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Commentary on Proposition VI.30¶

It will be observed that the construction in the text is a direct application of the preceding Prop. 29 in the particular case where the excess of the parallelogram which is applied is a square. This fact coupled with the position of VI. 30 is a sufficient indication that the construction is Euclid’s.

In one place Theon appears to have amplified the argument. The text above says “But FE is equal to AB,” while the MSS. B, F, V and p have “But FE is equal to AC, that is, to AB.”

The MSS. give after ὅπερ ἔδει ποιῆσαι an alternative construction which Heiberg relegates to the Appendix. The text-books give this construction alone and leave out the other. It will be remembered that the alternative proof does no more than refer to the equivalent construction in II. 11.

“Let AB be cut at C so that the rectangle AB, BC is equal to the square on CA. [II. 11]

Since then the rectangle AB, BC is equal to the square on CA,

[VI. 17]therefore, as BA is to AC, so is AC to CB.

Therefore AB has been cut in extreme and mean ratio at C.”

It is intrinsically improbable that this alternative construction was added to the other by Euclid himself. It is however just the kind of interpolation that might be expected from an editor. If Euclid had preferred the alternative construction, he would have been more likely to give it alone.