Suppose now that DA standing on BAC makes the two angles DAB, DAC equal, so that each is a right angle by the definition.

Similarly, let LH form with the straight line FHM the right angles LHF,

LHM.

Let DA, HL be equal; and suppose the whole of the second figure so laid upon the first that the point H falls on A, and L on D.

Then the straight line FHM will (by the third lemma) not touch the straight line BC at A; it will either

(a) coincide exactly with BC, or

(b) cut it so that one of its extremities, as F, will fall above [BC] and the other, M, below it.

If the alternative (a) is true, we have already proved the exact equality of all rectilineal right angles.

Under alternative (b) we prove that the angle LHF, being equal to the angle DAF, is less than the angle DAB or DAC, and a fortiori less than the angle DAM or LHM: which is contrary to the hypothesis.

[Hence (a) is the only possible alternative, so that all right angles are equal.]

Saccheri adds that it makes no difference if the angle DAF diverges infinitely little from the angle DAB. This would equally lead to a conclusion contradicting the hypothesis.

A

It will be observed that Saccheri speaks of "the exact equality of all rectilineal right angles." He may have had in mind the remark of Pappus, quoted by Proclus (p. 189, 11), that the converse of this postulate, namely that an angle which is equal to a right angle is also right, is not necessarily true, unless the former angle is rectilineal. Suppose two equal straight lines BA, BC at right angles to one another, and semi-circles described on BA, BC respectively as AEB, BDC in the figure. Then, since the semi-circles are equal, they coincide if applied to one another. Hence the "angles" EBA, DBC are equal. Add to each the "angle"

E

B

ABD; and it follows that the lunular angle EBD is equal to the right angle ABC. (Similarly, if BA, BC be inclined at an acute or obtuse angle, instead of at a fight angle, we find a lunular angle equal to an acute or obtuse angle.) This is one of the curiosities which Greek commentators delighted in.

Veronese, Ingrami, and Enriques and Amaldi deduce the fact that all right angles are equal from the equivalent fact that all flat angles are equal, which is either itself assumed as a postulate or immediately deduced from some other postulate.

Hilbert takes quite a different line. He considers that Euclid did wrong in placing Post. 4 among "axioms." He himself, after his Group III. of Axioms containing six relating to congruence, proves several theorems about the congruence of triangles and angles, and then deduces our Postulate.

As to the raison d'être and the place of Post. 4 one thing is quite certain. It was essential from Euclid's point of view that it should come before Post. 5, since the condition in the latter that a certain pair of angles are together less than two right angles would be useless unless it were first made clear that right angles are angles of determinate and invariable magnitude.

POSTULATE 5.

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.

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Although Aristotle gives a clear idea of what he understood by a postulate, he does not give any instances from geometry; still less has he any allusion recalling the particular postulates found in Euclid. We naturally infer that the formulation of these postulates was Euclid's own work. There is a more positive indication of the originality of Postulate 5, since in the passage (Anal. prior. 11. 16, 65 a 4) quoted above in the note on the definition of parallels he alludes to some petitio principii involved in the theory of parallels current in his time. This reproach was removed by Euclid when he laid down this epoch-making Postulate. When we consider the countless successive attempts made through more than twenty centuries to prove the Postulate, many of them by geometers of ability, we cannot but admire the genius of the man who concluded that such a hypothesis, which he found necessary to the validity of his whole system of geometry, was really indemonstrable.

From the very beginning, as we know from Proclus, the Postulate was attacked as such, and attempts were made to prove it as a theorem or to get rid of it by adopting some other definition of parallels; while in modern times the literature of the subject is enormous. Riccardi (Saggio di una bibliografia Euclidea, Part IV., Bologna, 1890) has twenty quarto pages of titles of monographs relating to Post. 5 between the dates 1607 and 1887. Max Simon (Ueber die Entwicklung der Elementar-geometrie im XIX. Jahrhundert, 1906) notes that he has seen three new attempts, as late as 1891 (a century after Gauss laid the foundation of non-Euclidean geometry), to prove the theory of parallels independently of the Postulate. Max Simon himself (pp. 53-61) gives a large number of references to books or articles on the subject and refers to the copious information, as to contents as well as names, contained in Schotten's Inhalt und Methode des planimetrischen Unterrichts, 11. pp. 183-332.

