Abstract
The “common notions” prefacing the Elements of Euclid are a very peculiar set of axioms, and their authenticity, as well as their actual role in the demonstrations, have been object of debate. In the first part of this essay, I offer a survey of the evidence for the authenticity of the common notions, and conclude that only three of them are likely to have been in place at the times of Euclid, whereas others were added in Late Antiquity. In the second part of the essay, I consider the meaning and uses of the common notions in Greek mathematics, and argue that they were originally conceived in order to axiomatize a theory of equivalence in geometry. I also claim that two interpolated common notions responded to different epistemic needs and regulated diagrammatic inferences.
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Notes
For a first appraisal of the number of common notions in various manuscripts and Euclidean editions, see my De Risi (2016).
Elements I, 1, 2 and 13.
They are employed, for instance, in Data 3 and 4 respectively. It may be noted, however, that the spelling of CN3 is clearer than that of CN2 (and recurs again in Data 12): see below note 20.
Plato Theaet. 155a: “And as we consider them, I shall say, I think, first, that nothing can ever become more or less in size or number, so long as it remains equal to itself. Is it not so? … and secondly, that anything to which nothing is added and from which nothing is subtracted, is neither increased nor diminished, but is always equal” (transl. Fowler). The correspondence with Euclid’s common notions is admittedly quite vague, and the Platonic principles have a much more metaphysical aspect. It is remarkable, however, that they appear in a dialogue on scientific knowledge conducted by a young mathematician. If we suppose that Euclid’s common notions were developed in the generation of Theaetetus and Eudoxus, it may perhaps be considered a stroke of Platonic irony to imagine that they were notions advanced by an aged Socrates while a receptive stripling well versed in mathematics nodded in approval. See also Plato Parm. 154b: “For adding equals to unequals, in time or anything else whatever, always makes the difference equal in the amount by which the unequals originally differed” (transl. Allen). We are informed by Proclus (In Euclidis 197) that Pappus had proposed to add a similar common notion to the principles of Euclid, but we have no trace of it in any Greek manuscript of the Elements. It first appears as an interpolation in Gerardo da Cremona’s Latin translation (from the twelfth century) and passed from there over into several early modern editions of the Elements, such as Clavius’ (1574).
The most important passages are in An. Post. Α 2, 72a14–24; An. Post. Α 10, 76a41–42; Metaph. Γ 3, 1005b12–20.
See An. Pr. Α 24, 41b22–23; An. Post. Α 10, 76a41–42; Α 10, 76b21; Α 11, 77a26–31; Metaph. Κ 4, 1061b20. On Aristotle’s theory of axioms see my De Risi (forthcoming [a]).
For a few Aristotelian passages on a universal mathematical science, see An. Post. Α 5, 74a17–25; Α 24, 85a37–85b1; Metaph. Γ 2, 1004a2–9; Ε 1, 1026a25–27; Κ 7, 1064b7–9. For a discussion of the topic, see Rabouin (2009).
Alexander’s reference to CN1 is in In metaph. 265. Proclus’ testimony regarding Apollonius is to be found in In Euclidis 194–195. Galen’s reference to Carneades is to be found in De optima doctrina, §2 (ed. Kuhn, vol. 1, p. 45).
The lack of any reference to CN1 in Aristotle seems to be an especially telling detail, since the latter’s logical works surely offered many opportunities for such a principle to be spelled out, for example in the context of the exposition of syllogistic. A similar case seems to obtain with regard to Elements V, 11, which proves that, if two ratios are equal to a third, they are also equal to one another, and does so without recurring to CN1. It is possible that CN1 had not yet been formulated at the time at which Eudoxus composed the theory of this book; it is also possible, however, that the status of ratios in ancient mathematics did not allow them to be treated as “things” in the sense of other mathematical entities referred to by the Euclidean common notions.
Translation in Heath (1925, vol. 1, pp. 247–248). The Greek reads: τὸ Β σημεῖον ἐπὶ τὸ Ε διὰ τὸ ἴσην εἶναι τὴν ΑΒ τῇ ΔΕ … ἐφαρμόσει καὶ ἡ ΑΓ εὐθεῖα ἐπὶ τὴν ΔΖ διὰ τὸ ἴσην εἶναι τὴν ὑπὸ ΒΑΓ γωνίαν τῇ ὑπὸ ΕΔΖ… ἐφαρμόσει ἄρα ἡ ΒΓ βάσις ἐπὶ τὴν ΕΖ καὶ ἴση αὐτῇ ἔσται.
