Nine geometricall exercises, for young sea-men and others that are studious in mathematicall practices: containing IX particular treatises, whose contents follow in the next pages. All which exercises are geometrically performed, by a line of chords and equal parts, by waies not usually known or practised. Unto which the analogies or proportions are added, whereby they may be applied to the chiliads of logarithms, and canons of artificiall sines and tangents. By William Leybourn, philomath.

To project the SPHERE Upon the Plain of the MERIDIAN BY A LINE OF CHORDS. Whereby the Sides and Angles of Sphericall Triangles are naturally laid down in Plano, as they are in the Sphere it self; By which the nature of them is dis∣covered, and their Sides and Angles measured with speed and exactness. The Fourth EXERCISE.

BEing to treat of the Projection of the Sphere in Plano, I suppose the Reader to be acquainted with the Doctrine of the Sphere or Globe, and with the Circles thereof; the nature of them, how they are there situate one from another in respect of distance, and to what use each of them is appro∣priate. But if there be any that hath a desire to make use of this

Treatise, and is ignorant of the Sphericks, let them reade the Books of such as have written of the Sphere or Globe. In Latin there are divers, as Theodocius, Orontius, Clarius, &c. In English there is Record's Castle of Knowledge, Hill's School of Skill, Blundevil's Exercises, Hews of the Globes, Newton, Moxon, &c. But that this Treatise may not be accounted de∣ficient in that which is so absolutely necessary for the understan∣ding and practice of what is herein contained; I will in this place give you a brief and plain Description of the Names, Pro∣perties, Distances, &c. of such Circles of the Sphere as in this Book we shall have occasion to project for the delineating of Sphericall Triangles in Plano, and that in a correspondent Po∣sition to their situation on the Globe or Sphere it self.

The Circles therefore chiefly made use of in this Projection are these:

• 1. The Meridian
• 2. The Horizon
• 3. The Aequinoctial
• 4. The Ecliptick
• 5. The Prime Verticall, or Circle of East and West
• 6. The Hour-Circles
• 7. Azimuths, or Verticall Circles
• 8. The two Tropicks
• 9. Parallels or Circles of Declination
• 10. Circles or Parallels of Altitude.

Of these Circles all but the three last are great Circles of the Sphere, which divide it into two equal parts; and the two Tro∣picks, the Circles of Altitude and Declination, are smaller Circles, and divide the Sphere into two parts unequally.

Besides these great and small Circles, there are severall Points of note upon the Globe: as (1.) The Zenith, which is the Point in the Heavens directly over our heads, in what part of the Earth soever we be; (2.) The Nadir, which is the Point directly un∣der our feet; (3.) The Poles of the World, about which the Heavens are moved; (4.) The Poles of the Ecliptick; (5.) The Poles of all other Circles.

I. Of the MERIDIAN.

THE Meridian is a great Circle of the Sphere, which pas∣seth through both the Poles of the World, and also through the Zenith and Nadir Points, and crosseth the Horizon in the North and South Points thereof. Unto this Circle (any Day in the year) when the Sun cometh it is Noon or Mid-day; and when the Moon, Stars or Planets, in the Night come to touch this Circle, they are then said to be upon the Meridian, or at the highest they will be that Night. This Circle in the Scheme of this Projection is noted by the Letters Z H N O.

II. Of the HORIZON.

THE Horizon also is a great Circle of the Sphere, and it is that Circle which divideth the visible part of the Heavens which we see from the not visible, that is, it divideth the Sphere into two Hemispheres, the lower and the higher. To this Circle when either the Sun, Moon, Stars or Planets, come on the East part, they are then said to rise; and when they have passed from the Easterly Point, by the Meridian, and descended to the Western part of this Circle, they are then said to set. This Circle is represented in the Projection by the right Line H A O.