This note will include some account of or allusion to a few of the most noteworthy attempts to prove the Postulate. Only those of ancient times, as being less generally accessible, will be described at any length; shorter references must suffice in the case of the modern geometers who have made the most important contributions to the discussion of the Postulate and have thereby, in particular, contributed most towards the foundation of the nonEuclidean geometries, and here I shall make use principally of the valuable Article 6, Sulla teoria delle parallele e sulle geometrie non-euclidee (by Roberto Bonola), in Questioni riguardanti la geometria elementare (pp. 143-222).

Proclus (p. 191, 21 sqq.) states very clearly the nature of the first objections taken to the Postulate.

"This ought even to be struck out of the Postulates altogether; for it is a theorem involving many difficulties, which Ptolemy, in a certain book, set himself to solve, and it requires for the demonstration of it a number of definitions as well as theorems. And the converse of it is actually proved by Euclid himself as a theorem. It may be that some would be

deceived and would think it proper to place even the assumption in question among the postulates as affording, in the lessening of the two right angles, ground for an instantaneous belief that the straight lines converge and meet. To such as these Geminus correctly replied that we have learned from the very pioneers of this science not to have any regard to mere plausible imaginings when it is a question of the reasonings to be included in our geometrical doctrine. For Aristotle says that it is as justifiable to ask scientific proofs of a rhetorician as to accept mere plausibilities from a geometer; and Simmias is made by Plato to say that he recognises as quacks those who fashion for themselves proofs from probabilities. So in this case the fact that, when the right angles are lessened, the straight lines converge is true and necessary; but the statement that, since they converge more and more as they are produced, they will sometime meet is plausible but not necessary, in the absence of some argument showing that this is true in the case of straight lines. For the fact that some lines exist which approach indefinitely, but yet remain non-secant (ảσúμπтшτо), although it seems improbable and paradoxical, is nevertheless true and fully ascertained with regard to other species of lines. May not then the same thing be possible in the case of straight lines which happens in the case of the lines referred to? Indeed, until the statement in the Postulate is clinched by proof, the facts shown in the case of other lines may direct our imagination the opposite way. And, though the controversial arguments against the meeting of the straight lines should contain much that is surprising, is there not all the more reason why we should expel from our body of doctrine this merely plausible and unreasoned (hypothesis)?

"It is then clear from this that we must seek a proof of the present theorem, and that it is alien to the special character of postulates. But how it should be proved, and by what sort of arguments the objections taken to it should be removed, we must explain at the point where the writer of the Elements is actually about to recall it and use it as obvious. It will be necessary at that stage to show that its obvious character does not appear independently of proof, but is turned by proof into matter of knowledge."

Before passing to the attempts of Ptolemy and Proclus to prove the Postulate, I should note here that Simplicius says (in an-Nairīzi, ed. BesthornHeiberg, p. 119, ed. Curtze, p. 65) that this Postulate is by no means manifest, but requires proof, and accordingly "Abthiniathus" and Diodorus had already proved it by means of many different propositions, while Ptolemy also had explained and proved it, using for the purpose Eucl. 1. 13, 15 and 16 (or 18). The Diodorus here mentioned may be the author of the Analemma on which Pappus wrote a commentary. It is difficult even to frame a conjecture as to who "Abthiniathus" is. In one place in the Arabic text the name appears to be written "Anthisathus" (H. Suter in Zeitschrift für Math. und Physik, xxxvIII., hist. litt. Abth. p. 194). It has occurred to me whether he might be Peithon, a friend of Serenus of Antinoeia (Antinoupolis) who was long known as Serenus of Antissa. Serenus says (De sectione cylindri, ed. Heiberg, p. 96): "Peithon the geometer, explaining parallels in a work of his, was not satisfied with what Euclid said, but showed their nature more cleverly by an example; for he says that parallel straight lines are such a thing as we see on walls or on the ground in the shadows of pillars which are made when either a torch or a lamp is burning behind them. And, although this has only been matter of merriment to every one, I at least must not deride it, for the respect I have for the author, who is my friend." If Peithon was known as "of Antinoeia" or "of Antissa," the two forms of the mysterious name might perhaps be an attempt at an equivalent; but this is no more than a guess.

Simplicius adds in full and word for word the attempt of his "friend" or his "master Aganis" to prove the Postulate.

He

Proclus returns to the subject (p. 365, 5) in his note on Eucl. 1. 29. says that before his time a certain number of geometers had classed as a theorem this Euclidean postulate and thought it matter for proof, and he then proceeds to give an account of Ptolemy's argument.

Noteworthy attempts to prove the Postulate.

Ptolemy.

