# Algebra History

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ALGEBRA 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 4 'What was the progress of its improvement 1 And lastly, by what means and at what period was the science diffused over Europe 4 It was a common opinion in the 17th century that the ancient Greek mathematicians must have possessed an analysis of the nature of modern 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 geometers 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 I 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 thne ; 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 illfated 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 undertak.en ; but the text of Diophantus was so much injured, that he was frequently obliged to guess ihe 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 Almarnoun.

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 title-page 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 Ahnamoun, 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 sonic 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 Diophantits 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 Cie 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. lie 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 Rule o.) 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 b)- Lucas Paciolus, or Lucas de Burgo, a minorite friar. It was first printed in 1494, and again in 1523. The title is Summa de Arithmetiea, Geometria, Proportioni, et Proportionalita.

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, in. deed, it is from this work that one of his lost treatises ha; been restored.

The power of algebra as an instrument of research is it a very great degree derived from its notation, by which al the quantities under consideration are kept constantly it view ; but in respect of convenience and brevity of expres sion, the algebraic analysis in the days of Lucas de Burg( was very imperfect : the only symbols employed were a fe's abbreviations of the words or names which occurred it the processes of calculation, a kind of short-hand, whicl formed a very imperfect substitute for that compactness of expression which has been attained by the moder. 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 bad of their science was also circumscribed : it extended only to the resohition 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 Ilalley'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 lie 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."

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, be 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 been overlooked, while the honours due to them have been transferred to others having only secondary pretensions. The formula; for the resolution of cubic equations are now called Cardan's rules, notwithstanding the prior claim of Tartalea. It must be confessed, however, that he evinced considerable selfishness in concealing his discovery ; and although Cardan cannot be absolved from the charge of bad faith, yet it must be recollected that by his improvements in what Tartalea communicated to him, he made the discovery in some measure his own ; and he had moreover the high merit of being the first to publish this important improvement in algebra to the world.

The next step in the progress of algebra was the discovery of a method of resolving equations of the fourth order. An Italian algebraist had proposed a question which could not be resolved by the newly invented rules, because it produced a biquadratic equation. Some supposed that it could not be resolved at all; but Cardan was of a different opinion. He had a pupil named Lewis Ferrari, a young man of great genius, and an ardent student in the algebraic analysis : to him Cardan committed the solution of this difficult question, and he was not disappointed. Ferrari not only resolved the question, but he also found a genera] method of resolving equations of the fourth degree, by making them depend on the solution of equations of the third degree.

This was another great improvement ; and although the precise nature of an equation was not then fully understood, nor was it indeed until half a century later, yet, in the general resolution of equations, a point of progress was then reached which the utmost efforts of modern analysis have never been able to pass.

There was another Italian mathematician of that period who did something for the improvement of algebra. This was Bombelli. He published a valuable work on the subject in 1572, in which he brought into one view what had been done by his predecessors. He explained the nature of the irreducible case of cubic equations, which had greatly perplexed Cardan, who could not resolve it by his rule ; he showed that the rule would apply sometimes to particular examples, and that all equations of this case admitted of a real solution ; and he made the important remark, that the algebraic problem to be resolved in this case corresponds to the ancient problem of the trisection of an angle.

There were two German mathematicians contemporary with Cardan and Tartalea, viz., Stifelius and Scheubelius. Their writings appeared about the middle of the 16th century, before they knew what had been done by the Italians. Their improvements were chiefly in the notation. Stifelius, in particular, introduced for the first time the characters which indicate addition and subtraction, and the symbol for the square root.

The first treatise on algebra in the English language was written by Robert Recorde, teacher of mathematics and practitioner in physic at Cambridge. At this period it was common for physicians to unite with the healing art tha studies of mathematics, astrology, alchemy, and chemistry. This custom was derived from the Moors, who were equally celebrated for their skill in medicine and calculation. In Spain, where algebra was early known, the title of physician and algebraist were nearly synonymous. Accordingly, in the romance of Don Quixote, when the bachelor Samson Carasco was grievously wounded in his rencounter with the knight, an algebrista was called in to heal his bruises.

