to rule, to begin
The verb ἄρχω carries in Plato's vocabulary two intertwined senses: "to begin, be first in something" and "to rule, hold authority." From the same root grows a cluster of central terms: beginning, principle (ἀρχή); ruler, magistrate (ἄρχων); to begin (ἄρχομαι in the middle voice). Plato develops both senses separately — and at the same time keeps their link: the one who rules is the one closer to the beginning, the one who knows ἀρχή.
In the sense "to rule," the question is not "who in fact rules" but "who ought to rule and on what ground." In the Republic, ruling is first exposed as the self-interest of the stronger in Thrasymachus's speech (Resp. I 338c), then redefined: the true ruler looks not to his own advantage but to the benefit (συμφέρον) of the ruled (Resp. I 342e); no one wants to rule willingly except the knower under compulsion (Resp. I 347d) — and from this comes the thesis of philosopher-rulers. In the Statesman the art of ruling is set apart as a distinct kingly technique, different from herd-tending and from household management (Plt. 292e). In the Laws seven titles to rule are listed, and the dispute among them is settled in favor of practical wisdom (φρόνησις) (Lg. III 690a). Alongside runs an inward line: to rule means above all "to rule oneself" (ἑαυτοῦ ἄρχειν, Grg. 491d) — a thought Plato unfolds into the doctrine of the three parts of the soul, in which the rational part (λογιστικόν) rules over the spirited (θυμοειδές) and the appetitive (ἐπιθυμητικόν) (Resp. IV 441e, 442a).
In the sense "to begin," the verb sets a methodological move. In the Phaedrus Socrates demands: an argument "must begin" with what the thing under discussion is (Phdr. 237d). The soul itself is described as the "principle of motion" (ἀρχὴ κινήσεως, Phdr. 245c). In the Republic what is sought is an "unhypothetical beginning" (ἀρχὴ ἀνυπόθετος, Resp. VI 510b), to which dialectic ascends by abandoning conditional grounds (ὑποθέσεις). Substantively the two branches converge: the one entitled to rule is the one who knows the beginnings; and conversely, the beginning of an argument is its inner "ruler." Close verbs: to reign (βασιλεύω, of kingly power), to be master (δεσπόζω), to lead (ἡγέομαι).