viz. common notions (Kowai ěvvoiai), and there is no reason to suppose it to be a substitution for the original term due to the Stoics: cf. Proclus' remark that, according to Aristotle and the geometers, axiom and common notion are the same thing'.

Aristotle discusses the indemonstrable character of the axioms in the Metaphysics. Since "all the demonstrative sciences use the axioms," the question arises, to what science does their discussion belong? The answer is that, like that of Being (ovoía), it is the province of the (first) philosopher. It is impossible that there should be demonstration of everything, as there would be an infinite series of demonstrations: if the axioms were the subject of a demonstrative science, there would have to be here too, as in other demonstrative sciences, a subject-genus, its attributes and corresponding axioms; thus there would be axioms behind axioms, and so on continually. The axiom is the most firmly established of all principles. It is ignorance alone that could lead any one to try to prove the axioms'; the supposed proof would be a petitio principii. If it is admitted that not everything can be proved, no one can point to any principle more truly indemonstrable. If any one thought he could prove them, he could at once be refuted; if he did not attempt to say anything, it would be ridiculous to argue with him: he would be no better than a vegetable. The first condition of the possibility of any argument whatever is that words should signify something both to the speaker and to the hearer: without this there can be no reasoning with any one. And, if any one admits that words can mean anything to both hearer and speaker, he admits that something can be true without demonstration. And so on".

It was necessary to give some sketch of Aristotle's view of the first principles, if only in connexion with Proclus' account, which is as follows. As in the case of other sciences, so "the compiler of elements in geometry must give separately the principles of the science, and after that the conclusions from those principles, not giving any account of the principles but only of their consequences. No science proves its own principles, or even discourses about them: they are treated as self-evident.... Thus the first essential was to distinguish the principles from their consequences. Euclid carries out this plan practically in every book and, as a preliminary to the whole enquiry, sets out the common principles of this science. Then he divides the common principles themselves into hypotheses, postulates, and axioms. For all these are different from one another: an axiom, a postulate and a hypothesis are not the same thing, as the inspired Aristotle somewhere says. But, whenever that which is assumed and ranked as a principle is both known to the learner and convincing in itself, such a thing is an axiom, e.g. the statement that things which are equal to the same thing are also equal to one another. When, on

1 Proclus, p. 194, 8. 3 ibid. 996 b 26.

• ibid. 1005 b 11—17. 9 ibid. 1006 a 10.

• Metaph. 997 a 10.
▲ ibid. 1005 a 21-b 11.
7 ibid. 1006 a 5.
10 ibid. 1006 a 11-15.

ibid. 997 a 5-8.

8 ibid. 1006 a 17.
11 ibid. 1006 a 18 sqq.

the other hand, the pupil has not the notion of what is told him which carries conviction in itself, but nevertheless lays it down and assents to its being assumed, such an assumption is a hypothesis. Thus we do not preconceive by virtue of a common notion, and without being taught, that the circle is such and such a figure, but, when we are told so, we assent without demonstration. When again what is asserted is both unknown and assumed even without the assent of the learner, then, he says, we call this a postulate, e.g. that all right angles are equal. This view of a postulate is clearly implied by those who have made a special and systematic attempt to show, with regard to one of the postulates, that it cannot be assented to by any one straight off. According then to the teaching of Aristotle, an axiom, a postulate and a hypothesis are thus distinguished1."

We observe, first, that Proclus in this passage confuses hypotheses and definitions, although Aristotle had made the distinction quite plain. The confusion may be due to his having in his mind a passage of Plato from which he evidently got the phrase about "not giving an account of" the principles. The passage is: "I think you know that those who treat of geometries and calculations (arithmetic) and such things take for granted (vπoléμevo) odd and even, figures, angles of three kinds, and other things akin to these in each subject, implying that they know these things, and, though using them as hypotheses, do not even condescend to give any account of them either to themselves or to others, but begin from these things and then go through everything else in order, arriving ultimately, by recognised methods, at the conclusion which they started in search of." But the hypothesis is here the assumption, e.g. 'that there may be such a thing as length without breadth, henceforward called a line',' and so on, without any attempt to show that there is such a thing; it is mentioned in connexion with the distinction between Plato's superior' and 'inferior' intellectual method, the former of which uses successive hypotheses as stepping-stones by which it mounts upwards to the idea of Good.

