Euclid’s Fourth Postulate: Its authenticity and significance for the foundations of Greek mathematics

Argument

The Fourth Postulate of Euclid’s Elements states that all right angles are equal. This principle has always been considered problematic in the deductive economy of the treatise, and even the ancient interpreters were confused about its mathematical role and its epistemological status. The present essay reconsiders the ancient testimonies on the Fourth Postulate, showing that there is no certain evidence for its authenticity, nor for its spuriousness. The paper also considers modern mathematical interpretations of this postulate, pointing out various anachronisms. It further discusses the validity of the ancient proof by superposition of the Fourth Postulate. Finally, the article proposes an interpretation of the history of the concept of angle in Greek geometry between Euclid and Apollonius, and puts forward a conjecture on the interpolation of the Fourth Postulate in the Hellenistic age. The essay contributes to a general reassessment of the axiomatic foundations of ancient mathematics.

The theory of angles in Euclid’s Elements (third century BCE) is grounded on three definitions and a postulate. The definitions state that an angle (γωνία) is an inclination between lines, a rectilinear angle (ϵὐθύγραμμος γωνία) is an angle formed by straight lines, and a right angle (ὀρθὴ γωνία) is a rectilinear angle formed by two straight lines making equal adjacent angles. ¹

The definitions are complemented by a postulate stating that:

All right angles are equal to one another.

ᾑτήσθω … πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις ϵἶναι.

This Fourth Postulate (henceforth P4) has elicited mixed reactions over the centuries. On the one hand, modern mathematicians recognize in it an important principle of geometry, and praise Euclid highly for having correctly pinpointed a fundamental assumption underlying elementary geometry. On the other hand, philosophers are confused by its ascription to the list of the five postulates: the others seem to be constructive principles (e.g. “to draw a straight line from any point to any point”), whereas P4 spells out a state of affairs.

This philosophical perplexity is not new, and as early as the first century BCE Geminus suggested that this principle was ill-placed among the postulates and that it should have been repositioned among Euclid’s common notions (κοιναὶ ἔννοιαι): a different set of principles that, according to Geminus, included assertions rather than constructive principles (e.g. “if equals be added to equals, the wholes are equal”). A further reason for perplexity was occasioned by the fact that Geminus also reported a proof of the Fourth Postulate, as if this alleged fundamental statement had no real foundational import and could easily be taken as a theorem in the Elements.

Geminus’ opinions were often debated in the following Euclidean tradition. The early modern editions of the Elements, for instance, unconcerned with philological accurateness, listed P4 among the common notions, or proved it among the theorems, or even accepted it as a common notion equipped with a demonstration. Present-day historians struggle with it, making attempts to show that the postulate is constructive after all, or arguing why Euclid could not accept its proof. The mathematical meaning of the principle is also debated, as is its foundational role in the deductive structure of the Elements. The outcome of this research is a general puzzlement on P4, and most of today’s interpreters simply settle on a non liquet.

In the present paper, I attempt a general reassessment of the significance of the Fourth Postulate in Greek mathematics and address the question of its authenticity. I begin with a textual analysis in section 1, where I offer a discussion on the embedding of P4 in the text of the Elements and present ancient testimonies on it. I show that, contrary to what happens with the other principles in the Elements, we have no positive evidence for either the authenticity or the spuriousness of P4. The two following sections are devoted to mathematical analyses: in section 2, I challenge various modern interpretations of the meaning of P4, and in section 3 I further examine the ancient proof of P4 and its alleged shortcomings. The conclusions of these three sections are largely aporetic, but nevertheless establish a few important negative results regarding what the Fourth Postulate cannot be, and what we do not know about it. I consider these negative results as the main outcome of the present essay, but I also complement them with a positive conjecture in section 4, according to which P4 was not spelled out by Euclid himself and was rather added in a further Hellenistic stratification of the text of the Elements. I argue that this may have happened in the generations immediately after Euclid, taking a cue from Apollonius’ new definition of angles. While this conjecture is destined to remain unproven, it may solve several historical puzzles surrounding the Fourth Postulate, including the vexata quaestio of its incongruous ascription to the list of postulates.

The present essay may be seen as a particular contribution to a broader study of the historical foundations of Euclidean geometry. The conjecture advanced in the last section, indeed, would allow for a unitary reading of what I argue are Euclid’s four original postulates (the first three, and the fifth). It is not my intention to advance here a general interpretation of the epistemological meaning of Euclid’s postulates. I would like to stress, nevertheless, that by conjecturing that Euclid himself did not state P4, I do not intend to suggest that he had overlooked an important principle that was later recognized as essential for the foundations of geometry. Rather, I would argue that Euclid’s “ideal of proof” was not a gapless sequence of propositional inferences licensed by axioms and previous theorems. In a context that allowed for diagrammatic reasoning and other unaxiomatized demonstrative techniques (such as superposition), postulates did not have the modern function of licensing inferences. As a consequence, Euclid had no need to spell out the Fourth Postulate. He did acknowledge that all right angles are equal, and he did prove theorems requiring this assumption, but he did not believe that it had to be made explicit as a postulate or a common notion—these principles simply played a different role in his geometry.

1. Textual evidence for the Fourth Postulate

The authenticity of the Fourth Postulate may seem, at first, to be beyond question. No ancient source ever doubted its authenticity, and all the manuscripts of the Elements that we have—be they in Greek, Arabic or Latin—as well as all the indirect sources mentioning Euclid’s principles, include P4 in the list of postulates. The reconstruction of the textual traditions of the Elements, therefore, has to include P4 in all the lines of transmission of the ancient text, and there is no question of there being an archetype of the Elements which lacks this principle. The philological question is easily settled.

Yet, it must also be acknowledged that the tools of philology do not allow us to go back to the early stratifications of the text: the earliest extant manuscripts containing the Elements date to the ninth century CE, some twelve centuries after Euclid. Assuming that Proclus’ reference to Geminus is correct, we can state that P4 was listed among the postulates in the first century BCE, and it may have been added to the Elements in the two centuries intervening between Euclid and Geminus—during the flourishing of Hellenistic mathematics. Such an early interpolation is very difficult to ascertain, since the first centuries of the textual tradition of the Elements are shrouded in darkness and beyond the possibility of philological restoration.

A few interpreters in the past have indeed suggested that P4 was interpolated during this period. They did not advance any textual argument to this effect, though, merely using this possibility as a desperate escape route in order to account either for the odd presence of a non-constructive principle among the postulates or for the fact that P4 had an ancient proof. ² I do not think that such preconceived views about the nature of postulates and proofs should guide historical research and count as real arguments for the excision of a principle from a text that otherwise offers no further reason for doubt.

At the same time, the textual situation is not as clear as one would like. While it is true that we have no textual reasons to put the authenticity of P4 in question, it is also true that we have no positive evidence whatsoever that it was counted among the principles in the Elements before the times of Geminus. This marks a sharp difference with the other principles in the Elements.

Johan Ludvig Heiberg’s critical edition of the Elements, based on Byzantine manuscripts, lists some nine common notions and six postulates among Euclid’s principles (their exact number changes from manuscript to manuscript). Among these, Heiberg himself identified four common notions and one postulate as spurious. Heiberg offered convincing, positive evidence for his claim: these principles were either explicitly reported as interpolations by ancient sources such as Proclus and Simplicius, or were missing from several other lines of transmission of the text of the Elements (such as Boethius, Censorinus, or the Aristotelian commentators). I myself have argued that two more common notions (on superposition and on the whole and the part), which had already been regarded as spurious by some other commentators, are indeed so, and I have provided positive evidence for their late interpolation in the text. ³

The remaining three common notions are, by contrast, not just “possibly non-spurious” but positively authentic. The first common notion (“things which are equal to the same thing are also equal to one another”) is quoted three times in the course of the demonstrations in the Elements, and is referred to by Hellenistic authors. The second and the third (on the addition and subtraction of equals) are quoted explicitly in Euclid’s Data, and the third is mentioned several times by Aristotle as the main example of a mathematical axiom. ⁴

Similar supporting evidence exists for four postulates out of five. The Parallel Postulate is quoted by Euclid four times in the course of the demonstrations, and discussions on the theory of parallels were referred to by Aristotle and other ancient scholars. ⁵ The first three constructive postulates, licensing ruler-and-compass constructions, are seldom explicitly mentioned by Euclid (they are so pervasive in the Elements that it would be highly impractical to refer to them every time a construction is performed), but they are nonetheless mentioned time and again. ⁶ Note that we have no reason to doubt these internal quotations: they are present in all the lines of transmission of the text of the Elements and we have no clues that they may have been interpolated. In sum, we do have positive evidence for the spuriousness of a certain number of principles, and we do have positive evidence for the authenticity of the remaining principles.

