Commentary on Proposition VI.32¶
Back to Proposition VI.32.
It has often been pointed out (e.g. by Clavius, Lardner and Todhunter) that the enunciation of this proposition is not precise enough. Suppose that ABC is a triangle. From C draw CD parallel to BA and of any length. From D draw DE parallel to CA and of such length that
$$CD : DE = BA : AC.$$
Then the triangles ABC, ECD, which have the angular point C common literally satisfy Euclid’s enunciation; but by no possibility can CE be in a straight line with CB if, as in the case supposed, the angles included by the corresponding sides are supplementary (unless both are right angles). Hence the included angles must be equal, so that the triangles must be similar. That being so, if they are to have nothing more than one angular point common, and two pairs of corresponding sides are to be parallel as distinguished from one or both being in the same straight line, the triangles can only be placed so that the corresponding sides in both are on the same side of the third side of either, and the sides (other than the third sides) which meet at the common angular point are not corresponding sides.
Todhunter remarks that the proposition seems of no use. Presumably he did not know that it is used by Euclid himself in XIII. 17. This is so however, and therefore it was not necessary, as several writers have thought, to do away with the proposition and find a substitute which should be more useful.
De Morgan proposes this theorem: “If two similar triangles be placed with their bases parallel, and the equal angles at the bases towards the same parts, the other sides are parallel, each to each; or one pair of sides are in the same straight line and the other pair are parallel.”
Dr Lachlan substitutes the somewhat similar theorem, “If two similar triangles be placed so that two sides of the one are parallel to the corresponding sides of the other, the third sides are parallel.”
But it is to be observed that these propositions can be proved without using Book VI. at all; they can be proved from Book I., and the triangles may as well be called “equiangular” simply. It is true that Book VI. is no more than formally necessary to Euclid’s proposition. He merely uses VI. 6 because his enunciation does not say that the triangles are similar; and he only proves them to be similar in order to conclude that they are equiangular. From this point of view Mr Taylor’s substitute seems the best, viz.
“If two triangles have sides parallel in pairs, the straight lines joining the corresponding vertices meet in a point, or are parallel.”
Simson has a theory (unnecessary in the circumstances) as to the possible object of VI. 32 as it stands. He points out that the enunciation of VI. 26 might be more general so as to cover the case of similar and similarly situated parallelograms with equal angles not coincident but vertically opposite. It can then be proved that the diagonals drawn through the common angular point are in one straight line. If ABCF, CDEG be similar and similarly situated parallelograms, so that BCG, DCF are straight lines, and if the diagonals AC, CE be drawn, the triangles ABC, CDE are similar and are placed exactly as described in VI. 32, so that AC, CE are in a straight line. Hence Simson suggests that there may have been, in addition to the indirect demonstration in VI. 26, a direct proof covering the case just given which may have used the result of VI. 32. I think however that the place given to the latter proposition in Book VI. is against this view.