The actual Greek term that is used in the dialogue is καλόν, which as an adjective often means fine or noble as well as beautiful.
As in Charmides, Lysis and Euthyphro, Hippias Major has an "anatreptic" purpose, that is, the result of the dialogue is to defeat commonly held opinions, without necessarily offering a resolution.
The astuteness of Socrates in taking refuge under the authority of a supposed third protagonist in order to direct biting criticism at Hippias, endows the dialogue with humour.
Hippias, whose business had kept him away from Athens for a long time, arrives in the city to give a lecture at Phidostratus' schoolroom[1] in the next few days.
He meets Socrates, and the latter asks him why such a precious and wise man as Hippias has deprived the Athenians of his presence for so long.
It is, explains the great Sophist, because his native Elis was so in need of his services, and entrusted him with several important diplomatic missions to different cities; notably in Sparta.
One such example was the small town of Inycus, in Sicily, where the modest inhabitants sacrificed a good part of their savings to see their children die educated.
However, Socrates emphasized, the law is precisely made for use and happiness of the citizens, two things to which Hippias would have been greatly able to contribute.
It is not, Hippias answers, for his knowledge of arithmetic or astronomy, but rather "They are very fond of hearing about the genealogies of heroes and men, Socrates, and the foundations of cities in ancient times and, in short, about antiquity in general...[these being] beautiful pursuits".
Thus embarrassed by this exposure, Socrates claims to be delighted that finally one as competent as Hippias will be able to provide his opinion on the nature of beauty.
The great Sophist, flattered, does not object; and is goaded on by Socrates, who offers to reprise the discussion, playing the part of the harasser.
Tiring of the errors of Hippias, Socrates offers a definition in his turn, which he holds came from his famous harasser: the beautiful is simply that which is appropriate.
Which in turn requires that the definition be refocused; beauty is only usefulness applied to good ends, or those that are "favourable".
This hypothesis, while appealing, contains according to Socrates himself a fundamental flaw; that it ignores the beauty of the more noble pleasures, drawn from the studious occupations or the study of laws.
"[2] Socrates, taking his leave, pretends to feel bad about the situation, cornered between the attacks of Hippias and those of his mysterious opponent.
Although some works previously attributed to Plato have been determined to be inauthentic, this is one where authorship has still not been firmly established, though academic consensus tends toward its authenticity.
"[3] Sider, writing in 1992 states that G. R. Ledger, in Re-counting Plato (Oxford 1989) carried out a computer text analysis and though not conclusive "On balance the evidence for genuineness is fairly convincing".
He goes on to state that amongst other recent works, P. Woodruff, Plato: Hippias Major (Oxford 1982) also argues for authenticity and dates the document to "around 390" BC.