received this knowledge from the Greeks, or the Greeks from the Hindus, or, what is perhaps the more probable, whether each originally received it from some independent source. There is, however, one fact which distinguishes Hindu mathematical science from that of the Greeks, and which has given both perfection and simplicity to the Hindu system of calculation. It is that which consists in "the device of place" in their arithmetical notation; a contrivance which gives to each figure employed, a power depending on the place it occupies, in addition to the value it is assumed to have by itself.

The Hindus themselves can give no historical account of the origin or discovery of the sciences of arithmetic and algebra, nor even of the inventor of the denary scale of numeral notation, one of the most simple, and at the same time the most perfect of inventions.

In the year 1817 Mr. Colebrooke published a translation of four

Mesha, the Ram; Sinha, the Lion; Dhanus, the Bow; Vrisna, the Bull; Canya, the Virgin; Macara, the Sea Monster; Mithuna, the Pair; Tula, the Balance; Cumbha, the Ewer; Carcata, the Crab; Vrishchica, the Scorpion; Mina, the Fish." The figures of the twelve asterisms thus denominated with respect to the sun are specified in Sanscrit verses, of which the following is a verbal translation :"The Ram, Bull, Crab, Lion, and Scorpion have the figures of those five animals respectively the Pair are a damsel playing on a vina, and a youth wielding a mace; the Virgin stands on a boat in the water, holding in one hand a lamp, in the other an ear of rice-corn; the Balance is held by a weigher with a weight in one hand; the Bow, by an archer, whose hinder parts are like those of a horse; the Sea Monster has the face of an antelope; the Ewer is a water-pot borne on the shoulder of a man, who empties it; the Fish are two, with their heads turned to each other's tails; and all those are supposed to be in such places as suit their several natures."

The Greek names of the constellations of the zodiac as recorded in the Phenomena of Aratus, are:-1. Kpids, the Ram: 2. Taupos, the Bull: 3. Aídvuoi, the Twins: 4. Kapkívos, the Crab: 5. Aéwv, the Lion: 6. Пaрlévos, the Virgin : 7. XnAal, the Claws: 8. Zкopíos, the Scorpion: 9. Togevτhs, the Archer: 10. Alyoxepes, the Horned Goat: 11. "Topoxóos, the Man that holds the watering urn: 12. 'Ixoves, the fishes. Eudoxus, from whose work Aratus composed his poem entitled "The Phenomena," made Xŋλal, the claws, to occupy the seventh, and the remaining portion of the Scorpion, the eighth division of the zodiac. Aratus gives

to this sign the name of μeya0ńptov (line 82), the great beast. After the time of Aratus, Libra, the Bulance, was made to occupy the eighth division of the zodiac; and Virgil (Georg. i. 32-36) suggests that the Scorpion had drawn in his claws to make room for the constellation Libra, in honour of Augustus. In Gen. xxxvii. 9, can there be any allusion to the eleven constellations of the zodiac?

In the Hindu system of astronomy the planets are: Surya, the Sun; Budha, Mercury; Sucra, Venus; Mangala, Mars; Vrihaspati, Jupiter; Sani, Saturn; Chandra, the Moon. These are supposed to move in their respective orbits at the same rate; the dimensions of the moon's orbit being known, those of other planets are determined, according to their periodical revolutions, by proportion. The Hindus have observed that the moon revolves once on her axis in a lunar month, and has the same side always opposed to the earth. They have also noticed the difference of her apparent magnitude on the horizon and on the meridian.

ancient treatises on Arithmetic and Algebra,' written in the Sanscrit language, with a learned dissertation on those subjects. Two of these treatises, one on Arithmetic and the other on Algebra, constitute the twelfth and eighteenth chapters of one of the systems of Hindu Astronomy, entitled "Brahma-Siddhanta." The fact of these treatises being found in the midst of a system of astronomy, where the principles of one science are employed to aid in the development of another, affords at least a presumption that they were not of recent invention, but had existed in a former age, and had passed through stages of improvement. This presumption is supported by the fact that these treatises are composed in Sanscrit verse.

It may also be remarked that almost all the examples in the treatise on Algebra relate to astronomy. The author has given his own date in the Brahmasphuta Siddhanta in the following form: "In the reign of Sri Vyaythramukha, of the Sri Chapa dynasty, 550 years (A.D. 628) after the Saka king (Salivahana) having passed, Brahmegupta, the son of Jishnu, at the age of thirty, composed the Brahmegupta Siddhanta, for the gratification of mathematicians and astronomers." 2

