Descartes & Fermat

11.1 René Descartes (1596–1650)

René Descartes was born on March 31, 1596, in La Haye en Touraine, France (the town was later renamed Descartes in his honor). His mother died when he was an infant, and he was raised by his grandmother and then educated at the Jesuit college of La Flèche, one of the finest schools in Europe. There he received a thorough grounding in classical languages, philosophy, and mathematics.

After taking a law degree at the University of Poitiers (1616), Descartes chose not to practice law but instead embarked on a life of travel and intellectual inquiry. He served briefly as a gentleman soldier in the armies of Maurice of Nassau and the Duke of Bavaria. On the night of November 10, 1619, while stationed in a heated room (a poele) in southern Germany, Descartes had a series of three vivid dreams that he interpreted as a divine calling to reform all knowledge on a new, unified foundation. This vision would ultimately produce both his philosophy and his mathematics.

In 1628, Descartes settled in the Netherlands, where he lived for twenty years in relative seclusion, working on his great philosophical and scientific project. The result was the Discourse on the Method of Rightly Conducting One's Reason and of Seeking Truth in the Sciences (1637), commonly known as the Discours de la méthode. This work laid down Descartes' famous method of systematic doubt, arriving at the bedrock certainty Cogito, ergo sum (“I think, therefore I am”).

The Discourse was published with three appendices demonstrating the power of the method in specific domains: La Dioptrique (optics), Les Météores (meteorology), and La Géométrie (geometry). It is this last appendix that concerns us — for in its pages, Descartes created analytic geometry, one of the most consequential inventions in the history of mathematics.

11.2 La Géométrie (1637)

La Géométrie is divided into three books. The first shows how geometric problems can be translated into algebraic equations. The second classifies curves by the degree of their equations. The third applies the method to solve classical problems from ancient geometry. Together, they establish the fundamental principle of analytic geometry: geometric curves can be represented by algebraic equations, and algebraic equations can be interpreted as geometric curves.

Descartes' key innovation was to introduce a system of coordinates. Given two intersecting lines (axes), any point in the plane can be specified by two numbers: its distances along each axis. Descartes did not use the perpendicular axes that we now associate with “Cartesian coordinates” — he typically used oblique axes — but the essential idea was the same.

Descartes also introduced many of the algebraic conventions we still use today. He was the first to use letters from the end of the alphabet ($x, y, z$) for unknowns and letters from the beginning ($a, b, c$) for known quantities. He introduced the superscript notation for powers:$x^2, x^3, x^4, \ldots$ (before Descartes, mathematicians wrote $xx$ or used various abbreviations). He also used the $\sqrt{\phantom{x}}$ symbol consistently.

The philosophical significance of La Géométrie extended beyond specific results. By showing that geometry and algebra were two faces of the same coin, Descartes demonstrated the power of his philosophical method: break complex problems into simple parts, solve the simple parts, and recombine. The union of algebra and geometry created a tool far more powerful than either alone.

11.3 The Power of Coordinates

The bridge between geometry and algebra works in both directions. Geometric objects become equations, and equations become geometric objects. Let us see this correspondence in action.

Lines. A line through two points can be described by a linear equation. The most familiar form is the slope-intercept equation:

$$y = mx + b$$

where $m$ is the slope (the tangent of the angle the line makes with the $x$-axis) and $b$ is the $y$-intercept. Two lines $y = m_1 x + b_1$ and $y = m_2 x + b_2$ are parallel if and only if $m_1 = m_2$, and perpendicular if and only if $m_1 m_2 = -1$.

Circles. A circle with center $(h, k)$ and radius $r$ is described by:

$$(x - h)^2 + (y - k)^2 = r^2$$

This is simply the Pythagorean theorem applied to the distance from $(x, y)$ to $(h, k)$.

Conic sections. The ancient Greeks studied the parabola, ellipse, and hyperbola as cross-sections of a cone. In Cartesian coordinates, all conic sections are described by a single general equation of degree two:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

Intersection of curves. Perhaps the most powerful consequence of analytic geometry is that finding the intersection of two curves reduces to solving a system of equations. For instance, finding where a line meets a circle:

$$y = mx + b \quad \text{and} \quad x^2 + y^2 = r^2$$

Substituting the first into the second:

$$x^2 + (mx + b)^2 = r^2 \quad \Longrightarrow \quad (1 + m^2)x^2 + 2mbx + (b^2 - r^2) = 0$$

This is a quadratic in $x$, giving zero, one, or two intersection points depending on the discriminant. What had been a difficult geometric construction becomes routine algebra.

