Speculum nauticum a looking glasse for sea-men : wherein they may behold a small instrument called the plain scale whereby all questions nautical and propositions astronomical are very easily and demonstratively wrought ... / by Iohn Aspley.

Speculum nauticum a looking glasse for sea-men : wherein they may behold a small instrument called the plain scale whereby all questions nautical and propositions astronomical are very easily and demonstratively wrought ... / by Iohn Aspley.

Table of contents

Title page

TO THE WORSHIPFVLL THE MASTER, WARDENS, And Assistants, of the Trinity House in Deptford Strand.

To the Reader.

Speculum Nauticum. OR THE SEA-MANS GLASSE.

THE FIRST BOOK.

CHAP. I.

CHAP. II.

CHAP. III.

CHAP. IV.

CHAP. V.

CHAP. VI.

CHAP. VII.

CHAP. VIII.

CHAP. IX.

CHAP. X.

CHAP. XI.

CHAP. XII.

CHAP. XIII.

CHAP. IV.

CHAP. XV.

CHAP. XVI.

THE SECOND BOOK OF THE PLAIN SCALE, WHERIN is declared the definition of the Sphear, a description of the six great Circles, and also of the foure lesser Circles, and last of all, certaine Questions Astronomicall, performed by the said Scale.

CHAP. I.

CHAP. II.

CHAP. III.

CHAP. IV.

CHAP. V.

CHAP. VI.

CHAP. VII.

CHAP. VIII.

CHAP. IX.

CHAP. X.

CHAP. XI.

CHAP. XII.

CHAP. XIII.

CHAP. XIV.

CHAP. XV.

CHAP. XVI.

CHAP. XVIII

CHAP. XVIII.

A Description of some peculiar things, fit to be considered, by such as intend to practise the Art of Navigation, or Astronomie.

§

The use of the Table of the Tides.

§

§

[Page]

Speculum Nauticum.

A LOOKING GLASSE FOR SEA-MEN: Wherein they may behold a small Instrument called the Plain SCALE▪ whereby all Questions Nauticall, and Propositions Astronomicall are very easily and demonstratively wrought.

Published for the use and benefit of such as will make good use of it.

By Iohn Aspley, Student in Physick, and Practitioner of the Mathematicks, in the Citie of London.

The fourth Edition corrected.

LONDON, Printed by Thomas Harper, and are to be sold by George Hurlock, at his shop at Saint Magnus corner.

1647.

TO THE WORSHIPFVLL THE MASTER, WARDENS, And Assistants, of the Trinity House in Deptford Strand.

[Page]

[Page]

ALbeit (Right Worshipfull) there be more essential natures contained in each part both of this Macrocosme, and Microcosme, existing in their absolute being, than the understanding of man can fully apprehend; yet I doubt not, but those which may with labour and diligence be known, & those also which of ingenious spirits and notable wits have been invented, and by them artificially & methodically taught, (tending not onely to manifest profit in [Page] the Common-wealth, but also to the great increase and setting forth of Gods divine power, wisdome, goodnesse, providence, and increase of Virtue ought of all men to be imbraced, (and especially of those which have any government, publike charge, or authority in the Common wealth.) In regard that the neerer men approach to such excellent Vertues, the neerer (without doubt) do they come unto goodnesse, to felicyty, and to God himselfe. Hence saith the Prophet (unto men seated in eminent places) Dixi vos dii estis, so that those which either through arrogance, or ignorance deride, and contemne those Arts (which with great dexterity, care, and industry have bin found out, and left unto us by the love of our Predecessors) doe both offer contempt unto the goodnesse of God, and do much endamage and anoy all humane society. So on the contrary part, they that do by all means further [Page] those so profitable Disciplines doe both render true honour unto GOD, and do greatly advance the good of the Common-wealth wherin they live. Now in regard there is no study (Divinitie excepted) wherein the wit of man may be better employed then in the motion of the stars, and in the knowledge of their situation, place, and being, together with their wonderfull effects: In regard wherof I was incited to imploy some of my time in the study therof, and at last considering that of the Orator, that Non solum nobis nati sumus, &c. We are not born for our selves onely, but our friends challenge a part in us, and our Countries come in for a share, especially those honours and graces of our Countrey, those that traffique in the deepe, and have their businesse in the great waters, those that are unto this Island as a woodden wall, the Sea-chariots, and the horses of England: these, I say, may claime [Page] justly to the fruits of our labours, whatsoever they be which have not altogether been abhorrent from the Mathematicall studies: considering this, I could do no lesse than bring in this my mite into their treasury, and Si quid Ars mea efficere possit, (if my skill can stand them in any stead) to further that so much deserving Science of Navigation. Accept therefore, I beseech you, this yong sonne of my studies, this little handfull of paper, wherin is contained not Anacreons wanton Odes, or Ovids lascivious Elegies, the incendiaries of lust, but a pure spark of chast Vestalfire, a small part of the Mathematicks dedicated from a serviceable affection to your VVorships, that under the shield of your protections it may live secure, from the desperate stab of of criticall persons, and envious spirits, who not onely like snarling Satyres deride and contemne those so liberall Sciences, but [Page] swallow up with despite (if it were possible) the professours thereof. For rescue from such malignant spirits my Book flies to the shadow of your favour, which if you shall afford unto it, my labours shall be all sacrificed unto you, and I rest.

Your Worships bound servant, IOHN ASPLEY.

To the Reader.

[Page]

HAving ever since I came unto understanding (courteous Reader) practised myselfe in the Mathematicall studies: and haveing attained unto my desires therein, I am willing to impart some of that knowledge which God hath bestowed upon me, unto the open view of the World: for the manifestation wherof, I have freely given unto thee this smal Book, being the first fruit of my labours, containing such Astronomicall and Nauticall Questions as are wrought by the plaine Scale: which if I should finde to receive as free acceptance from thee, expect more of my labours in the same kind, and untill then Irest.

Thy friend in affection, IOHN ASPLEY.

Speculum Nauticum. OR THE SEA-MANS GLASSE.

[Page]

THE FIRST BOOK.

CHAP. I.

BEing intended in this Treatise of the playne Scale, to declare the manner of projection of the Sphear, in plano, I have thought fitting first, to shew unto you some terms of Geometry which are necessary for the unlearned to know (for whose sake chiefly I write this Treatise) before they enter [Page] into the definition of the Spheare. First, therfore I intend to relate unto you, what a point or prick is, and afterward a Line both right and crooked, and such sorts thereof as are appertinent unto the operations and use of this Scale.

Punctum, or a Point, is the beginning of things, or a prick supposed indivisible, void of length, breadth, and depth; as in the figure following is noted by the Point or Prick A.

Linea, or a Line, is a supposed length, or a thing extending it selfe in length, not having breadth nor thicknesse, as is set forth unto you by the Line B AD.

dem: 8.

Parallela, or a Parallell Line, is a Line drawne by the side of another Line, in such sort that they may [Page] be equidistant in all places. And of such Parallels two only belong unto this work of the plain Scale, that is to say, the right lined parallell, and the circular Parallell.

Right lined Parallells, are two right lines equidistant one from another, which being drawn forth infinitely, would never touch or meete one another, as you may see in the figure where the line H. I. is Parallell unto the line C. E. and the line G. F. is Parallell unto them both.

A circular Parallell is a Circle drawn eyther within or without another Circle upon the same Centre, as you may plainly see by the two Circles B. C. D. E. and X. V. Y. W. These Circles are both drawne upon the Centre A. and therefore are Parallell the one unto the other. There is another kinde of Parallell also, which is called a serpentine Parallell, but because it is not belonging unto the use of this Scale, I will omit it and so proceede unto the rest.

Perpendiculum, or Perpendicular is a Line raysed from, or let fall upon another Line, making equall angles on both sides, as you may see declared in the figure, wherin the Line A. C. is perpendicular unto the line B. A. D. making equall angles in the point A.

Diameter Circuli, or the Diameter of a Circle, is a right Line drawne thorow the Centre of any circle, in such sort that it may divide the Circle into two equall parts, as you may see the line B. A. D. is the Diameter of the Circle B. C. D. E. because it passeth thorow the Centre A. and the two ends therof do divide the Circle into two equall parts, in the two extrems B. & D making the semicircle B. C. D. equall unto the semicircle D. E. B.

Semidiameter circuli, or the Semidiameter of a Circle, [Page] is halfe of the Diameter, and is contayned betwixt the Centre, and the one side of the Circle, as the Line A D, in the Semidiameter of the Circle B C D E.

Semi-circulus, or a Semi-circle, is the one halfe of a Circle, drawne upon his Diameter, and is contained upon the Superficies, or Surface of the Diameter, as the Semicircle B C D, which is halfe of the Circle B C D E, and is contayned above the Diameter B A D.

