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Landmark Writings in Western Mathematics 1640-1940

I. Grattan-Guinness, editor
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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What a great idea for a book! The plan is to survey the history of mathematics in the period 1640–1940 by focusing on the most important writings (mostly books, but in some cases also papers or series of papers). Each such writing is covered by a specialist in a separate article. The articles are quite extensive and detailed, including tables of contents, information about the context in which they were written, and an analysis of the contents.

Most mathematicians, if they are at all interested in the history of their subject, will have heard of the works discussed in this book. But most of us do not have the time (or, perhaps, the interest) to read them in detail. Landmark Writings offers us a way to know a little more about each of these works. It will doubtless convince us that some of these are not worth reading. It will perhaps entice us into further study of some others. It may motivate publishers to reprint (or, in some cases, to translate and print in English for the first time) some of these classics.

In his introduction, Grattan-Guinness gives the broad outlines of the project. The initial date of 1640 was chosen because of a feeling that the publication of Descartes' Géométrie marks the beginning of modern mathematics in Europe. Previous works, such as those by Cardano, Regiomontanus, Stevin, and Viète, are felt to represent the culminating moment of a previous period, rooted in the great medieval translations and the rediscovery of the classics.

The ending date, 1940, seems more arbitrary. A round 300 years seems like a good idea, and the second world war provides a useful excuse. But in the end, it feels a little artificial. Perhaps 1914, marking what some historians have described as "the real end of the 19th century," would have made a better choice. In any case, by the mid-20th century it is clear that books are no longer "where the action is." Most new advances would be found in journal articles, and in such abundance as to preclude this kind of treatment.

The articles cover 89 different works published during the 300 year period covered by the book. Grattan-Guinness says that these were chosen from a list that was twice as long; one can certainly believe that. (It's a pity that this list of "honorable mentions" wasn't included!) He also tries to preempt criticism of the actual choices, saying, "If Your Favorite Writing is missing, dear reader, then we mortals have offended." He needn't have worried, really. Though I shall do a little bit of this kind of nitpicking below, the truth is that for the most part the picks are clearly the right ones.

So what writings are covered? A glance at the table of contents shows that most of the well-known and important writings are there, at least up to the end of the 19th century. One might argue about one or two of them, perhaps. The selections from the late 19th and early 20th century are a little bit easier to dispute. Given the editor's known interests, it's not surprising to see an emphasis on foundations and on applied topics. It's easy, however, to point out important books that are not included. For example, shouldn't Banach's 1932 Théorie des opérations linéaires be here? Or Camille Jordan's 1870 Traité des substituitions? Both seem more significant to me than, say, Rayleigh's Theory of Sound. And what about the massive rewriting of the theory of algebraic functions by Dedekind and Weber?

The authors of the articles on the individual works have been given quite a bit of leeway, so we get a variety of approaches. Some emphasize the location of the work in the life and times of its author. Some go into great detail on the publication history. Others focus mostly on the mathematics. Authors treating shorter works have a much better chance of giving us a detailed picture of the contents, of course, than those dealing with massive volumes such as Laplace's two huge treatises (on celestial mechanics and on probability).

A few things that seem to be production glitches remain. The table of contents accessible through the link above was taken from the publisher's web site. It lists, in almost every case, the actual titles of the book(s) discussed in each article. The book itself, however, sometimes does not. For example, the title of chapter 12 is "Leonhard Euler, book on the calculus of variations (1744)." Did someone suppose that readers would not know what the Methodus Inveniendi was about?

Most of the articles attempt to give information on existing modern editions, but the results are not always complete. For example, M. Serfati gives three different French editions of Descartes' Géométrie, but mentions only one English translation (and, to my mind, not the most useful one). Of course, such information quickly becomes out of date. Note, for example, the recent English translation of Bernoulli's Art of Conjecturing, just out from Johns Hopkins.

A final complaint: what a ridiculous price! To be sure, this is a massive volume. But setting the price at $252.00 automatically limits the audience and availability of the book. I hope many libraries will get copies, but at this price I have little hope that many individuals will. Too bad!

