*Excerpts from Dharampalji's book Indian Science and Technology in the 18th Century (1971)*

These tables go far back into antiquity. Their epoch coincides with the famous era of the Calyougham, that is with the beginning of the year 3102 before Christ. (In order to calculate for a given time, the place of any of the celestial bodies, three things are requisite. The first is, the point of the body in some past instant of time, ascertained by observation; and this instant, from which every calculation must set out, is usually called the epoch of the tables.) When the Brahmins of Tirvalore would calculate the place of the Sun for a given time, they begin by reducing into days the interval between the time, and the commencement of Calyougham, multiplying the years by 365 days, 6 hours, 12 minutes, 30 seconds; and taking away 2 days, 3 hours, 32 minutes, 30 seconds, the astronomical epoch having begun that much later than the civil. They next find, by means of certain divisions, when the year current began, or how many days have elapsed since the beginning of it, and then, by the table of the duration of the months, they reduce these days into astronomical months, days, etc. which is the same with the signs, degrees and minutes of the Sun's longitude from the beginning of the zodiac. The Sun's longitude, therefore, is found.

The tables of Tirvalore, however, though they differ in form very much from those formerly described, agree with them perfectly in many of their elements. They suppose the same length of the year, the same mean motions, and the same inequalities of the Sun and Moon and they are adapted nearly to the same meridian. But a circumstance in which they seem to differ materially from the rest is, the antiquity of the epoch from which they take their date, the year 3102 before the Christian era. We must, therefore, enquire, whether this epoch is real or fictitious, that is, whether it has been determined by actual observation, or has been calculated from the modern epochs of the other tables.. For it may naturally be supposed, that the Brahmins having made observations in later times or having borrowed from the astronomical knowledge of other nations, have imagined to themselves a fictitious epoch, coinciding with the celebrated era of the Calyougham, to which through vanity or superstition, they have referred the places of the heavenly bodies and have only calculated what they pretend that their ancestors observed.

In doing this, however, the Brahmins must have furnished us with means, almost infallible, of detaching their imposture. It is only for astronomy, in its most perfect state, to go back to the distance of forty-six centuries and unto ascertain the situation of the heavenly body yet so remote a period. The modern astronomy of Europe, with all the accuracy that it derives from the telescope and the pendulum could not venture on so difficult a task, were it not assisted by the theory of gravitation, and had not the integral calculus, after a100 years of almost continual improvement, been able, at last, to determine the disturbances in our system, which arise from the action of the planets on one another.

Unless the corrections for these disturbances be taken into account, any system of astronomical tables, however accurate at the time of its formation, and however diligently copied from the heavens, will be found less exact for every instant, either before or after that time, and will continually diverge more and more from the truth, both for future and past ages. Indeed, this will happen not only from the neglect of these corrections, but also from small errors unavoidably committed in determining the main motions, which must accumulate with time and produce an effect that becomes every day more sensible, as we

retire, on either side, from the instant of observation. For both reasons it may be established as a maxim, that if there be given a system of astronomical tables, founded on observations of an unknown date, that date may be found, by taking the time when the tables represent the celestial motions most exactly.The moon's mean place for the beginning of the Calyougham (that is, for midnight between the 17th and 18th of February 3102, B.C. at Benaras), calculated from Mayer's tables, on the supposition that her motion has always been at the same rate as at the beginning of the present century, is 10s, 0 degrees, 51 minutes, 16 seconds. But, according to the same astronomer, the moon is subjected to a small, but uniform acceleration, such, that her angular motion, in any one age, is nine seconds greater than in the preceding, which, in an interval of 4801 years, must have amounted to 5 degrees, 45 minutes, 44

seconds. This must be added to the preceding, to give the real mean place of the moon, at the astronomical epoch of the Calyougham, which is therefore 10 seconds, 6 degrees, 37 minutes. Now, the same, by the Tables of Tirvalore, is 10s, 6 degrees, 0 minutes; the difference is less than two thirds of a degree, which for so remote a period, and considering that acceleration of the moon's motion for which no allowance could be made in an Indian calculation, is a degree of accuracy that nothing but actual; observation could have produced.

seconds. This must be added to the preceding, to give the real mean place of the moon, at the astronomical epoch of the Calyougham, which is therefore 10 seconds, 6 degrees, 37 minutes. Now, the same, by the Tables of Tirvalore, is 10s, 6 degrees, 0 minutes; the difference is less than two thirds of a degree, which for so remote a period, and considering that acceleration of the moon's motion for which no allowance could be made in an Indian calculation, is a degree of accuracy that nothing but actual; observation could have produced.