Mentions of similar sentences are to be found in a great number of propositions, and while Tannery, Heath or Mueller only quote Elements I, 6, they can easily be read also in Elements I, 39, in Elements III, 2, 4, 5, 6, 11, and many other propositions. Cf. also Vitrac (1990–2001, vol. 1, pp. 182–84).
At least starting from Proclus, In Euclidis 264. Modern editions of the Elements generally refer back to CN5 every time Euclid employs the sentence on the greater and the less.
Autolycus, De sphaera, prop. 3 (p. 12 Hultsch). Cf. Vitrac (1990–2001, vol. 1, p. 182 fn. 18).
It happened, sometimes, that explications were interpolated into the text making reference to something like CN5. But not even in these cases does the exact formulation of CN5 ever appear. See, for instance, a textual interpolation into Elements XII, 12 quoted by Heiberg (ad loc.): here Euclid says that a cone is bigger than a pyramid, and an interpolated passage adds “since the former contains the latter”. No mention is made, however, of the notions of “whole” or “part”.
See for instance Tannery (1884), but the claim is endorsed by Heath, von Fritz, and very many other scholars.
In Greek mathematics, numbers are collections of objects and these may possibly “coincide” with one another. Notice, in any case, that the formulation of CN4 is such as to only mention “things” in general (the Greek uses the neuter pronoun), just as occurs with CN1–CN3. If CN4 was to apply only to figures, or geometrical magnitudes, whoever stated it could have just said so. For a similar opinion, see Einarson (1936, p. 43 fn. 52, and p. 47 fn. 67).
I will not belabor the point here, but it is remarkable that the only principle that seems to be spelled out more clearly in the arithmetical books is, precisely, CN3. It is quoted in some recognizable way, even if not literally, in both Elements VII, 7 and 8. It must be borne in mind that this is the only principle quoted by Aristotle as common to arithmetic and geometry. He may have had in mind some specific theorems to this effect and possibly a pre-Euclidean version of one of these two propositions.
van der Waerden (1965) has especially insisted that the content of Books VII–IX must be understood to precede the mathematical work of Theaetetus and we have seen that the latter may be considered a terminus post quem for the formulation of the common notions (see note 7). There is no doubt, in any case, that the material of the arithmetical books was reworked in later times. For a recent appraisal of the matter, see Saito (2019).
Archimedes’ only use of something like CN4 is in Conoids and Spheroids 18, where he shows that the two halves of a spheroid cut by a plane passing through its center are “equal”, by showing that they are congruent. No consideration of measure, however, is involved here, and even though in the last sentence of the demonstration Archimedes does employ the words ἐφαρμόζειν and ἴσος, the equality of the two halves is not even mentioned in the statement of the proposition. Archimedes is rather interested, throughout, only in congruence, and equality of size is never at issue. In Eq. plan. Α 9, on the other hand, Archimedes says that two figures coincide (ἐφαρμόζειν) when they are “equal and similar” (ἴσα καὶ ὁμοῖα). Again, however, this statement does not suggest the reference to any principle entailing equality from congruence. Neither CN4 and CN5 are mentioned by Proclus in relation to Geminus’ many considerations on axiomatics. Zenodorus’ theorems on isoperimetric figures (second century BCE) would have easily allowed inferences through CN4 and CN5, but they are nowhere mentioned. These theorems are to be found in Theon’s commentary on Ptolemy’s Almagest and again in Pappus’ Collectiones Ε 1–19 (see also Hultsch 1875–1878, vol. 3, pp. 1190–1211). Pappus himself makes an inference from congruence to equality in Collectiones Δ 39 but I am not aware of any further passage in the corpus and the latter seems to be too isolated to make a case for the existence of an explicit axiom to this effect. The passage reports a demonstration by Nicomedes, and we cannot say whether the reference to superposition was already in Pappus’ source.
I may note that Apollonius’ proof of CN1, based on sameness of place, seems to take this common notion as a geometrical axiom. This may fit with a general interpretation of them as principles for a theory of equivalence (see below, Sect. 3).
For Apollonius’ proof, see above, note 11. Ibn al-Haytham offered a proof of CN1 grounded on superposition and CN4: see Ighbariah and Wagner (2018) and Rashed (2019). It is not clear whether Al-Haytham may have been aware of Apollonius’ attempt (he seems not to have known Proclus’ commentary on Euclid), but his proof of CN1, so similar to Apollonius’, clearly points to the possibility of envisaging a foundation of CN1 by CN4.
