*THE MATHEMATICAL PRINCIPLES*

business as the science of exchange. This science, which can be summed up in a table found everywhere, should not distract our attention. In other words, we have nothing to do with nominal exchange, but only with actual exchange, *i.e.* the ratio between the values in exchange of the same weight of fine silver according as it is payable in different places. It is plain also that the cost of exchange, or the difference of the ratio of exchange from unity, cannot exceed the cost of transportation of this weight of fine silver from one place to the other, when free trade in the precious metals is allowed between the two places, or the cost of transportation plus the expense of smuggling when this trade is embarrassed by prohibitory laws. To find the *equations of exchange*, we will suppose, to begin with, that the cost of exchange is less than the cost of transportation, or that the exchange takes place without any real transportation of money, without any change in the distribution of the precious metals between the two commercial centres.

14. Let us suppose at first only two centres of exchange. Let us designate by *m*_{1,2} the total of the sums for which the centre (1) is annually indebted to the centre (2); and by *m*_{2,1} the total of the sums for which the place (2) is annually indebted to the place (1); by *c*_{1,2} the rate of exchange at the place (1) on the place (2), or the amount of silver given at the place (2) in exchange for a weight of silver expressed by 1 and payable at the place (1).

Adopting this notation, and starting from the hypothesis that the two places balance their account without transporting money in either direction, it is plain that we shall have