1. Ancient foundations of mathematics. Plan of the article

How to define the meaning of “foundational” in modern mathematics is not immediately clear. The meaning might amount to looking back at the principles of a well-developed mathematical discipline (definitions of basic entities, axioms) in order to set out them in a more orderly, rigorous fashion, and on this basis reshape portions of the whole deductive building. Hilbert's Grundlagen der Geometrie is a paradigmatic example of this way of defining the meaning of “foundational” in modern mathematics. It might also amount to providing new, non-formalized discussions of basic mathematical objects, as in Frege's Grundlagen der Arithmetik. Finally, the meaning may lie somewhere midway between these two extremes, for example take Dedekind's Was sind und was sollen die Zahlen. Doing all of this may entail developing new mathematical techniques: it may even happen that entire domains, like set theory or (formal) logic, would serve as tools or conceptual reference frames to investigate foundational themes in other domains, for instance analysis – and that such tools, if not invented for, were at least partly developed toward this end.

This article aims to explore what foundational issues in Greek mathematics might have possibly meant “in the field.” The last qualification, “in the field,” is important. As a matter of fact, “professional” mathematicians were normally silent about the foundations of their own domain, whereas “professional” philosophers appear (or, better said, are made to appear by modern, philosophically-oriented scholarship) to focus mainly on the restricted issue of the ontological status of mathematical objects. My goal is to show that there was a wide-ranging and articulated mathematical field of discourse in “foundations of mathematics,” a fact that is by no means obvious. Further, I aim to display the variety of philosophical, scholarly, and mathematical tools that were set up by the authors who were involved in the mathematical field of discourse as they developed their investigations.

One cannot assume that ancient authors had the same sensibility as modern mathematicians regarding what counts for a mathematical subject to be classified as “foundational”. What is more, Apollonius, for one, was surely convinced that he was doing nothing but basic geometry (and not something he would have termed “foundational”) when he undertook to simplify, or even to reform, so many notions that were the basic ground to what he received from the geometrical tradition.

There did not exist in Greek mathematics a technical domain that was devised for foundational purposes, as algebra had to become in the hands of Arabic mathematicians. The equations were explicitly and preliminarily introduced by al-Khwārizmī as a reference paradigm unifying the treatment of problems pertaining to geometry, to arithmetic, or to one of the subordinate sciences (geodesy, logistics, . . .), namely, as a kind of super-mathematics that would be able to validate other branches of mathematics. He first lists, discusses, and solves the six normal forms of the equations of the second degree; he then justifies his procedure, effected by completion of the square, by means of a geometrical argument applying the results obtained from El. II; finally, he passes to their applications. Very faint traces of a similar approach, breaking the borders of the canonical division between the sciences, can be found in Greek sources. One has to turn to the “general mathematics” (Rabouin Reference Rabouin2009) identified by Proclus with proportion theory or with the theorems of El. II (iE, 7–10, 60–1), or to Hero of Alexandria, who in the Metrica develops systematically the idea of a geometrical justification, by means of the language of the “givens” mutuated from Euclid's Data, of the algorithms for the calculation of areas and volumes of plane and solid figures, while at the same time, according to Pappus, Coll. VIII.1, regarding “rational mechanics” as a meta-discipline subsuming, and hence logically preceding, arithmetic, geometry, astronomy, and “physics.”

For this reason, even though I shall show that foundational questions were pursued in a systematic way by Apollonius (he also wrote two treatises mainly concerned with such questions), nevertheless, he did not approach these questions in the same way he approached questions about, say, conic sections. That part of what Apollonius did can be termed “foundational” is more a matter of his assuming an attitude with a technical import than it is the subject-matter inserted into a “domain of research.” The latter term is inadequate simply because foundational themes in Greek mathematics never reached the status of a well-established field of research, with its own principles, techniques, and open problems. The reason for this becomes clear if we finally turn to the problem of the meaning of “foundational” in Greek mathematics.

Of course, this meaning can only be settled by scrutinizing the extant evidence, however scant that may be, but the very silence of our sources concerning motivations and aims of what they are doing forces us to set a priori some preliminary points. My basic assumption, which has sound historical support and will be repeatedly corroborated in what follows, is that Greek mathematics itself constitutes a tradition: looking back at, and building on, what has been done by past mathematicians was not only a matter of course but mandatory. The starting point of the tradition was unanimously taken to be Euclid and his variegated corpus (cf. Acerbi Reference Acerbi, Bartocci and Odifreddi2007). On this assumption, my preliminary statement is minimal and surely not unexpected: the Greek geometers would have agreed to call “foundational” what pertains to the principles set out in (the first book of) Euclid's Elements: the definitions, the postulates, and the common notions (the latter frequently called “axioms”).

My formulation explicitly sets out the idea that what is at issue is not what we could regard as “foundational” in Greek geometry, but what the Greek geometers would have regarded as such, were they asked to explicate it. The obvious trick of my statement will delude no one: what meaning do I attach to the verb “pertains”? Well, the rest of this paper does nothing but sharpen this meaning. Without qualification, it allows a certain latitude for the techniques of investigation. On the one hand, Greek geometry did not admit an independent and formalized meta-theory: the foundations of geometry were made by using geometrical techniques as only technical tools. On the other hand, even though the subject-matter is mathematical, it does not imply that the tools set forth to treat it were exclusively mathematical. One of my aims is to show that the foundations of Greek mathematics were made “in the field” also by means of philosophical, philological, scholarly tools. Scholarly tools are, for instance, Apollonius’ fondness for crypto- and self-quotations, his subtle variations on, and re-deployment of, proof techniques of his own or those handed down from the tradition. Scholarly tools (I insist on this qualification, which is crucial in my approach) are Geminus’ linguistic and theoretical analyses, his deploying the philosophical background of the principles. The two approaches interact and complete one another, each contributing in its own right to shaping a rich field of discourse.

From this perspective, the writings of a foundational character (there were such writings, as we shall see) can safely be included, very much in the Greek way, in a literary genre. These foundational writings are a very specialized branch of the genre “mathematical commentary,” and perhaps they were the initial germ from which the genre itself was conceived and its rules shaped by the hands of Apollonius and Geminus and, after them, the hands of Hero of Alexandria. Such rules included the following: adding missing cases of lemmas and corollaries or adding mathematical complements to received mathematical texts; proposing alternative proofs suited to strengthen their deductive structure or to simplify it; replacing indirect proofs, such as reductions to impossible, with direct proofs or, less frequently, the opposite; elaborating counter-examples (apparently a specialty of Porphyry, cf. iE, 297.1–298.3, 347.20–352.14); making structural adjustments, such as inverting the order of certain propositions, suppressing them, or adding definitions and axioms; even rewriting entire segments of the text, as in the case of El. II.2–10, set in the format of analysis and synthesis by Hero (Anaritius, 74–86). The authors who engaged in commentaries were concerned with issues of logical coherence and deductive structure, and they adapted demonstrative techniques from other fields. This reminds us of the Alexandrian commentaries of the literary genre, where difficult passages of the edited author were explained but substantial modifications to the text were also proposed.

I have mentioned above the characters who will play key roles in this article. They are not customarily ranked among the most prominent thinkers in the domain of foundations. They are the late third-century B.C. top-ranked mathematician Apollonius, the author of the Conica, the main extant treatise on conic sections from Greek antiquity, and the first-century B.C. scholar and polymath Geminus, author of a lost multi-volume encyclopedia Theory of Mathematics. Discussing their respective contributions is made difficult by the fact that Geminus is almost surely the primary source of most of the foundational material originating with Apollonius – and the fact that we have access to the doctrines of both via a third source: the Commentaries on the first book of Euclid's Elements, written by the fifth-century Neo-Platonist Proclus, who presents substantial extracts from Geminus’ encyclopedia. One must face, then, a delicate doxographical problem in disentangling the respective contributions. In what follows, I shall show that Apollonius entertained a consistent and wide-ranging research program aimed at reforming “from within” many key aspects of the geometry he had received from the mathematical tradition.

