The trittys was one third of the tribe. Consequently there were twelve trittyes, corrsponding to the four tribes. After the reforms of 507 B.C. there were thirty trittyes for the ten tribes. But what was new in Cleisthenes' plan, which envisaged the fragmentation of the tribes' local power, was the division of these thirty trittyes: into ten peri to asty (surrounding the Town), ten paraliai (coastal), and ten mesogeiai (midland). The lot was used to assign three (one from each category) to each tribe, and to these all the demes were allocated. The demes peri to asty were the largest. Thus under certain circumstances each of them could be a trittys on its own; while elsewhere a trittys comprised up to eight or nine demes.

The heads of the trittyes were the trittyarchoi. They were responsible for road works; for constructing walls and ships; and for matters concerning recruiting. They probably also took on the role of celebrant of the ritual in certain cults, as a sacrifice by the name of trittys is recorded. The connection between the trittys and cult duties is also apparent from the single trittys name we possess, 'Leukotainioi', meaning "those who wear a white ribbon", and it denotes a trittys of the Geleontes tribe.

The naukraria was the first stage of the Athenian body politic's administration and taxation system. Each trittys comprised four naukrariai; in other words, before the time of Cleisthenes there were forty-eight naukrariai altogether. Each naukraria, according to Pollux, was under an obligation to provide a ship and two horsemen. It maintained its own treasury, from which the public pay-clerks (kolakretae) paid the expenses of state embassies (theoriae) that the city sent to Delphi. The principle of the naukraria was retained by Cleisthenes, and their number was increased to fifty (five for each tribe). Their number and their participation in the Athenian naval force are consistent with the piece of information from Herodotus that before the Persian Wars Athens had fifty ships. The person in charge of the naukraria was called the naukraros and was responsible for collection contributions and incurring expenses. The naukraros, or 'chief of ship', was also the captain of his naukraria's ship and was under the command of the polemarch. With Cleisthenes' reforms, the duties of the naukraroi passed to the demarchs. Both the fact that the naukraria was replaced in the political field by the demos, and the fact that the names of the naukrariai coincide with Attic place names (e.g. Colias) indicate that in certain cases the institution had a local base. Recent research lays particular emphasis on the religious and cult function of the institution of the naukraria.

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