in the following Treatise. These are to be sought for, in the introductions, which are commonly prefixt to books of logarithms. The doctrine of the corresponding variations of the sides and angles of triangles, and that of the projections of the sphere, have, also, been omitted: because the former, of those two subjects, seems more properly to belong to the method of fluxions, or to the differential method; and the latter, if it had been strictly and fully investigated, would have furnished matter sufficient for a separate Tract.

To the Vice-Chancellor, and the rest of the Syndics of the University Press, the Author begs leave to offer his sincere thanks, for the aid which they have afforded him, in his present undertaking: the greater part of the expence attending it, having been defrayed from the funds, which are at their disposal. He has also another debt, of another kind, to acknowledge; of which he can only acquit himself, by subjoining the following list of Authors, who have been consulted, with more or less advantage, in the course of his Work, Theodosius, Menelaus, Ptolemy, Muller, Copernicus, Clavius, Euler, Bertand, Cagnoli, Legendre, Lagrange, Delambre.

Trin. Coll.

Oct. 21, 1816.

ERRATA.

P. 9. 1. 5. from the bottom, for a sphere's read the sphere's. 13. 1. 9. from the bottom, for though read through.

17. Note, for 30 read 29.

19. 1. 13. from the bottom, for Cor. read Cor. 1.

42. In the figure, the straight lines EG and EF are wanting.

139. In the last line, place a comma after sphere. 169. 1. 1. dele the comma after constructed.

175. 1. 6. from the bottom, dele therefore.

176. 1. 14. from the top, place a comma after sines.

180: 1. 8. from the top, at the beginning insert (25).

185. 1. 4. from the bottom after triangle, read L being the perpendicular

188. 1.5. from the bottom, after co-tangent put a comma.

225. 1. 3. from the top, in the denominator of the fraction at the end

of the line, for B' read B.

238. 1. 8. from the top, for tan (S+) S′′ read tan 1⁄2 (S+S′′)

1. 12. from the top, dele less.

253. 1. 2. from the top, for toublesome read troublesome.

256. 1. 7. from the top, for Manduit read Mauduit.

257. 1. 4. in the note, before the word another insert the word one. 259. 1. 1. for tan ¦ (A~A) read tan 1⁄2 (A~A').

A

TREATISE

ON

Spherics.

PART I.

SPHERICAL GEOMETRY.

"Nullum autem est dubium, quin, si symptomata linearum curvarum, in superficie spherica descriptarum, eodem modo evolverentur, ac curvarum, in plano descriptarum, affectiones explicitæ fuerunt, nova Geometriæ pars prodiret, quæ non solum insigni varietate, sed elegantia quoque, inventorum se commendaret."

LEXELL.

A

PART I.

THE ELEMENTS OF

Spherical Geometry.

SECTION I.

ON THE COMMON SECTIONS OF A SPHERE

AND A PLANE.

DEFINITIONS.
ITION,

ART. 1. A Sphere is a solid figure contained by

one surface, and is such that all straight lines, drawn from a certain point within the figure to the surface, are equal to one another:

(2.) And this point is called the Center of the Sphere.

(3.) A Diameter of a Sphere is a straight line, drawn through the center, and terminated, both ways, by the surface of the sphere.

(4.) COR.

The center of a sphere bisects (Art. 1, 2.) any of its diameters: and, therefore, all the diameters of a sphere are equal to one another.