Full phrase from Iamblichus,
On the Mathematics of Nicomachus of Gerasa 124.21ff.: ὡς ἔνεστι κατὰ τὸ φιλόκαλον δι' αὑτοῦ ἕκαστον πειραθέντα κατανοῆσαι, “… for each (student) on his own to try and recognize (those differences) based as much as possible on τὸ φιλόκαλον.”
The aorist passive participle πειραθέντα is masculine accusative singular to agree with ἕκαστον. Though the form is passive, the voice is middle — “having tried”; cf. LSJ s.v. πειράω:
Att.ἐπειράθην [ᾱ], found in med. sense, Il. 19.384, al.,Hdt. 3.152,al., Th. 2.5 (v.l.),33, 6.92, and in later Prose.
It’s hard to understand how LSJ, in citing this Iamblichus passage, can give a word based on ideas of “love” and “beauty” (τὸ φιλόκαλον) such a dryly neutral rendering. The perfect scientific (or mathematical) solution, the shortest and simplest, is proverbially the “neatest” or “prettiest” one. As the poet Edna St. Vincent Millay writes, “Euclid alone has looked on Beauty bare.”
LSJ do the same thing with the related noun φιλοκαλία when they cite its meaning in Vettius Valens (Book 9 ch. 19, 361.21ff,) as “calculation, working out.” The actual passage in Vettius (Τῶν δὲ περιπεπονημένων μοι μηδὲν ἀποκρύπτειν βουλόμενος καὶ ἑτέραν
φιλοκαλίαν ἐπεισφέρω τοῖς περὶ τὰ τοιαῦτα ἐσπουδακόσιν ἀφθόνως ἅτε δὴ γνησίοις παισίν) gets a far better treatment in Mark Riley’s recently published translation:
I have not wished to hide any of the methods which I have previously worked out, and now I am generously bestowing another scholarly gift on you devotees of such matters, as if you were my children. (
https://www.csus.edu/indiv/r/rileymt/Vettius%20Valens%20entire.pdf)
It’s very difficult to find an English equivalent for ὡς ἔνεστι κατὰ τὸ φιλόκαλον (perhaps Greek can do better) in the Iamblichus passage. Taking a cue from the 1688 Latin translation by Samuel Tennulius (
quemadmodum quis poterit per se animadvertere, quo elegantiae studuerit), the best I can do for ὡς ἔνεστι κατὰ τὸ φιλόκαλον δι' αὑτοῦ ἕκαστον πειραθέντα κατανοῆσαι is this: “… for each (student) on his own to try and recognize (those differences) based on the most elegant proof possible.”