Mechanics

Shuffling the roots: Lagrange's theorem

In 1770 algebra had a problem that had been stuck for more than two centuries. There were formulas for solving equations of degree 2, the quadratic formula, and also of degrees 3 and 4, published by Cardano in 1545. For degree 5, nobody could find anything. The best mathematicians had tried, and every method that worked up to degree 4 jammed at 5. Lagrange did not try yet another method. He asked a different question: why do the old methods work?

October 14, 2026 Read

The principle of virtual work

In every equilibrium of any forces whatever, in whatever way they are applied, the sum of the positive energies will be equal to the sum of the negative energies. — Johann Bernoulli, letter to Pierre Varignon (26 January 1717) Since Archimedes, each simple machine had had its own explanation. The lever obeyed the law of the arms; the inclined plane, a decomposition of forces invented by Stevin; the pulley, an argument about the ropes; the screw, yet another. In 1717, in a letter to his French friend Pierre Varignon, Johann Bernoulli showed that all these laws are cases of a single principle.

October 13, 2026 Read

The derivative without infinitesimals and the mean value theorem

At the end of the eighteenth century, calculus worked very well and nobody could properly explain why. Leibniz’s $dx$ were “infinitely small” quantities: sometimes they were treated as if they were different from zero, so that one could divide by them, and right afterwards as if they were zero, so that one could discard them. Bishop Berkeley had mocked this in 1734, calling them “ghosts of departed quantities”. Lagrange was bothered by it. In 1797 he published a book whose subtitle is a whole programme:

October 14, 2026 Read

The legacy of Johann Bernoulli

Johann Bernoulli did not write a great book, as his brother wrote the Ars conjectandi or Newton the Principia. His most famous book came out under someone else’s name. What he left was a way of working: turning every problem of physics or geometry into an equation and solving it with Leibniz’s calculus. And he left pupils. Leibniz’s champion In the dispute over who had invented the calculus, Newton or Leibniz, Johann was the most combative soldier on Leibniz’s side. When the Royal Society, presided over by Newton, published at the beginning of 1713 a report giving priority to Newton, Leibniz replied with an anonymous broadsheet, the Charta volans, which quoted the judgement of “an eminent mathematician”: it was Johann. In the years that followed, Johann challenged the English mathematicians with hard problems, which he thought impossible with Newton’s method. Leibniz died in 1716, and Johann carried on the fight alone for years. The dispute cut English mathematics off from that of the Continent for almost a century.

October 13, 2026 Read

Lagrange multipliers: optimising under a constraint

Almost every real decision is a maximum or minimum problem with a constraint. A factory wants the greatest profit but has a limited amount of raw material. A can must hold 350 ml using as little aluminium as possible. A pendulum moves however it likes, as long as the bob stays attached to the string. Lagrange found a single method for all these problems. It appears in the Analytical Mechanics of 1788, to deal with the ties of mechanical systems, and in the Theory of Analytic Functions of 1797, as a general rule of calculation.

October 14, 2026 Read

Image credits

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Johann Bernoulli, engraving in the Rijksmuseum collection: source page · CC0. Portrait of Guillaume de l’Hôpital: source page · public domain. Title page of the Analyse des infiniment petits, Paris, 1696: source page · public domain. Portrait of Daniel Bernoulli, engraving by Johann Jakob Haid after Johann Rudolf Huber: source page · public domain. The chain and the arch, drawings by Giovanni Poleni, Memorie istoriche della gran cupola del Tempio Vaticano, 1748: source page · public domain. Blog figures The cover and the diagram of the cycloid and the straight line between the same points are original illustrations, with versions in each of the site’s languages. The diagram is to scale: both axes use the same unit. The table of function notations, in the post on the legacy, was redone from a table in this site’s old course on functions.

October 13, 2026 Read

Lagrange's legacy

Lagrange is the lofty pyramid of the mathematical sciences. — Napoleon Bonaparte, according to his contemporaries Lagrange left no anecdotes of precocious genius like Pascal, nor a body of work the size of Euler’s. He left something else: a way of doing mathematics. Where others drew, he calculated; where others found a trick for each problem, he looked for the rule that explained all the tricks. Almost everything he did is still in the textbooks under his name: the Euler–Lagrange equation, the Lagrangian, the Lagrange points, Lagrange’s theorem, the Lagrange remainder, Lagrange multipliers.

October 14, 2026 Read

The screw, the claw and the mirrors

Plutarch writes that Archimedes considered engineering a lesser activity, and that he left no book about his machines. Even so, it was they that made him famous among his contemporaries. The ancient sources describe several, with very different degrees of reliability. The screw of Archimedes An inclined cylinder, with a helix inside or a tube wound spirally around the outside, has its lower end dipped in water. As the cylinder turns, each portion of water is trapped in a fold of the helix and rises, turn by turn, until it comes out at the top. The historian Diodorus of Sicily, in the first century BC, attributes the invention to Archimedes, during a stay in Egypt, where the machine was used to irrigate the fields along the Nile and to drain water from mines.

October 29, 2026 Read

The legacy of William Rowan Hamilton

The Hamiltonian principle has become the cornerstone of modern physics, the thing with which a physicist expects every physical phenomenon to be in conformity. — Erwin Schrödinger (1943) Hamilton believed quaternions would be his great work, the language of the physics of the future. He was wrong in the detail and right in the essentials. Physics adopted Gibbs’s and Heaviside’s vectors, but Hamilton’s mechanics, which he thought less important, became the basis of almost everything that came after.

October 27, 2026 Read

Image credits for Lagrange

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Joseph-Louis Lagrange, engraving by an unknown artist: source page · public domain. Medallion with Lagrange’s profile, bronze by Pierre-Jean David d’Angers, Musées d’Angers: source page · CC0. Title page of the Méchanique analitique, Paris, 1788: source page · public domain. Statue of Lagrange in Turin, photo by Twice25, 2009: source page · CC BY-SA 3.0. Blog figures The cover, the diagram of the five Lagrange points and the graph of curves of equal area with the constraint line are original illustrations, with versions in each of the site’s languages. The diagram of the points is not to scale: L1 and L2 are in reality a hundred times closer to the Earth than the Sun is.

October 14, 2026 Read