One of Descartes' mathematical achievements is the unification of…

One of Descartes' mathematical achievements is the unification of the dimension of quantity.

http://www.fi.uu.nl/publicaties/literatuur/ichme/Dig_where_you_stand-5.pdf

On French heritage of Cartesian geometry in Elements from Arnauld, Lamy and Lacroix Évelyne Barbin LMJL UMR 6629 & IREM, Université de Nantes, France Abstract When Descartes wrote La géométrie in 1637, his purpose was not to write "Elements" with theorems and proofs, but to give a method to solve "all the problems of geometry". However, in his Nouveaux Éléments de Géométrie in 1667, Antoine Arnauld included two important Cartesian conceptions. The first one is the systematic introduction of arithmetical operations for geometric magnitudes and the second one is what he called "natural order", that means Cartesian order which goes from the simplest geometric objects (straight lines) to others. This last conception led Arnauld to numerous novelties, mainly, a chapter on "perpendicular and oblique lines", and new proofs for Thales and Pythagoras theorems. In 1685, Bernard Lamy followed Arnauld's textbook in his Éléments de géométrie, in which he also introduced Cartesian method to solve problems. Our first aim is to analyze incorporations of Cartesian conceptions and Cartesian method into Arnauld and Lamy's Éléments. Our second aim is to analyze their impact for the heritage of Cartesian geometry into mathematical teaching, especially the "natural order" coming from Arnauld and the "application of algebra to geometry" coming from Lamy. In this framework, we show that the geometric teaching of Sylvestre-François Lacroix played an important role in the 19th century and beyond. Keywords: René Descartes, Antoine Arnauld, Sylvestre-François Lacroix, Cartesian order, arithmetization of geometry Introduction: Cartesian order and arithmetization of geometry Towards the end of the 1620s, René Descartes wrote Règles pour la direction de l'esprit [Rules for the Direction of the Mind]. This text had never been achieved and published in his lifetime, but it is interesting to know that it had been read by Antoine Arnauld. In his Rules, Descartes criticized Aristotle's science based on syllogisms, because they can conclude with certainty but they banish obviousness (Rule X), and he gave his proper conception of science. Indeed, he wrote in Rule XII: "We can never understand anything beyond these simple natures and a certain mixture or composition of them with one another" (Descartes, 1998, p. 155). Hence, all human knowledge consists in this one thing, to wit that we distinctly see how these simple natures together contribute to the composition of the other things (Descartes, 1998, p. 161). In that way, he proposed to substitute an order of simplicity of things instead of a logical order of propositions. Descartes continued to call deduction the manner by which a composite nature can be obtained from simple ones. Thus, Aristotelian and Cartesian deductions are different because, in the first one, propositions are deduced from others by logical rules and, in the second one, composed things are deduced from simple ones by simple operations. Simple things and simple operations of geometry are introduced as soon as the f irst sentence of La géométrie (1637), where Descartes wrote: Any problem in geometry can easily be reduced to such terms that a knowledge of the length of certain straight lines is sufficient for its construction. Just as arithmetic consists of only four or five operations, namely addition, subtraction, multiplication and the extraction of roots [...] (Descartes, 1954, p. 2). So, simple things are straight lines and simple operations are arithmetic operations. This 'arithmetization' of geometry, leans on the introduction of one line called "unit" by Descartes, by analogy with arithmetic. Indeed, this unit permits us to obtain a product of two lines BD and BC, not as a rectangle, like in Greek geometry, but as a simple line. If AB is the unit, then BE is the product of BD and BC (figure 1 left). It also permits us to divide two segments and to obtain a segment. To consider a square root of a line, has no meaning in Greek geometry, but in Cartesian geometry, if FG is the unit then GI is the square root of GH (figure 1 right). Fig. 1. Product of two segments and square root of a segment (Descartes, 1954, p. 4) Descartes pointed out that often, it is not necessary to draw lines and it is sufficient to designate them by single letters, to which symbols of arithmetic will be applied. Moreover, thanks to the unit, it is possible to consider for instance a3 or b2 as simple lines, and, for instance, to consider the cube root of a2b2 – b without taking into account the geometric meaning of this formula. Descartes' purpose was to provide a systematic method to solve problems of geometry by deducing unknown lines from known lines. This method consists of translating problems by equations on lines and to solve these ones. In the First Book of La géométrie, Descartes used his method to solve, not elementary problems, but a difficult problem left to us by Pappus. In the Second Book, in accordance with his general conception and thanks to the unit line, he considered curves as composed by simple lines by means of arithmetic operations, when for a given line AG and for each point C of the curve, there exists a single equation linking CM and MA. These lines are called "geometric" and the others "mechanical". So, he did not introduce a "Cartesian coordinate system". He used his method to find normal lines CP to a "geometric curve" (figure 2). Fig. 2. Normal line to a "geometric curve" in La géométrie (Descartes, 1954, p. 97)

Bjarnadóttir, K., Furinghetti, F., Krüger, J., Prytz, J., Schubring, G. & Smid, H. J.(Eds.) (2019). "Dig where you stand" 5. Proceedings of the fifth International Conference on the History of Mathematics Education. Utrecht: Freudenthal Institute.