figure and falling short by a parallelogrammic figure similar to a given one" is at once followed by the necessary condition of possibility: "Thus the given rectilineal figure must not be greater than that described on half the line and similar to the defect."

Tannery supposed that, in giving the other description of the Siopioμós as quoted above, Proclus, or rather his guide, was using the term incorrectly. The Siopiouós in the better known sense of the determination of limits or conditions of possibility was, we are told, invented by Leon. Pappus uses the word in this sense only. The other use of the term might, Tannery thought, be due to a confusion occasioned by the use of the same words (deî dn) in introducing the parts of a proposition corresponding to the two meanings of the word Siopio pós1. On the other hand it is to be observed that Eutocius distinguishes clearly between the two uses and implies that the difference was well known. The duopioμos in the sense of condition of possibility follows immediately on the enunciation, is even part of it; the Scopiouos in the other sense of course comes immediately after the setting-out.

Proclus has a useful observation respecting the conclusion of a proposition3. "The conclusion they are accustomed to make double in a certain way: I mean, by proving it in the given case and then drawing a general inference, passing, that is, from the partial conclusion to the general. For, inasmuch as they do not make use of the individuality of the subjects taken, but only draw an angle or a straight line with a'view to placing the datum before our eyes, they consider that this same fact which is established in the case of the particular figure constitutes a conclusion true of every other figure of the same kind. They pass accordingly to the general in order that we may not conceive the conclusion to be partial. And they are justified in so passing, since they use for the demonstration the particular things set out, not quâ particulars, but qua typical of the rest. For it is not in virtue of such and such a size attaching to the angle which is set out that I effect the bisection of it, but in virtue of its being rectilineal and nothing more. Such and such size is peculiar to the angle set out, but its quality of being rectilineal is common to all rectilineal angles. Suppose, for example, that the given angle is a right angle. If then I had employed in the proof the fact of its being right, I should not have been able to pass to every species of rectilineal angle; but, if I make no use of its being right, and only consider it as rectilineal, the argument will equally apply to rectilineal angles in general."

La Géométrie grecque, p. 149 note. Where dei on introduces the closer description of the problem we may translate, "it is then required" or "thus it is required" (to construct etc.): when it introduces the condition of possibility we may translate "thus it is necessary etc." Heiberg originally wrote deî dè in the latter sense in 1. 22 on the authority of Proclus and Eutocius, and against that of the MSS. Later, on the occasion of XI. 23, he observed that he should have followed the MSS. and written deî dǹ which he found to be, after all, the right reading in Eutocius (Apollonius, ed. Heiberg, 11. p. 178). deî dǹ is also the expression used by Diophantus for introducing conditions of possibility. Proclus, p. 207, 4—25.

* See the passage of Eutocius referred to in last note.

§ 6. OTHER TECHNICAL TERMS.

1. Things said to be given.

Proclus attaches to his description of the formal divisions of a proposition an explanation of the different senses in which the word given or datum (dedoμévov) is used in geometry. "Everything that is given is given in one or other of the following ways, in position, in ratio, in magnitude, or in species. The point is given in position only, but a line and the rest may be given in all the senses1."

The illustrations which Proclus gives of the four senses in which a thing may be given are not altogether happy, and, as regards things which are given in position, in magnitude, and in species, it is best, I think, to follow the definitions given by Euclid himself in his book of Data. Euclid does not mention the fourth class, things given in ratio, nor apparently do any of the great geometers.

(1) Given in position really needs no definition; and, when Euclid says (Data, Def. 4) that Points, lines and angles are said to be given in position which always occupy the same place, we are not really

the wiser.

(2) Given in magnitude is defined thus (Data, Def. 1): "Areas, lines and angles are called given in magnitude to which we can find equals." Proclus' illustration is in this case the following: when, he says, two unequal straight lines are given from the greater of which we have to cut off a straight line equal to the lesser, the straight lines are obviously given in magnitude, "for greater and less, and finite and infinite are predications peculiar to magnitude." But he does not explain that part of the implication of the term is that a thing is given in magnitude only, and that, for example, its position is not given and is a matter of indifference.

(3) Given in species. Euclid's definition (Data, Def. 3) is: "Rectilineal figures are said to be given in species in which the angles are severally given and the ratios of the sides to one another are given." And this is the recognised use of the term (cf. Pappus, passim). Proclus uses the term in a much wider sense for which I am not aware of any authority. Thus, he says, when we speak of (bisecting) a given rectilineal angle, the angle is given in species by the word rectilineal, which prevents our attempting, by the same method, to bisect a curvilineal angle! On Eucl. 1. 9, to which he here refers, he says that an angle is given in species when e.g. we say that it is right or acute or obtuse or rectilineal or "mixed," but that the actual angle in the proposition is given in species only. As a matter of fact, we should say that the actual angle in the figure of the proposition is given in magnitude and not in species, part of the implication of given in species being that the actual magnitude of the thing given in species is indifferent; an angle cannot be given in species in this sense at all. The confusion in Proclus' mind is shown when, after saying that a right angle is given in species, he describes a third of a right angle as given in magnitude.

