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.

In this paper, we attempt a new interpretation of Euclid’s Elements, Book X - a hazardous endeavour! But there are two hints that, we think, should be brought to bear and – with few caveats – were substantially ignored by past scholarship into this question. For the first hint, we follow Saito: the Elements was a toolbox, that is: it was not a self-explanatory work of mathematics but instead acted as a collection of useful tools. If so, we should look for what the book was useful for, rather than for where and how its definitions and theorems emerged. We need to identify a research programme, for the sake of which Euclid felt that he needed to provide the propositions of Book X as useful tools. Our second hint is that, in an important sense, Euclid was wrong. Hardly anyone ever used Book X in antiquity.

This combination is helpful, then: we look for a research programme, for which book X is useful, popular in the fourth century but falling out of favour quickly thereafter. This is the goal of this paper. We begin and end with the two hints. In the introduction, we say a little more about the first hint, the Elements as toolbox; in the conclusion, we briefly return (among other things) to the second hint, and consider the possible research context typical of the fourth century, but not the third. In between, we engage primarily in mathematical reconstruction: analysing how a certain problem (or, rather, a family of problems) might account for choosing to include in book X those very definitions and theorems that we find there.

1. Methodological introduction

We start, as we argue one should, with Saito. His rejection of mathematical reconstructions in favour of textual studies of Greek mathematics (1998) follows, as a matter of logic, from Saito (1985, 31), where he claims:

[…] it is not reasonable to attempt a determination of the nature of Book II by inconfirmable conjectures regarding its origin. The significance of Elem. II should be sought by studying applications of the propositions there. [Our Italics]

For example, Elements II.5 and II.6 seem to assert nearly the same result. We have a line, bisected, and then cut at another arbitrary point. In the case of II.5, the other point is on the original line; in II.6, that other point is on the original line, extended. The two propositions state a certain equality between ensuing squares and rectangles; algebraically, the relations are analogous to familiar and closely related quadratic identities. The results of II.5 and II.6 are so nearly identical – and their proofs so similar – that one can be tempted to understand their duplication as an expression of an interest in detailing a complete list of algebraic identities as such.

This, Saito explains, misses the mark. Elements II is frequently applied in more advanced Greek mathematical works and one can follow how II.5 and II.6 are applied, specifically, in the Conics. Sometimes they are applied when a bisected line is cut internally; sometimes they are applied when a bisected line is cut externally. Both conditions arise frequently enough that it becomes easy to see why a compiler of a work such as the Elements would wish to include both. Based on this simple observation, Saito went on to study the Elements as the ‘tool-box’ for more advanced mathematics (rather than a repository or sediments of past achievements or motivations), quantifying the applications of Euclid’s results in works by Apollonius and Pappus (Saito 2013).

In fact, ‘toolbox’ comes close to the surface meaning of the Greek word stoicheia, ‘Elements’. The etymology and usage are complex, but the relevant focal use of stoicheia is clearly in reference to the letters of the alphabet (Burkert 1959). To refer, in Greek, to stoicheia is nearly the same as the modern expression of ‘the ABC’ of a field. Even the Late Ancient commentator Proclus – who puts Euclid on a pedestal higher than most earlier Greek readers did – was very much aware of this understanding (In Eucl. 71.9–21: Morrow 1970, 59, slightly revised):

The theorems that are simplest and most fundamental and nearest to first principles are assembled here in a suitable order, and the demonstrations of the other [theorems] take them as the most clearly known and proceed from them. In this way also Archimedes in his book On Sphere and Cylinder and likewise Apollonius and all other geometers appear to use the theorems demonstrated in this very work as generally accepted starting points.

Proclus seems to oversimplify, implying a single line of deduction from the first principles to the more complex propositions (perhaps just a manner of speaking; to be sure, the linear emanation from a single source is very much in accord with a Neoplatonist world view). But he gets Saito’s point.

Let us add, briefly and somewhat dogmatically, to the historical context of the Elements. First, one notes a very substantial agreement between the ‘taken-for-granted’ of Archimedes – results Archimedes never bothers to spell out – with (a subset of) the contents of the Elements. This suggests, to us, that Euclid ought to have been notably older than Archimedes. At the same time, the fact that Euclid also wrote what appears to have been an Elements-like work in Conics¹ suggests, by the same token, that Euclid’s activity came notably later than the beginning of activity in the study of conics, that is, likely, rather later than Menaechmus, in turn probably active at about the mid-fourth century at the earliest.² Thus, there are good enough reasons to put Euclid, chronologically, into the box implied by both Pappus and Proclus – near the end of the fourth century or the beginning of the third. If the chronology of our sources was not wrong, there is no special reason to doubt the location that they attribute to Euclid, which, anyway, is likely enough: Alexandria.³ With time and place comes an entire context. We can put Euclid next to Callimachus and his Pinakes, cataloguing something like all past literature; or next to the effort, clearly underway in Alexandria of the time, to collect a ‘Hippocratic Corpus’; more metaphorically, we can put Euclid next to Herophilus’ and Erasistratus’ project of mapping the human body, Aristyllus’ and Timocharis’ project of mapping the sky and (somewhat later) Eratosthenes’ project of mapping the earth.⁴ It was a time, and place, of grand surveys. We can imagine Euclid invited, perhaps, to produce a survey of all past mathematics? (What a lost opportunity! But then again, would such a work survive?) At the end, he produced an especially comprehensive Elements, a full survey of the toolbox frequently used by all past mathematicians.

And – we now reach our point – if this is Euclid’s project, it is clear why ‘Mathematical Reconstructions’ should have gone out of fashion. They have been based, since time immemorial, on a completely different reading of Euclid. This indeed goes back to Proclus himself and indeed even further back, to his sources (In Eucl. 250.20–251.2: Morrow 1970, 195):

We are indebted to old Thales for the discovery of this and many other theorems. For he, it is said, was the first to notice and assert that in every isosceles the angles at the base are equal, though in somewhat archaic fashion he called the equal angles similar.

The final note provides this passage with false authority, as if Proclus or his sources had access to an actual relevant piece of writing by Thales (almost certainly, Thales wrote nothing; there is no point speculating how Proclus’ false report could come about⁵). The important point is that Proclus imagines here that the logical order of emanation – from the simplest to the more complex – also unfolded in time as the process of discovery. Scholarship, until quite recently, tried, Proclus-like, to reconstruct the history of early Greek mathematics essentially as the process of coming to write Euclid’s Elements. The capstone of this tradition – Knorr’s The Evolution of the Euclidean Elements, from 1975 – has this approach inscribed into its very title. In 1998, Saito could simply pronounce that these kinds of Mathematical Reconstructions were out. Which they remained: Sidoli 2014 shows clearly how far scholarship has moved away from concerns such as those of Knorr 1975. The only studies of Euclid that Sidoli finds worth mentioning are new translations – and studies of the Medieval tradition (Sidoli 2014, 30). The few works of scholarship focused on the early traditions of mathematics are piecemeal, on such topics as Pythagoreanism or mathematics in Plato, without any attempts of larger reconstructions and with no reference at all to Euclid! (Sidoli 2014, 38; indeed, this paper is written, in part, because we believe the pendulum may have swung too hard, in this regard).

