Commentary on Proposition I.4¶
Back to Proposition I.4.
else, e.g. of a pedestal or column; but when, as in I. 5, two triangles were compared occupying other than the normal positions which gave rise to the name, and when two sides had been previously mentioned, the base was, as Proclus says, necessarily the third side.
In the note on Common Notion 4 I have already mentioned that Euclid obviously used the method of superposition with reluctance, and I have given, after Veronese for the most part, the reason for holding that that method is not admissible as a theoretical means of proving equality, although it may be of use as a practical test, and may thus furnish an empirical basis on which to found a postulate. Mr Bertrand Russell observes (Principles of Mathematics I. p. 405) that Euclid would have done better to assume I. 4 as an axiom, as is practically done by Hilbert (Grundlagen der Geometrie, p. 9). It may be that Euclid himself was as well aware of the objections to the method as are his modern critics; but at all events those objections were stated, with almost equal clearness, as early as the middle of the 16th century. Peletarius (Jacques Peletier) has a long note on this proposition (In Euclidis Elementa geometrica demonstrationum libri sex, 1557), in which he observes that, if superposition of lines and figures could be assumed as a method of proof, the whole of geometry would be full of such proofs, that it could equally well have been used in I. 2, 3 (thus in I. 2 we could simply have supposed the line taken up and placed at the point), and that in short it is obvious how far removed the method is from the dignity of geometry. The theorem, he adds, is obvious in itself and does not require proof; although it is introduced as a theorem, it would seem that Euclid intended it rather as a definition than a theorem, “for I cannot think that two angles are equal unless I have a conception of what equality of angles is.” Why then did Euclid include the proposition among theorems, instead of placing it among the axioms? Peletarius makes the best excuse he can, but concludes thus: “Huius itaque propositionis veritatem non aliunde quam a communi iudicio petemus; cogitabimusque figuras figuris superponere, Mechanicum quippiam esse: intelligere verò, id demum esse Mathematicum.”
Expressed in terms of the modern systems of Congruence-Axioms referred to in the note on Common Notion 4, what Euclid really assumes amounts to the following:
On the line DE, there is a point E, on either side of D, such that AB is equal to DE.
On either side of the ray DE there is a ray DF such that the angle EDF is equal to the angle BAC.
It now follows that on DF there is a point F such that DF is equal to AC.
And lastly (3), we require an axiom from which to infer that the two remaining angles of the triangles are respectively equal and that the bases are equal.
I have shown above (pp. 229—230) that Hilbert has an axiom stating the equality of the remaining angles simply, but proves the equality of the bases.
Another alternative is that of Pasch (Vorlesungen über neuere Geometrie, p. 109) who has the following “Grundsatz”:
If two figures AB and FGH are given (FGH not being contained in a straight length), and AB, FG are congruent, and if a plane surface be laid through A and B, we can specify in this plane surface, produced if necessary, two points C, D, neither more nor less, such that the figures ABC and ABD are congruent with the figure FGH, and the straight line CD has with the straight line AB or with AB produced one point common.
I pass to two points of detail in Euclid’s proof:
The inference that, since B coincides with E, and C with F, the bases of the triangles are wholly coincident rests, as expressly stated, on the impossibility of two straight lines enclosing a space, and therefore presents no difficulty.
But (2) most editors seem to have failed to observe that at the very beginning of the proof a much more serious assumption is made without any explanation whatever, namely that, if A be placed on D, and AB on DE, the point B will coincide with E, because AB is equal to DE. That is, the converse of Common Notion 4 is assumed for straight lines. Proclus merely observes, with regard to the converse of this Common Notion, that it is only true in the case of things “of the same form” (ὁμοειδῆ), which he explains as meaning straight lines, arcs of one and the same circle, and angles “contained by lines similar and similarly situated” (p. 241, 3—8).
Savile however saw the difficulty and grappled with it in his note on the Common Notion. After stating that all straight lines with two points common are congruent between them (for otherwise two straight lines would enclose a space), he argues thus. Let there be two straight lines AB, DE, and let A be placed on D, and AB on DE. Then B will coincide with E. For, if not, let B fall somewhere short of E or beyond E; and in either case it will follow that the less is equal to the greater, which is impossible.
Savile seems to assume (and so apparently does Lardner who gives the same proof) that, if the straight lines be “applied,” B will fall somewhere on DE or DE produced. But the ground for this assumption should surely be stated; and it seems to me that it is necessary to use, not Postulate 1 alone, nor Postulate 2 alone, but both, for this purpose (in other words to assume, not only that two straight lines cannot enclose a space, but also that two straight lines cannot have a common segment). For the only safe course is to place A upon D and then turn AB about D until some point on AB intermediate between A and B coincides with some point on DE. In this position AB and DE have two points common. Then Postulate 1 enables us to infer that the straight lines coincide between the two common points, and Postulate 2 that they coincide beyond the second common point towards B and E. Thus the straight lines coincide throughout so far as both extend; and Savile’s argument then proves that B coincides with E.