As for Geminus, he practiced a form of lexical and doctrinal eclectism that makes it impossible to rank him either as a Stoic (granted, but which kind of Stoicism?) or as a Peripatetic. Modern scholarship is remarkably uncertain in assessing his role. He has even been depicted as an inept third-rank actor both in mathematical and philosophical matters, a slavish compiler of the most influential early first-century B.C. Stoic Posidonius (Blass Reference Blass1883), whose Meteorologica he had discussed in compendiary form, still preserving for us a Posidonian key text on the relationship between natural philosophy and the technical disciplines. Historians of mathematics know him predominantly as the main source of Proclus on foundational themes (Tannery Reference Tannery1887). Historians of ancient astronomy appreciate the uncommonly sober style and the sound rational judgment offered in the only extant work of his, the intermediate-level manual of descriptive astronomy Introduction to the Phenomena. Perhaps he was neither a philosopher nor a mathematician, but simply an enlightened rationalist trying to coordinate several mathematical sciences under an antiquarian, scholarly, technical project mid-way between setting firm philosophical foundations and producing high-level popularization. If one, then, has to set up a fictitious, long-distance debate between two authors as Apollonius and Geminus, this can by no means be reduced to an exchange between a “professional mathematician” and a “professional philosopher,” with the historiographical simplifications entailed by such a sharp divide. More generally, I would even doubt that one can sensibly speak of philosophically oriented schools in the domain of foundations of mathematics, where technical constraints may easily become predominant.

Plato's negative position in Rsp. VII is well-known: mathematics receives its principles from dialectic. A positive statement that philosophy is the meta-discipline dealing with the principles of mathematics (in the sense given above) is advanced by the early first-century Middle-Platonist and Stoicizing Biblical exegete Philo of Alexandria.Footnote ¹ He asserts that philosophy lays the “foundations” (themelia) of mathematics by providing definitions of the very basic geometric entities (points, lines), by means of which geometry will build more complex objects. Philo's claim can be backed by Posidonius’ prescription in the passage mentioned above, drawing a sharp divide between the tasks of the natural philosopher and those of the astronomer (as a mathematician) – making the latter subordinate to the former. As a matter of fact, Philo's statement is misguided: the distinction between mathematics and philosophy gets blurred, and is at bottom unimportant in dealing with our subject and with its sources. What is important is the use our sources make of what they report.

The paper is organized as follows. In Section 2 I shall quickly review some major foundational themes developed by authors other than Apollonius and Geminus. In Section 3 the documents concerning Apollonius’ foundational aims are collected and discussed; several new Apollonian fragments are added to those currently accepted. In Section 4 some documents concerning Geminus’ foundational aims are presented and analyzed. In Section 5 the role played by homeomery in directing foundational researches, and the respective contributions of Apollonius and Geminus in their study, are finally assessed. Apollonius’ aims, as well as his use of superposition in technical contexts, will support the thesis that his investigations about homeomeric lines extended even wider than the sources actually attest.

2. Foundational issues. An outline

In order to broadly outline a context to the main sections of this paper, I will briefly list, without aiming to be exhaustive, authors other than Apollonius and Geminus who were involved in debating foundational themes. I distribute these authors under a number of headings, according to their attitude towards the principles.

3. Apollonius' foundational aims

Apollonius is one of the most celebrated Greek mathematicians. He is the author of the Conica, the main extant treatise on conic sections from Greek antiquity (an overall assessment of it is found in Fried and Unguru 2000). He redacted a number of advanced monographs (in fact often intended to solve one single problem) in the “domain of analysis,” which Pappus describes in Collectio VII (cf. Pappus of Alexandria Reference Jones1986, 510–46). These are now lost in Greek; only an Arabic translation of the De sectione rationis survives. But this is not the whole of his production. He was also a renowned astronomer, maybe the one who proved that the description of the planetary motions by means of epicycles and that by means of eccentric circles are equivalent (cf. Almagest, III.3 and XIII.1). More importantly, Proclus ascribes to him a treatise About the screw, where he studied problems connected with geometrical homeomery (iE, 104.26–105.18). Most importantly, the late fifth-century Neo-Platonist, and pupil of Proclus, Marinus of Neapolis asserts that he wrote a General Treatise where he proposed modifications to the language of the “givens” employed in Euclid's Data (EOO VI:234.15–6). Pappus, in his turn, says that Apollonius proposed a classification of locus theorems “before his own [scil. Apollonius’] Elements (stoicheia)” (Coll. VII.21). Maybe we should not take this as the (only) mention of Apollonian Elements, but as a reference to his General treatise, or to elementary statements or propositions prefaced, for instance, to his Plane loci. Actually, Marinus’ denomination “general treatise” might simply be a characterization of a particular Apollonian work, and not its title. Admittedly, the same must be said of Pappus’ stoicheia. The problems with the title do not affect the fact that Apollonius wrote a treatise on basic features of mathematics; the surviving documents will allow us to get an idea of its contents (of course, one must make the minimal assumption that all the material comes from that work, but there would be no point in multiplying treatises preater necessitatem).

A complete picture of the foundational interests of Apollonius can only be obtained through a detailed discussion of the extant evidence, and the reader interested in a summary may jump to the end of this section. However, it is necessary to stress now why his role is so special. The fact is simply that he is surely the first, and by and large the only, Greek mathematician who did (basic and advanced) mathematics as a consequence of reflecting on mathematics, and not simply as a consequence of feeling himself to be part of a tradition, a tract, as I have suggested in the opening section, that generally qualifies the whole of Greek mathematics. Apollonius’ approach is a second-order one, pursued with first-order mathematical tools – setting definitions, proposing (alternative) proofs, etc. – but also with philosophical and scholarly methods. What is more, his approach takes concrete shape in a wide-ranging project of reform and coordination of the basis of geometry: if Apollonius is interested in problems of deductive economy, he explicitly treats them as such, and as the primary issues on which the focus of his investigation is directed. To understand this point, think of El. I.1–3. The presence of these problems clearly meets the requirements of deductive completeness, but we have no grounds to claim that their presence was dictated by such requirements. Well, many Apollonian achievements are set in a satisfactory context exactly because they were dictated by such and analogous requirements. Finally, his approach had momentous consequences, such as, if I am right in my suggestion, setting the standards of the literary genre of “mathematical commentary.” All of this will be documented below. A comparable approach is to be found maybe (and only as far as technical matters are concerned) in the first-century geometer and astronomer Menelaus, who wrote Geometrical Elements (Hogendijk Reference Hogendijk2000), and surely but very partly in the early fourth-century mathematician Pappus, who, however, does so because he is part of a tradition of commentators (but he produced some new mathematics, contrary to his colleagues from late antiquity). If we exclude Geminus, the only other Greek author displaying wide-ranging foundational interests with a conspicuous technical import is the first-century geometer and scholar Hero of Alexandria, but this would be the matter of another paper (preliminary statements in Acerbi Reference Acerbi and Koertge2008 and Vitrac, forthcoming).

The Apollonian documents about homeomery have been discussed in the companion paper (Acerbi Reference Acerbi2010) and will be summarized in section 5 of this article. Apollonius’ concern for foundational themes is mainly attested, as our nearest source, by Proclus in his commentary on Elements I. The latter reports a few arguments by means of which Apollonius tried to modify or explain notions and proofs in the Elements (an asterisk precedes items whose ascription to Apollonius can be argued but which are not assigned to him by Proclus).Footnote ⁶

1) A comment on Euclid's definition of line at El. I.def.2 (iE, 100.5–19): A clarification of the conception of line, of which

The two arguments correspond to the characterizations of a line in El. I.def.2 and 6; the example of the wall is Aristotelian.Footnote ⁸ The same explanation, slightly expanded and with the two component arguments interchanged, was compiled in Definitiones 2 (HOO IV:16.5–16). It is not obvious that the argument about perception is Apollonian, even if the astronomical simile seems to entail some technical competence in its proposer.

