470 | 0000065 | 0008400 | 0000033 |
| 21 | 0008893 | 0000068 | 0008820 | 0000035 |
| 22 | 0009316 | 0000071 | 0009240 | 0000036 |
| 23 | 0009736 | 0000074 | 0009660 | 0000038 |
| 24 | 0010156 | 0000077 | 0010080 | 0000039 |
| 25 | 0010582 | 0000080 | 0010500 | 0000041 |
| 26 | 0011008 | 0000083 | 0010920 | 0000043 |
| 27 | 0011434 | 0000086 | 0011340 | 0000044 |
| 28 | 0011860 | 0000089 | 0011760 | 0000047 |
| 29 | 0012287 | 0000092 | 0012180 | 0000049 |
| 30 | 0012714 | 0000095 | 0012600 | 0000051 |
| ☉ | Sig. o. &. 6 | 1 & 7 | 2 & 8 | ☉ | |||
| a | AEqu. ☊ | Inclin. | AEqu. ☊ | Inclin. | AEqu. ☊ | Inclin. | a |
| ☊ | Addi | limetis | Addi | limitis | Addi | limitis | ☊ |
| 0 | 0.00000 | 30000 | 1.06500 | 20000 | 1.08500 | 15000 | 30 |
| 1 | 0.00000 | 30000 | 1.12888 | 25722 | 1.01888 | 14527 | 29 |
| 2 | 0.00055 | 30000 | 1.19277 | 25472 | 0.95305 | 14055 | 28 |
| 3 | 0.00194 | 29972 | 1.25222 | 25166 | 0.88666 | 13583 | 27 |
| 4 | 0.0041 [...] | 29944 | 1.30833 | 24888 | 0.82055 | 13138 | 26 |
| 5 | 0.00888 | 29888 | 1.36166 | 24583 | 0.55333 | 12666 | 25 |
| 6 | 0.01472 | 29833 | 1.41055 | 24277 | 0.68694 | 12194 | 24 |
| 7 | 0.02305 | 29777 | 1.45666 | 23972 | 0.62416 | 11 [...]22 | 23 |
| 8 | 0.02416 | 29722 | 1.49916 | 23638 | 0.56444 | 11250 | 22 |
| 9 | 0.04805 | 29638 | 1.53666 | 23305 | 0.50555 | 10750 | 21 |
| 10 | 0.06500 | 29555 | 1.58000 | 22972 | 0.44666 | 10250 | 20 |
| 11 | 0.08555 | 29444 | 1.60027 | 22638 | 0.38888 | 09750 | 19 |
| 12 | 0.10944 | 29361 | 1.62527 | 22277 | 0.34000 | 09250 | 18 |
| 13 | 0.13666 | 29250 | 1.64472 | 219 [...]6 | 0.28972 | 08750 | 17 |
| 14 | 0.16833 | 29111 | 1.65771 | 21 [...]55 | 0.25083 | 08250 | 16 |
| 15 | 0.20250 | 28972 | 1.66277 | 21222 | 0.20972 | 07750 | 15 |
| 16 | 0.2411 [...] | 28833 | 1.65805 | 20833 | 0.17388 | 07250 | 14 |
| 17 | 0.27472 | 28667 | 1.64527 | 20444 | 0.14138 | 06750 | 13 |
| 18 | 0. [...]2944 | 28567 | 1.62638 | 20055 | 0.11305 | 06250 | 12 |
| 19 | 0.37916 | 28361 | 1.60194 | 19666 | 0.08805 | 05722 | 11 |
| 20 | 0.43277 | 28194 | 1.58222 | 19279 | 0.06722 | 05222 | 10 |
| 21 | 0.48888 | 18027 | 1.53972 | 18861 | 0.04916 | 04694 | 9 |
| 22 | 0.54833 | 27833 | 1.50333 | 18444 | 0.03416 | 04166 | 8 |
| 23 | 0.60694 | 27611 | 1.46222 | 18027 | 0.02333 | 03638 | 7 |
| 24 | 0.66833 | 27416 | 1.41799 | 17614 | 0.01500 | 03138 | 6 |
| 25 | 0.73555 | 27104 | 1.37027 | 17194 | 0.00888 | 02611 | 5 |
| 26 | 0.79805 | 26972 | 1.3186 [...] | 14750 | 0.00416 | 02083 | 4 |
| 27 | 0.8641 [...] | 26722 | 1.26527 | 16333 | 0.001 [...]4 | 01555 | 3 |
| 28 | 0.93083 | 26500 | 1.20833 | 15888 | 0.00055 | 01027 | 2 |
| 29 | 0.99611 | 26250 | 1.14750 | 15444 | 0.00000 | [...]0527 | 1 |
| 30 | 1.06500 | 26600 | 1.08500 | 14000 | 0.00000 | 00000 | 0 |
| Sntract | Subtract. | Subtract. | |||||
| Sig. 5. & 11 | 4 & 10 | 3 & 9 | |||||
| Aug. | Sig. o. N. | Incr. | 1. North | Incr. | 2. North | I ncr. | |