Logically this state of affairs can be brought back to Saito 1985, but of course such intellectual transformations are overdetermined. Knorr was looking to see what to do next, having completed his first masterpiece. There was much more to read and do with the tradition later than Euclid, so much of it extant: Knorr wrote extensively on Archimedes, on Hellenistic mathematics more generally, on the adventures of mathematical texts in Late Antiquity and the Middle Ages. Knorr’s individual trajectory shaped much of the scholarly work in the history of Greek mathematics in the 80’s and the 90’s, but as Saito documented, Knorr was not alone. Perhaps this is best understood in parallel with developments elsewhere in the study of the ancient world: a general tendency to expand beyond the classical era into later periods and even into reception studies. (Knorr’s The Evolution of the Euclidean Elements came out in 1975; Peter Brown’s The World of Late Antiquity came out in 1971). In the case of mathematics, however, this shift in chronology also implied a shift in methodologies. Pre-Hellenistic mathematics could be studied only by fitting together fragmentary evidence; from the Hellenistic era onwards, one could raise questions based on extant texts. Mueller, in 1981, studied Euclid not from the diachronic perspective of how the Elements were put together but from the synchronic perspective of the structure of Euclid’s work. The horizons opened up by this new kind of engagement with the extant texts made the labour, and meagre results, associated with reconstructing a pre-Euclidean past, much less attractive. There are of course plenty of exceptions to this, but, generally speaking, not much has been added in this particular endeavour since Knorr’s early work. Indeed, the recent author on Book X with whom we are in closest agreement is Bernard Vitrac, and we owe a lot to his comments on Book X in volume III of his magisterial edition of the Elements (above all, we follow one of his proposals: an emphasis on the role of proportion). In contrast to past authors, Vitrac is simply very cautious about his interpretations, offering multiple possible contexts without offering any single historical interpretation. Vitrac 1998 was cautious – because it was 1998 (and, of course, because it was Vitrac: ever the prudent scholar).

Scholarship has moved away from adventurous reconstructions. And yet, of course, pre-Euclidean mathematics did exist; the best evidence for it does remain in Euclid’s Elements themselves. As historians, we are duty-bound, among other things, to consider this evidence for this purpose. This article is offered, then, as a speculative exercise in the reconstruction of early Greek mathematics, based on Euclid’s Elements. Our one methodological observation is, by now, obvious. As we use the Elements to reconstruct its antecedents, the question is not the one pursued by Eudemus, Proclus or Knorr – how the various bits of the Elements came to be written. The question is, what were the early results, for which Euclid’s Elements would have been a good tool.

The force of Saito’s methodological assumption – that the Elements were put together out of the tried-and-true toolbox – admittedly varies from book to book. The final Book XIII, in particular, has always struck readers – once again, beginning with Proclus⁶ – as a goal within the Elements, rather than a tool meant for other applications. Indeed, we have some evidence for the reception of this book through Hypsicles’ Book XIV, and what we see there are Euclid’s results serving not as a tool but as a foil. (Effectively: ‘I, Hypsicles, achieve this concerning the regular solids, which is different and better than what Euclid and Apollonius achieved before me’). Book XIII may well have been the exception, the one non-elementary book in the Elements.

Furthermore, the task of curating a toolbox does not determine its structure, and it is quite clear that Euclid imposed some rather rigid editorial norms. The norms that guide the organization of tools in the Elements-box include some well-known symmetries: if one proves a convertible result, one is to prove its converse as well; stepping-stones are added in as required, as are all the constructions, whether they are useful for subsequent authors or not. Thus, many results are in the Elements not so much because they are generally useful tools, but because they are related in specific ways to other, genuinely required tools (we will see this in detail in Section 4). Note that what is operative here is not some vague notion of completeness or architectural perfection, whose dimensions are never clearly determined – we are concerned here with a set of concrete and specific editorial norms.

2. Interpretations of Book X

In a nutshell (we will expand on this as the details become relevant), Book X deals primarily with combinations of so called ‘rational’ lines, that is, lines which are commensurable or commensurable in square with a given reference line.⁷ The sum of rational lines that are commensurable with each other only in square is called a binomial, and the corresponding difference is called an apotome. Further, the book discusses lines that we will call ‘sides’, because they form the side of a square that equals a rectangle contained by a binomial (or apotome) and a rational line.

From the point of view of a toolbox conception of the Elements, the function of Book X has always been the one major mystery. This mystery, however, was much less acute for past scholars who did not approach the Elements as a toolbox. If the Elements are simply something like the ‘greatest hits’ of fourth century mathematics, the reason for including book X becomes less of a problem, and the burden of the question becomes, instead, what made fourth century mathematicians gradually come up with just those theorems included in Book X. The origins of the book are usually traced to Theaetetus’ work on means and Eudoxus’ work on polygons (Knorr 1983, 42–49). However, the question remains: why were these bits and pieces processed and expanded into the unique behemoth that we call Book X?

The most prominent interpretation – indeed the one which has dominated for more than a millennium – is the reading of book X as an arithmetical and algebraic treatise, obliquely concerned with surds as numbers and the solution of quadratic equations. This interpretation can be traced at least as far back as medieval Arabic treatises (Matvievskaya 1987), was celebrated by Stevin (1585), and is clearly a fruitful interpretation from the point of view of later mathematics. By contemporary historiographical standards, however, this interpretation is far too weak to merit a discussion due to its rampant anachronism (e.g. Taisbak 1982, 66–69; Fowler 1992, 255–258; Mueller 1981, 270).

Fowler suggested that the motivation for studying the irrational lines of Book X was the fact that binomials and apotomes yield a sequence of periodic results when subjected to the anthyphairesis algorithms for finding the greatest common divisor (Fowler 1999, 189–191). Even if we believe that the Greeks were aware of this fact (whether or not they could prove it), this interpretation explains the interest in rational, binomial and apotome lines, but does not explain anything about the choice of propositions and the classificatory aspects of the book. It is therefore, again, a discussion of the origin, rather than the content of Book X.

Most other recent commentators saw Book X as an exercise or tour-de-force in classification and deductive organization (Mueller 1981, 270–271; Taisbak 1982, 27, 58; Knorr 1983, 59–60). Taisbak (1982, 27), for example claims that.

Apart from the fun one can have (and some Greek surely had) from manipulating such objects, we shall not hesitate to maintain that the game, bar very few details, has no mathematical importance … All the same, it is a fascinating, nay haunting, piece of literature, and its composition reminds one of epic or dramatic poetry, with its long sequences of uniform statements apt to be learnt by heart and transmitted orally.

Taisbak supports his view by a clever color-coding of the various lines and rectangles. But the only motivation he suggests for the book beyond aesthetic pleasure is to prove that one cannot simplify the presentation of some lines in pentagons and decagons beyond their expression as binomials (1982, 62).

However, the aesthetics of classification as a goal in itself cannot be assumed without invoking a cultural or pedagogical context that would encourage it as a goal in itself and provide it with specific characteristics. Indeed, unless we have clear criteria for such goals as ‘classification’ or ‘completeness’, supported by their use in an ambient context, it is hard to use them to account for the specific structural features of Book X. Moreover, these goals do not explain why one should classify those specific lines and why include those specific propositions that are actually included. Why stop where Book X does, rather than continue as Apollonius later did (Heath 1956, 255–259)? Why include some propositions, such as X.112–115, dangling, making them look like a later addition, as Heiberg contended (Heath 1956, 246), or as remnants of Theaetetus’s work, detached from the rest of the book (Knorr 1983, 56)? Why is the proof structure of the book much less economical and unified than it could be (Knorr 1985)?

The vagueness of the above interpretation is further manifest by its co-existence with interpretations that invoke other cultural-pedagogical goals as explanations for Book X, such as inspiring awe, deliberately hiding underlying ideas, and presenting a model of formal rigour (Knorr 1985, 34–35; the esoteric interpretation is attributed to Taisbak and van der Waerden). Moreover, some authors see Book X as far from perfect (‘greatly in contrast with the usual view of Euclid’s mathematical genius’; Knorr 1985, 34). The bottom line is that, unlike Book XIII, the interpretation of Book X as a goal in itself is not well motivated. Indeed, Knorr (1985) explains Book X as a rigorous, but problematic, stitching together of several pre-existing, structurally conflicting treatments of his so-called ‘fundamental problem’: when is the side of a binomial line a binomial as well?

While the latter question will indeed turn out to be pivotal for the organization of Book X in our interpretation as well, for the time being, it is entirely unmotivated. Why this question? Another prominent interpretation that explains Book X in terms of an application – its role in comparing sides of regular polyhedra in Book XIII – suffers from the same motivational difficulty. Indeed, Book X is not used in Book XIII to actually find the sides of some regular polyhedra. Instead, it is used to conclude that the ratios between the sides of some regular polyhedra and between these sides and the circumscribing radius are not rational. The issue is again, why this question? Indeed, Mueller writes: ‘I do not know why this problem of characterization was considered to be mathematically significant’ (1981, 295). This interpretation is further plagued by two additional problems. First, in order to transition from the specific lines used in Book XIII to the full scope of Book X, one needs to rely again on comprehensive classification as a (vaguely defined) goal. Second, if Book XIII is indeed the intended application, how come there are glitches between the propositions of Book X and what is actually applied in Book XIII (Mueller 1981, 283; Fowler 1999, 188)? We will return to this last interpretation in the conclusion, where we will comment on the approach closest to our own, that of Vitrac (1998).

