The Circulation of Money in Exchange
The need of money for exchange is dependent not only on the amount of goods in exchange, but the frequency and volume at which exchange takes place.
Consider the difference in rents paid to landlords: the farmer may pay a landlord 1200 ounces of silver upon the sale of his crop, but the landlord may pay out this sum at a rate of 100 ounces per month to the baker, butcher, and others in buying the goods his household requires - such that the amount of money needed is 100 ounces with the rest remaining in reserve.
It rarely happens that landlords spend their rents in so fixed and regular a manner, when their own income is so irregular. Unless they have significant wealth to sustain them until the harvest, the landlord has borrowed money to support his household and repays the loan at the time of harvest.
The farmer, too, is often a debtor, having borrowed money to purchase the items and pay the workers necessary to produce his crop, a significant portion of the large sum he receives on its sale is already owed to creditors. As such, the community has been using some of the proceeds of the harvest since even before the ground was furrowed.
Even when there are no debts to repay, the landlord who receives a large sum of rent at harvest does not keep the money in a vault, but instead entrusts it to a banker, who loans it out at interest until the depositor needs it. In that way the sum held in reserve is not idle, but in circulation.
In this way, the mass of coin necessary for a large-scale transaction is not needed on an ongoing basis - and it is not so much amassed as it is gathered together temporarily, and soon enough dispersed.
Direct barter of goods also decreases the amount of money needed for circulation. That is, if the tailor desires wheat, then the farmer may trade some of his wheat for clothing directly with the tailor, rather than selling his wheat for money and spending money at the tailor. This further decreases the need of money to be used in commerce.
Consider the commerce between village and town, in which the produce of farmers of the village is wanted in the town for sale, and goods made by craftsmen in the town are wanted in the village. A single trader facilitating the exchanges between village and town must use money to purchase the crop, recovers his money when he resells the crop in the town, spends the money back out to obtain goods, and collects the money when the town goods are delivered to the village.
If money used in all four exchanges should not merely be added together to determine the amount needed in aggregate. Consider that in the village, the farmers who purchase town goods are giving back to the trader the same coins he paid them for their crop; and in the town, the merchant buys goods with the same coins that the merchants paid him for the village product.
Granted, it is not perfectly the same people with perfectly the same coin, but in aggregate it makes little difference: the same amount of currency is used in circulation. Even that which may be kept in profit is banked temporarily, eventually to be spent, and even while it is banked it is loaned out for others to use.
It's also observed that in large-scale transactions, money is used less frequently than bills of credit and notes of deposit. So long as the reputation of the issuer is maintained, these notes may circulate in the place of money for a considerable amount of time before they are redeemed.
All of these considerations are focused on the collection and disbursal of large sums of money, but there are many instances in which a large sum need not be gathered. Consider that it may cost in total 100,000 ounces of silver to build a house - but this sum is never gathered together. Instead, the masons, carpenters, and other tradesmen whose labor is enjoined are paid in small sums each week over the duration of construction, and these sums are dispersed further and in smaller amounts to make the daily purchase of their households.
Even for goods of modest price, which is indeed paid all at once, the aggregation of funds is temporary. The customer pays a carriage builder 400 ounces of silver for a carriage - who then immediately pays his laborers and suppliers if they worked on credit - though it is more common that the builder has already paid them and instead buys the materials to build his next carriage.
For the vast majority of people, consumption is done in small amounts daily, and it is not their habit to purchase a stockpile a year's worth of food and drink at once, but to make such purchases in small increments, daily or weekly. Neither do they purchase clothing for several years all at once but instead buy a few items each season.
Circulation in cities is very frequent and in small amounts, especially among the working classes whose wages are paid daily or weekly and who purchase goods daily or weekly in many small transactions, such that money is seldom ever aggregated. The income of a skilled craftsman may exceed that of a country farmer, but unlike the farmer the craftsman does not receive the profit all at once, at the end of the season. And as such little actual money is needed.
The need for cash is increased by the tendency of people to hoard it, though there is generally some reason for money to be collected (small amounts are saved over time toward a large purpose), the fulfillment of which will again return the cash to circulation. It is not unheard of, but not entirely unusual, for miserly people to hoard great amounts of cash for long periods of time - and even among them common practice is to maintain it in a bank (whence it is loaned for others to use) than to keep hard currency.
In all, there is such great variety in the commercial behavior of the various people within a nation that it would be impossible to arrive at a precise calculation of the amount of money truly needed for circulation. But by these examples it should be clear it is far less than might be supposed.