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.

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.