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Commentary on Proposition IV.5¶

Simson points out that Euclid does not prove that DF, EF will meet, and he inserts in the text the following argument to supply the omission.

“DF, EF produced meet one another. For, if they do not meet, they are parallel, wherefore AB, AC, which are at right angles to them, are parallel [or, he should have added, in a straight line]: which is absurd.”

This assumes, of course, that straight lines which are at right angles to two parallels are themselves parallel; but this is an obvious deduction from I. 28.

On the assumption that DF, EF will meet Todhunter has this note: “It has been proposed to show this in the following way: join DE; then the angles EDF and DEF are together less than the angles ADF and AEF, that is, they are together less than two right angles; and therefore DF and EF will meet, by Axiom 12 [Post. 5]. This assumes that ADE and AED are acute angles; it may, however, be easily shown that DE is parallel to BC, so that the triangle ADE is equiangular to the triangle ABC; and we must therefore select the two sides AB and AC such that ABC and ACB may be acute angles.”

This is, however, unsatisfactory. Euclid makes no such selection in III. 9 and III. 10, where the same assumption is tacitly made; and it is unnecessary, because it is easy to prove that the straight lines DF, EF meet in all cases, by considering the different possibilities separately and drawing a separate figure for each case.

Simson thinks that Euclid’s demonstration had been spoiled by some unskilful hand both because of the omission to prove that the perpendicular bisectors meet, and because “without any reason he divides the proposition into three cases, whereas one and the same construction and demonstration serves for them all, as Campanus has observed.” However, up to the usual words $\ὅ\pi\varepsilon\rho\ \ἔ\delta\varepsilon\iota\ \pi\omicron\iota\tilde{\eta}\sigma\alpha\iota$ there seems to be no doubt about the text. Heiberg suggests that Euclid gave separately the case where F falls on BC because, in that case, only AF needs to be drawn and not BF, CF as well.

The addition, though given in Simson and the text-books as a “corollary,” has no heading $\pi\acute{\omicron}\rho\iota\sigma\mu\alpha$ in the best MSS.; it is an explanation like that which is contained in the penultimate paragraph of III. 25.

The Greek text has a further addition, which is rejected by Heiberg as not genuine, “So that, further, when the given angle happens to be less than a right angle, DF, EF will fall within the triangle, when it is right, on BC, and, when it is greater than a right angle, outside BC. (being) what it was required to do.” Simson had already observed that the text here is vitiated “where mention is made of a given angle, though there neither is, nor can be, anything in the proposition relating to a given angle.”