From Fermat to Minkowski

Joseph Louis Lagrange lived from 1736 to 1813. Born in Turino, he had both French and Italian ancestors. His family was well off but Lagrange’s father lost the family fortune in risky financial transactions. This is said to have prompted Lagrange to remark, “Had I inherited a fortune I would probably not have fallen prey to mathematics.” (cf. E. T. Bell, Men of Mathematics). As a youth Lagrange was more interested in classical languages than in mathematics, but his interest in mathematics was stirred by a paper by Halley, the friend of Newton. In a short time he acquired a deep knowledge of analysis; only 19 years old, he became Professor at the Royal School of Artillery in Turino. Lagrange stayed there for about 10 years. His reputation as a mathematician grew quickly, mainly by basic contributions to analysis, specifically the calculus of variations, the theory of differential equations, and mechanics. This combination of mathematics and mechanics or, more generally, theoretical physics, is typical of the eighteenth century. Mathematics was not viewed as an end in itself but mostly as a tool for understanding nature. In 1766, d’Alembert was instrumental in bringing Lagrange to succeed Euler at the Berlin Academy of Science. Financial conditions in Berlin were very good; moreover, he could devote himself exclusively to his mathematical work. Lagrange stayed there until 1787 when he moved to the Academie Française in Paris. At that time, soon after Euler’s death, he was recognized as the most important living mathematician. Though Lagrange had had close ties to the French royal family he was not persecuted during the French Revolution. Altogether, the sciences gained importance during the era of the French Revolution and Napoleon. Lagrange’s authority transcended the sphere of science. He was a Senator of the Empire and in fact received a state burial in the Pantheon.

Carl Friedrich Gauss lived from 1777 to 1855. In his lifetime he was known as “princeps mathematicorum.” His main number-theoretical work, Disquisitiones Arithmeticae, and several smaller number-theoretical papers contain so many deep and technical results that we have to confine ourselves to just a small sample. Other equally important results will not be mentioned.

Jean Baptiste Joseph Fourier (1768–1830) was not a number theorist. He would probably not even have called himself a mathematician, but a physicist. His main area of research was the mathematical theory of heat. He wrote several papers about the topic and one basic book, Théorie analytique de la chaleur (Paris, 1822; an English translation was published in 1878). Fourier was a professional politician; as prefect of the Départment d’Isère (at Grenoble), he was closely associated with Napoleon. He accompanied Napoleon on his campaign in Egypt and had the reputation of being quite knowledgeable about that country.

The main emphasis of this book has been on the theory of quadratic forms, and we have given special attention to reduction theory. The main question of reduction theory can be formulated in the following way. Let us consider the real-valued quadratic forms in n variables. We look for inequalities for the coefficients such that every form is integrally equivalent to one and only one reduced form, i.e., to a form which satisfies all these inequalities. (From now on, without again stating this explicitly, we will confine ourselves to positive forms.)