III. Of the AEQƲINOCTIAL.

THE Aequinoctial is a great Circle, and in the Sphere it is elevated above the Horizon (upon the Meridian Cir∣cle) so much as is the Complement of the Latitude of the Place. As at London, where the Latitude is 51 degr. 30 min.

there the Aequinoctial is elevated 38 degr. 30 min. (which is so much as 51 degr. 30 min. wants of 90 degr.) and it cutteth the Horizon in the Points of East and West. Unto this Circle when the Sun cometh (which is twice every year, namely, about the 10. of March and the 12. of September) it causeth the Daies and Nights to be of equal length all the World over. This Circle is noted in the Scheme with AE A ae, and cuts the Horizon in the Point A, which represents both the East and West Points thereof.

IV. Of the ECLIPTICK.

THIS also is a great Circle of the Sphere, and (in the Northern Hemisphere, where the North Pole is visible above the Horizon, and the South Pole not visible) is elevated above the Aequinoctial Circle so much as is the Sun's greatest Declination, which is 23 degr. and about 30 min. and is as much depressed below the Aequinoctial in the Southern Hemi∣sphere. This Circle is called by some The Way of the Sun, for that the Sun in his motion never swerveth or goeth out there∣of, and so his Longitude or Place is counted in this Line. It cutteth the Horizon in the East and West Point A, as the Aequi∣noctial did. It is represented in the Scheme by the Line ♋ A ♑, and hath charactered upon it the 12 Signs of the Zodiack; the six Northern Signs, ♈ ♉ ♊ ♋ ♌ and ♍ being on that half which is above the Horizon, and the six Southern Signs ♎ ♏ ♐ ♑ ♒ and ♓, on the other half, which is below the Horizon.

V. Of the PRIME VERTICALL.

THE Prime Verticall, or Circle of East and West, (general∣ly called the Aequinoctial Colure, and then (as the Sphere is here projected) the Meridian representeth the Solstitial

Colure) is a great Circle passing through the Zenith and Na∣dir Points, and also through the East and West Points of the Horizon. Unto this Circle when the Sun, Moon, Stars or Planets, do (in their Motions) arrive, they are then due East or West. It is in the Projection signified by the right Line Z A N, passing through Z, the Zenith, N, the Nadir, and A, the East and West Point of the Horizon: and also cutteth the Aequinoctial in the Points ♈ and ♎.

VI. Of the HOƲR-CIRCLES.

THE Hour-Circles are great Circles of the Sphere, mee∣ting together in the Poles of the World, and crossing the Aequinoctial at right Angles, dividing it at every 15 de∣grees; and then every of those Divisions is one Hour of time: but if they pass through other parts of the Aequi∣noctial, dividing it unequally, then do those Hour-Circles re∣present unequal Spaces of time, according to the distance they are from the Meridian, or one from another. Of these Cir∣cles in the Scheme of the Projection there are four, thus no∣ted P B S, P A S, P C S, and P D S.

VII. Of the AZIMƲTH CIRCLES.

THE Azimuth or Verticall Circles are great Circles of the Sphere, meeting together in the Zenith and Nadir Points, as the Hour-Circles do in the Poles of the World, and divide the Horizon at right Angles, either equally, or unequally, as the Hour-Circles did the Aequinoctial. In the Scheme of the Pro∣jection there are four of these Verticall Circles, thus noted, Z O N, Z F N, Z A N, and Z G N.

VIII. Of the TROPICKS.

THE Tropicks are lesser Circles of the Sphere, dividing it unequally, and are drawn parallel to the Aequino∣ctial, at 23 degr. 30 min. distance therefrom, equal to the Sun's greatest Declination on either side. That Tropick which is on the North-side is called The Tropick of Cancer, to which when the Sun cometh (which is but once in a year, about the 10. of June) it maketh the longest Daies to all the Northern In∣habitants of the World, and the shortest Nights. The other Tropick, which is on the South-side of the Aequinoctial, is cal∣led The Tropick of Capricorn, to which when the Sun cometh, which is about the 11. of December, it maketh the shortest Daies and the longest Nights to all Northern Inhabitants, and the con∣trary to all the Southern Inhabitants of the World. In the Projection the Tropick of Cancer is signfied by ♋ I ♋, and the Tropick of Capricorn by ♑ K ♑.