1 Proclus, p. 205, 13—15.

No better example of what is meant by given in species, in its proper sense, as limited to rectilineal figures, can be quoted than the given parallelogram in Eucl. VI. 28, to which the required parallelogram has to be made similar; the former parallelogram is in fact given in species, though its actual size, or scale, is indifferent.

(4) Given in ratio presumably means something which is given by means of its ratio to some other given thing. This we gather from Proclus' remark (in his note on I. 9) that an angle may be given in ratio "as when we say that it is double and treble of such and such an angle or, generally, greater and less." The term, however, appears to have no authority and to serve no purpose. Proclus may have derived it from such expressions as "in a given ratio" which are common enough.

2. Lemma.

"The term lemma," says Proclus1, "is often used of any proposition which is assumed for the construction of something else: thus it is a common remark that a proof has been made out of such and such lemmas. But the special meaning of lemma in geometry is a proposition requiring confirmation. For when, in either construction or demonstration, we assume anything which has not been proved but requires argument, then, because we regard what has been assumed as doubtful in itself and therefore worthy of investigation, we call it a lemma, differing as it does from the postulate and the axiom in being matter of demonstration, whereas they are immediately taken for granted, without demonstration, for the purpose of confirming other things. Now in the discovery of lemmas the best aid is a mental aptitude for it. For we may see many who are quick at solutions and yet do not work by method; thus Cratistus in our time was able to obtain the required result from first principles, and those the fewest possible, but it was his natural gift which helped him to the discovery.

1 Proclus, pp. 211, I-212, 4.

2 It would appear, says Tannery (p. 151 n.), that Geminus understood a lemma as being simply außarbuevov, something assumed (cf. the passage of Proclus, p. 73, 4, relating to Menaechmus' view of elements): hence we cannot consider ourselves authorised in attributing to Geminus the more technical definition of the term here given by Proclus, according to which it is only used of propositions not proved beforehand. This view of a lemma must be considered as relatively modern. It seems to have had its origin in an imperfection of method. In the course of a demonstration it was necessary to assume a proposition which required proof, but the proof of which would, if inserted in the particular place, break the thread of the demonstration: hence it was necessary either to prove it beforehand as a preliminary proposition or to postpone it to be proved afterwards (ὡς ἑξῆς δειχθήσεται). When, after the time of Geminus, the progress of original discovery in geometry was arrested, geometers occupied themselves with the study and elucidation of the works of the great mathematicians who had preceded them. This involved the investigation of propositions explicitly quoted or tacitly assumed in the great classical treatises; and naturally it was found that several such remained to be demonstrated, either because the authors had omitted them as being easy enough to be left to the reader himself to prove, or because books in which they were proved had been lost in the meantime. Hence arose a class of complementary or auxiliary propositions which were called lemmas. Thus Pappus gives in his Book VII a collection of lemmas in elucidation of the treatises of Euclid and Apollonius included in the so-called "Treasury of Analysis" (Tówos dvaλvóμeros). When Proclus goes on to distinguish three methods of discovering lemmas, analysis, division, and reductio ad absurdum, he seems to imply that the principal business of contemporary geometers was the investigation of these auxiliary propositions.

Nevertheless certain methods have been handed down.

The finest is

the method which by means of analysis carries the thing sought up to an acknowledged principle, a method which Plato, as they say, communicated to Leodamas1, and by which the latter, too, is said to have discovered many things in geometry. The second is the method of division', which divides into its parts the genus proposed for consideration and gives a starting-point for the demonstration by means of the elimination of the other elements in the construction of what is proposed, which method also Plato extolled as being of assistance to all sciences. The third is that by means of the reductio ad absurdum, which does not show what is sought directly, but refutes its opposite and discovers the truth incidentally."

"The case (TTσis)," Proclus proceeds", "announces different ways of construction and alteration of positions due to the transposition of points or lines or planes or solids. And, in general, all its varieties are seen in the figure, and this is why it is called case, being a transposition in the construction."

4. Porism.

"The term porism is used also of certain problems such as the Porisms written by Euclid. But it is specially used when from what has been demonstrated some other theorem is revealed at the same time without our propounding it, which theorem has on this very account been called a porism (corollary) as being a sort of incidental gain arising from the scientific demonstration"." Cf. the note on I. 15.

