The Shaping of Arithmetic after C.F. Gauss's Disquisitiones by Catherine Goldstein, Norbert Schappacher, Joachim Schwermer

By Catherine Goldstein, Norbert Schappacher, Joachim Schwermer

Considering its ebook, C.F. Gauss's Disquisitiones Arithmeticae (1801) has obtained a virtually legendary recognition, status as an awesome of exposition in notation, difficulties and strategies; as a version of corporation and concept construction; and as a resource of mathematical notion. Eighteen authors - mathematicians, historians, philosophers - have collaborated during this quantity to evaluate the effect of the Disquisitiones, within the centuries considering the fact that its ebook.

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57 The Disquisitiones Arithmeticae had in fact been mentioned at the French Academy at least as early as January 1802: Citizen Legendre communicates a geometrical discovery, made in Germany by M. 59 The project of a French translation was supported by arguably the most prominent mathematician of the time, Pierre-Siméon Laplace, and on May 31, 1804, Joseph-Louis Lagrange wrote to Gauss: Your Disquisitiones have put you at once among the first mathematicians, and I consider the last section as one which contains the most beautiful analytic discovery made in a long time.

35. 22 I. , Gauss directly addressed this issue: as said above, he defined the domain of his book as that of general investigations of integers (and of fractions as expressed by integers),72 more precisely their advanced part, as opposed to the elementary part which deals with the writing of numbers and the usual operations. , the theory of algebraic equations. In other words, arithmetic provides the general theoretical framework for the investigation of equations in integers or fractions. Gauss thus claimed quite an important status for his domain, parallel to analysis and rich in its own applications, as illustrated by secs.

4. 2: the theory of elliptic functions. At the beginning of sec. , Gauss put this section, and indeed higher arithmetic as a whole, in a much wider perspective by mentioning “many other transcendental functions” besides the circular functions, to which the methods and results of sec. 7 could be extended. But he gave only one example of such functions: “those which depend on the integral d x/ 1 − x 4 ,”107 and never published the 104. [Dirichlet 1889–1897], vol. 1, p. 536: [L’expression de la loi] est d’une nature plus composée et en quelque sorte mixte, puisque, outre les éléments arithmétiques dont elle dépend, elle en renferme d’autres qui ont leur origine dans certaines équations auxiliaires qui se présentent dans la théorie des équations binômes, et appartiennent par conséquent à l’Algèbre.

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