We pass now to Proclus' account of the difference between postulates and axioms. He begins with the view of Geminus, according to which "they differ from one another in the same way as theorems are also distinguished from problems. For, as in theorems we propose to see and determine what follows on the premisses, while in problems we are told to find and do something, in like manner in the axioms such things are assumed as are manifest of themselves and easily apprehended by our untaught notions, while in the postulates we assume such things as are easy to find and effect (our understanding suffering no strain in their assumption), and we require no complication of machinery."..." Both must have the characteristic of being simple

1 Proclus, pp. 75, 10-77, 2.

Republic, VI. 510 C. Cf. Aristotle, Nic. Eth, 1151 a 17.

H. Jackson, Journal of Philology, vol. x. p. 144.

Proclus, pp. 178, 12-179, 8. In illustration Proclus contrasts the drawing of a straight line or a circle with the drawing of a "single-turn spiral" or of an equilateral triangle, the

and readily grasped, I mean both the postulate and the axiom; but the postulate bids us contrive and find some subject-matter (un) to exhibit a property simple and easily grasped, while the axiom bids us assert some essential attribute which is self-evident to the learner, just as is the fact that fire is hot, or any of the most obvious things'." Again, says Proclus, "some claim that all these things are alike postulates, in the same way as some maintain that all things that are sought are problems. For Archimedes begins his first book on Inequilibrium2 with the remark 'I postulate that equal weights at equal distances are in equilibrium,' though one would rather call this an axiom. Others call them all axioms in the same way as some regard as theorems everything that requires demonstration"."

"Others again will say that postulates are peculiar to geometrical subject-matter, while axioms are common to all investigation which is concerned with quantity and magnitude. Thus it is the geometer who knows that all right angles are equal and how to produce in a straight line any limited straight line, whereas it is a common notion that things which are equal to the same thing are also equal to one another, and it is employed by the arithmetician and any scientific person who adapts the general statement to his own subject."

The third view of the distinction between a postulate and an axiom is that of Aristotle above described.

The difficulties in the way of reconciling Euclid's classification of postulates and axioms with any one of the three alternative views are next dwelt upon. If we accept the first view according to which an axiom has reference to something known, and a postulate to something done, then the 4th postulate (that all right angles are equal) is not a postulate; neither is the 5th which states that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles. On the second view, the assumption that two straight lines cannot enclose a space, "which even now," says Proclus, "some add as an axiom," and which is peculiar to the subject-matter of geometry, like the fact that all right angles are equal, is not an axiom. According to the third (Aristotelian) view, "everything which is confirmed (Tiσтоûтai) by a sort of demonstration spiral requiring more complex machinery and even the equilateral triangle needing a certain method."For the geometrical intelligence will say that by conceiving a straight line fixed at one end but, as regards the other end, moving round the fixed end, and a point moving along the straight line from the fixed end, I have described the single-turn spiral; for the end of the straight line describing a circle, and the point moving on the straight line simultaneously, when they arrive and meet at the same point, complete such a spiral. And again, if I draw equal circles, join their common point to the centres of the circles and draw a straight line from one of the centres to the other, I shall have the equilateral triangle. These things then are far from being completed by means of a single act or of a moment's thought" (p. 180, 8—21).

1 Proclus, p. 181, 4—11.

It is necessary to coin a word to render aviopporŵr, which is moreover in the plural. The title of the treatise as we have it is Equilibria of planes or centres of gravity of planes in Book I and Equilibria of planes in Book II.

Proclus, p. 181, 16—23.

▲ ibid. p. 182, 6—14.