*2) A comment on Euclid's definition of surface at El. I.def.5 (iE, 114.20–25): A clarification of the conception of surface, of which

The structure of the argument is very similar to the one of the preceding item, and the wording is at several places identical. I decidedly ascribe it to Apollonius, and suggest that Proclus already found the argument anonymous in his sources (notice that Proclus attributed the clarification in the preceding item to “the followers of Apollonius”).Footnote ⁹

3) A comment on Euclid's definition of plane angle at El. I.def.8 (iE, 123.15–19 and 124.17–19): A general definition of angle as “contraction of a surface or of a solid into one single point under an inflected line or surface” is ascribed to Apollonius. Proclus observes that “in this way he seems to define every angle in general,” that is, both plane and solid angles, whereas in the Elements the definitions are kept distinct.Footnote ¹⁰ The same definition we find in Definitiones 12 and 14 (plane angle) and 22 (solid angle).Footnote ¹¹ Proclus immediately (iE, 123.19 ff.) reports Syrianus’ objections to this and Euclid's definition:Footnote ¹² an angle does not fall under the domain of relation (so Euclid), of quantity (Apollonius among others, and Carpus),Footnote ¹³ or of quality (Eudemus, mentioned at ibid., 125.7), but is rather a combination of all of these. Ranging Apollonius among the partisans of quantity does not strictly follow from the above definition, but from Plutarch's claim (ibid., 125.15–18) that Apollonius agreed in saying that an angle is “the first interval under the point.”Footnote ¹⁴ The claim is followed by an explanation; it is not immediately clear whether we should ascribe it to Apollonius or to Plutarch:

The logic of the argument suggests that the explanation is by Apollonius, endorsed by Plutarch, rather than Plutarch's interpretation of Apollonius’ definition above. The former, standard objection is by Proclus or rather by his source, the latter by Proclus himself. What is bewildering in the definition of angle as “the first interval under the point” is its manifest Epicurean ring: without assuming a theory of geometric minima, the definition is easily shown to be meaningless, as Proclus does. It is implausible that Apollonius had actually endorsed the definition, which is of course at variance with the one directly ascribed to him. However, we cannot exclude that he had discussed it, maybe providing an interpretation of the expression “first interval,” from which Plutarch extracted the reported explanation. The presence of Plutarch and Syrianus among the mentioned authorities, and the reservations about assessing the generality of Apollonius’ definition, suggest that Proclus probably drew his material concerning angles in form of excerpta and directly from his school's teaching tradition. This would explain why the name of Geminus is not appended to a classification of angles Proclus himself reports (iE, 126.7–127.16) and that must very likely be ascribed to him. Geminus himself might well have appended a number of definitions of angle to his classification, being thus the ultimate source of a large portion of the material presented by Proclus.

4) A proof of Euclid's Common Notion 1 (iE, 183.13–14 and 194.9–195.10): In the very first mention Proclus asserts that “Apollonius set out in vain to provide proofs of the axioms,” maybe implying that he proved also axioms other than the first. However, the subsequent mentions expressly refer to the first axiom only,Footnote ¹⁵ and we are given the proof too:

Two points qualify such a proof. First, Apollonius took El. I.cn.7, namely, “things which coincide with one another are equal to one another,” not as a one-way statement but as an equivalence, thereby replacing the relation of equality with “to occupy the same place as.” The move appears to be licensed, as far as straight lines are concerned, by El. I.4, namely, the first proposition where Common Notion 1 is applied – and in both directions (cf. Proclus’ short discussion at iE, 240.24–241.17). Second, he took the transitivity of this relation as more evident than the one of equality. As for the first point, Geminus, and Proclus with him, retort that the notion of “occupying the same place as” is “far less clear” than equality and that the sense in which things occupying the same place are equal is not clear. K. von Fritz (Reference von Fritz1955, 65) proposed that Apollonius aimed simply at proving transitivity of congruence of straight lines, as in fact the principle is applied in the Elements to straight lines only. Even if this appears to be too reductive an interpretation of the Apollonian text (Common Notion 1 is not restricted to lines, nor is Apollonius’ proof), unless the general import of the proof is a Proclean figment, von Fritz had some point. The ancient notion of equality extends to equivalent figures: the more restricted notion of congruence was formulated by further qualifying equality in the standard form “being equal and similar” and not by introducing new terms such as, for instance, “coinciding.”Footnote ¹⁶ The same happens with the only occurrence of the latter term in the extant Apollonius in Greek (Conica IV, preface, at AGE II:2.12), but of course the loss of Conica VI in Greek makes this datum unreliable. Given the extant evidence, the problem raised by the apparent Apollonian reference to the restricted notion of “equality” in his proof of Common Notion 1 cannot be solved. This material is very likely taken from Geminus, whom Proclus (iE, 183.14–20) mentions as raising a direct objection against the idea of proving axioms.

It is interesting to remark that the semantic domain of “occupying the same place as” served an analogous purpose, this time connected with symmetry of a relation and not with transitivity, in later expositions of the Aristotelian theory of the syllogism. The argument shows that universal negative premises convert simpliciter on the basis of the manifest symmetry of the relation “being disjoined from.” The proof is ascribed to Eudemus and Theophrastus by Alexander and, drawing from Alexander and in a less neat way as usual, by Philoponus (Alexander, in APr., 31.4–10 and 34.13–15, and Philoponus, in APr., 48.11–18 = fr. 90A-B Fortenbaugh et al.). Let us read Alexander's version:

There is no ground to suppose that this and Apollonius’ argument were correlated.

The remaining items in our list are alternative proofs. It is unlikely that Proclus drew them from the same sources as the passages concerning the principles, but we are not in a position to determine which were his (new) sources.

5) An alternative proof of El. I.10, showing how to find the midpoint of a segment (iE, 279.16–280.11): Apollonius did not resort, as is done in the Elements, to I.9 (how to bisect an angle) applied to an equilateral triangle having the given segment as base. He constructed instead two equilateral triangles having the given segment as their common base and then joined the remaining vertices. The triangles are neither posited nor mentioned as equilateral but their very construction according to the procedure in El. I.1 is performed.

6) An alternative proof of El. I.11, showing how to draw the perpendicular to a straight line from a point on it (iE, 282.8–282.24): The only difference is that the equilateral triangle posited in I.11 is again explicitly constructed according to the procedure in I.1.

7) An alternative proof of El. I.23, showing how to construct on a given straight line an angle equal to a given one (iE, 335.16–336.8): The proof is very different from that in the Elements, and reads:Footnote ¹⁷

The proof uses El. III.27 and avoids resorting to I.22 (applied directly in the Elements only in I.23), a proposition akin to I.1 since it shows how to construct a triangle from three given lines.

*8) An anonymous alternative proof of El. I.2 (“To place at a given point a straight line equal to a given straight line”) that according to Heiberg (Reference Heiberg1884, 488–9, a short account of Tannery Reference Tannery, Heiberg and Zeuthen1881) should be ascribed to Apollonius:

The proof assumes that the interval determining the radius of a circle does not “collapse” when the divider is lifted from the plane, and again refrains from using El. I.1. The first move amounts to a less restrictive interpretation of El. I.post.3, different from the one implied by the original proof of I.2; it is not a question-begging move as Proclus maintains. As a matter of fact, the proof is rather pointless unless a wider strategy is assumed on the side of its author; the strategic character, as well as the name of the author, was already lost in Proclus’ source, where the proof was very likely compiled and assessed in isolation. I think Heiberg was right in his guess and rightly identified the second move as authenticating the proof as Apollonian (see also below).

