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The observed weights of ancient coins are usually less than the nominal "ideal" weights of the coin standards to which they belong because state authorities took a fee—"seigniorage"—for minting coins to cover costs and to make a profit. The basis for calculating the amount taken by the state and the way it administered manufacture are not well understood. Here we analyze the weights of 1344 of the earliest coins of Athens (c. 550–479 bce). We reveal a parabolic relationship between the cost of the silver and the weights of the coins whereby a progressively higher proportion was taken as the denomination decreased, meaning that the smaller the coin, the larger was the proportion of silver taken from it. There was tight control of the minting process and mathematical sophistication in precisely adjusting the silver content from the first introduction of coinage. It also made minting a profitable business. Changes in minting practice can be detected with the introduction of the Athenian "owl" coins, when the percentage of silver taken by the state increased and the spread of weights widened to include coins weighing more than the nominal weight. The latter indicates a significant shift toward monetization of the economy. [ABSTRACT FROM AUTHOR]