This is part 1 of 2. Part 2 is here.

Ordo atque ratio Pythagorae ac deinceps familiae et successionis eius recipiendi instituendique discipulos huiuscemodi fuisse traditur: iam a principio adulescentes, qui sese ad discendum obtulerant, ἐφυσιογνωμόνει. id verbum significat mores naturasque hominum coniectatione quadam de oris et vultus ingenio deque totius corporis filo atque habitu sciscitari. tum qui exploratus ab eo idoneusque fuerat, recipi in disciplinam statim iubebat et tempus certum tacere: non omnes idem, sed alios aliud tempus pro aestimato captu sollertiae. is autem, qui tacebat, quae dicebantur ab aliis, audiebat, neque percontari, si parum intellexerat, neque commentari, quae audierat, fas erat; sed non minus quisquam tacuit quam biennium: hi prorsus appellabantur intra tempus tacendi audiendique ἀκουστικοί. ast ubi res didicerant rerum omnium difficillimas, tacere audireque, atque esse iam coeperant silentio eruditi, cui erat nomen ἐχεμυθία, tum verba facere et quaerere quaeque audissent scribere et, quae ipsi opinarentur, expromere potestas erat; hi dicebantur in eo tempore μαθηματικοί, ab his scilicet artibus, quas iam discere atque meditari inceptaverant: quoniam geometriam, gnomonicam, musicam ceterasque item disciplinas altiores μαθήματα veteres Graeci appellabant; vulgus autem, quos gentilicio vocabulo “Chaldaeos” dicere oportet, “mathematicos” dicit. exinde his scientiae studiis ornati ad perspicienda mundi opera et principia naturae procedebant ac tunc denique nominabantur φυσικοί.
(Aulus Gellius, Noct. Att. 1.9.1-7)

It is said that the order and method followed by Pythagoras, and afterwards by his school and his successors, in admitting and training their pupils were as follows: At the very outset he “physiognomized” the young men who presented themselves for instruction. That word means to inquire into the character and dispositions of men by an inference drawn from their facial appearance and expression, and from the form and bearing of their whole body. Then, when he had thus examined a man and found him suitable, he at once gave orders that he should be admitted to the school and should keep silence for a fixed period of time; this was not the same for all, but differed according to his estimate of the man’s capacity for learning quickly. But the one who kept silent listened to what was said by others; he was, however, religiously forbidden to ask questions, if he had not fully understood, or to remark upon what he had heard. Now, no one kept silence for less than two years, and during the entire period of silent listening they were called ἀκουστικοί or “auditors.” But when they had learned what is of all things the most difficult, to keep quiet and listen, and had finally begun to be adepts in that silence which is called ἐχεμυθία or “continence in words,” they were then allowed to speak, to ask questions, and to write down what they had heard, and to express their own opinions. During this stage they were called μαθηματικοί or “students of science,” evidently from those branches of knowledge which they had now begun to learn and practise; for the ancient Greeks called geometry, gnomonics, music and other higher studies μαθήματα or “sciences”; but the common people apply the term mathematici to those who ought to be called by their ethnic name, Chaldaeans. Finally, equipped with this scientific training, they advanced to the investigation of the phenomena of the universe and the laws of nature, and then, and not till then, they were called φυσικοί or “natural philosophers.” (tr. John C. Rolfe)

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