The master of them all

Leonhard Euler: Mathematical Genius in the Enlightenment. By Ronald Calinger. Princeton University Press; 669 pages; $55 and £37.95.

LEONHARD EULER is hardly a household name. Born in Switzerland in 1707, he was one of the most productive and influential mathematicians ever, yet surprisingly little has been written about him. Ronald Calinger’s major new biography aims to set this right.

Sir Isaac Newton (and independently Gottfried Wilhelm Leibniz) had introduced calculus a generation or so earlier. But it was Euler whose work really established calculus as the basic tool of the mathematical sciences. Euler also carried forward another aspect of Newton’s legacy, by showing that Newtonian theories of motion and gravitation gave incredibly accurate predictions of the motions of the Moon and other planetary phenomena. He made advances across an astonishing range of subjects, from the very pure to the very applied, in a way that would be impossible today. His publications covered physics, astronomy, acoustics, ballistics and gunnery, cartography, navigation and shipbuilding, optics and the theory of music, as well as number theory and the foundations of calculus.

Euler’s gifts were remarkable. He had great energy and an exceptional memory; he could recite the entire text of Virgil’s “Aeneid” by heart. His productivity was equally amazing. Over his career, he wrote more than 850 publications, including 18 books. His collected works run to more than 80 large volumes, and have been appearing steadily since 1910 (a few volumes remain to be published).

Euler was born in Basel, but his career followed the great royal powers of Europe. At the age of 19, he made the seven-week journey to Russia to take up a post at the St Petersburg Imperial Academy of Sciences, which the emperor, Peter the Great, had set up as part of his plan to modernise Russia. Euler was heavily involved in practical scientific matters, such as the construction of accurate maps of the Russian Empire and studies of floods, fire and shipbuilding. But he also pursued interests including number theory, infinite series and the shape of the Earth. A notable breakthrough happened in 1735, when he announced his solution to the famous Basel Problem. This asked for the sum of the infinite series 1 + 1/4 + 1/9 + 1/16 +... and had resisted the efforts of mathematicians for nearly a century (the answer is pi²/6).

While in St Petersburg, Euler married. He and his wife had 13 children, though only five made it past early childhood. Euler also started suffering from headaches, and his eyesight deteriorated steadily. He lost the sight of one eye in his early 30s, and was nearly blind by the age of 60.

In 1741 he was hired away to Berlin. Frederick the Great wanted to build a new Royal Prussian Academy, populated by the superstars of science and philosophy. Euler had hoped to lead it, but he did not fit in. His scientific reputation was undeniable, but he had neither the refined manners nor the sparkling wit that flourished at Frederick’s court. Writing to Voltaire, Frederick played both on Euler’s eminence and his disability, calling him the “great Cyclops of geometry”.

As relations with Frederick soured, Euler decided to move again. In 1766 Catherine the Great (it was an age of Greats) hired Euler back to St Petersburg. Despite being almost blind, he became a central figure in the academy there, publishing more than 400 articles, a major three-volume work on lunar motion and “Letters to a German Princess”, one of the earliest and most successful popularisations of science for a general audience.

Mr Calinger’s book is an impressive work of scientific biography. It is long, but it gives a fascinating portrait of Euler, his work and the world around him. For the reader who seeks more, a huge array of Euler’s writings can be found online at the Euler Archive. As Pierre-Simon Laplace, another 18th-century mathematician, reportedly said: “Read Euler, read Euler, he is the master of us all.”