these treatises that Brahmegupta compiled his treatise on Astronomy. Also from these five treatises Varaha-Mihira, a writer of an earlier

opinions, whether Adam be derived from adim, which in Sanscrit means the first, or Menu from Nuh, the true name of the patriarch; . . . . whether the two Menus can mean any other persons than the great progenitor and the restorer of our species."

In the third volume of the Asiatic Researches, Lieutenant Wilford printed an essay on Egypt and the Nile, from the ancient books of the Hindus, with some brief remarks by Sir William Jones. In those remarks is given the following literal translation of a passage of the Padma-Puran :--" To Satyavarman, that sovereign of the whole earth, were born three sons: the eldest, Sherma; then Charma; and thirdly, Jyápeti by name. They were all men of good morals, excellent in virtue and virtuous deeds, skilled in the use of weapons to strike with or be thrown; brave men, eager for victory in battle. But Satyavarman, being continually delighted with devout meditation, and seeing his sons fit for dominion, laid upon them the burden of government, whilst he remained honouring and satisfying the gods, and priests, and kine. One day, by the art of destiny, the king having drunk mead, became senseless, and lay asleep naked; then was he seen by Charma, and by him were his two brothers called, To whom he said: What now has befallen? In what state is this our sire?' By those two was he hidden with clothes, and called to his senses again and again. Having recovered his intellect, and perfectly knowing what had passed, he cursed Charma, saying, 'Thou shalt be the servant of servants; and, since thou wast a laugher in their presence, from laughter shalt thou acquire a name.' Then he gave to Sherma the wide domain on the south of the snowy mountain, and to Jyápeti he gave all on the north of the snowy mountain; but he, by the power of religious contemplation, attained supreme bliss." On this he remarks: "Now you will probably think that even the conciseness and simplicity of this narrative are excelled by the Mosaic relation of the same adventure; but, whatever may be our opinion of the old Indian style, this extract most clearly proves that the Satyavrata, or Satyavarman of the Purans, was the same personage as the Noah of Scripture, and we consequently fix the utmost limit of Hindu chronology; nor can it be with reason inferred that the divine legislator borrowed any part of his work from the Egyptians. He was deeply versed, no doubt, in all their learning, such as it was; but he wrote what he knew to be truth itself, independently of their tales, in which truth was blended with fables; and their age was not so remote from the days of the patriarch, but that every occurrence in his life might naturally have been preserved by tradition from father to son."

If the longevity of human life during the period before the deluge be admitted, and the traditional genealogies in the book of Genesis be correct, it may be shown that the early traditions therein recorded from the time of Adam to Moses passed through seven intermediate persons. See Gray's Key to the Old Testament, pp. 80, 81.

In a supplementary note, Sir William Jones adds :- "But whatever may be the comparative antiquity of the Hindu Scriptures, we may safely conclude that the Mosaic and Indian chronologies are perfectly consistent; that Menu, son of Brahma, was the Adima, or first created mortal, and consequently our Adam; that Menu, child of the sun, was preserved with seven others in a bahitra, or capacious ark, from an universal deluge, and must therefore be our Noah; that Hiranyacasipu, the giant with a golden axe, and Vali cr Bali, were impious or arrogant monarchs, and most probably our Nimrod and Belus; that the three Ramas, two of whom were invincible warriors, and the third, not only valiant in war, but the patron of agriculture and wine, which derive an epithet from his name, wero different repre

age, is understood to have compiled his astronomical treatise named Pancha-Siddhanta.

sentations of the Grecian Bacchus, and either the Rama of Scripture, or his colony personified, or the sun first adored by his idolatrous family; that a considerable emigration from Chaldea into Greece, Italy, and India happened about twelve centuries before the birth of our Saviour; that Sacya or Sisak, about two hundred years after Vyasa, either in person or by a colony from Egypt, imported into this country [Bengal] the mild heresy of the ancient Bauddhas; and that the dawn of true Indian history appears only three or four centuries before the Christian era, the preceding ages being clouded by allegory or fable."

Lieutenant (since Captain) Wilford employed a learned Brahmin to make extracts from Sanscrit books to aid him in a second essay he was writing on the Sacred Isles of the West. In the course of collating the Sanscrit authorities quoted or referred to, Captain Wilford discovered some discolorations in the manuscripts, which led him to suspect deception, which examination fully verified. This discovery naturally excited his apprehension that a similar imposition had been practised upon him, both with respect to the extracts for his former essay on Egypt and the Nile, and that the Purana, in which he had actually and carefully read the passage which he communicated to Sir William Jones, as an extract from it,does not contain that passage; and that it was interpolated by the dextrous introduction of a forged sheet, discoloured, and prepared for the purpose of deception, and which, having served this purpose, was afterwards withdrawn. The forgeries of the pundit (Captain Wilford observes) were of three kinds : 1. A word or two was only altered. 2. Such legends as had undergone a more material alteration. 3. All those which he had written from memory. With regard to those of the first class, when he found that I was resolved to make a collation of the manuscript, he began to adulterate and disfigure his own manuscript, mine, and the manuscripts of the college, by erasing the original name of the country, and putting that of Egypt or of Swetam in its place.

