"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)
"[...] the Greeks, Descartes, Newton, Euler, and many others believed mathematics to be the accurate description of real phenomena and that they regarded their work as the uncovering of the mathematical design of the universe. Mathematicians did deal with abstractions, but these were no more than the ideal forms of physical objects or happenings." (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)
"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)
"Descartes’ idea to use numbers to describe points in space involves the choice of a coordinate system or coordinate frame: an origin, together with axes and units of length along the axes. A recurring theme of all the different geometries [...] is the question of what a coordinate frame is, and what I can get out of it. While coordinates provide a convenient framework to discuss points, lines, and so on, it is a basic requirement that any meaningful statement in geometry is independent of the choice of coordinates. That is, coordinate frames are a humble technical aid in determining the truth, and are not allowed the dignity of having their own meaning." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)
"Descartes’ invention of coordinate geometry is another key ingredient in modern science. It is scarcely an accident that calculus was discovered by Leibnitz and Newton (independently, alphabetical order) in the fifty years following the dissemination of Descartes’ ideas. Interactions between the axiomatic and the coordinate-based points of view go in both ways: coordinate geometry gives models of axiomatic geometries, and conversely, axiomatic geometries allow the introduction of number systems and coordinates." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)
"Descartes’ algebra could be used to classify line segments by length only. The fundamental geometric notion of direction of a line segment finds no expression in ordinary algebra. The modification of algebra to have a fuller symbolic representation of geometric notions had to wait some 200 years after Descartes, when the concept of number was generalized by Herman Grassmann to incorporate the geometric notion of direction as well as magnitude. With a proper symbolic expression for direction and dimension came the broader concept of directed numbers, now known as multivectors." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"In 1884, just 40 years after the publication of Grassmann’s 'Algebra of Extension', Gibbs developed his vector algebra following the ideas of Grassmann by replacing the concept of the outer product by a new kind of product known as vector product and interpreted as an axial vector in an ad-hoc manner. This, in fact, went against the run of natural development of directed numbers started by Grassmann and completely changed the course of its development in the other direction. Grassmann’s outer product reveals the fact that the Greek distinction between number and magnitude has real geometric significance. Greek magnitudes, in fact, added like scalars but multiplied like vectors, asserting the geometric notions of direction and dimension to multiplication of Greek magnitudes. This revealing feature is a reminiscence of the distinction, carefully made by Euclid, between multiplication of magnitudes and that of numbers. Thus, Herman Grassmann fully accomplished the algebraic formulation of the basic ideas of Greek geometry begun by Renė Descartes." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Though from the very beginning algebra was associated with geometry, Descartes first developed it systematically in geometric language. Three steps are of fundamental importance in this development. First, he assumed that every line segment could be uniquely represented by a number that endowed the Greek notion of magnitude a symbolic form. Second, he labeled line segments by letters representing their numeral lengths. This resided in the fact that the basic arithmetic operations of addition and subtraction could be described in a completely analogous way as geometric operations on line segments. Third, in order to get rid of the apparent limitations of the Greek rule for geometric multiplication, he invented a rule for multiplying line segments, yielding a line segment in complete correspondence with the rule for multiplying numbers." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
"Vieta used letters to denote numbers, whereas Descartes introduced letters to denote line segments. Vieta studied rules for manipulating numbers in an abstract manner, and Descartes accepted the existence of similar rules for manipulating line segments and greatly improved symbolism and algebraic technique. Thus, it seemed that numbers might be put into one-to-one correspondence with points on a geometric line, leading to a significant step in the evolution of the concept of number." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)
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