The only exception is the Fourth Postulate. In all the demonstrations in the thirteen books of the Elements, P4 is never quoted as such, and Euclid never states that all right angles are equal—except in the list of principles.

The equality of all right angles is implicitly assumed innumerable times in the course of the Elements. It is very common, indeed, that in a demonstration Euclid assumes the equality of right angles in proving the congruence of two triangles having further sides or angles that are equal, or that he adds equal angles to right angles to obtain equal angles or, again, that he states that an angle sum of two right angles (e.g. the interior angle sum of a triangle) is indeed equal to the sum of another pair of right angles. Sometimes, these inferences do not require P4. There are occasions where, for instance, the compared right angles are adjacent to one another: in these cases, a simple appeal to the definition of a right angle (i.e. an angle which is equal to its adjacent angle) would establish their equality (a similar case is in the proof of Elements I, 13). But in many other cases the definition would not suffice, and Euclid would really need a stronger principle, such as P4, in order to justify the inference he makes.

In most cases in which a proof needs the equality of certain right angles, Euclid tacitly assumes their equality, making no explicit reference to the fact that the said angles are equal insofar as they are right. Euclid simply writes that “since angle ABC is equal to angle DEF”—both being right angles—then some conclusion follows. The definition of a right angle is also never mentioned in the cases in which it would suffice, and this fact may show that Euclid did not attach any special importance to the assumptions licensing the inference.

There are only a handful of instances (e.g. Elements III, 1, 3 and 4; Elements IV, 4 and 13), in which Euclid says something to the effect that a certain angle, which is explicitly referred to as a right angle, is equal to another right angle. Even these formulations do not seem to be allusions to P4: in these cases, too, the equality of the angles is not explicitly grounded on their being right angles, and the reference to the fact that these angles are right seems to be an addition and an explanation of something otherwise obvious. A slightly more explicit (but not decisive) stance is in Elements III, 18, in which Euclid proves, as a reductio argument, that both an angle and a proper part of it are right, concluding that this is impossible since two right angles are equal to one another.

There are, however, five instances in the text of the Elements, in which the equality of two right angles is inferred by reason of their being right angles. These are Elements I, 47 (Pythagoras’ Theorem); Elements III, 25; Elements III, 30; Elements VI, 8; and Elements XI, 8. ⁷ In all these demonstrations, the received text states that two angles are equal and adds “indeed, each of them is right” (ὀρθὴ γὰρ ἑκατέρα). The Greek formula recurs identically in the five passages. Such a statement is not a straightforward reference to P4 (it does not say that all right angles are equal), but it may represent a textual marker pointing back to an explicit postulate licensing the inference. The ὀρθὴ γὰρ ἑκατέρα clause may be the only trace of P4 in the text of the Elements. ⁸

Even this textual clue, however, can be seriously questioned. An inference followed by a short, independent clause explaining the grounds for it may easily be suspected of interpolation. That this is indeed the case seems to be confirmed by the Latin-Arabic tradition of the Elements, which is partially based on sources different from the extant Greek manuscripts. An important part of this tradition was clearly a translation of a Greek text that did not have the ὀρθὴ γὰρ ἑκατέρα clause. In particular, the Euclidean commentary by Al-Nayrīzī (tenth century) shows a translation of the Elements that does not include the clause in these five demonstrations. Similarly, the Latin translations (made from different Arabic sources) by Adelard of Bath (twelfth century) and Campano da Novara (thirteenth century) also lack the clause. Other translations made from the Arabic, such as those by Gerardo da Cremona and John of Tynemouth (twelfth century) have some scanty references to the fact that the angles are right in some (but not all) of these five propositions. In any case, these Latin texts do not translate an Arabic version of the ὀρθὴ γὰρ ἑκατέρα clause. ⁹

While the details of the Arabic-Latin textual tradition of the Elements are still very obscure, the fact that some of these manuscripts report a translation of the five demonstrations that skip the clause on right angles seems to be sufficient evidence that a Greek text without such a clause was circulating in Late Antiquity and grounded a branch of the transmission of the Elements. It is very unlikely that the clause was erased from the text by a scribe, and we may conclude that the ὀρθὴ γὰρ ἑκατέρα formula was probably added as an explanatory scholium and later interpolated.

This conclusion finds further confirmation in the tradition of Book XIV of the Elements. This was a treatise on regular solids written by Hypsicles in the second century BCE, and later (possibly in the sixth century CE) appended to Euclid’s Elements. The textual history of Book XIV is partially different from the rest of the Elements, and in the extant Greek manuscripts of it we find two examples of a similar clause on the equality of right angles (in the form “indeed, they are right”, ὀρθαὶ γάρ) as a marginal scholium rather than incorporated into the main text. I would suggest that the manuscripts of Hypsicles’ book witness an early stage of the process of interpolation of the clause, and shed some light on the making of the text of the above-mentioned five theorems of the Elements. ¹⁰

The situation does not change if we look at other works by either Euclid or other ancient authors.

A survey of Euclid’s Data and other minor works shows that in none of their demonstrations is any reference made to P4, or even to the fact that two angles are equal insofar as they are both right. The only exceptions come from Euclid’s Optics. We have two rather different Greek versions of this work, and also different Arabic translations of it. According to all modern interpreters, these texts are quite composite and full of modifications coming from the subsequent tradition. In some passages (Proposition 47 of the so-called Α version of the Greek text, and Propositions 19 and 47 of the so-called Β version), we find the standard clause on the equality of right angles, ὀρθὴ γὰρ ἐστιν ἑκατέρα. The main Arabic text of the Optics offers, in the proof of Proposition 47 (here numbered as 55), a reference to right angles in order to explain their equality, but does not translate the Greek clause as such. It omits any reference to right angles in Proposition 19 (here Proposition 20). Moreover, the Greek text has a very explicit reference to the equality of right angles (αἱ δὲ ὀρθαὶ ἴσαι) in Proposition 34 of the Α version only, which is echoed in a scanty reference to right angles in the proofs of the Greek Β version (Proposition 35) and the Arabic version (Proposition 37). Since we do not know the exact relationships between these versions, it is difficult even to fathom a story about these variants. But the external interventions on these texts are so numerous that nothing can be drawn from the reference to right angles in the proofs, which are very likely post-Euclidean revisions made by someone well aware of P4. ¹¹ I think that, given the situation of the texts, and even more so given that P4 was not quoted in the proofs in the Elements, we may safely conclude that these other references were interpolated as well.

Other Greek mathematicians predating Geminus do not seem to be aware of the Fourth Postulate. The extant works by Autolycus, Aristarchus and Theodosius do not refer to P4 or the equality of right angles. Archimedes did employ, in the course of his demonstrations, the fact that all right angles are equal, just as Euclid did; and like him, he never bothered to suggest that such an equality might be grounded on a particular principle. Still later, Apollonius too did not mention P4, and he usually tacitly assumed the equality of right angles throughout the Conics. This notwithstanding, two isolated passages in Book V and Book VI of this work validate an inference on angular equality by means of a reference to right angles: “…but the angles ΖΒΔ and ΑΛE are equal because they are right” (Conics V, 43); “…and the angles at τ and γ are equal because they are right”(Conics VI, 18). ¹² Such inferences might suggest a grounding on a principle such as P4. These two books of the Conics, however, have only survived in Arabic, and it is hard to say whether their original text contained any reference to P4. Since no mention of P4 is found in any of Apollonius’ books extant in Greek, it remains perfectly possible that the Arabic translator, knowing Euclid’s work, took the liberty of adding a reference to P4.

Non-mathematical works written in the Classical or Hellenistic period also did not refer to P4. Philosophers who discussed mathematical definitions and principles, such as Plato or Aristotle, never mentioned the equality of all right angles. ¹³

In conclusion, if we leave aside the likely interpolations, it appears that the Fourth Postulate was never referenced in the course of the demonstrations in the Elements, and there is no explicit reference to it in any extant Greek text predating Geminus. This cannot count, of course, as a mark of spuriousness for this postulate. It is possible that Euclid, who was bringing together several works written by previous mathematicians, added P4 among the principles of geometry and did not bother to add the references to it in the course of the demonstrations. It is also quite possible that authors such as Archimedes, Apollonius and other Hellenistic mathematicians did not quote P4 just because they did not find it useful, or appropriate, to refer to the basic assumptions of another treatise. ¹⁴ The surviving evidence is just too thin to make any positive statement on the inauthenticity of P4.

We have, however, reached an important negative result. Euclid’s works and the other early sources provide no evidence whatsoever that P4 was originally included among the postulates in the Elements. This textual situation is unique as far as Euclid’s postulates and common notions are concerned, and it opens the possibility of conceiving different stories about the epistemology underlying Euclid’s principles or the reasons that may have brought Euclid, or someone after him, to include P4 among the postulates.

