14 Irving Fisher — Mathematical investigations 



purchaser at the time of purchase the quantity of the commodity 

 purchased m^idtiplied by its marginal utility equals the like pyroduct 

 for the commodity sold. Or again: for a given purchaser the utili- 

 ties of A and B, though actually unequal icould be equal if every 

 portion of A (and also of B) ^ere rated at the same degree of utility 

 as the last infinitesimal. This hypothetical equality underlies, as 

 will subsequently appear, the notion of the equality of values of A 

 and B. 



§ -^• 



But the tTTO definitions (1) and (2) do not fully determine the 

 sense in which utility is a quantity. To define when the " grades " 

 of two parts of a highway are equal or unequal (viz : when they 

 make equal or unequal angles with a horizontal), does not inform us 

 when one shall be ticice as steep as the other. It does not oblige us 

 to measure the '* grade" by the sine of the angle of elevation, or 

 by the tangent, or by the angle itself. If the two highways were 

 inclined at 10" and 20° respectively, the "grades" have a ratio of 

 1-97 if measured by sines, of 2*07 by tangents, and exactly 2 by 

 angles. For a long time philosophers could define and determine 

 when two bodies were equally or unequally hot. But not till the 

 middle of this century* did physicists attach a meaning to the phrase 

 " twice as hot." 



It is here especially that exactitude has been hitherto lacking in 

 mathematical economics. Jevons freely confesses that *• TTe can 

 seldom or never affirm that one pleasure is an exact multiple of 

 another."! 



Kow throughout Part I the assumption is made that the utility of 

 any one commodity (or service) depends on the quantity of that 

 commoditj^ or service, but is independent of the quantities of other 

 commodities and. services. This assumption is preliminary to the 

 definition we seek. 



Onr first problem is to find the ratio of two infinitesimal utilities. 

 If an individual I consumes 100 loaves of bread in a year the utility 

 of the last infinitesimal, or to fix our ideas, the utility of the last 

 loaf is (presumably) greater than what it would be if he consumed 

 150 loaves. What is their ratio/ It is found by contrasting the 

 utilities of the 100th and 150th loaves with a third utility. This 



* The first thermodynamic definition of one temperature as a multiple of 

 another was made hy W. Thomson in 1848. See Maxwell, Theory of Heat, p. 

 155. 



tp. 13. 



