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Book 3 – State of the Hindus in Later Times, continued

Few of the subjects which follow are noticed by Menu: we can, therefore, no longer attempt to mark the changes effected since his time, but must endeavour from other sources to trace the rise and describe the present state of each branch of inquiry as it occurs.

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Chapter 1 – Astronomy and Mathematical Science

The antiquity and the originality of the Indian astronomy form subjects of considerable interest469.

Antiquity of the Hindu astronomy

The first point has been discussed by some of the greatest astronomers in Europe; and is still unsettled.

Cassini, Bailly, and Playfair maintain that observations taken upwards of 3000 years before Christ are still extant, and prove a considerable degree of progress already made at that period.

Several men, eminent for science, (among whom are La Place and De Lambre,) deny the authenticity of the observations, and, consequently, the validity of the conclusion.

The argument is conducted entirely on astronomical principles, and can only be decided by astronomers: as far as it can be understood by a person unacquainted with science, it does not appear to authorise an award, to the extent that is claimed, in favour of the Hindus.

All astronomers, however, admit the great antiquity

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of the Hindu observations; and it seems indisputable that the exactness of the mean motions that they have assigned to the sun and moon could only have been attained by a comparison of modern observations with others made in remote antiquity470. Even Mr. Bentley, the most strenuous opponent of the claims of the Hindus, pronounces, in his latest work, that their division of the ecliptic into twenty-seven lunar mansions (which supposes much previous observation) was made 1442 years before our sera; and, without relying upon his authority in this instance, we should be inclined to believe that the Indian observations could not have commenced at a later period than the fifteenth century before Christ. This would be from one to two centuries before the Argonautic expedition and the first mention of astronomy in Greece.

The astronomical rule relating to the calendar, which has been quoted from the Vedas471, is shown to have been drawn up in the fourteenth century before Christ; and Parasara, the first writer on astronomy of whose writings any portion remains, appears to have flourished about the same time472.

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Its extent

In our inquiries into the astronomy of the Indians, we derive no aid from their own early authors. The same system of priestcraft, which has exercised so pernicious an influence on the Hindus in other respects, has cast a veil over their science. Astronomy having been made subservient to the extravagant chronology of the religionists, all the epochs which it ought to determine have been thrown into confusion and uncertainty; no general view of their system has been given; only such parts of science as are required for practical purposes are made known; and even of them the original sources are carefully concealed, and the results communicated as revelations from the Divinity473.

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From this cause, the data from which their tables were computed are never quoted; and there is no record of a regular series of observations among them.

If this system be an obstruction to our inquiries, it must have been much more so to the progress of science. The art of making observations was probably taught to few; still fewer would be disposed to employ an instrument which could not confirm, but might impair, the faith due to divine truths. They had none of the skill which would have been taught, nor of the emulation which would have

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been excited, by the labours of their predecessors; and when the increasing errors of the revealed tables forced them at length on observations and corrections, so far from expecting applause for their improvements, they were obliged, by the state of public opinion, to endeavour to make it appear that no alteration had been made474.

In spite of these disadvantages, they appear to have made considerable advances in astronomy. As they have left no complete system which can be presented in a popular form, and compared with those of other nations, they must be judged of by mathematicians from the skill they have shown in treating the points on which they have touched, The opinions formed on this subject appear to be divided; but it seems to be generally admitted that great marks of imperfection are combined, in their astronomical writings, with proofs of very extraordinary proficiency.

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The progress made in other branches of mathematical knowledge was still more remarkable than in astronomy. In the “Surya Sidhanta,” written, according to Mr. Bentley, in AD 1091, at the latest, but generally assigned to the fifth or sixth century475, is contained a system of trigonometry, which not only goes far beyond any thing known to the Greeks, but involves theorems which were not discovered in Europe till the sixteenth century476.


Their geometrical skill is shown, among other forms, by their demonstrations of various properties of triangles, especially one which expresses the area in the terms of the three sides, and was unknown in Europe till published by Clavius (in the

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sixteenth century)477; and by their knowledge of the proportion of the radius to the circumference of a circle, which they express in a mode peculiar to themselves, by applying one measure and one unit to the radius and circumference. This proportion, which is confirmed by the most approved labours of Europeans, was not known out of India until modern times478.


The Hindus are distinguished in arithmetic by the acknowledged invention of the decimal notation; and it seems to be the possession of this discovery which has given them so great an advantage over the Greeks in the science of numbers479.


