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History of Greek Mathematics: Volume II, From Aristarchus to Diophantus

Thomas L. Heath
Dover Publications
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
S. Cuomo
, on

Thomas Little Heath (1861–1940) can definitely be called the product of a different era. He held degrees in both mathematics and classics, a combination which stood in his stead when in 1884 he applied to join the British Civil Service, at a time when Latin was still a pre-requisite to pass the entry examination. Heath subsequently embarked on a double career. A well-respected civil servant in the financial departments of Treasury and National Debt Office, he was also a scholar and later fellow of the British Academy, specializing in the history of the ancient Greek exact sciences. Heath’s combination of skills, a valuable asset even at the time, has now become an absolute rarity; a fact which might go some way towards explaining why his works are still in print.

It is a rare textbook that manages to survive for almost a hundred years: the two volumes under review have accomplished exactly that. Heath’s place on the list of steady sellers is also secured by his English versions of Greek mathematical classics: Aristarchus of Samos, Apollonius, Diophantus, Archimedes and, in particular, Euclid’s Elements. His translation of Euclid’s thirteen books, supplemented by commentaries detailing the pre- and post-history of many of the Elements’ propositions, has remained standard in the English-speaking world. While recent examples in Italian and French show how the new historiography of ancient science can be applied to translations, no attempt has yet been made to replace Heath’s Euclid. [1]

But let’s turn to the work at hand. In two volumes, supplemented by a good index, numerous diagrams and some illustrations, A History of Greek Mathematics has a mixed structure (discussed by Heath: I, vii). Some chapters are thematic, and they stretch over several centuries: for instance, chapters on arithmetical operations (including extraction of square and cube roots), Pythagorean arithmetic, ‘special problems’ (the traditional trio of squaring the circle, trisecting the angle, and doubling the cube), trigonometry. The remaining chapters tend to follow a chronological order. Heath’s conception of the chronological span of Greek mathematics is rather generous: he includes material on Egyptian and Mesopotamian mathematics before Thales, and concludes with Byzantine authors, such as Maximus Planudes.

After almost a hundred years, the big question for us today is, how well has Heath’s History stood the test of time? I will focus on three areas — Heath’s selection criteria, his translation criteria, and his approach to the sources — and aim to account both for the contents of the History, and for developments in the field since it was first published.

Although there is a sort of established canon in the history of Greek mathematics, authors still make informed choices as to what, and whom, should be left out or given a prominent place. Heath’s main criterion for inclusion, never made explicit but evident from many throwaway remarks, appears to have been the advancement of science. He gave priority to those mathematical contributions from antiquity which, if properly translated, were still valid and valuable in his day. For example, of Apollonius he enthuses: ‘it will have been realized how entirely scientific and general the method is. […] Apollonius is able to develop the rest of the subject on lines more similar to those followed in our text-books.’ (II, 148, cf. also e.g. Eudoxus: I, 327) He is happy to point out continuities between ancient and modern mathematics, and his belief that there are continuities falters very rarely. For example, he doubts that one could properly say, as implied by Bertrand Russell, that Zeno anticipated Weierstrass: ‘this, I think, a calmer judgement must pronounce to be incredible. If the arguments of Zeno […] contain ideas which Weierstrass used to create a great mathematical theory, it does not follow that for Zeno they meant at all the same thing as for Weierstrass’ (I, 274).

More specifically, Heath tends to include or highlight pieces of mathematics because he thinks that they contribute towards what he takes to be the culmination of a particular field or enterprise. Thus, a whole chapter (I, VI) is entitled: ‘Progress in the Elements down to Plato’s time’; those authors who discuss conics, although their definitions, famously, vary, and although they are interested in conics for different reasons, are nevertheless teleologically organized on the basis of how they converge towards Apollonius. The same goes for early attempts to square the circle, all seen from the perspective of the definitive, and final, contribution by Archimedes (e.g. I, 224).

The above is not to say that matters are not sometimes included purely because of historical interest, but Heath tends to apologize for that (e.g. II, 237, 239). He also exhibits relatively small interest in the context of ancient mathematics, including an ancient author’s declared aims. This can again be understood against the assumption that there are strong continuities between past and present. If the mathematical enterprise is fundamentally the same across the centuries, we do not need to be overly concerned with the circumstances in which Euclid put together the Elements, or why. It should be almost self-explanatory, or so similar to what we do today, that we should be able to understand it on the basis of our values, criteria for rationality, and common sense.