Given the available interpretations and our methodological commitments, the question arises in its full urgency: just what was the mathematical project, within which tools such as those of Book X could have been regularly applied? It is the absence of any evidence for such a body of work that makes the book, to us, so mysterious. Our goal is therefore to find a well motivated family of problems, whose tools, together with the concrete Euclidean editorial norms (rather than a vague notion of completeness or classification) would account for the content of Book X.

3. Finding an application

The mathematics of Plato’s Theaetetus (147d–148b) is all about incommensurability. This already suggests a connection to the historical Theaetetus and Pappus obliges us with more specific historical evidence: in his introduction to his Commentary to Book X, now extant in Arabic (Thomson 1930, 63–64), he mentions two mathematicians: Theaetetus and Apollonius. (We shall return below to discuss Apollonius’ possible contribution). Concerning Theaetetus, Pappus begins by discussing his overall talent as vouched by Plato’s dialogue and then continues to assert that he introduced the distinction between commensurable and commensurable in square only (this seems worryingly close to the contents of the Theaetetus dialogue itself, and so makes one wonder if Pappus might simply spin history out of the dialogue), but then concludes by claiming, on Eudemus’ authority, that Theaetetus classified the irrational lines according to the means: medial and geometry; binomial and arithmetic; apotome and harmony. This evidence from Eudemus is very hard to interpret but it is clearly independent of Plato and so it suggests that, already in the fourth century, the classification of irrationals was primarily associated with Theaetetus.⁸ Finally, there are three major names we have for fourth century mathematicians – Archytas, Eudoxus and Theaetetus. We have some basis for seeing, in books VII-IX, the implications of Archytas’ work⁹; in Books V and XII, the implications of Eudoxus’.¹⁰ It would be natural to have some of the contents of the Elements reflect Theaetetus, and the likeliest candidate, then, is, indeed, Book X (as well as Book XIII which – in the form Euclid chose to give it – is clearly associated with Book X¹¹).

As we seek the function of Book X we ask, then, a very specific historical question: what was Theaetetus trying to accomplish? The point needs stressing again: we mean by this that we ask, not ‘what was Theaetetus trying to accomplish by writing Book X itself’. Instead, we are asking ‘what was Theaetetus trying to accomplish by proving some unknown results, for which Book X-like tools could have been useful’. The ‘unknown results’ could be much wider in scope than anything to do, directly, with irrationals, and so, when we ask, ‘what was Theaetetus trying to accomplish’, we literally mean, what were the key open questions and fields of research in any field of mathematics he was likely to pursue. If we find a convincing answer to this question, which can provide a good account of the specific content of Book X, Saito’s methodological approach, which we adopt here, will prove fruitful.

Wide as this question is, and sparse as our evidence for fourth century mathematics might be, we do have some starting points. In particular, the evidence for Theaetetus’ closeness to Plato is compelling, and the mature philosopher’s view of mathematics could well have been informed by what he had heard from the younger mathematician. Mathematics, as seen from Plato’s perspective, was above all the study of proportion. This is most clearly seen in the Timaeus where the mathematical principles of the universe are introduced in 31b-36b, bringing together two variations on proportion. One is the idea of organizing solids (‘body’) through the structure of four lines in continuous proportion, the other is the idea of organizing more abstract structures (‘soul’) through the numerical ratios of musical harmony. The proposal of this paper is to take this Platonic combination seriously, as a guide for the context in which Theatetus’ mathematics could have been developed.

The suggestion is that we look for a philosophically as well as mathematically significant combination of ideas concerning proportion. Indeed, for both variations on proportion that we noted from the Timaeus – four lines in proportion, musical numerical ratios – we have reports of major contributions, specifically, by Archytas – likely to have been the most significant presence in the mathematics known to Theatetus.¹² Archytas was the first to offer a solution to what we would call here the four proportional lines problem: where, given two lines A, D, we seek to find two more lines B, C, to produce a series of four lines in continuous proportion A: B ∷ B: C ∷ C: D. (This is often referred to as ‘finding two mean proportionals’ or, for historical reasons that need not detain us here, ‘the duplication of the cube’, see Huffman 2005, 342–401). Archytas may also have been the first to develop a theory of the conditions constraining numerical ratios in musical harmony. A key concept in music theory is that of the epimoric ratio, that is the ratio of the form $(n+1):n$ (a standard assumption is that only epimoric ratios represent harmonic combinations, such as 2:1, 3:2 or 4:3 – the octave, the fifth and the fourth, respectively). We have a single result reported from Archytas – which, however, seems to have been of central importance in the music theory of his generation. To paraphrase in simplified, modern terms, this result reduces to: there is no whole number $q$ such that $a(n+1):q::q:an$, where $a$ and $n$ are whole numbers (Huffman 2005, 451–470). We call this the epimoric result. From the epimoric result we can see that it is impossible to divide a ratio that produces harmonies into two equal constituent parts. (There is a reason, then, why the octave does not divide into two fifths and is instead the combination of a fifth and a fourth).

The two – the epimoric result and the four proportional lines problem – were certainly paradigmatic for the generation of Theaetetus. We also note that, while the two emerge in distinct domains, they are structurally analogous. In both cases, we consider how it is possible to insert terms, in between two given extremes, so as to construct a continuous proportion. In the case of music theory, the terms are whole numbers, the continuous proportion has three terms, and the task is to find whether the problem is solvable at all. In the case of geometry, the terms are lines, which would make things much easier, since the problem of finding a geometric mean between two given lines was already solved; however, since, following the Timaeus, we are looking for two mean proportionals, the problem turns out to be much harder. The question is not simply whether the mean proportional lines exist as such – this is evident through simple assumptions of continuity – but how to find a good geometrical mechanism with which to find the required lines. This task, indeed, would become one of the problems confronted most creatively in the history of Greek mathematics.

Now, since the two kinds of problems (the arithmetical and the geometrical) are structurally analogous, mathematicians would likely try to think of the one in terms of the other. There is not much conceptual new ground to be gained by thinking of the epimoric result on analogy of the four proportional lines problem. (Indeed, if by this extension we mean the more general problem of the conditions under which four – better still, any number of – whole numbers may be set in continuous proportion, this general problem is in fact handled in some detail in Euclid’s Elements Book VIII and quite possibly was already pursued by Archytas). The more difficult direction of analogy would be from the epimoric result to the four proportional lines problem. This means asking, in analogy to the impossibility of a whole number as mean proportional between two terms in an epimoric ratio, whether there are any geometric constraints that would exclude some kinds of lines from fitting as two mean proportionals between two given lines.

Note carefully: if we follow the guidance of the Timaeus, this is not some arbitrary, self-motivating analogy. Indeed, as noted above, the world of the Timaeus is composed of elements (earth, water, air and fire) shaped like regular solids from the Elements Book XIII and somehow forming a proportional sequence. Figuring out constraints on such proportional sequences, especially in the context of lines involved in regular polyhedra, becomes then a deep metaphysical quest within a meaningful philosophical context: are there constraints, analogous to those of music theory, governing the structure of the physical world? How could a Platonist not ask himself such a question? Theaetetus, we propose, did.

It bears clarifying further the force of our argument. Here and in what follows, we do not take Book X of the Elements as evidence for the significance of music theory for mathematicians such as Theaetetus, active in Plato’s milieu. To the contrary: the paradigmatic significance of music theory for authors of this milieu is well established, independently (it is as strongly established as it could be, given the state of our evidence quoted above). Given this paradigmatic significance, then, it is only natural to ask how this could serve as an inspiration for a geometrical study giving rise, ultimately, to Book X – which is the argument we develop in this paper.

