Page images
PDF
EPUB

of their tables, the chords are expressed in sexagesimals of the radius, and the arches in sexagesimals of the circumference. They had therefore two measures, and two units; one for the circumference, and another for the diameter. The Hindu mathematicians, again, have but one measure and one unit for both, viz. a minute of a degree, or one of those parts whereof the circumference contains 21,600. From this identity of measures, they derive no inconsiderable advantage in many calculations, though it must be confessed, that the measuring of a straight line, the radius, or diameter of a circle, by parts of a curve line, namely, the circumference, is a refinement not at all obvious, and has probably been suggested to them by some very particular view, which they have taken, of the nature and properties of the circle. As to the accuracy of the measure here assigned to the radius, viz. 3,438 of the parts of which the circumference contains 21,600, it is as great as can be attained, without taking in smaller · divisions than minutes, or 60ths of a de

gree. It is true to the nearest minute, and this is all the exactness aimed at in these trigonometrical tables. It must not however be supposed, that the author of them meant to assert, that the circumference is to the radius, either accurately or even very nearly, as 21,600 to 3,438. I have shewn, in another place, from the Institutes of Akber, that the Brahmins knew the ratio of the diameter to the circumference to great exactness, and supposed it to be that of 1 to 3.1416, which is much nearer than the preceding. Calculating, as we may suppose, by this or some other proportion, not less exact, the authors of the tables found, that the radius contained in truth 3437′ 44" 48", &c.; and as the fraction of a minute is here more than a half, they took, as their constant custom is, the integer next above, and called the radius 3438 minutes. The method by which they came to such an accurate knowledge of the ratio of the dia

[ocr errors][merged small]

meter to the circumference, may

founded on the same theorems which were subservient to the construction of their gonometrical tables."

These tables are two, the one of sizes and the other of versed sizes. The size of an arch they call eramajon, er farindo, and the versed sine uttromaiya, They also make use of the cosine or bhujajua. These terms seem all to be derived from the word ja which signifies the chord of an arch, from which the name of the radius, or sine of 90a, viz, trijya, is also taken. This regularity in their trigonometrical language, is a circumstance not unworthy of remark. But what is of more consequence to be observed. is, that the use of sines, as it was unknown to the Greeks, who calculated by help of the chords, forms a striking difference between the Indian trigonometr and theirs. The use of the sine, instead of the chord. is an improvement which our modern trigonometry owes, as we have hitherto been taught to believe. to the Arabs. But whether the Arabs are the

authors of this invention, or whether they themselves received it, as they did the numerical characters. from India, is a question. which a more perfect knowledge of Hindă literature will probably enable us to resolve."

No mention is made in this trigonometry. of tangents or secants: a circumstance not wonderful. when we consider that the use of these was introduced in Europe no longer ago than the middle of the sixteenth century, It is, on the other hand, not a little singular, that we should find a table of versed sines in the Surya Siddhanta; for neither the Greek nor the Arabian mathematicians, had any such."

After giving an ample explanation of the tables, and the mode of calculating by them, Mr. Playfair says: Now, it is worth remarking, that this property of the table of sines, which has been so long known in the east, was not observed by the mathematicians of Europe till about two hundred years ago. The theorem, indeed, concerning the circle, from which it is deduced,

under one shape or another, has been known from an early period, and may be traced up to the writings of Euclid, where a proposition nearly related to it forms the 67th of the Data: If a straight line be drawn cithin a circle given in magnitude, cutting 67 a segment containing a ghen angle, and if the angle in the segment be bisected by a straight kine produced till it meet the circumference straight lines, which contain the ghen angle shall both of them together have a giten

to the straight line which bisects the angle.

It

This is not precisely the same with the theorem which has been shewn to be the foundation of the Hinda rule, but differs from it only by afirming a certain relation to hold among the chords of arches, which the other affirms to hold of their sines. is given by Euclid as useful for the construction of geometrical problems; and trigonometry being then unknown. he probably did not think of any other applica tion of it. But what may seem extraordinary is, that when, about 400 years after

« PreviousContinue »