Given all this, it is perhaps surprising that Heath makes space for mathematicians who were competent, but not the absolute peak of mathematical genius. Possibly because he was a mathematician by training, but not someone who spent his life discovering theorems, Heath is not judgemental of greater or lesser mathematical achievement. I certainly could find no trace in History of his contemporary G. H. Hardy’s rather absolute purist mathematical ethos. [2] Thus, I particularly appreciated Heath’s appreciative treatment of Pappus and of Eutocius. Even the wacky neo-Pythagorean Iamblichus, usually the butt of modern historians, is credited with ‘a proposition of the greatest interest’ (I, 114). On the other hand, while thankfully no mention is made of his ‘steam-engine’, Hero of Alexandria’s formula for the area of the triangle is dismissively attributed to Archimedes (II, 322), and Heath has rather harsh words for Nicomachus, guilty above all, it would seem to me, of not being Heath’s beloved Euclid. Even then, the History devotes a significant amount of space (I, 99–112) to the contents of Nicomachus’ Introductio Arithmetica.

What would we make of Heath’s inclusion criteria today? The field is very much in flux. Different definitions are being given of what ‘mathematician’ meant, and that obviously leads to rather different pictures of mathematical practice in antiquity: from minimalist ones, where a mathematician is someone who, basically, has actively contributed to the field, to maximalist ones, aiming to include people who used simple mathematics in their day job, but were not necessarily able to understand Archimedes’ proofs. [3]

On the whole, because more and more historians of mathematics today have a background not in mathematics, but in history, we no longer feel the need to apologize for our interest in figures about whom Heath had little to say. The history of Greek astronomy in particular has become very open to so-called ‘minor’ figures such as Geminus or Cleomedes (of whose work Heath says: ‘its mathematical interest is almost nil’ (II, 235)), or poets like Aratus, as well as to astrology and the study of material culture. [4] The assumption of continuity is still debated, especially in relation to the issue of translation, as we shall see later. And yet, we would be hard put to find anything in recent historiography like Heath’s remark: ‘the theorems of Euclid II.9, 10 were invented for the purpose of finding successive integral solutions of [a certain type of] indeterminate equations’ (I, 167). In other words, many historians today (albeit by no means all) think that the past is a different country, and to a large extent, that is the very point of studying it.

Heath’s assumption of continuity was also at the root of his decision to rewrite ancient mathematics into modern notation. If mathematicians were ultimately engaged in the same enterprise, the universe they were referring to is the same and unalterable, therefore perfect translation is achievable. This approach extends to his translations, which in some cases it would be more appropriate to call ‘versions’. Moreover, Heath says in the preface that he was keen to reproduce at least some of the procedures (I, viii). Modern translations, in being shorter and snappier, make that an easier task. That is not to say that Heath was unaware that there was a problem: often, he refers to modern notation as an ‘equivalent’ of the ancient mathematical text. Occasionally, he expands on terminological differences between past and present (II, 265).

Something is always lost in translation, especially from a dead language into a live one. The problem is exacerbated in the case of mathematics because decisions have to be made regarding the nature of the language used (was it ‘technical’ or everyday, and how can we tell? Was it metaphorical or concrete, and how can we tell?), and the nature of the objects being talked about. Have the referents changed, or are they still the same — did the Greek mean the same thing by ‘one plus one is two’ as we do by ‘1+1=2’? And is it feasible for a translator to engage in controversies about the philosophy of language, mathematics and history, while trying to produce something readable by as wide an audience as possible?

The debate around translating ancient mathematics has flared up on several occasions, most recently in connection with Plimpton 322 and Archimedes’ works. [5] Each translator’s decision, pragmatically, seems to be about what price one is prepared to pay — what one is prepared to lose. If you translate into modern notation, the mathematics just does not seem ‘ancient’ any more. We lose the fact that Greek mathematics was significantly less abstract, for instance, or that nouns referring to geometrical objects could often be elided and referred to only by the letters of the diagram accompanying a proposition. On the other hand, a literal translation will still not be ancient Greek, and the sense of alienation generated by having something vaguely familiar (the volume of a sphere, say) expressed in completely different language may obfuscate, without producing any dividends.

