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Am I right in saying that the Archaemenid Empire was the first empire of Greater Persia and named after a pretend king? It was founded by Cyrus the Great in Lydia? Was Lydia Greek at the time?
The Achaemenid Empire, also known as the Medo-Persian Empire, was not the first empire in the region - though it was one of the most internally coherent, and was also the first to have it's capital city and origins in what is now Iran.
The ancestry of the ruling hose of the Empire is the subject of much debate, largely because the records of the Persians themselves have mostly been lost and the few that survive are of questionable reliability. It is best to describe Achaemenes as "mythical" rather than "pretend" - in the same sense that King Arthur is "mythical".
Cyrus the Great did not "found" the Persian state, but he did take it from a small tribal city-state and turned it into an Empire - much as Philip II and Alexander III did for Macedonia a couple of centuries later. Lydia, in what is now western Turkey, was just one of numerous areas he conquered. Prior to the Persian invasion, Lydia was inhabited by a mixture of native peoples (mainly the descendants of Hittites that had conquered the place a thousand years earlier, known as "Lydians" or "Sfard" in their own language) and Greek colonists.
The Persians did not strike any coinage of their own until they conquered Lydia, which was already striking coins prior to the conquest. While the gold darics are found throughout the eastern Mediterranean and beyond, the siglos is rarely found outside of western Turkey, meaning it was primarily intended as a replacement for the Lydian coinage, rather than as an empire-wide monetary unit.
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What is the difference between a daric and a siglos, how many siglos make a daric and what are the materials of the coins in question.
The name "daric" for the gold coin is well attested; it is even found in the Bible (in Hebrew form) as "adarkonim". Recall that in those days, gold coins were traded by weight, so "daric" was the name of both the coin itself and the amount of gold that was needed to make 1 coin. The word was not used to measure any other substance; the daric weighed 1 old Babylonian shekel and "shekel" would have been used as the name of the unit of weight.
"Siglos" (plural "sigloi"), the name given to the silver coins, is somewhat more convoluted. The Lydians had a unit of weight called the "siglos", which at about 10 grams was actually twice the weight of the Persian coin that now bears that name; more properly, the coins should perhaps then be called "half-sigloi", though getting people to change things at this late stage seems redundant.
The designs of the two coins were the same, even made from the same dies. The weights were different, of course; the daric is heavier because gold is denser than silver. The rate of exchange was fixed, at 20 sigloi (that's the coin, not the weight) to the daric.
The "siglos" was a different weight to the greek "drachm"; the drachm weighed slightly less. The exchange rate was 25 drachms to the daric.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis