milk must be, then, commensurable quantities, i. e., must have a common quality, present in each in definite quantitative degree. This quality is value, the exchange ratio will vary with the extent to which the common quality is present in each of the goods. We can have no quantitative ratios between unlike things. And yet, we must have terms for our ratios." The logic seems conclusive. If the value relation is a ratio this fact implies quantities of something homogeneous, and that homogeneous something becomes very important, so important as to demand an important name. To call it "value" is the obvious conclusion, leaving the term "price" free to express, if desired, the ratios of exchange, monetary and non-monetary, from which the existence of the quantity, "value" was inferred.

But this whole logical structure rests on the fact that the adversaries have used terms which assume the conclusion for which Professor Anderson is fighting. And even this is only true on condition that it can be shown that the word "ratio " is to be taken throughout in one very limited meaning: that of a quotient of two commensurable quantities. But the term is often used loosely, as all terms are in common speech, and this looser usage has even gained the dignity of recognition by dictionaries. Nor need this be regretted; men must use words loosely. If they did not, all literature would be reduced to mathematics and nobody would read it. This looseness does no special harm so long as neither writer nor reader shifts over to the strict usage, and begins drawing conclusions based on one usage from statements based on the other.

To illustrate: a rate of speed, say ten miles an hour, has sometimes been called a ratio. Suppose now some one were to attempt to prove from this that distance

1 See, for example, Webster's New International Dictionary, 1909.

and time have a common quality, and that an hour has just ten times as much of this quality as a mile has? Or even that a mile of a given road has a quality in common with an hour of Mr. Kolehmainen's running, and has just one-tenth as much of it? The answer is severely simple. Ten miles an hour is clearly a rate, but it is not so clear that it is proper to call it a ratio. Certainly it is not a ratio in the sense of a quotient between miles and hours. It can be stated so as to involve a ratio, it is true, but the result only shows how much more than a mere ratio it is. We are talking of a rate of speed

such that the number of miles covered is to the number of hours elapsed as the number ten is to the number one. Here we have a pair of ratios, not one, and both are ratios between abstract numbers, as indeed all ratios must be in the strict mathematical sense. We have not divided distance by time: we cannot, any more than we can divide apples by potatoes. We have merely divided one number by another. If we call the whole expression "ten miles an hour" a ratio, we are merely defining "ratio" loosely, not implying any theorems as to the oneness of time and space.

Zeno proves that Achilles cannot move to catch a tortoise. What has he proved? Simply that Zeno's conception of motion is artificial and false, since he has conceived it as something Achilles cannot do. Man moves, then finds a word to express his action, then frames syllogisms about it. The final appeal is from Zeno to Achilles. And from those who call prices ratios and from those who draw conclusions based on this usage, the final appeal is to facts stripped bare of all that may have been read into them.

Suppose now that Smith gives Brown forty gallons of milk and receives in exchange ten bushels of wheat, or

perhaps a warehouse receipt for 258 grains of standard gold bullion. Milk exchanges for wheat at the rate of four gallons per bushel and for gold at 25 cents per gallon or, as the farmer is quite as likely to say, four gallons for a dollar. The writer contends that these phrases express ratios in just the same sense that "ten miles an hour" does, and in no other - that is, they do not strictly express ratios at all, but rates. A rate tells us that for every unit of one thing so many units of something else may be achieved or obtained; for every hundred dollars of principal five dollars of interest, for every thousand dollars' worth of real estate, eighteen dollars of taxes, for every hour, ten miles, for every dollar, four gallons of milk. The terms of a rate may or may not be commensurable with each other.

Again, cigars may sell for ten cents apiece, three for a quarter, or three dollars and a half for a box of fifty. Strange that there should be three different ratios existing at once between the same two terms! This is supermathematics with a vengeance. But if we are talking about rates, not ratios, there is no more inconsistency about a "rate" of exchange which varies with quantity than there is about a runner who covers one mile at a faster rate than he could keep up for five times that distance.

Indeed, there are various ways in which, whether it is price or value that one is talking about," rate seems a more handy word than "ratio" for the latter term introduces an element of arithmetical unreality which must be explained away, or illustrated away, before the discussion can move on. For example: 1 1 When a certain quantity of wealth of one kind is exchanged for a certain quantity of wealth of another kind, we may divide either of the two quantities by the other and

1 Irving Fisher, Elementary Principles of Economics, pp. 13–14.

obtain what is called the price of the latter. That is, the price of wealth of one kind in terms of wealth of another kind is the ratio of exchange between the two, i. e., the ratio of the number of units of the latter to the number of units of the former which will be given in exchange. Thus, if 200 bushels of wheat are exchanged for 100 ounces of silver, the price of the wheat in terms of silver is 200 ÷ 100 or two bushels per ounce. Thus, there are always two prices in any exchange. Practically, however, we usually speak only of one, viz., the price in terms of money, obtained by dividing the number of units of money by the number of units of the article exchanged for that money. It follows that the price of any particular sort of wealth is the amount of money for which a unit of that wealth is exchanged."

Reading this passage in the light of the foregoing, does it not appear that the author is put to much trouble simply because he twice brings in the idea of a mathematical ratio, or quotient, between abstract numbers and then twice has to make clear that he really means something different a much more complex relationship between quantities of concrete things? As a price,

200 100 by itself means nothing, and 200 bu.: 100 oz. would mean exactly as much. The true ratio here is only an intermediate step in the process of finding the price, and disappears when its function is performed. The process in its painful fulness is as follows: 200 bushels of wheat buy 100 ounces of silver. How many ounces of silver does each bushel buy? In getting the answer, oz., we really use, not one ratio, but a proportion of two ratios, one between wheat and wheat, the other between silver and silver, thus: 1 bu.: 200 bu.::

oz.: 100 oz. Why not say " wheat buys silver at the rate of oz. per bu.," and then forget that the method of proportion was used to find the answer?

The case is much the same with the briefer statement: 1 "Value is a ratio of exchange between two goods, quantitatively specified." The troublesome word once having been inserted, the statement must at once be qualified in order to show that the ratio is not between the commodities, but between the abstract numbers of the units of measure which each commodity contains.

Now if "ratio" does not really mean ratio, but rate, then all this trouble is needless, and the sources of our terminological discord may be diminished, if ever so little. And besides, we should economize one syllable. Those who hold that value is a relation should be the last to adopt a term which delivers them needlessly into the hand of the doctrinal enemy. And if the enemy, holding that value is a quantitative thing, chooses to define price as a ratio between values, why he has thereby assumed the truth of his conclusion, but not strengthened the evidence in its favor. Indeed, he is in danger of proving too much; of proving that value is not merely a quality but a jelly, and of running foul of the experience common to all men who have ever debated whether or not to buy their cigars by the box. It would seem that the use of the term rate" would avoid some embarrassment, ambiguity, and sterile dialectic. To pay for this we should merely incur a slight awkwardness when speaking of things like works of art which are unique; since the expression "rate of exchange" suggests a considerable number of sales.

So much for the matter of terminology. Meanwhile, the question remains unsettled whether value is a mere relation between goods and derived from the fact of

1 Davenport, Value and Distribution, p. 569. In his later book Davenport substitutes the word "relation" for "ratio." Economics of Enterprise, p. 236.