Quadrans circuli, is the fourth part of a Circle, and is contayned betwixt the Semidiameter of the Circle, and a line drawn Perpendicular, unto the Diameter of the same Circle, from the Centre thereof, dividing the Semicircle into two equall parts, of the which parts, the one is the quadrant, or fourth part of the same Circle. As for example, the Diameter of the Circle B C D E, is the line B A D, dividing the Circle into two equall parts: then from the Centre A, raise the Perpendicular A C, dividing the Semicircle likewise, into two equall parts, so as A B C, or A C D, the quasired.

[Page]

CHAP. II.

The manner how to raise a Perpendicular from the midle of a Line given.

DRaw first a ground line whereupon you would have a Perpendicular raised, then open your Compasses unto any distance (so it exceed not the end of your line,) placing one foot of the said Compasses, in the point from whence the Perpendicular is to be raised, and with the other foot make a mark in the line on both sides, then removing your Compasses, unto any other distance that is greater, and set one foot therof, in one of the marks, and with the other foot make a mark over the middle point, then with the same distance of your compasse, set one foot in the other mark upon the Line, and with the other foot make another arch of a Circle over the middle point, so that it may crosse the first arch, and from the meeting of these two arches, draw a right Line unto the middle point, from which the Perpendicular was to be raised, which Line shall be the Perpendicular desired.

Example, suppose your Base or ground Line wherupon a Perpendicular is to be raised by the Line F L K, and from L, the Perpendicular is to be raised, set one foot of your Compasses in the point L, and with the other, make the marks G, and M, on both sides of the point L, then opening your Compasses wider, set one foot in the point M, and with the other draw the arch S, over the point L, then with the same distance of your Compasses, set one foot in G, and with [Page] the other make the arch R. crossing the arch S. in the point T. then from T. draw the Line T. L. which Line is Perpendicular unto the Line F. L. K. from the point L, which is the Perpendicular desired.

CHAP. III.

To let a Perpendicular fall from any Point assigned, unto the middle of a Line.

LEt the line whereupon you would have a Perpendicular let fall, be the Line L F K, and the Point assigned, to be the point T, from whence you would have a Perpendicular let fall upon the Line F L K, first set one foote of your Compasses in T, and open your Compasses unto any distance, so that it be more than the distance T L, which here wee suppose to bee the distance T M, then make in the Line F L K, the marks G, and M, then with your Compasses, take the one halfe of G M, which is in the point L, then from L, draw a Line unto the point T, so the Line T L, shall be the Perpendicular, which was desired to be let fall from the assigned point T, unto the middle of the Line F L K.

[Page]

[figure]

CHAP. IV.

To rayse a Perpendicular upon the end of a Line.

SUppose the Line whereupon you would have a Perpendicular raysed, be the Line F L K, and from the point F, a Perpendicular is to be raysed: first open your Compasses unto any distance, which here we put to be the distance F G, & set one foot of your compasses in the point F, and with the other draw the arch D E G, then set one foote of your Compasses in the point G, and with the other draw the arch E, then placing one point of your Compasses in F, with the other draw the arch D B, then place your Compasses in D, and with the same distance, draw the arch A, cutting the arch D B, in C, then draw a Line from C, unto the end of the Line F L K, unto the assigned point F, so shall the Line G F, be a Perpendicular, raysed from the end of the Line F L K, and from the assigned point F.

[Page]

CHAP. V.

To let a Perpendicular fall from any point assigned unto the end of a Line.

LEt the Line F L K, bee the Base or ground Line, and from the point I, a Perpendicular is to bee let fall upon the end of the Line K, first from the assigned point I, draw a Line unto any part of the Base, which shall be the Line IHM, then finde the middle of the Line IM, which is at H, place therefore one foote of your Compasses in the point H, and extend the other unto I, with which distance draw the arch INK, upon the Centre H, cutting the Base or ground-line in the point K, then draw the Line KI, which Line shall be the Perpendicular desired.

NOW I doubt not but you understand the way to let fall, or to rayse, any manner of Perpendicular Line, eyther from, or upon any part of a Line: therefore now I intend to proceede unto the maine point here aymed at, which is, to declare, and make known unto you the several operations performed by the plain Scale, which though it be in use with very few, yet it is most necessary with Sea-men, because all questions in Navigation are thereby easily and plainly wrought. And also all questions in Astronomy (belonging unto the expert and industrious Sea-men) may both speedily and easily be wrought by the same Scale: in regard whereof I have declared in this little Book, that knowledge (which God hath beene pleased [Page] to bestow upon me) concerning the necessary use and practice therof; hoping that you will as kindly accept it, as it is freely offered unto your courteous considerations.

The figure of the plaine Scale.

CHAP. VI.

Of the description of the Scale.

THis Scale usually is divided into three parts, the first wherof is a Scale of equall Leagues, divided into Degrees, or Leagues from 1 unto 100. and upwards, at your pleasure, and numbred with 10 20 30 40, and so forth unto the end. All these divisions are equall one unto another, and is in use for to measure the leagues that any ship hath run upon any course, or the leagues that she hath raised or depressed the Pole, or departed the Meridian, as in the worke hereafter shall be more fully declared.

The second part of the Scale, is the single Corde [Page] of a Circle, or the Cord of 90, and in dividing into 90, unequall divisions, representing the 90, deg. of the Quadrant: and are numbred with 10, 20, 30, 40, &c. unto 90. This Cord is in use to measure any part, or arch of a Circle not surmounting 90 degrees: The number of these degrees from 1 unto 60, is called the Radius of the Scale, upon which distance, all Circles are to be drawne, whereupon 60 of these degrees are the Semidiameter of any Circle, that is drawn upon that Radius.

The third part of the Scale is divided into 8 parts, representing the 8 points of the Mariners Compasse, contained in one quarter of a Circle, if the Circle bee drawn upon the same Radius, and every one of the aforesaid points, is (for exactnesse sake) subdivided into 8 smaller parts.

I have likewise caused two other lines to be placed upon the back side of the Scale, which I doe call the first and second Lines of Longitudes: the first is divided into 20 unequall parts, or leagues, which 20 Leagues are equall unto the Cord of 90. The use of this first Line of Longitude, is to shew how many Leagues and Miles in any Parallell, doe answere unto one degree of the Equinoctiall.

The second Line of Longitude is divided into 100 proportionable parts, or into 100 unequall Leagues; and every league is subdivided into miles, and halfe miles. The use of this Line is thus, when you have found by the first Line of Longitudes, how many Leagues and Miles doe answere unto a degree of the Equinoctial in any latitude you desire: this second line will shew you how many degrees any number of leagues in that Parallell, will answere unto a degree in the Equinoctiall Circle.

[Page] Thus having shewed you the parts of the Scale, and unto what use they doe generally serve, I will proceede to declare the particular use thereof, in the Art of Navigation, as followeth.

CHAP. VII.

To finde how much any Ship hath raised or depressed the Pole, knowing the course she hath made, and the leagues she hath sayled.

THe Course is Southwest and by South, the Leagues sayled are 100, the difference of Latitude is demanded.

In the first Demonstratiou, draw first the Line A B, and upon the Centre A, raise a Perpendicular A F. Then opening your Compasses unto the Radius of your Scale, and set one Foot therof in the Center A. and with the other draw the Arch K C B, then in regard your course is Southwest & by South, that is three points from the South, take three of the eight points of the Compasse with your Compasses, and place them from K, unto C, then draw the Line A C D, and place the distance of the Leagues you have sayled (which) are 100. upon the Line A C D, from A, unto D. Then from D, draw the Line D F. Parallell unto A B, cutting the Meridian A K F, in the point F, then take the distance of F A, and apply it unto the Scale of equall Leagues, and you shall find it just 83 Leagues, or 4 Deg. 9, Min. which are the degrees you have altered the Latitude, which degrees and minutes (if the Latitude from whence you departed, be South) must bee added unto the Latitude [Page] from whence you departed, and you shall have the Latitude that you are in: contrariwise, substract them (if the Latitude from whence you departed, be North) and you have likewise the Latitude that you are in.

CHAP. VIII.

The distance of Latitude and Leagues sayled being given, to finde the distance Meridionall, and consequently the difference of Longitude.

Sayling from the North Parallell of 56. deg. and 5. min. 100 Leagues betwixt South and West, untill the Pole be depressed 4 deg. 9 min. the difference of Langitude is demanded.

dem. prim.

