principles: For the successive differences of the sines of the arcs A-B, A, and A+B, are S,A—S,A—B, and S,A+B-S,A; and consequently the difference between these again, or the second difference of the sines, is S,A+B+S,A—B—2S,A=(Prop. 3. cor. 2. T.) — 2VS,B x S,A. The second differences of the progressive sines are hence subtractive, and always proportional to the sines themselves. Wherefore the sines may be deduced from their second differences, by reversing the usual process, and recompounding their separate elements. Thus, the sines of A-B and A being already known, their second and descending difference, as it is thus derived from the sine of A, will combine to form the succeeding sine of A+B, which is −2VS,B× S,A+(S,A−S,A−B) +S,A. It only remains then, to determine, in any trigonometrical system, the constant multiplier of the sine, or twice the versed sine of the component arc. Suppose the quadrant to be divided into 24 equal parts, each containing 3° 45', or 225'. The length of this arc is nearly 168 I twice its versed sine = the successive sines, corresponding to the division of the quadrant into 24 equal parts, be therefore continually multiplied by the == in approximate terms. If 233 I fraction or divided by the number 233, the quotients thence 233, arising will represent their second differences. But, since 233 is nearly equal to 225, or the length in minutes of the primary or component arc, and which differs not sensibly from its sine,-this last may be assumed as the divisor, the small aberration so produced being corrected by deferring the integral quotients. In this way, the following Table is constructed : The number 225, which expresses the length of the compo nent arc, and consequently represents very nearly its sine, is here employed as the constant divisor. Thus, 225, divided by 225, gives a quotient 1, and this, subtracted from 225, leaves 224, which, being joined to 225, forms 449, the sine of the second arc. Again, 449 divided by 225, gives 2 for its integral quotient, which taken from 224, leaves 222; and this, added to 449, makes 671, the sine of the third arc. In this way, the sines are successively formed, till the quadrant is completed. The inte gral quotients, however, are deferred; that is, the nearest whole number in advance is not always taken. Thus the quotient of 38 1315 by 225, is 5, which approaches nearer to 6, and yet 5 is still retained. These efforts to redress the errors of computation are marked with asterisks. It should be observed, that each of the 'three composite columns really forms a recurring series. In the second quadrant, the first differences become subtractive, and the same numbers for the sines are repeated in an inverted order. By continuing the process, these sines are reproduced in the third and fourth qua drants, only on the opposite side. Such is the detailed explication of that very ingenious mode which, in certain cases, the Hindu astronomers employ, for constructing the table of approximate sines. But, ignorant totally of the principles of the operation, those humble calculators are content to follow blindly a slavish routine. The Brahmins must, therefore, have derived such information from people farther advanced than themselves in science, and of a bolder and more inventive genius, Whatever may be the pretensions of that passive race, their knowledge of trigonometrical computation has no solid claim to any high antiquity. It was probably, before the revival of letters in Europe, carried to the East, by the tide of victory. The na tives of Hindustan might receive instruction from the Persian astronomers, who were themselves taught by the Greeks of Constantinople, and stimulated to those scientific pursuits by the skill and liberality of their Arabian conquerors. The same principles lead to an elegant construction of the approximate sines, entirely adapted to the decimal scale of numeration, and the nautical division of the circle. Suppose a quadrant to contain 16 equal parts, or half points; the length of each arc and consequently twice its versed sine is 22 I is nearly X II I ', or, in round numbers, 102 It will be sufficiently accu The rate, therefore, to employ 100 for the constant divisor. sine of the first arc being likewise expressed by 100, let the nearer integral quotients be always retained, and the sine of the whole quadrant, or the radius itself, will come out exactly 1000. The first term being divided by 100 gives 1 for the second difference, which, subtracted from 100, leaves 99 for the first difference, and this joined to 100, forms the second term. Again, dividing 199 by 100, the quotient 2 is the second difference, which, taken from 99, leaves 97 for the first difference, and this, added to 199, gives the third term. In like manner, the rest of the terms are found. |