Descartes and Fermat — First Half of the 17th-Century Scientific Revolution

The First Half of the Scientific Revolution

After Fibonacci and Seki Takakazu in the previous article, European mathematics enters a phase of rapid modernization. The first half of the 17th century belongs to Descartes (1596–1650) and Fermat (1607–1665) — one a philosopher-mathematician, the other a magistrate and mathematical amateur.

Their shared achievement is the independent discovery of coordinate geometry. Individually, Fermat gives number theory, probability, and optics; Descartes gives symbolic algebra and the classification of curves. And one marginal line of Fermat’s will torment mathematicians for 358 years — the greatest drama in the history of mathematics, until Wiles’s proof in 1995.

René Descartes — Mathematician of “I Think, Therefore I Am”

René Descartes (1596–1650) was a French philosopher, mathematician, and scientist. Known for “Cogito, ergo sum” and remembered as the founder of modern philosophy, he also plays a decisive role in the history of mathematics as the founder of coordinate (analytic) geometry.

Life

• 1596 — Born at La Haye in Touraine (now “Descartes”)
• 1606–1614 — Jesuit college of La Flèche
• 1616 — Law degree at Poitiers
• 1618 — Served in the Dutch army under Prince Maurice; met the mathematician Isaac Beeckman, who awakened his mathematical and physical interests
• 10 November 1619 — At Neuburg in Germany, has his famous “three dreams” and conceives a new plan for the sciences
• 1628–1649 — Lives in Holland for more than 20 years, writing his major works
• 1637 — Discourse on Method with appendices Optics, Meteorology, and Geometry
• 1649 — Invited to Stockholm by Queen Christina of Sweden
• 11 February 1650 — Dies in Stockholm of pneumonia, aged 53

Christina and the Early-Morning Lessons

Christina required Descartes to deliver philosophy lessons daily at 5 a.m. The predawn commute and unfamiliar Nordic climate led to pneumonia, and he died within months. “The queen killed Descartes,” people said at the time — a scene from 17th-century Europe in which a royal summons proves fatal.

The Geometry — The Birth of Coordinate Geometry

Descartes’s principal mathematical work is the three-book appendix La Géométrie (1637):

• Book I — Problems solvable using straight lines
• Book II — Properties of curves; classification of curves
• Book III — Higher-degree equations

Coordinate Geometry — The Algebraization of Geometry

Descartes’s greatest mathematical innovation was to translate geometrical problems into algebraic equations:

• Take two orthogonal axes on the plane (later “Cartesian coordinates”)
• Represent an arbitrary point as (x, y)
• Express figures — lines, circles, conic sections — by equations f(x, y) = 0

The three classical problems of antiquity (doubling the cube, trisecting the angle, squaring the circle) could now be analyzed by algebraic manipulation. Figures and equations became expressible in the same language.

Symbolic Algebra

Descartes shaped modern algebraic notation in ways still used today:

• Unknowns — end of the alphabet (x, y, z)
• Knowns — beginning of the alphabet (a, b, c)
• Exponents — superscripts (x², x³)

The convention of “using x for the unknown” traces back here.

Descartes’s Rule of Signs

In Book III: the number of positive real roots of a polynomial is at most the number of sign changes in its coefficients. Example: x³ − 6x² + 11x − 6 = 0 has sign pattern +, −, +, − — three sign changes, so at most three positive roots (in fact 1, 2, 3).

Still one of the basic tools of numerical computation.

Pierre de Fermat — King of Amateur Mathematicians

Pierre de Fermat (1607–1665) was a French magistrate and mathematician. His profession was counselor at the Parlement of Toulouse (regional judge); mathematics was a complete amateur pursuit — and yet his output surpasses many professional mathematicians.

Life

• August 1607 — Born at Beaumont-de-Lomagne in southwestern France; his father Dominique was a leather merchant and consul
• Studies law and classics in Toulouse
• 1631 — Law degree from Orléans; marries; begins service at the Parlement of Toulouse
• 1631 – life — Counselor at the Parlement, later promoted to criminal chamber judge
• From 1636 — Correspondence with the Mersenne circle in Paris; exchanges with Descartes, Pascal, Roberval, Torricelli
• 12 January 1665 — Dies at Castres, near Toulouse, aged 57

A Mathematician Who Did Not Publish

Fermat held no university post and published almost nothing in his lifetime. His mathematics survived mainly in letters and in marginal notes in books he loved.

Posthumously, his son Clément-Samuel edited his papers and in 1670 published an annotated edition of Diophantus’s Arithmetica, making public 48 marginal notes — including Fermat’s Last Theorem.