Recorde published a treatise on arithmetic, which was dedicated to Edward VI. ; and another on algebra, with the title, The Whetstone of Wit, he. Here, for the first time, the modern sign for equality was introduced.

By such gradual steps did algebra advance in improvement from its first introduction byLeonardo, each succeeding writer making some change for the better ; but with the exception of Tartalea, Cardan, and Ferrari, hardly any one rose to the rank of an inventor. At length came Vieta, to whom this branch of mathematical learning, as well as others, is highly indebted. His improvements in algebra were very considerable ; and some of his inventions, although not then fully developed, have yet been the germs of later discoveries. He was the first that employed genera I characters to represent known as well as unknown quant, ties. Simple as this step may appear, it has yet led to important consequences. He must also be regarded as the first that applied algebra to the improvement of geometry. The older algebraists had indeed resolved geometrical problems, but each solution was particular; whereas Vieta, by introducing general symbols, produced general formula:, which were applicable to all problems of the same kind, without the trouble of going over the same process of analysis for each.

This happy application of algebra to geometry produced great improvements : it led Vieta to the doctrine of angular sections, one of the most important of his discoveries, which is now expanded into the arithmetic of sines or analytical trigonometry. He also improved the theory of algebraic equations, and he was the first that gave a general method of resolving them by approximation. Ati he lived between the years 1540 and 1603, his writings belong to the latter half of the 16th century. He printed them at his own expense, and liberally bestowed them on men of science.

The Flemish mathematician Albert Girard was one of the improvers of algebra. He extended the theory of equations somewhat further than Vieta, but he did not completely unfold their composition ; he was the first that showed the use of the negative sign in the resolution of geometrical problems, and the first to speak of imaginary quantities. He also inferred by induction that every equation has precisely as many roots as there are units in the number that expresses its degree. His algebra appeared in 1629.

The next great improver of algebra was Thomas Harriot, an Englishman. As an inventor he has been the boast of this country. The French mat1Mmaticians have accused the British of giving discoveries to him which were really due to Vieta. It is probable that some of these may be justly claimed for both, because each may have made the discovery for himself, without knowing what had been done by the other. Harriet's principal discovery, and indeed the most important ever made in algebra, was, that every equation may be regarded as formed by the product of as many simple equations as there are units in the number expressing its order. This important doctrine, now familiar to every student of algebra, developed itself slowly. It was quite within the reach of Vieta, who unfolded it in part, but left its complete discovery to Harriot.

We have seen the very inartificial form in which algebra first appeared in Europe. The improvements of almost 400 years had not given its notation that compactness and elegance of which it is susceptible. Harriot made several changes in the notation, and added some new signs : he thus gave to algebra greater symmetry of form. Indeed, as it came from his hands, it differed but little from its state at the present time.

Oughtreed, another early English algebraist, was a contemporary with Harriot, but lived long after him. He wrote a treatise on the subject, which was long taught in the universities.

In tracing the history of algebra, we have seen, that in the form under which it was received from the Arabs, it was hardly distinguishable as a peculiar mode of reasoning, because of the want of a suitable notation; and that, poor in its resources, its applicability was limited to the resolution of a small number of uninteresting numeral questions. We have followed it through different stages of improvement, and we are now arrived at a period when it was to acquire additional power as an instrument. el analysis, and to admit of new and more extended applications. Vieta saw the great advantage that might be derived from the application of algebra to geometry. The essay he made in his theory of angular sections, and the rich mine of discovery thus opened, proved the importance of his labours. He did not fully explore it, but it has seldom happened that one man began and completed a discovery. He had, however, an able and illustrious successor in Descartes, who, employing in the study of algebra that high power of intellect with which he was endowed, not only improved it as an abstract science, but, more especially by its application to geometry, laid the foundation of the great discoveries which have since so much engaged mathematicians, and have made the last two centuries eve/ memorable in the history of the progress of the humar mind.

