← euclid / The Thirteen Books of Euclid’s Elements

Commentary on Proposition II.14¶

Heiberg (Mathematisches zu Aristoteles, p. 20) quotes as bearing on this proposition Aristotle’s remark (De anima II. 2, 413 a 19: cf. Metaph. 996 b 21) that “squaring” (τετραγωνισμός) is better defined as the “finding of the mean (proportional)” than as “the making of an equilateral rectangle equal to a given oblong,” because the former definition states the cause, the latter the conclusion only. This, Heiberg thinks, implies that in the text-books which were in Aristotle’s hands the problem of II. 14 was solved by means of proportions. As a matter of fact, the actual construction is the same in II. 14 as in VI. 13; and the change made by Euclid must have been confined to substituting in the proof of the correctness of the construction an argument based on the principles of Books I. and II. instead of Book VI.

As II. 12, 13 are supplementary to I. 47, so II. 14 completes the theory of transformation of areas so far as it can be carried without the use of proportions. As we have seen, the propositions I. 42, 44, 45 enable us to construct a parallelogram having a given side and angle, and equal to any given rectilineal figure. The parallelogram can also be transformed into an equal triangle with the same given side and angle by making the other side about the angle twice the length. Thus we can, as a particular case, construct a rectangle on a given base (or a right-angled triangle with one of the sides about the right angle of given length) equal to a given square. Further, I. 47 enables us to make a square equal to the sum of any number of squares or to the difference between any two squares. The problem still remaining unsolved is to transform any rectangle (as representing an area equal to that of any rectilineal figure) into a square of equal area. The solution of this problem, given in II. 14, is of course the equivalent of the extraction of the square root, or of the solution of the pure quadratic equation [ x^2 = ab. ]

Simson pointed out that, in the construction given by Euclid in this case, it was not necessary to put in the words “Let BE be greater,” since the construction is not affected by the question whether BE or ED is the greater. This is true, but after all the words do little harm, and perhaps Euclid may have regarded it as conducive to clearness to have the points B, G, E, F in the same relative positions as the corresponding points $A$, $C$, $D$, $B$ in the figure of II. 5 which he quotes in the proof.