Commentary on Proposition VI.2¶
Back to Proposition VI.2.
Euclid evidently did not think it worth while to distinguish in the enunciation, or in the figure, the cases in which the parallel to the base cuts the other two sides produced (a) beyond the point in which they intersect, (b) in the other direction. Simson gives the three figures and inserts words in the enunciation, reading “it shall cut the other sides, or those sides produced, proportionally” and “if the sides, or the sides produced, be cut proportionally.”
Todhunter observes that the second part of the enunciation ought to make it clear which segments in the proportion correspond to which. Thus e.g., if AD were double of DB, and CE double of EA, the sides would be cut proportionally, but DE would not be parallel to BC. The omission could be supplied by saying “and if the sides of the triangle be cut proportionally so that the segments adjacent to the third side are corresponding terms in the proportion.”