Book I: On the relationship between Harmonic Ratios and Proportions.

Pythagoreans believed that these harmonic ratios gave music powers of healing which could "harmonize" an out-of-balance body.

This kind of scaling dictates that the ratio of string lengths should be equal to the harmonic ratio of their pitches.

Based on engraved plans of Meroitic King Amanikhabali's pyramids, Nubians had a sophisticated understanding of mathematics and an appreciation of the harmonic ratio.

The Pythagorean theory of harmonic ratios is basis of studies on music theory in the Islamic world, for example the al-Farabi's Kitab al-Musiqa al-kabir.

The inharmonicity disappears when strings are bowed because the bow's stick-slip action is periodic, so it drives all of the resonances of the string at exactly harmonic ratios, even if it has to drive them slightly off their natural frequency.

Above all, he was the first to notice the recurrence of cross-ratios and harmonic ratios in the lemmas, and because these concepts suffuse most of his restoration, it is probable that many of his inventions coincide with some of Euclid's, even if we cannot now tell which they are.