*9) An alternative proof of El. I.5, that Proclus ascribes to Pappus (iE, 249.20–250.19): the inversion in the order of the denotative letters is a convenient way to formalize the idea of superposing a figure with its own mirror image.

The proof can be viewed either as a limiting case of the original El. I.5 or as an adaptation of the idea of I.6 to the converse case. Its obvious drawback is the use of I.26, but we have learnt from item 7 that Apollonius was eager to perturb the deductive order established in the Elements, and at any rate it is not said that he rewrote their entire initial segment. The reader may recall that in the preceding paper I used this proof as model for proving Geminus’ second lemma. Its connection with the domain of homeomeric lines is readily seen. One has simply to realize that a characteristic property of homeomeric lines could be stated by suitably generalizing a very simple property valid for one such line, namely the straight line. The idea can be put to effect by changing the proof of I.5 attested in the Elements, in such a way as to offer an argument minimizing the number of geometrical elements involved, thereby making its adaptation to a more general context easier. This is what Pappus’ proof does. In this perspective, it fits well the general aims underlying the other Apollonian proofs alternative to those transmitted in the Elements. What is crucial is that Proclus presents it just before the fundamental reference to homeomery where he stresses that the isosceles triangle is not the figure to which the result of El. I.5 applies in the most general way. The right figure is in fact an isosceles trilateral whose base is a portion of a homeomeric line. This I could not take as a mere coincidence, and I would be inclined to assume that the same link was already established in Pappus or in his source. What is more, putting emphasis on superposition might well have been the original motivation for proposing the alternative proof of I.5. If the proof is to be ascribed to Apollonius, we must assume that Pappus compiled the proof without mentioning, as he very often does, his source (which, at any rate, might well be an earlier compilation in which the name of Apollonius had already dropped out).

*10) A short tract on the fact “that if two lines are drawn at less than two right angles, they meet,” that I would tentatively ascribe to Apollonius and that the tradition hands down in Arabic and under the name of Thābit ibn Qurra.Footnote ¹⁸ I shall quickly expound some circumstantial arguments; a detailed analysis of the terminology employed in the Arabic text is mandatory and might possibly settle the issue. First, the Fihrist (Reference Dodge1970, 637–8) reports that a tract on the same subject and with the same title is explicitly ascribed to Apollonius by Thābit ibn Qurra himself. Second, the title expressly refers to the abridged reference to the fifth postulate contained in El. I.29. We find an almost identical title in a Ptolemaic tract on the same subject (iE, 365.7–9). It is different from any other title found in Arabic authors, who normally refer to the “fifth postulate” itself or to Euclid's postulates in general (cf. Jaouiche Reference Jaouiche1986). Third, Thābit is credited for another tract on the same subject.Footnote ¹⁹ The tract is quite inept, and does not stand comparison with the other one; some interpreters regard it as a youthful tentamen by Thābit. Its title in the ms. Aya Sofya 4832, f. 51r, quotes in full the fifth postulate, whereas in the ms. Cairo, National Library n. 40 Riyāḍa Mīm, f. 200v, mention is made of “the well-known postulate of Euclid.” Of course, Thābit might well have written two treatises on the same subject, but it is by no means implausible that a translation of his could be miscredited to him. Fourth, the envoy that usually closes writings of Muslim authors is absent, or at least reduced to a very short one that is very likely an addition of the copyist (cf. the much richer envoy in the other tract ascribed to Thābit).

The seven propositions contained in the short treatise have several striking features, on which I shall not insist here. What is important in our perspective are the introductory remarks and the proof of prop. 1 (Sabra Reference Sabra1968, 19–21). The former contain a careful discussion of the role and the use of superposition techniques in geometry. Such operation, defined as “moving one of the two things to be measured by one another and pushing it from its place and transferring it in our imagination without changing its shape by movement so as to place it with its shape [unchanged] upon that which is to be measured by it,” serves in fact to ground the basic operations of measurement as a comparison between geometrical objects: coincidence, exceeding, and falling short. The author goes on to observe that such an operation is necessarily at work in El. I.4 and I.8, the former being “independent of the preceding proposition, a principle and a first in relation to other [propositions following it].” As for the propositions preceding I.4, the author says, the clue of the proof of I.1 lies in the “equality of the straight lines drawn from the centre,” and, therefore, in our understanding the “construction and definition of a circle,” namely its mode of generation by means of a rotating line with one extreme fixed.Footnote ²⁰ Such a rotation involves a mathematical object and happens in our imagination. That the rotating line remains unchanged “is something evident that has been made familiar by custom.” On the contrary, a translation of a straight line “such that one of the extremes continues to touch a fixed straight line” would be something unfamiliar. In particular, it is not obvious that the moving line keeps its “shape and position” unchanged. This entails incertitude as to the equality of the line at different positions. In order to ensure this condition, the author conceives of the movement of the line as being performed as if the line were contained in a solid, that is, as a rigid translation, according to the following assumption:

Of course, the move amounts to assuming a definition of parallelism in terms of equidistance, but it is remarkable how far the assumption is made to reach on very basic conceptions. Such a principle, and prop. 1 where it is applied,Footnote ²¹ reminds us of the geometrical characteristic property of a straight line formulated as a locus problem in Apollonius’ Plane Loci (notice also the verb “to touch,” typical of the formulation of locus problems, here and in the quote above from the Arabic tract):Footnote ²²

It seems to me that several features of the discussion summarized above can be profitably set in parallel with some of the (possibly) Apollonian fragments listed above.Footnote ²³ These include resorting to mental representations of mathematical objects obtained by abstraction from experience (cf. items 1 and 2), establishing a definite connection between superposition and equality, set out from the very outset of the tract on parallels (cf. item 4), reducing “elements,” namely, starting points or propositions of the Elements, be they controversial or not, to notions perceived as more basic and entailing a rethinking of the meaning and applications of basic geometrical operations (cf. items 4–9). Echoes of some of the conceptions expounded in the tract, and maybe a more direct relation of doxographical dependence, can be found in a passage of the late mathematician Carpus quoted by Proclus (iE, 242.22–243.11, but Carpus’ doctrines are reported starting from 241.19). In the context of a discussion of the issue of the relative ordering of El. I.1–3 and I.4, probably as an appendix to the debate about the prominence of problems over theorems, Carpus expounds there his conception of “superposition” by expressly referring to I.4:

In a wider perspective, further evidence suggests that a critical rethinking of the whole mathematical tradition was one of the guiding lines of Apollonius’ attitude towards research. The following arguments can be adduced:

1. a) A scholium to Euclid's Data asserts that definitions 13–5 should be ascribed to Apollonius (EOO VI:264.2–3).Footnote ²⁴ In his Prolegomena to the Data themselves, Marinus of Neapolis relates that Apollonius introduced “ordered” tetagmenon as a substitute for “given” dedomenon both in the On Vergings and in the General Treatise (EOO VI:234.15–6). A possible connection can be set with the likely Apollonian coinage tetagmenōs “in an ordered way” in the theory of conic sections.

2. b) Apollonius wrote a treatise about Unordered irrationals, expressly devised as an extension of book X of the Elements.Footnote ²⁵

3. c) In the introduction to the so-called XIV book of the Elements, Hypsicles asserts that he composed the small treatise spurred by Apollonius’ research on the comparison between the dodecahedron and the icosahedron (EE V,1:1.4–13). The latter investigations should be regarded as a natural complement of Elements XIII.

4. d) Pappus asserts that “Apollonius filled out Euclid's four books of Conics and added on another four, handing down eight volumes of Conics” (Coll. VII.30). This is of limited value, however: the numerical coincidence is suspect and no other source mentions this Euclidean work.

