La Géométrie Quotes

“Any problem in geometry can be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction.”

“Descartes published his "Géométrie" as an application of his general method of unification, in this case the unification of algebra and geometry.”

“The gradual evolution of calculus was considerably stimulated by the publication of Descartes' "Géométrie"... which brought the whole field of classical geometry within the scope of algebraists.”

“It is evident from Descartes' explanation of his method that he had an intuitive grasp of the elusive concepts of 'variable' and 'function,' both of which are basic in analysis. Moreover, he intuited continuous variation.”

“Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties.”

“The reader must pretty much construct the method for himself from certain isolated statements. There are... figures... but in none do we find the coordinate axes explicitly set forth. This work was written with intentional obscurity... too difficult to be widely read.”

“If the square root of GH is desired, I add, along the same straight line, FG equal to unity; then, bisecting FH at K, I describe the circle FIH about K as the center, and draw from G a perpendicular and extend it to I, and GI is the required root.”

“This is also evident from what Pappus has done in the beginning of his seventh book, where... he refers to a question which he says that neither Euclid nor Apollonius nor any one else had been able to solve completely...”

“The considerations that forced ancient writers to use arithmetical terms in geometry, thus making it impossible for them to proceed beyond a point where they could see clearly the relation between two subjects, caused much obscurity and embarrassment, in their attempts at explanation.”

“Descartes' recognized that the points of intersection of two curves are given by solving their equations simultaneously. The last implies... a major advance over all who had previously used coordinates: Descartes saw that an infinity of distinct curves can be referred to one system of coordinates. In this... he was far ahead of Fermat...”

“The Latin term for "ordinate," used by Descartes comes from the expression lineœ ordinatœ, employed by Roman surveyors for parallel lines. The term abscissa occurs for the first time in a Latin work of 1659, written by Stefano degli Angeli...”

“Let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA; then BE is the product of BD and BC.
If it be required to divide BE by BD, I join E and D, and draw AC parallel to DE; then BC is the result of division.”

“Often it is not necessary thus to draw the lines on paper, but it is sufficient to designate each by a single letter. ...it must be observed that by a², b³, and similar expressions, I ordinarily mean only simple lines, which, however, I name squares, cubes, etc., so that I may make use of the terms employed in algebra.”

“Descartes separated all curves into two classes, the "geometrical" and the "mechanical" ...according as (in our terminology) dy/dx is an algebraic or a transcendental function. ...this classification was abandoned long ago... The current definition... [a curve] which intersects some straight line in an infinity of points was given by Newton in his work on cubics.”

“The essays of Descartes on dioptrics and geometry were sharply criticised by Fermat, who wrote objections to the former, and sent his own treatise on "maxima" and "minima" to show that there were omissions in the geometry. Descartes thereupon made an attack on Fermat's method of tangents. Descartes was in the wrong in this attack, yet he continued the controversy with obstinacy.”

“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 with general quantities.”

“In book II, he implicitly recognized as legitimate curves of construction only those that are algebraic, that is, expressible in equations. ...discussing in book III problems for which different constructions were possible, he mandated that the simplest curves be used, and he defined those curves as those with equations of the lowest algebraic dimension. This was algebra dictating the methodology of geometry.”

“The one book that turned out to be perhaps the most influential in guiding Newton's mathematical and scientific thought was none other than Descartes' La Géométrie. Newton read it in 1664 and re-read it several times until "by degrees he made himself master of the whole." ...Not only did analytic geometry pave the way for Newton's founding of calculus... but Newton's inner scientific spirit was truly set ablaze.”

“It was the use of algebra in geometry that he undertook to exploit. He saw fully the power of algebra and its superiority over the Greek geometrical methods in providing a broad methodology. He... stressed the generality of algebra and its value in mechanizing the reasoning processes and minimizing the work in solving problems. He saw its potential as a universal science of method. The product of his application of algebra to geometry was La Géométrie.”

“It is true that [analytic geometry] eventually evolved under the influence of Descartes' book, but the "Géométrie"... can hardly be considered the first textbook on this subject. There are no "Cartesian" axes and no equations of the straight line and of conic sections are derived, though a particular equation of the second degree is interpreted as denoting a conic section. Moreover, a large part of the book consists of a theory of algebraic equations, containing the "rule of Descartes" to determine the number of positive and negative roots.”