of the rules, much in the manner of GANESA; and there is a scholiast of a still later date, who appears to have flourished about the year 1621. If, therefore, it be true, that the Hindus of the present time understand nothing of their scientific books, the decline of knowledge among them must have been very rapid, as it is plain that, at the distance of less than two centuries from the present time, the light of science was shining in India with considerable lustre.

The correctness of the text has not only been ascertained by a comparison with the different commentaries just mentioned, but has since been further verified, as far as respects the Lilavati, by a circumstance unknown even to the translator. Another translation of the Lilavati, made by DR TAYLOR of Bombay, was printed there in the course of last year; and several copies of it arrived in Europe just about the time when MR COLEBROOKE'S translation was published; and its agreement with this last, is a proof of the accuracy of both. The only difference is in the language, and in the subdivisions of the work; but in what is most material, the rules and the examples, there is no variation of any amount. The translations of DR TAYLOR seems the more literal; that of MR COLEBROOKE more paraphrastic, so far as one ignorant of the original may presume to judge. The former has accompanied his translation with notes from the Indian commentators, and with many very useful observations of his own, in which he sometimes gives the investigations in the language of European Algebra. His translation appears to have been presented to the Literary Society at Calcutta in 1815, and to have been printed by the order, and at the expense of the Society. ‡

Mr Colebrooke proceeds to make a comparison between the Grecian, Hindu, and Arabian Algebras, as they existed at the earliest periods to which they can now be traced; and, for this, his knowledge both of the history and the principles of the mathematical sciences, render him fully qualified. Here the Notation, or Algorithm, as it is called, is the first thing to be taken notice of. The Hindu algebraists use for their symbols abbreviations and initials of words: they distinguish negative quantities by a dot set over the letter or letters that denote the quantity; but they have no mark for a positive quantity, except the absence of the negative sign. They have no symbol

It is entitled Lilawati, or a Treatise on Arithmetic and Geome try, by BHASCARA ACHARYA. Translated from the original Sanscrit by JOHN TAYLOR M. D. of the Honourable East India Company's Bombay medical establishment. Printed at Bombay, 1816.

that expresses addition, nor any that either signifies equality, or the relation of greater and less. A product of two quantities is denoted by the initial syllable of the word multiplication subjoined to those quantities, or sometimes by a dot interposed between them. A fraction is denoted by placing the divisor under the dividend, but without a line of separation. The two sides of an equation are ordered in the same manner, one under another; and thus it is by position, and not by a particular character, that equality is expressed; but as this method of ar ranging quantities is also used for other purposes, the context is necessary to enable us to determine exactly the import of the algebraic expression. The symbols of unknown quantity are not confined to a single one, but extend to any number; and the characters used are the initial syllables of the names of colours, excepting the first, which is expressed by the initials of the word yavat-tavat, (how much, or as much as), synonymous with tanto, as used for the same purpose by BOMBELLI and some of the early algebraists of Italy. We ventured, in our analysis of Mr STRACHEY'S extracts from the Bija Ganita, † to cffer what seemed an explanation of this singular use of the names of the colours; referring it to the state in which algebra may have employed palpable symbols, or counters, to denote the quantities that were to be subjected to computation. Characters also are here employed, not only for unknown, but for variable quantities, of which the value may be arbitrarily assumed; and in demonstrations, for both given and sought quantities. Initials of the terms square and solid, denote the 2d and 3d powers respectively; and are combined, not according to their sums, but according to their products, to indicate the higher powers. An initial syllable or letter is in like manner used to denote the root.

The terms of a compound quantity are written in a line, ordered according to the powers of one of the letters; and the absolute number always comes last, being distinguished by the initial syllable ru, the mark of a known quantity. Numeral coefficients are employed, including unity, and comprehending fractions, and are always written after the symbol of the unknown quantity; the dot which denotes minus being put over

In our review of Mr Strachey's extracts from the Bija or Vija Ganita, we made the mistake of saying that the Hindu algebra contained an expression of addition and of equality. The fact is as above stated.

+ Edinburgh Review, July 1813.

VOL. XXIX. No. 57.

K

the coefficient, and not over the literal part of the term. In stating an equation, the usual practice is to repeat every term which occurs on the one side, on the other also; setting down 0 for the coefficient of those that are in reality wanting. So, if it were required to state, according to the Hindu notation, that five times the cube of the quantity sought, diminished by three times its square, and augmented by four times the quantity itself, is equal to 90, or that 53-3x+4x=90, it might be done shas;

This algorithm is sufficiently distinct and precise, but it is prolix; and, though imperfect compared with ours in Europe, is greatly superior to those of DrOPHANTUS, and of the Arabians. It has abundant resources for the mere expression of quantities; but is deficient in the means of denoting the operations to be performed on them.