5. e) The peculiarities of some books of the Conica can be understood only as extremizations of corresponding peculiarities of books in the Elements.Footnote ²⁶ Cases in point are the following. First, the exclusive use of proofs by reductio in Conica IV brings a less pervasive but still obvious feature of Elements III to the extreme (a move that did not escape Eutocius, see AGE II: 354.8–13). Second, the treatise On tangencies generalizes and provides a complete solution to the particular problems El. IV.4–5 (to inscribe or circumscribe a circle in or around a given triangle). Third, certain properties of minimum and maximum lines in Conica V, such as, for instance, V.31–3, appear to have been exemplated from analogous properties of the radius of a circle (a particular conic section indeed); the whole book appears to be an amplification of El. III.7–8 to any conic section. As it turns out, Apollonius was fond of crypto- and self-quotations: subtle variations on, and re-deployment of, proof techniques are a major feature of the Apollonian corpus, in which the degree of entanglement between different works of his is apparent even in the actual, deplorable status of the evidence. For instance, some of his works in the domain of analysis (De sectione rationis, De sectione spatii, Sectio determinata), provide a complete and detailed solution of single problems, constituting the analytic pendant of important propositions in the theory of conic sections (Con. III.41, III.42–3, and the characterization of a conic section as the locus on three and four lines, respectively).

6. f) The systematic structure of Apollonius’ overall production is in line with the analogous character of the Euclidean corpus. Some of his analytical works are true lessons upon method, as opposed to the treatises in the same corpus redacted by Euclid and Aristeus; these (for instance the Porisms) are collections of disparate results simply belonging in the same genre of mathematical propositions. Apollonius systematized not only the theory of conic sectionsFootnote ²⁷ but also the domain of loci, most notably the plane ones. He provided a rather abstract classification of them and possibly reformed their format, as we know from Pappus: according to Apollonius, they can be “occupation” (ephektikoi) loci, as a point is locus of a point, “path” (diexodikoi) loci, as a line of a point or a surface of a line, “conversion” (anastrophikoi) loci, as a surface of a point (Coll. VII.21). The conception of such a classification is maybe reminiscent of the Eratosthenian definition of a line as a flux of a point (Theon of Smyrna, Expositio, 83.21–4, and Sextus, Adv. math. III.28).

7. g) He systematized also the variegated panorama of neusis-techniques, by reduction of a number of them to plane constructions (Pappus, Coll. VII.27–8). This is the first sustained attempt in Greek mathematics at solving problems by requiring minimality of the mathematical techniques employed.

8. h) Taken as a whole, the first propositions of the Plane loci constitute a meta-theorem (cf. Pappus, Coll. VII.23), where it is shown that particular geometric transformations preserve the character of plane locus (straight line or circumference) of the lines on which they act. The enormous potentialities of such a general approach were deployed only by Newton (Principia I.5).

9. i) Apollonius took up and refined at least two Archimedean achievements: the estimate of the ratio of a circumference to its diameter (Eutocius, In Dim. circ. 3, AOO III:258.16–8) and the system of notation for large numbers developed in the Arenarius (Pappus, Coll. II, passim).

10. j) Apollonius generalized the concept of “solid number,” which is properly speaking (El. VII.def.18) a number obtained by multiplying three numbers only, to a number obtained by simultaneous multiplication of any set of numbers (Pappus, Coll. II, passim).

From all that was said above, the following tentative summary of Apollonius’ foundational endeavor can be put forward:

1. i) A particular concern for deductive economy, by reduction of the number of undefined notions and basic propositions employed. The rationale behind Apollonius’ alternative constructions appears to have been to avoid using El. I.1 or, more generally, constructions of triangles;Footnote ²⁸ the proof of I.5 reduces to zero the number of auxiliary geometrical objects introduced. He also tried to reduce equality to superposition. The latter is simply one of the key notions of the whole geometry,Footnote ²⁹ and has obviously much to do with the peculiar “mode of existence” of geometrical objects that, according to the ancient terminology established in Euclid's Data, could be “given” in magnitude, shape, and position. To be given in shape is, in Data def. 3, made to depend on being given in magnitude. To be given in magnitude, in its turn, is quite an abstract notion and really amounts to reducing the two-term relation of equality to the single-term predicate “being given in magnitude” (Data def. 1). An ontological commitment of sorts, in geometrical matters, is reached only when the notion of position comes into play – and this problem is dealt with in Data def. 4 exactly by reducing the key predicate “given in position” to the notion of “occupying the same place” (verb epechein).

   A fundamental Apollonian move can be read in this double perspective, namely reducing, in his proof of El. I.cn.1, equality to the relation, endowed with a stronger geometrical connotation, of “occupying the same place as.” On the one hand, this move results in an attempt at coordinating the Elements, the Data and the classifications of locus theorems (the loci given in position are called ephektikoi, an adjective deriving from the verb epechein). On the other hand, the move necessarily entails such a wider interpretation of El. I.post.3 as permits to drastically simplify the proof of El. I.2, thereby greatly reducing the interest in a construction like El. I.1. It appears thus that the concept of “occupying the same place as,” a geometrical relation characteristic of the discipline, and the strictly related notion of “superposition,” were pivotal in the foundational efforts of Apollonius, especially if we accept the ascription to him of the tract on the proof of the fifth postulate and recall that the mathematical technique of superposition is the main technical tool at work in Con. VI.1–10, treating the subject of equal conic sections.

2. ii) An attempt to pursue the idea that an already full-fledged mathematical theory gains in being inserted in a larger system. This can be made by introducing classifications of loci (“Elements” and maybe Plane loci) and of lines (About the screw), by discussing and coordinating assumptions, by reforming the terminology of the Data and making it fit the one of the Elements, by rewriting portions of the Elements (General treatise), by reducing certain constructions to others (Neuseis). Also Apollonius’ general characterizations of the sumptōmata of the conic sections in Con. I.11–3 can be read in this perspective. His approach in this case can be defined “foundational” exactly because there already existed similar characterizations that he generalized by showing that they hold with respect to any diameter of a conic section (the so-called “oblique conjugation”), not only with respect to the axis (“orthogonal conjugation”). His generalization allowed him to reshape the whole domain, to “see” new properties that have become “invisible” in the modern Cartesian approach, such as the generalized sumptōmata themselves and the relation between the parameters of a conic section in changing conjugation from a diameter to another (Con. I.49–51; in the case of the parabola this was made more perspicuous by Newton, Principia I.3, lemma 13, by introducing the focal distance), to “discover” objects that were already there, such as the opposite sections (Con. I.14), and to solve long-standing problems, such as the locus on three and four lines (AGE I:4.13–7).

4. Geminus' foundational aims

Geminus, a younger contemporary of Posidonius, wrote an encyclopedic treatise on mathematical sciences, apparently focusing on historico-philosophical issues. The work is lost, and the sources preserve two slightly different titles: “ordering [taxis] of mathematics” in Pappus (Coll. VIII.3),Footnote ³⁰ and “theory [theōria] of mathematics” in Eutocius (AGE II:170.24–26).Footnote ³¹ Proclus is often, and mistakenly, taken to refer to it (iE, 177.24–5) with the title “love of beauty” (philokalia),Footnote ³² and mentions Geminus as his source a number of times.