| Sig. 6. S. | 7. South | 8. South | |||||
| Lat. | or Exc. | or Exc. | or Exc. | ||||
| Latit. | Latit. | Latit. | |||||
| 0 | 0.00000 | 00000 | 2.49750 | 15000 | 4.32888 | 26000 | 30 |
| 1 | 0.08722 | 00527 | 2.57277 | 15444 | 4.37166 | 26250 | 29 |
| 2 | 0.17444 | 01027 | 2.64722 | 15888 | 4.41361 | 26500 | 28 |
| 3 | 0.26166 | 01555 | 2.72083 | 16333 | 4.45388 | 26722 | 27 |
| 4 | 0.34861 | 02083 | 2.79361 | 16416 | 4.49277 | 26972 | 26 |
| 5 | 0.43555 | 02611 | 2.86555 | 17194 | 4.53055 | 27194 | 25 |
| 6 | 0.52222 | 03138 | 2.93638 | 17611 | 4.56666 | 27416 | 24 |
| 7 | 0.60888 | 03638 | 3.00666 | 18027 | 4.60166 | 27611 | 23 |
| 8 | 0.69527 | 04166 | 3. [...]7583 | 18444 | 4.63500 | 27833 | 22 |
| 9 | 0.78138 | 04694 | 3.14416 | 18861 | 4.66722 | 28027 | 21 |
| 10 | 0.86722 | 05222 | 3.21166 | 19277 | 4.69777 | 28194 | 20 |
| 11 | 0.95277 | 05722 | 3.27805 | 19666 | 4.72694 | 28361 | 19 |
| 12 | 1.03833 | 06250 | 3.34333 | 20055 | 4.75472 | 28527 | 18 |
| 13 | 1.12333 | 06750 | 3.41055 | 20444 | 4.78111 | 28666 | 17 |
| 14 | 1.10805 | 07250 | 3.47111 | 20833 | 4.80583 | 28833 | 16 |
| 15 | 1.29250 | 07750 | 3.53333 | 21222 | 4.82916 | 28972 | 15 |
| 16 | 1.37666 | 08250 | 3.59444 | 21555 | 4.85111 | 29111 | 14 |
| 17 | 1.46027 | 08750 | 3.65472 | 21916 | 4.87166 | 29250 | 13 |
| 18 | 1.54333 | 09250 | 3.71361 | 22277 | 4.89055 | 29361 | 12 |
| 19 | 1.62611 | 09750 | 3.77138 | 22638 | 4.90777 | 29444 | 11 |
| 20 | 1.70805 | 10250 | 3.82833 | 22972 | 4.92388 | 29555 | 10 |
| 21 | 1.78972 | 10750 | 3.88388 | 23305 | 4.93833 | 29638 | 9 |
| 22 | 1.87111 | 11250 | 3.93805 | 23638 | 4.95111 | 29722 | 8 |
| 23 | 1.95166 | 11722 | 3.99138 | 23972 | 4.96250 | 29777 | 7 |
| 24 | 2.03166 | 12194 | 4.04333 | 24277 | 4.97250 | 29833 | 6 |
| 25 | 2.11083 | 12666 | 4.29416 | 24583 | 4.98083 | 29888 | 5 |
| 26 | 2.18972 | 13138 | 4.14361 | 24888 | 4.98777 | 29944 | 4 |
| 27 | 2.26777 | 13583 | 4.19166 | 25166 | 4.99301 | 29972 | 3 |
| 28 | 2.34500 | 14055 | 4.23861 | 25472 | 4.99694 | 29972 | 2 |
| 29 | 2.42166 | 14527 | 4.28444 | 25722 | 4.99916 | 18000 | 1 |
| 30 | 2.49750 | 15000 | 4.32888 | 26000 | 5.00000 | 18000 | 0 |
| Sig. 11. S. | Sig. 19. S. | 9. South | |||||
| Sig. 5. N. | 4 N. | 3. North |
| As Lat. | Sig.0.6 | S.1.7 | S.2.8 | As. Lat. | |||
| Incr. | Incr. | Incr. | |||||
| Red. | Red. | Red. | |||||
| 0 | .00000 | 00000 | 09444 | 01166 | 09472 | 01166 | 30 |
| 1 | .00388 | 00055 | 09638 | 01194 | 09277 | 01138 | 29 |
| 2 | .00750 | 00111 | 09805 | 01222 | 09055 | 01111 | 28 |
| 3 | .01138 | 00166 | 09972 | 01250 | 08833 | 01083 | 27 |
| 4 | .01527 | 00222 | 10111 | 01277 | 08611 | 01055 | 26 |
| 5 | .01888 | 00250 | 10250 | 01277 | 08388 | 01027 | 25 |
| 6 | .02277 | 00305 | 10388 | 01277 | 08138 | 01000 | 24 |
| 7 | .02638 | 00333 | 10500 | 01305 | 07861 | 00972 | 23 |
| 8 | .03000 | 00361 | 10583 | 01305 | 07611 | 00944 | 22 |
