Previous articleNext article FreeRonald S. Calinger. Leonhard Euler: Mathematical Genius in the Enlightenment. xvii + 669 pp., illus., apps., bibl., index. Princeton, N.J./Oxford: Princeton University Press, 2016. $55, £40.95 (cloth). Leonhard Euler. Correspondence. Edited by Franz Lemmermeyer and Martin Mattmüller. 2 parts. (Opera Omnia, 4A: Commercium Epistolicum, 4.) xiii + 1,248 pp., illus., figs., bibl., index. Basel: Springer, 2015. $458, £148.50 (cloth).Jeremy GrayJeremy Gray Search for more articles by this author PDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinked InRedditEmailQR Code SectionsMoreLeonhard Euler, who lived from 1707 to 1783, poses a challenge to every historian of modern science—and indeed to every historian of the modern world. His life was deeply entangled with the rise of the eighteenth-century academies and, therefore, with the place of science in the creation of the modern state. It is also central to the creation of modern science after Newton and modern mathematics after Newton and Leibniz. Because Euler published so much, and corresponded so energetically, and because his works were translated into several languages, his influence reached everyone whose work overlapped with his. His Collected Works, begun in 1911 and now nearing completion with editions of his correspondence, will run to over eighty large volumes, and he dominated the mathematical world of his time, yet his name has appeared only seldom in Isis or in other journals, and many historians of the eighteenth century pass over him with barely a mention. If we can understand why this is, we can form a better impression of the books under review.In the last thirty years history of science has been outward looking. It has looked increasingly outside Western Europe; it has looked outside mathematical physics; it has looked beyond the big names; and it has looked for the significance of science in walks of life beyond science itself. The relationship between the history of science and the history of mathematics remains fraught and distant. Perhaps it was inevitable in all this expansion, and all the negotiations with other academic disciplines, that some core domains would be neglected, but there are echoes of the flight to the suburbs and the collapse of the inner cities. The books under review invite us to return to the potholed streets and rundown apartment blocks of the old town. Who lives there, and how have they managed?Euler rewrote the calculus in the language of functions and in so doing put Newton’s laws of mechanics into usable form, the way they have been used ever since. More than anyone, even d’Alembert, Euler created the theory of partial differential equations, the means by which mathematics was applied to the fields of hydrodynamics, to the production and propagation of sound, and, in the nineteenth century, to every branch of mathematical physics. He invented the calculus of variations that was rapidly improved by the young Lagrange and has been the inspiration for innumerable principles in physics ever since. He revitalized the theory of numbers and, with Lagrange, placed it centrally in mathematics, where it remains. He wrote on everything from optics and gunnery to the design of ships and at every level from the elementary (his Algebra of 1770, for example) to the philosophical and the most advanced.Ronald Calinger’s biography of Euler is strictly chronological: 538 pages take us from just before Euler was born to events shortly after his death. There are extensive endnotes, a good bibliography, a valuable register of principal names, and a general index. The text is clear; the book is a pleasure to read and will be easy to use. It weaves together Euler’s personal life, his education, and his activities in the intellectual life of the eighteenth century, noting the intrusion of events beyond his control. We get a good impression of the people Euler knew, those he worked with, and his numerous correspondents. Almost everything is included, from Euler’s simplest work and elementary teaching to his most insightful and influential papers and books. At every stage, we get a succinct account of what Euler did and why it mattered.There are no great surprises here but, rather, an abundance of fresh information drawn from contemporary sources. There are some reassessments of prevailing judgments—for example, over the Samuel König affair and the principle of least action, where Calinger sides quite firmly with Maupertuis and Euler and against Voltaire. The details on life in the academies of St. Petersburg and Berlin add considerably to our sense of how these places worked and how Euler operated within them, and we get an impression of a brilliant man who learned very quickly how to cooperate with others and look after his own interests.The discoveries that make Euler so important are described as they evolve. Interesting connections are brought to light between what can seem to be disparate activities. The motion of solid bodies, often presented as routine mathematics (if difficult to discover), is tied to the rotation of the Earth and the precession of the equinoxes, a key problem in the acceptance of Newtonian gravity. The design of ships is related to the calculus of variations; the vexed business of the Sans Souci fountains is tied to work on hydraulics and hydrodynamics.Newton stalks these pages. His optics generated debates about the nature of light, but also the attempts to