80 Irving Fisher — Mathematical investigations 



concern us. It seems unfortunate that only mathematicians should 

 be acquainted with this fact. 



§ 6. 



In this ra dimensional space make ni mutualh^ perpendicular axes 

 for the commodities A, B, C, . . . M. Fill the space with a total 

 utility density. Pass an m— 1 flat* through the origin giving it the 

 proper orientation in view of the prices. The indifference loci will 

 be (r/i— 1) spaces (curved). The point of tangency of the (m— ]) 

 flat with an (w— 1) indifference locus will indicate the total con- 

 sumption and production combination for an individual. A normal 

 to the [ni — l) flat and (r/i — 1) indifference locus at their tangency 

 shows his " maximum direction " and its components the marginal 

 utilities of all articles. ., 



These ideas are not so unfamiliar as they appear. This space is 

 simply the " economic world " in which we live. We often speak of 

 spending an income in this or that " direction," to express the rela- 

 tive amounts of commodities. When one speaks of the "point" 

 which a consumer or producer reaches, the use of the word is a 

 natural attempt to group in thought ra different magnitudes. This 

 is accomplished by regarding them as coordinates of a '* point " in 

 the " economic world." It is an application to economics of those 

 ideas of " multiple algebra " which have addedf so to the beauty 

 and simplicity of geometrj^ and mathematical physics. 



These conceptions will tend to a more compact comprehension of 

 the nature of economic equilibrium. In order to have equilibrium 

 in the whole system including production : 



(1) The utility distribution must be given for each individual. 



(2) The " maximum directions " must be alike among all indi- 

 viduals and between production and consumption. 



(3) The origin must be the centre of gravity of all the individ- 

 ual points : that is the sum of all A coordinates for consumption 

 must equal the sum for production and likewise for B, C, etc. 



(4) The common income and expenditure flat must pass through 

 the origin : that is the money values of each man's production and 

 consumption must cancel. 



*I. e. a Euclidean space of {m — 1) dimensions related to the ») -dimensional 

 space as a plane is to our space. 



f See J. W. Gibbs, Multiple Algebra, Proceedings Amer. Asso. Adv. Sci., vol. 



XXXV. 