2. Mathematical significance of the Fourth Postulate

In this section, I consider four standard modern interpretations of the significance of the Fourth Postulate in the foundations of geometry, which are related to the works of Klein, Clifford, Hilbert and Zeuthen respectively. While these interpretations were devised at the end of the nineteenth century and were integral to the foundational programs of the time, they are still widespread in the current literature. Taken as historical accounts of Euclid’s own ideas, however, they are grossly anachronistic. I will argue, indeed, that the importance of P4 in the foundations of geometry is eminently modern, and it depends on a conception of the nature and aims of geometry that was alien to Euclid and Greek mathematics in general. I will also offer, at the end of this section, a brief account of the historical developments that produced such modern interpretations.

The present section may be seen as a criticism of the use of the notion of “mathematical equivalence” in historical research: the equality of all right angles may well be mathematically equivalent to the isotropy of space or angular transportation according to modern standards, but its historical meaning in Greek mathematics could not have been this. ¹⁵ A byproduct of this line of research is to unpick the argument that P4 ought to be authentic since its importance in the foundations of mathematics is such that a system of geometrical axioms cannot dispense with it. The present section complements the negative results obtained in the previous section, by showing that we have neither textual nor mathematical arguments in favor of the authenticity of P4.

Modern interpretations of P4 generally connect it with the notion of space. According to these interpretations, the meaning of the postulate is that all right angles are equal in any spatial position whatsoever. Accordingly, Euclid’s concern would be to assume that a right angle constructed here must necessarily be equal to a right angle constructed there. Therefore, by means of this postulate Euclid would exclude from the models of his geometry many spatial structures in which such equality fails. These interpretations of P4 come in two variants, which are seldom distinguished in the literature.

A first interpretation of P4 was advanced (among others) by Felix Klein, who equated this principle with an axiom on the isotropy of space. ¹⁶ The idea is that P4 would state that a right angle does not change its shape when displaced in space, and continues to stay “right” at any point. It is not difficult to see that this property is equivalent to isotropy, so that P4 would indeed state, in modern terms, that the space of Euclid’s geometry has constant curvature. ¹⁷ This interpretation of P4 may be paired with the further interpretation of the Parallel Postulate as a principle stating the flatness of space. Euclid’s five postulates, therefore, would first license ruler-and-compass constructions (the first three), and then add two more “spatial principles” to the effect that space has constant curvature (the Fourth Postulate) and, more specifically, a vanishing curvature (the Fifth Postulate). In these nineteenth-century foundational programs, P4 thus became the most important axiom of geometry as a whole, since it expressed the very condition that a space must satisfy in order to become the object of a geometry. ¹⁸

A second, related interpretation of P4 was advanced by William Clifford. Rather than focusing on the preservation of angles through motions, Clifford equated P4 with the statement that one may construct equal right angles at any point (and direction) in space. This may fail in spaces with non-differentiable points. In particular, the geometry on a (semi)cone would be a perspicuous counterexample to P4. This geometry is locally Euclidean almost everywhere (since the cone has zero Gaussian curvature), but has a singular point at its vertex. An angle centered at the vertex would be right, according to Euclid’s definition, if it is equal to its adjacent angle. Such a right angle centered at the vertex, however, will not in general be congruent with a “Euclidean” right angle on the lateral surface of the cone, and therefore P4 fails in this geometry. Since cones were widely studied in ancient geometry (and indeed more studied than surfaces with variable curvature), it has been claimed that Euclid may have introduced P4 in order to make explicit that the background plane in which his geometry is performed is indeed a “standard” plane rather than a conical surface. ¹⁹

These two interpretations of P4 are badly anachronistic. We have no hint whatsoever that Euclid or any other Greek mathematician conceived postulates (or other principles) as statements ruling out certain spatial structures. We have no hint, for instance, that a denial of the Parallel Postulate would have been regarded by Greek mathematicians as the foundation of a geometry on the hyperbolic plane. Spherical geometry (the only kind of “non-Euclidean” geometry that the Greek world produced) was not axiomatized in this way, and it was not regarded as non-Euclidean at all. No mention of the geometry on the surface of a cone, or of right angles on the cone, is found in ancient texts. ²⁰ No Greek source ever suggested that a geometry in which P4 is false could be envisaged, and much less that it could be represented as the geometry on a non-flat surface.

In fact, the modern interpretation of P4 only makes sense insofar as geometry is regarded as the science of space: a theory describing the mathematical structure of space and its properties. This conception of geometry was first advanced in the early modern age and came to be fully developed (and widely accepted by the mathematical community) only in the nineteenth century. Greek geometry had nothing to do with space, and one would look in vain for any occurrence of the word τόπος in the thirteen books of Euclid’s Elements. Space and spatial properties were not discussed in ancient geometry, and the philosophy of mathematics (such as Plato’s or Aristotle’s) also excluded any relation whatsoever between geometrical figures and space. Τὰ μαθηματικὰ οὐ πού: mathematical objects are nowhere. ²¹ Geometry was regarded by Euclid, and indeed by all Greek mathematicians and philosophers, as a science of figures with the aim of investigating the properties of circles, triangles, squares, conic sections, and other similar objects. These were not conceived as being placed in any background space and, what is more important, surely not in a space possessing geometrical properties, curvature, or singular points. I have argued at length elsewhere that this is the most important divide between ancient and modern geometry, and tracing the distinction between a geometry of figures and a geometry of space is crucial for any historical understanding of mathematics. ²²

The idea of the isotropy of space simply makes no sense in this epistemic context, and the original foundational significance of P4 could not be the one that modern geometry ascribes to the equality of all right angles. The meaning of P4 was not concerned with the position of different right angles, but rather with their bare plurality. A right angle is considered to be an individual figure, and P4 plainly states that all these figures are equal to one another, in the same way in which, say, triangles with equal sides are equal to one another. Their positions are never at stake, since they were not conceived as being embedded in space.

There are further modern interpretations of P4 that, although still conceived in the framework of a geometry of space, do not explicitly refer to space in order to stress the foundational relevance of this postulate. By avoiding the reference to space, these interpretations may have at first a greater plausibility, but they are nonetheless historically problematic.

The Fourth Postulate was read as an axiom grounding the comparison of angles. In the Grundlagen der Geometrie, indeed, David Hilbert assumed an axiom to the effect that any angle may be uniquely reproduced in any other point and on any ray. This is a weaker axiom than those on the isotropy of space or the absence of singular points, and permits the comparison of angles that are at a distance by allowing one of them to be re-created in another place. Hilbert employed this axiom to prove that all right angles are equal. Since Euclid lacks an axiom on angle transportation, it may be claimed that P4 (together with the postulates allowing ruler-and-compass constructions) may be a good substitute for it. The assumption of the equality of all instances of a standard angle, stated by P4, would indeed allow to compare angles constructed in different positions. ²³

The problem with this interpretation is that angle transportation is not at all related to P4 in the deductive structure of the Elements. Euclid’s theorem on angle transportation is Elements I, 23, teaching how “on a given straight line and at a point on it, to construct a rectilinear angle equal to a given rectilinear angle.” The latter proposition, however, is itself based on the transportation of segments (Elements I, 2-3), and the transportation of triangles (i.e. the criteria of congruence). The transportation of segments is simply based on the constructive postulates on straight lines and circles (and the definition of a circle). The transportation of triangles (here, Elements I, 8) is proven by superposition, which is a technique not grounded in any more fundamental principle. In particular, P4 is not employed by Euclid for proving either the transportation of segments or the transportation of triangles (or the technique of superposition), and therefore the transportation of angles (Elements I, 23) does not really depend on it. Euclid could not have devised P4 in order to compare angles at a distance. ²⁴

A last important interpretation of the foundational meaning of P4 was offered by Hieronymus Zeuthen. He realized that the most important theorem in the Elements employing P4 at a foundational level is Elements I, 14. This proposition states that if any two adjacent angles are equal to two right angles, then they lie on a straight line. It is also the main theorem of the sequence of Elements I, 13–15 dealing with the elementary properties of angles. Of these, Elements I, 13 does not require P4 and only relies on the definition of a right angle; but Elements I, 14 does require the assumption of the equality of all right angles (as Elements I, 15 also does), and it is in fact the first proposition of the Elements that needs P4. ²⁵ The importance of Elements I, 14 is apparent by its application in such strategic places as Elements I, 45, concluding the theory of equivalence for polygons (possibly the most important result of Book I) and the related theorems in Book VI (e.g. Elements VI, 25), as well as Elements I, 47 (Pythagoras’ Theorem). ²⁶

Zeuthen made a further step in appreciating the foundational significance of Elements I, 14, and interpreted this proposition as stating the uniqueness of a perpendicular to a straight line from a point, and even as a theorem stating that two straight lines cannot have a segment in common without coinciding (these results follow, indeed, from Elements I, 14). Zeuthen took the cue from the latter statement and attributed to P4 the foundational meaning of stating the uniqueness of the extension of a straight segment. In Zeuthen’s view, the Second Postulate licensed the extendibility of straight segments, and the Fourth Postulate complemented it by asserting the uniqueness of such an extension. Zeuthen added that this interpretation has the consequence that P4 is itself constructive like the other postulates. ²⁷

There is no trace of any of this, however, in historical sources. The uniqueness of the perpendicular or the uniqueness of the straight extension are never mentioned by Euclid, who seems not to have regarded these statements as problematic or in need of proof. Even more, as we have seen, the Fourth Postulate is never mentioned in the course of the Elements, and in particular is not mentioned in the important proof of Elements I, 14—as if Euclid had not appreciated any strong connection between the postulate and this proposition.

