^{page 86 note *} Trans. R. S. Edin. vol. II. p. 185. Phys. Cl.

^{page 86 note †} See Note, § 6.

^{page 97 note *} By such continual bisections, the Hindoo mathematicians, like those of Europe before the invention of infinite series, may have approximated to the ratio of the diameter to the circumference, and found it to be nearly that of 1 to 3.1416 as above observed. A much less degree of geometrical knowledge than they possessed, would inform them, that small arches are nearly equal to their sines, and that the smaller they are, the nearer is this equality to the truth. If, therefore, they assumed the radius equal to 1, or any number at pleasure, after carrying the bisection of the arch of 30, two steps farther than in the above construction, they would find the sine of the 384th part of the circle, which, therefore, multiplied by 384, would nearly be equal to the circumference itself, and would actually give the proportion of 1 to 3.14159, as somewhat greater than that of the diameter to the circumference. By carrying the bisections farther, they might verify this calculation, or estimate the degree of its exactness, and might assume the ratio of 1 to 3.1416 as more simple than that just mentioned, and sufficiently near to the truth.

^{page 101 note *} This seems to me the most probable reason that can be assigned for the measuring of the radius, and the other, straight lines in the circle, in parts of the circumference. It is remarkable that the Hindoos should have been thus led, at so early a period, to put in practice a method, the same in the most material point, with one which has been but lately suggested in Europe as an important improvement in trigonometrical calculation. In the Phil. Trans. for 1783, Dr Hutton of Woolwich proposed to divide the circumference, not into degrees, as is usually done, but into decimals of the radius; and he has pointed out how the present trigonometrical tables might be accommodated to this new division, with the least possible labour, in a paper which displays that intimate acquaintance with the resources, both of the numerical and algebraic calculus, for which he is so much distinguished. His plan is, in one respect, the same with the Hindoo method, for it uses the same unit to express both the circumference and the diameter; in another respect it differs from it, viz. in making the radius the unit, while the other assumes for an unit the 360th part of the circumference. Dr Hutton's plan has never been executed, though it certainly would be of advantage to have, besides the ordinary trigonometrical tables, others constructed according to that plan.

^{page 102 note *} Asiatic Refearches, vol. II. p. III, &c.

^{page 102 note †} The obliquity of the ecliptic is stated at 24° in the Surya Siddhanta, as in all the other astronomical tables of the Hindoos which we are yet acquainted with. (Tranf. R. S. Edin. vol. II. p. 164.) Mr Davis concludes from this, (Asiatic Researches, vol. II. p. 238), that if the obliquity diminish, at the rate of 50″ in a hundred, years, the Surya Siddhanta is at present about 3840 years old, which goes back, nearly 2000 years before the Christian æra. But the diminution of the obliquity of the ecliptic, is supposed considerably too rapid in this calculation. According to Mayer it is 46″ in a century; and according to De la Grange, (Mem. Berlin 1782), at a medium no more than 30″. This last is most to be depended on, as it proceeds on an accurate inquiry into the law of the secular variation of the obliquity, that variation being by no means uniform. Let us however take the mean, viz. 38″, and the obliquity at the beginning of the present century having been 23°. 28′. 41″, we shall have 5000 years for the age of the Surya Siddhanta, reckoned from that date or about 3300 years before Christ, which is near the æra of the Caly Yur.

^{page 105 note *} The sphere of Chiron and Musæus was constructed, according to Newton, about the year 936 before Christ, (Newton's Chron. chap. i. § 30). According to the system generally received, the ancient sphere, described by Eudoxus, was constructed about 1350 years before Christ, (Dr Playfair's Chronology, p. 37). The medium is 1143.