22 



Irving Fisher — Mathematical investigation!^ 



equally sure to diminish gain. Herein lies the virtue of insur- 

 ance and the vice of gambling. Xeither alters (directly) the 

 amount of wealth. But insurance modifies and gambling intensi- 

 fies its fluctuations. Hence the one increases the other decreases 

 gain. 



§13. 



Again it may be objected to the foregoing definitions that the use 

 of infinitesimals is inappropriate since an individual does not and 

 cannot reckon infinitesimals. The same apparent objection attaches 

 to any application of the calculus. We test forces by weights but 

 cannot weigh infinitesimal masses nor do they probably exist; yet 

 the theory of forces begins in infinitesimals. We apply fluxions to the 

 varying density of the earth, though we know that if we actually 

 take the infinitesimal ratio of mass to volume we shall generally get 

 zero since matter is discontinuous. The pressure of a confined gas 

 is due to collisions of its molecules against the containing vessel. 

 As each molecule rebounds the chano-e of momentum divided bv the 

 infinitesimal time is the pressure. Yet at any actual instant the 

 value of this fluxion is quite illusory. But these facts do not mil- 

 itate against the use of fluxions for a thinkable theory of forces, 

 density and gaseous pressure. In cases of discontinuity fluxions 

 have important applications though infinitesimals may not exist. 

 The rate of increase of population at a point in time is an impor- 

 tant idea, but what does it mean ? It is convenient to define it as 

 infinitesimal increase of population divided by the infinitesimal time 

 of that increase though we know that population increases discon- 

 tinuously by the birth of whole individuals and not of infinitesimals. 



Practically we can find the approximate 

 marginal utility of a commodity just as 

 we approximately find the rate of increase 

 of population by taking small increments 

 in place of infinitesimals. 



In actual fact inequality of utilities is 

 the rule and absolutely equal utilities never 

 exist. Instead of a curve of utility we 

 should draw a belt (fig. 1) whose limits 

 are ill-defined and whose width in general 

 depends on the amount of antecedent atten- 

 tion which the individual has bestowed on the alternative amounts 

 and modes of consumption. 



