algebraic, was discovered in the course of the next century, and given to the world in a Latin translation, by Xylander, in 1575. This is the work of Diophantus of Alexandria, who had composed thirteen books of arithmetical questions, and who is supposed to have flourished about 150 years after the Christian era. The questions he resolves are often of considerable difficulty: and much address is displayed in stating them, so as to bring out equations of such a form as to involve only one power of the unknown quantity. The expression is that of common language abbreviated and assisted by a few symbols. The investigations do not extend beyond quadratic equations; they are, however, extremely ingenious, and prove the author to have been a man of talent, though the instrument he worked with was weak and imperfect.'

There has been considerable controversy respecting the invention of algebra, and whether that of Diophantus is to be regarded as wholly independent of the Hindoo algebra, or related to it as antecedent or consequent, progenitor or offspring. The reader who feels any curiosity on the subject may consult two able articles on Hindoo algebra in the forty-second and fiftyseventh numbers of the Edinburgh Review. The opinion expressed in Brewster's Encyclopædia is, that Diophantus has the merit of independent invention; and that the Arabs (from whom Camillus Leonard and Lucas Pacioli de Burgo received it and brought it into Europe), borrowed their algebra principally from the Greeks assisted by occasional communications with Hindostan. The opinion of the writer of this article is, that what is ascribed to Diophantus was, as well as the Pythagorean arithmetic, of foreign extraction; and that the Arabs received their algebra directly from the same quarter as they obtained by their own confession the decimal arithmetic. But the question is not worth the arguments which might be expended upon it; especially as they would amount after all only to probability, which will be always appreciated differently by different minds, and by the same minds in different circumstances.

It has been already intimated that algebra, as imported into Italy in the thirteenth or fifteenth century, was in a very rude and imperfect state, being little more than the raw material, so to speak, of what now bears the same name. It has been justly remarked that our present algebraic notation (the same holds of all the algebraic discoveries, inventions, methods, arts, or what ever we choose to call them) has arisen by almost insensible degrees, as convenience suggested different mark of abbreviation to different authors; and that perfect symbolic language which addresses itself solely to the eye, and enables us to take in at a glance the most complicated relations of quantity, is the result of a series of small improvements made from time to time, some of which have even been forgotten and re-invented; while in no case, till at least within a very short time from the present period, has any general and systematic view of the nature of symbols directed the choice of new ones.'

As Italy was the country which first received algebra in Europe, so there its first and greatest improvements were made. The resolution of equations speedily reached a point at which it still remains stationary. In 1505 Scipio Ferrea discovered the solution of a cubic. This soon led to other important discoveries and improvements.

Algebra owes much to Cardan Tartalea and Louis Ferrari, or rather to their bad passions; for there is but too much evidence that, if they had been better men, they would have been worse mathematicians. We have, in their envious rivalry and malignant contention, a striking instance of good being educed from evil, and proof as convincing as any demonstration, not only that there may be disputes among mathematicians, but that there is no necessary connexion between mathematical eminence and moral excellence. Cardan, the first named of the aforesaid triumviri, stands foremost in the fame which he coveted, or rather in the notoriety which he merived. He was born at Milan or at Pavia in 1501. He took his degree of M. D. at Padua in 1525, and at the age of thirty-three was appointed professor of mathematics at Milan, where he read lectures on medicine. In 1552 he went to Scotland, and cured the archbishop of St. Andrew's of an asthma which had baffled the skill of numerous physicians. He was next at the court of Edward VI., and caleulated the nativity of that prince; thence he rambled through various countries, and at Bologna was committed to prison. On recovering his liberty he went to Rome, where, in 1576, he starved himself to death to accomplish one of his own astrological predictions. He was a man,' remarks Mr. Playfair, in whose character good and ill, strength and weakness, were mixed up in singular profusion. With great talents and industry, he was capricious, insincere, and vain-glorious to excess. Though a man of real science, he professed divination, and was such a believer in the influence of the stars, that he died to accomplish an astronomical prediction. He remains, accordingly, a melancholy proof, that there is no folly or weakness too great to be united with high intellectual attainments.'

Having by importunate solicitations, and after binding himself by promises and oaths to secrecy, obtained some algebraical discoveries from Tartalea, Cardan soon published them as his own. However perfidious this might be, we cannot regret the mortification of Tartalea, who wished to make a mystery of knowledge, and to conceal his discoveries for purposes merely selfish. In consequence of the envious rivalry and mutual challenges between these old friends, now irreconcileable enemies, several important improvements were made in algebra. Louis Ferrari, who had been a pupil of Cardan, accomplished the general resolution of the biquadratic equation by a very beautiful process. The properties of surds too now became much better understood. With Cardan originated the idea of denoting general, or as commonly termed indefinite quantities, by letters of the alphabet. The same method was adopted nearly at the

same time by Stiefel, a German, and it was extended and rendered an essential part of algebra by Vieta in 1600. Stiefel introduced the same characters for plus and minus which are still employed. Soon after this Robert Recorde, an English mathematician, published the first English treatise on algebra, and introduced in it the same sign of equality which is still in use. Vieta may be regarded as the person with whom the language of algebra acquired that perfection which has since rendered it such a powerful instrument of investigation. With him too originated improvements in trigonometry, and his treatise on angular sections was an important application of algebra to investigate the theorems and resolve the problems of geometry.

