Between music and geometry: a proposal for the early intended application of Euclid’s Elements Book X

Abstract

This paper attempts a new interpretation of Euclid’s Elements Book X. This study of irrational lines has long been viewed as an anomaly within the Euclidean corpus: it includes a tedious and seemingly pointless classification of lines, known as ‘the cross of mathematicians’. Following Ken Saito’s toolbox conception, we do not try to reconstruct the book’s mathematical process of discovery, but, instead, the kind of applications for which it serves as a toolbox. Our claim is that the book provides tools for solving questions about proportional lines inspired by results in music theory and a context of Pythagorean-Platonic interest in proportions. We show that the entire content of Book X can indeed be accounted for as a set of tools for these questions, augmented by the general editorial norms that govern the Elements. We conclude by explaining why the purpose of Book X as reconstructed here has disappeared from mathematical memory.

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No potential conflict of interest was reported by the author(s).

Notes

1 The evidence is not conclusive, but it is clear that Apollonius saw his first four books as in competition with Euclid’s and also understood them to have been more elements-like. (For the evidence, and a discussion of the meaning of ‘elementary’ for Apollonius, see Fried and Unguru 2001, 370–386.). The key point is that there is, once again, a certain set of results in Conics that Archimedes takes for granted, a set summarized by Dijksterhuis 1987 [1938], 55–108. This set is in fact quite thin.

2 Menaechmus is said, perhaps on Eudemus’ authority, to have been Eudoxus’ pupil (Proclus in Eucl. 67.9–10: Morrow 1970, 55). Eudoxus’ chronology is bound by narrow parameters: almost certainly active mostly in the second quarter of the fourth century. (Lasserre 1966, 137–139 offers a precise birthdate 395, which is contentious; but the sources allow us to push the date no more than a decade or so earlier). That Menaechmus was pivotal for the rise of interest in conic sections is clear from his solution to the problem of finding two mean proportionals, to which we return below.

3 Ever since Schneider (1979), scholarship tended to emphasize our relative ignorance concerning Euclid’s date (Mueller 2008, 304: ‘Proclus makes Euclid a contemporary of the first Ptolemy (d. 282), but his evidence does not inspire confidence’). Our argument rests on probabilities rather than certainties; as probabilities go, they may be considered quite firm.

4 This follows Netz 2020, 298–302.

5 That Thales was not a prose author is not controversial; the crucial point is that archaic wisdom was a form of performance, for which see Martin (1993).

6 In Eucl. 70.23–25 (Morrow 1970, 58): ‘[…] the whole of the geometer’s discourse is obviously concerned with the cosmic figures’. Proclus has a Platonist axe to grind but it should be remembered that already in the fourth century BCE Plato was a major, canonical author, his works very widely read. It is likely enough that most educated readers, in Euclid’s time, who knew anything about advanced mathematics, knew it through its reflection through Plato’s works and in particular the Timaeus. Concluding a collection of the Elements with the construction of the regular solids was therefore a very reasonable choice.

7 We return to define the terms more formally below. The English word ‘rational’ is the one used by Heath to translate the Greek rhētos. In its non-mathematical sense, the adjective means ‘that can be spoken of’ (Vitrac, 1998, 34–36, uses ‘expressible’, very close to the non-technical meaning; perhaps a better translation for the mathematical usage as well). The modern mathematical use of ‘rational’ is related but not identical, and it is a misfortune that Heath’s choice has become entrenched in English translations: we stick to it but emphasize that this is an arbitrary and somewhat misleading translation.

8 In general for Pappus’ commentary to Elements X see Mansfeld 1998, 31–35 (we note in passing that the question of the authenticity of Pappus’ text – for which see Vitrac 1998, 418–421 – does not concern us here: whether Pappus or not, the author or authors are Late Ancient, with reasonable access to earlier mathematical works we no longer possess).

9 On the connection between the arithmetic books of the Elements and Archytas, see Huffman 2005 466–470 (the evidence is purely circumstantial and should be treated with caution – but the core proposal – that Euclid’s arithmetic books were written with music theory in mind, and that music theory was primarily associated with Archytas – seems likely enough).

10 For Book V, see Heiberg 1888/1977, 211. For Book XII, see e.g. Netz 2004, 32.

11 The Suda S.V. Theaetetus (Adler 1931, 689 Θ93) asserts that he was the first to write (or to prove) concerning the regular solids, while a scholion to book XIII asserts that the Pythagoreans were the first to find the pyramid, the cube and the octahedron while Theaetetus was the first to find the Dodecahedron and the Icosahedron (Heiberg 1888 = Stamatis 1977, 291; the Suda’s report may well be dependent on whatever formed the basis of this scholion). It is curious that both reports concerning Theaetetus ascribe to him, somewhat illogically, a part of a classificatory system; perhaps we see here the misleading traces of the emphasis – by Eudemus and others – on the question of first discovery (Zhmud 2006, chapter 1).

