Originally published by Yeshiva University in 1956 and reissued by Dover Publications in 2004, this may be the only book devoted solely to the history of analytic geometry. Within 276 pages, it provides wide-ranging coverage of this theme. Indeed, Carl Boyer seems to have researched the subject to the extent that few writings of importance have escaped his notice. Consequently, the detailed nature of the historical and mathematical analysis means that the book is as much a reference work as straightforward reading.

The rationale for the contents is reflected by two conflicting statements contained within it. The first came from J.L. Coolidge’s *A History of Geometrical Methods*. Coolidge defended the view that ‘analytic geometry was an invention of the Greeks’. The second is from Boyer himself, who maintained that analytic geometry was the independent and simultaneous invention of two men — Pierre de Fermat (1608–1665) and René Descartes (1596–1650).

This disparity of viewpoint emanates from different definitions for the term ‘analytic geometry’. From Classical Greece to the late medieval period, the basis of analytic geometry was the idea of ratio. Sometimes described as ‘geometrical algebra,’ this approach was based on ratios of line segments or ratio of areas rather than actual line lengths and specific areas. So, for example, it was the *ratio* of the area of the square on a diameter to the area of the circle that would be constant (but this constant ratio would not then be identified with the real number \(4\pi\)).

Therefore, within the first 75 pages, Boyer outlines relevant features of Greek and medieval geometric algebra, and he explains how such classical methods dominated mathematics up to the time of Viète, Fermat and Descartes. Even this eminent trio did not entirely free themselves from the ideas of synthetic geometry. Fermat used the ideas and notation of Viète. Descartes re-worked the innovative algebra of Viète and used it to formulate his version of analytic geometry. Descartes more or less presented classical algebra in the form we know today.

Neither Fermat nor Descartes used what we now call ‘coordinate systems.’ Their use of algebra was characterised by ‘ordinate’— rather than ‘coordinate’ — geometry, and the plotting of curves wasn’t then part of this new geometry. By such means, Descartes showed that, if any curve were mechanically constructible, one could derive its equation. Conversely, Fermat began with an equation and then established properties of the curve. In fact, the emphasis and presentation of their work was sufficiently different as to eliminate suspicion of interdependence. Heinrich Wieleitner is quoted as saying ‘In one respect, Descartes gave less than Fermat, in other respects, much more’.

Having described the transition from classical geometry to the methods of Fermat and Descartes, Boyer then outlines the stages through which, by the mid-19th century, analytic geometry evolved to become the subject recognizable by present day high school students.

The first stage is referred to as ‘The Age of Commentaries’, which meant clarifying and circulating the ideas underlying Cartesian geometry. The need for this was due to the inscrutable manner in which Descartes presented his work, and the fact that he wrote in French, rather than the universal scholarly language: Latin. Also, Fermat’s work was published posthumously, and at a much later date. Hence, the task of publicising the new geometry was undertaken by other mathematicians — such as Van Schooten and Roberval.

The first significant take-up of Cartesian geometry was when Newton employed it for the classification of cubic curves; and its status was confirmed when Euler made great use of it and succeeded in freeing it from any reliance upon synthetic methods.

Boyer’s ultimate destination is the ‘Golden Age’, which takes us up to the mid-19th century. By this time, the methods of analytic geometry had extended to the study of surfaces (Gaspard Monge), Plücker coordinates, barycentric coordinates (Möbius) and projective geometry. Cauchy and Gauss are the most notable participants by this stage.

This book is rather like one of those buses that let tourists hop on and hop off at various points of interest. Boyer, the driver, will let the reader stay on board for a general historical overview; or one can hop off at any stage to explore the intricate mathematical or historical byways. Either way, the journey is well worthwhile.

Peter Ruane taught mathematics to people from the age of 5 to 55 — that is, from early school arithmetic to transfinite arithmetic.