| 9 | .03361 | 00416 | 10666 | 01305 | 07305 | 00916 | 21 |
| 10 | .03722 | 00472 | 10750 | 01333 | 07027 | 00861 | 20 |
| 11 | .04083 | 00527 | 10805 | 01333 | 06722 | 00833 | 19 |
| 12 | .04444 | 00555 | 10861 | 01333 | 06416 | 00805 | 18 |
| 13 | .04777 | 00611 | 10888 | 01333 | 06111 | 00777 | 17 |
| 14 | .05111 | 00638 | 10916 | 01361 | 05805 | 00722 | 16 |
| 15 | .05444 | 00666 | 10916 | 01361 | 05472 | 00666 | 15 |
| 16 | .05777 | 00722 | 10916 | 01361 | 05138 | 00638 | 14 |
| 17 | .06111 | 00777 | 10862 | 01361 | 04805 | 00611 | 13 |
| 18 | .06416 | 00805 | 10861 | 01361 | 04444 | 00555 | 12 |
| 19 | .06722 | 00833 | 10805 | 0133 [...] | 04111 | 00527 | 11 |
| 20 | .07000 | 00861 | 10750 | 01333 | 03750 | 00472 | 10 |
| 21 | .07305 | 00916 | 10694 | 01305 | 03388 | 00416 | 9 |
| 22 | .07583 | 00944 | 10611 | 01305 | 03027 | 00361 | 8 |
| 23 | .07888 | 00972 | 10500 | 01305 | 02092 | 00333 | 7 |
| 24 | .08111 | 01000 | 10388 | 01277 | 02611 | 00305 | 6 |
| 25 | .08361 | 01027 | 10277 | 01277 | 02222 | 00250 | 5 |
| 26 | .08583 | 01055 | 10138 | 01277 | 01527 | 00222 | 4 |
| 27 | .08823 | 01083 | 10000 | 01250 | 01138 | 00166 | [...]3 |
| 28 | .09055 | 01111 | 09833 | 01222 | 00750 | 00111 | 2 |
| 29 | .09250 | 01138 | 09638 | 01194 | 00388 | 00055 | 1 |
| 30 | .09444 | 01166 | 09472 | 01166 | 00000 | 00006 | 0 |
| 11.5 | 10.4 | 9.3 |
| AEra | ☽ à ☉ in Years | ☽ à ☉ in Years | |
| Chr. | 56.8114797531 | 1 | 36.0063707331 |
| 1600 | 21.5206732464 | 2 | 72.012741466 [...] |
| 1620 | 58.5795367034 | 3 | 08.0191121993 |
| 1640 | 95.6384101604 | 4 | 47.4117836215 |
| 1660 | 32.6972836174 | 5 | 83.4181543546 |
| 1680 | 69.7561560744 | 6 | 19.4245250877 |
| 1700 | 06.8150305314 | 7 | 55.4308958208 |
| 1720 | 43.8739039884 | 8 | 94.8235672430 |
| 1740 | 80.9027774454 | 9 | 40.8298379761 |
| 1760 | 17.9916509024 | 10 | 76.8362087092 |
| 11 | 02.8426794423 | ||
| Motion of the | 12 | 42.2353508645 | |
| Moon from the | 13 | 78.2417215976 | |
| Sun in Months. | 14 | 14.2480923307 | |
| 15 | 50.2544630638 | ||
| 16 | 89.6471344860 | ||
| Ian. | 04.9758835440 | 17 | 25.6535052191 |
| Feb. | 99.7928106160 | 18 | 61.6598759522 |
| Mar. | 04.7686941600 | 19 | 97.6662466853 |
| April | 06.3582588800 | 20 | 37.0589181075 |
| 40 | 74.1178362150 | ||
| May | 11.3341424240 | 60 | 11.1767543225 |
| Iune | 12.9237071440 | 80 | 48.2356724300 |
| Iuly | 17.8995906880 | 100 | 85.2945905375 |
| Aug. | 22.8754742320 | 200 | 70.5891810750 |
| 300 | 55.8837716125 | ||
| Sept. | 24.4650389520 | 400 | 41.1783621500 |
| Octo. | 29.4409224960 | 500 | 26.4729526875 |
| Nov. | 31.0304872160 | 600 | 11.7675432250 |
| Dec. | 36.0063707331 | 700 | 97.0621337625 |
| ☽ à ☉ in Days. | ☽ à ☉ in Hours. | ||
| 1 | 03.3863188240 | 1 | 00.1410966176 |
| 2 | 06.7726376480 | 2 | 00.2821932352 |
| 3 | 10.1589564720 | 3 | 00.4232898530 |
| 4 | 13.5452752960 | 4 | 00.5643864706 |
