This first part of the Exercises contains the indeterminate analysis of the first degree and its natural applications to show what the theory of numbers consists of, and to give an idea of the methods which it employs.

If it is favourably received, a second part will contain the theory of binomial congruences, and in particular of the complete congruence of the second degree, which it immediately reduces to. Applications will be the resolution of the binomial equation and the exposition of some related theorems.

The various parts of these Exercises being brought together might, by means of some modifications, form an elementary treatise on the theory of numbers. The work of Legendre is no longer sufficient in spite of its extent, and by this very fact the author has not been willing to confine himself to the simple role of translator. Many memoirs published since 1830 make it much easier to write a treatise on number theory. Among these memoirs we must distinguish those of Lejeune Dirichlet, so remarkable for their relative clarity and simplicity. The science of numbers has been simplified and perfected by the eminent geometer whom it has just lost [Dirichlet died in May 1859]; but it will owe him still more, for it is to be hoped that the general researches on the application of numbers, called complex, to the theory of forms, research which M Kummer speaks of in one of his memoirs, will one day be published.

Readers who want to see more immediately and in a clear manner what number theory consists of and how useful it is, should read the Réflexions sur le principes fondamentaux de la théorie de nombres. This memoir, by M Poinsot, is found in Volume X of the Journal de mathématiques.

Just as I have taken advantage of the observations made by a friend, M Prouhet, to improve the drafting of this booklet in a few points, I shall profit later from remarks that one would address to me.

The excellent Recherches arithmétiques of Gauss and the translation of this work being completed, Legendre's Théorie des Nombres having also become rare, it seemed to me that it would be useful to write a new Treatise presenting roughly the present state of the science of numbers. As early as 1858, when I left my duties as a professor at the Faculty of Sciences at Bordeaux, I had resolved to devote my leisure to this work. I then published Exercices d'Analyse numérique, and I announced the publication by subscription of the book in question. The small number of subscribers did not allow this intention to be carried out, and I had relinquished it with regret, when prince Alphonse de Polignac (1826-1863), who is himself engaged in number theory research, as can be seen from the reports of the sessions of the Academy of Sciences, was kind enough to remove most of the obstacles that prevented me. My gratitude for this service will undoubtedly be shared by all those who, like the prince, think that a clear exposition of the present state of the theory of numbers, especially since it is well-established, cannot fail to serve for the advancement of pure mathematics.

The following introduction to the Théorie des Nombres contains elementary propositions on numbers composed of primes. The theory of the residues of numbers in arithmetic progression gives the means of constructing a table for the decomposition of numbers into prime factors. The theory of residues of numbers in geometric progression leads to the theory of primitive roots and indices or otherwise to the theory of logarithms for a given modulus (modular logarithms). From this is deduced the construction of the 'Canon arithmeticus' of Jacobi or logarithmic and anti-logarithmic tables. These tables are reduced by half and the use remains almost as easy. This introduction will be followed by memoirs on the various parts of the Theory of Numbers. These memoirs, united and enriched with all that the first two volumes of the complete edition of Gauss' works will present again on numbers, will form a Treatise in which the Memoirs of Jacobi, Lejeune Dirichlet, Eisenstein, Cauchy, etc. will be put to good use, as well as those of some geometricians who are still occupied with questions relating to the Theory of Numbers, not considering it other than a study of pure curiosity.

Sieve arithmetic is nothing more than a table giving the least prime divisor of a sequence of numbers in arithmetic progression. It is arranged in such a way that by suppressing in each page a certain number of columns, the multiples of such prime numbers are thus excluded. Burckhardt's table is a sieve that excludes the multiples of 2, 3, 5. The table we give in the following memoir is a sieve that excludes multiples of 2, 3, 5, 7. M Hoüel, Professor at the Faculty of Science of Bordeaux, was kind enough to take charge of the construction of this table, according to my indications. He has also lent me his help in coordinating the materials contained in this Memoir, and for the correction of the proofs.

The various means proposed here to reduce the tables to the smallest possible volume may not be new. It has, however, seemed useful to say a few words on the manner of their application.

Jules Hoüel was appointed to the chair of pure mathematics in the Faculty of Science at Bordeaux in 1859 and held this post until his death in 1886.

Eugène Prouhet (1817-1867) was a student of Charles-François Sturm at the École Polytechnique in Paris. He is best known for his Memoir on the Relationship between the Powers of Numbers which deals with a combinatorial problem now known as the Prouhet-Tarry-Escott problem. Although Prouhet's results were achieved in 1851, many years before Tarry and Escott rediscovered the solution to the problem, his Memoir was little known since although it was presented to the Academy of Sciences, only an excerpt appeared and the paper was never published.