In proving that an increase of money must proportionately increase prices, it is necessary to prove that the volume of trade is independent of the quantity of money and credit instruments by means of which trade is carried on. Money on the one hand, and quantity of goods to be exchanged on the other, are the two great independent magnitudes, whose equilibration mechanically fixes the average of prices. This notion, as to the essence of the quantity theory, finds expression in Taussig,[225] "The statement of a quantity theory in relation to prices assumes two independent variables: total money or purchasing power on the one hand, total supply of goods or volume of transactions on the other." Taussig, though he would maintain that this independence holds, so far as money and trade are concerned, admits that it breaks down so far as trade and elastic bank credit, bank-notes and deposits, are concerned. Trade and elastic bank-credit are largely _inter_dependent.[226] This concession on Taussig's part means virtually giving up the quantity theory for Western Europe and the United States and Canada, though Taussig still sees something left of the quantity theory tendency in view of the "irregular and uncertain" connection which he finds between money and bank-credit.[227] Fisher, however, makes no such surrender. He is quite as uncompromising as to the independence of _deposits_ and trade as he is with reference to the independence of _money_ and trade. He does, indeed, make the concession that increasing trade tends to increase deposits _indirectly_, by increasing the ratio of M' to M, by modifying the habits of the people as to the use of checks as compared with cash (p. 165),[228] but he denies stoutly that there is any _direct_ relation between them. (P. 168.) Trade acts only _via_ a modification of the ratio between M and M', and M still remains controlled, not by trade, but by quantity of money. As to any control over T by M', he repudiates it explicitly, (P. 163.) Increasing M', either through an increase of M, or through an increase in the normal ratio between M and M', will have no effect on T,--or, for that matter, on the V's. The introduction of credit, therefore, leaves the quantity theory intact: an increase of M, increasing M' proportionately, leaving the V's unchanged, and having no effect on T, must exhaust its influence on P, raising P proportionately, if the equation of exchange is to remain valid.