This book is concerned with the later years of the period 1830 to 1930, which witnessed enormous progress in the foundational aspects of mathematics. Emerging from that hundred-year span, there was the rigorisation of analysis and calculus, initiated by Cauchy and furthered by Dirichlet. Subsequently, there was the work of Peano and Dedekind on the foundations of arithmetic and number; and the newer disciplines of mathematical logic and set theory were put on a strong footing due to the influence of Peano, Frege, Russell, Cantor and Zermelo. Later still (1899), Hilbert provided a sound axiomatic basis for Euclidean geometry, whilst Felix Klein had previously set out principles that enabled the whole of geometry to link more fully with algebra. However, deeply imbedded (and somewhat lost) within the general developments of this era are the considerable achievements of Mario Pieri (1860–1913), whose life and work are the subject of this book.

In fact, the name Mario Pieri, if it appears at all, is easily overlooked in the majority of mathematical histories, and it is notably absent from two recently published books on 19^{th} and 20^{th} century Italian mathematics [1], [2 ]. But Pieri’s obscurity is even less explicable when, from this book, we learn of his extensive contributions in the fields of algebraic and differential geometry, vector analysis and the axiomatic foundations of geometry and arithmetic. For instance, his axioms for arithmetic were regarded by Peano as being superior to his own, and his work on the foundations of elementary geometry (based on the concept of point and sphere) was utilised by Alfred Tarski for his development of the foundations of the geometry of solids.

Seeking to compensate for almost ninety years of historical neglect, Marchisotto and Smith have embarked upon the writing a series of three books on Pieri’s life and work, and this volume is the first of that trio. The authors suggest that it will be of interest to graduate students, researchers and historians with a general knowledge of logic and advanced mathematics; but readers are not expected to have specialised experience in mathematical logic or the foundations of geometry. Book two will continue the examination of Pieri’s research on foundations by considering his axiomatisations for absolute and projective geometry, and the third and final volume will discuss his work on algebraic and differential geometry.

One of the reasons given for Pieri’s relative anonymity is that his name became lost amongst those of the many great Italian mathematicians of the period, with his work being attributed to the Peano school of mathematical logic, or being generally ascribed to the group of mathematicians who worked on algebraic geometry under the leadership of Segre. In fact, Pieri has been described as the true bridge between those two prestigious Italian schools of mathematics; but he suffered the misfortune of seeing his ideas on the foundations of geometry being eclipsed by the work of David Hilbert. Pieri’s inconspicuousness is also put down to the tumultuous effects of the First World War; although the fact that such circumstances were not so injurious to the reputations of many of his mathematical contemporaries makes one question their significance.

As for the contents of this book, the first sixty pages provide a biographical portrayal of Mario Pieri, together with an overview of his mathematical research. This narrative is then contextualized within a ninety-page historical discussion of the foundations of geometry (projective, inversive, absolute and Euclidean). Here, there is particular discussion of Pieri’s paper ‘Point and Motion’*,* wherein he took up Klein’s ideas by means of an axiomatisation of absolute geometry built upon twenty postulates.

However, at the heart of this book is one of Pieri’s most influential ‘memoirs’, titled ‘Point and Sphere’, which is a lengthy research paper translated from Italian with clarity and apparent accuracy. One of Pieri’s main aims in devising these ideas was to reveal Euclidean geometry as a hypothetical-deductive system in which all concepts and postulates are derived from the concept of equidistance of points on a sphere with respect to its centre. Incorporated within this scheme was the use of transformations, thereby linking the philosophy of the Erlanger programme to the methods of synthetic geometry. Pieri’s use of spheres, rather than circles, enabled him to establish the relationship between plane and solid geometry.

Pieri’s 120 page paper is frequently punctuated with explanatory footnotes by the authors, who later provide wider historical and critical analysis of such ideas. There are many interesting observations in this, such as the fact that Pieri’s postulates are also valid in Bolyai-Lobachevskian, as well as Euclidean geometry, suggesting that their common ground is absolute geometry. His ideas are then contrasted with those of Hilbert and Veblen on the foundations of geometry. However, apart from this paper, there is also indication of Pieri’s work on complex projective geometry, where it is explained that he was the first to base it on postulates that ensured its independence of real projective geometry; but deeper discussion of this will appear in the next volume in the series.

Another notable aspect of Pieri’s work included in this book is his paper on the foundations of arithmetic, which he based upon ideas from Dedekind, Peano and Padoa. His axioms for the natural number system were as follows:

- There exists at least one number.
- The successor of a number is a number
- Two numbers, neither of which are the successor of any number, are always equal.
- In any nonempty class of numbers, there is at least one number that is not the successor of any number in the class.

Although axioms II and III are the same as those used by Peano, they are regarded as being an improvement because, for one thing, Peano’s induction axiom has been replaced by IV, and the principle of mathematical induction is subsequently derived as a theorem in Pieri’s system.

However, such insights into the nature of Pieri’s mathematics, although they are the main feature of this book, are not its sole attraction. Another exemplifying attribute is the perspectives it offers on the mathematical developments in Italy and Europe in during, and shortly after, Pieri’s lifetime. We also read of his interactions with Peano, Tarski, and many other mathematicians of the period. There are observations on the work of others, such as Dedekind, Russell, Hilbert and so on.

This book portrays Pieri as an individual of exceptional ability, who responded to the influence of many of his contemporaries; and they, in turn, acknowledged the quality of Pieri’s work. In fact, there is a sixty-page section in the first chapter containing biographical sketches of the lives and work of almost a hundred such mathematicians. Moreover, the general historical analysis and interpretative commentary, whilst focussing particularly upon the achievements of Mario Pieri, also extends to many of the general mathematical developments of his era. The book is very thoroughly documented and contains an abundance of photographic portraits. It culminates with an extensive bibliography and an accurately compiled index, and it will undoubtedly appeal to those concerned with the history of ‘modern mathematics’, as well as being of interest to the general readership specified by the authors.

**References:**

[1] Italian Mathematics Between the Two World Wars, Angelo Guerraggio and Pietro Nastasi (Birkhäuser, 2005)

[2] The Volterra Chronicles, by Judith R. Goodstein (AMS, 2007)

Peter Ruane (ruane.pandh@blueyonder.co.uk) was Senior Lecturer in Mathematics Education at Anglia Polytechnic University, England. His research interests lie within the field of mathematics education and the history of geometry.