A further feature of Professor Fisher's system, to which especial attention must be given, is the large role played in it by the "equation of exchange." This device has been used by other writers before him, notably by Newcomb, Hadley, and Kemmerer, receiving at the hands of the last named an elaborate analysis. But Fisher, basing his work on Kemmerer's, has made even more extensive use of the "equation of exchange," and has given it a form which calls for special consideration.[137] The "equation of exchange," on the face of it, makes an exceedingly simple and obvious statement. Properly interpreted, it is a perfectly harmless--and, in the present writer's opinion, useless--statement. It gives rise to complications, however, as to the meaning of the algebraic terms employed, which we shall have to study with care. The starting point is a single exchange: a person buys 10 pounds of sugar at seven cents a pound. "This is an exchange transaction in which 10 pounds of sugar have been regarded as equal to 70 cents, and this fact may be expressed thus: 70 cents = 10 pounds of sugar multiplied by 7 cents a pound. Every other sale and purchase may be expressed similarly, and by adding them all together we get the equation of exchange _for a certain period in a given community_."[138] The money employed in these transactions usually serves several times, and hence the money side of the equation is greater than the total amount of money in circulation. In the preliminary statement of the equation of exchange, foreign trade, and the use of anything but money in exchanges are ignored, but later formulations of the equations are made to allow for them. "The equation of exchange is simply the sum of the equations involved in all individual exchanges in a year.... And in the grand total of all exchanges for a year, the total money paid is equal in value to the total value of the goods bought. The equation thus has a money side and a goods side. The money side is the total money paid, and may be considered as the product of the quantity of money multiplied by its rapidity of circulation. The goods side is made up of the products of quantities of goods exchanged multiplied by their respective prices." Letting M represent quantity of money, and V its velocity or rapidity of circulation, p, p', p'', etc., the average prices for the period of different kinds of goods, and Q, Q', Q'', etc., the quantities of different kinds of goods, we get the following equation: MV = pQ + p'Q' + p''Q'' + etc.[139] "The right-hand side of this equation is the sum of terms of the form pQ--a price multiplied by the quantity bought."[140] The equation may then be written, MV = [Greek: S] pQ (Sigma being the symbol of summation).