To express in few words what it is which effects the transition from the science of arithmetic into a new field is not easy. It will serve, probably, to convey some notion of the position of the boundary line, when it is stated that the operations of arithmetic are all capable of direct interpretation per se, whilst those of algebra are in many cases interpretable only by comparison with the assumptions on which they are based. For example, multiplication of fractions—which the older writers on arithmetic, Lucas de Burgo in Italy, and Robert Recorde in England, clearly perceived to be a new application of the term multiplication, scarcely at first sight reconcilable with its original definition as the exponent of equal additions,—multiplication of fractions becomes interpretable by the introduction of the idea of multiplication into the definition of the fraction itself. On the other hand, the independent use of the sign minus, on which Diophantus, in the 4th century, laid the foundation of the science of algebra in the West, by placing in the forefront of his treatise, as one of his earliest definitions, the rule of the sign minus, “that minus multiplied by minus produces/»?««”—this independent use of the sign has no originating operation of the same character as itself, and might, if assumed in all its generality as existing side by side with the laws of arithmetic, more especially with the commutative law, have led to erroneous conclusions. As it is, the unlimited applicability of this definition, in connection with all the laws of arithmetic standing in their integrity, pushes the dominion of algebra into a field on which the oldest of the Greek arithmeticians, Euclid, in his unbending march, could never have advanced a step without doing violence to his convictions.

In asserting that the independent existence of the sign minus, side by side with the laws of arithmetic, might have led to anomalous results, had not the operations been subject to some limitation, we are introducing no imaginary hypothesis, but are referring to a fact actually existing. The most recent advance beyond the boundaries of algebra, as it existed fifty years ago, is that beautiful extension to which Sir W. R. Hamilton has given the designation of Quaternions, the very foundation of which requires the removal of one of the ancient axioms of arithmetic, “that operations may be performed in any order.”

HISTORY.

At what period and in what country algebra was invented? is a question that has been much discussed. Who were the earliest writers on the subject? What was the progress of its improvement? And lastly, by what means and at what period was the science diffused over Europe? It was a common opinion in the 17th century that the ancient Greek mathematicians must have possessed an analysis of the nature of modem algebra, by which they discovered the theorems and solutions of the problems which we so much admire in their writings; but that they carefully concealed their instruments of investigation, and gave only the results, with synthetic demonstrations.

This opinion is, however, now exploded. A more intimate acquaintance with the writings of the ancient geo-meters has shown that they had an analysis, but that it was purely geometrical, and essentially different from our algebra.

Although there is no reason to suppose that the great geometers of antiquity derived any aid in their discoveries from the algebraic analysis, yet we find that at a considerably later period it was known to a certain extent among the Greeks.

About the middle of the 4th century of the Christian era, a period when the mathematical sciences were on the decline, and their cultivators, instead of producing original works of genius, contented themselves with commentaries on the works of their more illustrious predecessors, there was a valuable addition made to the fabric of ancient learning.

This was the treatise of Diophantus on arithmetic, consisting originally of thirteen books, of which only the first six, and an incomplete book on polygonal numbers, supposed to be the thirteenth, have descended to our times.

This precious fragment does not exhibit anything like a complete treatise on algebra. It lays, however, an excellent foundation of the science, and the author, after applying his method to the solution of simple and quadratic equations, such as to “find two numbers of which the sum and the sum or difference of the squares are given,” proceeds to a peculiar class of arithmetical questions, which belong to what is now called the indeterminate analysis.

Diophantus may have been the inventor of the Greek algebra, but it is more likely that its principles were not unknown before his time; and that, taking the science in the state in which he found it as the basis of his labours, he enriched it with new applications. The elegant solutions of Diophantus show that he possessed great address in the particular branch of which he treated, and that he was able to resolve determinate equations of the second degree. Probably this was the greatest extent to which the science had been carried among the Greeks. Indeed, in no country did it pass this limit, until it had been transplanted into Italy on the revival of learning.

The celebrated Hypatia, the daughter of Theon, composed a commentary on the work of Diophantus. This, however, is now lost, as well as a similar treatise, on the Conics of Apollonius, by this illustrious and ill-fated lady, who, as is commonly known, fell a sacrifice to the fury of a fanatical mob about the beginning of the 5th century.

About the middle of the 16th century, the work of Diophantus above referred to, written in the Greek language, was discovered at Rome in the Vatican library, having probably been brought there from Greece when the Turks possessed themselves of Constantinople. A Latin translation, without the original text, was given to the world by Xylander in 1575; and a more complete translation, by Bachet de Mezeriac (one of the earliest members of the French Academy), accompanied by a commentary, appeared in 1621. Bachet was eminently skilful in the indeterminate analysis, and therefore well qualified for the work he had undertaken; but the text of Diophantus was so much in jured, that he was frequently obliged to guess the meaning of the author, or supply the deficiency. At a later period, the celebrated French mathematician Fermat supplemented the commentary of Bachet by notes of his own on the writings of the Greek algebraist. These are extremely valuable, on account of Fermat’s profound knowledge of this particular branch of analysis. This edition, the best which exists, appeared in 1670.