The distance formula. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

This is simply the Pythagorean theorem in coordinate form, and it makes distance calculations that required elaborate geometric constructions into simple arithmetic.

11.4 Descartes' Rule of Signs

In La Géométrie, Descartes stated an elegant result about the roots of polynomials that now bears his name. The Rule of Signs provides an upper bound on the number of positive and negative real roots of a polynomial, based solely on the pattern of sign changes in its coefficients.

11.5 Pierre de Fermat (1601–1665)

Pierre de Fermat was born on August 17, 1601, in Beaumont-de-Lomagne, in southern France. His father was a wealthy leather merchant, and Fermat received an excellent education before entering the legal profession. He became a conseiller (councilor) at the Parlement de Toulousein 1631, a judicial position he held for the rest of his life.

Fermat was the consummate “amateur” mathematician — not in the sense of being unskilled, but in the original sense of the word: one who pursues a discipline for the love of it. He held no academic position and published almost nothing during his lifetime. Instead, he communicated his discoveries through an extensive network of correspondence with other mathematicians, including Mersenne, Pascal, Huygens, Roberval, and others. His letters were often terse announcements of results without proofs, accompanied by challenges to the recipient to provide their own demonstrations.

This style of mathematical communication, frustrating as it often was to his correspondents, was characteristic of a period before mathematical journals existed. Marin Mersenne in Paris served as a sort of human post office, collecting and distributing mathematical discoveries among the scholars of Europe. Fermat was one of the most prolific contributors to this network, and his results in number theory, analytic geometry, probability, and proto-calculus make him one of the most versatile mathematicians of the seventeenth century.

11.6 Fermat's Analytic Geometry

One of the most remarkable coincidences in the history of mathematics is that Fermat developed analytic geometry independently of Descartes, and in fact before Descartes published. Fermat's manuscript Ad Locos Planos et Solidos Isagoge (“Introduction to Plane and Solid Loci”) was written around 1636 and circulated among Parisian mathematicians via Mersenne, though it was not formally published until 1679, after Fermat's death.

Fermat's approach differed from Descartes' in an important way. Descartes started with geometric problems and showed how to translate them into algebra. Fermat started with equations and showed that they define geometric curves. Specifically, Fermat demonstrated that every equation of the form:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

represents a conic section (possibly degenerate), and conversely, every conic section can be expressed by such an equation. He classified the curves systematically:

• $Dx + Ey = F$: a straight line (locus planus, “plane locus”)
• $x^2 = Dy$: a parabola
• $x^2 + y^2 = r^2$: a circle
• $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: an ellipse
• $xy = k$: a hyperbola (referred to its asymptotes)

In some respects, Fermat's treatment was more systematic and more modern than Descartes'. While Descartes used only one axis (the $x$-axis) and measured the second coordinate as an oblique distance, Fermat used two perpendicular axes from the start, closer to our modern practice. However, Descartes' work was published first and had the enormous advantage of being printed, whereas Fermat's remained in manuscript form and was known only to a small circle.

The priority question is somewhat moot: both mathematicians arrived at the same fundamental insight independently, each approaching it from a different direction. Together, their work established analytic geometry as a permanent part of mathematics.

11.7 Fermat's Number Theory

Fermat's deepest and most lasting contributions are in number theory, a subject he essentially created as a modern discipline. His inspiration came from studying Diophantus' Arithmetica in the 1621 Latin edition by Bachet. In the margins of this book, Fermat recorded dozens of theorems and conjectures, many of which would occupy mathematicians for centuries.

Proof sketch. Consider the $p - 1$ numbers $a, 2a, 3a, \ldots, (p-1)a$. We claim these are all distinct modulo $p$. If $ia \equiv ja \pmod{p}$ for $1 \le i < j \le p-1$, then $p \mid (j - i)a$. Since $p \nmid a$ (by assumption) and $p$ is prime, we need $p \mid (j - i)$, which is impossible since $0 < j - i < p$.