Small hesitations and budgetary worries aside, this is a delightful and useful book. Choice magazine has named it one of the exceptional academic books of 2005. It definitely deserves the honor. I've had fun reading it through, but I suspect most of its readers will pick it up to read about a particular "landmark writing." It will serve them well. Ivor Grattan-Guinness, his editorial board (Roger Cooke, Leo Corry, Pierre Crépel, and Niccolò Guicciardini), and the many individual writers have produced a wonderful reference book.

Fernando Q. Gouvêa is professor of mathematics at Colby College.

Introduction (I. Grattan-Guinness)
1649 René Descartes, Geometria (Michel Serfati)
1656 John Wallis, Arithmetica infinitorum (Jackie Stedall)
1673 Christiaan Huygens, Horologium (Joella Yoder)
1684 G.W. Leibniz, first two calculus papers (Silvia Roero)
1687 Isaac Newton, Principia mathematica (Niccolò Guicciardini)
1713 James Bernoulli, De arte conjectandi (Ivo Schneider)
1718 Abraham De Moivre, Doctrine of chances (Ivo Schneider)
1734 George Berkeley, The analyst (Douglas Jesseph)
1738 Daniel Bernoulli, Hydrodynamica (Gleb Mikhailov)
1742 Colin MacLaurin, Treatise on fluxions (Erik Lars Sageng)
1743 Jean le Rond d'Alembert, Traité de dynamique (Helmut Pulte)
1744 Leonhard Euler, Methodus inveniendi (Craig Fraser)
1748 Leonhard Euler, Introductio ad analysin infinitorum (Karin Reich)
1755 Leonhard Euler, Differentialis (Sergei Demidov)
1763 Thomas Bayes and Richard Price, paper on probability theory (Andrew Dale)
1788 J.L. Lagrange, Méchanique analitique (Helmut Pulte)
1795 Gaspard Monge, Géométrie descriptive (Joel Sakarovitch)
1797 J.L. Lagrange, Fonctions analytiques (Craig Fraser)
1797-1800 S. F. Lacroix, Traité du calcul (João Caramalho Domingues)
1799-1802 Etienne Montucla and J.J. Lalande, Histoire des mathématiques , second edition (Pierre Crépel and Alain Coste)
1799-1805 P.S. Laplace, Exposition du système du monde, second edition, and Mécanique céleste (I. Grattan-Guinness)
1801 C.F. Gauss, Disquisitiones arithmeticae (Olaf Neumann)
1809 C.F. Gauss, Theoria motus corporum coelestium (Curtis Wilson)
1812, 1814 P.S. Laplace, Théorie analytique des probabilités and Essai philosophique (Stephen M. Stigler)
1821, 1823 A.-L. Cauchy, Cours d'analyse and Résumé of the calculus (I. Grattan-Guinness)
1825, 1827 A.-L. Cauchy, booklet and paper on complex-variable analysis (the late Frank Smithies)
1822 J.B.J. Fourier, Théorie analytique de la chaleur (I. Grattan-Guinness)
1822 J.V. Poncelet, Traité des propriétés projectives des figures (Jeremy Gray)
1826 N.H. Abel, paper on resolving the quintic (Roger Cooke)
1828 George Green, An essay on…electricity and magnetism (I. Grattan-Guinness)
1829 C.G.J. Jacobi, Fundamenta…functionum ellipticarum (Roger Cooke)
1844 Hermann Grassmann, Die lineale Ausdehnungslehre (Albert Lewis)
1847 K.G.C. von Staudt, Geometrie der Lage (Karin Reich)
1851 Bernhard Riemann, thesis on complex-variable analysis (Peter Ullrich)
1853 W.R. Hamilton, Lectures on quaternions (Albert Lewis)
1854 George Boole, Laws of thought (I. Grattan-Guinness)
1862 J.P.G. Lejeune-Dirichlet and Richard Dedekind, Vorlesungen über Zahlentheorie (Catherine Goldstein)
1867 W. Thomson and P.G. Tait, Treatise on natural philosophy (Norton Wise)
