40 Irving Fisher — Mathematical investigations 



To realize these two sets of conditions each cistern is divided into 

 two by a vertical partition of wood. The front compartments are all 

 of unit thickness one inch (say). All front compartments belonging 

 to the same front-and-back row are mutually connected by tubes (in 

 the tank but not in connection with the water of the tank) thus ful- 

 filling the first set of conditions. 



The thickness of the back compartments is adjustable but is (as 

 will soon appear) constrained to be always equal to the price, thus 

 if the price of A is |1, of B |3 and C $1.20, the thickness of all 

 cisterns in the A row will be 1 , in the B row 3 and in the C row 

 1.2 (inches). 



Since the thickness of the front compartment is unity, the con- 

 tents of each back compartment equals the contents of the front 

 multiplied by the number of inches of thickness of the back cistern, 

 that is the back compartment contains a volume of water equal to 

 the amount of the commodity multiplied by its price. It contains 

 therefore the money value of the commodity. The double cistern 

 represents the double light in which each commodity is commonly 

 regarded — so many pounds, yards, etc. and so many dollar's worth. 



All back compartments of the same right and left rows are 

 mutually connected by tubes — that is the sum of their contents is 

 given — thus fulfilling the second set of conditions. 



The back compartments can change their thicknesses, as the walls 

 at the right, left and bottom are of flexible leather; the back plane 

 is kept parallel to the wood partition by two double " parallel rules" 

 not diagramed. 



There remains to be described the system of levers. The purpose 

 of these levers is to keep the continuous ratio of marginal utilities, 

 the same for all individuals and equal to the ratio of prices. 



First there is a system of oblique* levers (Fl2, etc., fig. 9) con- 

 nected by sliding pivots with the tops of the cisterns and having 

 their lower extremities hinged to wooden floats F, the hinges being 

 on the level of the water of the tank. These floats are free only to 

 shift laterally. It is evident from the similar triangles FRl and 

 FR2 in fig. 9 that the ordinates of the two cisterns lA and IB are 

 proportional to the distances of the A and B rods R and K from 

 the hinge in the left float F. Likewise in the row behind, the ordi- 

 nates are proportional to the same distances. Hence the four 

 ordinates are proportional to each other and in general all the 



* A convenient angle for each lever can be assured by a careful selection of 

 commodity units. Thus if the marginal utility ■per pound gives inconvenient 

 ordinates in the A row, reconstruct the cisterns in that row so that the ordinates 

 are lengthened to represent marginal utility per ton or shortened for the ounce. 