And so: we look for music-theory-like constraints on the structure of four lines in continuous proportion. Obviously, this suggests that we should define some conditions on the four lines. In the musical case, all three terms must be whole numbers, and the given extremes are epimoric. The four proportional lines problem does not contain, in and of itself, such conditions and, as a consequence, appears to be unconstrained. The kernel of the thought of applying the analogy from the epimoric result, then, is as an exploration: assume certain conditions on the four proportional lines problem – what constraints follow? Specifically, the epimoric result suggests the following route: assume certain conditions on the given extremes and explore the ensuing constraints on the two lines to be found. This is a useful field of problems, because finding such constraints will help us identify or rule out cosmological harmonies, in the form of four proportional lines, among the lines of the regular solids that constitute the Timaean world. Instead of comparing lines arbitrarily, by means of trial and error, we will be guided by constraints that would inform us when and how two mean proportional lines can be obtained in principle. At this point, we set aside the philosophical interest in music theory, and concentrate on the mathematical narrative. However, we will return to philosophy and music in the conclusion.

It is worth noting that much of the argument can be sustained even without the philosophical context. Even within the terms of the mathematical research programme itself, after all, Archytas’ solution did involve a convoluted three dimensional construction and the same would be true, in a sense, of all future solutions (even apparently two-dimensional solutions, such as Menachmus’, discussed below, involve what was later called ‘solid curves’). This problem cannot be solved with the simplest geometrical tools (straight lines and circles in a plane), and so the proposed solutions always left ancient readers with the sense that one could usefully look for simpler, more concrete solutions. (This, indeed, is why so many solutions accumulated in the first place). In all likelihood, mathematicians felt the same even immediately following Archytas’ first proposed solution. This, then, seems like a very plausible motivation for the exploration: to map the terrain, so to speak, in advance, so as to identify the potential constraints limiting possible solutions.

Before we proceed, this should be added concerning the mathematical presuppositions for the exploration. The mathematical passage already mentioned from Plato’s dialogue Theaetetus envisages the mathematician as a young boy, studying the basic structures of commensurability and incommensurability. Past historians of mathematics, above all Knorr in 1975, tried to extract from this episode a history of the early Greek treatments of incommensurability. Following Burnyeat (1978) and the consensus of scholarship today, we find such efforts excessive. Plato did not set out to write a history à-clef. However, it is safe to conclude that Theaetetus did not invent the basic concepts and observations concerning incommensurability. Indeed, one does not need anything such as Book X to observe the incommensurability of the side and the diagonal in a square. (As is well known, this, in all likelihood, was not even originally in the Elements and was instead inserted as an appendix to Book X¹³). The idea that some lines are commensurable and others are not must have been a familiar observation, available to Theaetetus as he begins to ponder questions such as, for instance, the constraints on the four proportional lines problem.

Now, the first thought, perhaps, when looking for the application of the epimoric result to the four proportional lines problem, would be to make the extremes, quite simply, epimoric. It should be however quickly evident to a mathematician, pursuing this line of thought, that unless the inserted two lines are also assumed to be in a numerical ratio, there are no particular consequences associated with the epimoric as such – but if these are in a numerical ratio, we are simply back to the results already known from music theory. It becomes clear, then, that we should seek broader conditions: ‘epimoric’ should be replaced by something that would provide a properly geometrical, non-trivial and yet approachable, analogue. Evidently, the simplest analogue would be, simply, a commensurable ratio.

The question now is: what are the constraints following from the condition that the two extreme lines, in the four proportional lines problem, are commensurable?

Now, as the Theaetetus dialogue suggests, discussions of commensurability between lines – lines standing to each other in the ratio of number to number – go hand in hand with discussions of commensurability between areas – areas standing to each other in the ratio of number to number. One of the earliest observations must have been that some lines may be incommensurable, while their associated squares are commensurable: most famously, indeed, the side and the diagonal in the square. This is the basic observation underlying the mathematical passage in the Theaetetus and is so elementary as to be implicit in the lesson of the Meno (85b). We thus have the pair of concepts: lines that are commensurable as such, and – distinguished from them – lines that are commensurable in square only.

Once we have those two concepts available, and we look for constraints on the four-line problem, assuming that the extremes are commensurable, it is immediately apparent that we have found a strong constraint, which would exclude some candidates for the role of mean proportionals: the second term cannot be commensurable with the extremes in square only. Indeed:

• Let the lines $a:b:c:d$ be in continued proportion, where $a$ and $d$ are commensurable, and $c$ commensurable with them in square only.

• First, $\mathrm{rect}.(b,c)$ equals $\mathrm{rect}.(a,d)$.

  • o Therefore, $b$ and $c$ are commensurable. Indeed, $b:c::\mathrm{rect}.(b,c):\mathrm{sq}.(c)::\mathrm{rect}.(a,d):\mathrm{sq}.(c)$, and the latter two are commensurable by our hypothesis on $c$.

• Next, $\mathrm{sq}.(b)$ equals $\mathrm{rect}.(c,a)$.

  • o Therefore $b$ and $c$ are incommensurable. Indeed, $\mathrm{sq}.(b):\mathrm{sq}.(c)::\mathrm{rect}.(c,a):\mathrm{sq}.(c)::a:c$, and the latter two are incommensurable by our hypothesis on $c$. Therefore $b$ and $c$ are not commensurable either.

• So $b$ is both commensurable and incommensurable with $c$ – a contradiction to our assumption.

Theaetetus would then see, immediately, that, indeed, this exploration – the constraints, following from assuming the extremes commensurable – may lead to interesting exclusions of some lines from the pool of potential mean proportional between commensurable lines.

At this point one would need to think what further meaningful constraints one could look for. We find symmetrically that the second term in the four-line proportion could not be commensurable with the extremes in square only. Could it be, perhaps, some kind of combination of such lines? Say, the result of their addition, or subtraction? (Note that these kinds of lines are not just the next simplest candidates, they are also lines that are found in the regular solids of Book XIII, which makes them, along with possible proportional sequences involving them, clearly interesting for fourth century mathematicians inspired by the Timaeus). If the answer is positive, we should be able to come up with easy constructions for the two mean proportionals; if it is negative, we can easily exclude some dead end strategies or hack solutions. Such a question, therefore, would present itself very naturally as a potential extension of the result above. We have now motivated a well-defined question.

Question: Given four lines in proportion, where the two extremes are commensurable, could either of the two mean terms be the sum or difference of incommensurable lines that are commensurable in square with the extremes?

This, already, demands some real mathematical work.

What could guide our analysis of this question? Our best bet is to look for inspiration in the immediate context of the problem, namely in constructions of two mean proportionals. One of the best known constructions is that of Menaechmus (Netz 2004, 286–290). Menaechmus’ argument – produced, in all likelihood, in the generation following Theaetetus – proceeds as follows.¹⁴

• Given lines $a$ and $d$, Menaechmus assumes two mean proportionals, $b$ and $c$, between them.

• First, $\mathrm{rect}.(b,c)$ equals $\mathrm{rect}.(a,d)$

  • o This imposes one constraint, whose locus is a hyperbola.

• Second,$\mathrm{sq}.(b)$ equals $\mathrm{rect}.(a,c).$

  • o This imposes a second constraint, whose locus is a parabola.

• The intersection of the two curves determines $b$ and $c$.

The careful reader would note that we already used an analogous template in our proof above, which excluded lines commensurable only in square with the extremes from being either of the two mean proportionals.¹⁵ So let’s combine the two approaches. To facilitate the presentation, let’s fix the extreme commensurable lines as our reference lines, and call a line rational if it is commensurable with them, either simply or only in square only. Let’s call a binomial the sum of rational lines which are commensurable with each other only in square (if they are simply commensurable with each other, the sum is again rational, and the question reduces to the previous one). Now, applying the above argument template for discussing binomials as mean proportionals, we get the following:

• Given commensurable lines $a$ and $d$, and two mean proportionals, $b$ and $c$, assume that $c$ is a binomial.