Descartes' grand improvement was the application of algebra to the doctrine of curve lines. As in geography we refer every place on the earth's surface to the equator, and to a determinate meridian, so he referred every point of a curve to some line given by position. For example, in a circle, every point in the circumference might be referred to the diameter. The perpendicular from any point in the curve, and the distance of that perpendicular from the centre or from the extremity of a diameter, were lines which, although varying with every change of position in the point from which the perpendicular was drawn, yet had a determinate relation to each other, which was the same for all points in the curve depending on its nature, and which, therefore, served as a characteristic to distinguish it from all other curves.

The relations of lines drawn in this way could be readily expressed in algebraic symbols ; and the expression of this relation in general terms constituted what is called the equation of the curve.

This might serve as its definition; and from the equation by the processes of algebra, all the properties of the curve could be investigated.

Descartes' Geometry (or, as it might have been named, the application of algebra to geometry) appeared first in 16'37. This was six years after the publication of Harriot's discoveries, which was a posthumous work. Descartes availed himself of some of Harriot's views, particularly the manner of generating an equation, without acknowledgment; and on this account Dr Wallis, in his algebra, has reflected with considerable severity on the French algebraist. This spirit has engendered a corresponding eagerness in the French mathematicians to defend him. Montucla, in his history of the mathematics, has evinced a strong national prejudice in his favour ; and, as usually happens, in order to exalt him, he hardly does justice to Harriet, the idol of his adversaries.

The new views which the labours of Vieta, Harriot, and Descartes opened in geometry and algebra were seized with avidity by the powerful minds of men eager in the pursuit of real knowledge. Accordingly, we find in the 17th century a whole host of writers on algebra, or algebra combined with geometry.

Our limits will not allow us to enter minutely into the claims which each has on the gratitude of posterity. Indeed, in pure algebra the new inventions were not so conspicuous as the discoveries made by its applications to geometry, and the new theories which were suggested by their union. The curious speculations of Kepler concerning the solids formed by the revolutions of curvilinear figures, the Geometiy of Indivisibles by Cavalerius, the Arithmetic of Infinites of Wallis, and, above all, the Method of Fluxions of Newton, and the Differential and Integral Calculus of Leibnitz, are fruits of the happy union. All these were agitated incessantly by their inventors and contemporaries; by such men as Barrow, James Gregory, Wren' Cotes, Taylor, Halley, De l'iloivre, Maclaurin, Stirling, and others, in this country ; and abroad by Roberval, Fermat, Huyghens, the two Bernoullis, Pascal, and many others.

The first half of the 18th century produced little in the way of addition either to pure algebra or to its applications. Men were employed rather in elaborating and working out what Newton, Leibnitz, and Descartes had originated, than in exercising themselves in independent investigations. There are, indeed, to be found some names of eminence associated with the science of algebra, such as Maclaurin, but their eminence will be found to depend on their connection with the extensions of the science, rather than with the science itself. It was reserved forLagrange, in the latter part of the century, to give a new impulse to extension in pure algebra, in a direction which has led to most important results. Not only did he, in his Traite de la Resolution des Equations .Numeriques, lay the foundation on which Budan, Fourier, Sturm, and others, have built a goodly fabric after the pattern of the Universal Arithmetic of Newton, but in his Theorie des fonctions analytiques, and Calcul des fonctions. he endeavoured, and with a large amount of success, to reduce the higher analysis (the Fluxions of Newton), to the domain of pure algebra. Nor must the labours of a fellow-workman, Euler, be forgotten. In his voluminous and somewhat ponderous writings will be found a perfect storehouse of investigations on every branch of algebraical and mechanical science. Especially pertinent to our present subject is his demonstration of the Binomial Theorem in the .Novi Commentarii, vol. xix., which is probably the original of the development that Lagrange mak-es the basis of his analysis ( Calcul des fonctions, legon seconde), and which for simplicity and generality leaves nothing to be desired.

