La Géométrie
mathematical appendix to Descartes' Discourse on Method, published in 1637
La Géométrie, of René Descartes, was published in 1637 as an appendix to his Discours de la méthode. This ground-breaking appendix signaled the unification of algebra and geometry into the single subject of analytic or coordinate geometry. Its method transformed geometric lines and curves into algebraic equations, and emphasized the degree of an equation in x and y as a means of classification and as measure of complexity. As an example of the power of the method, Descartes displays his solution to "Pappus' problem." La Géométrie's improved method and notation was absorbed and utilized by mathematicians such as Newton and Leibniz, and served to remove some barriers to clearer thinking in the development of calculus.
Introductory page
Quotes
edit- Translation The Geometry of René Descartes (1925) by David E. Smith and Marcia L. Lantham, unless otherwise noted.
First Book
edit- 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.
- 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.- Note: although in the figure parallel lines DA and CE appear to be perpendicular to the baseline BD, that is not a requirement for the statement to hold true.
- 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.
- 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.
- If... we wish to solve any problem, we first suppose the solution already affected, and give names to all the lines that seem needful for its construction,—to those that are unknown as well as to those that are known. Then, making no distinction between the known and unknown lines, we must unravel the difficulty in a way that shows most naturally the relations between these lines, until we find it possible to express a single quantity in two ways. This will constitute an equation, since the terms of one of these two expressions are together equal to the terms of the other.
- Thus, all unknown quantities can be expressed in terms if a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth.
But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principle benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by anyone at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise.
- I have given these very simple [methods] to show that it is possible to construct all the problems of ordinary geometry by doing no more than the little covered in the four figures that I have explained. This is one thing which I believe the ancient mathematicians did not observe, for otherwise they would not have put so much labor into writing so many books in which the very sequence of the propositions shows that they did not have a sure method of finding all, but rather gathered together those propositions on which they had happened by accident.
- 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.
- The question, then, the solution of which... was completed by no one, is this:
Having three, four or more lines given in position, it is first required to find a point from which as many other lines may be drawn, each making a given angle with one of the given lines, so that the rectangle of two of the lines so drawn shall bear a given ratio to the square of the third (if there be only three); or to the rectangle of the other two (if there be four), or again, that the parallelepiped constructed upon three shall bear a given ratio to that upon the other two and any given line (if there be five); or to the parallelepiped upon the other three (if there be six);or (if there be seven) that the product obtained by multiplying four of them together shall bear a given ratio to the product of the other three, or (if there be eight) that the product of four of them shall bear a given ratio to the product of the other four. Thus the question admits of extension of any number of lines.- Note: this is a restatement of Pappus' Problem.
- Since there is always an infinite number of different points satisfying these requirements, it is also required to discover and trace the curve containing all such points. Pappus says that when there are only three or four lines given, this line is one of the three conic sections, but he does not undertake to determine, describe, or explain the nature of the line required when the question involves a greater number of lines. He only adds that the ancients recognized one of them which they had shown to be useful, and which seemed the simplest, and yet was not the most important. This led me to find out whether, by my own method, I could go as far as they had gone.
Second Book
edit- I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and them classify them in order, is by recognizing the fact that all points of those curves which we may call "geometric," that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by a single equation. If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely.
Quotes about La Géométrie
edit- The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum.
- La Géométrie... is... divided into three parts. The first... contains an explanation of some of the principles of algebraic geometry and shows a real advance over the Greeks. To the Greeks... the product of two variables [corresponded] to the area of some rectangle, and the product of three variables to the volume of some rectangular parallelepiped. Beyond this the Greeks could not go. To Descartes... x2 [is] the fourth term in the proportion 1 : x = x : x2... representable by an appropriate line length which can easily be constructed when x is known. ...
The second part... deals with... a now obsolete classification of curves and with an interesting method of constructing tangents to curves. ...
The third part...concerns... the solution of equations of degree greater than two. Use is made of what we now call "Descartes' rule of signs,"... determining limits to the number of negative and positive roots [of] a polynomial.- Howard Eves, An Introduction to the History of Mathematics (1953)
- 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.
- Howard Eves, An Introduction to the History of Mathematics (1953)
- Book 1 of Géométrie explains how to translate a geometrical problem into an equation ...by making a revolutionary break with the tradition of the theory of proportions to which Galileo adhered. ...Descartes interpreted algebraic operations as closed operations on segments. For instance, if a and b are segments, the product ab is not conceived by Descartes as representing an area, but rather another segment. Prior to the Géométrie the multiplication of two segments... would have been taken as the representation of the area of a rectangle... Descartes' interpretation of algebraic operations was a huge innovation... From Descartes' new viewpoint, ratios are considered as quotients, and proportions as equations. Consequently, the direct multiplication of ratios is allowed... The homogeneity of geometrical dimensions is no longer a constraint for the formation of ratios and proportions... multiplication of time and speed or division of weight by surface's area are all possible. Descartes' approach won great success.
- Niccolò Guicciardini, "Mathematics and the New Sciences," The Oxford Handbook of the History of Physics (2013) ed. Jed Buchwals, Robert Fox, pp. 238-239.
- 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.
