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.

Plot or Argument

• Book I: Problems that need only straight lines and circles

  Arithmetic operations are shown to correspond to ruler-and-compass constructions on segments, using a unit line, so that algebraic products remain lines. Descartes introduces his letter conventions and exponents, shows how a problem is solved by naming every line, known and unknown, and finding as many equations as unknowns, and states the Pappus locus problem from Pappus's Collection.

• Book I: The Pappus problem set up

  Taking one given line and one unknown segment along it as references, Descartes expresses the distances from a point to each given line in terms of two unknowns, so that the locus for three or four lines becomes an equation of the second degree in x and y, and for more lines an equation of higher degree.

• Book II: Which curves count as geometrical

  Descartes criticises the ancients for calling some curves mechanical, proposes linked instruments that trace curves by regulated motion, and classes admissible curves into genres by the degree of their equations. Curves such as the spiral and quadratrix, whose motions are not so related, are left out.

• Book II: The Pappus solution, normals and optical ovals

  He shows that the three- and four-line locus is always a conic section and discusses the five-line case. He gives a method for finding the normal at any point of a curve by a circle that touches it at a double root, and describes ovals that refract light to a point, linking the book to the Optics.

• Book III: The nature and roots of equations

  Equations are built by multiplying factors such as x - 2 = 0, so an equation can have as many roots as its degree. He distinguishes true (positive), false (negative) and imaginary roots, states the rule of signs, and shows how to increase, decrease or change the sign of all roots and remove the second term.

• Book III: Solid and supersolid problems

  Cubic and quartic equations are solved by intersecting a fixed parabola with a circle, applied to finding two mean proportionals and trisecting an angle. He extends the approach to equations of the fifth and sixth degree and closes by leaving further cases to the reader.

Characters

• Pappus of Alexandria, ancient source

  Fourth-century Greek geometer whose Collection reports the multi-line locus problem that Euclid and Apollonius had not fully solved; Descartes quotes him in Latin and makes the problem the test case of his method.

• Apollonius of Perga, ancient authority

  Author of the Conics, whose treatment of conic sections Descartes claims to complete by showing that the three- and four-line locus always yields one of them.

• The ancient geometers, foil

  Descartes argues that they lacked a general method, distrusted curves they could not construct with ruler and compass, and concealed their analysis, a charge historians have since qualified.

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.

  • Draw Descartes's construction for the product of two segments using a unit length and similar triangles. Why does the result have to be a line?
  • What could Greek geometers not express because they took a product of lines as an area?
  • Is the unit length a discovery or a convention, and does the answer matter?

• 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.

  • Why is it useful to pretend the problem is already solved?
  • Descartes chose early letters for knowns and late letters for unknowns. What makes a notation good?
  • Compare this method with Euclid's proofs, which begin from what is given. What does each make easy?

• 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.

  • What reason does Descartes give for admitting curves the ancients had called mechanical?
  • Why does he still exclude the spiral and the quadratrix?
  • Who should decide what belongs inside a science: its practitioners, its usefulness, or its first principles?

• 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.

  • Why does Descartes call negative roots 'false'? What would have to change for them to count as true?
  • Test the rule of signs on an equation you can factor. Does it give the exact number of positive roots or only an upper bound?
  • Is leaving problems for later readers generous, proud, or both?

Context

The Geometry belongs to the generation that turned algebra into a general language for mathematics. François Viète's analytic art (from 1591) had used letters for known and unknown magnitudes but kept the Greek rule that a product of lines has dimension; Descartes dropped that rule and simplified the notation. Pierre de Fermat reached a coordinate treatment of loci independently at about the same time in a manuscript that circulated in Paris but was printed only in 1679. Descartes's coordinates are not the perpendicular x- and y-axes of later textbooks: he usually measures from one reference line, with the second distance drawn at a fixed angle, and he rarely lets negative values stand. The 'Cartesian plane' as students meet it was shaped by later commentators.

Composition

Descartes had worked on the Pappus problem since about 1631-1632, when the Leiden orientalist and mathematician Jacob Golius drew his attention to it, and he wrote the Geometry while the Discourse was being printed. It was published anonymously at Leiden by Jan Maire in 1637 with the Discourse, the Optics and the Meteors, in French rather than Latin so that it could reach readers without university training. Frans van Schooten's Latin translation with commentary (1649; enlarged 1659-1661, with treatises by Hudde, de Beaune and others) made it usable across Europe.

Reception

Fermat and Roberval disputed Descartes's method of normals and his claims about the ancients, and many readers found the book obscure, which Descartes partly intended. Through van Schooten's Latin editions it became the starting point for the next generation, including Newton, who studied it closely in the 1660s, and the analytic geometry that underlies the calculus of Newton and Leibniz. The rule of signs and the notation for powers passed into standard algebra. The work was chosen for the Great Books of the Western World (2nd ed., 1990), volume 28.

Comparisons

• Discourse on Method

  The Geometry was printed after the Discourse as a demonstration of its method. Part II of the Discourse names the four rules (accept only what is evident, divide difficulties, proceed in order from simple to complex, enumerate completely) and says Descartes found them by reflecting on geometrical analysis and algebra; the Geometry shows those rules applied to curves and equations.

• Elements, Book I

  Euclid builds from definitions and postulates to constructions with ruler and compass and treats magnitudes geometrically, never multiplying lines into lines. Descartes keeps Euclid's constructions in Book I but redefines multiplication with a unit, so students who know Book I can see precisely where the old and the new geometry part ways.

• Rules for the Direction of the Mind

  The unfinished Rules, especially Rules XIV-XVI, recommend representing magnitudes as lines and rectangles and using short symbols to spare the memory. The Geometry carries out that programme a decade later, with the unit length removing the need to think in rectangles.

• The Method (Geometrical Solutions Derived from Mechanics)

  Archimedes's Method reveals the mechanical reasoning by which he found results he then proved geometrically, the kind of hidden analysis Descartes accused the ancients of concealing. It was rediscovered only in 1906, so Descartes never saw the evidence that would have both supported and complicated his charge.

• Scientific and Mathematical Treatises

  Pascal's work on conics and the arithmetical triangle comes from the same Parisian circle around Mersenne, but Pascal pursued projective and synthetic methods rather than Descartes's algebra, showing that the century's geometry did not move along only one road.

Quotes

Creative Prompts

• Classroom

  Using only a straightedge, a compass and a segment marked as the unit, construct the product, quotient and square root of two given segments as Descartes does in Book I, and write a short explanation of each step.

• Essay

  Descartes accuses the ancient geometers of hiding their method of discovery. Using Euclid Book I and what you know of Archimedes, argue whether the charge is fair.

• Dialogue

  Write a conversation in 1638 between Descartes and Fermat, who has his own way of finding tangents, in which each must explain his method to the other without using modern calculus.

• Classroom

  Take one problem from a current algebra textbook and solve it twice: once in Descartes's style, naming every line and finding equations, and once by a purely geometric construction. Report which was clearer and why.

Further Reading

• The Geometry of René Descartes, trans. David Eugene Smith and Marcia L. Latham

  The 1925 English translation printed facing a facsimile of the 1637 French; reprinted by Dover.

• Discourse on Method, René Descartes

  The preface to the Geometry and the source of its four rules.

• Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction, Henk J. M. Bos

  The standard scholarly study of what Descartes thought made a construction or curve acceptable.

• History of Analytic Geometry, Carl B. Boyer

  Places Descartes beside Viète, Fermat and their successors.