Although the revival of the writings of Diophantus was an important event in the history of mathematics, yet it was not from them that algebra became first known in Europe. This important invention, as well as the numeral characters and decimal arithmetic, was received from the Arabians. That ingenious people fully appreciated the value of the sciences; for at a period when all Europe was enveloped in the darkness of ignorance, they preserved from extinction the lamp of knowledge. They carefully collected the writings of the Greek mathematicians; they translated them into their language, and illustrated them with commentaries. It was through the medium of the Arabic tongue that the elements of Euclid were first introduced into Europe; and a part of the writings of Apollonius are only known at the present day by a translation from the Arabic, the Greek original being lost.

The Arabians ascribe the invention of their algebra to one of their mathematicians, Mahommed-ben-Musa, or Moses, called also Mahommed of Buziana, who flourished about the middle of the 9th century, in the reign of the Caliph Almamoun.

It is certain that this person composed a treatise on this subject, because an Italian translation was known at one time to have existed in Europe, although it is now lost. Fortunately, however, a copy of the Arabic original is preserved in the Bodleian Library at Oxford, bearing a date of transcription corresponding to the year 1342. The titlepage identifies its author with the ancient Arabian. A marginal note concurs in this testimony, and further declares the work to be the first treatise composed on algebra among the faithful; and the preface, besides indicating the author, intimates that he was encouraged by Almamoun, commander of the faithful, to compile a compendious treatise of calculation by algebra.

The circumstance of this treatise professing to be only a compilation, and, moreover, the first Arabian work of the kind, has led to an opinion that it was collected from books in some other language. As the author was intimately acquainted with the astronomy and computations of the Hindoos, he may have derived his knowledge of algebra from the same quarter. The Hindoos, as we shall presently see, had a science of Algebra, and knew how to solve indeterminate problems. Hence we may conclude, with some probability, that the Arabian algebra was originally derived from India.

The algebraic analysis, having been once introduced among the Arabians, was cultivated by their own writers. One of these, Mahommed Abulwafa, who flourished in the last forty years of the 10th century, composed commentaries on the writers who had preceded him. He also translated the writings of Diophantus.

It is remarkable, that although the mathematical sciences were received with avidity, and sedulously cultivated during a long period by the Arabians, yet in their hands they received hardly any improvement. It might have been expected that an acquaintance with the writings of Diophantus would have produced some change in their algebra. This, however, did not happen: their algebra continued nearly in the same state, from their earliest writer on the subject, to one of their latest, Behaudin, who lived between the years 953 and 1031.

Writers on the history of algebra were long under a mistake as to the time and manner of its introduction into Europe. It has now, however, been ascertained that the science was brought into Italy by Leonardo, a merchant of Pisa. This ingenious man resided in his youth in Barbary, and there learned the Indian method of counting by the nine numeral characters. Commercial affairs led him to travel into Egypt, Syria, Greece, and Sicily, where we may suppose he made himself acquainted with everything known respecting numbers. The Indian mode of computation appeared to him to be by far the best. He accordingly studied it carefully; and, with this knowledge, and some additions of his own, and also taking some things from Euclid’s Geometry, he composed a treatise on arithmetic. At that period algebra was regarded only as a part of arithmetic. It was indeed the sublime doctrine of that science; and under this view the two branches were handled in Leonardo’s treatise, which was originally written in 1202, and again brought forward under a revised form in 1228. When it is considered that this work was composed two centuries before the invention of printing, and that the subject was not such as generally to interest mankind, we need not wonder that it was but little known; hence it has always remained in manuscript, as well as some other works by the same author. Indeed it was not known to exist from an early period until the middle of the last century, when it was discovered in the Magliabecchian library at Florence.

The extent of Leonardo’s knowledge was pretty much the same as that of the preceding Arabian writers. He could resolve equations of the first and second degrees, and he was particularly skilful in the Diophantine analysis. He was well acquainted with geometry, and he employed its doctrines in demonstrating his algebraic rules. Like the Arabian writers, his reasoning was expressed in words at length; a mode highly unfavourable to the progress of the art. The use of symbols, and the method of combining them so as to convey to the mind at a single glance a long process of reasoning, was a much later invention.

Considerable attention was given to the cultivation of algebra between the time of Leonardo and the invention of printing. It was publicly taught by professors. Treatises were composed on the subject; and two works of the oriental algebraists were translated from the Arabian language into Italian. One was entitled the Pule of Algebra, and the other was the oldest of all the Arabian treatises, that of Mahommed-ben-Musa of Corasan.

The earliest printed book on algebra was composed by Lucas Paciolus, or Lucas de Burgo, a minorite friar. It was first printed in 1494, and again in 1523. The title is Summa de Arithmetica, Geometria, Proportioni, et Prο- portionalita.

This is a very complete treatise on arithmetic, algebra, and geometry, for the time in which it appeared. The author followed close on the steps of Leonardo; and, indeed, it is from this work that one of his lost treatises has been restored.

Lucas de Burgo’s work is interesting, inasmuch as it shows the state of algebra in Europe about the year 1500: probably the state of the science was nearly the same in Arabia and Africa, from which it had been received.