We learn from Proclus (p. 365, 7—11) that Ptolemy wrote a book on the proposition that "straight lines drawn from angles less than two right angles meet if produced,” and that he used in his "proof" many of the theorems in Euclid preceding 1. 29. Proclus excuses himself from reproducing the early part of Ptolemy's argument, only mentioning as one of the propositions proved in it the theorem of Eucl. 1. 28 that, if two straight lines meeting a transversal make the two interior angles on the same side equal to two right angles, the straight lines do not meet, however far produced.

I. From Proclus' note on 1. 28 (p. 362, 14 sq.) we know that Ptolemy proved this somewhat as follows.

Suppose that there are two straight lines AB, CD, and that EFGH, meeting them, makes the angles BFG, FGD equal to two right angles. I say that AB, CD are parallel, that is, they

are non-secant.

For, if possible, let FB, GD meet at K. Now, since the angles BFG, FGD are equal to two right angles, while the four angles AFG, BFG, FGD, FGC are together equal to four right angles,

the angles AFG, FGC are equal to two

right angles.

"If therefore FB, GD, when the interior angles are equal to two right angles, meet at K, the straight lines FA, GC will also meet if produced; for the angles AFG, CGF are also equal to two right angles.

"Therefore the straight lines will either meet in both directions or in neither direction, if the two pairs of interior angles are both equal to two right angles.

"Let, then, FA, GC meet at L.

"Therefore the straight lines LABK, LCDK enclose a space: which is impossible.

"Therefore it is not possible for two straight lines to meet when the interior angles are equal to two right angles. Therefore they are parallel."

[The argument in the words italicised would be clearer if it had been shown that the two interior angles on one side of EH are severally equal to the two interior angles on the other, namely BFG to CGF and FGD to AFG; whence, assuming FB, GD to meet in K, we can take the triangle KFG and place it (e.g. by rotating it in the plane about O the middle point of FG) so that FG falls where GF is in the figure and GD falls on FA, in which case FB must also fall on GC; hence, since FB, GD meet at K, GC and FA must meet at a corresponding point L. Or, as Mr Frankland does, we may substitute for FG a straight line MN through O the middle point of FG drawn perpendicular to one of the parallels, say AB. Then, since the two triangles OMF, ONG have two angles equal respectively, namely FOM to

GON (1. 15) and OFM to OGN, and one side OF equal to one side OG, the triangles are congruent, the angle ONG is a right angle, and MN is perpendicular to both AB and CD. Then, by the same method of application, MA, NC are shown to form with MN a triangle MALCN congruent with the triangle NDKBM, and MA, NC meet at a point L corresponding to K. Thus the two straight lines would meet at the two points K, L. This is what happens under the Riemann hypothesis, where the axiom that two straight lines cannot enclose a space does not hold, but all straight lines meeting in one point have another point common also, and e.g. in the particular figure just used K, L are points common to all perpendiculars to MN. If we suppose that K, L are not distinct points, but one point, the axiom that two straight lines cannot enclose a space is not contradicted.]

II. Ptolemy now tries to prove 1. 29 without using our Postulate, and then deduces the Postulate from it (Proclus, pp. 365, 14-367, 27).

The argument to prove 1. 29 is as follows.

The straight line which cuts the parallels must make the sum of the interior angles on the same side equal to, greater than, or less than, two right angles.

"Let AB, CD be parallel, and let FG meet them. I say (1) that FG does not make the interior angles on the same side greater than two right angles.

A

F

B

"For, if the angles AFG, CGF are greater than two right angles, the remaining angles BFG, DGF are less than two right angles.

"But the same two angles are also greater than two right angles; for AF, CG are no more parallel than FB, GD, so that, if the straight line falling on AF, CG makes the interior angles greater than two right angles, the straight line falling on FB, GD will also make the interior angles greater than two right angles.

"But the same angles are also less than two right angles; for the four angles AFG, CGF, BFG, DGF are equal to four right angles : which is impossible.

"Similarly (2) we can show that the straight line falling on the parallels does not make the interior angles on the same side less than two right angles. "But (3), if it makes them neither greater nor less than two right angles, it can only make the interior angles on the same side equal to two right angles."

III. Ptolemy deduces Post. 5 thus:

Suppose that the straight lines making angles with a transversal less than two right angles do not meet on the side on which those angles are.

Then, a fortiori, they will not meet on the other side on which are the angles greater than two right angles.

Hence the straight lines will not meet in either direction; they are therefore parallel.

But, if so, the angles made by them with the transversal are equal to two right angles, by the preceding proposition (= 1. 29).

Therefore the same angles will be both equal to and less than two right angles:

which is impossible.

Hence the straight lines will meet.