This brings the history down to the close of the last century. We have been as copious as our limits would permit on the early history, because it presents the interesting spectacle cf the progress of a science from an almost imperceptible beginning, until it has attained a magnitude too great to be fully grasped by the human mind.

It will be seen from what precedes, that we have not limited "algebra" to the pure science, but have retained the name when it has encroached on the territories of geometry, trigonometry, and the higher analysis. To continue to trace its course through all these branches during the present century, when it has extended into new directions within its own borders, would far exceed the limits of an introductory sketch like the present. We must, therefore, necessarily limit ourselves to what has been done in the Theory of Equations (which may be termed algebra proper), and in Determinants.

Theory of Equations. - That every numerical equation has a root - that is, some quantity in a numerical form, real or imaginary, which, when substituted for the unknown quantity in the equation, shall render the equation a numerical identity - appears to have been taken for granted by all writers down to the time of Lagrange. It is by no means self-evident, nor is it easy to afford evidence for it which shall be at the same time convincing and free from limitations. The demonstrations of Lagrange, Gauss, and Ivory, have for simplicity and completeness given way to that of Cauchy, published first in the Journal de l'Ecole Polytechnique, and subsequently in his Cours d' Anal yse Algebrique.

The demonstration of Cauchy (which had previously been given by Argand, though in an imperfect form, in Gergonne's Annales des illathematiques, vol. v.) consists in showing that the quantity which it is wished to provo capable of being reduced to zero, can be exhibited as the product of two factors, one of which is incapable of assuming a minimum value, or, iu other words, that a less value than one assigned can always be found, and therefore that it is capable of acquiring the value zero. This argument, if not absolutely free from objection, is less objectionable than any of the others. The reader may consult papers by Airy and De Morgan, in the tenth volume of the Transactions of the Cambridge Philosophical Society.

Admitting, then, that every equation has a root, it becomes a question to what extent are we in possession of an analysis by which the root can be ascertained. If the question be put absolutely, we fear the answer must be, that in this matter we are in the same position that v.-e have held for the last three centuries. Cubic and biquadratic equations can be solved, whatever they may be; but equations of higher orders, in which there exists no relation amongst the several coefficients, and no known or assunk 1 connection between the different roots, have baffled all attempts at their solution. Much skill and ingenuity have been displayed by writers of more or less eminence in the attempt to elaborate a method of solution applicable to equations of the fifth degree, but they have failed; whether it be that, like the ancient problems of the quadrature of the circle, and the duplication of the cube, an absolute solution is an impossibility, or whether it is reserved for future mathematicians to start in the research in some new path, and reach the goal by avoiding the old tracks which appear to have been thoroughly traversed in vain.

It is scarcely necessary to refer to such writers as Hoene de Wronski, who, in 1811, announced a general method of solving all equations, giving formulee without demonstration. In 1817, the Academy of Sciences of Lisbon proposed as the subject of a prize, the demonstration of Wronski's formulae. The prize was in the following year awarded to M. Torriani for the refutation of them.

The reader will find in the fifth volume of the Reports of the British Association, an elaborate report by Sir W. R. Hamilton on a Method of Decomposition, proposed by Mr G. B. Jerrard in his Mathematical Researches, published at Bristol in is work of great beauty and originality, but which Hamilton is compelled to conclude fails to effect the desired object. In fact, the method which is valid when the proposed equation is itself of a sufficiently elevated degree, fails to reduce the solution of the equation of the fifth degree to that of the fourth.

But although the absolute solution of equations of higher orders than the fourth remains amongst the things uneffected, and rather to be hoped for than expected, a very great deal has been done towards preparing the way for approximate, if not for absolute solutions.

In the first place, equations of the higher orders, when they assume certain forms, have been shown to be capable of solution. An equation of this kind, to all appearance of a very general and comprehensive form, had been solved by De Moivre in the Philosophical Transactions for 1737. Binomial equations had advanced under the skilful hands of Gauss, who, in his Disquisitiones Arithmeticae, which appeared in 1801, added largely to what had been done by Vandermonde in the classification and solution of such equations ; and subsequently, Abel, a mathematician of Norwegian birth, who died too early for science, completed and extended what Gauss had left imperfect. The collected writings of Abel published at Christiania in 1839, contain original and valuable contributions to this and many other branches of mathematics.