• First constraint: $\mathrm{rect}.(b,c)$ equals $\mathrm{rect}.(a,d)$.

  • o This means that $b$ is the kind of line that, together with the binomial $c$, contains a rational rectangle.

• Second constraint: $\mathrm{sq}.(b)$ equals $\mathrm{rect}.(a,c)$.

  • o This means that $b$ is the side of a square that equals a rectangle contained by a binomial and a rational line.

But what do these two constraints imply? Are they compatible? Contradictory?

Concerning the first constraint, we have evidence that Theaetetus may have been interested in it. As noted at the very beginning of this section, according to Pappus based on Eudemus, Theaetetus related the notions of arithmetic and harmonic means to sums and differences of rational lines, respectively. This is interpreted as relating the reciprocity of arithmetic and harmonic means to the fact that a rectangle is rational (i.e. commensurable with a rectangle contained by commensurable rational lines), if it is contained by a binomial and the cognate apotome (the difference of the components of the binomial). Inversely, if a rational rectangle is contained by a binomial and another line, the other line is commensurable with the cognate apotome. These results derive from the tools of Book II of the Elements,¹⁶ and provide a good grasp of the first constraint.

The second constraint is a bit more difficult to analyse. The starting point is to think of $b$ as the sum of lines and analyse the relation between these lines and the components of $c$ based on the equality $\mathrm{sq}.(b)=\mathrm{rect}.(a,c)$. But it turns out that, depending on the relative properties of the components of the binomial $c$, the components of the side $b$ may or may not be rational, and would or would not satisfy some relations (in particular, $b$ may or may not be a binomial itself). Our forest of lines is growing. We have binomials, apotomes, and the family of ‘$b$-sides’ that we have just discussed.

Now recall that eventually we need to figure out whether the apotome from the first constraint could or could not also be the side from the second constraint: if it could, then we have zoomed in on the structure of our mean proportional, and if it could not, then a binomial mean proportional between commensurable rational extremes does not exist! This means that we do not only have to sort out the specific properties of the lines that we have spawned, but also to follow closely which of these lines could or could not be equal to which of these other lines. In other words, we need a classification of binomials, apotomes and sides into mutually exclusive species.

We could go on with this analysis, but from this point on it becomes quite tedious, and would have limited historical value (rational reconstructions and history rarely coincide). And anyway, we have already analytically reconstructed the components and concerns of Book X of the Elements (binomials, apotomes, their sides and their classification) from a problem motivated by the interests of a mathematician of Theaetetus’ time.

So let us switch gears from analysis to synthesis. Indeed, as we argued in the previous section, our goal should not be a reconstruction of how book X came about, but an explanation of the actual content of Book X (which may be far removed from its original ‘evolution’) in terms of proposed applications. What we suggest is that Book X is a collection of tools for proving theorems such as the one that excludes binomials from being mean proportionals between commensurable extremes. Let us begin with fixing the terminology (allowing for some mild deviations from Euclid’s terminology; for overviews of the content of Book X see Mueller 1981, ch. 7.2; Knorr 1983; 1985; Fowler 1992):

• The assigned rational line¹⁷ is a given line, with respect to which we define rationality. In our context, this is either of the extreme lines in the proportion.

• Two lines are commensurable in square, if the squares built on them are commensurable.

• A rational line is a line commensurable in square (possibly, but not necessarily, also commensurable) with the assigned rational line.

• A rational area is an area commensurable with the square on the assigned rational line. Note that the meaning of ‘rational’ is more restricted for areas than for lines, from a modern point of view.

• A medial line is the side of an area contained by two rational lines commensurable only in square. The area itself is called a medial area.

• A binomial (resp. apotome) is the sum (resp. difference) of two rational lines commensurable only in square.

• A binomial (resp. apotome) side¹⁸ is the side of a square that equals a rectangle whose one side is a binomial (resp. apotome) and the other a rational line.¹⁹ Based on the relative properties of the binomial (resp. apotome) line and the rational line that generate the rectangle (which we omit), the binomial (resp. apotome), as well as the resulting side, belong to one of six kinds. Each kind of side is the side of the corresponding kind of binomial (resp. apotome). The binomial (resp. apotome) kinds are numbered ‘first’ to ‘sixth’, whereas the sides have individual names (set out in X.54–59 and X.91–96). Only the first of the six kinds of sides is itself a binomial (resp. apotome).

• Each kind of binomial (resp. binomial side) is identical with the corresponding kind of apotome (resp. apotome side) up to exchanging sum for difference. The cognate apotome of a binomial line is the line obtained from the latter by exchanging the sum for a difference. The cognate binomial of an apotome is defined symmetrically.

Now we can provide the suggested proof of our purported application for Book X. We will show that the second of the two mean proportionals in the continued proportion $a:b:c:d$between the commensurable lines $a$ and $d$ cannot be a binomial.

• Suppose the line $c$ is a binomial.

• Since $\mathrm{sq}.(b)=\mathrm{rect}.(a,c)$, and the line $a$ is rational, the line $b$ is one of the binomial sides (by propositions X.54–59).

• Since $\mathrm{rect}.(b,c)=\mathrm{rect}.(a,d)$, the former is a rational area. So the line $b$ is commensurable with the cognate apotome of the line $c$ (by X.112).

• Therefore the line $b$ is itself an apotome (by X.103).

• But a binomial side is not an apotome (by X.111 and the proposition following X.72) – contradiction.

Symmetric arguments extend the result to the first mean proportional, $b$, and to apotomes (using propositions X.91–96, X.113, X.66, X.111 and the remark following X.111 instead of the analogous propositions above).

4. Is Book X now motivated?

We recall that our task is to find convincing applications for Book X. To make our argument compelling, these applications should not only use a few isolated components of the book, but should, together with prevalent norms governing the organization of the Elements, account for all or almost all of the book. This would support our claim that Book X is not just a ‘for-itself’ classificatory systematization of past bits and pieces, but that its definitions and propositions were curated with the above and/or similar applications in mind.

The above argument uses the following propositions either directly or as components of the proofs of the directly used propositions.

• X.5–8 (arithmetic interpretations of commensurable lines)

• X.11–23, X.26 (building blocks concerning commensurability, ‘geometric algebra’ and medials)

• X.36–41, X.73–78 (binomial and apotome sides are irrational)

• X.42, X.79 (uniqueness of division of binomials and apotomes into a sum or difference of incommensurable rational lines)

• X.54–59, X.91–96 (the side of the square equal to the area contained by each kind of binomial or apotome and a rational line is a binomial or apotome side respectively)

• X.60–65, X.97–102 (the square on each kind of binomial or apotome side, when applied to a rational line, yields as breadth a binomial or apotome respectively)

• X.66, X.103 (a line commensurable with a binomial or apotome is, respectively, a binomial or apotome of the same kind)

• X.111, the remark that follows it and the proposition that follows X.72 (medial lines, binomials, apotomes, and the five kinds of sides of binomials and apotomes that are not themselves binomials or apotomes are all distinct from each other)

• X.112–113 (the application of a rational area to a binomial or apotome line yields as breadth a line that is commensurable with the latter’s cognate)

Some comments are in order for this reconstruction. First, among X.36–41 and X.73–78, only X.36 and X.73 are strictly necessary for our argument. The rest come in because the comment that follows X.111 and the proposition that follows X.72 bundle the results that we need together with the claim that all the sides are distinct from each other and from medial lines. However, even if we include only X.36 and X.73 in our list, Book X has good reasons to include the remaining propositions of these two blocks, as we will discuss below.

Second, a similar situation occurs with X.42 and X.79. Again, they are required only to show that all six binomial sides (resp. all six apotome sides) are different from each other. These claims are bundled into the remark and proposition following X.111 and X.72, but are not strictly required for our argument.²⁰ Nevertheless, again, we will provide independent good reasons to include these statements in Book X.

So we have several blocks of propositions that are not accounted for by our suggested application. In order to account for them, we will invoke several well-known norms that govern the organization of the Elements. These norms are not without exception in the Elements, but do hold to a large extent.

