---
title: Great Problems and the People Who Solved Them
module: Historical Notes and the Calculus of Variations
moduleNumber: 10
lessonNumber: 2
order: 1002
summary: >
  Differential equations grew out of specific problems, not a plan: the invention
  of calculus by Newton and Leibniz, the Bernoulli brachistochrone challenge,
  Euler's flood of methods, Lagrange's analytical mechanics, Gauss and Riemann's
  rigor, Laplace's celestial mechanics, and Poincaré's qualitative theory. Each
  method descends from a named problem, and reading the subject forward from those
  problems explains why its parts fit together.
topics: [Historical Notes and the Calculus of Variations]
sources:
  - book: Simmons
    ref: "Ch. 1 §6 The Brachistochrone; Fermat and the Bernoullis; Ch. 3 Appendix A Euler; Appendix B Newton"
  - book: Simmons
    ref: "Ch. 5 Appendix C Gauss; Ch. 9 Appendix A Laplace; Ch. 11 Appendix A Poincaré; Ch. 12 Appendix A Lagrange"
---

Differential equations did not arrive as a finished theory. Each method in this
course was invented to answer a concrete question — how a bead slides down a wire,
how the planets stay in their orbits, how a heated rod cools, how a rotating fluid
mass changes shape — and the general theory was abstracted afterward. Reading
the subject forward from its problems, rather than backward from its theorems,
explains why the parts fit together the way they do. The people and the problems
run from Newton to Poincaré.

$$
% caption: Two and a half centuries of the subject, from the invention of calculus
% to the qualitative theory of nonlinear equations, with each figure placed at the
% work that shaped what came after.
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## Newton and Leibniz: the calculus and its quarrel

The subject begins with the tool that made it possible. Isaac Newton (1642–1727)
did his foundational work in two plague years of rural solitude, 1665 to 1667,
when Cambridge closed and he returned to Woolsthorpe. In that stretch, between the
ages of 22 and 24, he developed the binomial series for fractional exponents, both
branches of calculus, universal gravitation as the mechanism of the solar system,
and the analysis of white light into the spectrum. He was secretive and disliked
publishing; his mathematical discoveries circulated by conversation and letter,
and his masterpiece, the _Principia_ (1687), was dragged out of him by Halley over
eighteen months of concentration. It deduced Kepler's laws from the inverse-square
law, treated wave motion and tides, and computed the masses of the sun and
planets.[^newton]

Gottfried Wilhelm Leibniz invented calculus independently in Germany; his papers
of 1684 and 1686 were the first publications on the subject. The two systems were
built from the same idea but different notation, and Leibniz's differential
notation proved the more fruitful. What began as mild insinuation escalated into
the most damaging priority dispute in the history of science.

- **Who was first.** Newton had the calculus eight to ten years earlier but did
  not publish; Leibniz published first.
- **The cost.** British mathematicians, loyal to Newton's geometric methods and
  clumsy notation, cut themselves off from the Continent. Leadership in analysis
  passed to the Bernoullis, Euler, Lagrange, Laplace, and Gauss, and British
  mathematics stagnated for most of the eighteenth and nineteenth centuries.

The analytic notation of the Continent, not the geometric style of the
_Principia_, became the language in which differential equations were actually
solved.

## The Bernoulli challenge

In 1696 Johann (John) Bernoulli posed a problem to the mathematicians of Europe:
given two points at different heights, find the shape of wire down which a bead
slides from the upper to the lower in the least time. He named it the
**brachistochrone**, from the Greek for _shortest time_. Galileo had guessed the
circular arc; the true answer, a cycloid, is far from obvious, and the problem
became the historical source of the [calculus of
variations](/differential-equations/history-variations/calculus-of-variations).[^bern]

$$
% caption: Johann Bernoulli's 1696 challenge and its solvers; the two brothers
% and the two founders of calculus all produced solutions, John's by an optical
% analogy and James's by a more general method.
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The challenge drew solutions from both Bernoulli brothers, from Leibniz, and from
Newton, who — by then Master of the Mint and out of the habit of research — solved
it in a single evening and published anonymously. Bernoulli reportedly recognized
the author at once: "I recognize the lion by his claw." The Bernoulli family was
to mathematics what the Bachs were to music, producing eight mathematicians in
three generations:

- **James Bernoulli (1654–1705).** Self-taught in the new calculus; worked on
  infinite series and special curves, introduced polar coordinates and the
  Bernoulli numbers, and stated the law of large numbers.
- **John Bernoulli (1667–1748).** James's younger brother; mastered calculus and
  applied it to geometry, differential equations, and mechanics; posed the
  brachistochrone and taught the young Euler at Basel.
- **Daniel Bernoulli (1700–1782).** John's son; his _Hydrodynamica_ founded fluid
  mechanics and gave the earliest kinetic theory of gases.