"To prevent my detecting those of the second class, which were not numerous, but of the greatest importance in their nature (and as books in India are not bound as in Europe, and every leaf is loose), he took out one or two leaves, and substituted others with an adulterous legend. In books of some antiquity, it is not uncommon to see a few new leaves inserted in the room of others that were wanting.

"To conceal the more numerous impositions of the third class, he had the patience to write two voluminous sections, supposed to belong, one to the ScandaPurana, and the other to the Bramanda, in which he connected all the legends together, in the usual style of the Puranas. These two sections, as he wrote them, consist of no less than 12,000 slocas or lines, the title of which he borrowed.

"The first imposition is a legend of the greatest importance, and is said to be extracted from the Padma-Purana. It contains the history of Noah and his three But unfortunately there is not a word

sons, and is written in a masterly style. of it to be found in that Purana."

The following is the passage referred to from Captain Wilford's Essay, pp. 312, 313" It is related in the Padma-Puran, that Satyavrata, whose miraculous preservation from a general deluge is told at length in the Matsya, had three sons, the eldest of whom was named Jyápeti, or lord of the earth. The others were Charma and Sharma, which last words are, in the vulgar dialects, usually pronounced Cham and Sham; as we frequently hear Kishu for Crishna. The royal patriarch (for such is his character in the Purans) was particularly fond of Jyápeti, to whom he gave all the regions to the north of Himalaya, or the Snowy Mountains, which extend from sea to sea, and of which Caucasus is a part. To Sharma he allotted the countries to the south of these mountains. But he cursed Charma; because, when

C

Aryabhatta, an astronomer of a still earlier age, is quoted by succeeding writers in support of the precession of the equinoxes. Though Brahmegupta in his treatise is silent on the subject, Bhascara and others have affirmed a periodical revolution of the places of the colures. The reason of the omission or denial of this periodical motion is attributed by Bhascara to the very small quantity of the deviation, and he states, in reference to Brahmegupta, that " "in mathematical science, holy tradition is authority so far only as it agrees with demonstration," and adds: "Such motion as results from the assigned revolutions must be admitted, when the places being calculated agree with those which are observed, whether taught by a holy sage or by a temporal teacher. If, then, the same places aro deducible from other revolutions, which of the assigned motions is the true one? The answer is, whichever agrees with the present observation must be admitted."

The author, Bhascara-Acharya,1 informs his readers that his work is a compilation, and that his own corrections and improvements are not very numerous nor important. In the Vija Ganita (Section 131) he has quoted a passage from Sridhara's Algebra, and another (Section 142) from Padmanabha's.

the old monarch was accidentally inebriated with a strong liquor made of fermented rice, Charma laughed; and it was in consequence of his father's imprecation that he became a slave to the slaves of his brothers. The children of Sherma travelled a long time, until they arrived at the bank of the Nile, or Cali; and a Brahmin informs me (but the original passage from the Puran is not yet in my possession) that their journey began after the building of the Padma-Mandira, which appears to be the Tower of Babel, on the bank of the River Cumudvati, which can be no other than the Euphrates."

1 The following extract is taken from the translation of an inscription discovered by Dr. Bhau Daji near Chalisgam, at the foot of the hills which contain the Peetulkhora caves. The inscription contains the names of several of BhascaraAcharya's descendants, who taught his works in a college endowed in the neighbourhood of Chalisgam. The original Sanscrit, and the translation, are printed in the Journal of the Asiatic Society for 1865:

"Full of good fame and merit was Bhascara. The learned Bhascara's son was Lakshmidhara, the first among the learned; acquainted with the meaning of the Vedas, the first among metaphysicians, and skilful in the knowledge of sacrificial ceremonies. Jaitrapala having recognised him as well versed in the meanings of all the Sastras, took him from this town and made him the chief of pundits. His son was Changadeva, the best of the astronomers and astrologers at the court of Singhana Chakravartin. Changadeva constructs the College Matha for the spread of the treatises composed by Bhascara-Acharya. His works, the chief of which is the Siddhanta Siromani, and the works of his ancestors and descendants, ought to be duly studied in my college. Sonhadeva granted ground, with gold, &c., to the college. Others also have made some grants. Future kings ought to protect this, for the increase of merit. In the year 1128 Saka, in the year Prabhava, in the Sravana month, full moon, on the occasion of a lunar eclipse, Sonhadeva, in the presence of the people, having thrown water into the hands, granted to the college of his preceptors as follows, &c."

