This book is a general survey of the history of mathematics, starting with the prehistory of mathematics, ancient Egyptian and Mesopotamian mathematics, and then progressing chronologically to the turn of the twentieth century, culminating with a chapter on Albert Einstein (1879–1955). This is a distinctive ending point, setting this book apart from other general surveys on the history of mathematics. Among the features of this book are eight appendices giving extra exposition on topics brought up in the main text. For example, there is an appendix on Gauss' Euclidean construction of the 17-sided regular polygon.
To get the most out of this book, readers will need at least a year of university mathematics. Some of the exposition giving the "more historically significant, elegant or unexpected theorems and procedures" [xii] can get quite involved and requires that the reader have some mathematical maturity.
This book is like most general surveys in that the tale told is that of the "great figures." Among those whose life and work are accorded significant portions of this book are Euclid, Apollonius, Archimedes of Syracuse, Descarte, Leibniz, Euler, Gauss, Hamilton, Boole, Dedekind, and Cantor. Newton and his work fill out two chapters, and Einstein has the concluding chapter. Let us look at a few of these chapters a little more closely.
Chapter 4 is on "Archimedes and the later Hellenistic period." After giving a sketch of Archimedes' life, the author moves on to his works. The first work to be discussed is the Quadrature of the Parabola. It contains 24 propositions, one of the main theorems being that the area of a parabolic segment bounded by a cord is equal to four thirds the area of the inscribed triangle. The proof of this theorem is sketched. The next work discussed is On the Sphere and Cylinder. This seems to have been one of Archimedes favourite works, so much so that it is said that he had a representation of the theorem that the volumes of a cone inscribed in a hemisphere, which itself is inscribed in a cylinder, are in the ratio 1:2:3, engraved on his tomb.
On Conoids and Spheriods is the next work examined. A proof of proposition 21 of this work is given in detail. The proposition states that the "volume of any segment of a paraboloid of revolution is half as large again as the volume of the cone which has the same base and the same axis" [pg. 70]. We are next treated to a discussion of the Archimedean spiral and Archimedes' Method, a work that was thought lost forever until its rediscovery in 1906 by J. L. Heiberg. The story of the palimpsest that Heiberg used is ongoing — I recommend visiting http://www.archimedespalimpsest.org to find out more. Midway through the chapter we are shown how to trisect an angle by using a "neusis" construction. The chapter finishes off with a brief discussion of Ptolemy of Alexandria and his work, Diophantus of Alexandria and his Arithmetica, which utilized a syncopated style of writing (i.e., using a mixture of special symbols verbal abbreviations to write equations, rather than simply writing them in ordinary prose, as was normal), and finally Pappus of Alexandria and his Mathematical Collection.
Chapter 10 discusses "Newton's Circle." This introduces some of the contributions of other mathematicians around Newton's time. John Wallis (1616–1703) was actually from the generation before Newton; he is the first to get attention in this chapter. Some of his contributions are discussed, for example, his book Arithmetica Infinitorum which utilized "the method of indivisibles" and his Algebra, which was one of the first books to systematically use algebraic symbolism. The next individual that is given attention is Isaac Barrow (1630-77), who would become Newton's teacher, colleague and friend. Barrow's geometrical lectures are examined, with a presentation of the first theorem from the tenth lecture, which at first appears to be "an unnecessarily complicated method of constructing tangents," but upon closer inspections "provides a geometrical demonstration of the basic formula for the arc length" of a curve [p. 238]. The chapter concludes with discussions of Edmond Halley (1656-1742), known for his cometary studies, and Roger Cotes (1682–1716), who edited the second edition of Newton's Principia and made other significant mathematical contributions.
The chapter that distinguishes this book from other general surveys is Chapter 10, on Einstein. Einstein's ideas are presented only in general terms, since, as the author points out, Einstein's "mathematical exposition of the General Theory of Relativity is a veritable tour de force, but one far beyond the scope of this book" [pg. 389]. What follows is a nice painting of Einstein's intellectual development and life. This is followed by a section on Einstein's pre-relativity papers of 1905: the paper on the photoelectic effect (which would eventually win him a Nobel prize) and the paper on Brownian motion. The main portion of the chapter examines Einstein's work on relativity, first the special theory and then the general theory. A derivation of the transformation equations for the special theory is given. After a discussion of the general theory, the chapter is finished off with a section on some of the experimental verifications of the the theory and a section on Einstein's later scientific work.
Like most Dover editions, this book is small enough to travel with and comes at a cost that is not prohibitive. I cannot claim that this is my favourite general survey of the history of mathematics, but it is definitely a solid survey. It is clear that the author took care to produce this volume. If you have ever wondered what mathematical practice was like in the past and wanted to learn more about some of the big historical figures mentioned in your texts, then Stuart Hollingdale's Maker of Mathematics is a good place to start.
Marcus Emmanuel Barnes is a graduate student studying the history of mathematics at Simon Fraser University in beautiful British Columbia, Canada. He can often be found passing the time in the company of books, especially those related to mathematics and science.