- Morris Kline, Mathematical Thought from Ancient to Modern Times (1972)
- Much of the obscurity was deliberate. Descartes boasted that few mathematicians in Europe would understand his work. He indicated the constructions and demonstrations, leaving to others to fill in the details. ...Many explanatory commentaries were written to make Descartes's book clear. ...He says that he omits the the demonstrations of most of his general statements because if one takes the trouble to examine systematically these examples, the demonstrations of the general results will become apparent, and it is of more value to learn them that way.
- Morris Kline, Mathematical Thought from Ancient to Modern Times (1972)
- 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.
- Mario Livio, Is God a Mathematician? (2009)
- His very first definition, that of the multiplication of two line segments, set the new geometry apart from the geometric demonstrations of algebraic results that Arabic mathematicians and Cardano had offered. In Descartes' geometry, multiplication of two line segments is a line segment. ...he stripped terms like "square" and "cube" of traditional geometric connotations but assigned them a new meaning that exploited a fundamental correspondence between arithmetic and geometry. ...he began with a problem stated geometrically; translated the problem into an equation; if necessary, reduced that equation into an irreducible one; and then geometrically constructed the root(s)... algebra was ...a means to an [geometric] end for Descartes.
- Helena M. Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton's Universal Arithmetick (2006)
- 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.
- Helena M. Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton's Universal Arithmetick (2006)
- Suppose 3, 4, 5 or a greater number of lines to be given in position, required a point from which, drawing lines to the given lines, each making a given angle with them, the rectangles of two lines thus drawn from the given point may have a given ratio to the square on the third, if there are three; or to the rectangle of the two others, if there are four; or again, if there are five lines, that the parallelopiped of the two remaining lines, together with a third given line, or to the parallelopiped composed of the three others, if there are six; or again, if there are seven, that the algebraic product of the three others and a given line, or to the four others, if there are eight, and so on.
This was a problem which very much perplexed the ancient geometricians. Pappus says that neither Euclid nor Apollonius could give a solution. He himself knew that when there are only three or four lines the locus was a conic section, but he could not describe it, much less could he tell what the curve would be when the number of lines were more than four. When the number of lines were seven or eight, the ancients could scarcely enunciate the problem, for there are no figures beyond solids, and without the aid of algebra, it is impossible to conceive what the product of four lines can mean.
It was this problem which Descartes successfully attacked, and which, most probably led him to apply algebra generally to geometry.
The following solution is that given by Descartes with a few abbreviations:
AB, AD, EF and GH (fig. 2) are the given lines, C the required point from which are drawn the lines CB, CD, CF and CH making given angles CBA, CDA, CFE, and CHG. AB (=x) and BC (=y) are the principal lines to which all the others will be referred. Suppose the given lines to meet CB in the points R, S, T, and AB in the points A, E and G. Let AE = c and AG = d... By the... method he found the equation to be; which he showed belonged to a circle.- Samuel Wilkes Waud, A Treatise on Algebraical Geometry (1835) p. 148
A History of Mathematics (1893, 1919)
edit- Florian Cajori, source of 1919, 2nd edition, revised and enlarged
- Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties.
- 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 sometimes 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.
- 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...
- 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.
- The first important example solved by Descartes in his geometry is the "problem of Pappus"; viz. "Given several straight lines in a plane, to find the locus of a point such that the perpendiculars. or more generally, straight lines at given angles, drawn from the point to the given lines, shall satisfy the condition that the product of certain of them shall be in a given ratio to the product of the rest." Of this celebrated problem, the Greeks solved only the special case when the number of given lines is four, in which case the locus of the point turns out to be a conic section. 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.
- Methods of drawing tangents [were] invented by Roberval and Fermat. Descartes gave a third method. Of all the problems which he solved by his geometry, none gave him as great pleasure as his mode of constructing tangents. It is profound but operose, and, on that account, inferior to Fermat's. His solution rests on the method of Indeterminate Coefficients, of which he bears the honour of invention. Indeterminate coefficients were employed by him also in solving bi-quadratic equations.
- 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.
Development of Mathematics (1940)
edit- Descartes' presentation differed from that now current. ...he used only an x-axis and did not refer to a y-axis. ...He considered only equations in the first quadrant, as it was thence that he translated the geometry into algebra. This... led to inexplicable anomolies in the translation back from algebra to geometry. As analytic geometry evolved and negative numbers were fearlessly used, the restriction was removed. ...
The new method was not fully appreciated by Descartes' contemporaries, partly because he had deliberately adopted a rather crabbed style. When geometers did see what analytic geometry meant, it developed with great rapidity. But it was only with the development of calculus that analytic geometry came into its own.
- 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.
- 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...
- 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.
A Concise History of Mathematics (1948)
edit- 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.
- Descartes published his "Géométrie" as an application of his general method of unification, in this case the unification of algebra and geometry.
- 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.
- Descartes' merits lie above all in his consistent application of the well developed algebra of the early Seventeenth century to the geometric analysis of the Ancients, and by this, in the enormous widening of its applicability. A second merit is Descartes' final rejection of the homogeneity restrictions of his predecessors which even vitiated Viète's "logica speciosa," so that x2, x3, xy were now considered line segments. An algebraic equation became a relation between numbers, a new advance in mathematical abstraction necessary for the treatment of algebraic curves.