Apollonius made use of superposition only in Book VI of the Conics. He sometimes employed the converse of CN4 by assuming that equal segments may be superposed (e.g. in Conica VI, 4); and in Conica VI, 5 (only) he may have used an implicit inference of the form of CN4 but in such vague terms that he did not seem to have in mind any axiom to this effect. On homeomeric lines in Apollonius, see Acerbi (2010).
Censorinus, De die natali liber, 62–63 Hultsch (it is a spurious fragment accompanying Censorinus’ main work). Martianus Capella, De nuptiis Philologiae et Mercurii, Book VI, §§ 722–723. For Boethius, see Folkerts (1970, pp. 117–118 and 184–185).
Philop. In an. post. 10–11, and 123 (Wallies). See below for Simplicius.
Proclus, In Euclidis 196: Καὶ μὴν καὶ τὸν ἀριθμὸν αὐτῶν οὔτε εἰς ἐλάχιστον δεῖ συναιρεῖν, ὡς Ἥρων ποιεῖ τρία μόνον ἐκθέμενος.
The most outstanding case for CN4 is probably Elements I, 4, in which Euclid’s alleged reference to this common notion is inferred by few words, whereas Proclus articulates at length its role in the demonstration (In Euclidis 240–241). As for CN5, Proclus offers an alternative demonstration of Elements I, 6 mentioning explicitly this principle (In Euclidis 257), adding that Euclid uses it in Elements I, 39 (In Euclidis 407), in Elements I, 40 (In Euclidis 411), and that it is required to prove the converse of Elements I, 41 (In Euclidis 414).
In particular, Heron’s Metrica would have been a perfect place to employ both CN4 and CN5 for a theory of measure. The only occurrence of coincidence (i.e. the verb ἐφαρμόζειν) in this work is in the Preface to the First Book, in which Heron tersely says that right angles and segments are congruent to one another. In the case of segments, Heron does not even mention that they have to be equal and may have simply had in mind that they are homeomeric (cf. Proclus, In Euclidis 237–238, for a similar reference). Nowhere in the work is mention made of the principle stating that the whole is greater than the part (or even of the Euclidean expression regarding the lesser and the greater). For a recent edition of this work, see Acerbi and Vitrac (2014). The same happens in Heron’s Definitiones, where the notions of whole and parts are dealt with at length (Def. 120), without mentioning CN5. In the same passages (Def. 117), equality among figures is defined as two figures being in conformity with one another (Heron uses the word ἁρμόττειν rather than Euclid’s ἐφαρμόζειν) either “according to the part” (i.e. if the parts of the figures are congruent one by one) or “according to the configuration” (i.e. if the figures are themselves congruent). I would guess that a clearer reference to CN4 would have been appropriate here, if Heron had known of it.
Al-Nayrīzī’s commentary reports several of Heron’s proofs. Nonetheless, none of the additional demonstrations mentioned in this commentary make use of CN5, and CN4 is only mentioned twice by Al-Nayrīzī (Besthorn and Heiberg 1893–1932, vol. 1, pp. 80–81 and 112–113) in proofs that are explicitly not ascribed to Heron (and that Heiberg took for Arabic).
Besthorn and Heiberg (1893–1932, vol. 1, pp. 38–39).
Gregg De Young has informed me that the hand in which this marginal scholium is written is the same hand as copied the whole text. Of course, it is possible that the scholium was already added in the source manuscript.
Capella was actually a contemporary of Theon. Clearly, however, he did not rely on recent or updated bibliography in mathematics but rather on derivative texts of (more ancient) editions of the Elements. Given the fragmentary character of Boethius’ text, I do not think that anything can be inferred from the fact that it does not refer to CN5.
It is possible, for instance, that the transition from papyrus to parchment (which may have occurred in the third-fourth century) could have given rise to new editions of the text of the Elements and modifications thereof.
Hilbert (1968). See also Amaldi (1912), which gives a modern reading of the role of Euclid’s five common notions in the theory of equivalence. A more recent formalization of the same theory, which is more faithful to Euclid’s Elements, is to be found in Robering (2016). It should be noticed that Greek mathematicians freely employed the word μετρεῖν, to measure, to designate what we call today a theory of content. According to this usage, a figure measures (μετρεῖν) another figure by being taken as a unit of measure of the latter. I have stuck to the modern terminology to avoid possible misunderstandings.
