There remains the further extension to what may be called “multiplicate proportion” (duplicate, triplicate, … n · plicate). This may be regarded as a limiting case of compound proportion, being a relation between variable quantities P, X of two different kinds such that measures p, x of these variables are in the proportional relation p α xn. (The definition can, of course, be expressed—like the definitions of direct and inverse proportion—in terms of ratios of any two quantities of the one kind and the two corresponding quantities of the other kind.)

“The constant reference to this barleycorn measure (which is seldom, if ever, omitted) induced me to try what it really would make. There is some difference between the breadths of barleycorns. A certain statement of Thevenot (cited in the History of Astronomy, Lib. Usef. Kn.) makes the breadths of 144 grains of oriental barley give 1½ French feet; at which rate 64 would only give 8 53 inches English. A sample from a London shop gave me (when the largest grains were picked out) 33 to just more than five inches. Some other samples, procured from two different parts of England as the finest which could be got, gave 33 to 5 inches, 33 to 5.1 inches, and 33 to 5.1 inches. The average of these is 9.8 inches to 64 grain-breadths : a result which coincides more nearly than could have been expected with the following determinations.