While Zeuthen did not refer to the episode, we are informed by Proclus of a famous dispute between the Epicurean Zeno of Sidon and the Stoic Posidonius (second-first century BCE). In this occasion, Zeno asserted that Euclid lacked a proposition on the uniqueness of the straight extension of a segment, offered a proof of it which is a slight variation on Elements I, 13 and 14, and further criticized his own proof in order to show that it (like any other proof of the same statement, we may guess) relies on an unavoidable petitio principii. In short, the uniqueness of extension cannot be proven. Posidonius replied that the demonstration offered by Zeno was his own fault, and that no such proof had ever appeared in a treatise of elementary geometry. ²⁸ This episode proves that there was indeed, as Zeuthen envisaged, an ancient discussion on the uniqueness of extension that revolved around the proof of Elements I, 14. But it also shows that, according to Posidonius, no one before Zeno (and, in particular, not Euclid) had argued that Elements I, 14 might be interpreted in this way. We may conclude that Zeuthen’s interpretation cannot account for the meaning of P4 in the Elements.

The previous remarks open a more general case concerning the significance of P4 in the foundations of ancient geometry. The lack of explicit references to this postulate in the proofs of the Elements obscures the mathematical goals for which it was devised. This situation remained unchanged in Late Antiquity, as if the commentators were not so sure about its foundational aims. Proclus, who dedicated many philosophical remarks to P4 in the section of his commentary dealing with principles, never quoted it again while explicating the propositions. His commentary on Elements I, 14 is particularly telling: Proclus pedantically mentions all the principles needed to prove this proposition, but completely overlooks P4—the only essential assumption on which the entire demonstration rests:

In the construction he uses one postulate, the second (that a finite straight line can be extended in a straight line), just as in the proof he uses the preceding theorem and two axioms (things equal to the same thing are equal to one another, and if equals be subtracted from equals the remainders are equal); and for the reduction to impossibility he uses the axiom that the whole is greater than the part, for when the one common angle had been subtracted, the whole was equal to the part, which is impossible. ²⁹

The importance of P4 in the deductive structure of the Elements, and the relevance of the equality of all right angles in the foundations of geometry in general, only began to be appreciated in the early modern age. In Christoph Clavius’ edition of the Elements from 1574, for instance, we find explicit references to P4 in the proof of Elements I, 14. ³⁰ In the following Euclidean tradition we see a mounting attention to this principle, which is quoted more and more frequently in the course of the demonstrations that make use of it. In the same period, the idea of a geometry of space began to supplant the Greek geometry of figures, and P4 explicitly assumed the meaning that right angles are equal to one another irrespective of their position in space. Mathematicians such as Claude Richard (1645) axiomatized the possibility of rigid motions of figures, Gerolamo Saccheri (1733) stressed the connection between the uniqueness of the extension of a segment with P4, while Thomas Simpson (1747) provided principles on the transportations of segments and angles that were not grounded on superposition. ³¹ The foundations of the theory of angles were developed widely, and Giovanni Alfonso Borelli (1658) stated new important theorems that became standard after Hilbert’s reworking of the theory. ³² A few years later, Nicolaus Mercator (1678) suggested a different foundational approach to the cluster of theorems of Elements I, 13-15. ³³

The increasing significance of P4 in modern mathematics reached a high point in the celebrated Éléments de géométrie by Adrien-Marie Legendre (1794), the last great reworking of Euclid’s book that still regarded it as a living mathematical treatise. Legendre’s book explicitly defines geometry as a theory of extension rather than figures. It begins with a proof of P4 (the first theorem in the volume), followed by proofs of Elements I, 13, then the statement that two straight lines do not enclose a space, then Elements I, 14 and Elements I, 15, thus grounding the whole of geometry on P4 and the Euclidean theory of angles depending on it. ³⁴ Only after presenting many theorems did Legendre introduce the Parallel Postulate (that he famously attempted to prove), thus clearly distinguishing a part of geometry grounded on this latter principle (“Euclidean” geometry) from another part, which was still undetermined in terms of the theory of parallels. The “absolute” part of geometry, however, was grounded on the Fourth Postulate just as the Euclidean part was grounded on the Fifth Postulate: Legendre appears to have thought that P4 was the most basic condition necessary to have any geometry whatsoever. It was this idea that, after the developments of non-Euclidean and Riemannian geometry, resulted in the nineteenth-century interpretations of P4 as a principle establishing isotropy and constant curvature as conditions for rigid motions and shape constancy.

Between the sixteenth and the nineteenth centuries, therefore, we witness the gradual ascent of the Fourth Postulate from being a neglected principle the utility of which no one could tell, to one of the core axioms in the foundations of geometry. This very history, however, hints that the important interpretations of P4 inspired by Klein, Clifford, Hilbert, and Zeuthen were indeed peculiarly modern, and that in Antiquity this principle may have been considered much less significant. We have no mathematical reasons to suppose that it ought to be assumed as an original postulate in Euclid’s Elements.

3. The proof of the Fourth Postulate

The idea that P4 played no important role in the Elements is strengthened by the fact, which sounded amazing to many modern interpreters, that Greek mathematicians claimed to have a proof of this statement. This is incompatible, of course, with any strong foundational reading of P4. It comes as no surprise that the ancient demonstration was harshly criticized by modern mathematicians, who praised Euclid for not having indulged in such a faulty proof, thus clearly understanding (they argued) the unprovability of the Fourth Postulate. ³⁵ A minority position was taken by other interpreters, who took the ancient proof as valid and claimed, as a consequence, that P4 must be spurious: for Euclid would have not accepted among the postulates a principle that could so easily be proven. ³⁶ I agree that the proof is valid according to Greek standards, and it contributes to the many puzzles raised by the Fourth Postulate.

The proof is reported by Proclus in the same context of Geminus’ epistemological objection to the inclusion of P4 among the postulates. Geminus, apparently, reported such a proof (which was already well known in the first century BCE) in order to argue that P4 was, after all, a theorem and not a postulate. Proclus tersely added to this that the proof “was given by other commentators and requires no great study.” The same demonstration is also mentioned by Simplicius, and everything seems to suggest that it was widely circulating and accepted in Antiquity. Since then, it has become a standard proof of the postulate in the Middle Ages and in the Early Modern Age, when it featured in most editions of the Elements. It was reworked and employed in the Grundlagen der Geometrie by Hilbert himself—who grounded it, of course, on his own axiom system. Only more recently has it been replaced, in some formal systems for elementary geometry, by different and more complex demonstrations. ³⁷

The ancient demonstration works through reductio: should right angle α be smaller than right angle β, then it would be possible to superpose them in such a way that angle α falls inside angle β, and then by extending their sides one should find two other adjacent right angles, α᾽ and β᾽, respectively equal (by the definition of a right angle) to α and β. Therefore, angle α᾽ should be smaller than angle β᾽, whereas it is clear from the diagram that the opposite relation is true (see figure 1). ³⁸

Figure 1. The ancient proof of the Fourth Postulate.

From a modern point of view, the proof is defective in at least two respects. First, it employs superposition, which is not a procedure that was ever explained or axiomatized in Greek mathematics. Second, it is entirely grounded on a diagrammatic inference, which merely shows that a straight line happens to cross another straight line—without spelling out explicit principles explaining this fact.

The issue of the use and abuse of diagrammatic inferences in mathematical proofs may only be discussed in the context of a general appraisal of these techniques in Greek geometry, a topic on which we have today a thriving literature. ³⁹ There is some consensus that ancient geometers regarded the equality of magnitudes as a quantitative (say, “exact”) relation among figures that cannot be established by inspecting the diagram. According to Greek mathematicians, namely, one cannot see in the diagram that the square drawn on the hypotenuse is equal to the sum of the squares on the other sides, but has rather to prove this fact through a chain of inferences. Similarly, one cannot read out from the diagram the equality of the magnitude of two angles, and needs a propositional way of ascertaining this. Greek geometry, however, permitted the drawing of diagrammatic inferences concerning mereological containment: e.g. one may see in the diagram that an angle is a part of another angle and therefore lesser than it (a “co-exact” attribution).