Thus have we enumerated no less than nine astronomical elements ( the inequality of 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 main motion of both these planets) to which the tables of India assigns 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 era. 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 Calyougham, 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 supposed invariable, and of which, had they supposed them to change, they had no rules to go by 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 no less 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 heavens, at the age in which they lived, have succeeded in describing one which took place several thousand years before they were born.

their tables to so remarkable an epoch as the Calyougham, 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 supposed invariable, and of which, had they supposed them to change, they had no rules to go by 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 no less 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 heavens, at the age in which they lived, have succeeded in describing one which took place several thousand years before they were born.

In another part of the calculation of eclipses, a direct application is made of one of the most remarkable propositions in geometry. In order to have the semiduration of a solar eclipse, they subtract from the square of the sum of the semi-diameters of the sun and moon, the square of a certain line, which is a perpendicular from the centre of the sun on the path of the moon; and from the reminder, they extract the square root, which is the measure of the semiduration. The same thing is practised in lunar eclipses. These operations are all founded on a very distinct conception of what happens in the case of an

eclipse, and on a knowledge of this theorem, that in a right-angled triangle, the square of the hypotenuse is equal to the squares of the other two sides. It is curious to find the theorem of Pythagoras in India, where, for aught we know, it may have been discovered, and from whence that philosopher may have derived some of the solid, as well as visionary speculations, with which he delighted to instruct or amuse his disciples.

eclipse, and on a knowledge of this theorem, that in a right-angled triangle, the square of the hypotenuse is equal to the squares of the other two sides. It is curious to find the theorem of Pythagoras in India, where, for aught we know, it may have been discovered, and from whence that philosopher may have derived some of the solid, as well as visionary speculations, with which he delighted to instruct or amuse his disciples.

The preceding calculations must have required the assistance of many subsidiary tables, of which no trace has yet been 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 would find it impossible to discover from them exactly, on account of 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 Ayeen Akbary, where we

are told, that the Hindoos suppose the diameter of a circle to be to its circumference as 1250 to 3927, and where the author, who knew that this was more accurate than the proportion of Archimedes (7 to 22), and 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.

are told, that the Hindoos suppose the diameter of a circle to be to its circumference as 1250 to 3927, and where the author, who knew that this was more accurate than the proportion of Archimedes (7 to 22), and 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 near approach to the quadrature of the circle; it differs little from that of Metius, 113 to 355, and is the same with one equally remarkable that of 1 to 3.1416. When found in the simplest and most elementary way, it requires a polygon of 768 size to be inscribed in a circle; an operation which cannot be automatically performed without the knowledge of some very curious properties of that curve, and, at least, nine extractions of the square root, each as far as ten places of decimals. All this must have been accomplished in India; for it is to be observed, that the above mentioned proportion cannot have been received from the mathematicians of the West. The Greeks left nothing on this subject more accurate than the theorem of Archimedes; and the Arabian mathematicians seem not to have attempted any nearer approximation. The geometry of modern Europe can much less be regarded as the source of this knowledge. Metius and Vieta (1540-1603 AD) were the first, who, in the quadrature of the circle, surpassed the accuracy of Archimedes; and they flourished at the very time when the Institutes of Akbar were collected to India.

*On the grounds which have been explained, the following general conclusions appear to be established.*

*1. The observations, on which the astronomy of India is founded, were made more than 3000 years before the Christian era; and, in particular, the places of the Sun and Moon at the beginning of the Calyougham, were determined by actual observation.*

This follows from the exact agreement of the radical places in the tables of Tirvalore, with those deduced for the same epoch from the tables of De La Caille and Mayer, and especially in the case of the moon, when regard is had to her acceleration.

Of such high antiquity, therefore, must we suppose the origin of this astronomy, unless we can believe, that all the coincidences which have been enumerated, are but the effects of chance or what is still more wonderful, that, some ages ago, there had arisen a Newton among the Brahmins, to discover that universal principle which connects not only the most distant regions of space, but the most remote periods of duration; and a De La Grange, to trace, through the immensity of both its most subtle and complicated operations.

[Dharampal notes that "the widespread prevalence of European ethnocentric bias" comes through in the above review by J. Playfair, professor of mathematics at the University of Edinburgh: "It became intellectually easier for him to concede this astronomy's antiquity rather than its sophistication and the scientific capacities of its underlying theories."]

*By K. Kesava Rao*

Department of Chemical Engineering

Indian Institute of Science

Bengaluru 560012, India

kesava@iisc.ac.in

Department of Chemical Engineering

Indian Institute of Science

Bengaluru 560012, India

kesava@iisc.ac.in

*...To be continued ...*

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