> COR. Hence, if the point B coincides with A, R: cof BC: fin BC: fin BD, or the radius is to the cofine of any arch, as the fine of the arch is to half the fine of twice the arch; or if any arch A›1⁄2 fin 2A fin A xcof A, or fin 2A 2fin A x cof A. Therefore alfo, fin 2' 2 fin 1' x cof 1'; fo that from the fine and co-fine of one minute the fine of 2' is found. Again, 1, 2, 3' being three fuch arches that the difference between the first and fecond is the fame as between the fecond and third, R: cof 1':: fin 2': (fin1' + fin 3′), or fin 1'+fin3'= 2 cof 1' x fin 2', and taking fin 1' from both, fin 3' fin 2'-fin I. In like manner, fin 4' fin 5' fin 6' 2 cof 1'x fin 3' — fin 2', 2 cof 1' x fin 4'— fin 3', 2 cof 1' x fin 5'-fin 4', &c. Thus a table containing the fines for every minute of the quadrant may be computed; and as the multiplier, cof 1' remains always the fame, the calculation is easy. For, computing the fines of arches that differ by more than I', the method is the fame. Let A, A+B, A+ 2B be three fuch arches, then, by this theorem, R: cof B :: fin (A+B) : (fin A+ fin (A+2B)); and therefore making the radius 1. fin A+fin (A+ 2B) = 2 cof B x fin (A+B), or fin (A+ 2B) = 2 cof B x fin (A+B) — fin A. By means of these theorems, a table of the fines, and confequently alfo of the co-fines, of arches of any number of degrees and minutes, from 0 to 90, may be conftructed. Then, the table of tangents is computed because tan A= fin A cof A' by dividing the fine of any arch by the co-fine of the fame arch. When the tangents have been found in this manner as far as 45°, the tangents for the other half of the quadrant may be found more eafily by another rule. For the tangent of an arch above 45° being the co-tangent of an arch as much under 45°; and the radius being a mean proportional between the tangent and co-tangent of any arch, (1. Cor. def. g.), it follows, if the difference between any arch and 45° be called D, that that tan (45°-D): 1 :: 1: tan (45°+D), fo that tan (45°+D) 1: Laftly, the fecants are calculated from Cor. 2. def. 9. where it is fhew that the radius is a mean proportional between the b co-fine and the fecant of any arch, fo that if A be any arch, The verfed fines are found by fubtracting the co-fines from the radius. 5. The preceding Theorem is one of four, which, when arithmetically expreffed, are frequently used in the application of trigonometry to the folution of the more difficult problems. imo, If in the last Theorem, the arch ACA, BC=B, and EC1, then AD A+B, and ABA-B; and by what has just been demonftrated, 1 1: cof B: fin A: fin (A+B) + 1⁄2 fin (A—B), : and therefore, 2 fin A x cof B = 1⁄2 fin (A+B) + 1⁄2 fin (A—B). 2do, Because BF, IK, DH are parallel, the ftraight lines BD and FH are cut proportionally, and therefore FH, the difference of the straight lines FE and HE, is bifected in K; and therefore, as was fhewn in the last Theorem, KE is half the fum of FE and HE, that is, of the co-fines of the arches AB and AD. But because of the fimilar triangles EGC, EKI, EC: EI :: GE : EK; now, GE is the co-fine of AC, therefore, R: cof BC :: cof AC: cof AD + 1⁄2 cof AB, or I : cof B :: cof A: cof (A+B) + 1⁄2 cos (A—B); and therefore, cof A x cof B 1⁄2 cof (A+B) + 1⁄2 cof (A—B). 3tio, Again, the triangles IDM, CEG are equiangular, for the angles KIM, EID are equal, being each of them right angles, and therefore, taking away the angle EIM, the angle DIM is equal to the angle EIK, that is to the angle ECG; and the angles DMI, CĞE are alfo equal, being both right Y 2 angles, angles, and therefore, the triangles IDM, CGE have the fides about their equal angles proportionals, and confequently, EC: CG:: DI: IM; now, IM is half the difference of the co-fines FE and EH, therefore, R: fin AC: fin BC: cof AB- cof AD, cof AB or I : fin A : fin B : : 1⁄2 cof (A—B)— 1⁄2 cos (A+B); fin A x fin B and also, cof (A—B) — — cos (A+B). 4to, Laftly, In the fame triangles ECG, DIM, EC: EG:: ID: DM; now, DM is half the difference of the fines DH and BF, therefore, 1 R: cof AC:: fin BC: fin AD - fin AB. or I : cof A::fin B : 1⁄2 fin (A+B)— 1⁄2 fin (A—B), and therefore, cof A x fin B fin (A+B) — fin (A—B). 6. If therefore A and B be any two arches whatsoever, the radius being supposed 1; 1 I. fin Ax cof B1⁄2 fin (A+B) + 1⁄2 fin (A—B). = II. cof A x cof B = 1⁄2 cof (A—B) + 1⁄2 cof (A+B). III. fin A x fin B1⁄2 cof (A-B)— 1⁄2 cof (A+B). IV. cof A x fin B fin (A+B) — ; fin (A—B). From these four Theorems are alfo deduced other four. For, by adding the first and fourth together, fin A x cof B + cof A x fin B= fin (A+B). Alfo, by taking the fourth from the first. fin Ax cof B- cof Ax fin B➡fin (A — B), Again, Again, adding the second and third, cof A x cof B + fin A x fin B = cof (A — B.); And, lastly, fubtracting the third from the fecond, cof A x cof B-fin Ax fin B cof (A + B). 7. Again, fince by the 1ft of the above theorems, fin Ax cof B = 1⁄2-1⁄2 fin (A + B) + 1⁄2 fin (A—B), if A+B=S; fin D. But as S and D may be any arches whatever, to preserve the former notation, they may be called A and B, which also exprefs any arches whatever : thus, In the fame manner, from Theor. 2. is derived A. In all thefe Theorems, the arch B is fuppofed less than 8. Theorems of the fame kind with respect to the tangents of arches may be deduced from the preceding. Because the tangent of any arch is equal to the fine of the arch Y 3 divided fin (A + B) But it has divided by its co-fine, tan (A + B)= cof (A+B)* 9. If the theorem demonftrated in Prop. 3. be expreffed in the fame manner with thofe above, it gives 10. In all the preceding theorems, R, the radius is fuppofed 1, becaufe in this way the propofitions are most concifely expreffed, and are alfo moft readily applied to trigonometrical calculation. But if it be required to enunciate any of them geometrically, the multiplier R, which has disappeared, by being made 1, must be restored, and it will always be evident from infpection in what terms this multiplier is wanting. Thus, Theor. 1., 2fin Axcof Bfin (A+B) +fin (A-B), is a true propofition, taken arithmetical ly i |