Commentary on Proposition VI.25¶
Back to Proposition VI.25.
This is the highly important problem which Pythagoras is credited with having solved. Compare the passage from Plutarch (Symp. VIII. 2, 4) quoted in the note on I. 44 above, Vol. I. pp. 343—4.
We are bidden to construct a rectilineal figure which shall have the form of one and the size of another rectilineal figure. The corresponding proposition of the $Data$, Prop. 55, asserts that, “if an area (χωρίον) be given in form (εἴδει) and in magnitude, its sides will also be given in magnitude.”
Simson sees signs of corruption in the text of this proposition also. In the first place, the proof speaks of the triangle ABC, though, according to the enunciation, the figure for which ABC is taken may be any rectilineal figure, εὐθύγραμμον “rectilineal figure” would be more correct, or εἶδος, “figure”; the mistake, however, of using τρίγωνον is not one of great importance, being no doubt due to the accident by which the figure was drawn as a triangle in the diagram.
The other observation is more important. After Euclid has proved that
(fig. ABC) : (fig. KGH) = (BE) : (EF),
he might have inferred directly from V. 14 that, since ABC is equal to BE, KGH is equal to EF. For V. 14 includes the proof of the fact that, if $A$ is to $B$ as $C$ is to $D$, and $A$ is equal to $C$, then $B$ is equal to $D$, or that of four proportional magnitudes, if the first is equal to the third, the second is equal to the fourth. Instead of proceeding in this way, Euclid first permutes the proportion by V. 16 into
(fig. ABC) : (BE) = (fig. KGH) : (EF),
and then infers, as if the inference were easier in this form, that, since the first is equal to the second, the third is equal to the fourth. Yet there is no proposition to this effect in Euclid. The same unnecessary step of permutation is also found in the Greek text of XI. 23 and XII. 2, 5, 11, 12 and 18. In reproducing the proofs we may simply leave out the steps and refer to V. 14.