Let us take the case of Apollonius. The Conics is particularly under-studied. I cannot help thinking that this must have to do with the fact that it contains incredibly long and convoluted propositions. Frankly, without the help of modern notation, Apollonius becomes such hard work that it is off-putting (see II, 175). And yet, one has to ask: how on earth did anyone in antiquity read Apollonius? He has several addressees; he mentions more than one edition; his manuscript has after all survived. The fact that Eutocius wrote commentaries on him is testament not simply to the difficulty of the Conics, but also to their reputation. A relatively significant number of people must have read Apollonius. However did they do that? To me, then, the problem becomes that translating Apollonius into modern notation leaves completely unsolved the mystery of how ancient readers made it through his books.

That may be seen as a purely historical problem. Sometimes, however, modern notation costs us some interesting mathematics. The practice of using a moving ruler to find a neusis (a line of given length, inserted between given lines and verging towards a given point) was a procedure peculiar to Greek mathematics. By writing that the solution of a particular neusis problem is equivalent to the solution of a cubic equation (I, 237; he only mentions the sliding ruler at II, 66), and then turning it entirely into algebra, Heath is in effect producing homogeneity out of a difference that the readers may have found stimulating or intriguing, had they been allowed to see it.

Connected to this is the much-disputed issue of geometric algebra. Some parts of Greek mathematics — the prime example is book II of the Elements — are, in the original language, about squares and rectangles and lines, but it has been observed that, if instead of Euclid’s ‘the square on the [line] AB’, one writes AB2, and instead of ‘the rectangle [formed] by the [lines] AB, BC’, one writes AB·BC, the proofs can be worked out as if they were simple algebra. Hence, geometric algebra: the idea is that the Greeks expressed in geometric language the same things that we express in algebraic terms. Given his assumptions of continuity, it is not surprising that Heath uses the term unproblematically and, one might say, innocently (e.g. I, 109, 150, 167, 379) — he only manifests a speck of doubt at II, 81. Heath assigns a central role to the method of application of areas, which he envisages as a continuous thread running from the Pythagoreans, to Euclid, to Apollonius (I, 150–4). In fact, the Greek restriction of algebra to geometric algebra, without a real development in the direction of greater use of abstract symbols, is one of the main factors, for Heath, in the decline of mathematics in late antiquity (II, 197).

But those were, as I said, innocent times. Ever since 1975, the very use of the term ‘geometric algebra’ outside of square quotes amounts to taking a stand in a rather fraught debate, which is not simply about whether some parts of Greek mathematics are really about geometry or about abstract magnitudes more generally, but also about the very possibility of a history of ancient mathematics in modern times. [6]

Heath’s prose is punctuated by ‘it can hardly be doubted that’, ‘evidently’, and similar phrases. His must have been the golden age of Quellenforschung, but his approach to the sources seems at times excessively trustworthy, except in the discussion of Hero’s date (II, 298-307), which is exemplary as a careful and ingenuous scrutiny of limited information. Even when he discusses their reliability, late sources like Proclus or early ones like Plato or Aristotle, each with their own agenda, are eventually treated as fairly unproblematic reservoirs of information. In fact, evidence is not always, each and every time, attributed to a specific source, and the precise reference for a source not always given in a footnote (e.g. I, 4). As we said, Heath does not spend too much time wondering why the authors he looks at did what they did just in the way that they did it. For him, Greek mathematics can be connected to the Greek love of philosophy, which in its turn is on the one hand quintessentially human, on the other hand, exquisitely Greek (I, 3). Even though Heath is not uninterested in biographical details, he does not bring them to bear significantly on what an author writes.

Thus, he hardly ever wonders about ancient audiences or readers. The library of Alexandria or the cultural context of Hellenistic kingdoms, now routinely foregrounded when talking about Euclid, Archimedes and Apollonius, are not mentioned once. Heath writes: ‘The imperishable and unique monuments of the genius of these two men [Euclid and Archimedes] must be detached from their surroundings’ (I, viii). Apart from different philosophical allegiances, the people contributing to Greek mathematics are seen as engaged in the same enterprise, albeit with a greater or smaller degree of success and originality.

This ‘ahistoricity’, already noted above, underlies the thematic chapters — most of the ancients would not have recognized a sub-area called ‘trigonometry’, but that does not stop Heath from putting together different authors, from different periods, under that heading. The uses and applications of calculation changed with time, and yet Heath reconstructs how multiplications were carried out throughout the whole of antiquity on the basis mostly of the examples in one sixth-century author, Eutocius, who reports multiplications for the specific purposes — never explored by Heath — of his commentaries (I, 57). Analogously, Heath often engages in reconstructions of trains of thought, or reasons why the ancient mathematicians may have done one thing rather than another thing (e.g. I, 16, 185: ‘[Hippocrates of Chios] would not have been capable of committing so obvious an error’, 186), but this is often based on what we could label mathematically-informed ‘common sense’, which later historiography has argued to be a cultural construct.