The picture we get of Geminus from the only extant work of his, the intermediate-level manual of descriptive astronomy Introduction to the Phenomena, is that of an enlightened rationalist, trying on occasion to disentangle science from myth in astronomical matters (for instance, he argues in chap. 17 that weather changes are not caused by but only indicated by the heliacal risings and settings of the stars). Geminus is important as a source: since Tannery (Reference Tannery1887),Footnote ³³ it is commonly (and very likely correctly) held that a good deal of the exegetical material of the historico-foundational sort found in Proclus is in fact drawn from Geminus, even when the former does not mention the latter as his source (one finds a dozen mentions). But, as we shall see, he became the source on foundational themes because he was engaged in an antiquarian, scholarly, technical project aimed at setting firm philosophical foundations to several mathematical sciences, a project whose extent we may only guess but that had no parallels in Greek antiquity. He is interested in classifications and definitions since these matters touch the crucial issue of the ontological status of mathematical objects and may have a cosmological import, as we have seen in the companion paper. He is eager to reform the structure of the founding principles of the Elements by throwing away two postulates. He is the first to have set out the long-lasting epistemological thesis of the heuristic value of the method of analysis and synthesis. He exploited quite technical notions like that of “characteristic property.” In a word, he is doing “philosophy of mathematics” in quite a modern way.

The results about homeomery that Proclus draws from Geminus will be summarized in the next section. The other documents attesting the foundational interests of Geminus are the following.

1) A classification of the mathematical sciences (iE, 38.2–42.18): Proclus sets out first a classification he ascribes to the Pythagoreans (ibid., 35.17–38.2), which he very likely lifts from authors such as Nicomachus or Iamblichus.Footnote ³⁴ It is a standard quadrivium entailing a doubled dichotomic partition into arithmetic, music (both dealing with “quantity” (poson),Footnote ³⁵ either in itself or in relation to some other number), geometry, spherics (both dealing with “qualified quantity,” (pēlikon),Footnote ³⁶ either at rest or in motion). Geminus’ classification is more articulated.Footnote ³⁷ It starts with a basic dichotomic distinction between disciplines dealing with intelligibles and disciplines attending to sensibles. The classification envisages eight primary domains: arithmetic, geometry (namely, those dealing with intelligibles), mechanics, astronomy, optics, geodesy, canonics, logistics (namely, those attending to sensibles). Further divisions are set out for geometry (into plane geometry and stereometry), arithmetic (theory of linear, plane, and solid numbers), optics (optics proper, catoptrics, scene-painting), mechanics (construction of war engines, wonder-working, science of equilibrim and centers of gravity, construction of celestial spheres, and in general everything pertaining to kynetics), astronomy (gnomonics, meteoroscopy, and dioptrics). We have access to Geminus’ classification from other compilations too. Material pertaining to optics is transmitted as Def. 135.(9)-13 ascribed to Hero and as single extracts accompanying the manuscript tradition of redation B of the Euclidean Optics and Damianus’ short tract on Optical Hypotheses.Footnote ³⁸ A definition of logistic can be found at Def. 135.5-(6) and, in a fuller form, as a scholium to Chrm. 165e6 (Scholium 27, 173 Cufalo); a definition of geodesy at Def. 135.7–8. These texts show that Proclus is in fact quite severely abridging Geminus.Footnote ³⁹ Maybe also a definition of geometry transmitted as Def. 135.1 and as a further scholium to Chrm. 165e6 was drawn from Geminus. In this case the text bears no resemblance with Proclus’ description at iE, 39.7–14.

2) A comment on the definition of straight line at El. I.def.4 (iE, 111.1–9, 111.9–113.3), and a comment on the definition of parallel straight lines at El. I.def.23 (iE, 176.18–177.25): The three passages present two classifications of lines.Footnote ⁴⁰ The first and third passages appear to refer to the same classification, whereas the second text seems to work out the Porphyrean tree in a different way.Footnote ⁴¹ Both classifications apply a dichotomic progression. The first classification starts with the dichotomy “incomposite”/“composite.” The second kind includes lines “that are inflected and make an angle.”Footnote ⁴² The first kind is further divided into lines “that are finite and make a figure”/lines “that can be produced indefinitely.” The former class includes the circular line, the contour of the ellipse,Footnote ⁴³ the cissoid.Footnote ⁴⁴ The latter class includes parabola, hyperbola,Footnote ⁴⁵ conchoid, straight line etc. [The first passage ends at this point. At the beginning of the third passage a side branching is set out within the latter category: some of these lines do not enclose any figure, such as the straight line and the said conic sections; others do enclose a figure and then extend indefinitely – one must suppose that a kind of conchoid is referred to. The third passage goes on as follows.] Of the latter, some are asymptotic, some are not.Footnote ⁴⁶ Of the former, some are in one single plane, some are not. Of the former, some are equidistant, as parallel straight lines are, some are progressively diminishing the distance between themselves, as an hyperbola does with respect to a straight line and as the conchoid does with respect to a straight line. The focus on asymptotic lines in the third passage is something we should expect. Proclus is in fact commenting on the definition of parallel lines, and buttresses his explanation by endorsing Posidonius’ definition in terms of equidistance (see below). We may safely assume that this was exactly Geminus’ approach.

The second classification starts with the dichotomy “simple”/“mixed.” Of the former, the circular makes a figure, whereas the straight line is indefinite. Among mixed lines, some are on a plane, some on a solid. Of the former, some intersect themselves as the cissoid, some can be produced indefinitely. Of the latter, some are conceived by means of sectioning solids, as toric and conic sections,Footnote ⁴⁷ some simply lie around solids, as the helix around a sphere or around a cone. Of the latter, some are homeomeric, as the cylindrical helix, some are not, as all the others. (The subsequent discussion is document b of the following section.)

3) A comment on the definition of plane surface at El. I.def.7 (iE, 117.23–120.12): Geminus first remarks that what is “mixed” must be intended to derive from what is “simple” in different ways if one refers to lines or to surfaces. Mixed lines are never obtained by composition or blending, contrary to what a certain Theodorus maintains.Footnote ⁴⁸ Mixed surfaces are instead generated by means of a blending of sorts: for instance, a conic surface reduces by section to a composition of circular, and hence simple, lines.Footnote ⁴⁹ Next (ibid., 118.23–120.12), a classification of surfaces is presented, surely to be ascribed to Geminus. The criterion is again a distinction between simple and mixed.Footnote ⁵⁰ Of the former kind are the plane and the spherical surfaces, of the latter kind are the three species of toroidal surfaces – of which quite a detailed description is givenFootnote ⁵¹ – and the four conoids obtained by rotating the three conic sections.Footnote ⁵²

4) General comments on postulates and axioms (iE, 178.1–182.4 and 183.14–184.10): The first passage is a long discussion of the nature of principles, culminating in a definition of postulate and axiom:

Only at the very end of the discussion are we told that the distinction is Geminus’, to whom very likely the entire discussion should be ascribed. Geminus, then, would definitely range the fourth and the fifth postulate of the Elements among the axioms. In the second passage, Geminus assails two categories of geometers with little philosophical awareness, among which Euclides and Apollonius find a prominent place:

Geminus remarks also that it is ridiculous to range theorems whose converses are demonstrated among unproved statements. As converses of the fifth and fourth postulates Geminus takes the enunciation of El. I.17 (“Two angles, substituted in any manner, of every triangle are less than two rights”) and, apparently, the clearly demonstrable statement “the angle equal to a right angle is not in general a right angle.”Footnote ⁵³

5) A comment on El. I.post.1–3 (iE, 185.6–25): The first three postulates really deserve to be regarded as such, because they are immediately evident and produce something. They derive in fact from the notions of line as the “flowing of a point,” and, accordingly, the notions of straight line as a “uniform and undeviating flowing” of a point, that is undergoing a motion that is “uniform and the shortest.” Recall the Eratosthenian definition of a line as a flux of a point (Theon of Smyrna, Expositio, 83.21–4, and Sextus, Adv. math. III.28).

6) A comment on El. I.post.4, stating that “all right angles are equal to one another” (iE, 188.3–189.10): According to Geminus’ definition, the fourth is not a postulate. Even worse, if it could be proved, “neither then will it be classed among the postulates.” Proclus then offers a proof; it is not clear whether his source for it is Geminus or not.