IX. Of the CIRCLES or PARALLELS of DECLINATION.

THESE also are smaller Circles of the Sphere, and are drawn parallel to the Aequinoctial, towards both the Tropicks, and up to them. Those that are on the North-side of the Aequinoctial are called Parallels of North Declination, and those that are on the South-side of the Aequinoctial are called Parallels of South Declination. Of these Parallels there are in the Scheme of the Projection two, one towards the Tropick of Cancer, the other towards the Tropick of Capricorn, and either of them 20 degrees distant from the Aequinoctial. The Nor∣thern Parallel of Declination is noted with ♊ ☉ ♌, and the Southern with ♒ E ♐.

X. Of the CIRCLES or PARALLELS of ALTITƲDE.

THE Circles of Altitude are likewise small Circles of the Sphere, and are drawn parallel to the Horizon, as the Circles of Declination were to the Aequinoctial. These Paral∣lels are drawn from the Horizon towards the Zenith Point, and upon occasion, in many Cases, quite up unto it. By these Parallels are measured the Altitude or Height of the Sun, Moon and Stars. In the Scheme there is onely one of them, and that is expressed by the Letters M E L.

Thus have I given you a brief and plain Description of the Circles, both great and small, which we shall have occasion to use in this following Treatise. And here note, that every Circle of the Sphere (both great and small) hath his proper Poles, which Poles (of all the great Circles) are 90 Degrees, or a Quadrant of a Circle, distant from the Circle it self. The Poles of the Circles in this Projection are as followeth.

• Z and N Are the Poles of H A O, the Horizon.
• P and S Are the Poles of AE A ae, the Aequinoctial.
• O and H Are the Poles of Z A N, the Prime Verticall.
• Q and R Are the Poles of the Ecliptick.
• AE and ae Are the Poles of P A S, the Axis of the World.

The Poles of these five Circles are all in the Meridian, and so there needeth no farther Precept for the finding of them; and the Pole of the Meridian is the Centre thereof.

But for the three Azimuth Circles, they fall in several Points of the Horizon; and the three Hour-Circles in certain Points in the Aequinoctial. How to finde which Points shall be shewed af∣terwards in due place.

• A Is the Pole of the Meridian, Z H N O.
• T Is the Pole of the Azimuth Circle, Z F N.
• G Is the Pole of the Azimuth Circle,Z ☉ N.
• ☉ Is the Pole of the Azimuth Circle,Z G N.
• X Is the Pole of the Hour-Circle P B S.
• Y Is the Pole of the Hour-Circle P D S.
• V Is the Pole of the Hour-Circle P C S.

The Poles of the World P and S are also the Poles of the Tro∣picks and of all the Parallels of Declination. And

The Zenith and Nadir, Z and N, are the Poles of all the Par∣allels of Altitude.

Having sufficiently acquainted the Reader with the several Circles, Lines, Points and Poles, belonging to every Circle, I will now proceed to my intended purpose; namely, to project (or lay down in Plano) all these Circles, Lines, Points and Poles, in their true Positions.

How to project the Sphere upon the Plain of the Meridian.

FIrst, take 60 degr. of your Line of Chords, and with that distance upon the Point A. (as a Centre) describe the Circle Z H N O, representing the Meridian, (within which Circle all the rest are to be projected) and cross it with the two Diameters H A O the Horizon, and Z A N the Prime Ver∣ticall.

Secondly, (because the Latitude of the place for which you draw your Projection, viz. London, is 51 degr. 30 min.) take 51 degr. 30 min. from your Line of Chords, and set them upon the Meridian from Z to AE, and from N to ae, and draw

the Line AE A ae for the Aequinoctial. Also set 51 degr. 30 m. from O to P, and from H to S, and draw the Line P A S, re∣presenting the Axis of the World and the Hour-Circle of 6 a clock.