IN the first demonstration draw the Quadrant A K C B, as is taught in the last Chapter. Then reduce your degrees of Latitude into Leagues, which is done by multiplying of them by 20 the product will be 83 leagues, which leagues being applied unto the Meridian Line A K F, they will end in the point F, Then from F, draw the Line D F, Parallell unto A B, Then open your Compasses unto the distance of 100. leagues of your Se [...]lt of equall parts, [Page] and set one foote of your Compasses in the point A, and with the other draw the arch G H, cutting the Line FD, in D, so shall the distance D F, bee the distance of the Meridian, from the Meridian, from whence you departed, which (being applyed unto the Scale) is 56 leagues. Then in regard you sayled from the North Parallell of 56 deg. and 5 min. untill you had depressed the Pole 4 deg. 5. min. Substract therefore 4 deg. 9. min. from 56 deg. 5 min. and there remayneth 51 deg. 56 min. which is the latitude of the place you are in, and in that Parallell have you departed the first Meridian 56 leagues. Then opening your Compasses unto 51 deg. 56 min. of your Cord, and apply it unto the first Line of Longitudes, and you shall finde that 12 leagues and one mile (in that Parallell) doe alter one degree of Longitude. Then set one foote of your Compasses in the second Line of Longitude, at 12 Leagues, one Mile, and extend the other unto one degree of that Line; then with that distance set one foot of your Compasses in 56 leagues of the aforesaid Line, and the other will extend unto 4 degrees 33 min. which is the distance Meridionall desired.

[Page]

CHAP. IX.

distance of Latitude and distance Meridionall, given to finde the Rhombe.

SAyling from the North Paralell 69 degrees 20 min. untill the Pole be depressed four degrees and 9 min. and the distance Meridionall, or difference of Longitude, six degrees to finde the Rhombe is required. By the first Demonstration, draw the Quadrant A K C B, then turne your four degrees nine minutes into Leagues, it maketh eighty three Leagues; which place upon the line A K F, from A, unto E, then substract the difference of the two Latitudes, from the number of the first Latitude, and it leaves the second Latitude 62 deg. 11 min. Then opening the Compasses unto the middle Latitude, which is sixty four degrees, and fifteen minutes of the Cord, applying it unto the first Line of Longitudes, and you shall finde eight Leagues, two miles, and four seconds to answere unto one degree: then set your Compasses in one degree, in your second Line of Longitudes, and extend the other foote unto eight Leagues, and two miles, and four seconds: then with that distance of the Compasses, place the one foot at six degrees of that line, and turne the other upward, and it will extend unto fifty six leagues, therefore open your Compasses to the distance of fifty six Leagues, in the line of equall leagues, and apply them from the point P, upon the line F D, parallell unto A B, from F, unto D, then from the point D, draw the [Page] line D A. cutting the Quadrant K C B, in C, so shall K C, be the distance of the Rhombe from the South Westward, which is just thirty three degrees, forty five minutes from the South, which is Southwest and by South, the Rhombe desired.

CHAP. X.

By the Latitude of two places, and distance upon the Rhombe to finde the leagues sailed.

The Pole depressed three degrees thirty minutes, the Rhombe the fourth from the Meridian.

IN the second Demonstration draw the line A E, then from A, raise the Perpendicular A C, then opening the Compasses to the distance of the Radius, placing one foot thereof in the Centre A, and with the other draw the Quadrant B D E, Then reduce your three degrees, thirty minutes into leagues, counting for every degree twentie Leagues, and for thirty minutes ten Leagues, so they make seventy Leagues; then open your Compasses unto seventy degrees in the line of equall parts, and place them upon the line A B C, from A, unto C, then from C, draw the line C F, Parallell unto A E. Then in regard your Rhombe was the fourth from the South, take foure of the eight points of the Compasse, and place them upon [Page] the Quadrant from B, unto D, then from A, by the point D, draw the line A D F, cutting the line C F, in F. So shall the distance betwixt A, and F, bee the number of leagues (upon the fourth Rhombe) before you can either raise or depresse the Pole three degrees thirty minutes, which is here found to be ninety nine Leagues.

CHAP. XI.

To finde the distance of any Island from you, that you may discerne by two Stations, knowing the point of the Compasse, the Island beareth unto each of the Stations.

Suppose (being at Sea) you discover an Island bearing South-west of you, which place let it be your first Station, and seventy Leagues sayling South observing the Island to beare West of you, which let be the second Station, the demand is to finde the Island from both the Stations.

IN the second Demonstration let A, be the first Station, and upon the Centre A, draw the quadrant A B D E, then in regard you found the Island to beare South-west from you, therfore take four of your eight points of Compasse, and place them upon your Quadrant from E, unto D, then from the Centre A, [Page] by the point D, draw the Line A D F, representing the visuall Line passing betwixt your sight and the Island, being in the first Station. Then sayling seventy leagues South, which is from A, your first Station, unto C, the second Station: then observing the Island to beare West of you, therefore from the point C, the second Station, draw the Line

dem: 2

C F, parallell unto A E, cutting the Line A D E, in point F, so shall the point F, be the place of the Island desired, and the distance A F, is the distance of the Island from the first Station, which is just ninety nine Leagues off the Line of equall parts. And likewise the distance from C, unto F, is the distance of the Island from the second Station, which is here found to be just seventy Leagues: and by this manner of worke you may finde the distance of any Island from you, which you may discerne either by Sea or Land.

[Page]

CHAP. XII.

Sailing from the South Longitude of 60 degrees, 51 minutes, and from Latitude 25 degrees, 24 min. 99 Leagues, upon a South-west course: the Latitude and Longitude of the second place is demanded.

IN the second Demonstration, draw the Quadrant A B C D E, as is formerly taught: then in regard you sail South-west, take foure points of the Compasse from your Scale, & place them from B, unto D, then by the point D, draw the line A D F, then place your ninety nine Leagues upon the Line A D F, from A, unto F, so shall E, be the place of your Ship. Then from F, draw the Line F C, parallell unto A E, cutting the line A B C, in C, so let the distance C A, be Leagues that you have run South, which is 70 Leagues. or 3 deg. 30 min. which being added to the latitude from whence you departed, makes sixty foure degrees and twenty one minutes for the Latitude of the second place: then take the distance C F, and apply it unto the Line of equall parts, and you shall finde it likewise seventy Leagues: then opening your Compasses unto the middle Latitude 62 degrees, 36 minutes in the Line of Cordes, and apply it unto the first Line of Longitudes, you shall finde that nine leagues and 0 miles, and 38 seconds, doe alter a degree of longitude, then placing one foot of your Compasses in the second line of longitudes, [Page] at 9 leagues and thirty eight seconds, and extend the other to one, then keeping the distance of the Compasses, set one foot in the seventy leagues of the same line, and the other foot will extend unto 7 degrees 37 min. which being substracted from the longitude from whence you departed, leaves seventeen degrees, and forty seven minutes for the Longitude of the second place.

CHAP. XIII.

A Ship sayling from the North Parallell of fifty degrees, having an hundred Leagues to saile South-west, and by West, by the way is inforced by contrary windes, to saile upon severall points of the Compasse, first sayling thirty leagues upon a direct course, then West North-west twenty Leagues, then South sixty Leagues, the question is to finde the Latitude of the second place, how farre it is to the place whereunto you are bound, the distance of the Rhombe that is betwixt them, the distance that you are from your first Meridian, and thereby the difference of Longitude.

IN the third Demonstration, draw the Line A D, and from the point A, raise the perpendicular A B, then open your Compasses unto the Radius of your Scale, and place one foot therof in the centre A, & with the other draw the Quadrant B C D, then take three points of the Compasses, and [Page] and place them upon the Quadrant from D, unto C, then from the Centre A, by the point C, draw the line ACL, 100 Leagues in length, which is the true course you are to saile. Then in regard you sayled 30 leagues direct, take thirty degrees from your Scale of equall parts, and place them upon the line AEC, extending from A, unto E, then in regard you turned your Course, West, Northwest, from the Centre E, draw the Line EG, parallel unto AD, and againe from the Centre E, draw the Line EH, Perpendicular to EG, and parallell to AB, then with the distance of the Radius, set one foot of your compasses in the Centre E, and with the other draw the quadrant G MH, and in regard you sayled West, Northwest, which is 2 points from the West,

