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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+E, which is-2versBsinA+ (sinA-sin(A-B)) + sinA. 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

22 1 11 7'48

233

11

168, and consequently twice its versed sine =()

168

2=

(1) in approximate terms. If the successive sines, corresponding to the division of the quadrant into 24 equal parts, be therefore continually multiplied by the fraction, or divided by the number 233, the quotients thence 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.

It will be seen that the number 225, which expresses the length of the component arc, and therefore 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 integral quotients, however, are deferred; that is, the nearest whole number in advance is not always ta

ken. Thus the quotient of 1315 by 225, is 5

38

45'

which ap

proaches nearer to 6, and yet 5 is still retained. These efforts to redress the errors of computation are marked with asterisks.

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Each of the three composite columns, we may observe, 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 quadrants, 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 natives 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.-This opinion seems to derive strong confirmation from the Lilawati, a very meagre and defective practical treatise of arithmetic and geometry, which I had some time since an opportunity of examining, with the kind assistance of the learned Dr Wilkins, at the library of the India House. Of that singular performance, a translation from the original Sanscrit by Dr John Taylor, printed at the expence of the Literary Society at Bombay, has just reached us, and will enable the European mathematicians, who are acquainted with the state of science at the revival of letters in Italy, to reduce the lofty pretensions of the Brahmins to their just level. They will perceive the utter nakedness of a sys ́tem, which, in the language of ignorance and oriental exaggeration, the Hindus represented as endued with a sort of magical virtue, that would enable the person who understands it "to tell, in the twinkling of an eye, the number of leaves on a tree, or of blades of grass in a meadow, or the number of grains of sand on the sea shore."

The principles before stated 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 is nearly

11

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versed sine is(- )2, or, in round numbers,

112

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sufficiently accurate, therefore, to employ 100 for the constant divisor. The sine of the first 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.

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The errors occasioned by neglecting the fractions accumu. late at first, but afterwards gradually diminish, from the effect of compensation. The greatest deviation takes place, as might be expected, at the middle arc, whose sine is 707 instead of 717. Reckoning the error in excess as limited by 10, and declining uniformly on each side, the correct sines are finally deduced. The numbers thus obtained seldom differ, by the thousandth part, from the truth, and are hence far more accurate than the practice of navigation ever requires. This simple and expeditious mode of forming the sines is not merely an object of curiosity, but may be deemed of very consider

able importance, as it will enable the mariner, altogether independent of the aid of books, to the loss of which he is often exposed by the hazards of the sea, to construct a table of departure and difference of latitude, sufficiently accurate for every real purpose.

12. In trigonometrical investigations, it is often requisite to determine the proportion which the difference of an arc bears to that of its related lines. With this view, let ▲ denote the increment or finite difference of the quantity to which it is prefixed.

1. In art. 29. of NO. 3. change A into A+AA, and B into A; then will

Asin A=2sin▲Acos(A+▲A).

2. Make the same change in art. 31. of that number, and AcosА=—2sin▲Asin(A+‡^A),

3. In art. 2. of NO. 7. let a similar change be made, and sinA A

Atan A.

4

cos Acos(A+AA)

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5. In art 22. of NO. 3. make a like substitution, and

Asin Asin▲Asin(2A+▲A).

6. Let the same change be made in art 23., and

Acos Asin Asin(2A+▲A).

7. Do the same thing in art. 16. of NO. 7. and

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s. Lastly, let a similar change be made in art. 17. of that number, and

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If the differences be conceived to diminish indefinitely and pass into differentials, these expressions, in coming to denote only limiting ratios, will drop their excrescences and acquire a much simpler form. Thus, adopting the characteristic d

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