Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/137

Rh The end of all study, says Descartes in one of his earliest writings, ought to be to guide the mind to form true and sound judgments on every thing that may be presented to it. 1 The sciences in their totality are but the intelligence of man ; and all the details of knowledge have no value save as they strengthen the understanding. The mind is not for the sake of knowledge, but knowledge for the sake of the mind. This is the re-assertion of a principle which the Middle Ages had lost sight of that knowledge, if it is to have any value, must be intelligence, and not erudition. But how is intelligence, as opposed to erudition, possible 1 The answer to that question is the method of Descartes. That idea of a method grew up with his study of geometry and arithmetic, the only branches of know ledge which he would allow to be &quot; made sciences,&quot; those which the Jesuits best taught, and which he himself cultivated most zealously in early life. But they did not satisfy his demand for intelligence. &quot;I found in them,&quot; he says, &quot; different propositions on numbers of which, after a calculation, I perceived the truth ; as for the figures, I had, so to speak, many truths put before rny eyes, and many others concluded from them by analogy ; but it did not seem to me that they told my mind with sufficient clearness why the things were as I was shown, and by what means their discovery was attained.&quot; 2 The mathematics of which he thus speaks included the geometry of the ancients, as it had been handed down to the modern world, and arithmetic with the developments it had received in the direction of algebra. The ancient geometry, as we know it, is a wonderful monument of ingenuity a series of (ours de force, in which each problem to all appearance stands alone, and, if solved, is solved by methods and principles peculiar to itself. Here and there particular curves, for example, had been obliged to yield the secret of their tangent; but the ancient geometers apparently had no consciousness of the general bearings of the methods which they so successfully applied. Each problem was something unique ; the elements of transition from one to another were wanting ; and the next step which mathe matics had to make was to find some method of reducing, for instance, all curves to a common notation. When that was found, the solution of one problem would immediately entail the solution of all others which belonged to the same series as itself. The arithmetical half of mathematics, which had been gradually growing into algebra, and had decidedly established itself as such in the Logistica Spedosa of Vieta (1540-1603), supplied to some extent the means of generalizing geometry. And the algebraists or arithmeti cians of the 16th century, such as Lucas de Borgo, Cardan, and Tartaglia, had used geometrical constructions to throw light on the solution of particular equations. But progress was &quot;made difficult, in consequence of the clumsy and irregu lar nomenclature employed. &quot;With Descartes the use of exponents as now employed for denoting the powers of a quantity becomes systematic; and without some such step by which the homogeneity of successive powers is at once recognized, the binomial theorem could scarcely have been detected. The restriction of the early letters of the alphabet to known, and of the late letters to unknown quantities is also his work. In this and other details he crowns and completes, in a form henceforth to be dominant for the language of algebra, the work of numerous obscure pre decessors, such as Etienne de la Roche, Stiefel, and others. Having thus perfected the instrument, his next step was to apply it in such a way as to bring uniformity of method into the isolated and independent operations of geometry. 1 Regulse, (Euvr. xi. 202. &quot; I had no intention,&quot; he says in the Mefltod,&quot;* of attempt ing to master all the particular sciences commonly called mathematics ; but as I observed that, with all differences in their objects, they agreed in considering merely the various relations or proportions subsisting among these objects, I thought it best for my purpose to consider these relations in the most general form possible, without refer ring them to any objects in particular except such as would most facilitate the knowledge of them. Perceiving further, that in order to understand these relations I should some times have to consider them one by one, and sometimes only to bear them in mind or embrace them in the aggregate, I thought that, in order the better to consider them individually, I should view them as subsisting between straight lines, than which I could find no objects more simple, or capable of being more distinctly represented to my imagination and senses ; and on the other hand that, in order to retain them in the memory or embrace an aggregate of many, I should express them by certain characters, the briefest possible.&quot; Such is the basis of the algebraical or modern analytical geometry. The problem of the curves is solved by their reduction to a problem of straight lines ; and the locus of any point is determined by its distance from two given straight lines the axes of co ordinates. Thus Descartes gave to modern geometry that abstract and general character in which consists its superi ority to the geometry of the ancients. In another question connected with this, the problem of drawing tangents to any curve, Descartes was drawn into a controversy with Fermat (1590-1663), Roberval (1602-1673), and Desargues (1593-1662). Fermat and Descartes agreed in regarding the tangent to a curve as a secant of that curve with the two points of intersection coinciding, while Roberval regarded it as the direction of the composite movement b)^ which the curve can be described. Both these methods, differing from that now employed, are interesting as preliminary steps towards the method of fluxions and the differential calculus. In pure algebra Descartes expounded and illustrated the general methods of solving equations up to those of the fourth degree (and believed that his method could go beyond), stated the law which connects the positive and negative roots of an equation with the changes of sign in the consecutive terms, and introduced the method of indeterminate coefficients for the solution of equations. 4 Attempts have been recklessly made to claim some of these innovations for the English algebraists Oughtred and Harriot, and others for the mathematicians of the Continent ; but such assertions are based upon no proof, and, if true, would only illustrate the genius of the man who could pick out from other works all that was productive, and state it with a lucidity which makes it look his crwn discovery. The Geometry of Descartes, unlike the other parts of his essays, is not easy reading. It dashes at once into the middle of the subject with the examination of a problem which had baffled the ancients, and seems as if it were tossed at the heads of the French geometers as a challenge. An edition of it appeared subsequently, with notes by his friend De Beaune, calculated to smooth the difficulties of the work. All along mathematics was regarded by Descartes rather as the envelope than the foundation of his method ; and the &quot; universal mathematical science &quot; which he sought after was only the prelude of a universal science of all-embracing character. 5 The method of Descartes rests upon the proposition that all the objects of our knowledge fall into series, of which the members are more or less known by means of one 3 Disc, de Methode, part ii. Geometric, book iii. VII. rfi
 * (Euvr. xi. 219.
 * CEuvres, xi. 224

