In the last article in this series we studied the twenty-third and final definition in Book 1 of Euclid’s Elements. We turn now to the Five Postulates—Αἰτήματα [Aitēmata]—which follow the definitions (Fitzpatrick 7):
| Greek | English |
|---|---|
| αʹ. ̓Ηιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν. | 1. Let it have been postulated to draw a straight-line from any point to any point. |
| βʹ. Καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ εὐθείας ἐκβαλεῖν. | 2. And to produce a finite straight-line continuously in a straight-line. |
| γʹ. Καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράφεσθαι. | 3. And to draw a circle with any center and radius. |
| δʹ. Καὶ πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις εἶναι. | 4. And that all right-angles are equal to one another. |
| εʹ. Καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ, ἐκβαλλομένας τὰς δύο εὐθείας ἐπ ἄπειρον συμπίπτειν, ἐφ ἃ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες. | 5. And that if a straight-line falling across two (other) straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side). |
What does Euclid mean by the word postulate, or, in the original Greek, αἴτημα [aitēma]? Today, the word postulate, when used as a technical term in logic or mathematics, is generally understood to be a synonym for axiom or assumption:
a. A proposition demanded or claimed to be granted, esp. something claimed, taken for granted, or assumed, as a basis of reasoning, discussion, or belief; hence, a fundamental condition or principle.
b. Sometimes with special reference to its undemonstrated or hypothetical quality : An unproved assumption, a hypothesis.
c. Sometimes with special reference to the self-evident nature of a proposition of fact : hardly distinct from Axiom.
d. Something required as the necessary condition of some actual or supposed occurrence or state of things; a pre-requisite. (Oxford English Dictionary, Postulate II.2)
Etymologically speaking, both words—the Greek and the English—mean a request, a demand, a thing demanded or claimed. A postulate is a proposition we are asked to concede without its being proven.
It appears that the ancient geometers, including Euclid, maintained an important distinction between a postulate (αἴτημα) and an axiom (ἀξίωμα [axiōma]). According to some commentators, the latter referred specifically to an assumption whose truth was taken to be self-evident or obvious—definition c above. The former, however, referred to an arbitrary assumption one makes as the basis for further reasoning—definition a. According to another view, however, postulates and axioms were both unproven assumptions, self-evident or not as the case may be : the distinction between them was that postulates were peculiar to geometry, while axioms were common to all the sciences.
Euclid never actually uses the word ἀξίωμα in the Elements, but his so-called Common Notions (Κοιναὶ ἔννοιαι [Koinai ennoiai.]) are clearly axioms in the sense of self-evident propositions. In fact, Proclus refers to these Common Notions as axioms (ἀξίωματα) in his Commentary on the First Book of Euclid’s Elements (Morrow 140 : Heiberg 10).
The Five Postulates, therefore, are not necessarily obviously true or even universally true. They are the fundamental principles that underpin the geometry we now call Euclidean. If we wish to do Euclidean geometry, we must accept these Five Postulates.
Euclid and his immediate successors probably did regard the Five Postulates as universally true. That is to say, Euclid believed that they described the sort of geometry we find in the actual world, which for Euclid was the only geometry. It was only in the 19th century that modern geometers such as Carl Friedrich Gauss, János Bolyai and Nicolai Lobachevsky realized that one could generate non-Euclidean geometries by adopting different postulates. It has even been hypothesized in recent years that the geometry of our Universe is actually non-Euclidean—though its deviation from a Euclidean, or flat, Universe can only be observed on cosmic scales.
One of the great ironies in the history of geometry is that spherical geometry, which Euclid and his contemporaries studied in great detail, is actually a non-Euclidean geometry. The ancient geometers, however, always studied the sphere as a two-dimensional surface existing in a three-dimensional Euclidean space, so they never recognized it as a separate geometry in its own right. If, however, we restrict ourselves entirely to the two-dimensional surface of the sphere, then it becomes clear that the fundamental principles underpinning this geometry are quite different from Euclid’s Five Postulates. In spherical geometry, for example, the role of straight lines is taken by great circles, which requires the postulate: There are no parallel lines in spherical geometry.
