514 ALGEBRA [HISTORY. This was another great improvement ; and although the precise nature of an equation was not then fully under stood, 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. jombelli. 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 IGth 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. list The first treatise on algebra in the English language was Inglish written by Robert Recorde, teacher of mathematics and J f practitioner in physic at Cambridge. At this period it was ambridge. comn ion for physicians to unite with the healing art the 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, &c. Here, for the first time, the modern sign for equality was introduced. By such gradual steps did algebra advance in improve ment from its first introduction by Leonardo, each succeeding writer making some change for the better ; but with the exception of Tartalea, Cardan, and Ferrari, hardly any one icta. 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 quantities. 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 pro blems, 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 pro duced great improvements : it led Vieta to the doctrine of angular sections, one of the most important of his dis coveries, 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 j general method of resolving them by approximation. As 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 Girard 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, Harrio an Englishman. As an inventor he has been the boast of this country. The French mathematicians 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. Harriot 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 sim ple 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 con- Oughti temporary 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 reason ing, 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 ol analysis, and to admit of new and more extended applica tions. 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 Las seldom happened that one man began and completed a dis covery. He had, however, an able and illustrioiis successor in Descartes, who, employing in the study of algebra that Dcscart 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 mathe maticians, and have made the last two centuries ever memorable in the history of the progress of the huniar niind. 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
