The Great Art of Artillery, Part One

...from 9, 3 remain.

Let it suffice to have said this much briefly concerning the extraction of the Cubic Root and the knowledge of Cubic numbers; for the use of the aforementioned will indeed be better revealed in what follows.

A table of Cubic Roots ascending from unity to infinity must be constructed. Therefore, first assume any number at pleasure as a root, which, when cubed in itself, produces the first Cubic number. Its Cubic Root, or the number you assumed as the root, will be placed in the first position in the table. For example, if the number 100 is taken as the root and is cubed in itself, 1,000,000 arises, which is the first Cube; its Cubic Root is 100, and thus it should be placed in the first position in the table as the first root. If you now desire the root of a doubled cube, double the first cube, and 2,000,000 will arise; seek the Cubic Root of this number, and the result will be approximately 125, to be transferred into the table as the second root. If you wish to have the roots of a tripled, quadrupled cube, and so on increased to infinity, triple and quadruple the first cube, and multiply and increase it to infinity, and from those numbers extract the Cubic Roots, and arrange them in order in the table, noting the numbers ascending in a natural series from unity to infinity. We have constructed the table placed below using this same art. By its help, if you now wish to construct a Regula Calibræ Caliber Rule, it is necessary first to have the diameter of a one-pound ball made from that metal for which you wish to construct the Caliber Rule. For instance, if you wish to prepare a Caliber Rule for calibrating iron balls, divide the diameter of a one-pound iron ball, taken from an iron globe (I shall teach how this is done below), into as many equal parts as the first root has units in the table of Cubes. As here in our table the first root consists of 100 units, divide that diameter of the one-pound iron ball which you have at hand into 100 equal parts, with the aid of the parallelogram delineated under No. 1. Then, using a common pair of compasses, transfer the particles taken from this scale into the Caliber Rule in the order in which the numbers are arranged in the table of Cubes. For example, if 100 parts are assumed from the scale above for the diameter of a 1 lb. iron ball, 125 must be placed for the diameter of a 2 lb. ball—that is, 25 parts must be added to the first diameter. For the diameter of a 3 lb. ball, 144 parts must be taken, or 44 parts must be added to the first diameter; these, joined together, will constitute the diameter of a 3 lb. iron ball. In a similar manner, you will easily transfer the diameters of other balls into the Caliber Rule. This growth of the diameters, and of the circumferences described upon them according to the ratio of increased solids, appears clearly in the figure under No. 2, in which the first circumference is the circumference of a globe whose diameter is the first root, and its solidity is the first Cube. The second circumference is the circumference of a globe whose diameter is the second root, and its solidity is the doubled cube, or double the first. The same must be understood regarding the remaining circumferences of circles, their diameters, and their solidities in the same figure.

What has been said just now concerning iron balls must also be understood regarding those made of lead and stone, and likewise of other metals, for the calibration of which a Caliber Rule will be easily constructed according to the premises above.

We present to view the figure of this Rule, and the diameters of iron balls described on one surface and leaden balls on the other, under No. 3.