In his Commentary on the First Book of Euclid’s Elements, the 5th-century philosopher Proclus discusses the Five Postulates at length. His terminology is a little confusing because he uses different terms to those found in Euclid:
| Euclid | English | Proclus | English |
|---|---|---|---|
| ὄροι | definitions | ὑποθέσεις | hypotheses |
| αἴτηματα | postulates | αἴτηματα | postulates |
| κοιναὶ ἔννοιαι | common notions | ἀξίωματα | axioms |
To begin with, Proclus adopts—or, perhaps, adapts—the distinction Aristotle makes in his Posterior Analytics between hypotheses (or definitions), postulates, and axioms (or common notions):
When a proposition that is to be accepted into the rank of first principles is something both known to the learner and credible in itself, such a proposition is an axiom: for example, that things equal to the same thing are equal to each other. When the student does not have a self-evident notion of the assertion proposed but nevertheless posits it and thus concedes the point to his teacher, such an assertion is a hypothesis [definition]. That a circle is a figure of such-and-such a sort we do not know by a common notion in advance of being taught, but upon hearing it we accept it without a demonstration. Whenever, on the other hand, the statement is unknown and nevertheless taken as true without the student’s conceding it, then, he says, we call it a postulate: for example, that all right angles are equal. (Morrow 62-63)
Aristotle’s discussion is a little more complicated than Proclus suggests. For example, Aristotle also distinguishes between a hypothesis and a definition.
Definitions are not hypotheses, because they make no assertion of existence or non-existence. (Aristotle, Posterior Analytics 76b34 ff : Tredennick 72-73)
Aristotle also distinguishes between two kinds of fundamental assumptions: those that are specific to one particular science, such as geometry : and those that are common to all the sciences (Tredennick 68-69). As we saw above, it has been argued that this is precisely the distinction Euclid makes between his Five Postulates (assumptions peculiar to geometry) and his Common Notions (assumptions common to other branches of mathematics as well as to geometry). This argument certainly explains why Euclid refers to his axioms as Common notions (Introduction to Euclid’s Geometry).
Following Aristotle, Proclus distinguishes between the three types of proposition using two principal criteria:
| Category of Proposition | Self-Evident? | Student’s Response |
|---|---|---|
| Hypothesis (Definition) | No | Conceded |
| Postulate | No | Not Conceded |
| Axiom (Common Notion) | Yes | Conceded |
Proclus later reiterates the distinction between postulates and axioms:
It is a common character of axioms and postulates alike that they do not require proof or geometrical evidence but are taken as known and used as starting-points for what follows. They differ from one another in the way in which theorems have been distinguished from problems. Just as in a theorem we put forward something to be seen and known as a consequence of our hypotheses but in a problem are required to procure or construct something, so in the same way axioms take for granted things that are immediately evident to our knowledge and easily grasped by our untaught understandings, whereas in a postulate we ask leave to assume something that can easily be brought about or devised, not requiring any labor of thought for its acceptance nor any complex construction. Hence clear knowledge without demonstration and assumption without construction distinguish axioms and postulates, just as knowing from demonstration and accepting conclusions by the aid of constructions differentiate theorems from problems. (Morrow 140-141)
As Proclus proceeds to remark, some early geometers used the term postulate for both, while others used the term axiom for both, just as some mathematicians do not distinguish between theorems and problems but designate all alike as theorems (Morrow 142).
Proclus concludes his general discussion of postulates and axioms by adducing a third criterion by which they may be distinguished:
According to the third, the Aristotelian method, everything that can be made convincing by a proof will be a postulate, and whatever is indemonstrable an axiom. (Morrow 143 : Tredennick 32-33, 72-73)
Whether Aristotle’s remarks on these different types of proposition are actually relevant to the Elements is debatable. Johan Ludvig Heiberg, whose edition of the Elements is generally considered definitive, argues that there is no trace of Euclid’s Five Postulates in Aristotle, where the term postulate (αἴτημα) has a different meaning. Thomas Heath, however, whose translation of the Elements is based on Heiberg’s text, disagrees (Heath 119-120). I’m with Heiberg. I do not think our understanding of Euclid is enhanced by bringing Aristotle into the discussion. For Euclid, postulates and common notions are simply unproven assumptions that one is obliged to accept if one wishesd to prove the propositions in the Elements. The postulates are peculiar to geometry, while the common notions are common to all branches of mathematics. Whether or not they are self-evident is irrelevant. Whether or not the student is willing to concede them without proof is also irrelevant.