| 5 | 16.9315941200 | 5 | 00.7054830882 |
| 6 | 20.3179129440 | 6 | 00.8465797060 |
| 7 | 23.7042317680 | 7 | 00. [...]876763236 |
| 8 | 27.0905505920 | 8 | 01.1287729412 |
| 9 | 30.4768694160 | 9 | 01.2698695588 |
| 10 | 33.8631882400 | 10 | 01.4109661766 |
| 11 | 37.2495070640 | 11 | 01.5520627942 |
| 12 | 47.6358258880 | 12 | 01.6931594120 |
| 13 | 44.0221447720 | 13 | 01.8342560296 |
| 14 | 47.4084635360 | 14 | 01.9753526472 |
| 15 | 50. [...]947823600 | 15 | 0 [...].1164492648 |
| 16 | 54.1811011840 | 16 | 02.2575458824 |
| 17 | 57.5674200080 | 17 | 02.3986425000 |
| 18 | 60.9537388320 | 18 | 02.5397391176 |
| 19 | 64.3400576560 | 19 | 02.6808357354 |
| 20 | 67.7263764800 | 20 | 02.8219323520 |
| 21 | 71.1126953040 | 21 | 02.9630289708 |
| 22 | 74.4990141280 | 22 | 03.1041255884 |
| 23 | 77.8853329520 | 23 | 03.2452222062 |
| 24 | 81.2716517760 | 24 | 03.3863188240 |
| 25 | 84.6579706000 | ||
| 26 | 88.0442804240 | ||
| 27 | 91 4306082480 | ||
| 28 | 94.8169270720 | ||
| 29 | 98.2032458960 | ||
| 30 | 01.5895647200 | ||
| 31 | 04.9758835440 |
| ☽ à ☉ in Minutes. | |
| 1 | 00.0023516102 |
| 2 | 00.0047032205 |
| 3 | 00.0070548308 |
| 4 | 00.00 [...]4064411 |
| 5 | 00.0117580513 |
| 6 | 00.0141096617 |
| 7 | 00.0164612719 |
| 8 | 00.0188128822 |
| 9 | 00.0211644924 |
| 10 | 00.0235161029 |
| 11 | 00.0258677131 |
| 12 | 00.0262193233 |
| 13 | 00.0305709335 |
| 14 | 00.0329225437 |
| 15 | 00.0352741539 |
| 16 | 00.0376257644 |
| 17 | 00.0399773746 |
| 18 | 00.0423289848 |
| 19 | 00.044680 [...]950 |
| 20 | 00.0170322052 |
| 21 | 00.0493838154 |
| 22 | 00.0517354256 |
| 23 | 00.0540870 [...]58 |
| 24 | 00.0564386460 |
| 25 | 00.0587902562 |
| 26 | 00.0611418664 |
| 27 | 00.0634934766 |
| 28 | 00.0658450868 |
| 29 | 00.0681966970 |
| 30 | 00.0705483080 |
| ☽ à ☉ in Minutes. | |
| 31 | 00.0728999183 |
| 32 | 00.0752515088 |
| 33 | 00.0776031390 |
| 34 | 00.0799547492 |
| 35 | 00.0823063594 |
| 36 | 00.0846579696 |
| 37 | 00.0870095798 |
| 38 | 00.0893611900 |
| 39 | 00.0917128002 |
| 40 | 00.0940644104 |
| 41 | 00.0964160206 |
| 42 | 00.0997676308 |
| 43 | 00.1011192410 |
| 44 | 00.1034708512 |
| 45 | 00.1058224614 |
| 46 | 00.1081740716 |
| 47 | 00.1105256818 |
| 48 | 00.1128772920 |
| 49 | 00.1152289022 |
| 50 | 00.1175805124 |
| 51 | 00.1199321226 |
| 52 | 00.1222837328 |
| 53 | 00.1246353430 |
| 54 | 00.1269869532 |
| 55 | 00.1293385634 |
| 56 | 00.1316901736 |
| 57 | 00.1340417838 |
| 58 | 00.1363933940 |
| 59 | 00.1387450050 |
| 60 | 00.1410966152 |
| ☽ à ☉ in Seconds | |
| 1 | 00.0000391935 |
| 2 | 00.0000783870 |
| 3 | 00.0001175805 |
| 4 | 00.0001567740 |
| 5 | 00.0001959675 |
| 6 | 00.0002351610 |
| 7 | 00.0002743545 |
| 8 | 00.0003135480 |
| 9 | 00.0003527415 |
| 10 | 60.0003919350 |
| 11 | 00.0004811285 |
| 12 | 00.0004703220 |
| 13 | 00.0005995155 |
| 14 | 00.0005487090 |
| 15 | 00.0005879025 |
| 16 | 00.00 [...]6270960 |
| 17 | 00.0006662895 |
| 18 | 00.0007954830 |
