as the importance of this inquiry is more generally felt, that we shall, ere long, obtain entire translations of the originals, or such accounts of them as may enable us to determine, with more certainty, to whom we are indebted for a science, which, in its present state, may be justly considered as doing the highest honour to the reasoning and inventive powers of man.

This information is the more desirable, as it is acknowledged by Mr. Strachey, that the Persian translations of these works, though highly valuable in many respects, do not convey so correct an idea of the original performances as could be wished; as they chiefly consist of an undistinguished mixture of text and commentary, and, in some places, even refer to Euclid; though Mr. Davis, who made the extracts from the original Beja Ganita, mentioned in the last note, is decidedly of opinion that the main part of the Persian translation is taken from the Sanscrit work, and has no doubt that the science it contains is of Hindoo origin; the references above spoken of, and other discrepancies, being considered by him as interpolations of the translator.

But, not to dwell upon this part of the subject, which, it must be confessed, is still attended with some obscurity, it is well known, that, in whatever age or country algebra was invented, both the name and the science were first made known to us, about the end of the eleventh century, by the Arabians,

or Moors, who were then settled in Spain; to which country a number of eminent men, of that time, repaired, in order to obtain, from these people, a knowledge of the several branches of literature and science which they had brought with them from the East; and among the rest of algebra, which soon afterwards began to be cultivated in most parts of Europe with great ardour and success (d).

Italy, however, appears to have taken the lead in this respect; the art, according to the testimony of Signor Cossali (History of Algebra, 2 vols. 4to., 1797) having been first introduced into that country, as early as in the beginning of the thirteenth century, by Leopard Bonacci, commonly called Leonard of Pisa, an Italian merchant, who, by visiting the sea-ports of Africa and the Levant, acquired a knowledge of such branches of the mathematical sciences as were then known; which he afterwards disseminated among his countrymen, by

(d) The etymology of the name algebra has been variously given by different authors; but it appears evident, from Mr. Strachey's account, that the word is purely Arabic, being compounded of the article al, and jebr, restoration; which denotes one of the modes of reducing equations, by transposing or adding the negative terms, so as to make them all affirmative. The same writer is also of opinion, that, although we received this science from the Arabs, neither that people nor the Persians can be considered as the inventors, as their knowledge of the subject appears to have been of a very limited kind, compared with that of the Hindoos.

composing, in the year 1208, a work on arithmetic, and in 1228 another on arithmetic. and algebra; which performances the author above quoted considers as very regular and orderly treatises, for the time in which they were written.'.,

are

Many other manuscript writings, of a similar nature, also appeared in Tuscany and other parts of Italy, soon after the time above mentioned; but the first printed works on this subject, those of Lucas Pacciolus, or Lucas de Burgo, a Cordelier or Minorite friar, commonly called Luca del Borga di san Sepolcro; who published several treatises on arithmetic, algebra and geometry, in the years 1470, 1476, and 1481; but his principal work is that entitled Summa de Arithmetica, Geometria, proportioni et proportionalita, printed in folio, at Venice, 1494,

From this performance, which exhibits the state of the science at the beginning of the sixteenth century, it appears that the earliest European analysts, as well as the Arabs, from whom they derived the greater part of their knowledge, used no symbols or signs for either quantities, or the operations that it was necessary to perform on them, excepting a few contractions for the words or names themselves; and that the art, at that time, extended no farther than to the resolution of numeral equations of the first and second degree; in the latter of which, they used the double values of the unknown quantity, in the case where they are both

positive; but take no notice of such of the roots as are negative or imaginary (e) von hd

The passage, indeed, to the higher orders of equations, was a matter of considerable difficulty, which, in all probability, had never hitherto been made; but this was in part effected, soon after the publication of the work last mentioned, by Scipio Ferreus, professor of mathematics at Bologna, in Italy, who first discovered, about the year 1505, a rule for resolving one of the cases of a compound cubic equation; and about thirty years afterwards, one of his disciples, of the name of Florido, to whom he had shown his method, having proposed, to Nicholas Tartalea, a celebrated mathematician of Brescia, by way of challenge, several questions, the solutions of which depended upon this formula, Tartalea not only discovered the rule for resolving them, but also those for the other cases; which ena bled him, in his turn, to confound his opponent, by proposing to him some questions, of a different kind, which the latter could not answer. "g

But this, as well as several other branches of mathematical knowledge, was more particularly culti

(e) The most remarkable circumstance in the algebra of this author is, that in the denoting of powers, he compounds the names or indices according to the multiplication of the numbers 2, 3, 4, &c. and not by their addition, as was done by Diophantus; which induced Dr. Wallis to adopt the opinion that the knowledge we first obtained of this science was not derived from the Greeks.

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vated and improved, soon after the time above men→ tioned, by Hieronymus Cardan, of Bononia, an eminent Italian writer, of great scientific acquirements, who first published, in the year 1589,"a work on arithmetic and algebra, in nine books, inthe Latin language, at Milan, where he practised physic and read mathematical lectures; and in a new edition of the same performance, printed in F545, he gave a tenth book, containing the whole doctrine of cubic equations, which had been chiefly: revealed to him by Tartalea, under an oath of secrecy, about the time of the publication of his first nînê books (f). Boubare aldaeus

*In this work," which is a very regular and masterly performance, Cardan lays down rules for the solution of all the forms and varieties of cubic equations, whether complete or incomplete, which he demonstrates geometrically: he also treats of nearly all the transformations and methods of managing equations that are known to use at present; showing, among other things, that he was acquainted with the number and nature of their

(f) For a farther account of this transaction, which was highly resented by Tartalea, see Montucla's Histoire des Mathematiques, vol. i, p. 591, et seq. second edition; also Hutton's Mathematical Dictionary, Art. Algebra; where â full detail is giveh of all the most remarkable circumstances relating to this curious business, as well as a copious analysis of Cardan's and other performances of that time; to which article I am indebted for several particulars respecting the improvements and discoveries of some of the earlier European writers on this subject,