A familiar norm of the Euclidean Elements is that one provides an explicit construction for the mathematical objects under discussion.²¹ This accounts for the constructions X.9–10 (lines commensurable and commensurable in square only), X.29–35 and X.73–78 (the six kinds of binomial and apotome sides) and X.48–53 and X.85–90 (the six kinds of binomials and apotomes). Next, if an inverse proposition to a given proposition holds, the Elements normally include it as well. This accounts for the presence of X.114, the inverse of X.112–113.

The next norm that we mention states that, if the same result is true for a family of analogous mathematical objects, it should be proved for all of them. This provides a motivation for including the blocks X.43–47 and X.79–85, assuming that X.42 and X.79 are required for our argument. Similarly, the necessity of X.36 and X.73 motivates the inclusion of the full blocks X.36–41 and X.73–78 (irrationality of all binomial and apotome sides), and the necessity of X.66 and X.103 motivates the inclusion of the full blocks X.66–70 and X.103–107 (lines commensurable with whatever kind of binomial or apotome side are sides of the same kind).

The presence of the blocks X.66–70 and X.103–107 is also motivated from the point of view of application: they provide tools for identifying binomial and apotome sides, which in turn may be useful for identifying lines that cannot serve as two mean proportionals between commensurable lines. This would also account for the presence of X.71–72 and X.108–110 (characterization of the binomial and apotome sides as sides of squares that equal the sum or difference of certain kinds of areas): they too provide tools for identifying binomial and apotome sides.

Moving on, the block X.1–4 is there, even though it is not used in Book X. X.1 (a variant of Archimedes’ principle and informal preliminary for X.2) is used in the context of exhaustion arguments, and X.3–4 (finding greatest common measures of commensurable lines) are quite useful (and are used, for example, in Theodosius’ Sphaerics). In the specific context of Book X, however, these statements serve an additional purpose. While it would be wrong to characterize Book X as arithmetical – irrational lines are simply not numbers (Taisbak 1982; Fowler 1992, 255–258) – this popular but wrong characterization is well motivated. Indeed, propositions X.2–4, are modelled after VII.1–3, which discuss greatest common divisors of numbers. This creates an explicit analogy between Book X and the preceding arithmetical books. The analogy is explored further in propositions X.5–8, which are, however, used later on in book X.²² This analogy could be there to make salient the cross-breeding between the arithmetic problem of a mean proportional in the epimoric ratio and the geometric construction of two mean proportionals between commensurable lines – which is where our story began.

Finally, our reconstruction naturally requires discussing binomials and apotomes as well as their sides (recall that the proof uses the fact that line $b$ is the side of the binomial $c$ and that it therefore could not also be an apotome). So classifying and contrasting irrationals is a well motivated concern, rather than just an aesthetic-pedagogical goal in itself. More generally, however, when mathematical objects are defined in the Elements, they should normally either supervene each other (like number and even number) or exclude each other (like even number and odd number). Situations where two definitions intersect without one containing another are shunned or resolved. For example, in the Elements – as opposed to modern definitions – squares, oblongs and rhomboids are distinct by definition, parallelograms are their disjoint union, and the latter is distinct by definition from rhombuses and trapezia.²³ And while even-even and odd-even numbers intersect without subsuming each other, they are resolved into distinct, non-intersecting sub-categories in IX.32–34. So the interest in coming up with non-intersecting categories of lines is motivated both in terms of our suggested application and editorial norms. As noted in footnote 20, this requires X.42 and X.79.

What is not explained, however, by our reconstruction, is why divide binomials, apotomes and their sides into those specific categories that Euclid actually articulates. (The polyhedral sides in Book XIII in themselves do not justify the division, because the specific irrational lines used there can be generalized into an overall division in various different ways.) Now, independently of any specific division, asking when a side of a binomial (resp. apotome) is a binomial (resp. apotome), which Knorr (1985, 19) christens the ‘fundamental problem’ of Book X, is already well motivated by the previous paragraph. In order to have a binomial for its side, a binomial $a+b$, where $a$ is the larger component, is characterized as having the side of the area $\mathrm{sq}.(a)-\mathrm{sq}.(b)$ commensurable with $a$, and as having $a$ commensurable with an assigned rational line (the first definition in the second definition block of Book X and proposition X.54). The first characterization imposes a twofold division (whether the two lines are or are not commensurable), and the second characterization a threefold division (whether $a$, $b$ or neither is commensurable with the assigned rational line). Altogether, we have six kinds of sides (see Knorr 1985, 20; other approaches to the classification are available elsewhere in the same paper and in Vitrac’s 1998, 51–62). Again, exploring all possible combinations in such a situation is normal for Euclid: in the context of quadrilaterals, for example, he defines right-angled equilaterals, non-right-angled equilaterals, right-angled non-equilaterals, and two kinds of non-right angled non-equilaterals, even though the latter three categories are never used as such. So once the question of overlap between binomials and their sides is raised and answered as in X.54, the entire division is well motivated.

This leaves us only with X.24–25, X.27–28 and X.115 as unmotivated propositions. But before we motivate their inclusion in terms of another (analogous) application of Book X, we make a methodological comment and raise a possible critique of our argument. So far, we tried to account for the entire Book X by means of the tools necessary for the single application that we suggested and by norms that would merit the inclusion of supplementary propositions. In a sense, we treated our proposed application of Book X as motivating the reconstruction of Book X, analogously to the way in which Mueller (1981, 16–26) motivated Book I based on the single proposition I.45. But this is, in fact, an overkill. We do not expect that Book X was designed with a single application in mind. There may be other related applications that could justify some components of Book X. However, the fact that we can account for so much of Book X with just a single application and some accompanying Euclidean norms serves, at the very least, as a proof of concept: Book X could arise as a set of tools useful for problems of this kind.

Of course, there are always other tools available for a given problem. A possible critique of our approach is that in fact there are simpler arguments for proving our main application. A modern, or an Arabic or Latin medieval algebraist, would probably argue along the following anachronistic lines. In the above setting, given a proportion $a:b:c:d$ with commensurable extremes, if the line $b$ were the binomial $\sqrt{x}+\sqrt{y}$ with rational (in the modern sense!) $x$ and $y$, then$d$ would equal $a\cdot {\left(\frac{b}{a}\right)}^3=\frac{{\left(\sqrt{x}+\sqrt{y}\right)}^3}{a^2}=\frac{x+3y}{a^2}\sqrt{x}+\frac{3x+y}{a^2}\sqrt{y}$. This reduces the original problem to showing that the latter binomial is irrational, namely to proposition X.36. However, this line of proof assumes a very non-Greek algebra, and even if the argument were to be ‘Greekified’ by dissecting cubes and establishing a theory of commensurability and rationality for boxes (defined analogously to the commensurability and rationality of rectangles), we would end up with a very eccentric argument for the context of Greek mathematics.

And yet, an analysis similar to our own would still allow for an easier proof. As before, $a:b:c:d$ is a continued proportion, with $a$ and $d$ commensurable lines.

• Suppose the line $c$ is a binomial.

• Since $\mathrm{rect}.(b,c)=\mathrm{rect}.(a,d)$, the former is a rational area. So the line $b$ is commensurable with the cognate apotome of the line $c$ (by X.112). Therefore the line $b$ is itself an apotome (by X.103).

• Since $\mathrm{sq}.(b)=\mathrm{rect}.(a,c)$, the line $c$ is the breadth of the $\mathrm{sq}.(b)$ applied to the rational line $a$, so the line $c$ is an apotome (by X.97).

• But a binomial line is not an apotome (by X.111) – contradiction.

Symmetric arguments extend the result to the first mean proportional $b$ and to apotomes (using X.113, X.66, X.60 and X.111 instead of the cognate propositions).

Note that this argument completely circumvents the need to discuss binomial and apotome sides, greatly simplifying the necessary toolbox. The existence of such a simpler proof may appear to pull the rug from under our argument, unless we make a leap of faith and assume that our initial argument was the actual historical route to Book X, which we do not. Our task after all is doubly obscured: we are not sure which problem Theaetetus was trying to solve, and we are not sure which tools he (or his followers, from whose work Book X was derived as a toolbox) actually used to solve it. Instead, we will identify a wider, related set of problems, for which Book X plausibly provides a set of tools. What matters is not a single result, but an entire field. Looking at the entire field of application, we will be able to motivate the discussion of sides and the few remaining, unaccounted propositions.