7) A comment on El. I.post.5 (iE, 192.5–29): Against those who think that the fifth postulate was placed among the postulates because making fewer than two right angles immediately convinces us that the two straight lines involved meet, Geminus warns us that plausible arguments cannot be regarded as compelling in the identification of acceptable assumptions in geometry. Geminus offers the existence of lines that converge indefinitely but still do not meet as a counterexample.

8) Beginning of the comments on El I.1 (iE, 202.9–202.25; take notice of the Aristotelian jargon): Against Amphinomus, Geminus argued that in geometry too one seeks for the cause, the “reason why” (dia ti). What else is, for instance,

In addition, general proofs make manifest the “reason why,” while in reductiones or proofs by division into cases the cause is not evident (a typically Aristotelian doctrine). It is almost surely to be ascribed to Geminus also the preceding passage (ibid., 200.21–202.8).Footnote ⁵⁴ This concerns the double nature of geometry, giving rise to the distinction between problems and theorems. Of such a common matter, the geometer investigates the “what is” (ti esti) in two alternative ways: either by definition or by determination of the extension of the notion at issue. The proposed example is the notion of homeomeric line. Further, the geometer investigates the “if it is per se” (ei estin auto kath’ auto), when he works out the diorisms, and the “which kind of object is” (hopoion ti estin), when he studies the per se attributes of the triangle, the circle, and the parallel lines.

9) A comment on El. I.4 (iE, 243.21–25): At the end of a long excursus on Carpus’ doctrine of the prominence of problems over theorems, Proclus asserts that

10) Beginning of the commentary on El. I.10 (iE, 277.25–279.11): About the indefinite divisibility of lines Geminus asserts that the geometers merely assume that what is continuous is divisible, and not that it is indefinitely so. To secure this, the proof of the existence of incommensurable magnitudes is enough. Accordingly, Euclid bisects a given finite line, but in doing this he only assumes that the line is divisible, and not that it is indefinitely divisible.

11) Beginning of the commentary on El. I.35 (iE, 395.13–18)Footnote ⁵⁵: Chrysippus sets a parallel between locus theorems, that set out the conditions identifying a class of geometrical objects sharing a well-defined property (in Greek mathematical jargon, the sumptōma), and Platonic forms, since

One should then ascribe to Geminus also the definition of locus theorems as

and a subsequent classification of loci as being set up on lines or on surfaces (ibid., 394.19–395.2). The former are subdivided into plane and solid, on the basis of a parallel subdivision of lines into plane (“those which lie in a plane and whose generation (noēsis) is simple”) and solid, that is, “whose generation is made evident by a suitable section of a solid figure,” among which the cylindrical elix is erroneously included. To Geminus must be ascribed also the subsequent reference to the statement “the parallelograms inscribed between the asymptotes and the hyperbola are equal” as an example of a solid locus theorem (ibid., 395.8–12).

12) Definition of mathematical analysis as “discovery of a proof,” as transcribed in a very short allusion of Ammonius (in APr. I, 5.28): It appears, then, that Geminus is the first author believing in the heuristic power of the method of analysis and synthesis. The same ideology of discovery we shall find implicitly in Galen, Pecc. dign. II.5, and explicitly in Proclus (iE, 43.1–2, 242.14–6) and in his pupil Marinus (EOO VI:252.24–254.4), the latter reporting exactly the same clause as Ammonius. Most notably, it is the ideology that informs the celebrated introduction to the analytical corpus by Pappus, Coll. VII.1–3, where he most scholarly offers as an implicit support to his contention a crypto-citation of the Aristotelian statement of the aims of the Topica (compare Coll. VII.1 with Top. 100a18–20 and SE 183a37–38).

Before briefly trying to assess Geminus’ methods and aims, a warning is mandatory. It is by no means said, in fact, that the passages just listed offer a representative sample of the contents of his treatise: as we have seen in Sect. 2, Proclus made a selection of the material, that should be more properly taken as reflecting his own aims. Moreover, if Geminus’ treatise had also antiquarian aims, the compilatory parts appear to be directed by quite an aggressive scholarly attitude. Geminus attacks directly, and by name, scholars or mathematicians who developed their foundational doctrines in a “wrong” perspective. More generally, the prescriptive side appears to greatly overcome the descriptive one. Such an aggressive and prescriptive attitude is in line with Posidonius’ approach to the same subjects. It is enough to recall (a passage we owe to Geminus at an intermediate step of the chain of transmission) the extract from Posidonius’ Meteorologica where he draws a sharp divide between the tasks of the natural philosopher and those of the astronomer (as a mathematician) – making the latter subordinate to the formerFootnote ⁵⁶ – and his writings against Zeno of Sidon, who had pointed out serious flaws in the conception itself of a mathematical system based on unproved statements (iE, 199.3–200.6 and 214.15–218.11 = frs. 46–7 Edelstein-Kidd = fr. 463 Theiler). Moreover, Posidonius offered the same kind of definitory/classificatory contributions whose prominence in (Proclus’ choice of) Geminus’ fragments is obvious (iE, 143.5–144.5, 170.13–171.4 and 176.5–17 = frs. 196, 198 and 197 Edelstein-Kidd = frs. 465–467 Theiler, respectively; the classification of triangles, not included among the fragments, is at 168.3–25):

1. i) An alternative definition of figure as “enclosing limit.”

2. ii) An alternative classification of quadrilaterals, hinging upon 7 species instead of the 5 species listed in El. I.def.22: 4 species of parallelograms, coinciding with those listed in the Elements, and 3 species of non-parallelograms: isosceles and scalene trapezia, and trapezoids. From this passage it can be inferred that also an alternative classification of triangles must have its source in Posidonius. This too lists 7 species: equilateral triangles; 3 species of isosceles triangles (rectangle, acutangle, obtusangle); 3 corresponding species of scalene triangles.Footnote ⁵⁷

3. iii) An alternative definition of parallels, as “those [straight lines] in a single plane that neither converge nor diverge, but keeping them equal all [straight lines] drawn perpendicular from points on either of them as far as the remaining one.”

Two other documents attest Posidonius’ foundational interests. First, a characterization of theorems and problems containing a proposal of reform of their enunciations (notice the Aristotelian jargon):

Second, Posidonius introduced a kind of relational syllogism “concludent by virtue of the power of an axiom,” that was apparently shaped on schemes of relations of proportionality among numbers.Footnote ⁶⁰

Yet, it is doubtful that Geminus could be blindly affiliated with Stoicism, even to the peculiar Stoicism we find in Posidonius. Scholars have repeatedly pointed out that several doctrines endorsed by Geminus, both in physical and astronomical matters, are at variance with basic Stoic tenets.Footnote ⁶¹ The issue is very difficult to settle, as one must bear in mind that particular features a priori peculiar of traditional philosophical schools had become a common background of Greco-Roman scholarly investigations. For instance, in Geminus’ language, if it has not been made-up by Proclus, the presence of Aristotelian terminology is massive, and constitutes both the background against which all of Geminus’ conceptions find their natural formulation and a polemical target. For instance, document 8 looks very much like a sort of glossary of Aristotelian terms, exemplified by selected mathematical notions such as definition and diorism, or by common mathematical practices.