dem: 3

Northward, take from your Scale two points of the Compasse, and place them upon the Quadrant GM, H, from G, unto M, then from the Centre E, unto the point M, draw the line EFM, then take twenty leagues with your Compasses from the Scale of equal parts, and place them upon the line EFM, from E, unto F, then is your Ship in the point F. Lastly, in regard [Page] you run South sixty Leagues, from F, draw a Line Parallell unto the Meridian AB, which is the line FI, then take from your Scale of equall parts 60 Leagues, and place them from F, unto I, then is your Ship in the point I, then last of all is to be found how far it is to the place whereunto you were bound, the distance of the Rhombe that is betwixt you, the degrees and minutes you have raised the Pole, the distance of departure from the first Meridian, and thereby the difference of Longitude, and that you may so doe, first draw the Line OIK. Perpendicular unto the line IF, in the point I, and with your Compasses opened unto the distance of the Radius, set one foot of your Compasses in the Centre I, and with the other draw the Quadrant KNF, then in regard your ship is in the point I, and the place whereunto you are bound in the point L, therefore from I, thorow the point L, draw the Line ILN, cutting the Arch KNF, in the point N, therefore let IL, be the Leagues you have unto the place whereunto you are bound, which is forty one Leagues and a halfe, and the Rhombe the distance KN, which is West, and by North, and three degrees unto the Northward, so likewise is the line AO, the number of Leagues you have run due South, which is 68 Leagues and one Mile, or 3 degrees and 25 minutes, which being taken from 50 degrees, the parallell from which you departed, leaves 46 degrees and 35 minutes for the Parallell you ate in. Last of all, shall the line IO, be the Leagues that you have departed your first Meridian, which are 42 leagues and one mile: then open your Compasses unto 48 degrees 17 minutes and 30 seconds of the line of Cords, which is the middle Latitude, and apply it unto the first line of Longitudes, you shall finde that [Page] 13 Leagues 0, miles, fifty six seconds do answer unto a degree of Longitude in that Parallell, then setting one foot of your Compasses in 13 Leagues, and 56 seconds in your second line of Longitudes, extending the other unto one degree, then with the same distance of your Compasses, set one foot in 42 Leagues and one mile of the same line, and the other will shew you 3 degrees and 13 minutes, which is the difference of the Longitude desired.

CHAP. IV.

Two Ships departing from one Parallell, and Port, the one in saying eight Leagues betwixt the North, and the West, hath raised the Pole two degrees, the other in suiling a hundred Leagues betwixt the North, and West, hath raised the Pole foure degrees, I demand by what Rhombes the said Ships have saild, and the Rhombe, and distance that is betwixt them.

IN the fourth Demonstration, draw the Quadrant ABC DE, then in regard the first Shippe hath raised the Pole two degrees, which is forty leagues, take forty leagues of your Scale, and apply them unto the Meridian line AG L, from A, unto G, then from the point G, draw the line GF, parallell unto AB, then opening your compasses unto 80 Leagues set one foot in the Centre A, [Page] with the other, make a marke in the line GF, which will bee at F, so shall F, bee the place of the first shippe: the second shippe hath raised the Pole foure degrees, which is eighty Leagues, therefore place eightie Leagues upon the Meridian line AGL, from A, unto L, and from the point L, draw the line LM, parallell unto GHF, then open your Compasses unto the distance of a hundred Leagues, which are the Leagues the second shippe did runne, and set the foot of your Compasses in the Centre A, and with the other make a marke in the line LM, which will be at M, then draw the line MA, which is the course of the second Ship, and the line FA, is the course of the first shippe, then from F, let a Perpendicular fall, being Perpendicular to the line GF, which is the line FK, then opening your Compasses unto the Radius of your Scale, set one foot in the Centre F, and with the other draw the Quadrant HIK, likewise from F, the place of the first ship, draw a line by the point M, the place of the second ship: cutting the quadrāt KHI, in I, so let IK, be the course that is betwixt them, that is, if you will saile from the first ship unto the second, you must saile North, and by East, and 41 minutes to the Eastward, likewise let FM, be the distance that is betwixt them, which in this Demonstration is 40 Leagues, two miles, so shall BC, be the course of the first shippe from the West Northward, which here is found to be 30 degrees and one minute frō the West Northward, or Northwest, by West & 3 deg. and 44 min. to the Westward. Lastly, the Arch E D, is in the distance of the course that the second ship made from the North Westward, which is found by this Demonstration to be Northwest, and by North, and three degrees five minutes to the Westward.

[Page]

CHAP. XV.

Two Ships departing from one Parallell and Port, in the Parallell of 47 degrees 56 minutes, the first in sayling 80 Leagues betwixt the North and West, hath raised the Pole two degrees, I demand by what course the second ship must run, and how much shee shall alter in her first Meridian or Longitude, to bring her selfe 40 Leagues and two Miles North and by East, and forty one Minutes to the Eastward of the first ship.

IN the fourth Demonstration draw the Quadrant ABCD E, then multiply your two degrees, you have altered your latitude by twenty, and it maketh forty Leagues; which forty Leagues set upon the line AEL, from A, unto G, then from the point G, draw the line GF, parallel unto AB, then open your Compasses unto the distance of 80 Leagues, which are the leagues your first ship did run, and place one foot of your Compasses in the Centre A, and with the other make a marke in the line GF, which will be at the point F, then from the Centre A, unto the point F, draw the line AF, representing the distance of the course of the first ship 80 Leagues: Then from F, let fall a perpendicular FK, and upon the Centre F, upon the Radices of the Scale draw the Arch HIK. Then in regard you must bring the second shippe [Page]

dem: 4

North and by East, and 41 minuts Eastward of the first ship, take 11 degrees, 56 minutes from your Scale of Coards, and place them from K, unto I upon the Quadrant KIH, Then from F, draw the line IF, and upon the line FI, place the distance that you must bring the second ship from the first (which is 40 leagues and two miles) from F, unto M, So is M, the place of your second ship. Then from M, draw the line ML, parallell unto FG, cutting the line AGL, in L, then draw the line MA, cutting the Quadrant BDE, in D, So shall the arch DE, be the course that the second ship must run, to bring her selfe 40 leagues, and two miles North and by East, and 41 minuts East of the first ship. Then to know what you have altered the Latitude, first take the distance LA, and apply it unto the Scale of equall parts, and you shall finde it to be 80 leagues, which is just 4 degrees, which you have altered your latitude, or Poles elevation: which 4 degrees added unto the Latitude you departed from it, makes 51 degrees, 56 minutes for the Latitude that your second Ship is in, then take the distance LM, and apply it to the Scale, it gives 60 leagues; then open your Compasses unto the distance of the middle Latitude, [Page] which is 39 degrees, 51 minuts of the Coard, and apply it unto the first line of Longitudes, and it gives 12 Leagues, and two miles and a halfe, to alter one degree of Longitude in that Parallell: Then set one foot of your Compasses in 12 Leagues, two miles and an halfe in the second line of Longitudes, extending the other foot unto one degree, and with the same distance upon the same Line, set one foot of the Compasses in 60 leagues (the leagues that you are from the Meridian) and the other foot will extend unto foure degrees forty minuts, which is the difference of the Longitude.

CHAP. XVI.

To finde by Demonstration how many miles or minutes of any Parallell, doth answer unto one degree of the Equinoctiall.

LEt the Latitude given be 58 degrees 54 minuts, therefore having drawne the Quadrant ABC, from B, upon the Coard BEC, set the Latitude of the place 58 degrees 54 minuts unto the point D. Then from the point D, draw the line DF, parallell unto BA, So shall the length of the line DF, be the number of miles which answer unto one degree of Longitude in the Parallell of 58 degrees 5 minuts, which being

dem: 5

[Page] applyed unto your Scale of equall parts, gives thirty one miles. So likewise the Arke BE, being the Latitude of 52 degrees, ending in the point E, give the lines EG. For the miles that answere unto one degree of Longitude in the parallell of 52 degrees were found by the Scale to be 36 miles, and 56 seconds, or 56/60 of a mile.

[Page]

[Page]

THE SECOND BOOK OF THE PLAIN SCALE, WHERIN is declared the definition of the Sphear, a description of the six great Circles, and also of the foure lesser Circles, and last of all, certaine Questions Astronomicall, performed by the said Scale.

CHAP. I.

De Sphaera.

The figure of the plaine Scale.

A Spheare according to the description of Theodosius, is a certain solid Superficies, in whose middle is a Point, from which all lines drawne unto the Circumference are equall; which Point is called the Centre of the Sphear, by which Centre a right Line being drawn, and extending himselfe on either [Page] side unto that part of the Circumference wherupon the Sphear is turned, is called Axis Sphaera, or the Axletree of the World.

A Sphear accidentally is divided into two parts, that is to say, in Sphaeram rectam, & Spharam obliquam.

Sphera recta, or a right Sphear, is onely unto those that dwell under the Equinoctiall, Quibus neuter Polorum magis altero elevatur: that is, to whom neither of the Poles of the World are seen, that lie hid in the Horizon.

Sphaera obliqua, or an oblique Sphear, is unto those that inhabit on either side of the Equinoctiall, unto whom one of the Poles is ever seen, and the other hid under the Horizon.

The Circles wherupon the Sphear is composed are divided into two sorts: that is to say, in Circulos majores & minores.

Circuli majores, or the greater Circles, are those that divide the Sphear into two equall parts and they are in number six: viz. the Equinoctiall, the midle of the Zodiacke, or the Ecliptique Line, the two Colures, the Meridian and the Horizon.

