Five-minute digest
The Geometry
Summary
La Géométrie is the third and last of the essays that Descartes published at Leiden in 1637 after the Discourse on Method, following the Optics and the Meteors. It is meant to show the method at work in mathematics, and it does so by joining two disciplines that had been kept apart since antiquity: the geometry of Euclid and Apollonius, which constructed figures, and the algebra of the sixteenth-century 'analysts', which solved equations. Its first sentence claims that every problem of geometry can be reduced to terms such that one needs only to know the lengths of certain straight lines to construct it. Book I makes good on that claim by showing that adding, subtracting, multiplying, dividing and extracting roots can all be carried out on line segments once a unit length is chosen, so that a product of two lines is another line rather than an area. Descartes names known quantities with early letters (a, b, c) and unknowns with late ones (x, y, z), writes powers with raised numerals, and so fixes much of the notation still taught. He then attacks the problem of Pappus, a locus problem about distances to three, four or more lines that the ancients had left unfinished, and shows how to write it as an equation in two unknowns. Book II asks which curves geometry should admit. Descartes rejects the old division into plane, solid and linear problems and accepts any curve traced by a continuous motion in which every part is fixed by one prior motion, which amounts to curves expressible by an algebraic equation; he excludes the spiral and the quadratrix as 'mechanical'. He completes the Pappus problem, gives a method for drawing normals to a curve, which he calls the most useful problem he knows, and describes the ovals needed for lenses in the Optics. Book III treats equations themselves: an equation has as many roots as its degree, some 'false' (negative) and some 'imaginary', and the rule of signs estimates how many are true or false. He solves cubic and quartic problems, among them the duplication of the cube and the trisection of the angle, by intersecting a circle with a parabola. The book is compressed and sometimes deliberately incomplete; Descartes ends by saying he has left some things for later readers to discover. For students, it shows the source of the algebra they use and the confidence of a mind that believed order and measure could reach every part of geometry.
Top Lessons
A line times a line is a line
Greek geometry treated the product of two lengths as an area and of three as a solid, which made four or more factors meaningless. By choosing a unit segment, Descartes shows that multiplication and division can be performed with similar triangles and yield a segment. That small change frees algebra from the dimensions of space and lets equations of any degree describe geometric problems.
Naming the unknown before you know it
Descartes's rule is to treat the problem as already solved, give names to every line needed, unknown as well as known, and then find as many equations as there are unknowns. This is the analytic method that the Rules for the Direction of the Mind had described in general terms. Students who set up word problems in algebra are following it.
Which curves belong in geometry
Book II argues about the boundaries of a discipline. Descartes admits any curve traced by motions whose relations are exactly determined and excludes curves like the spiral, whose motions involve an unknown ratio of straight to curved. The argument concerns exactness and what counts as knowing a curve, and later mathematicians, including Leibniz, would redraw the line he drew.
Roots, signs and the honest unfinished ending
Book III shows that an equation is built from its roots and can have as many as its degree, and it gives names to negative and imaginary roots even while treating them as less than true. Descartes closes by leaving some work undone for readers to find. Students can admire the reach of the method while noticing what its author did not yet understand and what he chose to leave out.