This may be expressed in mathematical language, where V is the value of money, Q is the quantity in circulation, and R the number expressing the rapidity of circulation, as follows: V = 1 / (Q × R). The phrase, rapidity of circulation, requires some comment. It must not be understood to mean the number of purchases made by each piece of money in a given time. Time is not the thing to be considered. The state of society may be such that each piece of money hardly performs more than one purchase in a year; but if this arises from the small number of transactions—from the small amount of business done, the want of activity in traffic, or because what traffic there is mostly takes place by barter—it constitutes no reason why prices should be lower, or the value of money higher. The essential point is, not how often the same money changes hands in a given time, but how often it changes hands in order to perform a given amount of traffic. We must compare the number of purchases made by the money in a given time, not with the time itself, but with the goods sold in that same time. If each piece of money changes hands on an average ten times while goods are sold to the value of a million sterling, it is evident that the money required to circulate those goods is £100,000. And, conversely, if the money in circulation is £100,000, and each piece changes hands, by the purchase of goods, ten times in a month, the sales of goods for money which take place every month must amount, on the average, to £1,000,000. [The essential point to be considered is] the average number of purchases made by each piece in order to affect a given pecuniary amount of transactions. “There is no doubt that the rapidity of circulation varies very much between one country and another. A thrifty people with slight banking facilities, like the French, Swiss, Belgians, and Dutch, hoard coin much more than an improvident people like the English, or even a careful people, with a perfect banking system, like the Scotch. Many circumstances, too, affect the rapidity of circulation. Railways and rapid steamboats enable coin and bullion to be more swiftly remitted than of old; telegraphs prevent its needless removal, and the acceleration of the mails has a like effect.” “So different are the commercial habits of different peoples, that there evidently exists no proportion whatever between the amount of currency in a country and the aggregate of the exchanges which can be effected by it.”(229) § 4. Explanations and Limitations of this Principle.