It is often said that the propositions—theorems—of Euclid’s Elements must be proved using only a straightedge (unmarked rule or ruler) and a compass. As we saw in an earlier article—Euclid’s Constructive Geometry—Euclid never actually mentions either of these instruments, and it is doubtful whether he himself ever saw his geometry as constructive at all. For Euclid, geometric objects like lines and circles are akin to the Platonic Forms or Ideas: abstract realities that exist only on a non-material plane. The diagrams that one might construct with a compass and rule are only crude imitations of this reality: Ceci n’est pas une pipe.
The idea that Euclid’s geometry is a constructive geometry of the straightedge and compass stems from the belief that these are the only instruments required by Euclid’s Five Postulates. Some commentators, however, have gone further than this and have even argued that the whole point of the Five Postulates was to define Euclidean geometry as a straightedge-and-compass constructive geometry. Thomas Heath rejects this notion:
There is of course no foundation for the idea, which has found its way into many text-books, that “the object of the postulates is to declare that the only instruments the use of which is permitted in geometry are the rule and compass.” (Heath 124)
Ernst Ferdinand August’s 1826 edition of Euclid’s Elements lists six postulates, not five. The Sixth Postulate reads (August 3):
| Greek | English |
|---|---|
| Καὶ δύο εὐθείας χωρίον μὴ περιέχειν | And two straight lines do not enclose a space. |
This postulate can also be found in Henry Billingsley’s English translation of 1582:
- That two right lines include not a superficies. (Billingsley Folio 7)
François Peyrard also includes this Sixth Postulate in his 1814 edition. His Latin and French translations read:
Et duas rectas spatium non continere.
Deux droites ne renferment point un espace. (Peyrard 5)
In his apparatus criticus, he notes:
Hoc postulatum in codice e exaratur eâdem manu in postulatis, et aliéna in not. com.; in codice f aliéna in postulatis, et eâdem in not. com.; in codicibus h, k in post. et in com. not. eâdem manu exaratur.
[This postulate is written in codex e in the same hand among the postulates, and in another hand among the common notions; in codex f it is written in another hand among the postulates, and in the same hand among the common notions; in codices h and k it is written in the same hand among the postulates and the common notions.] (Peyrard 454)
That this Sixth Postulate was already in existence in ancient times—albeit sometimes regarded as an axiom or common notion rather than as a postulate—is made clear by Proclus:
According to the second mode of distinguishing them [postulates and axioms], it will not be an axiom that two straight lines do not enclose an area, although some persons still list it as an axiom ... it is superfluous to include among the axioms “two lines do not enclose an area” if it can be established by demonstration. (Morrow 143 ... 144. See also 154).
Proclus himself does not include this among either the Postulates or the Common Notions. Another late commentator on the Elements, Simplicius of Cilicia, also knew of this Sixth Postulate. Simplicius was probably born shortly after the death of Proclus, and was one of the last philosophers to study at the Neoplatonic Academy in Athens before it was abolished by Justinian in 529. Simplicius’s Commentary on the Definitions, Postulates and Common Notions of Book 1 of Euclid’s Elements survives only in an Arabic translation by the Persian mathematician Al-Nayrizi (Anaritius). In the 12th century, this was translated into Latin by Gerard of Cremona:
Dixit EUCLIDES: Due recte linee non comprehendunt superficiem. Supra hoc SAMBELICHIUS: Hec petitio non invenitur in antiquis scriptis, que ideo fuit dimissa, quoniam est manifesta, et ideo dixerunt, quod petitiones sunt quinque.
[EUCLID said: Two straight lines do not encompass a surface. On this SIMPLICIUS: This postulate is not found in the ancient manuscripts. It was probably omitted because it is obvious, and therefore it is said that there are only five postulates.] (Curtze 35)
It is clear from Peyrard’s comment, however, that in the Middle Ages this proposition was generally included in manuscripts of the Elements, but there was never a consensus on whether it should be listed as a postulate or as a common notion.
In the 1880s, when Heiberg brought out his definitive edition of the Elements, he expunged it as having no authority, noting simply that it occurs in some manuscripts:
Today, some modern mathematicians add a Sixth Postulate of their own, one which asserts the rigidity of the triangle. This justifies Euclid’s principle of superposition, which he uses to prove Proposition 1:4. In that proof, Euclid “applies” one triangle to another: that is, he translates and rotates one triangle so that it lies neatly on top of another triangle, demonstrating that the two are identical. In his translation, Richard Fitzpatrick comments:
The application of one figure to another should be counted as an additional postulate. (Fitzpatrick 11 fn)
And that’s a good place to stop.
