the rectangle under the tangents of two arcs, is to the square of the radius, so is the difference of their tangents to the tangent of the difference of the arcs. If A and B denote the two arcs, then R2+tanA tan B: R2: :tan A-tan B: tan(A-B.) Cor. 2. Let the two arcs be equal; and R2-tan A2: R2:: 2tanA: tan 2A. Cor. 3. Let the greater arc contain 45o, whose tangent is equal to the radius, then R2 R.tanB : R2 :: R±tanB: tan(45°B), or Rtan B : R±tanB:: R: tan(45° B). Schalium. Assuming the radius equal to unit, expressions are hence easily derived for the tangents of multiple arcs. Let t denote the tangent of an arca; then 1-t2 : 1 : : 2t : These formulæ might also be derived from expressions for the sine and cosine of the multiple arc which involve the powers of the tangent. Thus, from (1), sin 2a = 2cs = = 3c2s-(1-c2)s = (2) S =c2.2t, and sin 3a $3 = 4c2s-s 3(35 3)=c(3t-t'); =c3 (3t-t3); again, from (2), cos 2a=2c2-1= C 1 = c2(1-t), and cos3a=4c3-8c= c3-3C(1-c2)=c3 (1-3)=c(1-3t*). In this way, the following tables are formed: Sin 2ac2.2t. Sin 3a = c3(3t-t3). (8.) Sin 4a = c4(4t-4t3). Sin 5a = c(5t-10t3 +t5). Sin 6a = c(6t-20t3+6t5), &c. &c. &c. Cos 2a = c2(1-t). Cos 3a = c3(1-3t). (9.) Cos 4a = c4(1-6t2+t4). Cos 5a = c(1-10t2+5t4). Cos 6a = c(1-15t2 + 15t4-to), The first set of expressions being divided by the second, will evidently give the same results for the tangent of the multiple arc. PROP. VI. THEOR. The supplemental chord of half an arc, is a mean proportional between the radius, and the sum of the diameter and the supplemental chord of the whole arc. This property, which is only a modification of cor. 2. to Pr. 2. will admit of a more direct demonstration. For draw the chord AB, the semichords AE and BE, and the supplemental chords CB and stand on equal arcs AE and EB; consequently AE : AB :: CO: CE. But, ACBE being a quadrilateral figure contained in a circle, CE.AB = AE.CB+EB.CA = AE (CA+CB), or AE: AB:: CE: CA+CB; wherefore CO: CE:: CE: CA+CB, or CE=CA(CA+CB). Cor. Hence, in small arcs, the ratio of the sine to the arc approaches that of equality. For, let the semiarcs AE and EB be again bisected in the points F and G; and, continuing their subdivision indefinitely, let the successive intermediate chords be drawn. The ratio of the sine BD to the arc AB may be viewed as compounded of the ratio of BD to the chord AB, of that of AB to the two chords AE and EB, of that of AE and EB to the four chords AF, FE, EG, and GB, and so forth. But these ratios, it has been shown, are the same respectively as those of the supplemental chords CB, CE, CF, &c. to the diameter CA. And since each of the ratios CB: СА, СЕ: СА, CF: CA, &c. approaches to equality, it is evident that their compounded ratio, or that of the sine to its corresponding arc, must also approach to equality. Scholium. Hence the ratio of the sine BD to the arc AB is expressed numerically, by the ratio of the continued product of the series of supplemental chords CB, CE, CF, &c. to the relative continued power of the diameter CA. The ratio may, therefore, be determined to any degree of exactness, by the repeated application of the proposition in computing those derivative chords. But a very convenient approximation is more readily assigned. Make CD to CI as CB to CA, CI to CK as CE to CA, CK to CL as CF to CA, and so forth, tending always towards the limit Z; then the ratio of CD to CZ, being compounded of these ratios, must express the ratio of the sine BD to its corresponding arc AB. Now CD:CB:: CB: CA; consequently CI=CB, and CD: CI:: CI: CA, or the point I nearly bisects DA. Again, CA+CB ), and therefore CE differs from CA, by nearly the fourth part of the difference between CB and CA. These differences being small in comparison of the quantities themselves, the series of supplemental chords may be considered as forming a regular progression, each succeeding term of which approaches four times nearer to the length of the diameter. Wherefore IK=DI, KLIK, and so continually. But (V. 21. El.) as the difference between the first and second term, is to the first, so is the difference between the first and last term, or DI itself, to the sum of all the terms, or the extreme limit DZ; that is, 3:4:: DI: DZ; and consequently DZ=DA. The ratio of the sine BD to the arc AB is, therefore, nearly that of CD to CD+DA, or of 3CD to CD+2CA. CE CA ( 2 This approximation may be differently modified. Since 3CD=6OA-3DA, and CD+2AC=6OA-DA, it follows that BD is to AB, as 60A-3DA to 6OA-DA. But this ratio, which approaches to equality, will not be sensibly affected, by annexing or taking away equal small differences. Whence the sine is to the arc, as 60A-6DA to 60A-4DA, or 3OD to OA+2OD. But OD is to OA, as the sine of AB is to its tangent; and consequently the triple of that arc is equal to its tangent together with twice its sine. Again, both terms of the ratio increased by the minute difference DA become 60A-2DA, and 60A; wherefore BD is to AB, as 30A-DA to 30A, or as 20C+OD to 3CO. Hence, if CP be ly equal to the intercepted portion AH. For BD:AH:: PD: PA, or 2OC+OD:3OC; that is, as the sine BD is to its arc AB. Another approximation, of much higher importance, may be hence derived; for PD : PA : : BD : AH, or as the sine to its arc nearly. But (V. 3. El.) PD.CD is to PA.CD in the same ratio, and PA.CD= PD.CD + AD.CD = (III. 26. cor. 1.) PD.CD+BD2; whence PD.CD is to PD.CD+BD2, as the sine to its arc nearly. If the arc be small, it is evident that OD will be very nearly equal to AO, and consequently PD may be assumed equal to 3AO, and CD equal to 2AO. Wherefore 6AO: 6AO2 + BD2 : : BD: AB nearly; or, the radius being unit, and a and s denoting a small arc and its |