When I first became interested in the history of mathematics, now literally a lifetime ago, it was dominated by a big events, big names approach to the discipline. It was also largely presentist, only interested in those aspects of the history that are still relevant in the present. As well as this, it was internalist history only interested in results and not really interested in any aspects of the context in which those results were created. This began to change as some historians began to research the external circumstances in which the mathematics itself was created and also the context, which was often different to the context in which the mathematics is used today. This led to the internalist-externalist debate in which the generation of strictly internalist historians questioned the sense of doing external history with many of them rejecting the approach completely.

As I have said on several occasions, in the 1980s, I served my own apprenticeship, as a mature student, as a historian of science in a major research project into the external history of formal or mathematical logic. As far as I know it was the first such research project in this area. In the intervening years things have evolved substantially and every aspect of the history of mathematics is open to the historian. During my lifetime the history of the book has undergone a similar trajectory, moving from the big names, big events modus to a much more open and diverse approach.

The two streams converged some time back and there are now interesting approaches to examining in depth mathematical publications in the contexts of their genesis, their continuing history and their use over the years. I recently reviewed a fascinating volume in this genre, Benjamin Wardhaugh’s *The Book of Wonder*: *The Many Lives of Euclid’s Elements*. Wardhaugh was a central figure in the Oxford-based *Reading Euclid* research project (2016–2018) and I now have a second volume that has grown out of two workshops, which took place within that project, *Reading Mathematics in Early Modern Europe*: *Studies in the Production, Collection, and Use of Mathematical Books*[1]. As the subtitle implies this is a wide-ranging and stimulating collection of papers covering many different aspects of how writers, researchers, and readers dealt with the mathematical written word in the Early Modern Period.

In general, the academic standard of all the papers presented here is at the highest level. The authors of the individual papers are all very obviously experts on the themes that they write about and display a high-level of knowledge on them. However, all of the papers are well written, easily accessible and easy to understand for the non-expert. The book opens with a ten-page introduction that explains what is being presented here is clear, simple terms for those new to the field of study, which, I suspect, will probably the majority of the readers.

The first paper deals with Euclid, which is not surprising given the origin of the volume. Vincenzo De Risi takes use through the discussion in the 16^{th} and 17^{th} centuries by mathematical readers of the Elements of Book 1, Proposition 1 and whether Euclid makes a hidden assumption in his construction. Risi points out that this discussion is normally attributed to Pasch and Hilbert in the 19^{th} century but that the Early Modern mathematicians were very much on the ball three hundred years earlier.

We stay with Euclid and his Elements in the second paper by Robert Goulding, who takes us through Henry Savile’s attempts to understand and maybe improve on the Euclidean theory of proportions. Savile, best known for giving his name and his money to establish the first chairs for mathematics and astronomy at the University of Oxford, is an important figure in Early Modern mathematics, who largely gets ignored in the big names, big events history of the subject, but quite rightly turns up a couple of times here. Goulding guides the reader skilfully through Savile’s struggles with the Euclidean theory, an interesting insight into the thought processes of an undeniably, brilliant polymath.

In the third paper, Yelda Nasifoglu stays with Euclid and geometry but takes the reader into a completely different aspect of reading, namely how did Early Modern mathematicians read, that is interpret and present geometrical drawings? Thereby, she demonstrates very clearly how this process changed over time, with the readings of the diagrams evolving and changing with successive generations.

We stick with the reading of a diagram, but leave Euclid, with the fourth paper from Renée Raphael, who goes through the various reactions of readers to a problematic diagram that Tycho Brahe used to argue that the comet of 1577 was supralunar. It is interesting and very informative, how Tycho’s opponents and supporters used different reading strategies to justify their standpoints on the question. It illuminates very clearly that one brings a preformed opinion to a given text when reading, there is no tabula rasa.