But it is in algebra that the Bramins appear to have most excelled their contemporaries. Our accounts of their discoveries in that science are obtained from the works of Brahma Gupta (who lived in the sixth century), and Bhascara Acharya (in the twelfth century)480, but both drew their materials

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from Arya Bhatta, in whose time the science seems to have been at its height; and who, though not clearly traced further back than the fifth century, may, in Mr. Colebrooke’s opinion, not improbably have lived nearly as early as Diophantus, the first Greek writer on algebra; that is, about AD 360.

But, whichever may have been the more ancient, there is no question of the superiority of the Hindus over their rivals in the perfection to which they brought the science. Not only is Arya Bhatta superior to Diophantus, (as is shown by his knowledge of the resolution of equations involving several unknown quantities, and in a general method of resolving all indeterminate problems of at least the first degree481,) but he and his successors press hard upon the discoveries of algebraists who lived almost in our own time. Nor is Arya Bhatta the inventor of algebra among the Hindus; for there seems every reason to believe that the science was in his time in such a state, as it required the lapse of ages, and many repeated efforts of invention, to produce482. It was in his time, indeed, or

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in the fifth century, at latest, that Indian science appears to have attained its highest perfection483.

Originality of Hindu science

Of the originality of Hindu science some opinions must have been formed from what has been already said.

In their astronomy, the absence of a general theory, the unequal refinement of the different

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portions of science which have been presented to us, the want of demonstrations and of recorded observations, the rudeness of the instruments used by the Bramins, and their inaccuracy in observing, together with the suspension of all progress at a certain point, are very strong arguments in favour of their having derived their knowledge from a foreign source. But on the other hand, in the first part of their progress, all other nations were in still greater ignorance than they; and in the more advanced stages, where they were more likely to have borrowed, not only is their mode of proceeding peculiar to themselves, but it is often founded on principles with which no other ancient people were acquainted; and shows a knowledge of discoveries not made, even in Europe, till within the course of the last two centuries. As far as their astronomical conclusions depend on those discoveries, it is self-evident that they cannot have been borrowed; and, even where there is no such dependence, it cannot fairly be presumed that persons who had such resources within themselves must necessarily have relied on the aid of other nations.

It seems probable that, if the Hindus borrowed at all, it was after their own astronomy had made considerable progress; and from the want of exact resemblance between the parts of their system and that of other nations, where they approach the nearest, it would rather seem as if they had taken up hints of improvement than implicitly copied the doctrines of their instructors.

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That they did borrow in this manner from the Greeks of Alexandria does not appear improbable; and the reason cannot be better stated than in the words of Mr. Colebrooke, who has discussed the question with his usual learning, judgment, and impartiality. After showing that the Hindu writers of the fifth century speak with respect of the astronomy of the Yavanas, (by whom there is every reason to think that, in this instance, they mean the Greeks,) and that a treatise of one of their own authors is called “Romaka Sidhanta,” very possibly in allusion to the system of the western (or Roman) astronomers, he goes on to say, “If these circumstances, joined to a resemblance hardly to be supposed casual, which the Hindu astronomy, with its apparatus of eccentrics and epicycles bears in many respects to that of the Greeks, be thought to authorise a belief that the Hindus received from the Greeks that knowledge which enabled them to correct and improve their own imperfect astronomy, I shall not feel inclined to dissent from the opinion. There does appear ground for more than a conjecture that the Hindus had obtained a knowledge of Grecian astronomy before the Arabs began to cultivate the science.”

In another place484 Mr. Colebrooke intimates his opinion that it is not improbable that the Hindus may have taken the hint of their solar zodiac from the Greeks, but adapted it to their

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own ancient division of the ecliptic into twenty-seven parts. Their astrology, he thinks, is almost entirely borrowed from the West485.

From what has been already said, it seems very improbable that the Indian geometry and arithmetic have been borrowed from the Greeks, and there is no other nation which can contest the priority in those sciences. The peculiarity of their method gives every appearance of originality to their discoveries in algebra also.

In this last science, the claims of the Arabs have been set up against them; but Mr. Colebrooke has fully established that algebra had attained the highest perfection it ever reached in India before it was known to the Arabians, and, indeed, before the first dawn of the culture of the sciences among that people486.

Whatever the Arabs possessed in common with the Hindus, there are good grounds for thinking that they received from the latter nation; and however great their subsequent attainments and

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discoveries, it is to be remembered that they did not begin till the eighth century, when they first gained access to the treasures of the Greeks.