Thirdly, take 23 degr. 30 min. the quantity of the Sun's greatest Declination, and also of the distance of the two Tro∣picks from the Aequinoctial, and set them upon the Meridian from AE to ♋, above the Aequinoctial, and also from AE to ♑, below the Aequinoctial. In like manner set the same distance of 23 degr. 30 min. from ae to ♋ above the Aequinoctial, and from ae to ♑ below it. This done, lay a Ruler upon the Points AE and ♋, and it will cut the Axis of the World P A S in the Point I. So a Circle drawn which shall pass through these three Points, ♋ I ♋, shall be the Tropick of Cancer. Again, lay a Ruler to AE and ♑, and it will cut the Axis in the Point K. So a Circle drawn through ♑ K ♑ shall be the Tropick of Capricorn. But to shew how you may find the Centres upon which these Tropical Circles are to be described, I must make this

Diversion.

Ʋpon a long piece of stiff Paper, or rather fine Pastboard, with 60 Degrees of your Line of Chords describe a Quadrant, as A B C, and upon the Point C erect the Perpendicular C D; in doing whereof you must be very carefull, for a small errour committed in that will produce a great one in finding of the true Centres.—Be∣ing thus prepared, you may readily find the Centres of the two Tropicks, and of any other Parallels of Declination or Altitude. But (1.) for the Tropicks: Being the Tropicks are distant from the Aequinoctial 23 degr. 30 min. on either side, take 23 degr. 30 min. out of your Line of Chords, and set them upon the Qua∣drant from B to f, and through the Point f draw the Line A f g, cutting the Perpendicular C D in the Point g. This done, take

[illustration] geometrical diagram

in your Compasses the di∣stance C g, and setting one foot in the Point I in your Projection, where the other Point falleth upon the Line I P, it being sufficiently ex∣tended, will be the Centre of the Tropick of Cancer; and a Circle described with this distance of the Compas∣ses, C g, will pass exactly through the Points ♋ I ♋, and so describe your Tro∣pick of Cancer. The like you must doe for the Tro∣pick of Capricorn, by taking the distance C g in your Compasses, and setting one foot in the Point K of your Projection, and where the other reacheth upon the Line K. S, being extended, there is the Centre of the Tropick of Capricorn. But (2.) to finde the Centres of any of the other Parallels of Decli∣nation, as of the two in the Projection, namely, ♊ a ♌ and ♒ c ♐, either of which are 20 d. distant from the Ae∣quinoctial; the Centres of them are also found in the same manner as the Centres of the Tropicks were. For take 20 degr. out of your

Line of Chords, and set them upon the Quadrant from B to e, drawing the Line A c h, the distance C h shall be the Semidia∣meter of the Parallel of 20 degr. of Declination; which, being set upon your Projection from a, upon the Line a P, (being ex∣tended) will there give you the Centre of the Parallel; and the Compasses at the distance of C h, being there set, will describe the Parallel of 20 degr. of Declination ♊ a ♌. And in the same manner that the Centre of this Parallel was found, so may you find the Centre of that on the other side of the Aequinoctial, ♒ c ♐. But (3.) you have in your Projection a Parallel or Circle of Altitude, namely, M E L, which is 12 degr. from or above the Horizon, the Centre whereof is found in the same man∣ner as were the Parallels of Declination. For if you set 12 degr. of your Chord upon the Quadrant A B C, they will reach to k: draw a Line therefore from A through k, extending it till it meet with the Line C D being also extended; so shall the distance between this Point, where the two Lines meet, and the Point C, be the Semidiameter of the Parallel of Altitude of 12 degr. Where∣fore that distance set from the Point m of your Projection, up∣on the Line m Z, being extended, will be the Centre of the Par∣allel, and the Compasses, at that distance, will describe the Par∣allel of Altitude M m L.

