Book 3 The Origin Of The Harmonic Proportions Although philosophy has embraced many skills, and so has mathematics, yet he writes the following about one part of the latter, called harmony, and about Numbers (believed to be the principles of harmonies). Proclus Diadochus, Commentary on Book I of the Elements of Euclid, Book I;i Mental endeavor is the preparation for theology. For those features which to the uninitiated in the truth of divine matters seem difficult to grasp and lofty are by mathematical reasoning shown to be trustworthy, manifest and uncontroversial, by means of certain images. For they show proof of the supernatural properties in numbers; and they make clear the powers of the intelligible forms in reasoning. Thus Plato teaches us many remarkable things about the nature of the gods through the appearance of mathematical things; and the Pythagorean philosophy disguises its teaching on divine matters with these, so to speak, veils. For of this kind is the whole of that sacred writing,^ both Philolaus on the Bacchae,^ and the whole Pythagorean system of teaching about God. Again, it perfects us in moral philosophy, implanting in our behavior order, propriety and harmony in social relations. It also informs us what figures, what songs, what motions are appropriate to virtue; and also the teaching by which the Athenian"* would have those who will pay attention to moral virtues from their youth cultivated and perfected. Furthermore he makes plain the proportions of numbers which are associated with the virtues, some in arithmetic, some in geometry and others in harmony; and he shows the excesses and deficiencies of the vices. By all of these we are guided to the middle way in behavior and in morals