The diagrammatic inference in the proof reported by Geminus nicely exemplifies this strategy. We cannot tell from looking at the first diagram whether angles α and β are equal—this is an exact attribution. However, after we superimpose them and extend their sides, we can infer, by inspecting the diagram, that angle β᾽, being contained in angle α᾽ (being a part thereof), is smaller than the latter—this is a co-exact attribution. This gives rise to a contradiction with the opposite statement we had propositionally established from the reductio hypothesis. The demonstration of P4, then, is not entirely trivial. Its construction and argument are used to transform a statement about properties subject to exact measurements (all right angles are equal) into an equivalent statement about properties that may be diagrammatically established (angle β᾽ is contained in angle α᾽). According to Greek standards, a diagrammatic inference may legitimately play a role in establishing P4.

The issue of superposition can only be formulated in the modern framework of a geometry of space. We have seen, indeed, that many modern interpretations of P4 connect it with principles on isotropy or angle transportation, but these modern axioms are simply pleonastic if one accepts, as Euclid and other Greek mathematicians did, the technique of superposition. Consequently, modern mathematicians rejected the proof of P4 as a petitio principii. Since P4 states the equality of right angles in different positions in space, so their reading goes, we cannot prove this statement by bringing the angles to the same position in space through the technique of superposition.

All difficulties vanish, however, as soon as we realize that the ancient proof was not concerned with the position of the right angles at all, and that space was never under consideration. We may usefully compare this proof with the other demonstrations employing superposition. In Elements I, 8, for instance, superposition is used to prove that any two triangles with equal sides are equal. Their position is irrelevant, and for this very reason they may be freely superposed in order to prove the theorem. In the modern interpretation, the proof of P4 would represent an essentially different use of superposition, since the very aim of the proof would be to show the equality of right angles in different positions, and the latter result cannot be attained by moving and superposing them. I claim we should, by contrast, conceive of the ancient proof of P4 on the same footing as the proof in Elements I, 8 (and others): they both achieve their goal since they are not concerned with position at all. Once we refrain from considering Greek geometry as a science of space, the ancient proof of P4 is perfectly fine, and we easily understand why no ancient commentator reported any concerns about it.

I will further belabor the latter point, since Nathan Sidoli has recently advanced an important interpretation of P4 and the technique of superposition. According to Sidoli, this technique was only considered viable, in Greek geometry, insofar as it was applied to objects (triangles, right angles, etc.) that are not given in position (τῇ θέσϵι δϵδόσθαι). This last expression, coming from Euclid’s Data, means that the object in question has a fixed position in relation to other objects in a certain configuration. I agree with this view, which expresses an important foundational stance employing Euclid’s own mathematical terminology. Sidoli maintains, however, that the ancient proof of P4 has, for this reason, only limited scope: it works when the right angles under consideration are not “given in position”, but it cannot work in all those cases in which the right angles are part of a larger configuration, and therefore are in a fixed position relative to one another. Thus, for instance, the equality of the right angles in the proof of Elements I, 46 could not be proven by superposition insofar as these angles are given in position as the four angles of a square. For this reason, Sidoli concludes, even though the ancient proof of P4 is sound, it does not apply to all possible cases, and a theorem such as Elements I, 46 would still need the assumption of P4 as a further principle. This would explain Euclid’s choice of putting it among the postulates. ⁴⁰

Sidoli’s argument is fascinating, but I think it proves too much. Take again Elements I, 8: in this theorem, two triangles may be superposed since they are not given in position, and Euclid may conclude that two triangles having equal sides are equal. When Euclid further applies this theorem (for instance in Elements I, 9), the triangles under consideration are given in position in a certain configuration, and nonetheless Euclid does not hesitate to make use of Elements I, 8 to conclude the demonstration—and similarly in all other cases. The point seems to be that once Euclid has proven by superposition that any two triangles with equal sides (which are not given in position) are equal, he assumes that all triangles with equal sides are equal—even those which are given in position. The equality of triangles does not depend on the way in which this is proven, and in particular it does not depend on whether the triangles are given, or not, in position. Similarly, then, if one accepts the ancient proof of P4, in which superposition may be employed since the right angles are not given in position, the same result (according to Euclid) would have universal validity and apply to all right angles—even in those cases in which they are given in position (such as in Elements I, 46). To assume that this generalization does not work in the special case of P4 is tantamount to saying that P4 states the invariance of right angles by position and accepting the modern reading of it as an axiom of space.

I would claim, therefore, that the ancient proof of the Fourth Postulate was sound according to Euclid’s standards and that it had universal applicability. Its early formulation and general acceptance by the Greek mathematical community (that we know of), may explain the minor and ancillary role that P4 played in the foundations of geometry in the ancient Euclidean tradition.

I would like to further advance the conjecture that the proof of P4 reported by Geminus was originally devised by his main source on matters of mathematics: the philosopher Posidonius. We know that the Parallel Postulate had been discussed at the time of Posidonius and that he had probably offered a proof of it. It seems very possible that Posidonius supplemented his proof of the Parallel Postulate with a proof of P4. We know, after all, that Geminus considered neither P4 nor the Parallel Postulate to be constructive in the sense of the first three postulates, and that he therefore wanted to delete both of them from the list of principles. If this was the opinion of his source as well, Posidonius may have attempted to prove both statements in order to adhere to Euclid’s three constructive principles as the only postulates. ⁴¹

Some interpreters, such as Thomas Heath and Oskar Becker, regarded P4 as a prerequisite for the Parallel Postulate, insofar as the very enunciation of the latter makes reference to right angles. This thesis has been used as an argument in favor of the authenticity of P4. The latter conclusion is, however, very weak, and not supported by any ancient text. I doubt we may establish that Euclid conceived P4 as a premise for the Parallel Postulate. On the other hand, should we accept that the proof of P4 reported by Proclus and Simplicius was actually conceived of by Posidonius in his essay proving the Parallel Postulate, then Posidonius may have himself pointed out a connection between the two principles and considered P4 (and his proof thereof) as a premise for the Parallel Postulate (and his proof thereof). Should this be the case, Heath’s and Becker’s opinion might be partially vindicated by applying it to Posidonius rather than Euclid. ⁴²

We have a few more clues that the P4 demonstration may have been produced by Posidonius. We have seen that Zeno and Posidonius discussed the statement that two straight lines cannot have a common segment, and connected this to the equality of right angles. In this connection, it is likely that the role and validity of P4 was also under consideration. We further know that Zeno’s arguments were probably based on Epicurean conceptions of minima in geometry. Zeno’s belief in the existence of minima resulted, for instance, in the objection that a straight line might not be susceptible to being bisected into two equal parts, since it may happen to be composed of an odd number of minima. ⁴³ We are also aware that minimal angles were discussed in Antiquity, even though the exact contexts of such discussions are quite obscure. ⁴⁴ Proclus tells us that Zeno and Posidonius also debated whether Euclid presupposed, or not, that any two straight lines may always be put at a right angle. Proclus does not mention here any issue about minima, but it may be fathomed that atomistic objections against Euclid’s definition of equal adjacent right angles may have been raised in the same vein as those denying the possibility of bisecting a segment: an odd number of minimal angles may prevent the possibility of having equal angles at the meeting of two lines. Posidonius’ debate with Zeno may have been the background for the ancient proof of P4. ⁴⁵

The ascription of the proof to Posidonius remains conjectural. Nonetheless, it may provide a foundational reason for either (1) the addition of P4 to the Elements in the centuries following Euclid, when the atomistic objections to geometry were first raised; or (2) the proof of such a postulate by Posidonius in case that P4 had been already formulated (either by Euclid or someone in between Euclid and Zeno) and later attacked by the Epicureans. I consider the second option more probable, since skeptical doubts would not be assuaged by bluntly stating a postulate, but in the lack of any further textual evidence I will not belabor the point, and will instead conclude with a wider historical conjecture.

4. A Conjecture: Euclid and Apollonius on angles

At this point, we must address the question of the authenticity of the Fourth Postulate. However legitimate, this question must nonetheless be carefully formulated and contextualized, since the Elements may be considered as a multi-authored work. ⁴⁶ They are composed of thirteen different essays, probably written by different mathematicians in different times, all of them in turn drawing on further, possibly scattered, older results. At some point these essays were collected together into the Elements, and a mathematician called Euclid is credited for having first created this unitary work. We do not know whether this collection already contained all thirteen books, or the extent of Euclid’s editorial work and adaptation of previous material. We do not know if several versions of the same treatise were produced by Euclid himself during his lifetime, and we can easily imagine that his students and followers continued to rewrite the Elements for many years after his death. The treatise was probably not considered to be an “authorial” work in the first period of its diffusion, and Archimedes, for instance, refers to some theorems that we find in the Elements by attributing them to Eudoxus rather than Euclid. ⁴⁷

Later authors commented and further modified the text of the Elements. Heron’s commentary (first century CE), for instance, may itself have been considered an edition of the text in Antiquity. Later on, Theon of Alexandria (fourth century) apparently systematized the book, and added some demonstrations. The relation between authorship and commentary changed in the course of the centuries, and Theon, for instance, while he felt authorized to modify the received text, also marked out (albeit in another work) some of the interventions he made—thus distinguishing the work of an author (Euclid) and that of an editor (himself). ⁴⁸ This was clearly not the case, however, in the first centuries after Euclid, and in this respect the very notion of authorship may be questioned. As far as we understand, indeed, the Fourth Postulate belongs to the original, opaque and multi-authored core of the Elements that was formed in the first three centuries BCE: this is as much as we can say about the “authenticity” of P4.