Minores vero Circuli, or the lesser Circles, are such as divide the Sphear into two parts, unequally: and they are foure in number; as the Tropick of Cancer, the Tropick of Capricorn: the Circle Artike, and the Circle Antartike.

[Page]

CHAP. II.

Of the four greater Circles.

THe Equinoctiall is a Circle that crosseth the Poles of the World at right angles, and divideth the Sphear into two equall parts, and is called the Equinoctiall, because when the Sunne commeth unto it, which is twice in the yeer, viz. In principio Arietus & Libra, that is, March and September) the dayes and nights are equall throughout the whole World, whereupon it is called, Equator dici & noctis, the equall proportioner of the day and night artificiall: and in the figure is described by the Line CAE.

The Meridian is a great Circle, passing thorow the Poles of the World, and the Poles of the Horizon, or Zenith point over our heads, and is so called, because that in any time of the yeare, or in any place of the World, when the Sunne (by the motion of the Heavens) cometh unto that Circle, it is noone or 12 of the Clock. And it is to be understood, that all townes and places that lie East and West one of another, have every one a severall Meridian: but all places that lie North and South one of another, have one and the same Meridian. This Circle is declared in the figure following by the Circle BCDE.

The Horizon is a Circle, dividing the superiour Hemisphere from the inferiour, wherupon it is called Horizon. that is to say, the bonds of sight, or the farthest distance that the eye can see, and therefore it is also called Circulus Hemispheri. The Horizons are divided into two sorts, viz. Rectus & obliquus, a right [Page] and an obliqne, or a declining Horizon: whereof those have a right Horizon which have the Equinoctiall for their Zenith, and the Poles of the World in their Horizon: Because the Horizon (hiding both the Poles of the World) is a Circle supposed to be drawne by the Poles of the World, dividing the Equinoctiall at right angles, as in the figure following you may plainly see. First, imagining the Circle XVYW, to be the earth, and those that inhabit at the Point V, have the line BD, for their Horizon, cutting the Equinoctiall CAE, at right angles in A, and therefore is called Horizon rectus & Sphaerarecta, a right Horizon, and a right Spheare. Those have an oblique Spheare, or an oblique Horizon to whom one of the Poles are visuall, or elevated above the Horizon, and have the other hid under the Horizon, and in regard such a Horizon doth crosse the Equinoctiall at oblique angles, it is called Horizon obliquns, or a declining Horizon, as for example: Those that inhabit at the point S, have T, for their Zenith, and KAL, for their Horizon, dividing the Equinoctiall CAE, at oblique angles, making the angle contained betwixt the Horizon AK, and the Equinoctiall A C, an angle of thirty eight degrees, and 28 minutes, and the angle contained betwixt the Horizon AL, and the Pole AD, an angle of 51 deg. 32 minutes, which is the elevation of the Pole for those that inhabit at S, those and all other have an oblique Spheare, except they inhabit just under the Equinoctiall Circle, unto whom onely doth a right Sphear belong.

Colurus Solstitiorum, or the Summer Colure is a Circle passing by the Poles of the World, and by the Poles of the Ecliptique, and by the head of Cancer [Page] and Capricorn, whereupon the first scruple of Cancer where the Colure crosseth the Ecliptique Line, is called Punctus Solstitiae aestivalis, or the point of the Summer Solstice: to which place when the Sun commeth: he can approch no neerer unto our Zenith, but returneth unto the Equator again. Arcus vero Coluri, The Arke of the Colure contained betwixt the Summer Solstice and the Equator, is called the greatest declination of the Sun, which Ptolemie found to be 23 degrees, 31 minutes: but by the observation of Copernicus it was found to vary, for he found the declination sometimes to be 23 degrees 52 minutes, and in the processe of time, to be but 23 degrees, 28 minutes. And in these our daies (by the observation of Ticho de Brake, and that late famous Mathematician, Master Edward Right) it is found distant from the Equinoctiall 23 degrees 31 minutes 30 seconds.

The other Colure passeth by the Poles of the World, and by the first point of Aries and Libra, whereupon it is called, Colurus distinguens Equinoxia. These two Colures doe crosse each other at right angles in the Poles of the world, whereupon these Verses were made.

Haec duo Solstitia faciunt Cancer, Capricornus,

Sed noctes aequant Aries & Libra diebus.

The Zodiack is another of the greater Circles, dividing the Equinoctiall into two equall parts, by the head of Aries and Libra, the one halfe of it doth decline unto the North, the other into the South, the greatest of which declination is 23 degrees, 31 minutes, and 30 seconds. Note also, this Circle is divided into 12 equall parts, which parts are attributed [Page]

dem: 6

unto the 12 Signes, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagitarius, Capricornus, Aquarius, & Pisces. And everyone of these Signes are divided into 30 equall parts, which are called degrees, so the whole Zodiake containeth 160 degrees. Likewise every degree is divided into 60 equall parts, which parts are called minutes, and are in number 21600 minutes, and as 21600 minutes is the whole circumference of the Heavens, so is 21600 miles the whole circuit of the Earth.

[Page]

CHAP. III.

Of the foure lesser Circles.

THe Sun having ascended unto his highest Solstitiall Point doth describe a Circle, which is the neerest that he can approach unto the North Pole, wherupon it is called, Circulus Solstitii aestivalis, the Circle of the Summer Solstice, or the Tropick of Cancer, and is noted in the figure before, by the line H. Y. I.

The Sun also approching unto the first scruple of Capricornus, or the Winter Solstice, describeth another Circle, which is the utmost bounds that the Sun can depart from the Equinoctiall Line towards the Antartike Pole, whereupon it is called, Circulus Solstitii hyemalis, sive Tropicus hyemalis, vel Caepricorni: the Circle of the Winter Solstice, the Winter Tropick, or the Tropick of Caepricorn, and is described in the figure by the Line G. X. F.

So much as the Ecliptick declineth from the Equinoctiall, so much doth the Poles of the Ecliptick decline frō the Poles of the World, wherupon the Pole of the Ecliptick, which is by the North Pole of the World, describeth a certain circle as it passeth about the Pole of the World, being just so farre from the Pole as the Tropick of Cancer, is from the Equator, and it is the third of the lesser Circles, and is called, Circulus Arcticus, or the Circle of the North Pole, and is described in the Diagram, in the second Chapter by the Line P. O.

The fourth and last of the lesser Circles, is described in like manner, by the other Pole of the Ecliptick, about the South Pole of the World, and therefore [Page] called Circulus Antarcticus, the Antartick Circle, or the Circle of the Antartick or South Pole, and is demonstrated in the former figure, by the line N. M.

CHAP. IV.

Certain questions Astronomicall performed by the plaine SCALE.

The true place of the Sun, given to finde his declination.

THe Sun being in the head of Taurus, his declination is desired: by the seventh Demonstration, draw the line A. D. then upon the Centre A. raise the Perpendicular A. B. thē opening your Compasses to the Radius of your Scale, and place one foot in the Centre A. and with the other draw the Quadrant B. C. D. then opening your Compasses unto the greatest declination of the Sun, and place it upon the Quadrant from D. unto K. then from the point K. draw the Line K. H. cutting the line A. B. in H. then with the distance A. H. draw the small Quadrant G. E. H. and in regard the Sun is in the head of Taurus, which is thirty degrees from the beginning of Aries, let A. D. be the Equator,

dem: sep:

[Page] and D the beginning of Aries, D. C. 30 degrees, or longitude of the Sunne, then from the point C. draw the line C. A. cutting the Quadrant G. E. H. in E. then from E. draw the line E. L. parallell to A. D. cutting the Quadrant B. C. D. in L. so shall the arch L. D. be the declination of the Sun desired, which in this demonstration is found to be 11 deg. and 31 minutes.

CHAP. V.

The declination of the Sun, and quarter of the Eclipse that he possesseth being given, it is desired to finde his true place.

The declination is 10 degrees 31 minutes, the first quarter that he possesseth, is betwixt the head of Aries and Cancer.

FIrst by the seventh Demonstration draw the Quadrant A. B. C. D. as is taught in the former Chapter, then set the greatest declination of the Sun upon the Cord from D. unto K. which is 23 deg. and 31 minutes, then from K. draw the line K. H. parallell unto the Equator D. A. cutting the line B. A. in the point H. So shall H. A. be the signe of the Suns greatest declination, then with the distance A. H. draw the Circle, G. E. H. then from D. upon the Cord D. B. C. set the declination of the Sun, which is 11 degrees, 31 min. from D. unto L. then draw the line L. E. parallell unto A. D. cutting the Quadrant G. E. H. in E. Then from the Centre A. by the point E. draw the line A. E. C. cutting the Quadrant B. C. D. in C. So shall the arke [Page] C. D. be the distances of the Sunne from the head of Aries, which is here found to be just 30 deg. which is in the beginning of Taurus.

CHAP. VI.

