1. Between arithmetic, geometry and music there is a difference in finding the means. In arithmetic in the first place you find it in this way. You add the extremes and divide and find the half; as for example, suppose the extremes are VI and XII, you add them and they make XVIII. You divide and get IX, which is the mean of arithmetic (_analogicum arithmeticae_), since the mean is surpassed by the last by as many units as it surpasses the first. For IX surpasses VI by three units, and XII surpasses it by the same number. 2. According to geometry you find it this way. The extremes multiplied together make as much as the means multiplied, for example, VI and XII multiplied make LXXII; the means VIII and IX multiplied make the same. 3. According to music you find it in this way: The mean is exceeded by the last term by the part by which it exceeds the first term, as for example, VI is surpassed by VIII by two units, which is a third part, and by the same part the mean VIII is surpassed by the last term which is XII. Chapter 9. That infinite numbers exist. 1. It is most certain that there are infinite numbers, since at whatever number you think an end must be made I say not only that it can be increased by the addition of one, but, however great it is, and however large a multitude it contains, by the very method and science of numbers it can not only be doubled but even multiplied. 2. Each number is limited by its own proper qualities, so that no one of them can be equal to any other. Therefore in relation to one another they are unequal and diverse, and the separate numbers are each finite, and all are infinite. ON GEOMETRY INTRODUCTION In spite of the high development of geometry among the Greeks it never took root as a pure science in the western Roman world,[249] and neither the various practical applications of its principles nor its use as a disciplinary educational subject sufficed to fasten thoughtful attention upon it; in consequence, it lost almost its entire content. As it appears in the four writers who treat of it in later Roman and early medieval times, Martianus Capella, Boethius,[250] Cassiodorus, and Isidore, it furnishes a striking commentary upon the intellectual conservatism that could retain without a suspicion of criticism a subject that was no longer anything but empty form. [249] Cantor, _Vorlesungen über Geschichte der Mathematik_, vol. i, p. 521. [250] The authenticity of the work on geometry that has been handed down under Boethius’ name is questioned. (See Cantor, _ibid._, pp. 536 _et seq._) It contains the complete proof of only three of Euclid’s propositions. It also contains calculations of areas of geometrical figures. See edition of Friedlein (Leipzig, 1867).