But it is not in the solution of equations of certain forms that the greatest advance has been made during the present century. The basis of all methods of solution must evidently be found in the previous separation of the roots, and the efforts of mathematicians have been directed to the discovery of methods of effecting this. The object is not so much to classify the roots into positive and negative, real and imaginary, as to determine the situation and number of the real roots of the equation. The first writer on the subject whose methods appeared in print is Budan, whose treatise, entitled Nouvelle methode pour la resolution des equations numeriques, appeared in 1807. But there is evidence that Fourier had delivered lectures on the subject prior to the publication of Budan's work, and consequently, without detriment to the claims of Budan, we may admit that the most valuable and original contribution to the science is to be found in Fourier's posthumous work, published by Navier in 1831, entitled Analyse des equations d&erminees. The theorem which Fourier gave for the discovery of the position, within narrow limits, of a root of au equation, is one of two theorems, each of which is known by mathematicians as "Fourier's Theorem." The other is a theorem of integration, and occurs in the author's magnificent work THorie de la Chalcur. During the interval between the publication of Budan's work and that of Fourier, there appeared a paper in the Philosophical Transactions of the Royal Society for 1819, by W. G. Horner, upon a new method of solving arithmetical equations. From its being somewhat obscurely expressed, the great originality of the memoir did not at once appear. Gradually, however, Mr Horner's method came to be appreciated, and it now ranks as one of the best processes, approaching, in some points, to Fourier's. In the 3ffmoires des savans etrangers for 1835, appears a memoir, which, if it does not absolutely supersede all that had been previously done in assigning the positions of the real roots of equations, yet in simplicity, completeness, and universality of application, surpasses them all. The author, M. Sturm, of French extraction, but born at Geneva, has in this memoir linked his name to a theorem which is likely to retain its place amongst the permanent extensions of the domain of analysis as long as the study of algebra shall last. It was presented to the Academy in 1829.

Determinants. - The solution of simultaneous equations of the first degree may be presented under the form of a set of fractions, the numerators and denominators of which are symmetric products of the coefficients of the unknown quantities in the equations. These products were originally known as resultants, a name applied to them by Laplace, and retained as late as 1841 by Cauchy in his Exercices d'analyse et de physique mathematique, vol. ii. p. 161, but now replaced by the title determinants, a name first applied to certain forms of them by Gauss. In his Cours d'analyse algebrique, Cauchy terms them alternate functions. The germ of the theory of determinants is to be found in the writings of Leibnitz, who, indeed, was farseeing enough to anticipate for it some of the power which, about a century after his time, it began to attain. More than half that period had indeed elapsed before any trace of its existence can be found in the writings of the mathematicians who succeeded Leibnitz. The revival of the method is due to Cramer, who, in a note to his Analyse des lignes courbes algebriques, published at Geneva in 1750, gave the rule which establishes the sign of a product as plus or minus, according as the number of displacements from the typical form has been even or odd. Cramer was followed in the last century by Bezout, Laplace, Lagrange, and Vandermonde. In 1801 appeared the Disquisitiones Arithmeticae of Gauss, of which a French translation by M. Poullet-Delisle was given in 1807. Notwithstanding the somewhat obscure form in which this work was presented, its originality gave a new impetus to investigations on this and kindred subjects. To Gauss is due the establishment of the important theorem, that the product of two determinants both of the second and third orders is a determinant. Binet, Cauchy, and others followed, and applied the results to geometrical problems. In 1826, Jacobi commenced a series of papers on the subject in Crelle's Journal. In these papers, which extended over a space of nearly twenty years, the subject was recast and made available for ordinary readers; and at the same time it was enriched by new and important theorems, through which the name of Jacobi is indissolubly associated with this branch of science. Following the steps of Jacobi, a number of mathematicians of no mean power have entered the field. Pre-eminent above all others are two British names, those of Sylvester and Cayley. By their originality, by their fecundity, by their grasp of all the resources of analysis, these two powerful mathematicians have enriched the Transactions of the Royal Society, Crelle's Journal, the Cambridge and Dublin Mathematical Journal, and the Quarterly Journal of Mathematics, with papers on this and on kindred branches of science of such value as to have placed their authors at the head of living mathematicians. The reader will find the subject admirably treated in Baltzer's Theorie and Anwenclung der Determinenten ; and more briefly in Salmon's Higher Algebra. Elementary treatises have also been published by Spottiswoode in 1851, by Briosehi in 1854, by Todhunter in his Theory of .Equations in 1861, and by Doclgson in 1867.

