The Age of Calculus: 17th and 18th Centuries

The 17th and 18th centuries are when mathematics becomes modern. In 1600, mathematics was a technical craft with roots in antiquity and medieval Arabic texts. By 1800, it was the discipline we recognize today: symbolic, analytic, rigorous, and in productive dialogue with physics.

The scientific revolution

Descartes and analytic geometry

René Descartes (1596–1650) published La Géométrie in 1637 as an appendix to his Discourse on the Method. In it he laid out a way to convert geometric problems into algebraic ones by assigning coordinates to points. Every curve became an equation; every equation, a curve. This analytic geometry unified two previously separate strands of mathematics and made calculus possible a generation later.

Pierre de Fermat, independently, developed similar ideas at about the same time.

Pierre de Fermat

Fermat (1607–1665) was a lawyer by profession and a mathematician by passion. He contributed to analytic geometry, number theory, and probability. His margin note in Diophantus’s Arithmetica — claiming a “marvellous proof” that $x^n + y^n = z^n$ has no integer solutions for $n > 2$ — set the problem that would stand for 358 years until Wiles.

Together with Blaise Pascal, Fermat founded modern probability theory in their correspondence about gambling problems.

Pascal

Blaise Pascal (1623–1662) contributed to probability, projective geometry, and hydrostatics, and built a mechanical calculator. The Pascal’s triangle of binomial coefficients bears his name (though it was known in China and India centuries earlier).

Calculus

Newton

Isaac Newton (1643–1727) developed his “method of fluxions” — calculus — during the plague years of 1665–1666 while at home in Woolsthorpe. He used it to derive Kepler’s laws of planetary motion, publishing the full synthesis in the 1687 Principia.

Leibniz

Gottfried Wilhelm Leibniz (1646–1716) developed calculus independently in the 1670s. His notation — $dy/dx$ , $\int$ — is what we still use. The priority dispute between the Newtonians and Leibnizians was a full-scale intellectual war, with the Royal Society appointing an “impartial” committee in 1712 (written, it turned out, largely by Newton himself).

The result was a split: England stayed loyal to Newton’s notation and methods and fell behind. The Continent adopted Leibniz’s and raced ahead. Only in the 19th century did English mathematics recover.

The Bernoullis and Euler

The Bernoulli family of Basel produced three generations of first-rate mathematicians. Jakob Bernoulli (1654–1705) pioneered the use of calculus for probability (Bernoulli trials, the law of large numbers). Johann Bernoulli (1667–1748) developed the calculus of variations. Daniel Bernoulli (1700–1782) contributed to fluid dynamics.

Johann Bernoulli’s greatest student was Leonhard Euler (1707–1783), the most prolific mathematician in history. Euler worked at the Saint Petersburg Academy and the Berlin Academy, wrote something like 850 papers and books, and essentially created the modern notation of mathematics: $e$ , $i$ , $\pi$ , $\Sigma$ , $f(x)$ . His identity $e^{i\pi} + 1 = 0$ connects five fundamental constants in a single line.

Euler contributed to analysis, number theory, graph theory (which he founded), combinatorics, physics, and astronomy. Laplace later told his students: “Read Euler, read Euler, he is the master of us all.”

Lagrange and Laplace

Joseph-Louis Lagrange (1736–1813) brought calculus to bear on mechanics, writing the influential Mécanique analytique (1788), which reformulated Newton’s physics in terms of abstract mathematical quantities. His work on the calculus of variations is foundational.

Pierre-Simon Laplace (1749–1827) worked on celestial mechanics, probability, and mathematical physics. His five-volume Mécanique céleste (1799–1825) showed that the solar system is stable (assuming Newtonian gravity), essentially closing the main physical problem opened by Newton.

The end of the century

By 1800, mathematics had tools — calculus, analytic geometry, differential equations, probability — that made it the master discipline of Western science. The 19th century would use those tools to solve concrete problems in physics, but also turn critical attention to the foundations themselves: what do infinitesimals really mean, what is a function, what makes a proof rigorous?

That story belongs to the 19th century.

Frequently asked

Why is this period so important?

Because it gave mathematics its modern form: symbolic algebra, analytic geometry, calculus, and the beginnings of analysis as a rigorous discipline. Nearly every subsequent development — physics, engineering, probability, economics — depends on the tools built in these two centuries.

Who 'really' invented calculus?

Both Newton and Leibniz, independently. Newton first (ca. 1666), Leibniz published first (1684). A bitter priority dispute between their followers poisoned Anglo-Continental relations for over a century. History has assigned them shared credit — and we use Leibniz's superior notation.