| 19 | 00.0007446765 |
| 20 | 00.0007838700 |
| 21 | 00.0008230635 |
| 22 | 00.0008622570 |
| 23 | 00.0009014505 |
| 24 | 00.0009406440 |
| 25 | 00.0009798 [...]75 |
| 26 | 00.0010190310 |
| 27 | 00.0010582245 |
| 28 | 00.0010974180 |
| 29 | 00.0011366115 |
| 30 | 00.0011758050 |
| ☽ à ☉ in Seconds | |
| 31 | 00.0012149985 |
| 32 | 00.0012541920 |
| 33 | 00.0012933855 |
| 34 | 00.0013325790 |
| 35 | 00.0013717725 |
| 36 | 00.0014109660 |
| 37 | 00.0014501595 |
| 38 | 00.0014893530 |
| 39 | 00.0015285465 |
| 40 | 00.0015677400 |
| 41 | 00.0016069335 |
| 42 | 00.0016461270 |
| 43 | 00.0016853205 |
| 44 | 00.0017245140 |
| 45 | 00.0917637075 |
| 46 | 00.0018029010 |
| 47 | 00.0018420945 |
| 48 | 00.0018812880 |
| 49 | 00.0019204815 |
| 50 | 00.0019596750 |
| 51 | 00 0019988685 |
| 52 | 00.0020380620 |
| 53 | 00.0020772555 |
| 54 | 00.0021164490 |
| 55 | 00.0021556425 |
| 56 | 00.0021948360 |
| 57 | 00.0022340295 |
| 58 | 00.0022732230 |
| 59 | 00.0023124165 |
| 60 | 00.0023516100 |
| The Names of the Stars | Longit. | Latit. |
| The first Star of Aries. | 07.671 ♈ | 7. 8. N 4 |
| The bright Star in the top of the head of Aries. | 00.583 ♉ | 9. 57. N 3 |
| The South Eye of Taurus. | 01.169 [...] | 5. 31. S 1 |
| The North Eye of Taurus. | 00.801 [...] | 5. 31. S 1 |
| The bright Star of the Pleiades. | 06.620 ♉ | 2. 6. S 3 |
| The higher head of Gemini. | 04.078 [...] | 4. 11. N 5 |
| The lower head of Gemini. | 04.921 ♋ | 10. 2. N 2 |
| The bright foot of Gemini. | 01.069 ♋ | 6. 38. N 2 |
| In the South Arm of Cancer. | 02.238 ♌ | 6. 48. S 2 |
| The bright Star in the neck of Leo. | 06.662 ♌ | 5. 8. S 3 |
| The heart of Leo. | 06.745 ♌ | [...].47. N 2 |
| In the extream of the tail of Leo. | 04.458 ♍ | 0. 26. N 1 |
| In Virgo's Wing; Vindemiatrix. | 01.217 ♎ | 12. 18. N 1 |
| Virgins Spike. | 05.074 ♎ | 16. 15. N 3 |
| South Ballance. | 02.643 ♏ | 1. 59. S 1 |
| North Ballance. | 03.833 ♏ | 0. 26. N 2 |
| The highest in the Forehead of Scorpio. | 07.388 ♏ | 8. 35. N 2 |
| The Scorpions heart. | 01.171 [...] | 1. 05. N 3 |
| Former of the 3 in the head of Sagittarius. | 02.203 [...] | 4. 27. S 1 |
| Northern in the former horn of Capricorn. | 07.861 [...] | 1. 24. N 4 |
| The left Shoulder of Aquarius. | 04.949 ♒ | 7. 22. N 3 |
| In the mouth of the South Fish. | 03.620 ♓ | 8. 42. N 3 |
| The Polar Star or last Star in the [...]ail of the lesser Bear. | 9. 4. N 5 | |
| 06.400 [...] | 66. 02. N 2 | |
| [Page] The last Star in the tail of the great Bear, | 05.888 ♍ | 54. 25. N 2 |
| The Tongu [...] of the Dragon. | 05.259 ♍ | 76. 17. N 4 |
| Arcturus in the skirt of his Garment. | 05.181 ♎ | 31. 2. N 1 |
| The bright Star of the North Crown. | 01.845 ♏ | 44. 23. N 2 |
| The Head of Hercules▪ | 02.921 [...] | 37. 23. N [...] |
| The bright S [...]r of the H [...]rp. | 0 [...].699 [...] | 61. 47. N [...] |
| The Head of Medusa. | 05.727 ♉ | 22. 22. N 3 |
