# Journey through Mathematics. Creative Episodes in Its History

The book grew out of a mathematical history course given by the author. It has a list of 39 pages with references to historical publications which are amply cited and from which many parts are worked out in detail. This is organized in 6 chapters describing the evolution of concepts from ancient times till the 18th century to what is now generally used in calculus courses.

González-Velasco has done a marvelous job by sketching this very readable historical tale. He stays as close as possible to the original way of thinking and the way of proving results. He is even using the notation and phrasing and explains how it would be experienced by scientists of those days. However at the same time he makes it quite understandable for us, readers, used to modern concepts and notation. A remarkable achievement that keeps you reading on and on. In my opinion, this is not only compulsory reading for a course on the history of mathematics, but everyone teaching a calculus course should be aware of the roots and the wonderful achievements of the mathematical giants of the past centuries. They boldly went where nobody had gone before and paved the road for what we take for granted today.

What follows is a brief summary of the subjects treated in each chapter.

The first chapter on *trigonometry* starts with the Greek, the Indian, and the Islamic roots (mostly geometric) of trigonometric concepts. One has to wait till the XVIth century when trigonometric tables were produced before the term sinus was used and 2 more centuries before the notation sin, cos,... was used.

The second chapter on the *logarithm* is a natural consequence of the trigonometric tables as an aid for computation. $\sin A \cdot \sin B=\frac{1}{2}[\cos(A−B)−\cos(A+B)]$ could be used to multiply numbers $x\approx \sin A$ and $y\approx\sin B$. Napier (1550-1617) and Briggs (1561-1630) worked out the concepts of the logarithm in base e and base 10. Later de St. Vincent (1584-1667) connected this to areas below (integrals of) $1/x$, and Newton (1642-1727) with infinite series, while Euler (1707-1783) generalized it to a logarithms in an arbitrary basis.

*Complex numbers* are introduced in chapter 3. This is tied up with the solution of a cubic equation (Cardano, 1501-1576) as square roots of negative numbers. Bombelli (1526-1572) described complex arithmetic and Euler even studied the logarithm of complex numbers, but it was only Wallis (1616-1703) and Wessel (1745-1818) who gave the geometric interpretation and made complex numbers accepted (if you can draw them, they must exist). To Hamilton (1805-1865) they were a couple of real numbers and Gauss (1777-1855) introduced the letter i for $\sqrt{-1}$.

Next chapter treats *infinite series*. Summation of numerical sequences was known to the Egyptians and the Greek, but it was Leibniz (1646-1716) who first summed the inverse of the triangular numbers $1/k(k−1)$ and Euler computed $\sum 1/k^2 = π^2/6$. As for function expansions, the Indians knew a series for $\sin(x)$ in the XIVth century, but in Europe one had to wait till the XVII-XVIIIth for Newton and Euler. However it was Gregory (1638-1675) with his polynomial interpolation formulas who later inspired Taylor (1685-1731) and Maclaurin (1698-1746) to develop their well known series.

Chapter 5 about *calculus* is the major part (about a quarter) of this book. Fermat (1601-1665), Gregory and Barrow (1630-1677) contributed but of course Newton and Leibniz are the main players here with the well known dispute of plagiarism as a consequence. Here the author gives a careful and detailed analysis of their contributions and concludes that they worked independently, but that Leibniz's publications lacked clarity, which made him difficult to understand for his contemporaries.

The last chapter is about *convergence*. Leibniz and Newton's ideas were still rather geometric: derivatives were tangents and integrals were quadratures, thus essentially finite. The notion of limit was lacking, which was only developed later in contributions by Fourier (1768–1830), Bolzano (1781-1848), Cauchy (1789-1857), Dirichlet (1805-1859), and others. This chapter has also a remakable original section on the less known Portugese mathematician da Cunha (1744-1787) whose much earlier contribution went largely unnoticed.

**Submitted by Adhemar Bultheel |

**9 / Mar / 2012