Mr. Colebrooke considers that these treatises are not the oldest composed by Hindu writers, and remarks that Ganesa, a learned commentator, has quoted a passage from a work of Aryabhatta (an older writer than Brahmegupta), and that another commentator has named him as one of the earliest writers on these sciences. Dr. Bhau Daji, in his essay, has shown that Aryabhatta was born A.D. 476, as given by himself in the Aryashtasata. He calls himself a native of Kusumapura, or Pataliputra. His work is written with great attention to conciseness. Varaha-Mihira flourished about A.D. 505, and died A.D. 587. He has cited Aryabhatta by name in a passage given by Bhatta Utpala, and quoted in the commentary on the Varahi Sanhita. It is highly probable that the Romaka Siddhanta was composed about A.D. 505, and Varaha-Mihira founded his Pancha Siddhantika Karana on the Romaka Siddhanta, and four other works. It is therefore clear that Varaha could not have lived before A.D. 505. It appears that Aryabhatta affirmed the diurnal rotation of the carth on its axis, and calculated its circumference, which, when reduced, was equivalent to 25,080 English miles; that he possessed the true theory of lunar and solar eclipses, affirming the moon and

1 Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhascara, translated by Henry Thomas Colebrooke, Esq., F.R.S., &c. London, 1817. (Reviewed in the "Edinburgh Review" for November, 1817.)

2 See Journal of the Asiatic Society, 1865. Brief notes on the age and authenticity of the works of Aryabhatta, Varaha-Mihira, Brahmegupta, Bhatta Utpala, and Bhascar-Acharya. By Dr. Bhau Daji.

3 See Journal of the Asiatic Society, 1863, vol. ix. On the Surya Siddhanta,

primary planets to be illuminated by the sun; that he noticed the motion of the solstitial and equinoctial points; and that he recognised a motion of the nodes and apsides of all the primary planets as well as of the moon. His work on Astronomy embraced treatises on Arithmetic, Algebra, and Geometry.

Mr. Colebrooke was led to consider that if Brahmegupta was not the oldest writer on these sciences, he improved and extended the knowledge of them as they existed in his time. In the age of Meya, the author of the Surya Siddhanta, the moon's horizontal parallax was made to be 53′ 20′′, which gave the mean distance from the earth 220,184 in English geographical miles, whereas European astronomers compute the mean distance of the moon about 240,000. The Hindus took no notice of the errors arising from refraction, and their taking the moon's motion as along the sine instead of its arc, may be noted. But they were not wholly ignorant of optics; they knew the angles of incidence and refraction to be equal, and computed the place of a star or planet as it would be seen reflected from water or a mirror.

There is also a perpetual commentary on the work of Brahmegupta, composed by Chaturveda Prithudaca Swami. He successively quotes at length every verse of the text, interprets it word by word, and subjoins elucidations and remarks. It is impossible to doubt that the knowledge of these sciences had a wide extension long before they assumed the form in which they are found exhibited.

Of the treatises of Brahmegupta forming the twelfth and eighteenth chapters of his Astronomy, the former begins with the declaration : "He who distinctly and severally knows addition and the rest of the twenty logistics, and the eight determinations, including measurement by shadow, is a mathematician."

It consists of ten sections, which are respectively entitled, Algorithm, Mixture, Progression, Plane Figure, Excavations, Stacks, Law, Mounds of Grain, Measure by Shadow, Supplement. The eighteenth chapter consists of eight sections, which are named Pulverizer, Algorithm, Simple Equation, Quadratic Equation, Equation

and the Hindu Method of Calculating Eclipses, by William Spottiswoode, M. A., F.R.S., &c.

In the second volume of the Asiatic Researches, printed at Calcutta, is an essay on the Astronomical Computations of the Hindus. This essay was drawn up by Samuel Davis, Esq., whose curiosity had been raised to know by what means the prediction of eclipses and other celestial phenomena, published in the Hindu Patra or Almanac, had been effected. He procured a copy of the Surya Siddhanta, an ancient treatise on Astronomy, and a commentary on the text, with other Sanscrit works on the subject. The object of Mr. Davis's essay is to exhibit a computation of an eclipse of the moon which was predicted in the Hindu Patra to happen in the month of November, 1789, not merely on the principles, but strictly by the rules of the Surya Siddhanta. His essay, with the computations, extends over sixty-three pages.

of several unknown, Equation involving a Factum, Square affected by a Coefficient, Problems. The author concludes the last section in these words: "These questions are stated merely for gratification. The proficient may devise a thousand others; or may resolve by the rules taught, problems proposed by others. As the sun obscures the stars, so does the proficient eclipse the glory of other astronomers in an assembly of people, by the recital of algebraic problems, and still more by their solution."

The treatises of Brahmegupta, on Arithmetic and Algebra, are two of the most ancient known to be extant; but they are not so complete nor extensive in the subjects they embrace as the other two which form the first portion of Mr. Colebrooke's publication. These are entitled the Lilavati and the Vija Ganita, the former a treatise on Arithmetic, and the other a treatise on Algebra. They constitute the introduction to a course of Astronomy' entitled Siddhanta Siromani. The author,

1 "Bhascara argues, that it is more reasonable to suppose the earth to be selfbalanced in infinite space, than that it should be supported by a series of animals, with nothing assignable for the last of them to rest upon; and Nerasinha, in his commentary, shows that by Rahu and Cetu, the head and tail of the monster, in the sense they generally bear, could only be meant the position of the moon's nodes, and the quantity of her latitude, on which eclipses do certainly depend; but he does not therefore deny the reality of Rahu and Cetu; on the contrary, he says that their actual existence and presence in eclipses ought to be believed, and may be maintained as an article of faith without any prejudice to astronomy."