Later historiography has also greatly increased our understanding of ancient mathematicians as individuals operating within specific contexts as well as within mathematical traditions which are now seen as diverse, rather than unified, and indeed possibly in competition with each other. Heath’s explanations drew on factors such as the Greeks’ instinctive love of knowledge for its own sake, ‘remarkable capacity for accurate observation’ (I, 5), and freedom from the fetters of institutionalized religion. Today, detailed and indepth analyses by scholars like G.E.R. Lloyd, informed by comparative anthropology, analyze the specific context of the Greek poleis, and Athens in particular, in order to understand scientific practices. As well as dealing a final blow to the use of the notion of ‘mentality’ as an explanatory category, Lloyd has drawn attention to the significance of democratic institutions such as assemblies, the habit of debate and, within that, of competition between arguments, the development of rhetoric. [7] The reasons why mathematicians did mathematics have been revealed to be more complex, if ultimately equally elusive. Certainly, the relationship between mathematics and philosophy, which Heath took to flow pretty much in the direction from the latter to the former (see e.g. I, 272, 283), has been questioned. [8] But also, biographical information is considered both more relevant, and viewed in a more complex way.

Take Heath’s treatment of Archimedes: the discovery of the Method is still fresh, and Heath is as much in awe of Archimedes’ achievements as some scholars are still today. Today, however, the circumstances of Archimedes’ life and death would be given much more space, to include probably an account of the reception and fame of Archimedes in, for instance, Roman culture. Plutarch’s depiction of Archimedes as an absent-minded genius (II, 17) would probably be presented more critically, and the order and chronology of Archimedes’ works revisited. The story of the Method would have taken more space, as would the question of what makes it non-rigorous as a demonstrative procedure (as opposed to a heuristic one). [9]

To conclude, is there still space for A History of Greek Mathematics on our bookshelves? Heath is dated, somewhat idiosyncratic, and not as good a reference tool as he is often made out to be. Some questions have been resolved by later evidence: a treatise by Diocles on burning mirrors (I, 264; II, 200), in Arabic, has since been edited and published twice. The date of Hero of Alexandria (discussed at II, 298–307), is now generally accepted to be around AD 62. The addressee of Pappus in book 3 of his Collectio (I, 268) has since been revealed to be a woman, not a man. [10]

More radically, the Italian translator of the Elements has written:

Heath’s works are all plagiarized, more or less blatantly, from contemporary works written by German-speaking historians or philologists: Nesselmann, Wertheim, Zeuthen, Heiberg […] Nonetheless, since hardly anybody, least of all in the English-speaking world, reads works written in languages other than the majority language, Heath’s surveys have become a reference point, especially for scholars working in neighbouring areas of research. [11]

Moreover, despite being rather relaxed about the Greeks learning things from Egypt or Mesopotamia (e.g. II, 255), Heath has inevitable moments of cultural imperialism: there are stabs at the fanciful Arabs (I, 355, 356) and the practically-minded Egyptians ((I, 122; II, 307); the Arabic manuscript tradition of the Elements is seen as inferior in authority (I, 362). [12]

And yet, there is nothing else quite like the History; the alternative, for some of the kind of things that its two volumes contain, is not to resort to more recent works covering similar ground, but to go straight to the sources (which may ironically mean resorting to yet more Heath). More than that, Heath is now valuable because he is out of fashion, a bit antiquated — in other words, he has become vintage. His unfaltering sense of the accumulation of knowledge over time, his almost patronizing generosity towards figures slightly outside the canon, his sheer delight at those mathematical procedures he finds interesting or elegant — all seem to hark back to simpler times, a mythological era when academic research was seen as implicitly valuable. There probably never was such a time — after all, Heath refers to his idea of writing the book as ‘quixotic’ (I, v) — but that is precisely what makes something vintage: not the reality, but what it represents and evokes. A History of Greek Mathematics is now the ancient mathematics equivalent of cookery books such as Mrs. Beeton’s: Too dated for practical everyday use, nonetheless one may like to have them around, and they are still useful for particular things — the recipe for jugged hare, or a breakdown of the contents of Apollonius’ Conics. Heath’s History is there, should one get nostalgic for a time when a historian could, seemingly un-selfconsciously, write things like ‘art, literature, philosophy, and science, […] the things which make life worth living to a rational human being’ (I, 1) and get away with it.