Therefore $\{a, 2a, \ldots, (p-1)a\}$ is a permutation of $\{1, 2, \ldots, p-1\}$modulo $p$. Multiplying all elements:

$$a \cdot 2a \cdot 3a \cdots (p-1)a \equiv 1 \cdot 2 \cdot 3 \cdots (p-1) \pmod{p}$$

$$a^{p-1} \cdot (p-1)! \equiv (p-1)! \pmod{p}$$

Since $\gcd((p-1)!, p) = 1$, we can cancel $(p-1)!$ to obtain $a^{p-1} \equiv 1 \pmod{p}$. $\square$

Fermat primes. Fermat conjectured that all numbers of the form:

$$F_n = 2^{2^n} + 1$$

are prime. The first five Fermat numbers are:

• $F_0 = 2^1 + 1 = 3$ (prime)
• $F_1 = 2^2 + 1 = 5$ (prime)
• $F_2 = 2^4 + 1 = 17$ (prime)
• $F_3 = 2^8 + 1 = 257$ (prime)
• $F_4 = 2^{16} + 1 = 65537$ (prime)

But in 1732, Euler showed that $F_5 = 2^{32} + 1 = 4294967297 = 641 \times 6700417$ is composite, disproving Fermat's conjecture. No further Fermat primes have been found, and it is now suspected (but not proven) that there are only finitely many.

Method of infinite descent. Fermat invented a powerful proof technique that he called his “method of infinite descent.” The idea is a variant of proof by contradiction: assume a solution to a problem exists with certain properties, then show that asmaller solution must also exist, contradicting the minimality of the first. This method is essentially the well-ordering principle in disguise, and Fermat used it to prove several of his number-theoretic results.

11.8 Fermat's Last Theorem

The most famous statement in all of number theory was scribbled by Fermat in the margin of his copy of Diophantus' Arithmetica, beside a discussion of Pythagorean triples. Fermat wrote (in Latin):

He then added the tantalizing note:

Fermat's proof for n = 4. The only case for which Fermat left a complete proof is $n = 4$. He actually proved a stronger result: that there is no Pythagorean triple in which the hypotenuse is a perfect square. Equivalently, the equation $x^4 + y^4 = z^2$ has no positive integer solutions.

Proof by infinite descent. Suppose $(x, y, z)$ is a solution of $x^4 + y^4 = z^2$ in positive integers, with $z$ minimal. We may assume $\gcd(x, y) = 1$ (otherwise divide through). Then $(x^2)^2 + (y^2)^2 = z^2$, so $(x^2, y^2, z)$ is a primitive Pythagorean triple. By the classification of Pythagorean triples, we may assume (possibly swapping $x$ and $y$):

$$x^2 = m^2 - n^2, \quad y^2 = 2mn, \quad z = m^2 + n^2$$

where $m > n > 0$, $\gcd(m, n) = 1$, and $m, n$ have opposite parity. From $y^2 = 2mn$ and $\gcd(m, n) = 1$, with $m$ odd and $n$ even (say $n = 2k$), we get $y^2 = 4mk$, so $(y/2)^2 = mk$. Since $\gcd(m, k) = 1$, both $m$ and $k$ must be perfect squares: $m = s^2, k = t^2$.

From $x^2 = m^2 - n^2 = s^4 - 4t^4$ we get... (continuing the descent). Through careful analysis of the parametrization, one arrives at a new solution $(x', y', z')$ with $z' < z$, contradicting the minimality of $z$. Therefore no solution exists. $\square$

Note that the case $n = 4$ implies all cases where $n$ is a multiple of 4, since if $x^{4k} + y^{4k} = z^{4k}$, then $(x^k)^4 + (y^k)^4 = (z^k)^4$. Similarly, it suffices to prove the theorem for $n = 4$ and for all odd primes $n = p$, since every $n > 2$ is either divisible by 4 or by an odd prime.

