The Great Art of Artillery, Part One

I

A mathematical calculation for extracting a cube root, showing a multi-digit number (34258630921) with several digits struck through, and a vertical list of calculated terms with their labels to the right.

This, therefore, is the sum of the whole operation. If, however, there remain more digits of numbers from which the Radix Cubica Cubic Root ought to be extracted, that operation differs not even by a dot from the canon already stated: that is, the whole quotient is tripled, the Triple is multiplied by the immediately found root, and the cube of the newly found root is added to the product. Finally, the aggregate is subtracted from the number above, and the remainder (if there is any) is noted above. As in our example; because more digits of numbers remain from which the Cubic Root is to be extracted, therefore, after instituting further operation according to the rules handed down by us above, you will have the whole Cubic Root of the number above 3 4 2 5 8 6 3 0 9 2 1 ( 3 2 4 7, with a remainder of 2 5 4 8 0 6 2 8.

If, after the operation is finished, an examination should be instituted, then the whole found root should be cubed; then the remainder from the operation should be added to the cube. If this aggregate corresponds to the number from which the root was extracted, no error has been committed in the operation. If not, the labor must be repeated, and another root must be sought, and the error must be corrected.

If something remains after the extraction, which often happens, the proposed number will be irrational or surdus surd/irrational, that is, it will lack a true Cubic Root. Therefore, to find the truly approximate root, let some sets of three ciphers be added to the remainder of the extraction, and let the operation be continued according to the method already handed down. To the found root as a numerator, a unit with as many ciphers should be subscribed as there were sets of three ciphers added to the proposed number from which the extraction was made.

Since it often happens that the Cubic Root must be extracted from a given number which does not have one exactly, in that case, so that time is not spent in vain, it has seemed best to add some rules, with the help of which you will have knowledge of such numbers that do not have a true Cubic Root exactly.

1. A number whose last digits are ciphers, and whose multitude the number three does not exactly measure, is not exactly a Cube. As the numbers: 3 4 2 0. 6 2 8 0 0. 4 5 3 0 0 0 0. are not Cubes.

2. A number whose last digit is 2 or 6, and the penultimate is other than an odd number, is not exactly a Cube. As the numbers: 3 4 2 2. 6 2 8 4 6. are not Cubes.

3. A number whose last digit is either 4 or 8, but the penultimate is other than a cipher or an even number, is not exactly a Cube. As the numbers 4 5 6 1 7 4. 1 1 0 0 3 8. are not Cubes.

4. A number whose proof by 9 is other than a cipher is not exactly a Cube. Thus, the number 1 2 0 0 0 will not be a Cube, since when cast out...