On René Descartes

"The analytical equations, unknown to the ancients, which Descartes first introduced into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they apply to all phenomena in general. There cannot be a language more universal and more simple, more free from errors and obscurities, that is to say, better adapted to express the invariable relations of nature." (Jean-Baptiste-Joseph Fourier, "The Analytical Theory of Heat", 1822)

"[…] with Newton's and Descartes' time, the whole Mathematics, becoming Analytic, walked so rapid steps forward that they left far behind themselves this study without which they already could do and which had ceased to draw on itself that attention which it deserved before." (Nikolai I Lobachevsky, 1829)

"The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared, for its importance, with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science." (James Clerk-Maxwell, "Remarks on the Mathematical Classification of Physical Quantities", 1871)

"Descartes' geometry was called 'analytical geometry', partly because unlike the synthetic geometry of the ancients it is actually analytical in the sense that the word is used in logic; and partly because the practice had then already arisen, of designating by the term analysis the calculus [i.e., symbolic calculation or computation] with general quantities." (Florian Cajori, "A History of Mathematics", 1893) 

"In mechanics Descartes can hardly be said to have advanced beyond Galileo. [...] His statement of the first and second laws of motion was an improvement in form, but his third law is false in substance. The motions of bodies in their direct impact was imperfectly understood by Galileo, erroneously given by Descartes, and first correctly stated by Wren, Wallis, and Huygens." (Florian Cajori, "A History of Mathematics", 1893) 

"It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs some times used algebra in connection with geometry. The new step that Descartes did take was the introduction into geometry of an analytical method based on the notion of variables and constants, which enabled him to represent curves by algebraic equations. In the Greek geometry, the idea of motion was wanting, but with Descartes it became a very fruitful conception. By him a point on a plane was determined in position by its distances from two fixed right lines or axes. These distances varied with every change of position in the point. This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree." (Florian Cajori, "A History of Mathematics", 1893)

"The first important example solved by Descartes in his geometry is the 'problem of Pappus' [...] Of this celebrated problem the Greeks solved only the special case [...] By Descartes it was solved completely, and it afforded an excellent example of the use which can be made of his analytical method in the study of loci. Another solution was given later by Newton in the Principia." (Florian Cajori, "A History of Mathematics", 1893)

"Descartes' method of finding tangents and normals [...] was not a happy inspiration. It was quickly superseded by that of Fermat as amplified by Newton. Fermat's method amounts to obtaining a tangent as the limiting position of a secant, precisely as is done in the calculus today. [...] Fermat's method of tangents is the basis of the claim that he anticipated Newton in the invention of the differential calculus." (Eric T Bell, "The Development of Mathematics", 1940)

"Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure - equivalent to the differential calculus - for maximizing and minimizing a function of a single variable. [...] Fermat applied his method [...] and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same, it will be noted that the hypothesis contradicts that of Descartes. Fermat assumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite." (Carl B Boyer, "History of Analytic Geometry", 1956)

"When the absolute concept of coordinate systems introduced by Descartes shifted to the relative concept of coordinate systems introduced by Gauss, a clear differences between continuity and differentiability emerged." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)