A convincing argument for taking Elements I, 45 as the culmination of Book I of the Elements is given by Mueller (1981, pp. 16–27). The deductive structure of Book I, in fact, seems to point to the proof of Elements I, 45, and Euclid may have arranged the previous propositions in order to prove this momentous result. In this respect, the following propositions of Elements I, 46–48, which together form Pythagoras’ Theorem, should not be regarded as the final aim of Book I (as it was commonly stated), but rather as an appendix of sorts. The Eudoxian paternity of the theory expounded in Elements I, 35–45 has been advanced by Neuenschwander (1973 [b]).
In the Timaeus, 53c, Plato exploits the fact that any surface may be decomposed into triangles in order to suggest that composition and decomposition of figures into triangles may explain the elemental changes. Aristotle further insisted that every figure may be decomposed into triangles in De an. Β 3, 414b20–32. The above mentioned Neuenschwander (1973 [b]) provides some arguments claiming that the decomposition of a figure into triangles is of Pythagorean origin. Book II of the Elements was possibly elaborated by Theodorus, from earlier Pythagorean sources, in the generation before Eudoxus: see Knorr (1975, pp. 193–203). Decomposing and recomposing equivalent figures in order to compare them is a widely-used practice with immediate practical uses and even recreational ones—the most outstanding example in Greek antiquity possibly being the game of Stomachion, discussed by Archimedes.
See Hilbert (1968, chap. 4, §19, pp. 72–73). Books V and XII of the Elements, heavily relying on the Axiom of Archimedes (and explicitly so), are also attributed to Eudoxus. Cf. also a reference to such a principle in Aristotle, Phys. Θ 10, 266b2–4.
The word παραλληλόγραμμος is employed as an adjective starting from Elements I, 34, and it is not defined by Euclid, who rather gives definitions of the actual figures (rectangle, square, and others) that are “parallelogrammic”. In the Data, Euclid applies the expression “given in magnitude” to plane areas only in the case of circles and “parallelogrammic regions”, thus showing that every polygonal figure, thanks to Elements I, 35–45, is subsumed under the latter label. In this respect, a parallelogrammic region is not much a figure, but rather a more abstract representation of content. The older Book II dealt only with shaped rectangles and squares, and this practice seems to be reflected in a passage in Aristotle, De an. Β 2, 413a16–17 (also only mentioning rectangles and squares). We witness a further development in Proclus, who defines the “area” (ἐμβαδόν) of a triangle as the region (χώριον) comprehended by the sides of the triangle, and “equal” triangles having equal ἐμβαδά (In Euclidis 236). The term ἐμβαδόν appears in Heron’s Metrica, but is never to be found in Euclid or Hellenistic mathematics.
Cf. the tables in Vitrac (1990–2001, vol. 1, pp. 514–515, and vol. 2, p. 557). A close number is to be found also in the reconstruction by Neuenschwander (1973 [a]). I may remark that the older theory of the application of areas expounded in Book II makes use almost exclusively of CN2, as do the theorems on equidecomposability of Book I, that may have predated those on equiampliability requiring CN3. It may be conjectured, then, that CN2 had been devised earlier than CN3, and that the latter was first conceived with the introduction of equiampliability during the era of Aristotle (who explicitly mentions CN3 as a newly-discovered principle). Should this be the case, and referring further to note 12, one may suppose that the historical order of introduction of the common notions was CN2, then CN3, then CN1.
Elements I, 4 is absolutely necessary for establishing a theory of content, and Hilbert proved in fact that a “non-Pythagorean” geometry may be built by assuming a weaker form of this theorem (i.e. Hilbert’s Axiom III 5). In such a geometry, CN4, CN5 and Elements I, 39 fail, and there are figures in part-whole relation having equal content. Cf. Hilbert (1968, appendix 2, pp. 152–53).
For CN5, see Hilbert (1968, chap. 4, §19, p. 74). De Zolt’s Postulate was first introduced in De Zolt (1881). Later on, the postulate became a theorem, insofar as it was proven by Schur from other assumptions, including the Axiom of Archimedes. Hilbert gave a complete proof of it independent of this latter axiom. See Hartshorne (2000, p. 210), stating that a proof of it without passing through a notion of measure has not yet been found. For modern treatments of De Zolt’s Postulate in relation to CN5, see the more historical Volkert (2010) and the more mathematical Giovannini et al. (2020). I may remark that from a modern, set-theoretical point of view, there are useful reformulations of the part-whole principle (CN5), as well as of the two “Aristotelian Principles” (CN2 and CN3). Their deductive relationships are quite complicated, and on some hypotheses it is possible to prove (modern versions of) CN2 and CN3 from (a modern version of) CN5. For some investigations in this direction, see Mancosu and Siskind (2019).