John's solution was the more ingenious, James's the more general, and the resulting
quarrel between the brothers was conducted in language better suited to a street
brawl. The rivalry was as productive as it was bitter: the general method James
reached for is the one Euler would systematize.

## Euler: the flood of methods

Leonhard Euler (1707–1783), a native of Basel and a student of John Bernoulli, was
the most prolific mathematician of any era, averaging some 800 printed pages a
year across a long life and continuing to increase his output after going blind in
his last seventeen years. His imprint on differential equations is everywhere.[^euler]

- **Calculus of variations.** He founded the subject in 1744 on the basic
  differential equation for a minimizing curve, the Euler equation of the previous
  lesson.
- **Notation.** The symbols $e$, $\pi$, $i$, $\Sigma$, the functional notation
  $f(x)$, and $\sin x$, $\cos x$ are his, along with the identity
  $e^{i\theta} = \cos\theta + i\sin\theta$ that ties the exponential and
  trigonometric functions together.
- **Special functions.** He discovered the integral defining the gamma function,
  $\Gamma(x) = \int_0^\infty t^{x-1}e^{-t}\,\d t$, and Euler's constant $\gamma$; he
  met the [Bessel functions](/differential-equations/series-solutions/bessel-and-special-functions)
  in the vibration of a stretched circular membrane, and applied Laplace transforms
  to solve differential equations — all before Bessel and Laplace were born.
- **Series methods.** He contributed much of the theory of second-order linear
  equations and the method of [solution by power
  series](/differential-equations/series-solutions/power-series-ordinary-points).

Euler's reach ran past differential equations into number theory, the Königsberg
bridges (the origin of graph theory), and the polyhedron formula $V - E + F = 2$.
Simmons ranks him, with Gauss and Riemann, among the three greatest mathematicians
of the modern era.

## Lagrange: mechanics made analytic

Joseph Louis Lagrange (1736–1813) was appointed professor at the Royal Artillery
School in Turin at nineteen. His earliest and most important work was in the
calculus of variations: in 1755 he sent Euler the **method of multipliers** for
isoperimetric problems that had baffled Euler for years, and Euler, with notable
generosity, withheld his own results so as not to diminish the younger man's
credit.[^lagrange]

Lagrange's masterpiece, the _Mécanique Analytique_ (1788), unified mechanics into
what Hamilton later called "a kind of scientific poem." It reduced the whole
subject to analysis, with no diagrams, resting on Hamilton's principle and
yielding:

- **Lagrange's equations of motion**, one per degree of freedom;
- **generalized coordinates**, freeing the analysis from any particular coordinate
  system;
- the concept of **potential energy**.

These are the tools that turn a mechanical [vibration
problem](/differential-equations/second-order-linear/mechanical-electrical-vibrations)
into a differential equation without ever drawing a free-body diagram. Lagrange
also led the establishment of the metric system.

## Gauss and Riemann: the age of rigor

Carl Friedrich Gauss (1777–1855) marks the boundary between the modern era in
mathematics and everything before it. A prodigy from Brunswick, he decided for
mathematics at eighteen on discovering which regular polygons are constructible by
ruler and compass, and filled a terse scientific diary with results he never found
time to publish.[^gauss]

| Work | Year | What it established |
| --- | --- | --- |
| Constructible 17-gon | 1796 | ruler-and-compass polygons after 2000 years |
| Doctoral dissertation | 1799 | first proof of the fundamental theorem of algebra |
| _Disquisitiones Arithmeticae_ | 1801 | modern number theory and the standard of rigor |
| Orbit of Ceres | 1801 | least squares applied to recover a lost planet |
| Hypergeometric series | 1812 | the archetypal [second-order equation with regular singular points](/differential-equations/series-solutions/regular-singular-frobenius) |
| _Disquisitiones circa superficies_ | 1827 | intrinsic differential geometry; the _theorema egregium_ |

Two consequences reach directly into this course. Gauss's insistence that a proof
must banish every doubt — "$\tfrac{1}{2}$ proof $= 0$" — reset the standard by
which the existence and convergence theorems of the subject would later be judged.
And his intrinsic geometry of curved surfaces, generalized by his student Bernhard
Riemann into the geometry of higher-dimensional spaces, supplied the language that
Einstein's general relativity would need. Gauss held many results back — non-Euclidean
geometry, elliptic functions, Cauchy's integral theorem — remarking that no
architect leaves the scaffolding standing after the building is done.