Algebra is allowed to have been indebted about this period to Albert Girard, a Flemish mathematician, whose principal work, Nouvelle Invention en Algebre, was published in 1669. He appears to have been the first who understood the use of negative roots in the solution of geometrical problems; and he is the author of the figurative expression which gives to negative quantities the name of quantities less than nothing; a phrase which has been severely censured,' says professor Playfair, by those who forget that there are correct ideas which correct language can hardly be made to express.' We will not stop at present to examine Girard's figurative expression, or Mr. Playfair's defence of it.

The person next in order, as an inventor or improver in algebra, is Thomas Harriot, an English mathematician, whose Artis Analytica Praxis was published after his death in 1631. In this work the author brought the whole theory of equations under one simple and comprehensive view, by his discovery of the composition of polynomials by the multiplication of simple factors; and he introduced the smaller letters of the alphabet instead of the capitals employed by Vieta. This distinguished author was employed in the second expedition sent out by Sir Walter Raleigh to Virginia, and he published an account of that country. He appears from his MSS. to have observed the spots of the sun in December, 1610, only a month later than Galileo. He also made observations on Jupiter's satellites, and on the comets of 1607 and 1618. He was born at Oxford in 1560.

The algebraic analysis was now brought to nearly its present state of perfection, and the remark of professor Playfair in reviewing its slow progress is so worthy of being noted and remembered as to merit transcription. I have been the more careful to note very particularly the degrees by which the properties of equations were thus unfolded, because I think it forms an instance hardly paralleled in science, where a succession of able men, without going wrong, advanced nevertheless so slowly in the discovery of a truth, which, when known, does not seem to be of a very hidden and abstruse nature. Their slow progress arose from this, that they worked with an instrument the use of which they did not fully comprehend, and employed language which expressed more than they were prepared to understand;-a language, which,

under the notion first of negative and then of imaginary quantities, seemed to involve such mysteries, as the accuracy of mathematical science must necessarily refuse to admit.' With all our respect and deference we must be allowed to doubt the soundness of the above paragraph as to conception, or its accuracy as to expression. But it has been put down here as a memorandum for future remark.

We have now reached a memorable stage in mathematical history; for we have come to the time of Descartes, and the application of algebra to geometry; which may be emphatically termed a new era. There is some diversity of opinion as to the person who should have the honor of having first applied algebra to geometry. Some would say Vieta, others our countryman Oughtred, whose Clavis was specially honored by the great Newton, and who, after having suffered much for his loyalty, is reported to have died of joy in 1660, on hearing that the king was about to be restored; but the greatest number of competent judges would pronounce in favor of Descartes. And when the most liberal allowance is made to the claims of Oughtred and Vieta as to absolute priority, there will be enough of distinction and pre-eminence left to the French philosopher. We will give first the statement of the case by professor Playfair, and then by the able writer of the article mathematics in Brewster's Encyclopædia, as we deem it of some importance to fix attention on this memorable epoch. The algebraic analysis being brought to nearly its present state of perfection, by the succession of discoveries above related, was thus prepared for the step which was about to he taken by Descartes, and which forms one of the most important epochs in the history of the mathematical sciences. This was the application of the algebraic analysis, to define the nature, and investigate the properties of curve lines, and consequently to represent the notion of variable quantity. The invention just mentioned is the undisputed property of Descartes,' says Mr. Playfair, and opened up vast fields of discovery for those who were to come after him. It is often said that Descartes was the first who applied algebra to geometry; but this is inaccurate; for such applications had been made before, particularly by Vieta in his treatise on angular sections.'Our countryman Oughtred,' says the writer in Brewster's Encyclopædia, ‘is usually considered the first who applied algebra to geometry so as to discover new properties. Vieta had, however, before demonstrated by algebraic processes a great number of properties of the chords of multiple arcs, which form to this day a very important part of the theory of angular sections. The use of the negative roots of equations in the solution of geometrical problems was clearly pointed out by Girard, though Montucla has (in express contradiction to his own words when speaking of that geometer) attributed it positively to Descartes. The latter, however, has claim enough on the admiration and gratitude of mathematicians, in his method of representing the characterising property of a curve by an equation between two variable magnitudes. This great step, which

brought at once the whole of geometry under the dominion of symbolic analysis, was made in his geometry, published in 1637. The revolution so produced, in the way of conceiving geometrical questions, can be compared to nothing but that which the invention of logic produced in the ancient methods. The comprehensive genius of Descartes immediately felt the whole force of his discovery, and hastened to apply it to problems then regarded of the greatest difficulty and generality, among which were the methods of drawing tangents to all sorts of curves, and the general theory of maxima and minima.

