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the genius of the people, that is diffused so widely among them, and diversified so much, has a right to be regarded, either as a native, or a very ancient inhabitant of the country where it is found.
4thly. The construction of these tables implies a great knowledge of geometry, arithmetic, and even of the theoretical part of astronomy, &c.
But what, without doubt, is to be accounted the greatest refinement, is the hypothesis employed in calculating the equations of the centre for the sun, moon, and planets; viz. that, of a circular orbit having a double eccentricity, or having its centre in the middle, between the earth and the point about which the angular motion is uniform. If to this we add the great extent of geometrical knowledge requisite to combine this and the other principles of their astronomy together, and to deduce from them the just conclusions, the possession of a calculus equivalent to trigonometry, and lastly, their approximation to the quadrature of the circle; we shall be as
tonished at the magnitude of that body of science, which must have enlightened the inhabitants of India in some remote age; and which, whatever it may have communicated to the western nations, appears to have received nothing from them."
Professor Playfair examines the construction of the tables contained in Brahminical trigonometry. After mentioning the circumference and division of the circle, he proceeds: "The next thing to be mentioned, is also a matter of arbitrary arrangement, but one in which the Brahmins follow a method peculiar to themselves. They express the radius of the circle in parts of the circumference, and suppose it equal to 3,438 minutes, or 60ths of a degree. In this they are quite singular. Ptolemy, and the Greek mathematicians, after dividing the circumference, as we have already described, supposed the radius to be divided into sixty equal parts, without seeking to ascertain, in this division, any thing of the relation of the diameter to the circumference: and thus, throughout the whole
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
* Ayeen Akbery.
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