"The so-called 'Treasury of Analysis' [also Heuristic] is, to put it shortly, a special body of doctrine for the use of those who, after having studied the ordinary Elements, are desirous of acquiring the ability to solve mathematical problems, and it is useful for this alone. It is the work of three men, Euclid, the author of the Elements, Apollonius of Perga, and Aristaeus the elder. It teaches the procedures of analysis and synthesis." (Pappus of Alexandria, cca. 4th century BC)
"Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent." (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)
"Perhaps nowhere does one find a better example of the value of historical knowledge for mathematicians than in the case of Fermat, for it is safe to say that, had he not been intimately acquainted with the geometry of Apollonius and Viéte, he would not have invented analytic geometry." (Carl B Boyer, "History of Analytic Geometry", 1956)
"We find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies - such as Apollonius' work on conic sections, or the non-Euclidean geometries, whose practical value became apparent only later." (Arthur Koestler, "The Sleepwalkers: A History of Man's Changing Vision of the Universe", 1959)
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