On Spirals is one of the most difficult texts by Archimedes. It presents challenging mathematics and interpretative problems, the most noteworthy being the origin of this piece of research by Archimedes. As such, N.’s translation – the first into English – is more than welcome. While the edition by T.L. Heath (first ed. 1897, second ed., including Method, 1912) has been of great use until now, it is a paraphrase of Archimedes’ text, not a translation. A similar assessment is valid for that of E.J. Dijksterhuis (1938, English translation 1956, second English ed. 1987), which consists of a radical transformation of the text. Moreover, both of these texts reflect the scholarly orientations of the last century, aiming to transpose Greek mathematics ‘in modern notation’, i.e. to reinterpret it in the light of contemporary mathematics.
From this point of view, N. makes a significant step, given that according to the criteria of his translation (as well as that of the On the Sphere and the Cylinder [2004]), it rigorously reproduces (in a kind of mathematical neo-English) as much of the syntactical construction of Greek mathematical phrasing as possible. N. holds the bar high for his readers: he assumes scholars intent on addressing the complexities of Archimedes’ text. Given the problems of interpretation that arise when reading On Spirals, N.’s approach definitely has its use, although it would be more desirable to have a translation that could be compared directly with the Greek text.
What is the model used for the translation? If it is J.L. Heiberg's edition (1910–15), why not offer the Greek text alongside an English translation more accessible to general readers? On the other hand, if N. (as he occasionally notes) has modified the text ope ingenii or ope codicum, why not also include an edited Greek text, with an apparatus in which he could make all his own conjectures or new readings evident?
Another interesting aspect of these labours by N. is his account of the iconographic tradition accompanying the text. The pages N. devoted to this in his The Shaping of Deduction in Greek Mathematics (1999) made a great contribution to the field. However, the choices N. makes in this translation are not as helpful to readers wishing to address the relationship between the textual and the iconographic traditions.
For example, N. writes about the diagram for proposition 13. The codices show two types of very different diagrams: let us call them type X and type Y. In particular, the most authoritative codex presents type X: it is the Laur. XXVIII.4, which slavishly imitates Codex A, the now lost ninth-century archetype. The same type is found in the Latin translation by William of Moerbeke in 1269 on the basis of the same Codex A. Most of the other codices (by far less authoritative) and the codex C (the famous ‘Archimedes Palimpsest’) have type Y. N. (p. 98) chooses type Y to be the main diagram and includes, in the margins, annotations of thumbnails from type X. He then offers a long commentary on proposition 13, in which he discusses the notion of ‘tangent’ in Greek mathematics and what he views as the logical difficulties of the Archimedean proof. It is not until p. 101 that a discussion of the diagram begins, although nothing is said about his choice to feature type Y diagrams.
N. prefers to tell us how he thought the figure should have been drawn, and does not discuss a fundamental problem: what are the origins of the two different types of diagrams? And if the diagram to be featured is that of type Y, on what basis does he discard type X provided by the Laurentian codex? Would type Y be the definitive one sketched by the hand of the Syracusan?
N. seems convinced that he knows how Archimedes drew his diagrams, how he thought, which rhetorical tropes he wanted to reproduce with his treatises and diagrams. His assumptions are not a trivial matter. His commentary on the translation is extensive. On Spirals is a short text, of only 28 propositions; the Greek text of Heiberg's edition occupies less than 60 pages in small format. The present volume is 180 pages in a much larger format. What is more, the introduction, notes and commentary are about four times the length of the text of Archimedes. The title of this volume could have been ‘A Study of Archimedes and his Spirals’. It would have been a more accurate representation of what the volume holds, especially given that this translation is the first one into English, and it will likely be used as a reference point for a long time.
N.’s footnotes are often very useful: it is not easy to read Archimedes, given how many steps are left to the reader. While praising N.’s overall work on this invaluable text, something I find disconcerting are the comments affixed after every single proposition. Upon first reading I had the impression that N. had more or less messed up his doubts and questions as he read every single sentence. On a second reading, however, I glimpsed a thread of consistency. All the commentary is based on very strong assumptions concerning Greek mathematics in general and that of Archimedes in particular and about the relationship between Archimedes and his contemporaries.
A particularly significant example is N.’s commentary on Archimedes’ letter to Dositheus. N. holds that Archimedes used refined rhetorical techniques to present his discoveries ‘at the edge of contestability’ (pp. 12–14). In other words, he believes that Archimedes aimed to challenge his contemporary mathematicians systematically. It is not clear what this assumption is based on, if not on a personal view of Greek and Archimedean mathematics. But on what, exactly, does N. base this portrait of Archimedes? In the prefatory letter there is a passage that can lead to different interpretations. Archimedes tells Dositheus that many years ago he had written to his friend Conon, sending him a list of theorems to prove. But at this point there is a corrupt passage. Codex A, as Heiberg has reconstructed it, reads: δυο τινα των εν αυτω μη κεχωρασμενα τελους δε ποτεσσομεν. Heiberg interpreted it as follows: δύο τινὰ τῶν ἐμαυτῷ μήπω πεπερασμένων διὰ τέλους ποτιτεθῆμεν, or, in his Latin translation: duo quaedam eorum, quae a me ipso nondum prorsus ad finem producta sunt, insuper addita sint. N. disagrees with the emendation suggested by Heiberg and proposes: δύο τινὰ τῶν ἐν αὐτῷ μὴ κεχωρισμένα τέλος δὲ ποθεσόμενα, a sentence that he translates: ‘<there are> a certain two of the <theorems> in it <=the letter to Conon>, not distinguished apart but added at the end’. The two ‘theorems’ happen to have been wrong. Archimedes, in fact, says (in N.’s translation): ‘so that those who claim to find all of them, but publish none of their <=the theorems’> proofs, would be refuted by promising to find solutions to impossible theorems’.
We have a choice between two narratives. Heiberg asserts that Archimedes sent Conon a list of statements (conjectures?) to prove, but that he himself had failed to prove two of them. N. says that there was a list of theorems and that two of them were placed at the end of the list. I do not want to discuss the soundness of the emendation proposed by N. I would instead like to underline how the passage was interpreted by Heiberg in a way that contrasts N.’s image of an ‘Archimedes-loner’. Armed with his apparently anodyne emendation, N. embarks on a reconstruction of the relations between Archimedes, Conon and the whole ‘republic of mathematical letters’ (pp. 25–33). He divines not only the content, but even the form of the ‘big letter’ that Archimedes wrote to Conon, deducing from his emendation an Archimedes who was cunning like a snake, worried that the trap (i.e. the sending of false statements) would be discovered, and for this very reason he resorted to strange expository tricks.
All of this is based on a conjectural and controversial emendation of a corrupt piece. That does not sound very philological. There are not many novels that have Archimedes as their protagonist. It is a pity that instead of writing one, N. chose to translate and comment on Spirals.