| The bright Star in the Goa [...]s left Shoulder. | 04.518 ♊ | 22. 50. N 1 |
| The middle of the Serp [...]nts Neck. | 04.583 ♍ | 25. 35. N 2 |
| The bright Star in the [...]agles Shoulder. | 07.264 ♑ | 29. 21. N 2 |
| The bright Star in the [...] Tail. | 02.370 ♒ | 29. 8. N 3 |
| The mouth of Pegas [...]s. | 07.3 [...]4 ♒ | 22. 7. N 3 |
| The head of And [...]omeda. | 0 [...].4 [...]0 ♈ | [...]5. 42. N 2 |
| In the top of the Triangle. | 00.366 ♉ | 16. 49. N 4 |
| In the Snout of the Whale. | 02.643 ♉ | 7. 50. S |
| The bright Star in the Whales Tail. | 07.481 ♓ | 20. 47. S 2 |
| Bright Shoulder of Orion. | 06.444 ♊ | 16.06 S 2 |
| Middlemost in the belt of Orion. | 04.972 ♊ | 24. 33. S 2 |
| The last in the tail of the Har [...]. | 0 [...].324 ♊ | 38. 26. S 4 |
| The great Dogs mouth Sirius. | 02.386 [...] | 38. 30. S 1 |
| The lesser Dog Procyon. | 05.641 [...] | 1 [...].57. S 2 |
| In the top of the Ships Stern. | 01.636 ♌ | 43. 18. S 3 |
| Brightest in Hydra's Heart. | 06.044 ♌ | 22. 24. S 1 |
THE CONTENTS OF THE First Part, CONTAINING The Practical Geometry or the Art of Surveying.
- CHapter 1. Of the Definition and Division of Geometry.
- Chap. 2. Of Figures in the General, more particularly of a Circle and the Affections thereof.
- Chap. 3. Of Triangles.
- Chap. 4. Of Quadrangular and Multangular Figures.
- Chap. 5. Solid Bodies.
- [Page] Chap. 6. Of the measuring of Lines both Right and Circular
- Chap. 7. Of the measuring of a Circle.
- Chap. 8. Of the measuring of plain Triangles.
- Chap. 9. Of the measuring of Heights and Distances.
- Chap. 10. Of the taking of Distances.
- Chap. 11. How to take the Plot of a Field at one Station, &c.
- Chap. 12. How to take the Plot of a Wood, Park, or other Champian Plane, &c.
- Chap. 13. The Plot of a Field being taken by an Instrument, how to compute the Content thereof in Acres, Roods, and Perches.
- Chap. 14. How to take the Plot of mountainous and uneven Ground, &c.
- Chap. 15. To reduce Statute measure [Page] into Customary, and the contrary.
- Chap. 16▪ Of the measuring of solid Bodies.
- A Table of Squares. Page. 99
- A Table for the Gauging of Wine Vessels. 114
- A Table for the Gauging of Beer and Ale Vessels. 120
- A Table shewing the third part of the Areas of Circles, in Foot measure and Deoimal parts of a Foot. 132
- A Table shewing the third part of the Area of any Circle in Foot measure, not exceeding 10 f. circumf. 136
- A Table for the speedy finding of the length or Circumference answering to any Arch in Degrees and Decimal parts. 151
- A Common Divisor for the speedy [Page] converting of the Table, shewing the Areas of the Segments of a Circle whose Diameter is 2 &c. 154
- A Table shewing the Ordinates, Arches, and A rea [...] of the Segments of a Circle, whose Diameter is [...] &c. 156
The Contents of the Second Part of this Treatise, of the Doctrine of the PRIMUM MOBILE.