By the elevation of the Pole, and declination of the Sun, to finde the amplitude of the Sun, or his true rising, or setling from the East or West point.

BY the eight Demonstration, first draw the line B. D. then upon the Center A. draw the Circle B. C. D. E. then from A. raise the perpendicular C. A. E. then is your Circle divided into foure equall parts: then suppose the elevation of the Pole to be 51 deg. 32 min. which must be placed upon the Circle, from D. unto F. then from the point F. by the Center A. draw the line F. A. G. representing the Pole of the World F. being the North Pole, and G. the South Pole, then substract 51 deg. 32 min. from 90. deg. and the remainder is the height of the Equinoctiall, which, is 38 deg. 28 min. which must be placed upon the Circle from the Horizon B. unto the point I: [...] then from I. by the Centre A. draw the line I. A. H. representing the Equinoctiall Circle. Then from I. unto M. set the declination of the Sun, being here supposed 14 deg. 52. min. North, then from the point M. draw the line, or Parallell of declination M. I. N. parallell unto the Equator I. A. H. cutting the Horizon B. D. in T. then from T. raise the perpendicular T. V. cutting the Circle B. C. D. E. in V, so shall the distance C. V. be the true amplitude of the Sunne desired, which here is found to be 24 deg. 21 min. North.

[Page]

CHAP. VII.

By the Amplitude of the Sun to finde the Variation of the Compasse.

HAving found the amplitude of the Sun by the last Chapter, first observe with a Compasse, or rather with a Semicircle upon what degree and minute the Sun riseth or setteth, beginning to reckon from the East or West, & ending at the North or South at 90 degrees: and when you have diligently observed the Magneticall rising or setting, by the Semicircle, or by some other like fitting Instrument: and also the true amplitude found, as is declared in the last Chapter, the difference of these two amplitudes, is the variation of the Compasse: But when the Sunne riseth upon the same Degree of the Compasse, as is found by the Scale, the variation is nothing, but the Needle pointeth directly unto the Poles of the World, which by Master Mulinux was affirmed to be at the Westernmost part of Saint Michaels, one of the Islands of the Azores, from whence hee will have the Longitude reckoned. Secondly when the Sunne is in the Equinoctiall Circle, where hee hath no amplitude, looke what distance the Compasse maketh the Sun to rise from the East or West of the Compasse, the same distance is the Compasses variation, from the North or South. Thirdly, if the Sunne rise more to the South of the Compasse, or setteth more to the North by the compasse, then is shewed by the Scale, the difference betwixt the amplitude given by the Scale, and the [Page] amplitude given by the Needle, is the Variation of the compasse from the North Westward. Fourthly, if the Compasse sheweth the Sun to rise more Northward, of set more Southward, then is shewed by the Scale, the difference is the variation of the Compasse, from the North Eastward. Fiftly, if the Scale shew the amplitude of the Sunne rising Southerly, and the Compasse shew it to be Northerly, adde both the Amplitudes together, and they shew you the variation Westerly.

CHAP. VIII.

The place of the Sun being given, to finde his declination by a whole Circle.

ACcording unto the right Demonstration, first draw the Circle B. C. D. E. then draw the Horizon B. A. D. and then the Equinoctiall I. A. H. as is before taught: and then the Tropick of Cancer K. L. 23 degrees and a halfe from the Equinoctiall: then draw the Tropick of Capricorn P. O. of like distance from the Equinoctiall, and after from K. to O. draw the Ecliptique Line K. A O. And when you have thus laid downe the Spheare, suppose the Sun to be in the tenth degree of Taurus, at which time his declination is desired. And in regard the Sunne is more neere unto the Tropicall point Cancer, then unto Capricorn; first finde how [Page] many degrees he is from the Tropick of Cancer, and you shall finde him to be 50 degrees; therefore take with your Compasses 50 degrees from the Coard, and apply it from the Tropicall point Cancer at K. unto V. upon one side, and upon P. on the other side: then draw the Line V. P. cutting the Ecliptick K. O. in the point R. then from R. draw the Line M. R. N. parallell unto the Equinoctiall I. A. H. and cutting the Quadrant B. C. in the point M. So shall the Arke M. I. be the declination of the Sun desired, which being applied unto your Scale, gives you 14 degrees and 52 minutes.

CHAP. IX.

The elevation of the Pole, and declination of the Sun, given to finde his height in the verticall Circle.

The Pole is elevated 51 deg. 32 minutes, the declination of the Sun is 14 degrees 52 minutes North, his height in the verticall Circle is found as followeth.

FIrst according unto the former Chapter draw the Circle B. C. D. E. then the Horizon B. A. D. and after the Verticall Line C. A. E. then the Pole of the World F. G. and likewise the Equator I. A. H. this being don, place the declination of the Sun 14 degrees, 52 minutes, upon the Circle from I. unto M. and also from H. unto N. then draw the Line M. N. cutting the line [Page] C. A. E. in S. then from S. draw the Line S. W. Parallell unto the Horizon B. A. D. cutting the Meridian Circle B. C. D. E. in W. So shall the distance D. W. be the height of the Sun in the verticall Circle, for the time demanded which by this proposition is found to be 19 degrees and 8 minutes.

CHAP. X.

The elevation of the Pole, and the Amplitude of the Sun, being given, to finde the declination.

The elevation of the Pole is 51 deg. 32 min. the Suns Amplitude is 24 deg. 21 minutes, the declination is found as followeth by the eight Demonstration.

dem: 8

[Page] FIrst upon the Center A. draw the Circle B. C. D. E. then draw the Line B. A. D. representing the Horizon, dividing the Circle into two equall parts: then draw the Line C A E, perpendicular to B A D. representing the East & West point of the Compasse, then placing the elevation of the Pole 51 degrees, and 32 minutes, from D unto F, from F, by the Centre A, draw the Line F A G, which let be the Pole or Axeltree of the World; then from B unto I, and from D unto H, set the complement of the Poles elevation: which shall represent the Equinoctiall, in regard it maketh right angles with the Pole of the World, in the Centre A. Then from C unto V, place the amplitude of the Sun, which is 24 degrees, and 21 minutes: then from V. let fall the perpendicular V T, cutting the Horizon B A D, in the point T: then from the point T, draw the Line M T N, parallell unto the Equinoctiall I A H: and cutting the Circle B C D E, in the point M and N, so let the distance of I and M, or H and N, be the declination of the Sun, which was desired: which being applied unto your Scale, gives you 14 degrees, and 52 minutes.

[Page]

CHAP. XI.

The elevation of the Pole, the declination of the Suum, and houre of the day being given to findethe Almicanter is desired.

The elevation of the Pole is 30 degrees, the declination of the Sun is 20 degrees North, the houre is nine in the morning, at which time the Almicanter is found, as followeth.

BY the ninth Demonstration, first upon the centre A, draw the circle BCDE, then draw the Line B. D. for the Horizon, then place your Poles elevation which is 30 degrees upon the circle from D, unto R, then from R by the Centre A, draw the Line RAS, representing the Pole of the World, then from B unto F, place the Complement of the Poles elevation, which is 60. degrees, and from the point F, by the Centre A, draw the line FAH, representing the Equinoctiall Line, and then set the declination of the Sun from F, unto L, and from L, draw the Line LPO, parallell unto the Equator FAH, cutting the Pole of the world in the point P, then set one foot of your Compasses in the point P, and extend the other either unto L, or unto O, and with the same distance of your Compasses, upon the centre P, draw the circle LNOQ, which is called the houre circle, so shall L, be the point of 12 a clock at noone, N, the place of six a clock afternoone, O the place of 12 a clock at midnight, and Q, the place of six a clock in the morning: Every one of the foure quarters, must [Page] be divided into six equall parts, or houres, making the whole circle to containe 24 parts, representing the 24 houres of the day and night, then in regard the houre of the day was 9 of the clock, which is 3 hours forenoone, take three of those 24 houres, and place them upon the circle LNOQ, from the Meridian point L, unto K, the nine a clock point in the morning, and unto M, the point of three a clock after noon, then draw the line MK, cutting the parallell of the Sun LO, in the point I, then from I, draw the line IG, parallell unto the Horizon BAD, which shall cut the Meridian circle BCDE, in the point G, so let the distance of G and B, be the Almicanter of the Sun which was desired, which in this Demonstration is found to be 48 degrees and 18 minutes.

dem: 9

[Page]

CHAP. XII.

The elevation of the Pole, the Almicanter, and declination of the Sun, being given to finde the houre of the day.

The elevation of the Pole is 30 degrees, the declination of the Sun is 20 degrees, the Almicanter of the Sun is 48. degrees, and 18 minutes, the houre of the day is found as followeth, by the ninth Demonstration.

