Plato defined man as a featherless biped [Statesman 266e], and was much applauded [for that]. [Diogenes heard this and so] he took a plucked chicken into the school and said, "here is Plato's man." So Plato's definition was changed with the addition "with broad nails."
Get it? I didn't think so. Let's try it one more time in Greek:
Πλάτωνος ὁρισαμένου, Ἄνθρωπός ἐστι ζῷον δίπουν ἄπτερον, καὶ εὐδοκιμοῦντος, τίλας ἀλεκτρυόνα εἰσήνεγκεν αὐτὸν εἰς τὴν σχολὴν καί φησιν, "οὗτός ἐστιν ὁ Πλάτωνος ἄνθρωπος." ὅθεν τῷ ὅρῳ προσετέθη τὸ πλατυώνυχον.
Still don't get it? Well, it's not really a very good joke, but it's a joke. And perhaps it's worth ruining it with an explanation.
The introduction is classic: Diogenes walks into a bar and hears that Plato defined a chicken as a featherless biped. The detail about the applause is evidently intended to amuse us (who could applaud such a ridiculous definition, ridiculous even within the dialogue, as the Stranger notes in the Statesman?), not to report a fact. The punch line of the joke is that τὸ πλατυώνυχον “the quality of having broad nails” sounds in Greek much like τὸ πλατώνικον, “the platonic thing;” the “improved” academic definition is thus “man is a ζῷον δίπουν ἄπτερον πλατυώνυχον/ πλατώνικον,” a “‘platonic’ featherless biped.”
Ok, I said it wasn't a great joke, but I think it's nice to know that people in antiquity took that part of the dialogue with a sense of humor.
I actually think it's a great joke, and I use it every time I teach Chaucer's Nun's Priest's Tale. The point being that humans and chickens are alike in every way but their toenails. Chaucer says it in umpteen hundred lines, but Diogenes Laertius says it in one.
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