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To realize these two sets of conditions each cistern is divided intotwo by a vertical partition of wood. The front compartments are allof unit thickness one inch (say). All front compartments belongingto the same front-and-back row are mutually connected by tubes (inthe 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 (aswill soon appear) constrained to be always equal to the price, thusif the price of A is $1, of B 83 and C $1.20, the thickness of allcisterns in the A row will be 1, in the B row 3 and in the C row1.2 (inches).
Since the thickness of the front compartment is unity, the con-tents of each back compartment equals the contents of the frontmultiplied by the number of inches of thickness of the back cistern,that is the back compartment contains a volume of water equal tothe amount of the commodity multiplied by its price. It containstherefore the money value of the commodity. The double cisternrepresents the double light in which each commodity is commonlyregarded—so many pounds, yards, etc. and so many dollar’s worth.
All back compartments of the same right and left rows aremutually connected by tubes—that is the sum of their contents isgiven—thus fulfilling the second set of conditions.
The back compartments can change their thicknesses, as the wallsat the right, left and bottom are of flexible leather; the back planeis kept parallel to the wood partition by two double “ parallel rules”not diagramed.
There remains to be described the system of levers. The purposeof 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 (F12, etc., fig. 9) con-nected by sliding pivots with the tops of the cisterns and havingtheir lower extremities hinged to wooden floats F, the hinges beingon the level of the water of the tank. These floats are free only toshift laterally. It is evident from the similar triangles FRl andFR2 in fig. 9 that the ordinates of the two cisterns IA and IB areproportional to the distances of the A and B rods R and K fromthe hinge in the left float F. Likewise in the row behind, the ordi-nates are proportional to the same distances. Hence the fourordinates are proportional to each other and in general all the
* A convenient angle for each lever can be assured by a careful selection ofcommodity units. Thus if the marginal utility per pound gives inconvenientordinates in the A row, reconstruct the cisterns in that row so that the ordinatesare lengthened to represent marginal utility per ton or shortened for the ounce.