1

COROLLARY II.

Hence, the fine of the double of either arch

2CS

(when they are equal) will be = and its co

fine =

C2 -S2
R

, R

whence it appears, that the fine of

the double of any arch, is equal to twice the rectangle of the fine and co-fine of the fingle arch, divided by radius, and that its co-fine is equal to the difference. of the fquares of the fine and co-fine of the fingle arch, alfo, divided by radius...

COROLLARY III.

མཚ༷ ༔ ་ *

X

Moreover, because Dmx CF OC × mv (≥OC × EGOC x OE-OG); and Om × OF OC x On (OC × 20n = OC × OE + OG), it follows, that the rectangle of the fines of any two arches. (AC, CD (BC) is equal to a rectangle under half the radius, and the difference of the co fines, of the fum and difference of thofe arches; and that the rectangle of their co-fines is equal to a rectangle under half the radius, and the fun of the co-fines, of the fum and difference of the fame arches.

PROP. III.

The tangents of two arches being given, to find the tangents of the fum, and difference, of those arches.

Let

Let AN and

AM be the two arches, and ABB and AC their N tangents; alfo

let NE be the

F

tangent of their

B

M

fum, in the first cafe, and the

tangent of their difference, in the fecond, and let CF, perpendicular to the radius DN, be drawn: then, becaufe of the equiangular triangles BAD and BFC, we fhall have

BD x CF = DAX BC by 24. 3.
(6y

BD x BF BA ×

Take each of the laft equal quantities from B and there will remain BD BD BF (BDDF) BDBA x BC: now BD × DF (BD'-BA x BC): BD x CF (DA × BC):: DF: CF :: DN (DA): NE :: DA2; DA × NE; whence, alternately, BD BA x BC: DA' (:: DA× BC: DA x-NE); BC: NE. But the first term, of this proportion, because BD2 — DA2 + BA', will also be expreffed by DA2 + BA2 — BA x BC, or by DA+BABA x AB+ AC; or, laftly, by, DA+BA × AC: therefore, the three first terms of the proportion being known, the fourth NE will likewife be known. 2. E. I.

COROLLARY.

Hence, if radius be fuppofed unity, and the tangents of two arches be denoted by T and t, it follows, that the tangent of their fum will be

But, it will be proper to take notice here (once for all) that, if in thefe, or any other theorems, the tangent, fecant, co-fine. co-tangent, &c. of an arch greater than 90 degrees be concerned; then, instead thereof, the tangent, fecant, co-fine, &c. of an arch, as much below 90 degrees, is to be taken, with a negative fign; according to the obfervation in page 5.

A

B

Thus, for inftance, let BA be an arch greater than 90°, and let the tangent of the fum of AB and AC be required; fuppofing T to represent

the tangent of AD (the fupplement of AB) and t the tangent of AC: then, by writing-T inftead of T, in the first of the foregoing theorems, we

T +

fhall have tang. of BC=

Tt

and therefore tang. DC (-tang. BC) =

which is the very theorem demonftrated in the 2d cafe.

PROP. IV.

As the fum of the tangents of any two angles BAC, BAD, is to their difference, fo is the fine of the fum of thofe angles, to the fine of their dif ference.

Let

that Ad

B

D

AD, and dAB DAB, and, confequently, that CAd is the difference of the two angles BAC and BAD.

Now, by reason of the fimilar triangles CDE and CdF, it will be, CD (CB+BD): Čd (CBBD) :: DE: dF; but DE and F are fines of DAE and dAF to the equal radii AD and Ad : whence the truth of the propofition is manifeft.

COROLLARY.

Hence it also appears, that the base (CD) of a plane triangle, is to (Cd) the difference of its two fegments (made by letting fall a perpendicular), as the fine of the angle (CAD) at the vertex, to the fine of the difference of the angles at the bafe.

PROP. V.

In any plane triangle ABC, it will be, as the fum of the two fides plus the bafe, is to the fum of the two fides minus the bafe, fo is the co-tangent of half either angle at the bafe, to the tangent of half the other angle at the bafe.

ABC),

2

that is, AC + BC+AB: AC+BC

AB:: co-tang. A tang. ABC. Q. E. D.'

PROP. VI.

In any plane triangle ABC, it will be, as the base plus the difference of the two fides, is to the base minus the fame difference, fo is the tangent of half the greater angle at the bafe, to the tangent of half the leffer.