1 This passage and another from Diogenes Laertius (III. 24, p. 74 ed. Cobet) to the effect that "He [Plato] explained (elonyhoaro) to Leodamas of Thasos the method of inquiry by analysis" have been commonly understood as ascribing to Plato the invention of the method of analysis; but Tannery points out forcibly (pp. 112, 113) how difficult it is to explain in what Plato's discovery could have consisted if analysis be taken in the sense attributed to it in Pappus, where we can see no more than a series of successive reductions of a problem until it is finally reduced to a known problem. On the other hand, Proclus' words about carrying up the thing sought to "an acknowledged principle" suggest that what he had in mind was the process described at the end of Book vi of the Republic by which the dialectician (unlike the mathematician) uses hypotheses as stepping-stones up to a principle which is not hypothetical, and then is able to descend step by step verifying every one of the hypotheses by which he ascended. This description does not of course refer to mathematical analysis, but it may have given rise to the idea that analysis was Plato's discovery, since analysis and synthesis following each other are related in the same way as the upward and the downward progression in the dialectician's intellectual method. And it may be that Plato's achievement was to observe the importance, from the point of view of logical rigour, of the confirmatory synthesis following analysis, and to regularise in this way and elevate into a completely irrefragable method the partial and uncertain analysis upon which the works of his predecessors depended.

Here again the successive bipartitions of genera into species such as we find in the Sophist and Republic have very little to say to geometry, and the very fact that they are here mentioned side by side with analysis suggests that Proclus confused the latter with the philosophical method of Rep. VI.

3 Tannery rightly remarks (p. 152) that the subdivision of a theorem or problem into several cases is foreign to the really classic form; the ancients preferred, where necessary, to multiply enunciations. As, however, some omissions necessarily occurred, the writers of lemmas naturally added separate cases, which in some instances found their way into the text. A good example is Euclid 1. 7, the second case of which, as it appears in our text-books, was interpolated. On the commentary of Proclus on this proposition Th. Taylor rightly remarks that "Euclid everywhere avoids a multitude of cases.'

4 Proclus, p. 212, 5—11.

• Tannery notes however that, so far from distinguishing his corollaries from the con

5. Objection.

"The objection (ĕvoraσis) obstructs the whole course of the argument by appearing as an obstacle (or crying 'halt,' aπavrŵσa) either to the construction or to the demonstration. There is this difference between the objection and the case, that, whereas he who propounds. the case has to prove the proposition to be true of it, he who makes the objection does not need to prove anything: on the contrary it is necessary to destroy the objection and to show that its author is saying what is false1."

That is, in general the objection endeavours to make it appear that the demonstration is not true in every case; and it is then necessary to prove, in refutation of the objection, either that the supposed case is impossible, or that the demonstration is true even for that case. A good instance is afforded by Eucl. I. 7. The text books give a second case which is not in the original text of Euclid. Proclus remarks on the proposition as given by Euclid that the objection may conceivably be raised that what Euclid declares to be impossible may after all be possible in the event of one pair of straight lines falling completely within the other pair. Proclus then refutes the objection by proving the impossibility in that case also. His proof then came to be given in the text-books as part of Euclid's proposition.

The objection is one of the technical terms in Aristotle's logic and its nature is explained in the Prior Analytics". "An objection is a proposition contrary to a proposition.... Objections are of two sorts, general or partial.... For when it is maintained that an attribute belongs to every (member of a class), we object either that it belongs to none (of the class) or that there is some one (member of the class) to which it does not belong."

6. Reduction.

This is again an Aristotelian term, explained in the Prior Analytics. It is well described by Proclus in the following passage:

"Reduction (amaуwyn) is a transition from one problem or theorem to another, the solution or proof of which makes that which is propounded manifest also. For example, after the doubling of the cube had been investigated, they transformed the investigation into another upon which it follows, namely the finding of the two means; and from that time forward they inquired how between two given straight lines two mean proportionals could be discovered. And they say that the first to effect the reduction of difficult constructions was Hippocrates of Chios, who also squared a lune and discovered many other things in geometry, being second to none in ingenuity as regards constructions."

clusions of his propositions, Euclid inserts them before the closing words "(being) what it was required to do" or "to prove." In fact the porism-corollary is with Euclid rather a modified form of the regular conclusion than a separate proposition.

Proclus, p. 212, 18-23.

2 Anal. prior. II. 26, 69 a 37.

3 ibid. 11. 25, 69 a 20.

Proclus, pp. 212, 24-213, 11. This passage has frequently been taken as crediting Hippocrates with the discovery of the method of geometrical reduction: cf. Taylor (Translation of Proclus, 11. p. 26), Allman (p. 41 n., 59), Gow (pp. 169, 170). As Tannery remarks (p. 110), if the particular reduction of the duplication problem to that of the two means is