Elements of mathematics: from Euclid to Gödel (Q2813040)

As Stillwell writes in his preface his book is thought to give a bird's eye view of elementary mathematics and its treasures. Having Felix Kleins `mathematics from an advanced standpoint' in mind [\textit{F. Klein}, Elementarmathematik vom höheren Standpunkte aus Erster Band. Arithmetik, Algebra, Analysis. Ausgearbeitet von E. Hellinger. 3. Aufl. Für den Druck fertig gemacht und mit Zusätzen versehen von Fr. Seyffarth. Berlin: Springer (1924; JFM 50.0041.04); 2. Band: Geometrie. 3. Auflage von F. Seyfarth. Berlin, Springer (1933; JFM 51.0055.13)], an `advanced standpoint' is often taken and even Hardy's review of \textit{R. Courant} and \textit{H. Robbins}' famous book [What is mathematics?. London, New York, and Toronto: Oxford University Press (1941; JFM 67.0001.05)] is cited: `a book on mathematics without difficulties would be worthless.' It seems clear that it lies in the eyes of the reader which parts of mathematics are more elementary than others and hence the degree of mathematical maturity varies within this book. There are eight topics treated: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic, and these topics are nicely defined in the first chapter.NEWLINENEWLINEThen the journey through elementary mathematics starts. Each of the chapters end with historical and philosophical remarks enlarging the perspective and placing the topics in history. Stilwell does never shun away from technical explanation, for example when he explains how the sum of a geometric series is derived on page 196 or when the differential quotient is defined on pages 198f. However, the explanations never really require more mathematical maturity than digested in high school (but perhaps forgotten already). If maths was once a beloved topic back at school the book should be a pleasure to read. Even proofs are provided and starred sections treat more complicated problems which may be skip by laymen. One of Stillwell's goals is to teach the reader a feeling of why some mathematical results lay `deeper' than others and he succeeds. Intentionally the chapter on logic appears very late in the book; here the hope prevails that a reader having moved on through the earlier chapters will then appreciate the role of formal logic.NEWLINENEWLINEThe book concludes with a tenth chapter on `Some Advanced Mathematics' where several nice issues as Wallis' product for \(\pi\), Ramsey theory, the Pell equation, and some others are discussed. The book is well suited for pupils finishing high school, teachers, and everyone interested to experience a fresh view on elementary mathematics.