What are the origins of Cambridge University? To what extent (and when) was mathematics included in its curriculum? Who were the mathematicians who studied or taught there? As one of the oldest schools of European mathematics, what was the extent of its influence up to the mid 19^{th} century?

In keeping with its title, this book addresses such questions. It also provides colourful insights into the origins and nature of other universities — not just in terms of mathematics, but with respect to the broader curriculum that was prevalent during the medieval period. Such universities (Cambridge, Oxford and Paris) were nothing like their later counterparts. They were, in fact, hardly institutions at all, since they had no set place of residence, and consisted of loosely formed groups of itinerant scholars located in humble dwellings near an appropriate major settlement. More of that, however, within the pages of this most enjoyable book

Covering the 700-year period from around 1150 to the mid 19^{th} century, the history of mathematics at Cambridge is considered chronologically:

- Medieval mathematics, from 1130 to the work of Cuthbert Tonstall in 1559.
- Mathematics of the renaissance (1510–1660), covering the period from Robert Recorde to William Oughtred.
- The period of ‘modern mathematics’, which is said to commence with the work of Fermat and Descartes’ on algebraic geometry. At Cambridge, the principle participants in this period where John Pell, John Wallis and Isaac Barrow (the giants upon whose shoulders Newton stood?)
- The life and work of Isaac Newton (1642–1727)
- The rise of the subsequent Newtonian schools of Cambridge mathematicians (1726–1828)
- The analytic school (1827-1870)

Subsequent chapters consider organizational matters, including the modes of assessment that were used at various points in history (some such methods were bizarre in the extreme). The book concludes with a chapter on the history of Cambridge University from broader educational perspectives.

Having read this book, my guess is that it would never have been written were it not for the achievements of one man. That man was Isaac Newton, whose origins and remarkable achievements are described in the fourth chapter. And yet, given the extent of Newton’s work, this short chapter effectively outlines his main contributions to celestial mechanics, mathematics and optics. In the process, Rouse Ball provides a nicely rounded biographical vignette of Newton that reveals his complex and interesting personality. It also portrays his relations with other mathematicians, such as John Pell and Isaac Barrow. There is also an indication of the nature of Newton’s correspondence with Leibniz.

For English mathematics in general, and Cambridge University in particular, the dominant influence of Isaac Newton was a mixed blessing. Although his achievements initially attracted universal approbation and enhanced the prestige of Cambridge University as a centre of learning, it led a stultification of British mathematical thought from the time of his death to the mid 19^{th} century. This was mainly due to the fact that, as far as calculus was concerned, English mathematicians persisted with the use of Newton’s outmoded fluxional notation, as opposed to the more flexible version introduced by Leibniz.

There was also stubborn adherence to Newton’s geometrical style of argument and a seeming lack of awareness of the superior analytic methods that were being developed in France and Germany. Indeed, modern analysis is derived from the writings of Leibniz and John Bernoulli, as interpreted by d’Alembert, Euler, Lagrange and Laplace. Even to the end, the English school of the latter half of the eighteenth century never brought itself in touch with these writers, and the isolation of the later Newtonian school accounts for a rapid decline of mathematical work at Cambridge.

The sixth chapter of this book (The Analytic School) discusses attempts that led to the reconnection of English mathematics to the mainstream. At Cambridge, George Peacock and Charles Babbage were principle movers in the goal of introducing Leibniz’s calculus, and their aim was stated as being the introduction of ‘the principles of pure d-ism as opposed to the dot-age of the university’. However, as with all educational reform, there was considerable opposition from quarters of vested interest (authors of textbooks, etc) and, in this case, from the professorial body and the senate of Cambridge University, who regarded any attempt at innovation as a sin against the memory of Newton.

Throughout this book, Rouse Ball explores the evolution of mathematical ideas within the micro-political circumstances of the times. There are hundreds of mathematicians who receive mention, and he succeeds in bringing them to life in most vivid terms. In the case of William Whiston, who followed Newton in the Lucasian chair, his description begins as follows:

Intolerant, narrow, vain, and with no idea of social proprieties, he was yet honest and courageous; and though not a specially distinguished mathematician himself, his services in disseminating the discoveries of others were considerable

In conclusion, I must express puzzlement as to why it’s taken 122 years for the arrival of the first reprint of Rouse Ball’s most interesting and enjoyable book

Peter Ruane spent his working years mainly on the training of teachers at the primary and secondary school levels.