On these subjects, however, as on all connected with the learning of the Bramins, the decisions of the most learned can only be considered as opinions on the facts at present before us; and they must all be regarded as open to question until our increased acquaintance with Shanscrit literature shall qualify us to pronounce a final judgment.

The history of science, after all, is chiefly interesting from the means it affords of judging of the character of the nation possessed of it; and in this view we find the Bramins as remarkable as ever for diligence and acuteness, but with the same want of manliness and precision as in other departments, and the same disposition to debase every thing by a mixture of fable, and by a sacrifice of the truth to the supposed interests of the sacerdotal order.


469. Much information on these subjects, but generally with views unfavourable to the Hindus, is given in the illustrations, by different hands, annexed to Mr. Hugh Murray’s Historical and Descriptive Account of British India, – a work of great ability and value.

470. See Pond’s La Place System of the World, vol. ii. p. 252.

471. In Appendix I. See also Asiatic Researches, vol. viii. p. 489.; vol. vii. p. 282.

472. This appears by his observation of the place of the Colures, first mentioned by Mr. Davis. (Asiatic Researches, vol. ii. p. 268.) Sir W. Jones, in consequence of some further information received from Mr. Davis, fixed Parasara in the twelfth century before Christ (1181, B. c.); but Mr. Davis himself afterwards explained (Asiatic Researches, vol. v. p. 288.) that, from the most minute consideration he could give the subject, the observation must have been made 1391 years before the Christian sera. Another passage quoted from Parasara shows that the heliacal rising of Canopus took place in his time at a period which agrees with the date assigned to him, on other grounds. (Colebrooke, Asiatic Researches, vol. ix. p. 356. See also Asiatic Researches, vol. v. p. 288., for the opinion of Mr. Davis.) Mr. Bentley, however, at one time suspected the whole of the works of Parasara to be modern forgeries (Asiatic Researches, vol. vi.. p. 581.); and when he admitted them afterwards (in his posthumous work), he put a different interpretation on the account of the rising of Canopus, and placed him, on that and other grounds, in the year 576 before Christ. (Abstract of Bentley’s History, Oriental Magazine, vol. v. p. 245.) The attempt made by Sir W. Jones to fix other dates, by means of the mythological histories into which the name of Parasara is introduced, does not appear successful. (Asiatic Researches, vol. ii. p. 399.)

473. Thus the “Surya Sidhanta,” the learned work of an astronomer of the fifth or sixth century, is only known to the Hindus as a revelation from heaven, received upwards of 2,164,900 years ago. Their enigmatical manner of communicating their knowledge is as remarkable in the other sciences as in astronomy. Professor Playfair speaks thus of their trigonometry: – “It has the appearance, like many other things in the science of those eastern nations, of being drawn up by one who was more deeply versed in the subject than may be at first imagined, and who knew more than he thought it necessary to communicate. It is probably a compendium formed by some ancient adept in geometry, for the use of others who were mere practical calculators.” Of their arithmetic the edinburgh Review” says (vol. xxix. p. 147.), “All this is done in verse. The question is usually propounded with enigmatical conciseness; the rule for the computation is given in terms somewhat less obscure; but it is not till the example, which comes in the third place, has been studied, that all ambiguity is removed. No demonstration nor reasoning, either analytical or synthetical, is subjoined; but, on examination, the rules are found not only to be exact, but to be nearly as simple as they can be made, even in the present state of analytical investigation.” The same observation is applied to their algebra, Ibid. p. 151.

474. The commentator on the “Surya Sidhanta” (Asiatic Researches, vol. ii. p. 239.) shows strongly the embarrassment that was felt by those who tried to correct errors sanctioned by religious authority. In the same essay (p. 257.) it appears that, although the rational system had been established from time immemorial, it was still thought almost impious to oppose it to the mythological one. A single writer, indeed, avows that the earth is self-balanced in infinite space, and cannot be supported by a succession of animals; but the others display no such controversial spirit, and seem only anxious to show that their own rational opinions were consistent with the previously established fables. In the edinburgh Review” (vol. x. p. 459.) there is a forcible illustration of the effect of the system of religious fraud in retarding the progress of science.

475. See Mr. Colebrooke (Asiatic Researches, vol. ix. p. 329. note) for the position of the vernal equinox when the “Surya Sidhanta” was written, and Sir W. Jones (Asiatic Researches, vol. p. 392.) for the period when the vernal equinox was so situated. Mr. Colebrooke thinks it contemporary with Brahma Gupta, whom he afterwards fixes about the end of the sixth century.