Mr COLEBROOKE observes,

'The notation which has just been described, is essentially different from that of Diophantus, as well as from that of the Arabian algebraists, and their early disciples in Europe. Diophantus employs the inverted medial of ἔλλειψις, defect or want, (opposed to υπαρξις, substance or abundance), to indicate a negative quantity; and prefixes the mark to such quantities. He calls the unknown quantity agues, representing it by the finals, which he doubles for the plural. The Arabians, again, apply the term number to the constant or given term; and the Hindus, on the other hand, apply the numerical character to the coefficient. Diophantus denotes unity, or the Monad, by μ°, and marks the powers by their initials; thus, is power simply, or the square; x" is the cube; is the biquadrate; dx the 5th power, &c.

The Arabian algebraists are almost entirely destitute of symbols. They have no marks, either arbitrary or abbreviated, for quantities known or unknown, positive or negative, or for the steps of an algebraic process, but express every thing by words at full length.'

e

Their European scholars introduced a few abbreviations, such as p and m for plus and minus; c, c, c" for the three first powers, &c.; from which, in time, has been produced the present language of algebra, the most perfect instrument of thought which has yet been contrived.

But we must take a nearer view of the Hindu treatises themselves.

The Lilavati, the first of them, treats of Arithmetic; and con

tains not only the common rules of that science (there reckoned eight), but the application of those rules to a variety of questions on interest, barter, mixtures, combinations and permutations, the sums of progressions, indeterminate problems; and, lastly, the mensuration of surfaces and solids. All this is done in verse; and the language, even when it is the most technical, seems often to be highly figurative. The question is usually propounded with enigmatical conciseness; the rule for the computation is next 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 not only found to be exact, but to be nearly as simple as they can be made, even in the present state of analytical investigation. The numerical results are readily deduced; and, if we compare them with the earliest specimens of calculation that have come to us even from Greece itself, the advantages of the decimal notation, and the algorithm arising from it, will be placed in a striking light.

*

But the peculiar character of the book, and of the Oriental style which often unites so ill with the severity of arithmetical computation, will be best understood by a few extracts from the work itself. It begins thus ;

Having bowed to the Deity, whose head is like an elephant's, whose feet are adored by gods; who, when called to mind, restores his votaries from embarrassment, and bestows happiness on his worshippers; I propound this easy process of computation, delightful by its elegance, perspicuous with words concise, soft and correct, and pleasant to the learned."

From this lofty and pious exordium, the author immediately descends to the common business of calculation, and enters on the explanation of such terms as are naturally placed at the beginning of a book of practical arithmetic, viz. the names of numbers, tables of coins, weights, measures, &c. In the table of measures, we remark the same attempt to fix on a natural standard of linear extent that was antiently made in our own country. Eight breadths of a barley-corn are said to make a finger or an inch; and it is added, in the commentary of GAK 2

Though we have said that the Lilavati contains no demonstrations, this is true only of the text. The commentators, in their annotations, have supplied this defect in many instances; and their corrections and amendments are to be found in the notes with which this translation is accompanied.

*

NESA, that the length of three grains of rice is held to be equal to the breadth of eight grains of barley. Much refinement, indeed, was not necessary to perceive the value of a standard. which the highest improvements both in art and in science have been found necessary to construct. The definitions are given in form of an introduction, and are followed by an invocation, 'Salutation to GANESA, resplendent as a blue and spotless lotus, and delighting in the tremulous motion of the dark serpent which is continually twining within his throat.' The rules of arithmetic are then delivered in verse, and addressed to Lilavati, a young and charming female, who appears to be receiving the instructions of the author, and to whom the examples of the rules are usually proposed, as questions to be resolved. After the elementary operations have been taught, the author proceeds to things less common. A section, consisting of several articles, is devoted to the subject Cipher, or the character which denotes Nothing, and to the effect of it when it enters into an arithmetical computation as a multiplier or a divisor. We meet here with a remark that is not very old even among the mathematicians of Europe. The text says, a definite quantity, divided by cipher, is a fraction having the definite quantity for the numerator and cipher for the denominator. This, however, is nothing but an identical proposition. The commentator GANESA gives the true answer, viz. that the said fraction or quotient is an infinite quantity; and the reason is also very rightly assigned, that while the numerator of a fraction remains constant, and the denominator diminishes, the quotient increases; and therefore cipher being in the utmost degree small, the quotient, when cipher becomes the denominator, must be in the utmost degree great, that is to say, infinite. This reasoning is perfectly sound; but involves in it ideas so considerably refined, that the conclusion from it was not recognised by the algebraists of modern Europe, till the new analysis had made them familiar with the notions of variable quantity,-of the law of continuity, and of infinity, as an extreme case of both.

In the beginning of a new chapter, the third, an operation is treated of, which is called Inversion; and nothing can be more unlike the scientific language of Europe, than the terms in which this rule is delivered.

To investigate a quantity, one being given, make the divisor a multiplier and the multiplier a divisor; the square a root, and the

The rules explained as elementary are, addition, subtraction, multiplication, division, squaring of numbers, cubing of numbers, extraction of the square root, extraction of the cube root.