## Laplace: the mechanics of the heavens

Pierre Simon de Laplace (1749–1827) was famous enough in his own time to be called
the Newton of France. His lifelong subjects were celestial mechanics and
probability.[^laplace]

- **_Mécanique Céleste_ (1799–1825).** Five volumes applying Newton's gravitation
  to the solar system as a whole, in which the planets perturb one another. Where
  Newton had thought divine intervention might be needed to keep the system from
  degenerating, Laplace proved the idealized system stable. Asked by Napoleon why
  God went unmentioned, he is said to have answered, "Sire, I had no need of that
  hypothesis."
- **Potential theory.** His development of the potential is so thorough that the
  fundamental equation of the theory has carried his name ever since — [Laplace's
  equation](/differential-equations/pdes-fourier-bvp/heat-wave-laplace-equations).
- **_Théorie Analytique des Probabilités_ (1812).** Here he used [Laplace
  transforms](/differential-equations/laplace/laplace-definition-ivps) and
  generating functions freely, decades before the transform became a standard
  method for solving initial value problems.

Laplace's political suppleness — he changed principles with each change of regime
and emerged each time with grander titles — has dulled his reputation, but his
mathematical legacy anchors two full modules of this course.

## Poincaré: from formulas to phase portraits

Jules Henri Poincaré (1854–1912) was recognized as the greatest mathematician of
his generation, and probably the last to hold creative command of the whole of the
subject. He worked in nearly every field and founded several.[^poincare]

His decisive move for differential equations came in celestial mechanics. In _Les
Méthodes Nouvelles de la Mécanique Céleste_ (1892–1899), confronting the
three-body problem, he found the equations unsolvable in closed form and changed
the question. Rather than seek formulas, he studied the **qualitative** behavior of
solutions — their stability, their periodic orbits, the geometry of their
trajectories in the [phase
plane](/differential-equations/nonlinear/phase-plane-autonomous-stability).

- **Qualitative theory of nonlinear equations.** The study of critical points,
  stability, and [periodic solutions and limit
  cycles](/differential-equations/nonlinear/competing-species-predator-prey-limit-cycles)
  begins here.
- **Topology.** His memoir _Analysis situs_ (1895) opened modern algebraic
  topology, the natural setting for the global structure of solution curves.
- **Recurrence and asymptotics.** The Poincaré recurrence theorem and his theory
  of asymptotic expansions grew from the same celestial-mechanics work.

The shift Poincaré made — from solving an equation to understanding its solutions
without solving it — is the intellectual content of the nonlinear and numerical
modules, where closed-form answers stop being available.

## The lineage of the course

Read as a whole, the modules of this course are the sediment of named problems. The
brachistochrone produced the calculus of variations; the motion of the planets
produced celestial mechanics, perturbation theory, and eventually the qualitative
theory of nonlinear systems; the vibrating string and the heated rod produced
Fourier series and the special functions of Bessel and Legendre.

$$
% caption: Each block of the course descends from a physical problem: the fastest
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The methods look tidy in a textbook because the problems that forced them into
existence have been quietly removed. Putting the problems back — the wire, the
planets, the string, the three bodies — is what turns a catalogue of techniques
into a history, and the history is what explains why the subject arrives, in the
end, at a bead sliding down a curve and a rotating mass of fluid changing its
shape.

[^newton]: **Simmons**, _Differential Equations with Applications and Historical Notes_, Ch. 3 Appendix B — Newton: the plague years, the _Principia_, the brachistochrone solved in an evening, and the priority controversy with Leibniz.
[^bern]: **Simmons**, Ch. 1 §6 — Fermat and the Bernoullis: the 1696 brachistochrone challenge, its solvers, "I recognize the lion by his claw," and the Bernoulli family.
[^euler]: **Simmons**, Ch. 3 Appendix A — Euler: his productivity, the founding of the calculus of variations in 1744, his notation, the gamma function and Euler's constant, and his anticipation of Bessel and Laplace.
[^lagrange]: **Simmons**, Ch. 12 Appendix A — Lagrange: the method of multipliers communicated to Euler in 1755, the _Mécanique Analytique_, generalized coordinates, and potential energy.
[^gauss]: **Simmons**, Ch. 5 Appendix C — Gauss: the constructible 17-gon, the fundamental theorem of algebra, the recovery of Ceres, the hypergeometric series, intrinsic differential geometry, and the standard of rigor carried on by Riemann.
[^laplace]: **Simmons**, Ch. 9 Appendix A — Laplace: the _Mécanique Céleste_ and the stability of the solar system, potential theory and Laplace's equation, and the transform methods of the _Théorie Analytique des Probabilités_.
[^poincare]: **Simmons**, Ch. 11 Appendix A — Poincaré: the three-body problem, the qualitative theory of nonlinear differential equations, _Analysis situs_, and the recurrence theorem.