4 Preface. 



ungen ueber die Theorie des Preises^ To the former I owe the idea 

 of marginal utility and of mathematical treatment in general, to the 

 latter the clear conception of the " symmetr}^" of supi^lv and (Jemand 

 and the use of rate of commodity in place of absolute commodity, 

 and to both many minor obligations. 



The equations in Chapter lY, § 10, were found by me two years 

 ago, when I had read no mathematical economist except Jevons. 

 They were an appropriate extension of Jevons' determination of 

 exchange of tico commodities between tico trading bodies to the 

 exchange of any number of commodities between any nutnher of 

 traders and were obtained as the interpretation of the mechanism 

 which I have described in Chapter IV. That is, the determinate- 

 ness of the mechanism was expressed by writing as many equations 

 as unknowns. These equations are essentially those of Walras in 

 his Elements d"* economic politiciue pure. The only fundamental 

 differences are that I use marginal utility throughout and treat it as 

 a function of the quantities of commodity, whereas Professor Walras 

 makes the quantity of each commodity a function of the prices. 

 That similar results should be obtained independently and by sepa- 

 rate paths is certainly an argument to be weighed by those skeptical 

 of the mathematical method. It seemed best not to omit these ana- 

 lytical portions of Part I, both because they contribute to an under- 

 standing of the other portions of the work and because they were in 

 a proper sense my own. 



Three days after Part II was finished I received and saw for the 

 first time Prof. Edgeworth's JMathematical Psychics. I was much 

 interested to find a resemblance between his surface on page 21 and 

 the total utility surfaces* described by me. The resemblance, how- 

 ever, does not extend far. It consists in the recognition that in an 

 exchange, utility is a function of both commodities (not of one only 

 as assumed by Jevons), the use of the surface referred to as an inter- 

 pretation thereof and the single phrase (Math. Psych., p. 28) "and 

 similarly for larger numbers in hyperspace" which connects with 

 Part 11,^ Ch. IT, § 5. 



There is one point, however, in which, as it seems to me, the 

 writer of this very suggestive book has gone far astraj'. Mathe- 



* His result, which translated into my notation is 



\dAj\dBj \cmJ\dAj 

 becomes by transposition and division identical with part of the continuous pro- 

 portion, Pai-t I, Ch. TV, ^3. 