We may nonetheless try to dissect the Elements in order to outline different stratifications of the text, and advance conjectures on how the various pieces stay together or were added to one another. In particular, we may conceive of many different stories about the motives and occasions of the interpolation of the Fourth Postulate in the years between Euclid and Geminus. I have already mentioned that Skeptic and Epicurean challenges to the equality of right angles may have provided reasons either to include P4 among the principles of the Elements, or to offer a proof of it. Simpler explanations are also possible. In the course of the proofs, Euclid happens to add together equal angles and right angles, concluding that the results are equal. The inference is drawn through the common notion on the additivity of measure (“if equals be added to equals, the wholes are equal”), but Euclid did not spell out that right angles are equal. It is possible that a commentator, a scholiast, or just a mathematician reworking the text of the Elements, introduced P4 as an explicit condition for the applicability of the common notion on addition. Should this be the case, the introduction of P4 may have been motivated, for instance, by the needs of teaching.

Philosophical disputes and pedagogical concerns were not, however, the only reasons to modify the Elements. We may conceive, indeed, also purely mathematical grounds for the addition of P4 to the list of postulates. In this section, I explore this possibility, and venture the conjecture that the Fourth Postulate was added to the text of the Elements in the second century BCE, following a transformation of the definition of angles initiated by Apollonius. The advantages of such a conjecture over the others (including the hypothesis that Euclid himself formulated P4) is its ability to cope with the many textual and mathematical difficulties that we have seen so far. The conjecture is based on an interpretation of the theories of angles in Greek mathematics, and exploits a tension between the Fourth Postulate and Euclid’s definitions of angles—two different strata, I claim, of the text of the Elements.

I begin by presenting the theory of angles that emerges from the definitions of the Elements and other ancient sources, and argue that it can do without P4 (section 4.1). I then discuss an objection and introduce a second conjecture to explain some puzzles about the treatment of angles in the Elements (section 4.2). Finally, I consider Apollonius’ theory of angles as a possible source of the interpolation of P4 in the Elements (section 4.3), before presenting some concluding remarks.

4.1. Euclid’s definitions of angles

In Elements I, def. 8, Euclid defines an angle as an inclination (κλίσις) between lines. The concept of “inclination” is taken as primitive, and it does not appear again in the Elements. ⁴⁹ The term κλίσις also recurs in relation to angles in the works of Autolycus of Pitane (fourth or third century BCE). Therefore, a definition of angles as inclinations may have been common in pre-Euclidean mathematics. ⁵⁰ More or less in the same period as Autolycus, Aristotle seems to have understood an angle as a quality of lines, and accordingly he called “similar” (ὅμοιαι) angles that Euclid would later call “equal” (ἴσαι) angles. Aristotle’s student Eudemus, who is credited by Proclus with an entire book on angles, also claimed that angles are qualities of broken lines. ⁵¹

While the details underlying these notions are quite obscure, all these early definitions of angles as qualities (Aristotle, Eudemus) or inclinations (Autolycus, Euclid) of lines concurred in taking angles to be shapes and configurations. This point of view is sharply contrasted by Proclus with subsequent definitions of angles as quantities or magnitudes. According to Proclus and other ancient commentators, Euclid’s definition of angles was essentially non-quantitative.

I maintain that we should take this idea seriously. By defining angles as inclinations between lines, Euclid was not thinking about the part of the plane delimited by such lines, nor the length of an arc between these lines, nor the width of their “opening”, but about a more abstract relation between two meeting lines. This relation between lines makes the configuration, or shape, of an angle. No measure is involved, but only the positional arrangement of a pair of intersecting lines. In the same vein, parallelism was not defined by Euclid as a determinate distance between straight lines (a metric notion that would become customary in the following Euclidean tradition), but rather as a positional arrangement of non-intersecting lines. ⁵²

If this is true, for Euclid a right angle was a unique, symmetrical configuration of lines. When the inclination of two meeting straight lines is the same on one side and the other, they make a right angle. A right angle is a peculiar geometrical shape. Accordingly, the equality of a right angle with its adjacent angle should not be intended as the equality in magnitude of these two angles, but rather as their equality in shape, viz. their congruence. It is indeed very common that Euclid calls congruent figures “equal” (ἴσα). ⁵³

I would claim that with such a conception of angles, the Fourth Postulate is neither needed nor meaningful. There is no room for doubt, indeed, that the relation (the κλίσις, inclination) obtaining between two perpendicular straight lines is the same relation as the one that obtains between any other pair of perpendicular straight lines. They are both configurations of straight lines, and they are both symmetrical: all perpendiculars are perpendicular in the same way. Asking about the equality of distinct pairs of right angles would be tantamount to asking whether two straight lines are straight in the same way, or whether two circles are circular in the same way. Euclid has no principles (besides the definitions) concerning the straightness of straight lines, or their reciprocal congruence and fitting, or their shape in general. The same is valid for a right angle: it is not a magnitude that might be greater or lesser, but a particular shape and a unique configuration. It cannot be different in different instances, otherwise it would no longer be a right angle.

This is, I would argue, the view vehiculated by the definitions of angles in the Elements. It may also be regarded as a consequence of Euclid’s geometry being a geometry of figures rather than space. In modern geometry we do accept, in fact, that two straight lines may be different from one another: geodesics may have different shapes and properties according to the curvature of space. But in a spaceless geometry, in which no underlying spatial background affects the properties of the figures—which are the only objects of investigation—there is no conceptual room for meaningfully asking if straight lines are identically “straight” (as if they might have different shapes here or there), or if all right angles are identically “right.” The self-identity of the (non-spatially located) figures is the main assumption of classical geometry, which lacks sufficient reason (the space structure) for ever doubting it. This also explains why all ancient commentators stressed that Euclid’s definition of a right angle spells out a unique configuration of lines, whereas Euclid’s definitions of acute and obtuse angles encompass a plurality of different angles. ⁵⁴ This apparently naïve observation is deeply rooted in the fact that a unique configuration cannot be different in different instances, so that the equality of all right angles is a straightforward consequence of their definition.

I find confirmations of this interpretation in the later Greek commentators. Heron of Alexandria adopted, in his Definitiones, a Euclidean perspective on the foundations of geometry and discussed at length the configurational properties of the perpendiculars forming two right angles, endorsing Euclid’s understanding of angles as inclinations. Heron, indeed, did more than this, placing the statement that all right angles are equal among the definitions, since he clearly saw it as an immediate consequence of the definition of a right angle. Heron did not even mention the fact that the same statement had once been regarded as a postulate. ⁵⁵ Elsewhere, Heron discussed the issue again, and just stated that all right angles are congruent to one another, without making reference to their equality. ⁵⁶

Further sources are Proclus and Simplicius. They seem to expand on Heron’s remark when they claim that there is a second proof of the Fourth Postulate (besides the one we saw in the previous section), which is purely logical. This proof is just that all right angles are formed by lines that “are not inclined” to one another (i.e. by perpendicular lines) and therefore must be equal to one another. The truth of P4 flows, therefore, from the definition of a right angle. This “logical” proof is clearly grounded on a conception of a right angle as a unique and symmetrical configuration of lines rather than a magnitude, and also elsewhere Proclus states that (according to Euclid) “when the inclination is one, there is one angle.” ⁵⁷

My hypothesis is that this was Euclid’s original understanding of the equality of all right angles—a statement following the definition of a right angle, that he did not deem necessary to spell out as a postulate. Had he wanted to spell it out, arguably, he would have proceeded just as Heron did, and added this remark to the definitions. Since we do not know the exact sources of Heron’s Definitiones and to what extent they rely on older Euclidean material, we are not able to further substantiate this conjecture and claim that this was indeed the case in an ancient version of the text. ⁵⁸

Finally, Proclus reports that Pappus had discussed at length the “converse” of P4, which states that if a right angle is equal to another angle, then the latter is a right angle as well. Pappus argued that this is not the case in general, since a curvilinear angle may be equal to a right angle (in the sense of having the same magnitude), without being itself a right angle. It is clear that Pappus’ qualms may only arise within the framework of a distinction between the shape of an angle and its magnitude, and would make no sense if angles were considered to be just magnitudes. A curvilinear angle may have the same magnitude as a right angle, but it is not itself a right angle—because the latter is a configuration of straight lines and not a magnitude. ⁵⁹

Ancient sources therefore agree that Euclid had a non-quantitative conception of angles and, moreover, that the equality of all right angles was for him an immediate consequence of the definition of a right angle as a configuration.

