COROLLÁRY II. 1 Hence, the fine of the double of either arch CS (when they are equal) will be = and its co R : whence it appears, that the fine of R the double of any arch, is equal to twice the rečtangle of the fine and co-fine of the single arch, divided by radius; and that its co-fine is equal to the difference. of the squares of the fine and co-fine of the single arch, also, divided by radius. , COROLLARY Ill. * Moreover, because Dm x CF=OC x mv (IOC * EGOCX OE-OG); and On ® OF=OCX On ({ OC x 20n = {OCXOE + OG), it follows, that the retangle of the fines of any two crches. (AC, CD.(BC) is equal to a rectangle under half the radius, and the difference of the co fines, of the sum and difference of those arcbes; and that the rectangle of their co-fines is equal to a refiangle under half the radius, and the sum of the co-fines, of the sum and dijerence of the same arches. PROP. III. The tangents of two arches being given, to find the tangents of the Jum, and difference, of ihoje arches. B M Let AN and AM be the two arches, and AB B and AC their N F M tangents; i alfo N let NE be the tangent of their fum, in the first D D = by COROLLARY. Hence, if radius be supposed unity, and the tangents of two arches be denoted by T and t, it follows, that the tangent of their sum will be = and the tangent of their difference = T tt I But, But, it will be proper to take notice here (once for all) that, if in these, or any other theorems, the tangent, fecant, co-fine. co-tangent, &c. of an arch greater than go, degrees be concerned; then, instead thereof, the tangent, fecant, co-fine, &c. of an arch, as much below go degrees, is to be taken, with a negative sign; according to the observation in page 5. Thus, for instance, let BA be an arch greater than go', and let the tangent of the sum of AB and AC be required; B fupposing T to represent the tangent of AD (the supplement of AB) and t the tangent of AC : then, by writing T instead of T, in the first of the foregoing theorems, we -T+ -T+ shall have tang. of BC = 1-tx-T I+tT TE and therefore tang. DC (--tang. BC) = It iT which is the very theorem demonstrated in the 2d case. D PROP. IV. As the sum of the tangents of any two. angles BAC, BAD, is to their difference , so is the fine of the sum of those angles, to the five of their difference. Let: А AB; Let BC and BD, be the two pro. posed tangents, to the radius take Bd. = BD, join Ad, and draw E DE and dF perpendicular to AC. B It is manifeft, because Bd = BD, that Ad = 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): Cd (CBBD) :: DE:df; but DE and df are fines of DAE and DAF to the equal radii AD and A2: whence the truth of the proposition is manifest. COROLLARY. Hence it also appears, that the base (CD) of a plane triangle, is to (Cd) the difference of its two legments (made by letting fall a perpendicular), as the line of the angle (CAD) at the vertex, to the line of the difference of the angles at the base, PROP. V. in any plane triangle ABC, it will be, as the sum of the two fides plus the base, is to the fum of the two fides minus the base, fo is the co-tangent of half either angle at the base, to the tangent of balf the other angle at the base. In ABC) that is, AC + BC + AB : AC + BC Ас +AB :: co-tang. A: tang. WABC. 2. E. D. PROP. VI. In any plane triangle ABC, it will be, as the base plus the difference of the two sides, is to the base ininus the same difference, so is the tangent of half the greater angle at the base, to the tangent of balf the lefser. In the lesser Gide CA, produced, take CD = CB, so that AD may be the difference of the two fides; and let BD be B drawn: then it is mani felt that the angle CBD will be equal to D: but (ly Theor. 5. p. $.) AB + AD: AB-AD: : D+ DBA CAB tangent (ly 9, 1,).: tangent 2 2.E.D. 2 2 2 Prop. |