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given in the Triangle acd the Angle e da 125 0, it being the Complement of c de to 180, Eucl. Lib. 1. Prop. 13. and the Side oppofite a c, and the Angle e ad 27.30 to find its oppofite Side cd; and when that is found, you have in the Triangle dce, one Side, and all the Angles to find the other Sides severally.

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In the right angled Triangle ab c, there is given ab 96, and be and ca in one Sum 220, to find the Sides and Angles feverally.

Draw ba 96, and perpendicular Plate 1. thereto at b, draw b d 220, and draw Fig. 19. a d, which biffect in e, and draw the Perpendicular ec, to cut b d in c, and draw a c, then is abc the Triangle required. Then for Calculation in the Triangle a bd, you have given bd 220, and a b 96, to find the Angle at d, which is equal to the Angle ca

(because their oppofite Sides are equal.) Likewise the Angled a b is the Complement of a db to 90; then from the Angle d'a b fubtract the Angle e a c, there remains the Angle ca b, whofe Complement is the Angle a cb, then in the faid Triangle a be, there is given the Base a b, and the Angles a and b, to find the Hipotenufe by Cafe 2. or the Perpendicular by Cafe 3. but I fhall leave the Operation to the Learner's Practice.

CHAP.

29

CHAP. II.

SECT. I.

Of the PROJECTION of the SPHERE.

HERE are feveral Ways of Projecting the Sphere, viz. Stereographick, Orthographick, and Gnomonical. I fhall only fpeak of the Stereographick at this time, as being most useful in Trigonometrical Operations, and fhall refer the other to their proper Places afterwards.

Stereographick Projection of the Sphere, is the Projection of the Sphere or Globe, upon a Plain, paffing through the Center thereof; the Eye being fuppofed to be placed upon the Superficies or Surface of the Globe, perpendicular to the Center of the faid Plain, or in the Pole of that great Circle, upon which

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the Projection is to be made, and on that Side of the Sphere oppofite to the Side to be projected; this Projection thus made, fhall truly reprefent all the Appearances of that half of the Sphere, oppofite to that, in whofe Pole the Eye is placed, as may be illuftrated by the following Similitude.

Suppofe a Sphere or Globe were made of Glafs, or any tranfparent Metal, on which were defcribed all the Points and Circles of the Sphere (as Zenith, Nadir, Equator, Eclip tick, &c.) in black Lines that might be dif cerned through the Globe; and admit there were a Glafs Plain paffing through the Center of the Globe, cutting it in two equal Parts in that great Circle, upon whofe Plain the Projection is to be made; as for Inftance, fuppofe I were to project the Western Hemifphere upon the Plain of the Meridian, having a tranfparent Globe, prepared as before defcribed, and the transparent Plain to pafs through the Center of the Sphere, and cut it in two Parts in the Meridian" (the great Circle upon which the Projection is to be made) if the Eye be fuppofed to be placed in the Eaft Point of the Horizon, upon the Surface of the Sphere; and obferve, where the Meridians, Azimuths, c. on the Weft Side of the Sphere cuts the Plain paffing through the Middle of the Sphere: Thefe Circles upon that Plain fhall truly reprefent the Projection of the Western Hemifphere upon the Plain of the Meridian; for the Projection of the Sphere, of what kind foever, is no more but

the

the Representation of thofe Circles and Points upon a Plain, which are imagined to be described in the Sphere, or upon the Celestial or. Terreftrial Globe.

tor.

The imaginary Points, are 1. The North and South Poles of the World, or of the Equa 2. The Poles of the Ecliptick. 3. The Equinoctial Points Aries and Libra, where the Ecliptick cuts the Equator. 4. The Zenith and Nadir.

The Circles of the Sphere, are either greater or leffer Circles. The greater Circles are thofe that divide the Sphere itfelf in two equal Parts; as 1. The Horizon which divides. that Part of the Sphere vifible to us, from that Part that is invifible. 2. The Meridian, which divides the Eaft half from the Weft. 3. The Equator, which divides the North Half from the South. 4. The Ecliptick. 5. The Prime Vertical, c. of all which fee a more particular Defcription afterwards.

The leffer Circles divide the Sphere in two unequal Parts, as the Almicanthar or Parallels of Altitude, and alfo the Parallels of Declination, fo called,, becaufe they are Parallel to the Equator, r.,

The Sphere may be Stereographically proje&ed upon the Plain of any great Circle; I have for the Learner's Information, inftanced in five Varieties, having projected the Sphere.

Meridian, Fig. 1.

Horizon, Fig. 2.

Upon the Plain Equator, Fig. 3. Plate 5.

of the

Ecliptick, Fig. 4.
Prime Vertical,
Fig. 5.

Upon the Plain of whatsoever great Circle the Sphere be projected, all the Circles in the Sphere will be Circles in the Plain of the Projection, and if continued to perfect Circles, will answer all the Representations and Conditions of Meafuring, ec. without the primitive Circle, as well as within it; and alfo the Angles made by the Interfection of two great Circles on the Surface of the Sphere, are equal to the Angles made by their Repre fentatives on the Plain of the Projection.

In any Projection, that Circle upon whofe Plain the Sphere is projected, is called the primitive Circle, and is always a perfect Circle defcribed about the whole Projection, as the Circle ZHN 0, Fig. 1. or the Circle N W SE, Fig. 2. c. in Plate 2.

All the rest of the great Circles in any Projection, are either right Circles, or oblique Circles.

A right Circle appears to the Eye to be a right Line, cutting the primitive Circle at op. pofite Points, and paffing through the Center thereof, thereby dividing the Superficies of it in two equal Parts, as in Fig. 1. the Lines Pa S and a 2 and Ea C are right Circles.

An