FIrst, upon the centre A. draw the circle B. C. D. E. then draw the Diameter B. D. representing the Horizon, then from D. unto R. set 30 degrees, the elevation of the Pole, then from R. by the Point A. draw the line R. A. S. representing the Pole of the World, then draw the line F. A. H. crossing the Pole in A. at right Angles, cutting the Meredian Line in F. then from F. set 20 degrees, the declination of the Sun unto L. and then from the point L, draw the Line LPO, representing the parallell of the Sunne, and cutting the Pole of the World in P, then placing one foot of your Compasses in P, extend the other unto L, with which distance of your Compasses, draw the houre circle I, NOQ, then from the Horizon at B, place the Suns Almicanter: (which is 48 degrees, and 18 minutes) upon the Quadrant BGL, from B, unto G, then from the point G, draw the line GI, parallell unto the Horizon BAD, cutting the line LO, in I, then from the point I, draw the line KIM, parallell to the Pole of the World QAN, cutting the circle LNO, in M, then let IN, be divided into 6 houres, whereof LM, are three: whereupon I conclude, that it is three houres from noon, that is, at nine a clock in the morning, or three in the afternoone.

[Page]

CHAP. XIII.

The Almicanter, or height of the Sun being given, to finde the length of the right shadow.

ACcording unto the tenth Diagram, draw the Line AF, and upon the centre A, raise the perpendicular AC; then upon the centre A, draw the Quadrant CDF. then suppose the height of your Gnomon, or substance yielding shadow be the Line AB, which is to be divided into 12 equall parts, which Gnomon I have heere made just 12 degrees of the equall League of the Scale: then from B to the top of the Gnomon, draw the Line BE, parallell unto AF, then set the Almicanter which is 45 degrees from F unto D, and from the point D, draw the Line DA, cutting the Line BE, in the point G, so shall BG be the length of the right shadow desired, which is heere found to be 14 degrees and 18 minutes, which is but just the length of your Gnomon, and 2/12 and ⅓ of a twelfth over: Note that the right shadow, is the shadow of any Post, Staffe, or Steeple, that standeth at right angles with the Horizon, the one end thereof respecting the Zenith of the place, and the other the Nadir.

[Page]

CHAP. XIV.

The Almicanter, or height of the Sun being given to finde the length of the contrary shadow.

BY the verse or contrary shadow, is understood the length of any shadow, that is made by a Staffe or Gnomon standing against any perpendicular wall, in such a manner that it may lie parallell unto the Horizon, the length of the contrary shadow, doth increase as the Sunne riseth in height: whereas contrariwise

dem: 10:

the right shadow doth decrease in length, as the Sun doth increase in height: the way to find the verse shadow is as followeth. First, draw your Quadrant as is taught in the last Chapter wherin let A, B, be the length of the Gnomon, likewise from B, draw the line B E, parallell unto A F, as before, then set your Almicanter from C, upon the Quadrant which is given to be 70 degrees, and it will extend from C, unto H, then from the point H, draw the line H A, cutting the line B E, in the point K, so shall K B, be the length of the contrary shadow, which here is found to be 34 degrees and 8 min. or twice so long as your Gnomon, and 10/12 and about ½ part of a twelfth more.

[Page]

CHAP. XV.

The latitude of the place, the Almicanter, and declination of the Sun being given, to finde the Azimuth.

The latitude of the place is 51 degrees, 30 minutes, the declination of the Sun 20 degreès North, the Almicanter 38 degrees 30 minutes, the true Azimuth of the Sun is desired.

FIrst upon the Centre Adraw the Circle B C D E, then draw the Diameter B A D, and from D, unto F, set the elevation of the Pole which is 51 degrees, and 30 minutes whose complement is 38 degrees and 30 minutes, which must be placed from B, unto H, then from H, draw the line H A I, representing the Equinoctiall Line, and from F, draw the line F A G, representing the Pole of the World, then from H, unto P, and from I, unto Q, set the declination of the Sun, which is 20 degrees, and by those two points draw the line P Q, for the Parallell of the Suns declination; then upon the Circle from B, unto H, set the Suns Almicanter, 38 degrees, and 30 minutes, then from H, draw the line H R, parallell unto the Horizon, cutting the Suns Parallell P O, Q, in O, then draw the line T V A E, Perpendicular unto the line B A D, in the Centre A, and cutting the line H V R, in V, then set one foot of your [Page] Compasses in the point V, extend the other unto R, and with the same distance draw the Semicircle H,

dem: 11:

L R, then draw the Concentrick Circle upon the Radius of the Scale M T N, and where the line P O Q, and the line M O N, doe meet in the point V, raise the Perpendicular O L, cutting the Semicircle H L R, in L, then lay the Scale from the Centre N, to the point L, and draw the Line L K, cutting the Semicircle M T N, in K, so shall K T, be the true distance of the Sun, from the East, or West point Southward, or [Page] the Suns true Azimuth, which is here found to be 72 degrees, and 10 minutes from the South part of the Meridian.

CHAP. XVI.

The place of the Sun being given to finde the right ascention is desired.

Suppose the Sun be in the 20 degree of Taurus, his right ascention is found as followeth.

FIrst draw the line B A F, for the Pole of the world, then upon the Centre A, draw the Circle B C D E, then from the Centre A, raise the Perpendicular C A E, for the Equator: then place your greatest declination frō C, unto Q, and from E, unto P, then draw the line Q A P, which doth represent the Ecliptick line, then in regard the Sunne is in the 20 degree of Taurus, which is 40 degrees, from the head of Cancer, which 40 degrees place from Q, unto L, and unto K, then draw the line K L, cutting the Ecliptick in I, then from the point I, draw the line H I, parallell unto C A E, cutting the Pole of the World in O, then set one foot of your Compasses in O, and extend the other unto G, with which distance draw the Semicircle H D G, then opening your Compasses unto the Radius of the Scale, and upon the Centre O, likewise draw the Circle H N F G, then draw the [Page] line I M, parallell unto A O D, cutting the Semicircle H M D G, in M, then lay your Scale from the

dem: 12:

Centre O, unto the point M, and draw the Line N M, cutting the Concentrick Circle in N, so shall the distance N F, be the right ascention, which is here found to be 42 degrees, 27 minutes.

[Page]

CHAP. XVIII

The elevation of the Pole, and declination of the Sun, given to finde the difference of the ascentions.

The Poles elevation is 51 degrees 32 minutes, the declination of the Sun is 21 degrees.

dem: 13:

FIrst draw the Line B A K, representing the Horizon, then upon the Centre A, draw the Circle B C D E F, Then from K, unto D, set the elevation of the Pole which is 51 degrees, and 32 minutes: then from the point D, by the Centre A, draw the line D A F, representing the Pole of the world: [Page] then from R, unto C, set the complement of the Poles elevation which is 38 degrees, and 26 minutes, then from C, by the Centre A, draw the line C A E, representing the Equinoctiall Line; then from C, unto G, and likewise from E, unto H, for the declination of the Sun, which is 21 degrees, then from G, unto H, draw the parallell of the Suns declination, cutting the Pole of the World in L, and the Horizon in I, then set one foot of your compasses in the point L, and extend the other unto G, then with that distance of your Compasses draw the Semicircle G M N H, then opening your Compasses unto the radius of your Scale, and upon the same centre draw a Concentrick circle, G X O H, then from I, where the declnation of the Sun doth cut the Horizon, draw the line I N, parallell unto the Pole of the World A M, cutting the circle G M H, in N, then lay your Ruler from the point I, unto the point N, and so draw the line N O, cutting the Concentrick circle G X O H, in O, so shall the distance of O, and X, be the difference of the ascentions, which is here found to be 28 degrees, and 54 minutes.

[Page]

CHAP. XVIII.

The right ascention of the Sun or Starre being given, together with the difference of their ascentions, to finde the oblique ascention.

