Geometry

Overview

The Geometry is not a book. It is the third of three appendices printed behind the Discourse on Method in 1637, along with essays on optics and on weather, and the Discourse itself was the preface to all three. Descartes’ point was demonstrative: here is a method, and here are three subjects in which it produces results nobody else has got. The Geometry is the one that worked.

It is about two hundred and forty pages, written in French rather than Latin, and it is one of the hardest books on this shelf. That is deliberate. Descartes says at the end that he has left things out on purpose, so that others may have the pleasure of discovering them, and the omissions are not small. Contemporaries complained. The book only became usable through Frans van Schooten’s Latin editions, which surrounded it with commentary running to several times the length of the original, and it was in that form that Newton read it.

A warning about the name. Cartesian coordinates are not in the Geometry. There are no axes, no origin in the modern sense, no negative coordinates and no plotting of a function. What Descartes has is a reference line and a set of relations between lengths, which does the same work by other means. The clean apparatus taught in school was assembled over the following century.

He was also not alone. Fermat had arrived at something very close, and in a form a modern reader would recognise more easily, several years earlier, though his work circulated only in manuscript.

The argument

Book I: problems that can be constructed using only circles and straight lines. The opening pages are the revolution, and they go past quickly.

Descartes begins by observing that any geometrical problem can be reduced to knowing the lengths of certain straight lines. Then he shows how to add, subtract, multiply, divide and take square roots of lengths by construction. Addition and subtraction are obvious. The others are not, because on the Greek understanding multiplying two lines gives an area, and there is no way back to a line.

His solution is to fix one line and call it unity. With a unit available, similar triangles let you construct a fourth line that stands to one factor as the other factor stands to unity, and that line is the product. The same device gives division and roots. Homogeneity is gone. You can now write an equation with terms of different degrees and every term still means a length, and there is no longer any reason to stop at the third power.

He then sets out the notation, and it is startling how modern it looks: known quantities are a, b, c, unknowns are x, y, z, and powers are written with a raised numeral. Almost everything in that sentence is his.

The rest of Book I applies this to the problem of Pappus, a locus problem from antiquity that had defeated the ancients in its general form. Descartes sets it up, names unknowns, derives an equation relating them, and shows that the case of three or four lines gives a conic section. The method is the point: an unmanageable geometrical problem becomes an equation, and the equation is tractable.

Book II: on the nature of curved lines. Having produced equations for curves, Descartes asks which curves count. He divides them into what he calls geometrical, those that an exact algebraic equation can express, and mechanical, those generated by processes involving motions that cannot be exactly related, such as the spiral and the quadratrix. He admits only the first into geometry, and classifies them by the degree of their equations into pairs of kinds.

This was a mistake in the long run, since the excluded curves turned out to be indispensable and are now called transcendental, but the classification by degree is exactly right and is still how we sort them.

The book also gives his method for finding the normal to a curve at a point, and therefore the tangent, by constructing a circle that meets the curve at two coincident points and requiring the resulting equation to have a double root. It is purely algebraic, it works, and it is one of the direct ancestors of differentiation.

Book III: the construction of solid and higher problems. This is a treatise on the theory of equations, and it is where much of the algebra syllabus comes from. Descartes shows that an equation is divisible by x minus a when a is a root; that the number of roots equals the degree; how to transform an equation to increase, decrease or change the sign of its roots, or to remove a term; and how to depress a quartic to a resolvent cubic. It contains the rule of signs, which relates the number of positive and negative roots to the number of sign changes in the coefficients, stated without proof.

He then solves cubics and quartics geometrically by intersecting a circle with a parabola, and insists throughout on a principle of economy: use the simplest curves that will do the job, and do not construct with conics what circles and lines can manage.

How to read it

Read the Discourse on Method first. The Geometry is presented as its proof, and reading it cold leaves you without the reason it exists.

Then read the first fifteen pages of Book I slowly, with a pencil, and make sure you can construct a product and a square root yourself. That is the whole revolution and it is over quickly.

After that, be selective unless you are a mathematician. The Pappus problem is worth following in outline; the classification of curves in Book II is worth reading for the argument; the equation theory in Book III is worth skimming to see how much of school algebra is here.

Use the Smith and Latham Dover edition, which prints a facsimile of the French with the English opposite and adds the notes that make the gaps survivable.

Expect to be stuck. Descartes meant you to be. If a step seems to be missing, it probably is, and the honest response is to try to supply it rather than assume you have misread.

The common misreading is that this book invented the graph. It did not, and seeing what it does instead is more interesting than the version in the textbooks.

Questions it raises

• The unit length makes products of lines into lines. Is that a discovery about quantity or a convention that turned out to be fruitful?
• Descartes excludes the spiral and similar curves from geometry. What is his criterion really tracking, and why did mathematics abandon it?
• He leaves out steps to give readers the pleasure of discovery. Is that pedagogy, vanity, or a way of protecting priority?
• Algebra can now solve geometrical problems. Does anything get lost when a construction becomes a calculation?
• Fermat had much of this first and published none of it. What does that tell you about how credit in mathematics works?
• The Discourse claims a method that applies to everything. Does the Geometry support that claim, or only show that the method suits one subject extremely well?