- CHap. 1. Of the General Subject of Astronomy.
- Chap. 2. Of the Distinctions and Affections of Spherical Lines and Arches.
- Chap. 3. Of the kind and parts of Spherical Triangles, and how to project the same upon the Plane of the Meridian.
- Chap. 4. Of the solution of Spherical Triangles.
- Chap. 5. Of such Spherical Problems as are of most general Vse in the Doctrine of the Primum Mobile, &c.
The Contents of the Third Part of this Treatise being an Account of the Civil Year with the reason of the difference between the Julian and Gregorian Calendars, and the manner of Computing the Places of the Sun and Moon.
- CHap. 1. Of the Year Civil and Astronomical.
- Chap. 2. Of the Cycle of the Moon, what it is, how placed in the Calendar, and to what purpose.
- Chap. 3. Of the use of the Golden Number in finding the Feast of Easter.
- Chap. 4. Of the Reformation of the [Page] Calendar by Pope Gregory the Thirteenth, &c.
- Chap. 5. Of the Moons mean Motion and how the Anticipation of the New Moons may be discovered by the Ep [...]ts.
- Chap. 6. To find the Dominical Letter and Feast of Easter according to the Gregorian Account.
- Chap. 7. How to reduce Sexagenary Numbers into Decimals, and the contrary.
- Chap. 8. Of the difference of Meridians.
- Chap. 9. Of the Theory of the Suns or Earths motion.
- Chap▪ 10. Of the finding of the Suns Apogaeon, quantity of Excentricity and middle Motion.
- Chap. 11. Of the quantity of the tropical and sydereal Year.
- Chap. 12. Of the Suns mean Motion otherwise stated.
- [Page] Chap. 13. How to calculate the Suns true place by either of the Tables of [...] middle Motion. I [...]
- Chap. 14. To find the place of the fixed Stars.
- Chap. 15. Of the Theory of the Moon and the finding the place of her Apogaeon, quantity of Excentricity, and middle motion.
- Chap. 16. Of the finding of the place and motion of the Moons Nodes.
- Chap. 17. How to calculate the Moons true place in her Orbs.
- Chap. 18. To compute the true Latitude of the Moon, and to reduce her place from her Orbit to the Ecliptick.
- Chap. 19. To find the mean Conjunctions and Opposition of the Sun and Moon▪
The Fourth Part, or an Introduction to Geography.
- CHap. 1. Of the Nature and Division of Geography.
- Chap. 2. Of the Distinction or Dimension of the Earthly Globe by Zones and Climates.
- Chap. 3. Of Europe.
- Chap. 4. Of Asia.
- Chap. 5. Of Africk.
- Chap. 6. Of America.
- Chap. 7. Of the description of the Terrestrial Globe, by Maps Vniversal and Particular.
A Table of the view of the most notable Epochas.
- The Iulian Calendar. Page. 461
- The Gregorian Calendar. 466
- [Page] A Table to convert Sexagenary Degrees and Minutes into Decimals and the contrary. 476
- A Table converting hours and minutes into degrees and minutes of the AEquator. 480
- A Table of the Longitudes and Latitudes of some of the most eminent Cities and Towns in England and Ireland. 482
- A Table of the Suns mean Longitude and Anomaly in both AEgyptian and Iulian Years, Months, Days, Hours and Minutes. 484
- Tables of the Moons mean motion. 493
- A Catalogue of some of the most notable fixed Stars, according to the observation of Tycho Brahe, rectified to the year 1601. 511
Books Printed for and sold by Thomas Passinger at the Three Bibles on the middle of London-Bridge.
THe Elements of the Mathematical Art, commonly called Algebra, expounded in four Books by Iohn Kersey, in two Vol. fol. A mirror or Looking-glass for Saints and Sinners, shewing the Justice of God on the one, and his Mercy towards the other, set forth in some thousands of Examples by Sam. Clark, in two Vol. fol.
The Mariners Magazine by Capt. Sam. Sturmy, fol.
Military and Maritime Discipline in three Books, by Capt. Tho. Kent, fol.
Dr. Cudworth's universal Systeme.
The Triumphs of Gods Revenge against the Crying and Execrable sin of wilful and premeditated Murther, by Iohn Reynolds, fol.
Royal and Practical Chymistry by Oswaldus Crollius and Iohn Hartman, faithfully rendred into English, fol.
