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Commentary on Proposition III.23¶

Clavius and the other early editors point out that, while the words “on the same side” in the enunciation are necessary for Euclid’s proof, it is equally true that neither can there be two similar and unequal segments on opposite sides of the same straight line; this is at once made clear by causing one of the segments to revolve round the base till it is on the same side with the other.

Simson observes with reason that, while Euclid in the following proposition, III. 24, thinks it necessary to dispose of the hypothesis that, if two similar segments on equal bases are applied to one another with the bases coincident, the segments cannot cut in any other point than the extremities of the base (since otherwise two circles would cut one another in more points than two), this remark is an equally necessary preliminary to III. 23, in order that we may be justified in drawing the segments as being one inside the other. Simson accordingly begins his proof of III. 23 thus:

“Then, because the circle ACB cuts the circle ADB in the two points A, B, they cannot cut one another in any other point: One of the segments must therefore fall within the other. Let ACB fall within ADB and draw the straight line ACD, etc.”

Simson has also substituted “not coinciding with one another” for “unequal” in Euclid’s enunciation.

Then in III. 24 Simson leaves out the words referring to the hypothesis that the segment AEB when applied to the other CFD may be “otherwise placed as CGD”; in fact, after stating that AB must coincide with CD, he merely adds words quoting the result of III. 23: “Therefore, the straight line AB coinciding with CD, the segment AEB must coincide with the segment CFD, and is therefore equal to it.”