two series whose law it is easy to discover, and by means of which the sine and cosine of a multiple arc A may be found, in a much speedier manner than by the operations indicated in Art. 24. * XXXIV. Since we have sin A=cos A tang A, these two series may be put under the form Suppose n=; by substituting this value, and still retaining A factor cos" A, we shall have cos x-cos"A(1-2.x-A tang Ax.x-A.x-2A.x-3A tang1 A 1.2 A2 1.2.3.4 A4 &c. In these formulas A may be assumed at pleasure: suppose A very small; then tang A will be very little different from unity, the tangent of a very small are being almost equal to the arc. Yet so long as the arc is not absolutely nothing, we have کے 1 Hence Cos A is always included between the Put A=0; we shall have cos A=1: hence is included between 1 and * AT (see the fig. of page 285) is greater than AM, because the triangle ATC is to the sector ACM: S ATX AC: AM× AC :: AT: AM. +AM (see the same fig.) is greater than MP, because the arc MAN is greater than its chord MN. exactly tang A = 1. Therefore making A=0, we shall have ენ cos x cos" A 1 1-1.2.3.4-1.2.3.4.5.6+ &c.) It remains to inquire what becomes of cos" A, supposing A to diminish more and more, and at length to vanish. Now we have 1 cos2 A =sec2 A=+tang2 A ; hence cos A=(1+tang2 A)—*, 2 n.n+2 hence cos”A=(1+tang2A) ̈13—1—2 tang2A+ tang1A &c. If we now suppose A to diminish more and more, while a remains the same, the value of cos" A will more and more approach to unity; and in fine, if A is made = 0 and shall have exactly cos" A=1. Hence we have the formulas by means of which, we are enabled to calculate the sine and the cosine of an arc, whose length is given, in parts of the radius taken as unit. * XXXV. These same values may be expressed more briefly, by means of exponentials. For this purpose, we must consider thate being the number whose hyperbolic logarithm is 1, we have If, in this formula, we make x=-1, we shall have By changing the sign of -1, we should in like manner have two series, whose second members are the values already found for cos x and sin x. Hence we have ex√1 = cos x—-1 sin x; hence dividing the one by 1--1 tang x' or taking the logarithms of each member, 2x-1=log.√1 tang x) 1+/-1 tang x 1 tang x). But we already know + 2 2 x * that log. (1±±±) = 2 + 3 + 3*+ &c.; hence putting √— tang x instead of %, and dividing both sides by 2 √ — 1, we shall have x=tang x-tang x+} tang x-tang7x+ &c. A very simple formula, which enables us to calculate the arc from its tangent when the latter is less than unity. * XXXVI. In order to apply the preceding formulas for determining the sine and cosine of an arc given in degress and parts of a degree, we must have the length of this arc expressed in parts of the radius, in other words, we must have the ratio of this arc to the radius. Now, the radius being 1, the semicircumference or the arc of 200° 3. 14159 26535 897932. m Put this number = π, the length of the arc 100° will be n ≥ m n 2' ; hence, if in the preceding formulas, we make x = and afterwards insert the value of 7, calculating the coefficients to sixteen places of decimals, we shall have the following formulas: + By merely changing 100° into 90°, this table might also be applied to the nonagesimal scale; though it would be less convenient for obtaining the sines of integer arcs, because n could not then so often be put equal to some power of 10.-ED. The sines and cosines of the arcs between 0° and 50° comprehend the cosines and sines of the arcs between 50° and 100°; for we have sin (50°+%) = cos (50° —z) and cos (50°+%) = sin (50° — %). Hence in the formulas exhibiting the values of sin 100°, and m n the series will converge so rapidly, that we shall never have need to calculate more than a small number of terms, especially when not many decimals are required. sin 80° cos 20° = 0.95105 65162 95154 sin 90° cos 10° = = 0.98768 83405 95138 sin 100° cos 0° 1.00000 00000 00000 which agree with the algebraical formulas of Art. 22. Also as was found in Art. 26; and the extreme facility with which these results are obtained is a proof of the excellence of the method. ON THE CONSTRUCTION OF TABLES OF SINES. * XXXVII. Those useful mathematicians, to whom we are indebted for the first construction of tables of sines, founded their calculations on methods which are ingenious in themselves, but difficult in their application. In subsequent times, Analysis has furnished us with far more expeditious methods of attaining this object; but the calculations being already performed, these methods would have remained without applica |