Geminus’ approach to mathematical matters puts emphasis on classificatory issues (documents 1–3), that appear to meet both the scholarly demands of the antiquarian and the prescriptive attitudes of the philosopher. We are offered classifications of the mathematical sciences, and of several kinds of mathematical objects: lines, angles, surfaces, figures, plane figures. In the latter cases a guiding criterion is the dichotomy simple/mixed (doc. 2). Most notably, the criterion of simplicity for lines relies upon their being conceived as generated by one or several motions (docs. 2 and 5). Homeomeric and asymptotic lines seem to have attracted Geminus’ attention more than others. The other pole of Geminus’ foundational endeavors is his concern for mathematical principles. He offers a definition of postulates and axioms and criticisms on postulates IV and V (docs. 4–7), a definition of a peculiar mathematical notion as locus theorems backed up with a striking parallel proposed by Chrysippus (doc. 11, with emphasis on asymptotic lines). Document 10 compares mathematical practices with a notorious conception of the continuum. Historical digressions on the generation (that is, definition) of the conic sections were offered in other books of his treatise. In a sense, Geminus’ general approach would seem to bear some resemblance to Sextus’ Against Geometers: a general, wide-ranging, philosophically-driven, scholarly survey of basic mathematical conceptions. The difference, of course, is that the goal is not a sceptical undermining of the mathematical building, but a renewed foundation on fresh, appropriate, dogmatic grounds.

5. Homeomeric lines as a foundational issue

Now it is clear why the study of homeomeric lines I have reconstructed in the preceding paper can be termed “foundational”:

1. 1) It involves the status of lines, most notably their definition, generation, and classification (cf. El. I.def.3–4). On the classification of lines was grounded one of the problems and of locus theorems, a conceptual reference frame that is pervasive in Greek mathematics.

2. 2) It involves the role and use of superposition in geometry (cf. El. I.cn.7). As we have seen with Apollonius, a critical examination of this notion naturally raises problems of deductive structure and demonstrative economy.

Let us finally come to the problem of assessing what Apollonius and Geminus contributed to the study of homeomeric lines: we can try to complete the picture delineated in the final section of the preceding article. I briefly recall what is ascribed to them on the subject. Geminus is called into play in the following documents, contained in Proclus’ commentary on Elements I:

1. a) The definition of homeomeric lines and the list of such lines (straight line, circumference, and cylindrical helix) set out as an example to explain the difference between an intensional and an extensional definition (iE, 201.18–202.1).

2. b) Geminus’ first and second lemmas, stating a characteristic property (sumptōma) of homeomeric lines. By means of these lemmas, he is said to have proven that there are only three such lines (iE, 112.22–113.6 and 251.2–19).

3. c) The direct reply to Xenarchus’ objection to the Aristotelian claim at Cael. A2 that there are only two simple magnitudes, the straight line and the circumference. Xenarchus contended that the cylindrical helix is also simple; Geminus replied that this line being generated by means of dissimilar motions prevents it from being simple: for only lines generated by similar motions can be simple (iE, 105.18–106.19).

The results about homeomery expressly ascribed to Apollonius are described in two documents:

1. i) Proclus, in his commentary on the definition of straight line in El. I.def.4, reports that someone maintained that there is a further simple line besides the straight line and the circular, namely, “the helix traced about the cylinder, whenever, a straight line moving around the surface of the cylinder, a point moves on it with uniform speed. It comes to be the helix, all parts of which coincide homeomerically with all, as Apollonius shows in the writing About the screw.” Proclus adds that, among the helices, only the cylindrical one enjoys such a property, and that the helix is homeomeric, “as Apollonius too had shown,” but not simple (iE, 104.26–105.18).

2. ii) In Pappus’ Collectio the generation of a cylindrical helix is described, followed by the statement “for this proved Apollonius Pergeus” (Coll. VIII.49). Pappus’ statement is puzzling: Apollonius could not have proved a definition.

We may safely assume that Proclus lifts from Geminus (as his ultimate source) all the material concerning homeomeric lines, often as a by-product of a selection of passages made to different purposes (these are mainly discussions of the status of principles and classification of geometrical objects). There is no reason, then, to suppose that all we read about homeomeric lines is original with Geminus, even when no other author is mentioned in his fragments; the character of his treatise suggests that the contrary was in fact the case. Moreover, Geminus might simply have omitted quoting his sources (as Proclus often does exactly with Geminus’ excerpts), or Proclus might simply have had access to Geminus only via later epitomes. Finally, Geminus might simply have been interested in downplying Apollonius’ contributions, since it is clear from what we have read above that Apollonius’ approach was regarded as utterly ill-directed by Geminus (and by Proclus too). The fact, then, that Apollonius’ name is absent in the most important technical fragments on homeomeric lines cannot be regarded as decisive. From the crucial Proclean passage containing the definition of homeomeric lines the context is clear in which the fragments of information on the homeomeric lines were inserted in Geminus’ exposition; the same must be said of Geminus’ lemmas, that were very likely aimed at giving a final touch to the classification of lines, and not presented as a self-contained technical development.Footnote ⁶² Therefore, it is not by chance that the single notion of homeomeric line exemplifying both intensional and extensional definitions comes from an antiquarian treatise with highly philosophical commitment. Exactly this kind of writing is the ideal source supplying stock examples to be excerpted by late commentators, in primis by Proclus. After all, we should not forget, Proclus almost surely draws from this same work the Apollonian fragments concerning the principles.

On these grounds, a “strong” mathematical reconstruction can be outlined to complete the development of the field of discourse centered on homeomery outlined in the companion article. If the identification of the two plane homeomeric lines was a pre-Euclidean achievement, as the Aristotelian passage at Phys. E4 suggests, Apollonius was the author to whom a rigorous study of plane homeomeric lines should be ascribed, including the elaboration of what I called Geminus’ lemmas. These set out a suitable characteristic property. Establishing characteristic properties of lines was in fact a prominent feature of the Apollonian foundational and systematic aims. The whole structure of the Conica attests to it, as well as such works as, for instance, the Plane loci.Footnote ⁶³ Given the role played by superposition in his reform of some Euclidean notions, it comes as no surprise that Apollonius became strongly interested in homeomeric lines.Footnote ⁶⁴ Viewed from inside the general Apollonian strategy, homeomery was also the last outcome of a whole rethinking of the notion of line, and a unified way to look at the properties of lines, eventually leading to the first non-generative definition of a class of lines. From this perspective, homeomery appears to have served as an effective motive force in shaping and directing foundational research, perhaps contributing to clarifying the very notions of equality and superposition (recall the subtle clause on concavity in the definition of homeomeric lines). Geminus simply compiled Apollonius’ lemmas as material related to the classification(s) of lines he proposes: he achieved nothing mathematically worth mention, and extended Apollonius’ classification to the solid case by a non-rigorous argument, after Apollonius had shown, perhaps with a polemical target as I have suggested in the preceding article, in his About the screw that the cylindrical helix is homeomeric.Footnote ⁶⁵ Proclus misunderstood, or perhaps was not in a position to determine,Footnote ⁶⁶ the authorship of the lemmas in Geminus as well as their real extent, insofar as referring to the general, solid, case and not to the plane case only.

If we deem Geminus as too shady a mathematical character to develop a rigorous systematization of geometrical homeomery and do not regard Apollonius’ candidature as sufficiently supported by the sources, we need only to recall the Proclean reference to “the moderns – among which is also Geminus” as the authors of the proof of Geminus’ second lemma. Now, another source attests that the field of special curves was extensively developed after the canonical authors of the Hellenistic period: these mathematicians might well deserve a collective denomination as “the moderns.” Pappus gives one such list, within a digression on the classification of problems (Coll. IV.58), and names Demetrius of Alexandria and his Linear institutions, Philo of Tyana, who resorted to “intersection of plectoids,” Menelaus and his “wonderful line.” Several kinds of helices, quadratrices, cochloids (or conchoids), and cissoids are mentioned without further specifications. As a matter of fact, we might also take Proclus’ denomination “the moderns” as a reference to their role only in the extension of the classification to the solid case. Other scenarios could be proposed, involving known authorities on special curves such as, for instance, Nicomedes, Dinostratos, Diocles, or Perseus, but it is unsafe to multiply the conjectures.