In those years from Archytas to Aristoxenus the evolution of Greek music had passed from the poet-musicians, the real masters of the lyre, into the hands of philosophers and disputants, men learned in all the subtleties of Pythagorean lore, who busied themselves with recondite demonstrations of the proportions of numbers, and applied them to the theoretical division of the octave, to an extent which transcended altogether the range of the practical art of the cithara players, nevertheless the labour was not wholly lost, since it went to the strengthening of the foundations of the _science of music_. A new era had arrived, Greece lost her position and became a dependency in the Macedonian empire. The centre of Greek life and thought had been transferred to Alexandria, and here at the great library which had been founded B.C. 332 by Alexander the Great, Eratosthenes was librarian, and his name figures largely in the mathematics of Music. His lifetime extended from 276 to 196 B.C. Two other Alexandrians complete the record so far as the present simple treatment of the development of the scale is concerned. They lived within the Christian era. Didymus, A.D. 60, introduced the minor tone into the scale, and consequently the practical major third. He demonstrated the lesser or _minor tone_ to be necessary to the right division of fourths and fifths. Claudius Ptolemy, A.D. 130, accepted the scheme, but altered the arrangement of the tones. Didymus and Claudius Ptolemy, the two latest philosophers who sought to perfect the diatonic scale, achieved highly important results by simple means; whereas the octochord as left by Pythagoras, comprised but two kinds of divisions, the tone and the hemitone (not exactly half a tone, it was the overplus after the measurement of the two whole tones in the tetrachord)—and these, taking C as the starting point for our convenience, may be represented thus:— C......D.......E......F.......G.......A.......B.....C major major hemi major major major hemi tone tone tone tone tone tone tone this was constructed from a series of fifths. Didymus shewed that the stricter mathematical division (not by fifths) required a lesser or minor tone in place of _one_ major, and the amount of decrease went to increase the hemitone to a semitone, thus:— C.......D.......E......F.......G.......A.......B......C minor major semi minor major major semi tone tone tone tone tone tone tone Claudius Ptolemy seventy years later altered this, transposing the minor tone to the second place,— C D E F G A B C I I I I I I I I major minor semi major minor major semi tone tone tone tone tone tone tone