Mr. Davis adds the following translation of a sentence from the Sanscrit on the controversy :

"Fruitless are all other Sastras; in them is contention only:

Fruitful is the Jyotish Sastra, where the sun and moon are two witnesses." "There are certain learned pundits who have truer notions of the earth and the economy of the universe than are ascribed to the Hindus in general; and that they must reject the ridiculous belief of the common Brahmins, that eclipses are occasioned by the intervention of the monster Rahu, with many other particulars equally unscientific and absurd. But as this belief is founded on explicit and positive declarations contained in the Vedas and Puranas, the divine authority of which writings no devout Hindu can dispute, the astronomers have, some of them, cautiously explained such passages in those writings as disagree with the principles of their own science; and, when reconciliation was impossible, have apologised, as well as they could, for propositions necessarily established in the practice of it, by observing that certain things, as stated in other Sastras, might have been so formerly, and may be so still; but for astronomical purposes, astronomical rules must be followed.' Others have, with a bolder spirit, attacked and refuted unphilosophical opinions."

Sir William Jones has given the following account of the two systems, which he received from a venerable Hindu mathematician :

"The Pauranics will tell you that our earth is a plane figure studded with eight mountains, and surrounded by seven seas of milk, nectar, and other fluids; that the part which we inhabit is one of seven islands, to which eleven smaller isles are subordinate; that a god, riding on a huge elephant, guards each of the eight regions; and that a mountain of gold rises and gleams in the centre; but we believe the

Bhascara Acharya, thus gives the date of his work: "In the year 1036 (A.D. 1114) of the Saka king I was born; and at the age of thirty-six I composed the Siddhanta Siromani."

In the age of Brahmegupta there appear to have existed five very ancient treatises on Astronomy, which also embraced Astrology, under the names of Paulisa-Siddhanta, Romaca-Siddhanta, VasishthaSiddhanta, Surya-Siddhanta,' and Brahma-Siddhanta. It was from

earth to be shaped like a cadamba fruit, or spheroidal, and admit only four oceans of salt water, all of which we name from the four cardinal points, and in which are many great peninsulas, with innumerable islands. They will tell you that a dragon's head swallows the moon, and thus causes an eclipse; but we know that the supposed head and tail of the dragon mean only the nodes, or points formed by intersections of the ecliptic and the moon's orbit. In short, they have imagined a system, which exists only in their fancy; but we consider nothing as true, without such evidence as cannot be questioned."

The following passage, which appeared in the Times of March 28, 1877, is quoted to show that faith in Rahu and Cetu, with other superstitions, such as judicial astrology, still hold a potent spell in the eastern parts of the world :

"On the recent eclipse of the moon, the Turks at Constantinople were reported to have fired guns, according to traditional custom, in order to frighten the dragon devouring that luminary into releasing its prey. A letter from Constantinople in the Journal de Genève records another illustration of Oriental beliefs. It states that the opening of the new parliament was postponed from the 13th to the 19th, not, as represented, to give the deputies time to arrive, but because the astrologer of the seraglio reported against the original date, and recommended not only the day, but the precise hour of the ceremony."

In the first Section of the Surya Siddhanta, it is stated that "Time of the denomination of Murta (sidereal time) is estimated by respirations; six respirations make a vicala (second); sixty vicalas a danda; sixty dandas a nacshatra (sidereal) day; and thirty nacshatra days a nacshatra month. The Savan (solar) month is that contained between thirty successive risings of Surya (the Sun), and varies in its length according to the Lagna Bhirja (right ascension). Thirty tithis (lunar) days compose the chandra (lunar) month. The Saura month is that in which the sun describes one sign of the zodiac, and his passage through the twelve sigus is one year, and one of those years is a Deva day, or day of the gods."

Sir William Jones, in an essay on the Chronology of the Hindus, written in 1788, and printed in the second volume of the Asiatic Researches, has shown how far the great antiquity of the Hindus, so firmly believed by themselves, is worthy of credit. He gives a concise account of Indian chronology, extracted from Sanscrit books, or conversations with pundits, and subjoins a few remarks on their system, without attempting to decide the question which he ventured to start-"Whether it is not in fact the same with our own, but embellished by the fancy of their poets and the riddles of their astronomers."

After relating incredible stories and wonderful periods of time, he writes :-"Let us compare the two Indian accounts of the Creation and the Deluge with those delivered by Moses. It is not made a question in this tract, whether the first chapters of Genesis are to be understood in a literal or merely allegorical sense. The only points are, whether the creation described by the first Menu, which the Brahmins call that of the Lotos, be not the same with that recorded in our Scriptures? and whether the story of the seventh Menu be not one and the same with that of Noah I propose the questions, but affirm nothing; leaving others to settle their