5. The field of applications

Once we show that binomials cannot serve as mean proportional between commensurable lines, a reasonable next step, in a context framed in terms of lines and areas, is to ask whether the sides of binomials can serve as mean proportional. Indeed, the toolbox that supports our original argument easily allows to show that sides of binomials and apotomes cannot serve as two mean proportionals between commensurable lines either. Indeed:

• Let the line $c$ be a binomial side.

• Since $\mathrm{sq}.(c)=\mathrm{rect}.(b,d)$, the line $b$ is a binomial (by X.60).

• According to our argument above, this is a contradiction.

Therefore, the discussion of sides extends the scope of applications by identifying further lines that cannot be two mean proportionals between commensurable lines. This extension is properly motivated as, recall, such sides are found in the context of regular polyhedral in Book XIII (the edge of the icosahedron) – the same polyhedra that appear in the Timaeus and aroused our interest in four mean proportional lines. Moreover, the discussion of binomial and apotome sides provides additional ways to identify binomials and apotomes (via X.60–65 and X.97–102), which extends the applicability of the main theorem.

In this spirit, then, let us explore some further potential applications. The remaining propositions not discussed yet, X.24–25, X.27–28, and X.115, are propositions and problems concerning medial lines and areas. Since X.27–28 are constructions that correspond to X.25, they would be justified once we justify the presence of the latter. Note that none of these is used elsewhere in Book X, so they stand out as an independent block.²⁴ Note also that X.25 and X.114 are analogous in the sense that they show that irrational lines can contain a rational area. But we doubt that this was the sole purpose that motivated X.25, as it is neither placed next to X.114 nor accompanied by a corollary similar to the one that follows X.114 that would make this fact explicit.

So in order to account for X.24–25 and X.115, we suggest the following application, analogous to the main application above, namely: medial lines cannot be first or second proportionals between commensurable rational lines (recall that medial lines implicitly occur as the line $b$ in our first proof in Section 3 and as components of some binomial sides, so they are already motivated in our context). There are several paths to a proof. There are simple intuitive proofs using only X.24²⁵ or only X.115.²⁶ We could even produce a proof that uses all three components (X.24–25 and X.115), but it is rather contrived, so we omit it. However, we suggest another reconstruction. It requires intervening in the text of Book X, but this intervention, as we will show below, appears to make sense.

The intervention is the insertion of an extra proposition into Book X, an inverse of X.24–25, which we will call X.extra. It states that a rational or medial area applied to a medial line yields as breadth a medial line. Once we assume this theorem, X.24–25 are justified as inverses of the new proposition (the coherence of this ‘extra’ proposition and its proof within Book X has already been established in Taisbak 1982, Theorem 215 and Vitrac 1998, 163–164). Moreover, propositions X.115 and X.extra together easily prove our application, which would exhaust the components of Book X. The proof is as follows:

• In the above setting, suppose $c$ is a medial line.

• $\mathrm{rect}.(b,c)=\mathrm{rect}.(a,d)$ is rational. Since $c$ is a medial line, $b$ must also be a medial line (by X.extra).²⁷

• $\mathrm{sq}.(b)=\mathrm{rect}.(a,c)$, so $b$ is a ‘medial-side’ line (the first new kind of irrational defined in X.115).

• But a ‘medial-side’ is not a medial (by X.115) – contradiction.

Note that unlike the proofs in the preceding footnotes, the format of the last proof fits Menaechmus’ template – four lines in continuous proportion determine equalities between two rectangles, and between a rectangle and a square – the template that guided us in the proof of our main argument. This, then, would provide a strong coherence to the logic of the Book.

Note that according to this reconstruction, X.24–25 and their missing (implied? lost?) inverse, X.extra,²⁸ play the same role for medial lines that propositions X.112–114 play for binomials and apotomes: they are the terminal points of sequences of tools for proving that these irrational lines cannot be either of the two mean proportionals between commensurable rational extremes. Thinking along these lines, a block discussing medial lines, beginning with X.19-23, followed by the theory of irrational ‘medial-side’ lines from X.115, would be the very brief medial analogue of the complicated theory of irrational sides of binomials and apotomes culminating in X.111. The missing X.extra, followed by X.24–25, would be analogous to the block X.112–114. The constructions X.27–28 (with or without X.26) would parallel the subsequent construction of binomials and apotomes.

Such a reconstructed block, used for proving that medial lines are not either of the two mean proportional between commensurable extremes, would form a miniature prelude to the elaborate sequence of tools for proving that binomials and apotomes cannot play the same role. So placing X.115 and X.extra after X.23 would generate a structural analogy that could guide the reader through the otherwise bewildering remainder of Book X. The proposition X.extra may have then been intentionally or accidentally discarded (see footnote 28), and the discussion of ‘medial-side’ (and further) lines, now rendered pointless, would be pushed all the way to the very end of the book (the current X.115), dangling there like a scholium or an afterthought.

This would make for a neat aesthetic and rational reconstruction of the structure of Book X, but remains a pure speculation, especially given the problematic transmission of these propositions (Vitrac 1998, 383–385). So let us return to well motivated applications. The solid polyhedra of the Timaeus and Book XIII involve various magnitudes and proportions (not just those implied by circumscribing all of them in the same sphere). The task of arranging them in sequences of four proportionals is made philosophically and mathematically interesting in the Timaeus. The toolbox of Book X allows us to exclude candidates for such proportional sequences by specifying the kinds of irrationals involved and applying the toolbox (Can a binomial side be one of two mean proportionals between a rational and a binomial? Can a rational be one of two mean proportionals between an apotome and a binomial?) This is clearly better than experimenting individually with all the possible combinations of four lines from this reservoir. Therefore, we have shown in this excursus the variety – and, to some extent, the internal coherence – of the field of applications of Book X defined by the question of which lines may, or may not, belong to a series of lines in continuous proportion.

6. Conclusion: the reception of Book X

Let’s return to previous interpretations of Book X. Much more persuasive than other previous accounts of Book X is that of Vitrac 1998 68–86. This account brings together two main strands of interpretation: the study of means (which is indeed very close to the issues we emphasize) and the study of regular figures. Ultimately, Vitrac’s theory is that a confluence between the study of means and of proportions of lines in regular figures, accompanied by the study of the application of areas to lines, gave rise to the variety of ways in which they can, and cannot, be commensurable with each other, hence to the classification of irrationals. This is a perfectly possible account, which has the great merit of grounding the origins of some propositions of Book X in the one application it does have in the extant Elements – the classification of the sides of regular solids. However, even this explanation accounts only for some of the lines considered in Book X. To account for the book as a whole Vitrac falls back on systematization:

Overall, one has to conclude that in the eyes of its author the classification was interesting in itself, justifying its inclusion in a treatise of Elements, because he was not content with results concerning the four categories [of lines] which were useful for him. (Vitrac 1998, 86)²⁹

To justify the generation of the actual list of lines (but not of the specific propositions of book X), Vitrac cleverly relies on a pedagogical-combinatorial recipe provided by Pappus (Vitrac 1998, 51–62).

Beyond the critiques mentioned in Section 2 concerning the ‘Book XIII application’ and ‘classification for itself’ accounts of Book X, Vitrac’s account has two further drawbacks, which our interpretation will confront in the following pages. One is obvious, the other – to our mind, the more important one – less so. The obvious drawback of this theory – that Book X was produced in the context of trying to find which sides of regular polygons are commensurable with which – is that it is somewhat mystifying why anyone would consider this a particularly burning question. The less obvious drawback is that there is no clear reason why, if this question was considered important, it would then recede from view. What stopped Greek mathematicians from producing studies classifying many different lines, arising in many configurations, according to the classifications of Book X? Why should this not be a live research programme of the third century? It is always possible to posit that such a research programme was in fact pursued but then was completely lost, but it should be clear that anyone making such a claim is, in fact, making a radical interpretative move, basing their historical theory on the belief that an entire mathematical project was once pursued but then was completely lost without leaving a trace. This, to repeat, is possible, but should not be seen as the more prudent interpretation.

