**Charles Méray** studied at the École Normale Supérieure in Paris. He began his studies in 1854 when he was eighteen years old, and graduated in 1857. After graduating, Méray taught at the Lycée of St Quentin for two years but then left teaching for seven years during which time he lived in a small village near Chalon-sur-Saône.

Following these seven years when he chose not to work, Méray took up a teaching position again in 1866, this time lecturing at the University of Lyon for a year before being appointed as Professor of Mathematics at the University of Dijon. He would continue to work in Dijon for the rest of his career.

In his time he was a respected but not a leading mathematician. Méray is remembered for having anticipated, clearly and with only minor differences of style, Cantor's theory of irrational numbers, one of the main steps in the arithmetisation of analysis.

So here we have a case of a mathematician who produced work which might have made him one of the leading mathematicians in the world. However, as happened many times throughout history, Méray was unlucky for the genius of his work was not recognised at the time. Others (we give details below) published the same ideas and it would be their work rather than that of Méray which influenced the direction of mathematics. All we can do now is to give Méray the credit he deserves for his remarkable work, even if fate did not allow Méray a role of importance in the development of the subject.

In 1869 Méray was the first to publish an arithmetical theory of irrational numbers in his paper *Remarques sur la nature des quantités définies par la condition de servir de limites à des variables données.* Others such as Martin Ohm (1829), Bolzano (1835) and Hamilton (1833) had published work on irrational numbers but none of these earlier authors gave a rigorous account. Méray's is the earliest coherent and rigorous theory of the irrational numbers to appear in print. His work was not influenced by Weierstrass (whose work was unpublished) or Dedekind who only published his theories after Cantor's important paper appeared in 1872. Méray followed Lagrange's earlier work but gave rigorous proofs of what Lagrange had only conjectured.

Méray published a second important work in 1872. This work is a book *Nouveau précis d'analyse infinitésimale* which aims to present the theory of functions of a complex variable using power series. It is another rigorous work and in fact between 1872 and 1894 Méray produced a series of papers which remove geometric considerations from analytic proofs. Méray's work consistently follows Lagrange in basing the whole of analysis on the concept of functions written as Taylor series.

We have noted above that Méray's work had no real influence on the development of mathematics despite being almost exactly the same as the work which would transform the direction of mathematics. It was not that Méray's work went unnoticed. His 1872 book *Nouveau précis d'analyse infinitésimale* was reviewed by Hermann Laurent in 1873. Hermann Laurent, in his review, ignored Méray's irrational numbers [1]:-

... while gently chiding the author for using too narrow a notion of a function and for being too rigorous in a supposed textbook. At that time there was not in France - as there was in Germany - a sufficient appreciation of the kind of problem considered by Méray, and not until much later was it realised that he had produced a theory of a kind that had added lustre to the names of some of the greatest mathematicians of the period.

**Article by:** *J J O'Connor* and *E F Robertson*

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