1867 Bernhard Riemann, thesis on trigonometric series (David Mascre)
1867 Bernhard Riemann, thesis on the foundations of geometries (Jeremy Gray)
1871 Stanley Jevons, Theory of political economy (Jean-Pierre Potier and Jan Van Daal)
1872 Felix Klein, essay on the Erlangen programme (Jeremy Gray)
1872 Richard Dedekind, Stetigkeit und Irrationalen Zahlen (Roger Cooke)
1872-1891 Oliver Heaviside, Electrical papers (Ido Yavetz)
1873 J.C. Maxwell, A treatise on electricity and magnetism (Franck Achard)
1877-1878 Lord Rayleigh, Theory of sound (Ja-Hyon Ku)
1881 Henri Poincaré, prize essay on the three-body problem (June Barrow-Green)
1883 Georg Cantor, essay on the foundations of set theory (Joseph Dauben)
1888 Richard Dedekind, Was sind ... die Zahlen? and 1889 Giuseppe Peano, Arithmetices principia (José Ferreiros)
1888-1893 Sophus Lie, Theorie der Transformationsgruppen (Peter Ullrich)
1892 W.W. Rouse Ball, Mathematical recreations (David Singmaster)
1893 A.M. Lyapunov, book on stability theory (Jean Mawhin)
1894 Heinrich Hertz, Die Prinzipien der Mechanik (Jesper Lutzen)
1894 W. Thomson, Baltimore lectures on dynamics and optics (Ole Knudsen)
1896 Heinrich Weber, Lehrbuch der Algebra (Leo Corry)
1897 David Hilbert, report on number theory (Norbert Schappacher)
1899 David Hilbert, Grundlagen der Geometrie (Michael Toepell)
1900 David Hilbert, lecture on mathematical problems (Michiel Hazewinkel)
1900 Karl Pearson, papers on statistics (Eileen Magnello)
1900-1908 Moritz Cantor, Geschichte der Mathematik , 1900s editions (Menso Folkerts)
1904-1905 Henri Lebesgue, books on Intégration and on Séries trigonométriques, and 1905 René Baire, Fonctions discontinues (Roger Cooke)
1909 Heinrich Lorentz, Theory of electrons (Anne Kox)
1910-1913 A.N. Whitehead and Bertrand Russell, Principia mathematica (I. Grattan-Guinness)
1915-1916 Albert Einstein, papers on general relativity theory (Tilman Sauer)
1915-1934 Federigo Enriques and Oscar Chisini, Teoria geometrica ... delle funzioni algebriche (Alberto Conte)
1917 d'Arcy Wentworth Thompson, On growth and form (Tim Horder)
1919-1923 L.E. Dickson, History of the theory of numbers (Della Fenster)
1921-1924 P.S. Urysohn and K. Menger, papers on dimension theory (Tony Crilly)
1924 David Hilbert and Richard Courant, Methoden der mathematischen Physik (Skuli Sigurdsson)
1925 R.A. Fisher, Statistical methods for research workers (Anthony Edwards)
1927 G.D. Birkhoff, Dynamical systems (David Aubin)
1930 Paul Dirac, Principles of quantum mechanics and 1932 Johann von Neumann, Quantenmechanik (Laurie Brown and Helmut Rechenberg)
1930-1931 B.L. van der Waerden, Moderne Algebra (Karl-Heinz Schlote)
1931 Kurt Gödel, paper on incompletability (Richard Zach)
1931 W.A. Shewhart, Economic quality control (Denis Bayart)
1931 Vito Volterra, Leçons sur…la lutte de la vie (Giorgio Israel)
1932 Solomon Bochner, Vorlesungen über Fouriersche Integralen (Roger Cooke)
1933 Andrei Kolmogorov, Wahrscheinlichkeitsrechnung (Jan von Plato)
1934 H. Seifert and W. Threlfall, Lehrbuch der Topologie, and 1935 P. Alexandroff and H. Hopf, Topologie (Alain Herreman)
1934, 1939 David Hilbert and Paul Bernays, Grundlagen der Mathematik (Wilfried Sieg)
List of contributors.