[20a]i. e. Number according to cause, which subsists at the extremity of the intelligible order. For number according to hyparxis or essence, subsists at the summit of the order which is intelligible and at the same time intellectual. See the 3d book of my translation of Proclus on the Theology of Plato. [21]The words περι πυθαγορειων are omitted in the original, but from the Protrept. of Iamblichus evidently ought to be inserted. [22]The same thing is said by the Pythagoreans to have befallen the person who first divulged the theory of incommensurable quantities. See the first scholium on the 10th book of Euclid’s Elements, in Commandine’s edition, fol. 1572. [23]Iamblichus, in this list of Pythagoreans, must not be supposed to enumerate those only who were contemporary with Pythagoras: since, if he did, he contradicts what he says of Philolaus in Chap. 31. viz. “that he was many ages posterior to Pythagoras;” but those in general who came from the school of Pythagoras, and were his most celebrated disciples. [24]From this passage it is evident that Iamblichus had many sources of information, which are unknown to modern critics; and this circumstance alone ought to check their pedagogical impertinence. [25]For αυτα here I read, conformably to the version of Obrechtus, αλλα. [26]For δηγμους here, I read οδυρμους; as I do not see what morsus has to do with this place. Obrechtus has in his version “pectorisque morsus;” but I have no doubt _lamentations_ is the proper word, which aptly associates with despondency. [27]“Well-instituted polities,” (says Proclus in MS. Comment. in Alcibiad. prior.) “are averse to the art of playing on wind-instruments; and therefore neither does Plato admit it. The cause of this is the variety of this instrument, the pipe, which shows that the art which uses it should be avoided. For instruments called Panarmonia, and those consisting of many strings, are imitations of pipes. For every hole of the pipe emits, as they say, three sounds at least; but if the cavity above the holes be opened, then each hole will emit more than three sounds.” [28]Odyss. lib. 4. [29]Iamblichus derived what he has said in this chapter about music, from Nicomachus. [30]The first part of this sentence in the original is ξενου τινος εκβεβληκοτος εν Ασκληπιειῳ Ζωνην χρυσιον εχουσαν, and in translating it I have followed the version of Obrechtus, because it appeared to me to convey the meaning of Iamblichus, though the translation is certainly forced, and not such as the natural construction of the words will admit. The translation of Arcerius is, “Cum hospes quidam in æde Æsculapii fœminam zonam auream habentem ejecisset;” and this is perfectly conformable to the natural construction of the words, but then it is void of sense. [31]This history is copiously narrated in chap. 33.