4.2. An objection, a reply, and a further conjecture: Angles and angular magnitudes

Despite this evidence, some modern interpreters have claimed that for Euclid angles were magnitudes. ⁶⁰ There is, indeed, a major objection against the interpretation of angles as relations: Euclid, sometimes, treats angles as if they were quantities and magnitudes (μϵγέθη). He compares angles according to the lesser and the greater, he adds and subtracts angles, and takes multiples of given angles. In the single instance of Elements VI, 33, Euclid even puts angles in a ratio, and according to Book V of the Elements only magnitudes may have ratios. ⁶¹ Moreover, in the Data Euclid states that angles may be “given in magnitude” (τῷ μϵγέθϵι δϵδόσθαι). ⁶² These discrepancies on the use of angles in the Elements and elsewhere should not be surprising. There is no doubt that ancient mathematicians used, and probably mixed together, different conceptions of angles in their mathematical practice. Greek geometers and astronomers had to measure and calculate angular magnitudes and, as a consequence, they considered ratios and proportions between angles and, in some cases, measured angles by numbers. Euclid’s Elements are a composite text and the different essays collected in the thirteen books display different mathematical practices concerning angles. ⁶³

However, I think that it is a mistake to take this variety of approaches to the notion of angles as an argument for claiming—against all ancient sources and textual evidence—that Euclid had a quantitative conception of them. This view stems from a confusion between the level of concrete mathematical practices and that of foundational analysis. The point is not how angles are used in the Elements, but how Euclid believed he could logically ground the theory of angles on a system of definitions—or a postulate. The question about P4 is not about mathematical practices or tacit knowledge (no Greek mathematician has ever doubted that all right angles are equal), but about the formulation of explicit foundations of geometry.

If we admit that Euclid had a design for the foundation of the theory of angles, we must show that it is at least possible to accommodate a definition of angles as relations with the mathematical practice of using angles as magnitudes. In the system of definitions of the Elements, there is no explicit trace of this sort of foundation, and so we can only speculate how Euclid, and perhaps the other mathematicians of the time, envisioned coordinating the two notions. ⁶⁴

The simplest conjecture to this effect, I suggest, is to assume that Euclid regarded angles as relations with the property of having magnitudes. Euclid distinguished between an angle and its magnitude, and the latter was conceived as a property and a consequence of the relation defining the angle. He considered angles to be inclinations and shapes at a foundational level, and nonetheless admitted that their magnitudes could be compared, summed, and put into ratios. The idea was not entirely unprecedented, because ratios, in the theory of Eudoxus developed in the Elements, are also relations at a foundational level, that nevertheless have certain quantitative properties (they are, for example, greater than, equal to, or less than other ratios). Similarly, an angle was for Euclid an inclination having a quantity rather than the quantity of an inclination. ⁶⁵

The distinction between angles and angular magnitudes is indeed required by the fact that Euclid considers only angles less than a straight angle. Since an angle is defined by him as an inclination between two lines, Euclid denies that a straight angle is an angle, and also excludes non-convex configurations of lines. This fact, which is everywhere confirmed by Euclid’s mathematical practice, rules out the possibility that angles can simply be identified with their magnitudes. The interior angle sum of a triangle, which Euclid proves to be equal to “two right angles,” for instance, is an angular magnitude which is not an angle. These angular magnitudes, distinguished from the angles from which they are derived, are summed, multiplied and put into ratios in the theorems of the Elements. ⁶⁶

This even occurs at the foundational level. The proposition of Elements I, 13, for instance, states that “if a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.” Should we take angles to be magnitudes, the theorem would be a triviality. It states, in fact, that angles forming together by hypothesis a straight angle are equal to two right angles, that is to say, to a straight angle. But, as we have seen, Euclid has no notion of a straight angle as a geometrical configuration, and therefore in this case he needs a propositional articulation (going beyond diagrammatic evidence) of the equality of two magnitudes, which is not an equality of two angles. The distinction between angles and angular magnitudes requires an explicit proof of this fact. ⁶⁷

The fact that angles are not magnitudes but have magnitudes is in sharp contrast with Euclid’s treatment of lines, plane figures and solid bodies: all these objects are considered to be magnitudes in the proper sense. The identification of figures and magnitudes is a pervasive and fundamental feature of ancient geometry and epistemology of mathematics, and it is at the foundations of the theory of content (viz. measure) in Greek geometry. I will not belabor the point here, but will note at least that the definability of a standard of measure for angles (i.e. the right angle) as opposed to the indefinability of such a standard for lengths (say, the “standard meter”) seems to be the core reason for the very different theories of angles and extensive magnitudes in classical geometry. An extensive magnitude may only be shown in intuition through a particular instance of it, and it is therefore natural to identify segments or squares with their own magnitudes. By contrast, an angular magnitude may be defined in a purely propositional way (e.g. “three right angles”), and it is therefore possible to refer to it without exhibiting it in intuition as a concrete figure. The distinction between angles and angular magnitudes is a natural byproduct of this more fundamental divide. ⁶⁸

The present conjecture about the distinction between angles and angular magnitudes explains the puzzle of the apparent incongruence of Euclid’s definitions with the mathematical practices found in the Elements. But even if this conjecture were not accepted, and other hypotheses were advanced as to the compatibility, or lack thereof, of Euclid’s definition of angles with the mathematical practices of the Elements, there would still be no need to infer from those practices that for Euclid angles were magnitudes. At the foundational level, as it is clear from the definitions and the ancient testimonies, an angle was for him a relation and a shape. This in turn, I maintain, makes the Fourth Postulate useless in Euclid’s own theory.

4.3. Apollonius on angles and the Fourth Postulate

Euclid’s conception may be usefully contrasted with Apollonius’ idea of angles as magnitudes in the full sense. Proclus, indeed, ascribed to Apollonius the first definition of angles as magnitudes in Greek mathematics. By defining angles as magnitudes, then, Apollonius consciously opposed Euclid and all other previous geometers. He knew, like Euclid before him, that mathematical practices on angles varied widely and that angles were treated as forms, qualities, relations, or quantities. But he believed that the correct foundational path for the theory of angles should begin with the notion of a magnitude.

Apollonius’ definition of angles is famously obscure: he called a plane angle “the contraction of a surface at a point under a broken line” (συναγογὴ ἐπιφανϵίας πρὸς ἑνὶ σημϵίῳ ὑπὸ κϵκλασμένῃ γραμμῇ). ⁶⁹ Apollonius’ definition is given by Proclus out of context and without any real mathematical application. Moreover, the definition itself does not make any reference to magnitudes, despite the fact that Proclus stated that Apollonius and his successors took angles to be magnitudes. ⁷⁰

I would say that the most obvious reason for introducing Apollonius’ definition may be found in the aim of comparing rectilinear and curvilinear angles with one another. Such a comparison may be needed in the treatment of tangents, and requires greater resources than Euclid’s own configurational definition, his conception of angular equality as congruence, and his diagrammatic inferences grounded on the intuitive notion of mereological containment. A curvilinear angle with the same vertex and side as a rectilinear angle, indeed, may neither contain nor be contained by the latter (see figure 2).

Figure 2. Diagrammatic comparison of curvilinear angles.

This makes their standard diagrammatic comparison impossible, since neither of them is a part of the other. The consideration of a small enough part of the plane around the vertex of the angles, however, allows us to tell which angle is inside the other. This would make available once again a criterion based on diagrammatic inspection, which would allow us to say that (for instance) the curvilinear angle is less than the rectilinear angle. I surmise, then, that Apollonius’ definition of an angle as “the contraction of a surface at a point under a broken line” had the aim of making these comparisons possible, and that the need for a consistent treatment of curvilinear angles in Greek geometry was the mathematical reason behind the switch from Euclid’s configurational definition to Apollonius’ quantitative definition. ⁷¹

It is important, indeed, to understand that Apollonius’ definition takes angles to be (finite, small) portions of the plane. We know, from Proclus’ report, that this was exactly the topic of discussion of the many further refinements and adjustments of Apollonius’ definition. Syrianus, Proclus’ master, seems to have corrected Apollonius by saying that an angle is a contracted surface, rather than the act of contraction itself. Another Neoplatonist, Plutarch of Athens, took this contraction to result in the first extension (πρῶτον διάστημα) under the point, possibly meaning a minimal, atomic extension. Carpus of Antioch maintained that a plane angle is a quantity extended in one dimension (rather than two), without it being a line. ⁷² Others argued whether a plane angle may have a dimension in between those of a line and a surface. ⁷³

Whatever we may think of these elliptic fragments of debate provided by Proclus, it seems clear that all these conceptions regarded an angle as a piece of a surface (or a line, or something in between), and moreover a finite piece of it (a contraction, or contracted surface, or minimal surface, according to various definitions). An angle is therefore a finite extended quantity of the same kind of the standard magnitudes—segments, plane figures, solid bodies. For this reason, an angle, according to Apollonius and his followers, is indeed a μέγϵθος in the full sense of Eudoxus, Aristotle and Euclid. The Apollonian definitions of angles as magnitudes may therefore have been regarded, in Antiquity, as a solid foundation of the use of proportions in the theory of angles.