The attention of the learned has, during the present century, been called to a branch of the history of algebra, in no small degree interesting ; we mean the cultivation of the science to a considerable extent, and at a remote period, in India.

We are indebted, we believe, to Mr Reuben Burrow for some of the earliest notices which reached Europe on this very curious subject. His eagerness to illustrate the history of the mathematical sciences led him to collect oriental manuscripts, some of which, in the Persian language, with partial translations, were bequeathed to his friend Mr Dolby of the Royal Military College, who communicated them to such as took an interest in the subject, about the year 1800.

In the year 1813, Mr Edward Strachey published in this country a translation from the Persian of the Baja Ganita (or Vija Ganita), a Hindoo treatise on algebra; and in 1816 Dr John Taylor published at Bombay a translation of Lelawati (or Lilavati), from the Sanscrit original. This last is a treatise on arithmetic and geometry, and both are the production of an oriental algebraist, Bhascara Aeharya. Lastly, in 1817, there came out a work entitled Algebra, Arithmetic, and Mensuration, from the Sanscrit of Brahmegupta and Mascara, translated by Henry Thomas Colebrooke, Esq. This contains four different treatises, originally written in Sanscrit verse, viz., the VVa Ganita and ..Lilavati of Bhascara Aeharya, and the Canitad'haya and Cuttacad'hyaya of Brahmegupta. The first two form the preliminary portion of Bhaseara's Course of Astronomy, entitled Sidd'hanta Siromani, and the last two are the twelfth and eighteenth chapters of a similar course of astronomy, entitled Brahma-sidd'hanta.

The time when Bhascara wrote is fixed with great precision, by his own testimony and other circumstances, to a date that answers to about the year 1150 of the Christian era. The works of Brahmegupta are extremely rare, and the age in which he lived is less certain. Mr Davis, an oriental scholar, who first gave the public a correct view of the astronomical computations of the Hindoos, is of opinion that he lived in the 7th century ; and Dr William Hunter, another diligent inquirer into Indian science, assigns the year 628 of the Christian era as about the time he flourished. From various arguments, Mr Colebrooke concludes that the age of Brahmegupta, was antecedent to the earliest dawn of the culture of the sciences among the Arabians, so that the Hindoos must have possessed algebra before it was known to that nation.

Brahmegupta's treatise is not, however, the earliest work known to have been written on this subject. Ganessa, a distinguished astronomer and mathematician, and the most eminent scholiast of Bhascara, quotes a passage from a much older writer, Arya-Bhatta, specifying algebra under the designation of Fija, and making separate mention of Cuttaca, a problem subservient to the resolution of indeterminate problems of the first degree. He is understood by another of Bhascara's commentators to be at the head of the older writers. They appear to have been able to resolve quadratic equations by the process of completing the square; and hence Mr Colebrooke presumes that the treatise of Arya-Bhatta then extant extended to quadratic equations in the determinate analysis, and to indeterminate equations of the first degree, and probably to those of the second.

The exact period when Arya-Bhatta lived cannot bo determined with certainty; but Mr Colebrooke thinks it probable that this earliest of known Ilindoo algebraists wrote as far back as the fifth century of the Christian era, and perhaps earlier. He lived therefore nearly as early as the Grecian algebraist Diophantus, who is reckoned to have flourished in the time of the emperor Julian, or about A.D. 360.

