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Lacroix and the Calculus

João Caramalho Domingues
Publication Date: 
Number of Pages: 
Science Network Historical Studies 35
[Reviewed by
Underwood Dudley
, on

Sylvestre (or Silverstre — both spellings were used during his lifetime) François Lacroix (1765–1843) was a French writer of expository mathematics. The people behind textbooks are seldom celebrated, as witness modern students who almost never know who wrote their mathematics texts; so it is with Lacroix. Outside the names of the schools at which he taught (there were many of them) and his professional achievements, few details of his life are known. He was a protégé of Gaspard Monge (1746–1818), he married, and stayed out of jail and kept his head attached to his shoulders despite the tumultuous times in which he lived. There is one surviving likeness of him, his profile on a medallion struck two years before his death, showing a bald-headed man with a beaky nose and, it seems to me, a keen intelligence.

Though Domingues does not mention it because it has nothing to do with calculus, Lacroix and Poisson (Siméon-Denis Poisson, 1781–1840) were appointed joint referees of a paper by Évariste Galois (1811–1832) in 1831. After three months, Galois asked about it, but got no reply. After a further three months, the paper was rejected because Lacroix and Poisson could not understand it. Had Lacroix been able to make it out, the course of mathematical history might have been different. As I will mention shortly, Lacroix influenced mathematical history in another way, this one all to the good.

Lacroix wrote texts in arithmetic, algebra, geometry, trigonometry, and probability, as well as a book on the teaching of mathematics and many biographical entries for mathematicians in a 52-volume Biographie Universelle.

It is for his calculus books, his three-volume Traité du calcul differential et du calcul intégral (1797–1800, second edition 1810–1819, others later) and its one-volume condensation, Traité élémentaire du calcul differential et du calcul intégral (1802, many later editions), that he is known. The first is not a text but a thorough summary of the subject, while the second was intended to be used in schools. Both were extremely successful, going thorough many editions and being translated into many languages. They were the standard works for more than fifty years.

Lacroix used pre-Cauchy rough and ready definitions: a limit is a “quantity which a magnitude cannot surpass as it increases or decreases, or even that it cannot achieve, but which it can approach as close as one might wish” and a function is a “quantity the value of which depends on one or more other quantities, whether or not it is known which operations are necessary to go from them to the former.” Exactly why the books were so successful cannot be determined, any more than can be determined the reasons why some present-day calculus texts succeed and others fail.

Domingues gives a detailed account of the contents of the three-volume Traité. In the course of this he treats the works of Euler and other earlier writers. As a result, the book gives a good view of how calculus was thought of, and written about, in the eighteenth century.

The Traité élémentaire changed mathematical history because three students at Cambridge University, Charles Babbage (1791–1871), John Herschel (1792–1872), and George Peacock (1791–1852) had the idea of translating it into English, which they did, having it published in 1816. The translation had the effect of at last driving out of English mathematics Newton’s inferior notation for calculus and replacing it with that which is today universal. English mathematics was thus freed to progress, which it commenced to do. No doubt some other text, translated by someone else, could have had the same effect, but we cannot know when that would have happened. Lacroix’ text had a very large effect.

Lacroix and the Calculus is an admirable book, admirably produced. I noticed only one typographical error. I could not see what principle the author was using when he refers to functions sometimes as “f” and sometimes as “f ” and has in some places “dx” and in others “dx”. He helpfully translates into English all the passages in French and Latin that he quotes. The book’s price of $169 will restrict its purchasers to libraries and serious scholars.

Woody Dudley once wrote a textbook, but hardly anyone noticed.


Introduction.- A short biography of Silvestre-François Lacroix.- An overview of Lacroix’s Traité.- The principles of the calculus.- Analytic and differential geometry.- Approximate integration and conceptions of the integral.- Types of solutions of differential equations.- Aspects of differences and series.- The Traité élémentaire.- The second edition of the Traité.- final remarks.- Two memoirs by Lacroix.- Lacroix’s historical appraisal of his own Traité.- Syllabi of Lacroix’s course of analysis at the École Polytechnique.- Biographical data on a few obscure characters.- Bibliography.