12 The role of proportion – and of Archytas – in the mathematics of the fourth century BCE is not controversial; it is elaborated further in Netz 2022, chapter 2.

13 The incommensurability of side and diagonal and square is transmitted in the manuscripts of the Elements as X.117. Heiberg (1888, lxxxiv – lxxxv) comments that this may well be a later interpolation (Knorr 1975, 228–229: probably motivated by Aristotelian commentary?). Unfortunately, this definitely does not mean that X.117 is a vestige of some earlier, pre-Euclidean theory. Note that Stamatis’ re-edition of Euclid does not include Heiberg’s introductory note that therefore must be consulted from the original 1888 printing, fortunately available online.

14 We sum up Menaechmus’ argument briefly. For a full study of this proof, see Sidoli (2018, 359–365) and references there.

15 This means that an earlier mathematician, like Theaetetus, does not have to know Menaechmus’ later argument to follow the same argument template, but rather that the success of this template may have motivated Menaechmus later efforts. Another possibility is that the specific route pursued here postdates Theaetetus and Menaechmus (closer in time to Euclid), and may have emerged from a reconsideration of the work of both mathematicians. Indeed, we are trying to reconstruct an account of the eventual formation of book X, not Theaetetus’s original process of discovery.

16 Indeed, if one constructs a rectangle contained by the binomial and apotome, then the rationality of the rectangle follows from Elements II.5, if the lines containing it are taken as the ‘unequal segments’ in the formulation of the proposition and the larger component of the binomial as identical to the ‘equal segments’. For the inverse, note that the given rational rectangle and the rational rectangle contained by the binomial and its cognate apotome are commensurable (as they are both rational), and their ratio is as the ratio between the cognate apotome and the non-binomial edge of the given rectangle, which are, therefore, also commensurable.

17 In Euclid’s terminology: the assigned straight line.

18 This is a term used often in the literature. Euclid does not subsume these lines under a common title, but names them individually.

19 In Euclid’s treatment, this is not the definition, but a consequence of the definition. See Fowler (1999, 187–188); Knorr (1983, 54–57); Mueller (1981, 288, 304–305 fn. 14) for a discussion of this issue.

20 That X.42 and X.79 are indeed necessary in this context seems to have been missed by Heath and Mueller. In order to show that all six kinds of binomial (resp. apotome) sides are distinct from each other, it is shown that when they are squared and applied to a rational line, the resulting breadths are the six kinds of binomials (resp. apotomes). It is obvious from the definition that all these kinds are distinct if their decomposition into incommensurable rational lines is given. But to show that the same line cannot be two kinds of binomial (resp. apotome) side with respect to two different decompositions, X.42 and X.79 are necessary.

21 This observation is familiar but is typically obscured by discussions of whether or not the role of constructions should be seen as proof of existence (and then, the debate is whether Euclid needs to show that his objects exist, based on a construction, or whether he simply realistically assumed their existence). In fact, it is clear that solving the problem of constructing objects is simply another tool in the Euclidean tool-box, hence the need for such constructions. (See Knorr 1982, Harari 2003 for the philosophical argument; a related claim is that of Sidoli and Saito 2009).

22 This analogy is only an analogy due to the gap between the Eudoxian definition of proportion and the arithmetic notion of proportion in terms of measurement by parts. This gap is never completely resolved in the Elements (see Mueller 1981, 136–138).

23 The question of the correctness of classifications is very significant to the philosophers of the fourth century BCE – from Plato’s method of division to Aristotle’s biological taxonomy. It is generally understood that Plato expected a ‘correct’ classification to be unique, all-encompassing and mutually exhaustive, while Aristotle in his biology was more accommodating of more fluid systems; but this was likely forced on Aristotle, so to speak, by the complexities of biological reality and there is no surprise in fourth century mathematicians preferring neater systems of classification (in mathematics, after all, classification is often up to one’s control). The locus classicus for this philosophical background is Pellegrin (1986).

24 Heath (1956, 161) considers X.28 as a construction required for X.75, but this assumes an interpretation of constructions as proofs of existence, and further assumes that conditional propositions in the Elements require a construction of the object involved in the antecedent, which is a debatable interpretation. If we accepted this interpretation, then together with the norms promoting the inclusion of analogous propositions and proposition-construction pairs, we could justify X.24–25 and X.27–28, leaving out only X.115.

