Having read and enjoyed Goulding's book, I realize how much I did not know about the histories of mathematics that were written and argued over in the sixteenth century. I have had my ignorance remedied in a scholarly and engaging way, which shows that the subtitle is a good guide to the contents of this very informative work. However, Hypatia, supposedly both the first famous female mathematician and also a pagan beauty who was scandalously abducted, stripped and dismembered by her Christian enemies in 415 CE, has to wait until the last three pages to make her appearance.

Hypatia is significant because she was the daughter of Theon (c.335–c.405 CE). He was the last of the Alexandrian librarians and, as Goulding shows, was believed during the Renaissance to have been a major contributor to, not merely an editor of, Euclid's canonical work The Elements. Euclid had, in fact, completed it some seven hundred years earlier. Given that he clearly drew on earlier mathematicians, Renaissance writers were able to debate an important question. Was the text of The Elements a holistic work and a realization of mathematics in pure Platonic form, or had it been assembled over centuries from disparate and disjointed parts – arguably reminiscent of Hypatia's corpse?

Of Goulding's two major figures, the Huguenot Petrus Ramus (1515–1572) took the latter view. His arguments were countered by the English autodidact Henry Savile (1549–1622). What interested Goulding, and should interest the reader, is that such writers could fashion diametrically opposite histories out of the same, albeit circumscribed, accounts of the earliest centuries of mathematics. Goulding locates his protagonists' histories in their institutional contexts. Both were trying to raise the status and standard of mathematics, Ramus in the schools of Paris until he was brutally murdered in the St Bartholomew's Day Massacre, and Sir Henry until his death as the esteemed provost of Eton College and founder of Oxford University's eponymous professorships of geometry and astronomy.

In Chapter 1, Goulding reviews the classical sources and their earliest interpretations. Diodorus Siculus (and later Proclus) provided support for a Greek origin of mathematics, while for Josephus (and much later Polydore Vergil) it went back to the Jewish patriarchs. As Renaissance authors rediscovered these histories, Regiomontanus emphasized the importance of mathematical practice as a motor of change over time, while Melanchthon represented those who stressed its Platonic ideals.

Chapters 2 and 3 expound Ramus's developing positions from his Dialecticae Institutiones of 1543 to his Prooemium Mathematicum of 1567. In 1543 mathematics had served as the exemplar of Ramus's belief in a natural human capacity to reason. It was only afterwards that he actually studied Euclid for the evidence. In Goulding's words, ‘[q]uite against his expectations, mathematics – the goal and paragon of his logical method – turned out to be hard’ (p. 30, original emphasis). The later Ramus had to add Euclid to Aristotle and the others who had polluted natural and easy knowledge with their ‘contrived’ systems.

Chapter 3 best exemplifies Goulding's commitment to context, showing how Ramus's changing histories of mathematics, and especially his later championing of Pythagoras as the model mathematician, were shaped by battles with his professional rival Jacques Charpentier, the ultra-orthodox Catholic and self-confessed ‘ageometrêtos’. According to Proclus, the great fifth-century CE Platonist and commentator on Euclid, Euclid had defended the difficulty of mathematics by telling Ptolemy I that there was no easy ‘royal road to geometry’ (p. 72). Ramus tried to ingratiate himself with Henry III by declaring that ‘Proclus appears for the defense, on behalf of Euclid's Elements. I have taken up the king's cause’ (p. 74).

Chapter 4 turns to Savile and shows how he drew on Ramus's histories whilst promoting opposite views. The context provided by Goulding for Savile is less overtly ideological, being primarily the lack of support for mathematics provided by Oxford University in general and by its system of ordinary lecturers in particular. Savile's conception of mathematics was very Platonic: he even held that God had created the stars for the purpose of exercising human reason via the discipline of astronomy. Again, Savile ignored practical mathematics because of what he called ‘a sort of disgust for external objects’ (p. 107). And he aimed to promote and recover a pure mathematics grasped by Noah, spoiled by the Chaldeans, recovered by Euclid and the Greeks, but lost again when the Platonic culture propagated in the Alexandria of Theon and Hypatia was overrun by ‘“Turkish cruelty” (i.e. the Islamic conquest of Egypt …)’ (p. 104).

These chapters prepare us for the otherwise baffling material presented in Goulding's final chapters. Chapter 5, ‘The puzzling lives of Euclid’, takes us through the confusions made possible by the persistent identification by Renaissance authors of Euclid the mathematician (c.300 BCE) with Plato's philosophical contemporary, Euclid of Megara, who lived a century earlier. Despite evidence to the contrary from Proclus, Bartolomeo Zamberti repeated the ‘one Euclid’ thesis in an influential biography that accompanied his 1505 edition of Euclid's works. Goulding shows how Ramus revived Proclus's correct ‘two Euclids’ thesis, but more because it fitted his historical narrative than because he understood the evidence. Savile emerges as the first to differentiate the two Euclids persuasively and effectively, albeit in unpublished form until 1621.

It comes as no surprise, therefore, to learn in the climactic Chapter 6, ‘Rending Hypatia’, that most Renaissance authors believed Proclus to have lived in the second century CE, as the tutor of Emperor Marcus Aurelius, and not after Theon in the fifth century CE. This misdating fuelled the ‘Theonine hypothesis’ that Euclid had written down the propositions of The Elements, whilst Theon had compiled the demonstrations seven or eight hundred years later. In his final example of variant interpretations, Goulding shows how Ramus accepted these dates in order to deny that Euclid alone wrote The Elements, while Savile accepted them precisely to argue that the work of The Elements was one beautiful body, like Hypatia.

That we are ignorant of these disputes about the history of mathematics, argues Goulding, is testimony to the success of Ramus, Savile and their contemporaries in providing mathematics with the classical origins it needed to be taken seriously in the Renaissance. Once fully established, mathematics could allow these flawed histories to fall into the neglect from which Goulding has rescued them.

In an era when some academic publishers demand good sales to a wide audience, it is good that Springer and the board of Archimedes continue to publish studies like this one. The print is small, there are no illustrations, and the argument depends upon a close reading of a limited number of Latin texts. As such it is more like a published PhD thesis than a monograph. The conclusion, occupying pp. 179–183, is rather short and undeveloped, and (perhaps as a result) does not get a separate entry in the table of contents. I would have liked some integration with other aspects of Renaissance historiography, for example the development of the sciences. In the case of Savile's manuscript lectures, I needed more help with how the parts discussed fitted into the whole. (What were his views on Ptolemy's Almagest, the subject that dominated his lectures, according to Appendix B?) I would have liked more discussion of the vitality or otherwise of mathematical studies in Oxford in Savile's time. Goulding's support for Mordechai Feingold's positive picture seems at odds with his account of the standard of teaching and Savile's own disdain: he even advised would-be patrons against endowing chairs of mathematics!

These minor cavils apart, I recommend this book as an enduring and intriguing contribution to scholarship. Defending Hypatia is an accomplished piece of scholarship which establishes significant conclusions about Renaissance histories of mathematics and, arguably, about the nature of history itself. In addition, one encounters what is, at the time of writing, a phrase unique to the book: the ‘protreptic exordium’. Try googling it: I have worked out what it means, but I'll keep that classical term of art to myself.