476. Such is that of Vieta, pointed out by Professor Playfair, in his question sent to the Asiatic Society. (Asiatic Researches, vol. iv. p. 152.) Professor Playfair has published a memoir on the Hindu trigonometry (Transactions of the Royal Society of Edinburgh, vol. iv.), which is referred to by Professor Wallace, with the following important observation of his own: –However ancient, therefore, any book may be in which we meet with a system of trigonometry, we may be assured it was not written in the infancy of science. We may therefore conclude that geometry must have been known in India long before the writing of the Surya Sidhanta.’” There is also a rule for the computation of the sines, involving a refinement first practised by Briggs, in the beginning of the seventeenth century. (British India, vol. iii. p. 403., in the edinburgh Cabinet Library.”)

477. Edinburgh Review, vol. xxix. p. 158.

478. The ratio of the diameter to the circumference is given in the “Surya Sidhanta,” probably written in the fifth century (Asiatic Researches, vol. ii. p. 259.), and even by Mr. Bentley’s account, in the eleventh. The demonstrations alluded to in the preceding lines are generally by Brahma Gupta in the sixth century.

479. A writer in the edinburgh Review” (vol. xviii. p. 211.), who discusses the subject in a tone of great hostility to the Hindu pretensions, makes an observation which appears entitled to much consideration. He lays down the position, that decimal notation is not a very old invention, and points out the improbability of its having escaped Pythagoras, if it had in his time been known in India.

480. Mr. Bentley, in his last work, wishes to prove, by his usual mode of computation, that Bhascara wrote in the reign of Akber (1556); but the date in the text is mentioned in a Persian translation presented to that very emperor by the celebrated Feizi, whose inquiries into Hindu science form the most conspicuous part of the literature of that age. (See Book IX. chap. iii.) Bhascara is likewise quoted by many authors anterior to Akber, whose authenticity Mr. Bentley is therefore obliged to deny.

481. Edinburgh Review, vol. xxix. p. 142.

482. Ibid. p. 143.

483. In the “Edinburgh Review” (vol. xxi. p. 372.) is a striking history of a problem (to find x so that a x2 + b shall be a square number). The first step towards a solution is made by Diophantus; it is extended by Fermat, and sent as a defiance to the English algebraists in the seventeenth century; but was only carried to its full extent by Euler; who arrives exactly at the point before attained by Bhascara in A. n. 1150. Another occurs in the same Review (vol. xxix. p. 153.), where it is stated, from Mr. Colebrooke, that a particular solution given by Bhascara (1150) is exactly the same that was hit on by Lord Brounker, in 1657; and that the general solution of the same problem was unsuccessfully attempted by Euler, and only accomplished by De la Grange, 1767; although it had been as completely given by Brahma Gupta in the sixth century of our tem. But the superiority of the Hindus over the Greek algebraists is scarcely so conspicuous in their discoveries as in the excellence of their method, which is altogether dissimilar to that of Diophantus (Strachey’s Bija Ganita, quoted in the “Edinburgh Review,” vol. xxi. pp. 374, 375.), and in the perfection of their algorithm. (Colebrooke, Indian Algebra, quoted in the “Edinburgh Review,” vol. xxix. p. 162.) One of their most favourite processes (that called cuttaca) was not known in Europe till published by Bachet de Mezeriac, about the year 1624, and is virtually the same as that explained by Euler. (Edinburgh Review, vol. xxix. p. 151.) Their application of algebra to astronomical investigations and geometrical demonstrations it also an invention of their own; and their manner of conducting it is, even now, entitled to admiration. (Colebrooke, quoted by Professor Wallace, ubi supra, pp. 408, 409.; and Edinburgh Review, vol. xxix. p. 158.)

484. Asiatic Researches, vol. ix. p. 347.

485. In addition to the points already mentioned, in which the Hindus have gone beyond the other ancient nations, Mr. Colebrooke mentions two in astronomy: one is in their notions regarding the precession of the equinoxes, in which they were more correct than Ptolemy, and as much so as the Arabs, who did not attain to their degree of improvement till a later period; the other relates to the diurnal revolution of the earth on its axis, which the Bramins discuss in the fifth century, and which, although formerly suggested in ancient times by Heraclitus, had been long laid aside by the Greeks, and was never revived in Europe until the days of Copernicus.

486. Colebrooke’s Algebra, Arithmetic, &c.

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