Commentary on Proposition I.43¶
Back to Proposition I.43.
In the text of Proclus’ commentary as we have it, the end of the note on I. 41, the whole of that on I. 42, and the beginning of that on I. 43 are missing.
Proclus remarks (p. 418, 15—20) that Euclid did not need to give a formal definition of complement because the name was simply suggested by the facts; when once we have the two “parallelograms about the diameter,” the complements are necessarily the areas remaining over on each side of the diameter, which fill up the complete parallelogram. Thus (p. 417, 1 sqq.) the complements need not be parallelograms. They are so if the two “parallelograms about the diameter” are formed by straight lines drawn through one point of the diameter parallel to the sides of the original parallelogram, but not otherwise. If, as in the first of the accompanying figures, the parallelograms have no common point, the complements are five-sided figures as shown.
When the parallelograms overlap, as in the second figure, Proclus regards the complements as being the small parallelograms FG, EH. But, if complements are strictly the areas required to fill up the original parallelogram, Proclus is inaccurate in describing FG, EH as the complements. The complements are really (1) the parallelogram FG minus the triangle LMN, and (2) the parallelogram EH minus the triangle KMN, respectively; the possibility that the respective differences may be negative merely means the possibility that the sum of the two parallelograms about the diameter may be together greater than the original parallelogram.
In all the cases it is easy to show, as Proclus does, that the complements are still equal.