The power of algebra as an instrument of research is in a very great degree derived from its notation, by which all the quantities under consideration are kept constantly in view; but in respect of convenience and brevity of expression, the algebraic analysis in the days of Lucas de Burgo was very imperfect: the only symbols employed were a few abbreviations of the words or names which occurred in the processes of calculation, a kind of short-hand, which formed a very imperfect substitute for that compactness of expression which has been attained by the modem notation.

The application of algebra was also at this period very limited; it was confined almost entirely to the resolution of certain questions of no great interest about numbers. No idea was then entertained of that extensive application which it has received in modern times.

The knowledge which the early algebraists had of their science was also circumscribed: it extended only to the resolution of equations of the first and second degrees; and they divided the last into cases, each of which was resolved by its own particular rule. The important analytical fact, that the resolution of all the cases of a problem may be comprehended in a single formula, which may be obtained from the solution of one of its cases, merely by a change of the signs, was not then known: indeed, it was long before this principle was fully comprehended. Dr Halley expresses surprise, that a formula in optics which he had found, should by a mere change of the signs give the focus of both converging and diverging rays, whether reflected or refracted by convex or concave specula or lenses; and Molyneux speaks of the universality of Halley’s formula as something that resembled magic.

The rules of algebra may be investigated by its own principles, without any aid from geometry; and although in some cases the two sciences may serve to illustrate each other, there is not. now the least necessity in the more elementary parts to call in the aid of the latter in expounding the former. It was otherwise in former times. Lucas de Burgo found it to be convenient, after the example of Leonardo, to employ geometrical constructions to prove the truth of his rules for resolving quadratic equations, the nature of which he did not completely comprehend; and he was induced by the imperfect nature of his notation to express his rules in Latin verses, which will not now be read with the kind of satisfaction we receive from the perusal of the well-known poem, “the Loves of the Triangles.”

As Italy was the first European country where algebra became known, it was there that it received its earliest improvements. The science had been nearly stationary from the days of Leonardo to the time of Paciolus, a period of three centuries; but the invention of printing soon excited a spirit of improvement in all the mathematical sciences. Hitherto an imperfect theory of quadratic equations was the limit to which it had been carried. At last this boundary was passed, and about the year 1505 a particular case of equations of the third degree was resolved by Scipio Ferreus, a professor of mathematics in Bononia. This was an important step, because it showed that the difficulty of resolving equations of the higher orders, at least in the case of the third degree, was not insurmountable, and a new field was opened for discovery. It was then the practice among the cultivators of algebra, when they advanced a step, to conceal it carefully from their contemporaries, and to challenge them to resolve arithmetical questions, so framed as to require for their solution a knowledge of their own new-found rules. In this spirit did Ferreus make a secret of his discovery: he communicated it, however, to a favourite scholar, a Venetian named Florido. About the year 1535 this person; having taken up his residence at Venice, challenged Tartalea of Brescia, a man of great ingenuity, to a trial of skill in the resolution of problems by algebra. Florido framed his questions so as to require for their solution a knowledge of the rule which he had learned from his preceptor Ferreus; but Tartalea had, five years before this time, advanced further than Ferreus, and was more than a match for Florido. He therefore accepted the challenge, and a day was appointed when each was to propose to the other thirty questions. Before the time came, Tartalea had resumed the study of cubic equations, and had discovered the solution of two cases in addition to two which he knew before. Florido’s questions were such as could be resolved by the single rule of Ferreus; while, on the contrary, those of Tartalea could only be resolved by one or other of three rules, which he himself had found, but which could not be resolved by the remaining rule, which was also that known to Florido. The issue of the contest is easily anticipated; Tartalea resolved all his adversary’s questions in two hours, without receiving one answer from him in return.

The celebrated Cardan was a contemporary of Tartalea. This remarkable person was a professor of mathematics at Milan, and a physician. He had studied algebra with great assiduity, and had nearly finished the printing of a book on arithmetic, algebra, and geometry; but being desirous of enriching his work with the discoveries of Tartalea, which at that period must have been the object of considerable attention among literary men in Italy, he endeavoured to draw from him a disclosure of his rules. Tartalea resisted for a time Cardan’s entreaties. At last, overcome by his importunity, and his offer to swear on the holy Evangelists, and by the honour of a gentleman, never to publish them, and on his promising on the faith of a Christian to commit them to cypher, so that even after his death they would not be intelligible to any one, he ventured with much hesitation to reveal to him his practical rules, which were expressed by some very bad Italian verses, themselves in no small degree enigmatical. He reserved, however, the demonstrations. Cardan was not long in discovering the reason of the rules, and he even greatly improved’'them, so as to make them in a manner his own. From the imperfect essays of Tartalea he deduced an ingenious and systematic method of resolving all cubic equations whatsoever; but with a remarkable disregard for the principles of honour, and the oath he had taken, he published in 1545 Tartalea’s discoveries, combined with his own, as a supplement to a treatise on arithmetic and algebra, which he had published six years before. This work is remarkable for being the second printed book on algebra known to have existed.

In the following year Tartalea also published a work on algebra, which he dedicated to Henry VIII., king of England.

It is to be regretted that in many instances the authors of important discoveries have...