Mr Colebrooke has instituted a comparison between the Indian algebraist and Diophantus, and found reason to conclude that in the whole science the latter is very far behind the former. He says the points in which the Hindoo algebra appears particularly distinguished from the Greek are, besides a better and more convenient algorithm, 1st, the management of equations of more than one unknown quantity; 2d, the resolution of equations of a higher order, in which, if they achieved little, they had at least the merit of the attempt, and anticipated a modern discovery in the resolution of biquadratics; general methods for the resolution of indeterminate problems of the first and second degrees, in which they went far indeed beyond Diophantus, and anticipated discoveries of modern algebraists; and 4th, the application of algebra to astronomical investigations and geometrical demonstration, in which also they hit upon some matters which have been re-invented in modern times.

When we consider that algebra made little or no progress among the Arabians, a most ingenious people, and particularly devoted to the study of the sciences, and that centuries elapsed from its first introduction into Europe until it reached any considerable degree of perfection, we may reasonably conjecture that it may have existed in one shape or other in India long before the time of AryaBhatta; indeed, from its close connection with their doctrines of astronomy, it may be supposed to have descended from a very remote period along with that science. Professor Play-fair, adopting the opinion of Bailly, the eloquent author of the Astronomie Indienne, with great ingenuity attempted to prove, in a Memoir on the Astronomy of the Brahmins, that the observations on which the Indian astronomy is founded were of great. antiquity, indeed more than 3000 years before the Christian era. The very remote origin of the Indian astronomy had been strongly questioned by many in this country, and also on the Continent; particularly by Laplace, and by Delambre in his IIistoire de l'Astronomie Ancienne, tome i. p. 400, &e., and again in Ilistoire de l'Astronomie du Moyen Age, Discours p. 18, &e., where he speaks slightingly of their algebra; and in this country, Professor Leslie, in his Philosophy of Arithmetic, pp. 225 and 226, calls the Lilavati "a very poor performance, containing merely a few scanty precepts couched in obscure memorial verses." We are disposed to agree with Professor Leslie as to the value, and with Professor Playfair as to the antiquity of this Hindoo algebra. That it should have remained in a state of infancy for so many centuries is accounted for by the latter author in the following passage : - " In India everything [as well as algebra] seems equally insurmountable, and truth and error are equally assured of permanence in the stations they have once occupied. The polities, the laws, the religion, the science, and the manners, seem all nearly the same as at the remotest period to which history extends. Is it because the power which brought about a certain degree of civilisation, and advanced science to a certain height, has either ceased to act, or has met with such a resistance as it is barely able to overcome 1 or is it because the discoveries which the Hindoos are in possession of are an inheritance from some more inventive and more ancient

people, of whom no memorial remains but some of their attainments in science T' Writers on, Algebra.

For the titles of works on Algebra, consult Murhard, Bibliotheca Mathematica ; and for Memoirs on Algebra, in Academical Collections, see Reuss, Repertorium Commentationum, tom. vii.; Smith (on the Theory of Numbers), Brit. Assoc. 1859-60,1862-63.

5ab are like quantities, but + ab and + abb are unlike.

There are some other characters, such as > for greater than, < for less than, for therefore, which will be explained when we have occasion to use them; and in what follows we shall suppose that the operations and notation of common arithmetic are sufficiently understood.

has been extended, we extend that of x and + to something like the following : x and 4- are cumulative symbols of operations the inverse of each other. We may now exhibit the most general definition of the four symbols in the following form : + and - are symbols of operations prefixed to algebraical symbols of quantity, and are such that +a - a= + 0 or - 0, where + 0 means simply or very nearly increased by 0 ; - 0, diminished by O. x and + are symbols of operations prefixed to algebraical symbols of quantity, and are such that x a a = x 1 or +1, where x 1 means simply or very nearly multiplied by 1; 4-1, divided by 1.

L.AW I. Quantities affected by the signs + and - are in no way influenced by the quantities to which they are united by these signs.