25 In the above setting, if $c$ is a medial line, then by $\mathrm{sq}.(c)=\mathrm{rect}.(b,d)$, X.24 and X.22, the line $b$ is rational. By $\mathrm{rect}.(c,a)=\mathrm{sq}.(b)$ and X.20, the line $c$ is rational as well – a contradiction to the remark following X.111.

26 If $c$ is a medial line, then by $\mathrm{rect}.(c,a)=\mathrm{sq}.(b),$the line $b$ is a ‘medial-side’ (the first new kind of irrational mentioned in X.115). By $\mathrm{sq}.(c)=\mathrm{rect}.(b,d)$, the line $c$ is ‘medial-side-side’ (the second new kind of irrational mentioned in X.115) – a contradiction to X.115.

27 Note that we only need a part of X.extra for this argument: the application of a rational area to a medial line yields as breadth a medial line. But then, the inverse is not necessarily correct. The full X.extra, with its full inverse X.24–25, exhausts the relevant possibilities, as Euclid tends to prefer.

28 There are several textual problems in the relevant portion of the text that indicate a (pre- or post-Euclidean) editorial intervention, which might have ended up discarding X.extra. First, X.24–25 lack an inverse, which is somewhat odd. Second, the term ‘medial’ referring to areas is never defined, but appears suddenly in the corollary to X.23. Third, X.19 and X.24–25 share a textual problem (the strange lines ‘commensurable in length in any of the aforesaid ways’; see Heath’s 1956, 48, 55 comments to these propositions). Fourth, proposition X.26 unnecessarily intervenes between propositions and their corresponding constructions.

Moreover, the proof of X.114 is overcomplicated (Mueller 1981, 290; Knorr 1983, 55), because it relies on the inverse X.112–113 rather than proceeding directly. This may suggest that X.114 may have been derivative with respect to X.112–113, which accords with our interpretation of the latter’s role as crucial elements of the toolbox. Now, X.114 is analogous to X.25 in that they both show how incommensurable irrationals lines contain a rational area. Therefore, assuming a structural analogy, X.25 (analogous to X.114), together with its complement X.24, may have been preceded by their inverse X.extra (analogous to X.112–113). An editor who thought that ‘a rational area out of irrational lines’ was the point of these propositions may have discarded the seemingly unnecessary X.extra, leaving the textual traces discussed above and breaking the structural analogy.

29 ‘Surtout il faut conclure qu’aux yeux de son auteur la classification avait un intérêt intrinsèque justifiant son inclusion dans un traité d’Eléments puisqu’il ne s’est pas contenté des résultats concernant les quatre catégories qui lui étaient utiles’.

30 Pappus, agonistic with philosophers and, for this purpose, positioning himself as a philosopher, is the subject of Cuomo (2000, Chapter 2).

31 If we start with rational lines and are interested in three lines in proportion, we obtain closure with medial lines, ‘second order’ medial lines, etc. (propositions X.19–25 and X.115). However, when we go to four lines in proportion, the absence of solutions can motivate investigating binomials as candidates.

32 Can we learn anything from the title? In its Alexandrian context, it perhaps could suggest the title of a seminal and famous work of philological scholarship, Philitas’ ataktoi glossai, which Bing (2003, 338–339) suggests we should translate ‘Disorderly Words’. (In this book – setting the tone for much of future Alexandrian scholarship – Philitas started from the explication of individual Homeric words to note, in each case, rare, surprising facts; providing, in the process, a panorama of scholarship that emphasized its own lack of order or finality). Better still, as Bing points out, the title ataktoi glossai could in turn allude to (or could simply be illuminated by) Simonides’ ataktoi logoi, ‘Disorderly Tales’. The title to Apollonius’ work as provided by Proclus includes an element of description (‘The books on … ’) followed by what may have been the core title, ataktoi alogoi, ‘Disorderly Non-Tales’. In Alexandria, near the end of the third century … We doubt such a title was chosen by accident.

33 Netz (2020, 402). For Eratosthenes as mathematician and Platonist, see now Panteri (2021).

34 This is the main claim of Netz (2020, Section 4.3).

35 Stevin (1585); incidentally, this was not Stevin’s own view! ‘La croix des mathématiciens’ becomes the title of Knorr’s influential treatment from 1983.

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Funding

The authors did not receive support from any organization for the submitted work.