Proclus does not provide us with any direct application of Apollonius’ new definition of angles in geometry. Nonetheless, in the course of his commentary on Book I of the Elements, he mentions a few other elementary theorems about angles attributed to Apollonius that may hint at Apollonius’ foundational endeavors in the theory of angles. We know, indeed, that Apollonius offered alternative demonstrations of Elements I, 10–11, teaching how to construct perpendiculars and thereby right angles. ⁷⁴ Another extant demonstration ascribed to him reformulates Euclid’s theorem of angle transportation (Elements I, 23). ⁷⁵ While the two former proofs on perpendiculars are quite similar to Euclid’s, the last one widely diverges from the material in the Elements, suggesting that Apollonius’ theory of angle transportation might have been itself very different from Euclid’s. We have seen, indeed, that Euclid grounded angle transportation on the congruence of triangles obtained by superposition (with no references to P4). Apollonius seems to follow an approach closer to the modern one, in which the transportation of angles is established before proving the congruence theorems for triangles. Proclus objected that, by proceeding in this way, Apollonius was obliged to assume a few results on circular arcs coming from Book III of the Elements. But it is not evident that this was the case, and Apollonius may have employed definitions (or postulates) on angle equality that made his proof more direct than Euclid’s. In particular, Apollonius (according to Proclus) seems to have assumed that two angles subtending equal arcs are equal, and this may well be a consequence (or just a reformulation) of his definition of an angle as a magnitude and the contraction of a surface. ⁷⁶

Apollonius’ definition of angles clearly has important consequences on the meaning of the Fourth Postulate and its foundational significance. According to this definition, indeed, P4 would not mean that all right angles are congruent, but would rather state that all right angles are equal magnitudes. In this interpretation, P4 means that (say) a small portion of the plane around the point of intersection of a straight line and its perpendicular is equal to every other such portion formed by other perpendicular lines. This is a non-trivial statement that cannot be grounded on the definition of a right angle and requires, indeed, an explicit propositional assumption.

It is impossible to establish, from Proclus’ overly terse account, whether Apollonius’ new theory made explicit use of P4. We have seen that there are two possible references to P4 in the Arabic translation of Apollonius’ Conics, and that it is difficult to ascertain whether they may have been interpolated in the text—and therefore whether Apollonius may have made use of P4 as a principle. The very fact that Apollonius reworked the entire foundations of the theory of angles shows that his new definition of angles as magnitudes was not a “local” amendment with the aim of fixing some philosophical difficulties, but required a reassessment of the entire axiomatic theory. It is possible, then, that P4 was first formulated as the outcome of Apollonius’ new definition of an angle, either by Apollonius himself or by some other mathematician of his time, to cope with the demands of this new, thoroughly quantitative understanding of angles.

It may still be asked why Apollonius or other mathematicians of his time did not prove P4 through superposition rather than assuming it as a postulate. Perhaps they did: while Euclid does not seem to assume any provable principle in the Elements, the philosophy of mathematics in the following centuries was sufficiently flexible to admit these as well. ⁷⁷ Alternatively, Apollonius may have had his own reasons not to employ superposition in this connection, and have thought that this technique could prove nothing concerning angles as magnitudes. After all, he seems not to have proven angle transportation (Elements I, 23) through triangle transportation and superposition. ⁷⁸ Had he done so, P4 may have been first formulated as an unprovable principle and later proven in a still different epistemological context (Posidonius’ replies to Zeno’s criticisms).

It is not implausible that, if P4 was first formulated in Apollonius’ circle, it was later added to the text of the Elements. Apollonius, we are told, was “a student of the students of Euclid,” ⁷⁹ and we know from Apollonius’ preface to the Conics that he was corresponding with a group of mathematicians working in Alexandria, who were most likely these followers of Euclid. Apollonius himself wrote some elements (στοιχϵία) of geometry, which may have been a reworking of Euclid’s, discussed the axiomatization of geometry and proposed demonstrations of Euclid’s common notions. ⁸⁰ More importantly, a Greek scholium tells us that three definitions in Euclid’s Data (two of which concern angles) had in fact been conceived by Apollonius. These definitions are never further employed in the Data (just as P4 is not explicitly employed in the Elements), and it is difficult to fathom their mathematical significance. The early interpolation of these Apollonian definitions into Euclid’s work is important evidence that other principles coming from Apollonius may easily have been added to the Elements. ⁸¹ In sum, it is wholly possible that Apollonius’ discussions on the foundations of mathematics, carried out with Euclid’s students in Alexandria, contributed to the modification of the text of the Elements.

In conclusion, we may take a second look at Geminus’ statement about P4 as we find it in Proclus, which is our earliest testimony of the existence of P4 in the Elements. Geminus’ aim is to show that P4 is not a postulate. The argument seems to run as follows: if P4 is a principle, then it must be a common notion, since it is not constructive in the sense that the other postulates are; and if it is not a principle but rather a provable proposition (according to the proof that Geminus expounded), then it is not a postulate either, since it should be considered a theorem. ⁸² It is surely possible that Geminus’ dichotomy (P4 is a principle or a provable proposition) was merely dialectic and constructed in order to better make his point. It is more likely, however, that Geminus was simply reporting existing discussions. In the mathematical literature with which he was familiar, someone had posited P4 among the postulates, while someone else (possibly Posidonius) had proposed its demonstration and placed it among the theorems. It is even possible that in Geminus’ time there was a debate on the need for P4 in the foundations of geometry, and Geminus was attempting (though failing) to have it excluded from the list of the postulates of the Elements. Should this be the case, the text of the Elements would show a remarkable fluidity even in the first century BCE. Given that the next historical source at our disposal is Heron (first century CE), who mentioned P4 among the definitions, this state of uncertainty may have persisted for some time.

This, then, is my conjecture on the addition of P4 to the Elements. It is grounded on an interpretation of Euclid’s notion of angles as non-quantitative geometrical configurations, and an inference to the uselessness of P4 in such a context. The conjecture also offers an interpretation of Apollonius’ theory of angles as magnitudes and an inference to the need for P4 in this different foundational context. Finally, it shows the historical plausibility that Apollonius’ ideas may have entered the system of principles of the Elements. The conjecture, therefore, provides positive arguments for the missing references to P4 in the demonstrations in the Elements, without claiming that Euclid simply overlooked this principle: Euclid did not need P4 at all. A later interpolation of P4 may account, in turn, for the non-constructive features of P4, since the epistemology of mathematics was rapidly changing in the centuries after Euclid. The same story also accounts for the puzzlement about P4 which we find among ancient interpreters, who clearly did not know what to do with it: the principle, indeed, came from a different mathematical tradition, and attempted to solve problems that were not in the Elements. Finally, this conjecture offers non-trivial mathematical and logical reasons for the addition of P4 to the Elements, thus giving an interpretation of the meaning and significance of P4 in ancient geometry.

The conjecture complements with positive arguments the negative textual and mathematical results of the previous sections, which had established that we have no evidence that the Fourth Postulate was in the Elements before the time of Geminus, and that its foundational aims could not possibly be those attributed to it by modern mathematicians. These latter conclusions are more certain, and set the stage for conceiving several different stories about the role and the interpolation of the Fourth Postulate in the text of the Elements. These stories tell of Hellenistic professors inventing the postulate to justify the use of common notions about the addition of angles; of sophists and atomists discussing angles that are almost right angles, but not quite right angles; of Posidonius, who may have believed that the equality of right angles was a condition for the Parallel Postulate; and many more can be dreamed up. It seems to me that the conjecture about the transformation of the definition of an angle between the times of Euclid and those of Apollonius is the most consistent with the scanty evidence we have, and our best guess for explaining several puzzles in the foundations of Greek mathematics.

We do not know who, between the third and the first century BCE, conceived the idea that the equality of all right angles should be a principle of geometry. Since then, the significance of this statement has changed many times over the centuries. Yet the Fourth Postulate, for more than two thousand years, has remained at the core of the foundations of mathematics.