* Pp. 118, 119.

will be a postulate, and what is incapable of proof will be an axiom1." This last statement of Proclus is loose, as regards the axiom, because it omits Aristotle's requirement that the axiom should be a selfevident truth, and one that must be admitted by any one who is to learn anything at all, and, as regards the postulate, because Aristotle calls a postulate something assumed without proof though it is "matter of demonstration" (aπоdeixтòv ov), but says nothing of a quasi-demonstration of the postulates. On the whole I think it is from Aristotle that we get the best idea of what Euclid understood by a postulate and an axiom or common notion. Thus Aristotle's account of an axiom as a principle common to all sciences, which is self-evident, though incapable of proof, agrees sufficiently with the contents of Euclid's common notions as reduced to five in the most recent text (not omitting the fourth, that "things which coincide are equal to one another "). As regards the postulates, it must be borne in mind that Aristotle says elsewhere' that, "other things being equal, that proof is the better which proceeds from the fewer postulates or hypotheses or propositions." If then we say that a geometer must lay down as principles, first certain axioms or common notions, and then an irreducible minimum of postulates in the Aristotelian sense concerned only with the subject-matter of geometry, we are not far from describing what Euclid in fact does. As regards the postulates we may imagine him saying: "Besides the common notions there are a few other things which I must assume without proof, but which differ from the common notions in that they are not self-evident. The learner may or may not be disposed to agree to them; but he must accept them at the outset on the superior authority of his teacher, and must be left to convince himself of their truth in- the course of the investigation which follows. In the first place certain simple constructions, the drawing and producing of a straight line, and the drawing of a circle, must be assumed to be possible, and with the constructions the existence of such things as straight lines and circles; and besides this we must lay down some postulate to form the basis of the theory of parallels." It is true that the admission of the 4th postulate that all right angles are equal still presents a difficulty to which we shall have to recur.

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"

§ 4 THEOREMS AND PROBLEMS.

Again the deductions from the first principles," says Proclus, "are divided into problems and theorems, the former embracing the

1 Proclus, pp. 182, 21-183, 13.
3 Cf. Lardner's Euclid: also Todhunter.

2 Anal. post. I. 25, 86 a 33-35.

generation, division, subtraction or addition of figures, and generally the changes which are brought about in them, the latter exhibiting the essential attributes of each1."

"Now, of the ancients, some, like Speusippus and Amphinomus, thought proper to call them all theorems, regarding the name of theorems as more appropriate than that of problems to theoretic sciences, especially as these deal with eternal objects. For there is no becoming in things eternal, so that neither could the problem have any place with them, since it promises the generation and making of what has not before existed, e.g. the construction of an equilateral triangle, or the describing of a square on a given straight line, or the placing of a straight line at a given point. Hence they say it is better to assert that all (propositions) are of the same kind, and that we regard the generation that takes place in them as referring not to actual making but to knowledge, when we treat things existing eternally as if they were subject to becoming: in other words, we may say that everything is treated by way of theorem and not by way of problem? (πάντα θεωρηματικῶς ἀλλ ̓ οὐ προβληματικώς λαμβάνεσθαι).

"Others on the contrary, like the mathematicians of the school of Menaechmus, thought it right to call them all problems, describing their purpose as twofold, namely in some cases to furnish (TOρíoaobai) the thing sought, in others to take a determinate object and see either what it is, or of what nature, or what is its property, or in what relations it stands to something else.

"In reality both assertions are correct. Speusippus is right because the problems of geometry are not like those of mechanics, the latter being matters of sense and exhibiting becoming and change of every sort. The school of Menaechmus are right also because the discoveries even of theorems do not arise without an issuing-forth into matter, by which I mean intelligible matter. Thus forms going out into matter and giving it shape may fairly be said to be like processes of becoming. For we say that the motion of our thought and the throwing-out of the forms in it is what produces the figures in the imagination and the conditions subsisting in them. It is in the imagination that constructions, divisions, placings, applications, additions and subtractions (take place), but everything in the mind is fixed and immune from becoming and from every sort of changes."

"Now those who distinguish the theorem from the problem say that every problem implies the possibility, not only of that which is predicated of its subject-matter, but also of its opposite, whereas every theorem implies the possibility of the thing predicated but not of its opposite as well. By the subject-matter I mean the genus which is the subject of inquiry, for example, a triangle or a square or a circle, and by the property predicated the essential attribute, as equality, section, position, and the like. When then any one

1 Proclus, p. 77, 7-12.

• ibid. pp. 78, 8—79, 2.

2 ibid. pp. 77, 15-78, 8.