The period at which Descartes lived was preeminently the age of distinguished philosophers. Bacon died in 1626, eleven years before the geometry of Descartes was given to the world; Kepler in 1630; Galileo (whom Descartes visited at Florence) in 1642; Gassendi in 1655, just five years after Descartes; and the intellectual Leviathan Hobbes, aged ninety-one, in 1679.

Of the mathematical contemporaries of Descartes who contributed their respective shares of discovery, invention, or improvement, were Cavaleri, a disciple of Galileo and professor of astronomy in Bologna, who published his Method of Indivisibles in 1647; Pascal the celebrated author of the Provincial Letters, pronounced by Boileau and Voltaire the finest productions in the French language; Roberval, and above all the others just named Fermat, the rival of Descartes. It could be wished that the rivalry which existed between these two great mathematicians had been less, but there was, unhappily for the dignity of philosophy and for the honor of the pure mathematics, too much jealousy on the one side and too much envy on the other. We would not liken Descartes and Fermat to Cardan and Tartelia; but we regret that they were in any measure actuated by the same spirit.

Opinion is divided concerning the amount of Fermat's claim to the admiration and gratitude of mathematicians. According to La Grange and La Place he is to be regarded as the inventor of the differential calculus. The former says, On peut regarder Fermat comme le premier inventeur des nouveaux calculs:' the latter, Il paroit que Fermat, le veritable inventeur du calcul differential, l'ait envisagé comme un cas particulier de celui des differences.' These are high authorities, but we haughty islanders, so proud of our Cambridge and boastful of our Newton, are unwilling to bow to their decision, and ascribe it to national partiality.

Before advancing to the next memorable stage in the history of the mathematics, and to the distinguished names of Newton and Leibnitz, it will be proper to notice an arithmetical invention, to which, however, we can afford less time and space than its importance merits. The reader must be aware that we allude to the invention or (if it be thought the more appropriate term) discovery of logarithms by Napier of Merchiston early in the seventeenth century. By this time calculations connected with astronomy, &c., had become excessively burdensome,

and Napier applied his inventive mind to dis cover a remedy for the evil. In the course of his attempts he perceived that, whenever the numbers to be multiplied or divided were terms of a geometrical progression, the product of the quotient must also be a term of that progression and must occupy a place in it indicated by the places of the given numbers, so that it might be found from mere inspection, if the progression were far enough continued. The resource of the geometrical progression was plainly sufficient when the given numbers were terms of that pro gression; but, if they were not, no advantag seemed derivable froin it. Napier, however, perceived, though it was by no means obvious that any numbers whatever might be inserted, and have their places assigned in the progression. The next difficulty was to discover the principle, and execute the arithmetical process, by which, the places were to be ascertained. It is in these two points that the peculiar merit of the inven tion consists; and when all the circumstances are considered it discovers such inventive power as has been rarely surpassed. The able assist ance of Briggs, Gresham professor of mathe matics, was willingly given to Napier, and the former has no small merit, and is entitled to i much praise in perfecting the invention of logarithms. Subsequent improvements in science, instead of offering any thing that could supersede this invention, have only enlarged the sphere of its utility. It was a most invaluable present to the calculator, and logarithmic tables have been ap plied to numberless purposes not contemplated when they were first constructed. To conclude this notice of Napier in the elegant language of his countryman Playfair, from whom indeed we have adopted the above with some slight alter tion and abridgment; Even the sagacity of Napier did not see the immense fertility of the principle he had discovered; he calculated his tables merely to facilitate arithmetical, and chiefly trigonometrical computation, and little imagined that he was at the same time constructing a scale whereon to measure the density of the strata of the atmosphere, and the heights of moun tains; that he was actually computing the areas and lengths of innumerable curves, and was preparing for a calculus, which was yet to be discover ed, many of the most refined and most valuable of its resources. Of Napier, therefore, if of any man, it may safely be pronounced, that his name will never be eclipsed by any one more conspicuous, or his invention superseded by any thing more valuable.'

