Commentary on Proposition VI.19¶
Back to Proposition VI.19.
Having combined Prop. 18 and the first part of Prop. 20 as just indicated, De Morgan would tack on to Prop. 19 the second part of Prop. 20, which asserts that, if similar polygons be divided into the same number of similar triangles, the triangles are “homologous to the wholes” (in the sense that the polygons have the same ratio as the corresponding triangles have), and that the polygons are to one another in the duplicate ratio of corresponding sides. This again, though no doubt an improvement of form, would necessitate the drawing over again of the figure of the altered Proposition 18 and a certain amount of repetition.
Agreeably to his suggestion that Prop. 23 should come before Prop. 14 which is a particular case of it, De Morgan would prove Prop. 19 for parallelograms by means of Prop. 23, and thence infer the truth of it for triangles or the halves of the parallelograms. He adds: “The method of Euclid is an elegant application of the operation requisite to compound equal ratios, by which the conception of the process is lost sight of.” For the general reason given in the note on VI. 14 above, I think that Euclid showed the sounder discretion in the arrangement which he adopted. Moreover it is not easy to see how performing the actual operation of compounding two equal ratios can obscure the process, or the fact that two equal ratios are being compounded. On the definition of compounded ratios and duplicate ratio, De Morgan has himself acutely pointed out that “composition” is here used for the process of detecting the single alteration which produces the effect of two or more, the duplicate ratio being the result of compounding two equal ratios. The proof of VI. 19 does in fact exhibit the single alteration which produces the effect of two. And the operation was of the essence of the Greek geometry, because it was the manipulation of ratios in this manner, by simplification and transformation, that gave it so much power, as every one knows who has read, say, Archimedes or Apollonius. Hence the introduction of the necessary operation, as well as the theoretical proof, in this proposition seems to me to have been distinctly worth while, and, as it is somewhat simpler in this case than in the more general case of VI. 23, it was in accordance with the plan of enabling the difficulties of Book VI. to be more easily and gradually surmounted to give the simpler case first.
That Euclid wished to emphasise the importance of the method adopted, as well as of the result obtained, in VI. 19 seems to me clearly indicated by the Porism which follows the proposition. It is as if he should say: “I have shown you that similar triangles are to one another in the duplicate ratio of corresponding sides; but I have also shown you incidentally how it is possible to work conveniently with duplicate ratios, viz. by transforming them into simple ratios between straight lines. I shall have occasion to illustrate the use of this method in the proof of VI. 22.”
The Porism to VI. 19 presents one difficulty. It will be observed that it speaks of the figure (εἶδος) described on the first straight line and of that which is similar and similarly described on the second. If “figure” could be regarded as loosely used for the figure of the proposition, i.e. for a triangle, there would be no difficulty. If on the other hand “the figure” means any rectilineal figure, i.e. any polygon, the Porism is not really established until the next proposition, VI. 20, has been proved, and therefore it is out of place here. Yet the correction τρίγωνον, triangle, for εἶδος, figure, is due to Theon alone; P and Campanus have “figure,” and the reading of Philoponus and Psellus, τετράγωνον, square, partly supports εἶδος, since it can be reconciled with εἶδος but not with τρίγωνον. Again the second Porism to VI. 20, in which this Porism is reasserted for any rectilineal figure, and which is omitted by Campanus and only given by P in the margin, was probably interpolated by Theon. Heiberg concludes that Euclid wrote “figure” (εἶδος), and Theon, seeing the difficulty, changed the word into “triangle” here and added Por. 2 to VI. 20 in order to make the matter clear. If one may hazard a guess as to how Euclid made the slip, may it be that he first put it after VI. 20 and then, observing that the expression of the duplicate ratio by a single ratio between two straight lines does not come in VI. 20 but in VI. 19, moved the Porism to the end of VI. 19 in order to make the connexion clearer, without noticing that, if this were done, εἶδος would need correction?
The following explanation at the end of the Porism is bracketed by Heiberg, viz. “Since it was proved that, as CB is to BG, so is the triangle ABC to the triangle ABG, that is DEF.” Such explanations in Porisms are not in Euclid’s manner, and the words are not in Campanus, though they date from a time earlier than Theon.