This book is closely related to courses of mathematics held for students at New Mexico State University and parts of its preliminary versions were used in courses on the history of mathematics. Four main subjects treated in the reproduced texts are: (1) The bridge between continuous and discrete, describing the transition from figural numbers, sums of powers, the Euler-Maclaurin formula, etc., to Euler’s solution of the Basel problem; (2) Solving equations numerically: finding our roots, from which we learn how the problem of finding roots of functions was tackled through centuries, at first for polynomials and later for more general functions; (3) Curvature and the notion of space, covering the story of Riemann’s inaugural lecture (the subject was chosen since it was a particular interest to Gauss) and later describing studies of Huygens, Newton and Gauss, culminating with the notion of higher-dimensional space; (4) Patterns in prime numbers: the quadratic reciprocity law, containing contributions by Fermat, Legendre, Euler and others, in particular Eisenstein and Cayley.
The introductory section of each chapter gives the reader an overview of material studied and then leads them step-by-step through more than twenty well-chosen parts of historical texts. The book contains many nice pictures including portraits of mathematicians, parts of old texts, drawings and schemes. An important aspect of the book is the numerous exercises, which should help students to gain a deeper insight into the presented material. Numerous references and well-organized indices make the book easy to use. It can be recommended for university libraries and students with an interest in the history of mathematics presented from a modern point of view.