The following is the conclusion of the Vija Ganita, and forms the ninth chapter of Mr. Colebrooke's translation :—

"On earth was one named Maheswara, who followed the eminent path of a holy teacher among the learned. His son, Bhascara, having from him derived the bud of knowledge, has composed this brief treatise of elemental computation.

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"As the treatises of Algebra by Brahmegupta, Sridhara, and Padmanabha are too diffusive, he has compressed the substance of them in a well-reasoned compendium for the gratification of learners. For the volume contains a thousand lines, including precept and example. Sometimes exemplified to explain the sense and bearing of a rule; sometimes to illustrate its scope and adaptation; one while to show variety of inferences; another while to manifest the principle. For there is no end of instances, and therefore a few only are exhibited. Since the wide ocean of science is difficultly traversed by men of little understanding; and on the other hand, the intelligent have no occasion for copious instruction. A particle of tuition conveys a science to a comprehensive mind; and having reached it, expands of its own impulse. As oil poured on water, as a secret entrusted to the vile, as alms bestowed upon the worthy, however little, so does science infused into a wise mind spread by intrinsic force.

"It is apparent to men of clear understanding, that the rule of three terms constitutes Arithmetic; and sagacity, Algebra. Accordingly I have said in the chapter on Spherics:

"The rule of three is Arithmetic; spotless understanding is Algebra. What is there unknown to the intelligent? Therefore, for the dull alone it is set forth.

"To augment wisdom and strengthen confidence, read, do read, mathematician, this abridgment, elegant in style, easily understood by youth, comprising the whole essence of computation, and containing the demonstration of its principles, replete with excellence and void of defect."

There are extant several commentaries on the work of Bhascara, of which the oldest appears to have been composed by Gangadhara about the year A.D. 1420.

The commentary of Ganesa (the most eminent scholiast of Bhascara) on the Lilavati bears a date which corresponds to A.D. 1545. It embraces a copious exposition of the Sanscrit text of the Lilavati, with demonstrations of the rules. Mr. Colebrooke informs his readers that he has used this commentary throughout his translation as the best interpreter of the original. Ganesa has quoted a passage of Aryabhatta, denoting the science of Algebra under the name of VIJA, and has made mention of a method for the solution of indeterminate problems.

Crishna wrote a commentary on the Vija Ganita, which bears the

date of A.D. 1602. This commentary contains a clear and copious exposition of the sense of the text, with ample demonstrations of the rules, in the same manner as the commentary of Ganesa on the Lilavati.

Suryadasa was the author of a Complete Commentary on the Siddhanta Siromani. His notes on the Lilavati bear a date corresponding to A.D. 1538, and those on the Vija Ganita to A.D. 1541. He considered Aryabhatta to be at the head of the older writers on Algebra. His rule for the solution of quadratic equations by means of completing the square is given in the Section 131 of Mr. Colebrooke's translation. Ranganatha, another scholiast, appears to have flourished about the year A.D. 1621.

These are the principal of the numerous commentaries in Sanscrit on the Lilavati and the Vija Ganita, which acquired in past times a high reputation, both in Hindustan and in countries beyond its borders, and superseded the use of all the preceding treatises on the subjects of Arithmetic and Algebra.

The Vija Ganita is more complete than the treatise of Brahmegupta, and consists of nine chapters. It contains an explanation of the nature and distinction between affirmative and negative quantities known and unknown, and of the ordinary operations of the science, also the arithmetic of surd numbers. It gives methods for the solution of simple and quadratic equations, and of indeterminate equations of the first and second degree. It also explains the application of algebra to geometrical figures.

It may be added, that the preceding treatise, the Lilavati,'

1 See the author's "Elementary Arithmetic," with some notices of its history, Section 1, pp. 9-12, where some account is given of the Lilavati, which was first published in English with the following title: "Lilawati, or a Treatise on Arithmetic and Geometry by Bhascara-Acharya, translated from the original Sanscrit by John Taylor, M.D., of the Honourable East India Company's Bombay Medical Establishment. Bombay, 1816."

In the preface Dr. Taylor states that the object of his translation is to furnish an authentic document, which, by exhibiting not only the actual degree of mathematical knowledge possessed by the Hindus in the twelfth century, but also, by showing their modes and principles of operation, may lead to a fair conclusion regarding their pretensions to originality in this department of science.

Dr. Taylor has printed an Appendix to his translation of the Lilavati, giving an account of the mode of teaching arithmetic in the Hindu schools of the Mahratta country and Guzerat. The following extract is taken from this Appendix :

"But what chiefly distinguishes the Hindu schools is the plan of instruction by the scholars themselves. When a boy joins the school, he is immediately put under the tuition and care of one who is more advanced in knowledge, and whose duty it is to give lessons to his young pupil, to assist him in learning, and to report his behaviour and progress to the master. The scholars are not classed as with us, but generally paired off, each pair consisting of an instructor and pupil. These pairs are so arranged that a boy less advanced may sit next to one who has made greater progress, and from whom he receives assistance and instruction. When, however,