Elements I, 39 is never used again in the theory of equivalence, nor anywhere else in the Elements, and no special foundational value seems to be attached to it; for the role of Elements I, 39 in the demonstration of the other theorems of the Elements, see Hartshorne (2000, p. 203). I have mentioned above (see note 29) that Proclus recognized the foundational role of CN5 in Elements I, 39. He surely did not interpret this proposition, however, as one stating the impossibility that all figures are equivalent to one another. In the Eudoxian essay on equiampliability, the formula “the less would be equal to the greater”, which is taken by some interpreters to signal an application of CN5, is also used in Elements I, 40. The latter proposition is, however, surely spurious: it was added for symmetry with Elements I, 39, and it is not contained in an old papyrus fragment. See Heiberg (1903, p. 50). Note that also in this Fayyūm papyrus (probably second-third century CE) the demonstration of Elements I, 39 makes use of the usual expression “the greater to the less” (as do all the Greek manuscripts that we possess), and that there is no mention of the whole and the part.
For example, that “the doubles of the same thing are equal” is an assumption used in the proof of Elements I, 47; that “the halves of the same thing are equal” is used in Elements I, 37 and 38; that “if equals are added to unequal things, the wholes are unequal” is used in Elements I, 19 and 21; that “two straight lines do not encompass a space” is used in Elements I, 4 and then in Elements XI, 3 and 7. Some of these principles may have been added to the main text of the Elements and to the list of common notions at the same time. If Euclid had already included the statement about “halves of the same thing” in the proofs of Elements I, 37 and 38, expressed twice with the same words, it seems that he would also have included it as a common notion from the very beginning. It is more likely that Euclid simply made an inference without mentioning the halves and a later editor (or two different editors) added a piece of text pointing to the need for assuming the principle on halves in these propositions and a corresponding common notion at the beginning of the treatise. This, at least, was the opinion of Heiberg; for a contrary opinion, see Mueller (1981, pp. 34–35). Should this be true in general, it is possible that no common notions were added to the Elements in antiquity by extracting them from Euclid’s text.
We find already in Proclus’ commentary (In Euclidis 184 and 240–241) the observation that in Elements I, 4 Euclid is employing the converse of CN4, an axiom which is false in general but true if restricted just to straight segments and rectilinear angles. The same is repeated in Campanus’ edition (mid-thirteenth century), and the principle was later to be stated as an axiom in the Euclidean editions by Claude Richard (1645), Andreas Tacquet (1654), Isaac Barrow (1655), Gilles-François de Gottignies (1669), Milliet Dechales (1672), Mercator (1678), Abraham Kästner (1758), and others. Clavius (1574) prefers to conclude the proof of Elements I, 4 through the axiom that two straight lines cannot have a common segment, also derived from Proclus (In Euclidis 214).
It is immaterial, for my interpretation, whether Euclidean superposition is interpreted as an actual displacement of figures or rather (as suggested by Zeuthen and others) as a re-construction of a given figure in another position. Note that thanks to the results of Elements II we may also just compare the sides of the squares (i.e. segments), again by means of CN4 and CN5.
This notwithstanding, I do not think we should accept the highly anachronistic interpretation of CN1–CN5 as giving an implicit definition of the relation of “being equal/greater in content”. In fact, we have no hint of any ancient discussion in epistemology (such as Aristotle’s) suggesting that a system of axioms could ever work as a definition. For this anachronistic interpretation, see Heath (1925, vol. 1, p. 325); and von Fritz (1955).
This first happened in modern-age editions of Euclid; Claude Richard’s Elements (1645), for example, have a postulate stating that any two figures may be displaced and superposed at will.
The distinction between diagrammatic and “exact” properties was introduced into scholarly debate by a paper by Kenneth Manders. Manders did not deal with the ancient theory of measure, nor did he mention CN5, but he already recognized parthood as a relation individuated by the diagram. See Manders (2008, pp. 91 and 112). Several formal treatments of these ideas has been offered in the literature. See for instance the important Avigad et al. (2009), which has a discussion on diagrams and common notions in p. 741.
See the occurrences of CN5 in Proclus’ commentary, always referring to part-whole relations read off from the diagrams: In Euclidis 257, 264, 296, 407, 408, 414.
