For two thousand years scholars repeated a saying of Aristotle’s: nature abhors a vacuum. If you suck the air out of a straw, the liquid rises because nature will not let the space stay empty. It seemed to explain everything. But the well-diggers of Florence had a problem. Suction pumps could not raise water more than about 10 metres, however hard they tried. Why would nature’s horror of the vacuum stop at 10 metres?
Newton was the greatest genius who ever lived, and the most fortunate, for there is only one universe to discover, and it can be discovered only once. — Joseph-Louis Lagrange, as reported by his contemporaries The calculus war Newton created the method of fluxions in 1665–1666 but did not publish it. Leibniz reached calculus independently in 1675 and published in 1684. For some years nobody quarrelled. In 1676 the two even exchanged cordial letters through Henry Oldenburg, and Newton hid his ideas in an anagram.
Sound is a compression wave: the air is pushed, compresses, pushes the next air and decompresses. The speed of this wave depends on two properties of the air: how much it resists being compressed and how heavy it is. Newton’s formula In the Principia, of 1687, Newton calculated the speed of sound for the first time. Assuming that, when compressed, air obeys Boyle’s law, that is, that pressure is proportional to density, he obtained
The heart has its reasons, which reason does not know. — Blaise Pascal, Pensées Pascal left few finished books. Almost everything he wrote on mathematics came out as loose sheets, in letters or after his death. Even so, few authors turn up in so many different places: in the triangle at school, in the weather forecast, in a car’s brakes, in an insurance calculation and even in the name of a programming language.
Historical images All of them come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Blaise Pascal, unknown artist, around 1690, Versailles: source page · public domain. Yang Hui’s triangle, in Zhu Shijie’s book, 1303: source page · public domain. First page of the Triangulus arithmeticus, copy held by Clermont Auvergne Métropole: source page · CC BY-SA 4.0. Pascaline at the Musée des Arts et Métiers, photo by David Monniaux, 2005: source page · CC BY-SA 3.0. Marker of the experiment on the Puy de Dôme, Agence Rol, 1923, Bibliothèque nationale de France: source page · public domain. Plaque of the grave in Saint-Étienne-du-Mont, Agence Meurisse, 1923, Bibliothèque nationale de France: source page · public domain. Autograph manuscript of the Pensées, exhibition at the Bibliothèque nationale de France, photo by ActuaLitté, 2016: source page · CC BY-SA 2.0. First page of the Pensées, counterfeit of the 1670 edition, copy held by Clermont Auvergne Métropole: source page · CC BY-SA 4.0. Blog figures The cover, the fountain of bowls, the carry diagram and the cycloid arch are original illustrations. The controls for the triangle, the split of the stakes and the hydraulic press were made for this trail. They are teaching aids, not reproductions of manuscripts or of the internal mechanism of the Pascaline.
A notation does not belong to all mathematicians in the same way. Each field, sometimes each school, has its shortcuts, and outsiders trip over them. This is not a flaw: it is what happens with any living language, which splits into accents and jargons. But it has consequences for readers and for learners. Einstein erases the summation sign In 1916, in the paper that presented general relativity, Albert Einstein had to write hundreds of expressions like