HISTORY. ALGEBRA 515 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 deter minate 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 1637. 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 generatingan 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 Harriot, 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 com bined 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 Moivre, 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 con nection with the extensions of the science, rather than with iagrange. 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 Numcriques, 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 Theoriedesfonctions analytiques, and Galcul des fonction-s, 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 Euler, 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 makes the basis of his analysis (Galcul des fonctions, lecon 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 interest ing spectacle of the progress of a science from an almost imperceptible beginning, until it has attained a mag nitude 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 dur ing 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 Theory has a root that is, some quantity in a numerical form, real Equatio 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 I Ecole Polytechnique, and subsequently in his Gours $ Analyse Algebrique. The demonstration of Cauchy (which had previously Cauchy. been given by Argand, though in an imperfect form, in Gergonne s Annales des Mathematiques, vol. v.) consists in showing that the quantity which it is wished to prove capable of being reduced to zero, can be exhibited as the product of two factors, one of which is incapable of assum ing a minimum value, or, in 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 be- General comes a question to what extent are we in possession of solutipi an analysis by which the root can be ascertained. If the C( ^ ] question be put absolutely, we fear the answer must be, an( | hio that in this matter we are in the same position that we orders, have held for the last three centuries. Cubic and biquad- a desicl ratic equations can be solved, whatever they may be ; but tum equations of higher orders, in which there exists no relation amongst the several coefficients, and no known or assumed
connection between the different roots, have baffled all