Independent Discovery of Coordinate Geometry

Contemporaneously with Descartes, Fermat also arrived at coordinate geometry (manuscript Introduction to Plane and Solid Loci, c. 1636). Their approaches differed:

	Descartes	Fermat
Starting point	Figure → equation	Equation → figure
Emphasis	Algebraization of figures	Analysis of loci
Publication	1637	Posthumous, 1679

Descartes was first to publish, so coordinate geometry is named after him. A classic case of near-simultaneous discovery decided by publication.

Fermat’s Principle — Precursor of Variational Principles

“Light traveling between two points takes the path of least time” (c. 1662) — a principle that unifies reflection and refraction.

It leads directly to Hamilton–Jacobi theory, the calculus of variations, and quantum-mechanical path integrals — a foundational idea. “Nature takes the most efficient path,” made mathematical.

A Precursor of Calculus — 30 Years Ahead

Thirty years before Newton and Leibniz, Fermat had a method for finding maxima and minima (c. 1637):

Set f(x) and f(x+e) approximately equal (adequation), divide by e, and let e → 0.

This is essentially the modern derivative. Newton and Leibniz both build on Fermat’s method — a hidden third candidate in the story of calculus.

Founding Probability — The Correspondence with Pascal

In 1654, the nobleman de Méré posed the “problem of points” — how to divide the stakes when a game of chance is interrupted. Fermat and Pascal exchanged letters, and in doing so laid the foundations of modern probability:

• Fermat solved it by counting cases (combinatorics)
• Pascal solved it by expected values (by induction)

Their letters led to Huygens’s De ratiociniis in ludo aleae (1657), Jacob Bernoulli’s Ars Conjectandi (1713), and Laplace’s Théorie analytique des probabilités (1812). A “problem of gambling” became the starting point of insurance, statistics, and modern financial mathematics.

Contributions to Number Theory

Fermat single-handedly revived 17th-century number theory. Highlights:

Fermat’s Little Theorem

If p is prime and a is not divisible by p, then a^(p-1) ≡ 1 (mod p). A foundation of modern RSA cryptography.

Fermat Primes

Primes of the form F_n = 2^(2^n) + 1. F_0=3, F_1=5, F_2=17, F_3=257, F_4=65537 are all prime. Euler showed in 1732 that F_5 is not prime — one of the rare cases in which Fermat’s conjecture was too hasty.

Infinite Descent

“Assuming a solution exists, construct a smaller one, and continue indefinitely — but positive integers cannot descend forever, so contradiction.” Fermat’s own invention, still a basic tool in number theory.

The Two-Square Theorem

Every prime of the form 4k+1 is uniquely a sum of two squares. Stated by Fermat; proved by Euler.

Fermat’s Last Theorem — a 358-Year Homework Problem

In the margin of Book II, Problem 8 of Bachet’s Arithmetica (1621) — “divide a square into two squares” — Fermat wrote:

“It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power beyond the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.”

That is: for any integer n ≥ 3, there are no positive integers x, y, z with x^n + y^n = z^n.

The Proof — a Lineage of 358 Years

Year	Mathematician	Progress
c. 1637	Fermat	Margin note
1670	Posthumous edition	Published
c. 1770	Euler	Case n=3
c. 1825	Dirichlet, Legendre	n=5
1839	Lamé	n=7
1847	Kummer	Proof for “regular primes”
1955	Taniyama Yutaka	Taniyama conjecture (later the modularity theorem)
1957	Shimura Gorō	Refined as Taniyama–Shimura
1986	Frey, Ribet	Proved: “the Last Theorem follows from Taniyama–Shimura”
1993–1995	Andrew Wiles	Proved Taniyama–Shimura; resolved the Last Theorem

Did Fermat Actually Have a Proof?

Almost certainly not, historians of mathematics now agree. Fermat did leave a proof for n=4, but his method does not generalize, and his margin note was likely too hasty. Wiles’s proof uses elliptic curves, modular forms, and Galois representations — 20th-century tools Fermat could never have known.

Yet that one marginal line drove 350 years of the world’s mathematicians and shaped modern number theory — the greatest practical joke in the history of mathematics.

From This Article to the Next

We have traced the first half of the 17th-century revolution — Descartes and Fermat, the birth of coordinate geometry, probability, and number theory. The next article turns to the second half: the invention of calculus by Newton (1643–1727) and Leibniz (1646–1716). Newton in the “annus mirabilis” 1665–66, Leibniz’s independent path, and the greatest priority dispute in mathematical history — 30 years in which the language of mathematics takes its modern form.