This is by all indications en excellent edition of Conics I-IV. Scholarly translated, meticulously edited, thoughtfully typeset. Green Lion Press is a wonderful publisher doing invaluable work. Their selfless and idealistic mission to produce quality editions of classic texts is a thing of beauty and a credit to academia and mankind.In this review, however, I shall only comment on the introductory editorial commentaries by Harvey Flaumenhaft and Michael N. Fried, which are a small and insignificant part of the book as a whole. I am not doing this beautiful edition justice this way, but, sadly, I have not studied the Conics itself in nearly enough depth to say anything wise about it. “Had I 20 heads, I would put one of them toward working out the theory of conics,” Leibniz once said. I can only concur.Flaumenhaft and Fried are both card-carrying adherents of a very distinctive interpretation of Greek geometry, which says in brief that Greek geometry must be read very literally and that a great conceptual abyss separates it from modern ways of thinking based on numerical and algebraic conceptions of geometrical figures.Thus, to “study Apollonius in a world transformed by Descartes” (xvii), “readers must, at least for a while, make themselves at home in a world … which gives an account of shapes in terms of geometric proportions rather than in terms of the equations of algebra. For a while, readers must stop saying ‘AB-squared’, and must speak instead of ‘the square arising from the line AB’; they must learn to compound ratios instead of multiplying fractions; they must not speak of ‘the square root of 2’.” (xvi)Grandiose conclusions are based on this: “It was because of this that Descartes’ Geometry was called the greatest single step in the progress of the exact sciences.” (xv) “By studying classical mathematics on its own terms, we prepare ourselves to consider Descartes’ … transformation not only of mathematics but of the world of learning generally——and therewith his work in transforming the whole wide world.” (xvi)These words by Flaumenhaft are extreme in their bombast, but in essence he is expressing what might be called the “St. John’s interpretation” of the history of mathematics. Generations of scholars associated with this “great books” college have espoused essentially this view, starting with Jacob Klein. Just as some believers always add “peace be upon him” whenever mentioning the founder of their faith, so adherents of the St. John’s interpretation seem incapable of mentioning the work of Jacob Klein without calling it “profound” (as Fried does on page 278).I for one do not subscribe to this interpretation. I believe the difference between ancient and modern mathematics was quite superficial. The Greeks did not stick to their geometrical way of writing because they were “conceptually” incapable of conceiving the numerical-algebraic point of view. Rather, they understood the latter perfectly well, but chose the former mode of writing for technical, methodological reasons, just as modern mathematicians, in their rigorous works, insist on deriving all theorems about a given object from one highly refined, formal definition of it, even though they of course almost always have several other (intuitive and informal) ways of thinking about that object.In defence of my view I observe, firstly, that no one in the 17th century claimed that mathematics had been profoundly “transformed” in the manner the St. John’s interpretation claims. On the contrary, 17th-century mathematicians constantly emphasised that their “new” way thinking was just the method of the ancients in a slightly different garb.Judge for yourself whether you find it plausible that all 17th-century mathematicians—who were intimately familiar with Greek mathematics, as well as exceptionally creative geometers themselves—failed to understand what they themselves were doing, and that this was only comprehended 300 years later by Jacob Klein—a philosopher and student of Heidegger.It is furthermore a fact that Apollonius can be interpreted in a manner consistent with my view that he understood the numerical-algebraic perspective perfectly well even though he wrote in geometric form. That is how mathematicians have read him for hundreds of years. It fits. It makes sense. There is no “smoking gun” evidence anywhere in Greek mathematics to show that they did not have such a numerical-algebraic understanding. Instead the St. John’s school must rely on this much weaker argument along these lines:“Heath wrote that Apollonius’s Conics Book IV ‘is on the whole dull, and need not be noticed at length’. A great thinker can sometimes write a dull book, and so it is possible that Heath is right in this estimation …. But, it is more likely that Heath’s dismissal … comes from deeply rooted prejudices … [according to which] Greek mathematical work[s] …[are] approached and rewritten as a modern mathematical text … [i.e.,] as an algebraic text disguised in geometric language. … I have tried, on the other hand, to give Apollonius a fair chance.” (Fried, 267)This, then, is the great price you have to pay for accepting the standard view among mathematicians: you must be willing to take a huge leap of faith on the outrageously unlikely hypothesis that out of Apollonius’s eight books on conics, one is more boring than the others; in fact, the one book where he says he is departing from older works on conics and doing something of his own devising (“I have not found [the topics of Book IV] even noticed by anyone”; 290), which he introduces with a preemptive preface implicitly acknowledging that people will question its value (“these things are … worthy of acceptance for the demonstrations themselves: indeed, we accept many things in mathematics for this and no other reason”; 291).