Commentary on Proposition VI.18¶
Back to Proposition VI.18.
Simson thinks the proof of this proposition has been vitiated, his grounds for this view being (1) that it is demonstrated only with reference to quadrilaterals, and does not show how it may be extended to figures of five or more sides, (2) that Euclid infers, from the fact of two triangles being equiangular, that a side of the one is to the corresponding side of the other as another side of the first is to the side corresponding to it in the other, i.e. he permutes, without mentioning the fact that he does so, the proportions obtained in VI. 4, whereas the proof of the very next proposition gives, in a similar case, the intermediate step of permutation. I think this is hypercriticism. As regards (2) it should be noted that the permuted form of the proportion is arrived at first in the proof of VI. 4; and the omission of the intermediate step of alternando, whether accidental or not, is of no importance. On the other hand, the use of this form of the proportion certainly simplifies the proof of the proposition, since it makes unnecessary the subsequent ex aequali steps of Simson’s proof, their place being taken by the inference [v. 11] that ratios which are the same with a third ratio are the same with one another.
Nor is the first objection of any importance. We have only to take as the
given polygon a polygon of five sides at least, as CDEFG, join one extremity of CD, say D, to each of the angular points other than C and E, and then use the same mode of construction as Euclid’s for any number of successive triangles as ABL, LBK, etc., that may have to be made. Euclid’s construction and proof for a quadrilateral are quite sufficient to show how to deal with the case of a figure of five or any greater number of sides.
Clavius has a construction which, given the power of moving a figure
bodily from one position to any other, is easier. CDEFG being the given polygon, join CE, CF. Place AB on CD so that A falls on C, and let B fall on D', which may either lie on CD or on CD produced.
Now draw D'E' parallel to DE, meeting CE, produced if necessary, in E', E'F' parallel to EF, meeting CF, produced if necessary, in F', and so on.
Let the parallel to the last side but one, FG, meet CG, produced if necessary, in G'.
Then CD'E'F'G' is similar and similarly situated to CDEFG, and it is constructed on CD', a straight line equal to AB.
The proof of this is obvious.
A more general construction is indicated in the subjoined figure. If CDEFG be the given polygon, suppose its angular points all joined to any point O and the connecting straight lines produced both ways. Then, if C'D', a straight line equal to AB, be placed so that it is parallel to CD, and C', D' lie respectively on OC, OD (this can of course be done by finding fourth proportionals), we have only to draw D'E', E'F', etc., parallel to the corresponding sides of the original polygon in the manner shown.
De Morgan would rearrange Props. 18 and 20 in the following manner. He would combine Prop. 18 and the first part of Prop. 20 into one, with the enunciation :
Pairs of similar triangles, similarly put together, give similar figures ; and every pair of similar figures is composed of pairs of similar triangles similarly put together.
He would then make the problem of VI. 18 an application of the first part. In form this would certainly appear to be an improvement; but, provided that the relation of the propositions is understood, the matter of form is perhaps not of great importance.