"can tell undoubted Raphaels from Gerard Dows and Zoffanies" — Raphael painted religious iconography heavy with symbolism while Dow and Zoffany painted photorealistc scenes from life. The difference is unmistakable.
"tell you every detail of Caractacus’s uniform" — The only depiction of Caractacus shows him in the nude. Also, Caractacus was a Gallic chieftain who lived around the birth of Christ, and would never have worn a uniform anything.
As well as some which are outright impossible:
"I can write a washing bill in Babylonic Cuneiform": Cuneiform was at the time understood to be a form of writing, but nothing else was known about it.
"And I can hum a fugue of which I've heard the music's din a-fore": Fugues are per definition polyphonic, i.e. have more than one melody running at the same time. Tuvan throat singers can accomplish something to this effect by singing one tune and humming another, but good luck humming two tunes at once.
"About binomial theorem I’m teeming with a lot o’ news" — No real advances had been made in the area since Newton generalized it more than 200 years before, hence there were no "news" for him to teem with.