As coming after Descartes and Fermat, and immediately preceding Newton and Leibnitz, some notice is due to the Dutch geometers Hudde, Huygens, and Sluse, and to our own countrymen Wallis and Barrow; though it does not appear that any thing very marked, or much of the nature of material improvement, is attributable to them, except Wallis, Savilian professor of geometry at Oxford, and who died in 1703. He was a mathematician of the highest rank, and of great originality. In his Arithmetica Infini torum is discovered that firm reliance on the law of continuity in analytical expressions, which has since conducted to so many brilliant genera

of, except in one branch, the integration of differential or of fluxional equations. In this our countrymen had fallen considerably behind; and the distance between them and their brethren on the continent continued to increase, just in proportion to the number and importance of the questions, physical and mathematical, which were found to depend on these integrations. The habit of studying only our own authors on these subjects, produced at first by our admiration of Newton, and our dislike to his rivals, and increased by a circumstance very insignificant in itself, the diversity of notation, prevented us from partaking in the pursuits of our neighbours; and cut us off in a great measure from the vast field in which the genius of France, of Germany and Italy, was exercised with so much activity and success. Other causes may have united in the production of an effect, which the mathematicians of this country have had much reason to regret; but the evil had its origin in the spirit of jealousy and opposition which arose from the controversies that have just passed under our review. The habits so produced continued long after the spirit

itself had subsided.'

Such were the results of the differential war, and of our high admiration of Newton. The whole continent was against us; and it appeared in the end that the continent was more independent of England than she of the continent; but with Newton as her prime minister, or commander-in-chief, she was prepared to contend against the whole world.

It is probably owing to the gigantic dimensions and overawing superiority of the intellect of Sir Isaac Newton, that we have had no mathematicians of remarkable stature after him. His immediate successors were scarcely tall enough to measure his height, or but just able, like good William Whiston, to read lectures on the Principia. The truth is, we have much reason to make frequent and emphatic mention of the immortal Newton; nor is there any thing to distract and divide our attention, or to prevent us from consecrating the whole force of our admiration upon one object. Brook Taylor secretary to the royal society, and Roger Cotes Plumian professor of astronomy and experimental philosophy at Cambridge, both contemporaries of Newton, are not to be passed by without some notice; but they are unknown to fame; and by the admission of the few who have been at the trouble to read them, or who are disposed to praise them as profound, they are both acknowledged to be obscure. The former published his Method of Increments in 1715, and the Harmonia Mensurarum of the latter appeared in 1722. Cotes died in 1716, Newton in 1726, Taylor in 1731. A [blank] theorem has the name of Taylor attached to it, and is deemed of some importance in mathematical advancement.

The change which had by this time been made in the mathematics amounted to a kind of revolution, though it is not yet determined who had the greatest hand in bringing it about. Much is doubtless due to Newton, much to Leibnitz; but more to time the greatest of all innovators.' If the mathematical world had not been already almost, if not altogether, in possession of what

is now called the differential calculus, it is hardly conceivable that there should have been so much controversy about its origin or inven tion, that it should be claimed by England, Germany, and France too, or that it could pos sibly be a moot point, whether Newton, Leibnitz, or Fermat has the best claim in this chancery suit, The differential calculus ought to be of much importance, for we not only find an extraordinary combustion of passion and commotion of controversy respecting its actual origin, but much opposition made to its very existence. The remarks of Mr. Playfair are so just and so appl cable, in reference to any revolution, reformation, or extraordinary improvement in matte matical science, that we niust again undertake to transcribe them.

It must not be supposed that so great a revolution in science, as that which was made by the introduction of the new analysis, could be brought about entirely without opposition, as in every society there are some who think themselves interested to maintain things in the condition wherein they have found them. The considerations are, indeed, sufficiently obvious, which, in the moral and political world tend to produce this effect, and to give a stability to human institutions, often so little proportionate to their real value, or to their general utility. Even in matters purely intellectual, and in which the abstract truths of arithmetic and geometry seem alone concerned, the prejudices, the selfishness, or the vanity of those who pursue them, not un frequently combine to resist improvements, and often engage no inconsiderable degree of talent in drawing back instead of pushing forward the machine of science. The introduction of me thods entirely new must often change the relative place of the men engaged in scientific pursuits; and must oblige many, after descending from the stations they formerly occupied, to take a lower position in the scale of intellectual advancement. The enmity of such men, if they be not animated by a spirit of real candor, and the love of truth, is likely to be directed against methods by which their vanity is mortified and their importance lessened.'

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The above quotation deserves to be inscribed not only on the porch of every academy (where Plato had his memento of the importance of geometry); but on the gates of every college, city, and town, on the entrance to every of justice and every court of judicature. The able and elegant writer adds: though such changes as this must have every where accompanied the ascendancy acquired by the new calculus, for the credit of mathematicians, it must be observed, that no one of any considerable eminence has had the misfortune to enrol his name among the adversaries of the new science; and that Huygens, the most distinguished and most profound of the older mathe maticians then living, was one of the most forward to acknowledge the excellence of that science (method would surely be the more appro propriate word), and to make himself master of its rules, and of their application.' This excellent Dutchman and eminent mathematician visited England in 1661, and was chosen fellow