René Descartes, *Geometria à Renato Des Cartes anno 1637 Gallicè edita; postea autem unà cum notis Florimondi de Beaune* (Amsterdam, 1659), portrait.

By 1618, René Descartes (1596-1650), was a soldier in the army of Stadhouder, Prince Maurice of Orange (Maurits van Oranje), at the military school in Breda in the Netherlands. It was there that his interest in mathematics began to flourish and he developed his ideas over the following decade while travelling throughout Europe. By the end of the 1620s he had settled in Holland, convinced that only mathematics provided a sufficiently certain basis for all knowledge. He wrote *La Géométrie* as an appendix to his* Discours de la méthode*. It was published in French in 1637. The spread of Cartesian philosophy was due in no small way to the work of Frans van Schooten the Younger (1615-1660), who was responsible for the first Latin edition of *La Géométrie*, published in 1659. It is this edition that is in the Edward Worth Library. Van Schooten’s name is prominent on the title page, moreover it is clear that this copy belonged to John Worth (1648-1688), for it is signed by him, in 1682, when his son, Edward (1676-1733), was only six years old.

René Descartes, *Geometria à Renato Des Cartes anno 1637 Gallicè edita; postea autem unà cum notis Florimondi de Beaune* (Amsterdam, 1659), title page.

The first complete edition of *La Géométrie* in English appeared in 1925 (translated by Smith and Latham, derived from both the French and Latin editions). It was republished by Dover in 1954 and this edition is accessible online. Descartes gives a very clear idea of the ambition of his project:

If, then, we wish to solve any problem, we first suppose the solution already effected, 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 known and unknown lines, we must unravel the difficulty in any 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.

*La Géométrie*, trans. Smith & Latham (1925/1954, p. 6)

As Smith and Latham point out in their commentary, the idea of first supposing the solution already effected goes back to Plato and Pappus; Descartes’ ground-breaking innovation was to assert that ‘any’ geometric problem can be expressed as an algebraic equation. Thus geometry and algebra were, in effect, brought together in what became known as analytic geometry, a topic usually introduced in the Irish Junior Cycle as ‘coordinate geometry’. Let us consider two examples from *Geometria* (1659 edition).

René Descartes, *Geometria à Renato Des Cartes anno 1637 Gallicè edita; postea autem unà cum notis Florimondi de Beaune* (Amsterdam, 1659), vol 1, p. 6.

Here Descartes discusses, a single geometrical diagram, the solution of the following three equations:

- z² = az + b²
- y² = –ay + b²
- x4 = –ax² + b²

In each case, the coefficient, a, is represented by the diameter of the circle shown, while b is the length of the side LM of the triangle. So, given a and b, to solve any of the three equations geometrically, draw a right-angled triangle, LMN (with hypotenuse MN), where LM = b and LN = a/2. Then z = OM, y = PM and x = √PM. All of these follow from Pythagoras’ theorem applied to LMN:

(z – a/2)² = (a/2)² + b² = (y + a/2)²

It is worth noting that the symbol looking like an incomplete ‘∞’ is what Descartes used for equality. The observant visitor will have noticed that this diagram is used in the wallpaper for the exhibition!

The second example comes from book II. The description of EC in Descartes’ text as ‘the intersection of the ruler GL and the rectilinear plane figure CNKL’ (‘*per intersectionem regulæ GL & plani rectilinei CNKL*’) is not so helpful. Rather, begin with angle GAK being a right-angle, where GA = a and AK = d are fixed lengths. Let L be a point on the line segment AK, and a ruler passes through G and L, allowed to pivot at G as the point L moves up and down.

René Descartes,

Now introduce a point N, so that NLK is a right-angle and NL = c is fixed. Join K to N and extend to meet the ruler (GL) at C. The problem is to find the equation defining the position of C (as L moves), and, in particular, the relationship between its coordinates, x = AB and y = BC. Descartes identifies two pairs of similar triangles: KBC is similar to KLN, and LAG is similar to LBC. Comparing vertical and horizontal sides gives:

b/c = (b + z)/y and z/y = (z + x)/a

These equations allow z to be expressed in two ways: z = by/c – b = xy/(a – y), from which it follows that y² = (a + c)y – (c/b)xy – ac, agreeing with Descartes conclusion (p. 23): ipsa nulla alia est quam Hyperbola (it is none other than a hyperbola). This would all be very well if a, b and c were constants (independent of x and y). Although a and c are constants, b depends on x. However d = x + z + b = AK is constant. Using the substitution b = (d – x)c/y allows us to eliminate b from Descartes’ equation, and conclude that y² = (a + c)y – [(a + c)/d]xy + (ac/d)x – ad, and indeed *ipsa nulla alia est quam Hyperbola*!

**Sources**

Descartes, René, *The Geometry*, trans. D.E. Smith and M.L. Latham, (Dover, New York, 1954).

Mastin, Luke, Descartes,* The Story of Mathematics* (2010).

O’Connor, J.J. and Robertson E.F., Frans van Schooten, *MacTutor History of Mathematics*, (University of St Andrews, Scotland, February 2009).

O’Connor, J.J. and Robertson E.F., René Descartes, *MacTutor History of Mathematics*, (University of St Andrews, Scotland, February 2014).

**Text:** Maurice OReilly and John Brady.