design achromatic lenses that Calinger describes in detail. His inverse-square law of gravity was much disputed by Clairaut, d’Alembert, and Euler, and the difficult mathematics as well as the intricate business of academy prizes are nicely tied together here. Leibniz is also present, but for all that Euler learned the calculus from Johann Bernoulli, a staunch Leibnizian, the scientific interests of the time promoted a Newtonian agenda, and Leibnizian philosophical questions (represented by Christian Wolff ) occupy less space.Leonhard Euler: Mathematical Genius in the Enlightenment is a biography without a thesis. It is not written to establish a claim; by and large it endorses the prevailing opinion of Euler’s achievements. But without insisting too much it does put Euler at the center of the struggle to establish the future of Newtonian science. It was one of Clifford Truesdell’s major themes that the eighteenth century was the century of rational mechanics, the attempt to mathematize nature on the basis of observation but little in the way of experiments, and that Euler led the way. Calinger does not disagree, although “rational mechanics” has only three entries in the general index. A more dramatic book might have shed a stronger light.The success of Newton’s Principia Mathematica established that advanced mathematics could explain the motion of the heavens. But much of the study of nature requires more. It requires that the calculus be extended from one to several variables and that the differential equations that are produced can also be reliably solved. By doing precisely that, Euler made possible the scientific analysis of nature that has continued ever since. In various overlapping ways this was the agenda of Euler, Daniel Bernoulli, Clairaut, and Lagrange, and it is described in detail here, although more should have been said about the value of Euler’s books of the 1770s on the differential and integral calculus.Another lasting difference between the scientific worlds of the seventeenth and eighteenth centuries is publication. The academies generated journals, and Euler, with his immense productivity and activities behind the scenes as well, contributed hugely to this. Because he wrote so clearly, and was not averse to publishing inconclusive articles if they were on their way to something better, he might well have played a major role in creating the public character of modern science. This Calinger leaves as a question for others to pursue.The book should be read by every historian of modern science, and it will form the basis for a new generation of Euler scholarship. The invaluable Euler Archive on the Web is also opening up Euler’s works for a new generation and putting many of them into English. There is comparable French attention being paid to d’Alembert. Perhaps the time has come for a return to the inner cities.The two volumes of the Euler–Goldbach correspondence show that new editors, Franz Lemmermeyer and Martin Mattmüller, have continued the sumptuous line of volumes that commemorate Euler. Before we get to the letters themselves, we are given a very helpful short biography of Christian Goldbach, who was seventeen years older than Euler and successfully navigated the upper reaches of Russian life for over thirty years. This enabled him both to support Euler and to correspond with him about matters of mutual mathematical interest. They seem to have gotten to know each other well only in 1729 and then kept in touch until shortly before Goldbach’s death in 1764.As the editors say, the most interesting topic that Euler and Goldbach discussed was number theory, and in these years Euler did much to rescue the subject from the lack of interest of his more science-minded colleagues and create the tools to make it a central component of mathematical research. The editors provide a very helpful introduction to the mathematics from a modern standpoint that leads from number theory to analysis and finally to the other topics that came up.Various other issues are also considered: the disputes over monads and the principle of least action, celestial mechanics, academic prizes, and then editorial matters to do with the sources and the letters themselves. Part 1 of this two-part volume concludes with 483 pages of the letters in their original languages; Part 2 consists of the English translations and the endnotes.The translated letters are a delight. There was never going to be a great surprise in them—P. H. Fuss produced an edition in 1843—but they offer all the pleasures one can expect from two intelligent people thinking out loud. They show that Euler took a considerable amount of time to come to some results that we now find elementary and that he produced a variety of fresh techniques for dealing with all sorts of questions. The endnotes that the editors provide do an excellent job of helping the reader through this complicated material. Notes Jeremy Gray is an emeritus professor of the history of mathematics at the Open University. His latest book was a scientific biography of Henri Poincaré, and he is now working on a historical approach to the philosophy of Hermann Weyl. Previous articleNext article DetailsFiguresReferencesCited by Isis Volume 108, Number 1March 2017 Publication of the History of Science Society Article DOIhttps://doi.org/10.1086/690803 © 2017 by The History of Science Society. All rights reserved.PDF download Crossref reports no articles citing this article.