pose, as in the case of the sun's equation, that the observations on which this determination is founded, were made 1200 years before the Kaly-Yug, we shall find that the obliquity of the ecliptic was 23° 57′ 45′′, and that the error of the tables did not much exceed 2'.

Thus do the measures, which the Brahmins assign to these three quantities, the length of the tropical year, the equation of the sun's centre, and the obliquity of the ecliptic, all agree, in referring the epoch of their determination to the year 3102 before our æra, or to a period still more ancient. This coincidence in three elements, altogether independent of one another, cannot be the effect of chance. The difference, with respect to each of them, between their astronomy and ours, might singly, perhaps, be ascribed to inaccuracy; but that three errors, which chance had introduced, should be all of such magnitude as to suit exactly the same hypothesis concerning their origin, is hardly to be conceived. Yet there is no other alterna

tive, but to admit this very improbable supposition, or to acknowledge that the Indian astronomy is as ancient as one or other of the periods abovementioned.

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In seeking for the cause of the secular equations, which modern astronomers have found it necessary to apply to the mean motion of Jupiter and Saturn, M. de la Place has discovered, that there are inequalities belonging to both these planets, arising from their mutual action on one another, which have long periods, one of them no less than 877 years; so that the mean motion must appear different, if it be determined from observations made in different parts of those periods. Now I find' (says he) by my theory, that at the Indian epoch of 3102 years before Christ, the apparent and annual mean motion of Saturn was 12° 13′ 14′′, and the Indian tables make it 12° 13′ 13′′. In like manner, I find that the annual and apparent mean motion of Jupiter at that epoch, was 30° 20′ 42′′, precisely as in the Indian astronomy."

"

Thus have we enumerated no less than

nine astronomical elements,* to which the tables of India assign such values as do by no means belong to them in these later ages, but such as the theory of gravity proves to have belonged to them three thousand years before the Christian æra. At that time, therefore, or in the ages preceding it, the observations must have been made from which these elements were deduced. For it is abundantly evident, that the Brahmins of later times, however willing they might be to adapt their tables to so remarkable an epoch as the Kaly-Yug, could never think of doing so, by substituting, instead of quantities which they had observed, others which they had no reason to believe had ever existed. The elements in question are precisely what these astronomers must have sup

"The inequality or the precession of the equinoxes; the acceleration of the moon; the length of the solar year; the equation of the sun's centre; the obliquity of the ecliptic; the place of Jupiter's aphelion; the equation of Saturn's centre; and the inequalities in the mean motion of both these planets."

posed invariable, and of which, had they supposed them to change, they had no rules to guide them for ascertaining the variations; since to the discovery of these rules is required, not only all the perfection to which astronomy is at this day brought in Europe, but all that which the sciences of motion and of extension have likewise attained. It is equally clear that these coincidences are not the work of accident; for it will scarcely be supposed that chance has adjusted the errors of the Indian astronomy with such singular felicity, that observers, who could not discover the true state of the heavens, at the age in which they lived, have succeeded in describing one which took place several thousand years before they were born.*

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The preceding calculations must have required the assistance of many subsidiary tables, of which no trace has yet been

* See Trans. of the Royal Society of Edinburgh, vol. ii. pp. 169, 170.

found in India,-besides many other geometrical propositions. Some of them also involve the ratio which the diameter of a circle was supposed to bear to its circumference, but which we should find it impossible to discover from them exactly, on account of the small quantities that may have been neglected in their calculations. Fortunately, we can arrive at this knowledge, which is very material when the progress of geometry is to be estimated, from a passage in the Ayin Akbery, where we are told that the Hindus suppose the diameter of a circle to be to its circumference, as 1250 to 3927; and where the author, who believed it to be perfectly exact, expresses his astonishment, that among so simple a people, there should be found a truth, which among the wisest and most learned nations had been sought for in vain.

The proportion of 1250 to 3927, is indeed a nearer approach to the quadrature of the circle; it differs little from that