Ibn al-Haytham’s interpretation and syllogistic proof of CN5 can be found in his famous Optics (II, iii, §32) and in his second commentary on Euclid (On the Resolution of Doubts in Euclid’s Elements). For an English translation of the Optics, see Sabra (1989; cf. vol. 1, 133). The commentary on Euclid has not been translated into English but the relevant discussion may be found in Ighbariah and Wagner (2018). Leibniz proved CN5 several times in his writings. The first instance may have been in the Demonstratio propositionum primarum from 1671 or 1672 (A vi, 2, n. 57, pp. 482–483; cf. also A ii, 1, n. 109, p. 355). On Leibniz’s proof see Fichant (1998). A similar stance on the analyticity of CN5 was taken in 1866 by Jean-Marie-Constant Duhamel, who is regarded as one of the first mathematicians interested in grounding an axiomatic theory of equivalence in elementary geometry. See Duhamel (1865-1873, vol. 2, p. 7); and the historical remarks in Volkert (2010).
I would formally express the meaning of CN5 as an axiom on the monotonicity of content: A⊏B → A≼B. It may be compared to the modern axiom of the monotonicity of measure, which is generally stated in the form A ⊂ B → m(A) ≤ m(B), where the order-relation of set inclusion entails a similar order of the measure of sets. The latter formula should be adjusted in several respects to fit with the Greek theory of content. First of all, the set-theoretical inclusion (⊂) should be transformed into the relation of parthood in a mereological framework (⊏), since CN5 is formulated in mereological terms. Second, the ordering between real numbers as outputs of the measure function (<) must be transformed into an ordering relation (≺) between geometrical figures. The latter ordering relates to the content of a figure, and may be applied, of course, also to figures that are not in a part-whole relation but are equiampliable to such figures. The formula is, however, not even restricted to polygonal figures (as, for instance, De Zolt’s Postulate), insofar as the Greek theory of equivalence had a broader scope than an elementary theory of equiampliability and admitted advanced techniques such as exhaustion. In Euclid’s proofs, CN4 would apply, for instance, to circular arcs in Elements III, 24, and CN5 would apply to circles and squares in Elements XII, 2.
The expression appears in Elements XI, def. 10 to define what we would call congruent polyhedra. Already in Book VI, however, Euclid had inferred that two polygons having equal area and being similar to one another must have equal sides (cf. Elements VI, 22).
It is well known that Euclid employs the expression ἴσος in the first thirty-four propositions of Book I (i.e. before the essay of Elements I, 35–45) to refer to congruent figures. I do not think that this expression should be read as enthymematic for “congruent and therefore equal in content”. On the contrary, equal triangles are here (e.g. Elements, I, 16, 33, 34) simply congruent triangles, and Euclid’s proofs would generally not work if these triangles were equal in measure but not also equal in shape. This is a general phenomenon. Equal angles are in fact congruent angles (as it was noted already in antiquity: cf. Proclus, In Euclidis 189–190). In Book III, equal circles are obviously congruent circles rather than circles equal in measure (even if the two classes have the same extension), and in Elements III, 23–24 Euclid calls “equal circular arcs” figures that must be congruent in order for the demonstration to succeed (cf. Heath 1925, vol. 2, p. 53). The examples might be further multiplied. On Euclid’s “flou” terminology of equality and congruence, see also Vitrac (1990–2001, vol. 1, pp. 502–512).
A general discussion of Euclid’s notions of equality and “congruence” may be found in von Fritz (1959).
For a modern axiomatic treatment of the subject, complementing Euclid’s common notions with further principles, see Mueller (1981, pp. 36–37).
I have claimed that Euclid’s definition of a point as the boundary of a line was interpreted (possibly already by Euclid, but possibly by Pappus) as a principle complementing a diagrammatic inference with a propositional statement, in order to ground a theory of intersections. I have also advanced the conjecture that the fourth postulate of the Elements, stating that all right angles are equal, was interpolated in the age of Apollonius in order to provide another similar principle bridging the gap between diagrams and propositions. See De Risi (2019).
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Acknowledgements
I thank Gregg de Young, Eduardo Giovannini, Marco Panza, and an anonymous referee, for their insightful remarks. I am especially grateful to Mattia Mantovani, whose many suggestions and relentless requests of better evidence substantially improved the final version of this essay.
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De Risi, V. Euclid’s Common Notions and the Theory of Equivalence. Found Sci 26, 301–324 (2021). https://doi.org/10.1007/s10699-020-09694-w
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DOI: https://doi.org/10.1007/s10699-020-09694-w