If a straight line be cut at random, the rectangle contained by the whole and one of the segments is equal to the rectangle contained by the segments and the square on the aforesaid segment. I leave it to the reader to prove it from the construction opposite. If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. I leave it to the reader to prove it from the construction opposite. Proposition 5 is discussed below.
Interested readers can investigate others in Heath .
Solving Equations by Proportion As mentioned previously, the early Pythagoreans defined ratio in terms of the Natural numbers. Their techniques for solving what we would call algebraic equations were done using proportions based on this definition. The discovery of the incommensurables caused this definition for ratio to be discarded. It wasn't until Eudoxus' outstanding work 150 years later that an effective definition for ratio was found and a rigorous theory of proportions was established. To overcome this problem in the meantime, the Pythagoreans resorted to geometrical proportion using the properties of similarity.