The 350-year quest. After Fermat, progress was painfully slow:

• Euler (c. 1770): Proved the case $n = 3$ (with a gap later filled by others)
• Dirichlet & Legendre (1825): Proved $n = 5$
• Lamé (1839): Proved $n = 7$
• Kummer (1850s): Proved it for all “regular” primes (a large class of primes)
• Computer-assisted proofs (1970s–1990s): Verified for all $n < 4{,}000{,}000$
• Andrew Wiles (1995): Proved the full theorem by establishing the modularity theorem for semistable elliptic curves

Wiles' proof, announced in 1993 and completed (with Richard Taylor) in 1995, runs over 100 pages and draws on the deepest parts of modern number theory and algebraic geometry. It is virtually certain that Fermat did not have a valid proof of the general case — the mathematics required simply did not exist in the seventeenth century.

11.9 Fermat and Calculus

Less well-known than his number theory, but equally important historically, are Fermat's contributions to what would become calculus. He developed a method he called adequality (from the Latin adaequalitas, inspired by Diophantus' notion of “approximate equality”) for finding maxima, minima, and tangent lines to curves.

The method of adequality for maxima/minima. To find the maximum or minimum of a function $f(x)$, Fermat proceeded as follows:

1. Replace $x$ by $x + e$ (where $e$ is a small quantity)
2. Set $f(x + e)$ “adequal” to $f(x)$: $f(x + e) \approx f(x)$
3. Subtract $f(x)$ from both sides
4. Divide through by $e$
5. Set $e = 0$
6. Solve for $x$

The tangent method. Fermat also used adequality to find tangent lines. Given a curve $y = f(x)$, the tangent at a point $(x_0, f(x_0))$ is determined by its slope. Fermat's procedure amounts to computing what we now call the derivative.

Fermat's method of adequality is, in modern language, the computation of the derivative by the limit definition (without the formal notion of limit). Historians have debated the precise logical status of Fermat's $e$ — is it an infinitesimal? A quantity that approaches zero? Fermat never clarified this, but the procedure was correct and highly effective. Newton and Leibniz would later provide more systematic frameworks, but Fermat's methods were an essential step toward calculus.

Fermat also computed areas under curves. He showed that for $y = x^n$ with $n$ a positive integer, the area under the curve from 0 to $a$ equals $\frac{a^{n+1}}{n+1}$. He even extended this to rational exponents, finding that for $y = x^{p/q}$ (with $p/q \ne -1$):

$$\int_0^a x^{p/q}\,dx = \frac{a^{p/q + 1}}{p/q + 1}$$

This is the power rule for integration, obtained decades before Newton and Leibniz. The single case Fermat could not handle was $n = -1$ (the integral of $1/x$, which gives the logarithm), as his geometric subdivision method broke down.

11.10 Legacy

The contributions of Descartes and Fermat, taken together, mark a watershed in the history of mathematics. Their achievements may be summarized along several dimensions:

The merger of algebra and geometry. Before Descartes and Fermat, algebra and geometry were separate disciplines with different methods, different standards of rigor, and different communities of practitioners. After their work, the two were permanently fused. Any geometric problem could be attacked algebraically, and any algebraic relation could be visualized geometrically. This fusion was the single most important structural change in mathematics since the axiomatic method of the ancient Greeks.

Foundations for calculus. Fermat's methods of adequality — for finding maxima, minima, tangent lines, and areas — were direct precursors to differential and integral calculus. Newton himself acknowledged his debt to Fermat. When Newton wrote “If I have seen further, it is by standing on the shoulders of giants,” Fermat was among the giants he had in mind.

Modern number theory. Fermat's investigations of primes, divisibility, and Diophantine equations established number theory as a vibrant field. His little theorem remains fundamental (it underpins the RSA cryptosystem that secures modern internet communications). His Last Theorem, though not proven for over three centuries, stimulated the development of algebraic number theory, the theory of elliptic curves, and modular forms — areas that remain at the frontier of mathematical research.

The power of notation. Descartes' notational innovations — using $x, y, z$ for unknowns, superscript exponents, the $\sqrt{\phantom{x}}$ sign — became the universal language of mathematics. Good notation is not merely convenient; it actively shapes mathematical thought by making certain ideas easy to express and others hard to overlook.

Together, these two men — one a philosopher-scientist who sought to rebuild all knowledge from first principles, the other a provincial lawyer who did mathematics for the sheer joy of it — transformed the mathematical landscape and set the stage for the explosive developments of the eighteenth century.