But for those Parallels of Altitude which fall near the Hori∣zon, those Circles or Parallels of Declination which fall near to the Aequinoctial, those Hour-Circles which fall near to the Axis of the World or Hour of Six, and those Azimuth Circles which are near to the Prime Verticall or Azimuth of East and West; those that make Mathematicall Instruments have an In∣strument called a Bow, which, by the help of one or more Screws, (according to the length of the Bow) may be extended to touch any three Points which lie near in a straight Line; by the edge of which Bow you may draw your Hour-Circles, Azimuths, Par∣allels of Declination and Altitude, as easily as you may draw a right Line by the edge of a Ruler.—But to return again to our Projection.

Fourthly, draw a right Line ♋ A ♑ between the two Tro∣picks, touching the Tropick of Cancer above the Horizon at ♋, and the Tropick of Capricorn below the Horizon at the Point ♑.—This Circle hath upon it the Characters of the 12 Signs of the Zodiac, which are to be put on in this manner.—Take 23 d. 30 min. out of your Line of Chords, and set them from P to Q, and from S to R: which Points, Q and R, are the two Poles of the Ecliptick. Then take 60 degr. from your Line of Chords, and set them from Q to 1, and from Q to 3. Also set the same distance from ♋ to 2, and from ♑ to 4.—This done, lay a Ruler to the Pole R and the figure 1, it will cut the Ecliptick in the Point ♊ and ♌: the Ruler laid to R and 2 will cut it in the Point ♉ and ♍; and laid to R and 4, in ♏ and ♓; and laid to R and 3, in ♐ and ♒. So have you the true Points for the Sun's entrance into every Sign. And if you would have every tenth degree of each Sign, divide every of the Spaces ♋ 1, 12, 2 Q, Q 4, 4 3, and 3 ♑, into three equal parts; so will each part contain 10 d. and a Ruler laid to each of them and the Point R shall give you the Points upon the Ecliptick answering to the 10. degr. of every Sign. And in the same manner may you (if your Projection be large) put on every Degree.

Fifthly, for the putting on of the Hour-Circles; consider how far the Circle you are to put on is distant from the Meri∣dian, and set so many degrees upon the Meridian from the Aequinoctial: a Ruler laid from Z to those degrees will cross the Aequinoctial, and through that Point in the Aequinoctial where the Ruler so crosseth, the Hour-Circle will pass.—Ex∣ample: The Hour-Circle P B S, in this Projection, is distant from the Meridian 62 d. 46 m. wherefore take 62 d. 46 m. from your Chords, and set them from a to b; then laying a Ruler from Z to b, it will cut the Aequinoctial in B, through which Point the Hour-Circle of 62 d. 46 m. must pass.—To find the Centre of this Hour-Circle, (and so of any other) repair to the former Scheme for finding of the Centres of the Parallels of Altitude and De∣clination;

and (because this Hour-Circle is distant from the Me∣ridian 62 degr. 46 min.) take 62 degr. 46 min. from your Line of Chords, and set them upon the Quadrant A B C, from C to l, and draw the Line A l m. So shall the Line A l m be the Semidiameter of the Hour-Circle P B S; which being taken in your Compasses, and set upon your Projection from B, upon the Line B AE, (being extended,) shall there give you the Cen∣tre of that Hour-Circle. And in the same manner may the Centres of all the rest be found.

Sixthly, the Azimuth Circles are to be drawn upon the Pro∣jection, and the Centres of them found in all respects as the Hour-Circles were.—So the Azimuth Circle Z ☉ N, being 56 degr. 40 min. from the Meridian, take 56 degr. 41 min. out of your Line of Chords, and set them upon the Meridian of your Projection from O to d; then laying a Ruler unto Z and d, it will cut the Horizon in the Point ☉ through which the Azimuth of 56 degr. 41 min. Z ☉ N, must pass.—Then, to find its Centre, repair to the former Scheme for finding of Centres, and upon the Quadrant A B C set 56 degr. 41 min. of your Chords, from C to n, and draw the Line A u o: so shall the Line A u o be the Semidiameter of the Azimuth Circle Z ☉ N; which being taken in your Compasses, and set upon your Projection from ☉, upon the Line ☉ H, (being extended,) shall there give you the Centre of the Azimuth Circle Z ☉ N. And in this manner may the Centre of any other Azimuth Circle be found.

