80 Irving Fisher—Mathematical investigations

concern us. It seems unfortunate that only mathematicians shouldhe acquainted with this fact.

§ 6 .

In this m dimensional space make m mutually perpendicular axesfor the commodities A, B, C, . . . M. Fill the space with a totalutility density. Pass an m— 1 flat* through the origin giving it theproper orientation in view of the prices. The indifference loci willhe (in— 1) spaces (curved). The point of tangency of the (m— 1)flat with an (m— 1) indifference locus will indicate the total con-sumption and production combination for an individual. A normalto the (»! — ]) flat and (ra—1) indifference locus at their tangencyshows his “ maximum direction ” and its components the marginalutilities of all articles.

These ideas are not so unfamiliar as they appear. This space issimply the “ economic world ” in which we live. We often speak ofspending 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 anatural attempt to group in thought m different magnitudes. Thisis accomplished by regarding them as coordinates of a “ point ” inthe “ economic world.” It is an application to economics of thoseideas of “ multiple algebra ” which have addedf so to the beautyand simplicity of geometry and mathematical physics.

§ 7 .

These conceptions will tend to a more compact comprehension ofthe nature of economic equilibrium. In order to have equilibriumin the whole system including production :

(1) The utility distribution must be giveu 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 individual points : that is the sum of all A coordinates for consumptionmust equal the sum for production and likewise for B, C, etc.

(4) The common income and expenditure flat must pass throughthe origin : that is the money values of each man’s production andconsumption must cancel.

* I. e. a Euclidean Bpace of (m — 1) dimensions related to the m-dimensionalspace as a plane is to our space.

fSee J. W. Gibbs, Multiple Algebra, Proceedings Amer. Asso. Adv. Sei., vol.

XXXV.