III.

TRIGONOMETRICAL THEOREMS.

(1) Lemma 1. The product of the radius by the difference of the verfed fines of two arcs is equal to twice the product of the fines of half the fum and half the difference of thofe arcs.

R (fin V, a-fin V, 6) = 2 finx fin.

(2) Corollary. The product of the radius by the verfed fine of an arc is equal to twice the fquare of the fine of half that arc.

R fin V,a=2 fin2 ļa.

(3) Lem. 2. The fum of the cofines of two arcs is to their difference as the cotangent of half the fum of thofe arcs is to the tangent of half their difference.

Cof a+cofb: cofa-cofb:: cot: tang.

(4) Lem. 3. The fum of the fines of two arcs is to their difference as the tangent of half the fum of thofe arcs is to the tangent of half their their difference.

Sina+fin b: fin a-fin b:: tang : tang.

(5) Lem. 4. The fum of the cotangents of two arcs is to their difference as the fine of the fum of thofe arcs is to the fine of their difference Cot a + cot b: cot a-cot b:: fin(b+a) : fin (b—a).

(6) Lem. 5. The product of the fine of the fum of two arcs and the tangent of half that fum, is to the product of the fine of their difference and the tangent of half that difference, as the fquare of the fine of half their fum is to the fquare of the fine of half their difference.

Sin (a+b) × tang : fin (a—b) × tang: fin': fin3 ex

(7) Lem. 6. The product of the fine of the fum of two arcs and the tangent of half their difference, is to the product of the fine of their difference and the tangent of half their fum, as the fquare of the cofine of half their fum is to the fquare of the cofine of half their difference. Sin (a+b) tang: fin (a—b) x tang :: cof±3: cof3 ~~'. (8) Lem. 7. In right angled spherical triangles the cofine of the hypothenufe is to the cotangent of one of the oblique angles as the cotangent of the other is to the radius.

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(9) Lem. 8. In right angled spherical triangles the cofine of the hypothenufe is to the cofine of one of the fides as the cofine of the other is to the radius.

(10) Lem. 9. In any spherical triangle the product of the fines of the two fides is to the fquare of the radius as the difference of the verfed fines of the bafe and the difference of the two fides is to the verfed fine of the vertical angle, Fig. XIV.

Sin ABX fin BC: R':: fin V, AC— fin V, (AB—BC) : fîn V, B *. (11) Lem. 10. In any spherical triangle the product of the fines of the two fides is to the fquare of the radius, as the difference of the verfed fines of the fum of the two fides and the bafe is to the verfed fine of the fupplement of the vertical angle, Fig. XIV.

Sin ABX fin BC: R':: fin V, (AB + BC)— fin V, AC: fin V, fup. B..

This is one of Regiomontanus' propofitions.

(12)

(12) The natural parts of a triangle are its three fides and its three angles.

(13) The circular parts of a rectangular (or quadrantal) fpherical triangle are the two natural parts adjoining to the right angle (or quadrant fide) and the complements of the other three.

(14) Any one of thefe five being confidered as a middle part, the two next to it are called the adjacent parts, and the other two the oppofite parts: Thus, in the triangle dAB (Fig. XV.) rectangular in A, if the complement of the angle d is taken as a middle part, the adjacent parts are the fide d A and the complement of the hypothenufe db; and the opposite parts the fide, b A and the complement of the angle b.

(15) Of five great circles of the fphere AB, BC, CD, DE, and EA (Fig. XV.) let the first interfect the second; the fecond, the third; the third, the fourth; the fourth, the fifth; and the fifth, the first; at right angles in the points B, C, D, E and A: there are formed, by the interfections mentioned and by those at the respective poles a, b, c, d and e of thefe great circles, five rectangular triangles dAb, bDe, eBc, cEa and aCd: and, if these poles are joined by the quadrantal arcs ab, bc, cd, de and ea, there are formed five quadrantal triangles adb, dbe, bec, eca, and cad. The circular parts in all these triangles are the fame: the pofition of these equal circular parts with respect to one another in each of these triangles is different: therefore

(16) What is true of the circular parts of a rectangular triangle is true of thofe of a quadrantal; and what is true of one middle part and its adjacent and oppofite parts is true of the other four middle parts and their adjacent and oppofite parts.

(17)

(17) The circular parts of an oblique fpherical triangle are its three fides and the fupplements of its three angles.

(18) Any one of these fix being confidered as a middle part, the two next to it may be called the adjacent parts; the one facing it, the remote part; and the other two, the oppofite parts: Thus, in the triangle ABC (Fig. XIV.), if the fide AC is taken as a middle part, the adjacent parts are the fupplements of the angles A and C; the oppofite parts, the fides AB and BC, and the remote part, the fupplement of the angle B.

ca,

(19) Of fix great circles of the fphere let the first three, AB, BC, and CA, interfect cach other at the poles, B, C and A, of the second three, ab and be the interfections, c, a and b, of the latter are the poles of the former: there are formed two triangles ABC and abc in which the circular parts are the fame; the position of these equal circular parts is different in both: therefore

(20) What is true of one middle part and its adjacent, opposite, and remote parts, is true of any other middle part and its adjacent, oppofite, and remote parts.

(21) If an arc bBDd pass through the vertices of these two triangles, it will be perpendicular to their bases CDA and cda, and the fegments at the bafe of the one triangle will be the complements of the fegments at the vertical angle of the other: that is, CD=90°-dba, AD=90°— dbc, cd=90°-ABD, ad=90°-DBC.

(22) If the radius of the sphere is fuppofed infinite, the fines and tangents of the fides of a triangle defcribed on its furface, become the fides themselves of a plane triangle. Confequently all the formulæ of spherical trigonometry, where the fines and tangents only of the fides enter, are applicable to plane trigonometry. Thofe, however, in which any

functions