And here note (I.) That the Centres of all Azimuth Circles fall in the Horizon H A D, being extended where need is. The Centres of all the Hour-Circles fall in the Aequinoctial Line AE A ae, being extended. The Centres of the Tropicks and Par∣allels of Declination fall in the Axis of the World P A S, ex∣tended. And the Centres of the Circles of Altitude fall in the Prime Verticall Circle Z A N.

Note (II.) That if the middle Point of any Hour-Circle do not fall just in the Aequinoctial, or any Azimuth Circle just in

the Horizon, but on either side of them; then you may find the Centres by the Geometricall Propositions at the beginning of this Book; though there be other waies to find the Centres upon the Projection it self, which I omit, for that I would not cumber the Scheme with unnecessary Lines.

Seventhly, Every Circle in the Projection hath its proper Pole, as was before intimated. Now for the finding of them, you are to note, that the Pole of every great Circle is 90 degr. or a Quadrant of a Circle, distant from the Circle it self, upon that Line which cutteth the Circle at right Angles. Thus the Poles of all the Hour-Circles are upon the Aequinoctial, and the Poles of all the Azimuths upon the Horizon.—Now if you would find the Pole of the Hour-Circle P D S, lay a Ruler up∣on P and D, and it will cut the Meridian Circle in e: then take 90 degr. of your Line of Chords, and set them from e to f, a Ruler laid from P to f will cut the Aequinoctial in Y: so is Y the Pole of the Hour-Circle P D S.

Lastly, The finding of the Poles of the Azimuth Circles is the same with the Hour-Circles. So if you would find the Pole of the Azimuth Circle Z G N, lay a Ruler upon Z and G, it will cut the Meridian Circle in g; then set 90 degr. of your Chord from g to d, so a Ruler laid from Z to d will cut the Horizon H A O in the Point ☉, which Point ☉ is the Pole of the Azimuth Circle Z G N. And thus have you found the Poles of one of the Hour and one of the Azimuth Circles. And by the same manner of Work you may find the Poles of all the rest. As

• The Pole of the Hour-Circle P D S will be found at Y
• The Pole of the Hour-Circle P C S will be found at V
• The Pole of the Hour-Circle P A S will be found at AE or ae
• The Pole of the Hour-Circle P B S will be found at X
• The Pole of the Azimuth Circle Z G N will be found at ☉
• The Pole of the Azimuth Circle Z A N will be found at H or O
• The Pole of the Azimuth Circle Z F N will be found at T
• The Pole of the Azimuth Circle Z ☉ N will be found at G

• The Poles of the Horizon H A O are Z and N, the Zenith and Nadir.
• The Poles of the Aequinoctial AE A ae are P and S, the Poles of the World.
• The Poles of the Ecliptick ♋ A ♑ are Q and R.

Thus have I given you at large a plain and easie method how to project the Sphere upon the Plain of the Meridian Circle, by help of the Line of Chords onely: Ʋpon which Projection, by the intersection or crossing of the severall Circles thereof, are constituted divers Sphericall Triangles; some Right-angled, and others Oblique-angled. By the resolving of which Triangles variety of Questions appertaining to Astronomie, Geographie and Navigation, may (with speed and exactness) be resolved. But before I come to shew the manner of working particular Que∣stions of any kind, it will be expedient that I shew you, (1.) how to measure or find the quantity of the Sides and Angles of a Sphericall Triangle, as they are here projected; and (2.) how to project or lay down an Angle or Side of any quantity that shall be required.

I. A Sphericall Triangle being projected, how to find the quantity of any Angle thereof.

LAY a Ruler to the angular Point, and the extremity of the Sides containing the Angle, they being continued to Quadrants; and note where the Ruler cuts the Meri∣dian or outward Circle; at both which places make marks up∣on the Meridian: the distance between those two marks, being measured upon your Line of Chords, shall give you the quan∣tity of the Angle required.

