Quadrantal, the Roman standard of capacity, a cubic vessel measuring one foot on each of its inside panels.
When it is seen that a difference of 1 in 2500 in the length of the foot taken as the inside measure of a quadrantal makes a difference of 3 cubic inches out of 1728 in its capacity, the material difficulties of constructing a vessel exactly cubical will be understood. However, a quadrantal being constructed, perhaps after many trials of sides as exactly equal as possible, and holding 1000 ounces of Roman ounces (= 437 grains) of water, the mean measure of its panels was taken as a foot, and the quadrantal as a cubic foot—the wine-bushel.
Let us take 1000 Roman ounces and divide the total number of grains weight by the statute number of grains in a cubic inch of water as determined by Captain Kater in 1824.
The dividend will be the number of cubic inches, and its cube root will be the foot—
of which the cube root is 12·0042 inches, a length differing by only 1/2400 from the actual Imperial foot.
I took the idea of this hypothesis from that by which Don V. V. Queipo inferred the Beládi cubit to be the double measure of the side of a cubical vessel containing a Ptolemaïc talent of water. Certainly it solves the question of the origin of our foot, and it happens that, applied to the equally obscure origin of the Rhineland foot, the results are equally satisfactory.
Let the same process of involution be applied to the side of a cubical vessel containing 1000 Troy ounces of water.
The standard of Troy weight varied very much, from the Danish value of a little over 481 grains in the ounce, to the French Troy value of 472·13 grains.
The Scots Troy weight, = 476·09 grains to the ounce, is nearly the same as the Amsterdam weight, = 476·68 grains.
These Troy weights may be taken at three main standards, high, medium, and low, represented by:
English Troy, its ounce = 480 grains Amsterdam „ „ „ = 476·68 „ French „ „ „ = 472·13 „
Let us apply to 1000 ounces of water, at the medium Amsterdam standard, = 10 Egyptian dirhems of 47·6 grains, the same measurement of a quadrantal made to contain them as exactly as possible.
and the cube root of the dividend gives 12·357 inches, exactly, to 1 in 20,000, the Rhineland foot as established in Prussia = 12·3564 inches. The Prussian standard of the Cologne pound (its ounce = 451·1 grains) was 1/66 of a Rhineland cubic foot of water at 65·75 F., and was fixed at 7217·9 grains. This was exactly 1/66 of 1000 Troy ounces of water at the standard of 476·38 grains. So 66 Prussian pounds were equal to 1000 Troy ounces, or to 62·5 Troy pounds at that standard.