[1] Apollonius of Perga, Treatise on Conic Sections (Cambridge 1896); Diophantus of Alexandria, a Study in the History of Greek Algebra (Cambridge 1910); The Works of Archimedes (Cambridge 1897 and 1912); Aristarchus of Samos, the Ancient Copernicus (Oxford 1913); The Thirteen Books of Euclid’s Elements (Cambridge 1926). French translation by Bernard Vitrac (Paris 1990–2001); Italian translation by Fabio Acerbi (Milano 2007).

[2] Articulated in A Mathematician’s Apology (Cambridge 1940).

[3] See e.g. R. Netz, “Greek mathematicians: a group picture”, in C.J. Tuplin & T. Rihll (eds.), Science and Mathematics in Ancient Greek Culture (Oxford 2002), 196–216.

[4] Just in the English language, and very selectively, see e.g. Geminus, Introduction to the Phaenomena (eds. J. Evans & L. Berggren, Princeton 2006); Cleomedes, On the Circular Motion of the Celestial Bodies (eds. A.C. Bowen & R.B. Todd, Berkeley 2004); two new translations of Aratus’ Phaenomena (by D. Kidd, Cambridge 1997, and A. Poochigian, Baltimore 2010); T. Barton, Ancient Astrology (London 1994); R. Beck, A Brief History of Ancient Astrology (Oxford 2007); J. Evans, “The material culture of ancient astronomy”, Journal for the History of Astronomy 30 (1997), 237–307; D. Lehoux, Astronomy, Weather and Calendars in the Ancient World (Cambridge 2007); A. Jones, Astronomical Papyri from Oxyrhynchus (Philadelphia 1999).

[5] See, respectively, E. Robson, “Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322”, Historia Mathematica 28 (2001), 167–206, and the introduction to the first volume of the new English translation of Archimedes’ works (by R. Netz, Cambridge 2004).

[6] See mostly S. Unguru, “On the need to rewrite the history of Greek mathematics”, Archive for History of Exact Sciences 15 (1975), 67–114; B. van der Waerden, “Defence of a ‘shocking’ point of view”, Archive for History of Exact Sciences 15 (1976), 199–210.

[7] See e.g. G.E.R. Lloyd, Magic, Reason, and Experience (Cambridge 1979); The Revolutions of Wisdom (Berkeley 1987); Demistifying Mentalities (Cambridge 1990).

[8] See the debate between A. Szabó, The Beginning of Greek Mathematics (English tr. Dordrecht 1978) and W.R. Knorr, “On the early history of axiomatic: the interaction of mathematics and philosophy in antiquity”, in J. Hintikka, D. Gründer, E. Agazzi (eds.), Theory Change, Ancient Axiomatics, and Galileo’s Methodology (Dordrecht 1981), 145–86.

[9] See e.g. E.J. Dijksterhuis, Archimedes (Princeton 1987, 2nd edition); M. Jaeger, Archimedes and the Roman Imagination (Ann Arbor 2008); Netz, introduction to The Works of Archimedes, cit.

[10] Diocles, On Burning Mirrors, ed. G.J. Toomer (Berlin 1976) and R. Rashed (Paris 2000); on Hero of Alexandria see e.g. K. Tybjerg, “Hero of Alexandria’s mechanical geometry”, in P. Lang (ed.), Re-Inventions: Essays on Hellenistic and Early Roman Science, special issue of Apeiron 37 (2004), 29–56; the introduction to A. Jones (ed.), Pappus of Alexandria. Book 7 of the Collection (New York 1986).

[11] Acerbi, op. cit. 326: “[…] le opere di Heath sono tutte plagi più o meno clamorosi di opere coeve scritte da storici o filologi di lingua tedesca: Nesselmann, Wertheim, Zeuthen, Heiberg […]. Nonostante ciò, visto che quasi nessuno, meno che mai nel mondo anglosassone, legge più opere scritte in lingue differenti da quella dominante, le sintesi di Heath si sono costituite a punto di riferimento, specialmente per tutti gli studiosi che operano in campi di ricerca contigui.”

[12] A point of view now superseded: see e.g. W.R. Knorr, “The wrong text of Euclid: on Heiberg’s text and its alternatives”, Centaurus 38 (1996), 208–76.

S. Cuomo teaches Roman history at Birkbeck College London and is the author of Ancient Mathematics and Pappus of Alexandria and the Mathematics of Late Antiquity.

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