Example I.

IN the Triangle P ☉ O, in the Projection, let it be requi∣red to find the quantity of the Angle ☉ P O.—First, lay a Ruler upon the angular Point P, and to the extreme ends of the Sides P ☉ and P O, they being extended to Quadrants, which is, to that Circle which measures that Angle: (as the Aequinoctial measures all the Angles at P, the Pole of the World; the Horizon all the Angles at Z, the Zenith, &c.) So the Ru∣ler laid from P to ae, will cut the Meridian in ae; and being laid from P to B, it will cut the Meridian in the Point b. The distance b ae, being taken in your Compasses and measured upon your Line of Chords, will be found to contain 62 degr. 46 min. which is the quantity of the Angle ☉ P O.—But if upon the Point P you were to project an Angle to contain 62 degr. 46 min. then take 90 degr. of your Chords, and set them from P to ae, and through the Centre A draw the Line AE A ae; then take 62 degr. 46 m. out of your Line of Chords, and set them from ae to b; and laying a Ruler from P to b, it will cut AE A ae in the Point B: the Circle P B S being drawn, the Angle at P will contain 62 degr. 46 min.

Example II.

LET it be required to find the quantity of the Angle Z E P.—Lay a Ruler to ☉, the Pole of the Circle Z E N, and the Point E, it will cut the Meridian Circle in M; from M set 90 degr. to z; a Ruler laid from ☉ to z will cut the Circle Z E N (it being extended beyond the Zenith Z) at the Point δ.

Again, Lay a Ruler upon Y, the Pole of the Circle P E S, and it will cut the Meridian Circle in v; set 90 degr. from v

to x upon the Meridian; a Ruler laid from Y to x will cut the Circle P E S in y.

This done, lay a Ruler from E to δ, and it will cut the Me∣ridian in θ; also lay the Ruler from E to y, it will cut the Me∣ridian in λ: the distance θ λ, being taken in your Compasses and applied to your Line of Chords, will be found to contain 21 degr. 45 min. And such is the quantity of the Angle Z E P.

These two sorts of Angles are the most troublesome to find their quantities, and therefore I have instanced in them. There are other Angles in the Projection which render their measures to the eye, without farther Instructions for finding their quan∣tities.

II. A Sphericall Triangle being projected, to find the quantity of any Side thereof.

A Ruler laid upon the Pole of the Circle which is to be measured, and to the extreme ends of the Side of the Triangle; note where the Ruler, so laid, cuts the Me∣ridian at both ends of the Side: that distance, taken in your Compasses and measured upon the Line of Chords, will give you the quantity of the Side of the Triangle.

Example I.

LET it be required to find the Side E Z of the Triangle Z E P.—Lay a Ruler to ☉ (the Pole of the Circle Z E N) and the angular Point E, it will cut the Meridian in M; and a Ruler laid to Z will cut the Meridian in Z. So the distance M Z, taken in the Compasses and measured upon the Line of Chords, will be found to contain 78 degr. And such is the quantity of the Side Z E.

Example II.

LET it be required to find the Side ☉ B of the Sphericall Triangle A ☉ B.—Lay a Ruler upon X, the Pole of the Circle P B S, and the Point B, it will cut the Meridian Cir∣cle in ae.—Also lay a Ruler from X to ☉, it will cut the Meri∣dian in the Point ♌. The distance between ae and ♌, being taken and measured on the Line of Chords, will contain 20 d. And such is the quantity of the Side ☉ B.

I could instance in divers other Examples concerning the Mea∣suring of the Sides and Angles of Triangles upon the Projection; but I here omit them, because in the resolving of the following Propositions they will come in practice, and the Manner of the performance is there plainly expressed: onely I deemed it conve∣nient here to give some taste thereof, as a Preparative to that which followeth.—But before I come to shew the Manner of resolving of particular Questions in Astronomie, Geogra∣phy, &c. I will declare the Variety of Sphericall Problems that will naturally arise out of every Sphericall Triangle, being pro∣jected.