To be clear, then: when we hypothesize that the key to the Elements was their potential application, and try to interpret it in this key, we make a claim concerning reception: historiography and mathematics are of a piece. So what evidence do we have for the way in which earlier Greek mathematicians interpreted the book? Obviously, the evidence is meagre. And yet it supports, we believe, two claims: one positive, the other negative.

Pappus’ Commentary is the most direct evidence we have for the ancient reception. It is, of course, a late reflection and perhaps coloured by later approaches. Through much of Book I of the commentary, Pappus returns again and again to Platonic philosophy, and it seems, from this discussion that, as far as Pappus is concerned, the goal of the Book is in its philosophical classification (as we saw above, most modern readers who consulted Pappus for his interpretations have taken this to heart). This, then, is the philosophical reading of Euclid, certainly relevant for many readers in Late Antiquity (when mathematics folded back into Platonist philosophy, so that even Pappus needs to position himself in such philosophical terms³⁰). In the remainder of the commentary, however, Pappus pursues certain consequences of the treatise. He emphasizes the infinite number of classes noted by Euclid himself, but above all Pappus notes various outcomes of ‘applications’ (in the sense of applying an area of a given kind on a line of a given kind). Here is Pappus’ own summary (Thomson 1930, 165):

We can investigate by means of the facts that have been presented, any problems that may be set, as, for example: – if a rational and an irrational line be given, which line is the mean proportional between them, and which line the third proportional to them […].

In his capacity as a philosopher, Pappus emphasizes Book X as a classification. In his capacity as a mathematician, he understands it as a set of tools, using which one can answer, primarily, the following question: given a series of lines in continuous proportion, and knowing the classification of some of the lines, what can be said about the classification of the remainder? This, obviously, is the question we have emphasized throughout, but in the more demanding context of four proportional lines.³¹ (Note, conversely, the absence of reference to the classification of sides of regular polygons!)

This is useful – Pappus, after all, probably did have access to more works from antiquity that relied on Book X, and he possibly would have reflected alternative mathematical applications, had those been known to him. Nothing of this sort is visible and so the direct evidence from Pappus provides some support – however tenuously – to our interpretation.

But the invisibility of any further evidence is, indeed, the most frustrating aspect of Pappus’ commentary. There’s simply very little history contained in this work (which makes it quite unlike Pappus’ Collection). This is suggestive, perhaps, not so much of Pappus’ diverging projects in the commentary to Book X (which after all is very meta-mathematical and indeed more philosophical than what we see from Pappus elsewhere) but of the nature of the evidence available already to Pappus himself. This is the main, negative observation concerning the ancient reception of Book X, repeated throughout this article: there was apparently so little of it. It appears that Euclid, producing his Elements by the end the fourth century, believed that the tool of the classification of irrationals was supremely important but that, in practice, very little use was made of this tool.

The only clear evidence we have for later work in the field of irrationals is that of Apollonius. The evidence tells us little regarding the contents of the work: all we learn from Pappus is that Apollonius ‘added some remarkable species of these [i. e. irrationals] after much laborious application’ (Thomson 1930, 63), while Proclus cites Apollonius’ results on irrationals as an example of what would be inappropriate for a work on elements ‘because they lead to great and unlimited complexity’ (In Eucl. 74.21–22: Morrow 1970, 61). Proclus continues to provide us with, perhaps, the title of Apollonius’ work, ta peri tōn ataktōn alogōn, ‘The [books] On Unordered Irrationals’.³² It is noteworthy that these two snippets of evidence – tiny as they are – concur on one thing: the treatment in Unordered Irrationals was especially laborious; apparently, even by the standards of Apollonius. It appears that Apollonius set himself the challenge of composing a book similar in spirit to Book X – but even more cumbersome, even more difficult to follow – proposing an even more difficult and self-defeating classification. (Speculatively: the evidence from Apollonius suggests not that Book X was not perceived as a cul-de-sac but rather that Apollonius found a creative way to build on the very cul-de-sac features of the book, emphasizing its nature as what has appeared, even then, as a frustrating classification). This may well fit within the carnival of calculation noted in Netz (2009) (and perhaps alluding, in its title, to the Hellenistic carnivals of erudition? See footnote 32). At any rate, this does not appear like an application of Book X and so, by implication, the conclusion is that very few, if any, such applications were in fact produced in antiquity.

The research programme, for the sake of which Book X could have been composed, must have mattered through the fourth century and then fell to the side in the third. The insight we ascribe to Theaetetus is the application, in geometry, of a research programme originally proposed for music theory. This would be fitting for a Platonist mathematician. Music theory, in fact, is what best fits our bill. The study of musical harmony loomed large in the fourth century: it may have been Archytas’ main achievement and it certainly inspired mathematical Platonism. It is remarkable, then, that with all the tremendous efflorescence of mathematics in the Hellenistic era, no attention at all seems to be paid to music theory (with the single exception of the work of Eratosthenes who was unique, among his fellow-mathematicians and fellow Alexandrians, in being, among other things, a Platonist philosopher³³).

Mathematics, in the Hellenistic era, seems to mark itself away from philosophy.³⁴ This creates a certain gap between the mathematics of the fourth century and that of later generations. The gap is most evident with music theory: foundational for authors such as Archytas and probably Theaetetus, trivial and ignored by authors such as Archimedes and Apollonius. Later on, music as a mathematical discipline would be revived and would become central, once again, to a Pythagorean and Platonist style of mathematics seen, e.g. in authors such as Nicomachus. But by the time music theory was brought back to the fold, the research programme implied by Book X would have been dead.

So let us recapitulate by spelling out concretely this research programme, based on the application of musical thinking to geometry. We have throughout discussed questions of the form: ‘is it possible to fit lines with characteristics X within a continuous proportion?’ There is an obvious sense in which, once such questions begin to accumulate, they become self-motivating. One becomes aware of the possibility of the question and a competent mathematician would naturally wish to extend it from one case to another. However, it is also very likely that such questions could have been independently motivated, because particular lines were seen as valuable and important on their own, and therefore it was considered important to see if those lines could be put together in the harmonious relation of a continuous proportion. We think that one obvious case where this question would present itself is the regular solids. An interest in the way in which the sides of the regular solids could fit together in a proportion would be a very natural one to ask by anyone reading – or writing – the Timaeus. In this sense, indeed, the study of regular figures could form an important background to the development of Book X.

This resolves, for our interpretation, our first objection to that of Vitrac. The problem would not be the unmotivated exploration of the commensurability of the sides of regular solids as such, nor a self-motivating quest to embed them in a masterful classification scheme. The problem would be that of finding whether lines involved in regular solids could fit into four-line proportional sequences. Indeed, the irrationality results in the Elements XIII.18 preclude the edges of the four Timaean solids – tetrahedron, octahedron, icosahedron and cube – from forming a proportional sequence when circumscribed in the same sphere. This raises the question: which relevant lines might form the proportion alluded to in the dialogue? Our account explains how negative propositions related to this question can be derived from, and effectively exhaust, the toolbox known as Book X. This would render questions imported from the theory of music to the theory of regular solids a direct application of Book X.

Which leads us to our second objection to Vitrac’s interpretation, which our interpretation resolves as well: why did the project disappear from view? One answer is that the results derived from Book X were all negative. They were never successfully used for constructing proportional sequences of lines, only for excluding candidates for such sequences, and therefore left no trace in the constructions that reached us. More generally, it is easy to see how – once mathematicians retreat from philosophy – such applications would lose their interest, indeed even their meaning. In the fourth century BCE, taking inspiration from music theory to geometry would have been self-evident. Beginning with the Hellenistic era, this application was, at first, actively shunned and then became, simply, incomprehensible. Hence, the trajectory of Book X, from self-evident ‘Elements’ to ‘The Cross of Mathematicians’.³⁵

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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.