Commentary on Proposition V.7¶
Back to Proposition V.7.
In this proposition there is a similar use of ὁ ἔτυχεν to that which has been discussed under Prop. 4. $Any$ multiple F of C is taken and then, four lines lower down, we are told that “F is another, chance, magnitude.” It is of course not any magnitude whatever, and Simson leaves out the sentence, but this time without calling attention to it.
Of the Porism to this proposition Heiberg says that it is properly put here in the best MS. ; for, as August had already observed, if it was in its right place where Theon put it (at the end of V. 4), the second part of the proof of this proposition would be unnecessary. But the truth is that the Porism is no more in place here. The most that the proposition proves is that, if A, B are equal, and C any other magnitude, then two conclusions are simultaneously established, (1) that A is to C as B is to C and (2) that C is to A as C is to B. The second conclusion is not established from the first conclusion (as it ought to be in order to justify the inference in the Porism), but from a hypothesis on which the first conclusion itself depends ; and moreover it is not a proportion in its general form, i.e. between four magnitudes, that is in question, but only the particular case in which the consequents are equal.
Aristotle tacitly assumes inversion (combined with the solution of the problem of Eucl. VI. 11) in $Meteorologica$ III. 5, 376 a 14—16.