1. The third division is _rhythmica_, having to do with strings and instruments that are beaten, to which are assigned the different species of cithara, the drum, and the cymbal, the sistrum, acitabula of bronze and silver, and others of metallic stiffness that when struck return a pleasant tinkling sound, and the rest of this sort.[262] [262] Other instruments mentioned are _psalterum_, _lyra_, _barbitos_, _phoenix_, _pectis_, _indica_, _aliae quadrata forma vel trigonali_, _margaritum_, _ballematica_, _tintinnabulum_, _symphonia_. 2. The form of the cithara in the beginning is said to have been like the human breast, because as the voice was uttered from the breast so was music from the cithara, and it was so-called for the same reason. For _pectus_ is in the Doric language called κίθαρα. Chapter 23. On the numbers of music. 1. You inquire into the numbers according to music as follows: setting down the extremes, as for example, VI and XII, you see by how many units VI is surpassed by XII, and it is by VI units; you square it; six times six make XXXVI. You add those first-mentioned extremes, VI and XII; together they make XVIII; you divide XXXVI by XVIII; two is the result. This you add to the smaller amount, VI namely; the result will be VIII and it will be the mean between VI and XII. Because VIII surpasses VI by two units, that is by a third of six, and VIII is surpassed by XII by four units, a third part [of twelve]. By what part, then, the mean surpasses, by the same is it surpassed. 2. Just as this proportion exists in the universe, being constituted by the revolving circles, so also in the microcosm—not to speak of the voice—it has such great power that man does not exist without harmony.[263] [263] The general sense of the passage: “ut sine ipsius perfectione etiam homo symphoniis carens non consistat.” 3, 23, 2. See p. 65. ON ASTRONOMY INTRODUCTION The science of astronomy, in its history from the great period of Greece down to the dark ages, furnishes almost as complete a spectacle of decay as does geometry. It is quite certain “that Aristarchus taught the annual motion of the earth around the sun, and both he and Seleukus taught the diurnal rotation of the earth,”[264] but the general scientific development of the age was not sufficient to assimilate this advanced theory, and astronomers went back to a geocentric universe. Strange to say, the later rise of practical astronomy at Alexandria, and the development of pure mathematics, did not secure a return to the more advanced theory, the efforts of the later astronomers being devoted, not to a reconsideration of the fundamental theses of the subject, but to putting the geocentric theory on a secure mathematical basis. The greatest of these astronomers, Ptolemy (second century A.D.), left in his _Syntaxis_ a comprehensive summing up of mathematical astronomy.