THe right ascention of any point of the Heavens being knowne, the difference of the ascention is either to be added thereunto or else to be substracted from it, according as the Starre is situate in the Northerne or Southerne Signes: As for example, if the Sun be in any of these six Signes, Aries, Taurus, Gemini, Cancer, Leo, or Virgo, then the difference of the ascentions is to be substracted from the right ascention, and the remainder is the oblique ascention. Suppose therefore the Sunne to be in the fourth degree of Gemini, where the right ascention is found to be foure houres, and 8 minutes, or 62 degrees, and the difference of ascention where the Pole is elevated 51 degrees, is found to be one houre 53 minutes otherwise 28 degrees 53 minutes, which being taken from the right ascention, leaves two houres, and 16 minutes, or 33 degrees, and 42 minutes, which is the oblique ascention of the Sun in the fourth degree of Gemini. But if the Sun be upon the South side of the Equinoctiall, either in Libra, Scorpio, Sagitarius, Capricornus, Aquarius, or Pisces, then the difference of the ascentions is to be added unto the right ascention, and the Product will be the oblique ascention. Suppose the fourth degree of Sagitarius, is given, for which signe and degree the oblique ascention of the Sun is desired, [Page] his right ascention being then found to be 242 degrees, or 16 houres, 8, minutes, the difference of the ascention is one houre, 53 minutes, or 28 degrees, 18 minutes, which being added unto the right ascention, makes 18 houres, and one minute; or in degrees 270 degrees, and 18 minutes: which is the oblique ascention of the Sun, when he is in the fourth degree of Sagitarius. And if you would finde the oblique descention, you must adde the difference of the ascentions unto the right ascention, when the Sun is in these six Signes, Aries, Taurus, Gemini, Cancer, Leo, Virgo: and contrariwise, when the Sunne is in the other six Signes, you must substract the difference from the right ascention, and you shal have the oblique descention of the Sun, or any Starre, whose right ascention and difference of ascentions is knowne. But it is to be understood, that this manner of operations doth serve no longer then you are upon the North side of the Equinoctiall. For if the South Pole be elevated, the worke is contrary: for so long as the Sun is in any of the Northerne Signes, the difference of the ascentions is to be added unto the right ascention, to finde the oblique ascention. And contrariwise, substracted to finde the oblique descention. Likewise if the Sun or Starre be in the Southerne Signes, then is the difference of ascentions, substracted from the right ascention, to finde the oblique ascention, and added, to finde the oblique descention.

[Page]

A Description of some peculiar things, fit to be considered, by such as intend to practise the Art of Navigation, or Astronomie.

THE Zenith is an imaginary point in the Heavens over our heads, making right Angles with the Horizon, as the Equinoctiall maketh with the Pole.

The Nader is a prick in the Heavens under our feet, making right Angles with the Horizon under the earth, as the Zenith doth above; and therefore is opposite unto the Zenith.

The declination of the Sun, is the Arke of a Circle contained betwixt the Ecliptick and the Equinoctial, making right Angles with the Equinoctiall. But the declination of a Starre, is the Arke of a Circle let fall from the Centre of a Starre, perpendicularly unto the Equinoctiall.

The Latitude is the Arke of a Circle contained bewixt the Centre of any Star, and the Ecliptick Line [Page] making right Angles with the Ecliptick, and counted either Northward, or Southward, according to the situation of the Sea, whether it be neerer unto the North or South Pole of the Ecliptick.

The Latitude of a Town or Country is the height of the Pole above the Horizon, or the distance betwixt the Zenith and the Equinoctiall.

The Longitude of a Star, is that part of the Ecliptick, which is contained betwixt the Starres place in the Ecliptick, and the beginning of Aries counting them from Aries according to the succession or order of the Signes.

The Longitude of a Towne or Countrey are the number of degrees, which are contained in the Equinoctiall, betwixt the Meridian that passeth over the Isles of Azores, (from whence the beginning of Longitude is accounted) Eastwards, and the Meridian that passeth over the Town or Country desired.

The Altitude of the Sun or Star, is the Arch of a circle contained betwixt the Centre of the Sun, or any Starre, and the Horizon.

The Amplitude is that part of the Horizon which is betwixt the two East or west points, and the point of the Compasse that the Sun or any Star doth rise or set upon.

Azimuthes are Circles, which meet together in the Zenith, and crosse the Horizon at right Angles, and serve to finde the point of the Compasse, which the Sun is upon at any houre of the day, or the Azimuth of the Sun or Star, is a part of the Horizon contained betwixt the true East or West point, and that Azimuth which passeth by the Centre of the same Starre to the Horizon.

The right asention of a Starre is that part of the [Page] Equinoctiall that riseth or setteth with the Star, in a right Sphear, or in an oblique Sphear, it is that portion of the Equinoctiall, contained betwixt the beginning of Aries, and that place of the Equinoctiall, which passeth by the Meridian with the centre of the Starre.

The oblique ascention is a part of the Equinoctiall, contained betwixt the beginning of Aries, and that part of the Equinoctiall that riseth with the centre of a Starre, in an oblique Sphear.

The difference ascentionall, is the difference betwixt the right and oblique ascention: or it is the number of degrees contained betwixt that place of the Equinoctiall that riseth with a Centre of a starre, and that place of the Equinoctiall that commeth unto the Meridian, with the Centre of the same star.

Almicanterahs, are Circles drawne parallell unto the Horizon, one over another untill you come unto the Zenith: these are Circles that do measure the elevation of the Pole, or height of the Sun, Moon, or Stars above the Horizon, which is called the Almicanter of the Sun, Moon, or Star: the Arke of the Sun or Stars Almicanter, is a portion of an Azimuth contained betwixt that Almicanter which passeth thorow the centre of the star, and the Horizon.

§

[Page]

A generall Table for the Tides in all places.

The Moons age. | Hours and Minutes to be added. | The Moons age. | Hours and Minutes to be added.
Dayes. | Degrees. | Minutes | Dayes | Degrees. | Minutes.
1 | 0 | 48 | 16 | 0 | 48
2 | 1 | 36 | 17 | 1 | 36
3 | 2 | 24 | 18 | 2 | 24
4 | 3 | 21 | 19 | 3 | 12
5 | 4 | 0 | 20 | 4 | 0
6 | 4 | 48 | 21 | 4 | 48
7 | 5 | 36 | 22 | 5 | 36
8 | 6 | 24 | 23 | 6 | 24
9 | 7 | 12 | 24 | 7 | 12
10 | 8 | 0 | 25 | 8 | 0
11 | 8 | 48 | 26 | 8 | 48
12 | 9 | 36 | 27 | 9 | 36
13 | 10 | 24 | 28 | 10 | 24
14 | 11 | 12 | 29 | 11 | 12
15 | 0 | 0 | 30 | 0 | 0

The use of the Table of the Tides.

FIrst it is to be understood, that by the swift motion of the first mover, the Moon and all the rest of the Stars and Planets, are turned about the World in 24 houres, upon which swift motion of the Moon, the dayly motions of the Sea do depend, which motion of the Sea falleth not out alwaies at one houre, [Page] the reason therof is, because of the swift motion of the Moon in regard she goeth almost 13 degrees in 24 houres, and the Sun mooveth scarce one degree: which gives every day 12 degrees, that the Moon commeth slower to any point in the Heaven then the Sunne: which 12 degrees makes 48 minutes of time for the difference of every full Sea, according unto the middle motion of the Moon, which difference is here set downe in this Table for every day of the Moons age. Therfore if you would know the full Sea at any place in the World, first you must know at what houre it is full Sea at the new or full Moon; which houres and minutes keep in minde, then seek the age of the Moon as is before taught, and with the number of her age enter this Table, under the title of the Moons age, and having found her age in the Table, against it you shall find the houres and minutes which are to be added unto the time that the Moon maketh full Sea in any place, and the whole number of houres and minutes is the time that the Moon maketh full Sea in any place upon the day desired. As for ensample, I desire to know the full Sea at London bridge upon the 13 of July 1624. the age of the Moon being found as before, is eight daies, then in the Table I finde eight daies, and against it 6 houres, and 24 minutes, which being added unto 3 houres, the full Sea upon the change day gives 9 a clock, 24 minutes for the time at the full Sea upon the 13 day of July 1624.

FINIS.

Information about this book

Title statement

Speculum nauticum a looking glasse for sea-men : wherein they may behold a small instrument called the plain scale whereby all questions nautical and propositions astronomical are very easily and demonstratively wrought ... / by Iohn Aspley.

Aspley, John.

Edition statement

1647

Publication

Publisher

Text Creation Partnership,

Place of publication

Ann Arbor, MI ; Oxford (UK) :

Date

2007-01 (EEBO-TCP Phase 1).

ID [DLPS]

A26045

ID [STC]

Wing A4011

ID [STC]

ESTC R27669

ID [EEBO-CITATION]

10113880

ID [OCLC]

ocm 10113880

ID [VID]

44506

Availability

This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal. The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission.

Series

Early English books online.

Notes

(EEBO-TCP ; phase 1, no. A26045)

Transcribed from: (Early English Books Online ; image set 44506)

Images scanned from microfilm: (Early English books, 1641-1700 ; 1375:19)

Source

— Speculum nauticum a looking glasse for sea-men : wherein they may behold a small instrument called the plain scale whereby all questions nautical and propositions astronomical are very easily and demonstratively wrought ... / by Iohn Aspley., Aspley, John.. The fourth edition corrected..

Extent

[6], 61 p. : ill.

Printed by Thomas Harper and are to be sold by George Hurlock,. London :: 1647.. (Reproduction of original in the British Library.)

Creation

Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford.

Editorial practices

EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO.

EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org).

The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source.

Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data.

Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so.

Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor.

The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines.

Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements).

Keying and markup guidelines are available at the Text Creation Partnership web site.