We change direction completely with Mordechai Feingold, who takes us through the reading of mathematics in the English collegiate-humanist universities. This is a far from trivial topic, as the Early Modern humanist scholars were, at least superficially, not really interested in the mathematical sciences. Feingold elucidates the ambivalent attitude of the humanists to mathematical topics in detail. This paper was of particular interest to me, as I am currently trying to deepen and expand my knowledge of Renaissance science.

Richard Oosterhoff, in his paper, takes us into the mathematical world of the relatively obscure Oxford fellow and tutor Brian Twyne (1581–1644). Twyne’s manuscript mathematical notes, complied from various sources open a window on the actual level and style of mathematics’ teaching at the university in the Early Modern Period, which is somewhat removed from what one might have expected.

Librarian William Poole takes us back to Henry Savile. As well as giving his name and his money to the Savilian mathematical chairs, Savile also donated his library of books and manuscripts to be used by the Savilian professors in their work. Poole takes us on a highly informative tour of that library from its foundations by Savile and on through the usage, additions and occasional subtractions made by the Savilian professors down to the end of the 17^{th} century.

Philip Beeley reintroduced me to a recently acquired 17^{th} century mathematical friend, Edward Bernard and his doomed attempt to produce and publish an annotated, Greek/Latin, definitive editions of the Elements. I first became aware of Bernard in Wardhaugh’s *The Book of Wonder*. Whereas Wardhaugh, in his account, concentrated on the extraordinary one off, trilingual, annotated, Euclid (Greek, Latin, Arabic) that Bernard put together to aid his research and which is currently housed in the Bodleian, Beeley examines Bernard’s increasing desperate attempts to find sponsors to promote the subscription scheme that is intended to finance his planned volume. This is discussed within the context of the problems involved in the late 17^{th} and early 18^{th} century in getting publishers to finance serious academic publications at all. The paper closes with an account of the history behind the editing and publishing of David Gregory’s Euclid, which also failed to find financial backers and was in the end paid for by the university.

Following highbrow publications, Wardhaugh’s own contribution to this volume goes down market to the world of Georgian mathematical textbooks and their readers annotations. Wardhaugh devotes a large part of his paper to the methodology he uses to sort and categorise the annotations in the 366 copies of the books that he examined. He acknowledges that any conclusions that he draws from his investigations are tentative, but his paper definitely indicates a direction for more research of this type.

Boris Jardine takes us back to the 16th century and the Pantometria co-authored by father and son Leonard and Thomas Digges. This was a popular book of practical mathematics in its time and well into the 17^{th} century. Jardine examines how such a practical mathematics text was read and then utilised by its readers.

Kevin Tracey closes out the volume with a final contribution on lowbrow mathematical literature and its readers with an examination of John Seller’s *A Pocket Book*, a compendium of a wide range of elementary mathematical topics written for the layman. Following a brief description of Seller’s career as an instrument maker, cartographer and mathematical book author, Tracey examines marginalia in copies of the book and shows that it was also actually used by university undergraduates.

The book is nicely presented and in the relevant papers illustrated with the now ubiquitous grey in grey prints. Each paper has its own collection of detailed, informative, largely bibliographical endnotes. The books referenced in those endnotes are collected in an extensive bibliography at the end of the book and there is also a comprehensive index.

As a whole, this volume meets the highest standards for an academic publication, whilst remaining very accessible for the general reader. This book should definitely be read by all those interested in the history of mathematics in the Early Modern Period and in fact by anybody interested in the history of mathematics. It is also a book for those interested in the history of the book and in the comparatively new discipline, the history of reading. I would go further and recommend it for general historians of the Early Modern Period, as well as interested non experts.

[1] *Reading Mathematics in Early Modern Europe*: *Studies in the Production, Collection, and Use of Mathematical Books*, eds. Philip Beeley, Yelda Nasifoglu and Benjamin Wardhaugh, Material Readings in Early Modern Culture, Routledge, New York and London, 2021

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