Practical Navigation by Iohn Seller. Quarto.
The History of the Church of Great Britain from the Birth of our Saviour until the Year of our Lord 1667. quarto.
The Ecclesiastical History of France from the first plantation of Christianity there unto this time, quarto.
The book of Architecture by Andrea Palladio, quarto.
[Page] The mirror of Architecture or the ground Rules of the Art of Building, by Vincent Scammozi quarto.
Trigonometry, on the Doctrine of Triangles, by Rich. Norwood, quarto.
Markham's Master-piece Revived, containing all knowledge belonging to the Smith, Farrier, or Horse-Leach, touching the curing of all Diseases in Horses, quarto.
Collins Sector on a Quadrant, quarto.
The famous History of the destruction of Troy, in three books, quarto.
Safeguard of Sailers, quarto.
Norwood's Seamans Companion, quarto.
Geometrical Seaman, quarto.
A plain and familiar Exposition of the Ten Commandments, by Iohn Dod, quarto.
The Mariners new Calendar, quarto.
The Seamans Calendar, quarto.
The Seamans Practice, quarto.
The honour of Chivalry do the famous and delectable History of Don Belianus of Greece, quarto.
The History of Amadis de Gaul, the fifth part, quarto.
The Seamans Dictionary, quarto.
The complete Canonier, quarto.
Seamans Glass, quarto.
Complete Shipwright, quarto.
The History of Valentine and Orson, quarto.
The Complete Modellist, quarto.
The Boat-swains Art, quarto.
Pilots Sea-mirror, quarto.
The famous History of Montelion Knight of the Oracle, quarto.
The History of Palladine of England, quarto.
[Page] The History of Cleocretron and Clori [...]ma, quarto.
The Arralgnment of lower, idle, froward and unconstant Women, quarto.
The pleasant History of Iack of Newb [...]y, quarto
Philips Mathematical Manual, Octavo.
A prospect of Heaven, or a Treatise of the happiness of the Saints in Glory, oct.
Etymologicunt parvum, oct.
Thesaurus Astrologiae, or an Astrological Treasury by Iohn Gadbury, oct.
Gellibrand's Epitome, oct.
The English Academy or a brief Introduction to the seven Liberal Arts, by Iohn Newton, D. D. oct.
The best exercise for Christians in the worst times, by I. H. oct.
A seasonable discourse of the right use and abuse of Reason in matters of Religion, oct.
The Mariners Compass rectified, oct.
Norwood's Epitome, oct.
Chymical Essays by Iohn Beguinus, oct.
A spiritual Antidote against sinful Contagions, by Tho. Doolittle, oct.
Monastieon Fevershamiense; or a description of the Abby of Feversham, oct.
Scarborough's Spaw, oct.
French Schoolmaster, oct.
The Poems of Ben. Iohnson, junior, oct.
A book of Knowledge in three parts, oct.
The Book of Palmestry, oct.
Farnaby's Epigramms, oct.
The Huswifes Companion, and the Husbandmans Guide, oct.
Jovial Garland, oct.
Cocker's Arithmetick, twelves.
The Path Way to Health, twelves.
Hall's Soliloquies, twelves.
[Page] The Complete Servant Maid, or the young Maidens Tutor, twelves.
Newton's Introduction to the Art of Logick, twelves.
Newton's Introduction to the Art of Rhetorick, twelves.
The Anatomy of Popery, or a Catalogue of Popish errors in Doctrine and corruptions in Worship, twelves.
The famous History of the five wise Philosophers, containing the Life of Iehosophat the Hermit. twelves,
The exact Constable with his Original and Power in all cases belonging to his Office, twelves.
The Complete Academy or a Nursery of Complements, twelves.
Heart salve for a wounded Soul, and Eye salve for a blind World, by Tho. Calvert. twelves.
Pilgrims Port, or the weary mans rest in the Grave, twelves.
Christian Devotion or a manual of Prayers, twelves.
The Mariners divine Mate, twelves.
At Cherry Garden Stairs on Rotherhith Wall, are taught these Mathematical Sciences, viz. Arithmetick, Algebra, Geometry, Trigonometry, Surveying, Navigation, Dyalling, Astronomy, Gauging, Gunnery and Fortification: The use of the Globes, and other Mathematical Instruments, the projection of the Sphere on any circle, &c. He maketh and selleth all sorts of Mathematical Instruments in Wood and Brass, for Sea and Land, with Books to shew the use of them: Where you may have all sorts of Maps, Plats, Sea-Charts, in Plain and Mercator, on reasonable Terms.