Thrasymăchus in Harpers Dictionary of Classical Antiquities (1898)
（Θρασύμαχος). A native of Chalcedon, was a Sophist, and one of the earliest cultivators of the art of rhetoric. He was a contemporary of Gorgias. He is one of the speakers in Plato's Republic.
Thrasymachus in Wikipedia
Thrasymachus (Θρασύμαχος) (ca. 459-400 BCE) was a sophist of Ancient Greece best known as a character in Plato's Republic.
The Historical Thrasymachus
Thrasymachus was a citizen of Chalcedon, on the Bosphorus. His career appears to have been spent as a sophist, at Athens as far as we know, though there is no concrete evidence that he was a sophist. He is credited with an increase in the rhythmic character of Greek oratory, especially the use of the paeonici rhythm in prose; also a greater appeal to the emotions through gesture.
Aristophanes makes what is the most precisely dateable of references to Thrasymachus, in a passing joke from a lost play dated to 427 BCE. Nils Rauhut of the Internet Encyclopedia of Philosophy concludes from this passage that Thrasymachus must have been teaching in Athens for several years before this point. A further fragment, this time from Clement of Alexandria provides some further context, by placing Thrasymachus contrary to the Macedonian Archelaus. "And while Euripides says in the Telephus, 'Shall we who are Greeks be slaves to barbarians?', Thrasymachus says in his speech For the People of Larisa, 'Shall we become slaves to Archelaus, Greeks as we are, to a barbarian?'" Rauhut therefore declares it evident that Thrasymachus became most prominent in the last three decades of the fifth century. Dillon and Gergel posit the alternate possibility that the speech was composed by the second-century CE Herodes Atticus, of whom we have extracts similar in spirit to Clement's fragment, and sound authentically fifth-century, exhibiting detailed knowledge of Thessalian politics.
A fragment of his work On Constitutions survives, which contains the maxim that the ancestral constitution is common to all. The meaning of this is debatable; one suggestion is that every orator can claim to speak for it, no matter what he is advocating.
There is a man by the same name mentioned in Aristotle's Politics who overthrew the democracy at Cyme, but nothing is known of this event, nor can it be said with any degree of certainty that they are the same man. Aristotle mentions a Thrasymachus again in his De Sophisticis Elenchis, where he credits him with a pivotal role in the development of rhetorical theory. Quoting the W. A. Pickard-Cambridge text: "For it may be that in everything, as the saying is 'the first start is the main part'... This is in fact what has happened in regard to rhetorical speeches and to practically all the other arts: for those who discovered the beginnings of them advanced them in all only a little way, whereas the celebrities of to-day are the heirs (so to speak) of a long succession of men who have advanced them bit by bit, and so have developed them to their present form, Tisias coming next after the first founders, then Thrasymachus after Tisias, and Theodorus next to him, while several people have made their several contributions to it: and therefore it is not to be wondered at that the art has attained considerable dimensions." Dillon and Gergel are cautious not to read this as stating that this makes Thrasymachus a student of Tisias, just as it does not make Theodorus a student of Thrasymachus.
Writing more specifically in the Rhetoric, Aristotle attributes to Thrasymachus a witty simile. "A simile works best when it is in effect a metaphor, for it is possible to say that a shield is like the drinking-cup of Ares, or that a ruin is like the tattered rag of a house, and to say that Niceratus is like a Philocreres bitten by Pratys - the simile made by Thrasymachus when he saw Niceratus, who had been beaten by Pratys in a recitation competition, still going around with his hair uncut and unkempt." A further reference to Thrasymachus in the Rhetoric finds Herodicus punning on Thrasymachus' name. "Herodicus said of Thrasymachus, 'You are always bold in battle (thrasymakhos)!'" Dillon and Gergel suggest that this might explain Plato's choice of Thrasymachus as the "combative and bombastic propounder of the 'might is right' theory" for his Republic.
Plato mentions Thrasymachus as a successful rhetorician in his Phaedrus, but attributes nothing significant to him. The Byzantine Suda gives a brief description of Thrasymachus affirming his position as a rhetorical theorist. "A Chalcedonian sophist, from the Chalcedon in Bithynia. He was the first to discover period and colon, and he introduced the modern kind of rhetoric. He was a pupil of the philosopher Plato and of the rhetor Isocrates. He wrote deliberative speeches; an Art of Rhetoric; paegnia; Rhetorical Resources." Dillon and Gergel state that the second sentence is a "preposterous statement, both as concerns Plato and Isocrates." They further declare that emending 'pupil' (mathętęs) for 'teacher' (kathęgętęs) is equally foolish. They themselves suggest a lacuna in the text, wherein Thrasymachus is declared the pupil of another, and a rival of Plato and Isocrates.
Dionysius of Halicarnassus praises Thrasymachus for various rhetorical skills in his On Isaeus, finding Thrasymachus "pure, subtle, and inventive and able, according as he wishes, to speak either with terseness or with an abundance of words." Dionysus but still found Thrasymachus a second-rate orator beside the "incisive" and "charming" Lysias, because he left no forensic speeches to posterity, only handbooks and display-speeches.
Thrasymachus' current importance derives mainly from his being a character in The Republic. He is noted for his unabashed, even reckless, defense of injustice and for his famous blush at the end of Book 1, after Socrates has tamed him. The meaning of this blush, like that of Socrates' statement in Book 6 that he and Thrasymachus "have just become friends, though we weren't even enemies before" (498c), is a source of some dispute.
There is a long philosophical tradition of exploring what exactly Thrasymachus meant in Republic I, and of taking his statements as a coherent philosophical assertion, rather than as Plato's straw man.
In the Republic I, Thrasymachus violently disagreed with the outcome of Socrates' discussion with Polemarchus about justice. Demanding payment before speaking, he claims that "justice is the advantage of the stronger" (338c) and that "injustice, if it is on a large enough scale, is stronger, freer, and more masterly than justice'" (344c). Socrates counters by forcing him to admit that there is some standard of wise rule - Thrasymachus does claim to be able to teach such a thing - and then arguing that this suggests a standard of justice beyond the advantage of the stronger. The rest of the dialogue is occasioned by Glaucon's dissatisfaction with Socrates' refutation.
His name means fierce fighter, which may have influenced his role in the dialogue.
In Leo Strauss's interpretation, Thrasymachus and his definition of justice represent the city and its laws, and thus are in a sense opposed to Socrates and to philosophy in general. As an intellectual, however, Thrasymachus shared enough with the philosopher to potentially act to protect philosophy in the city.