The Great Art of Artillery, Part One

2. Before the operation, the given number should be distinguished with dots from right to left, such that the first dot is placed under the first digit on the right, the second under the fourth toward the left, the third under the seventh, the fourth under the tenth toward the same left, and so on as far as the digits of the numbers suffice. Dots should always be marked with two digits skipped, as appears here: 3 4 2 5 8 6 3 0 9 2 1.

3. From the table above, take the root of the number intercepted from the first dot to the left, whether it consists of one, two, or three digits. That is, seek this number in the cubic table exposed above. If it is not found, take the one closest to it that is smaller, and place its cubic root outside the parenthesis. As in our example, seek the root of the number 34, which, since it is not found exactly in the table of cubes, take the closest smaller one, namely 27, and note its root, 3, in this way: 3 4 2 5 8 6 3 0 9 2 1 ( 3.

4. Subtract the cube of this root from the number intercepted under said first dot, namely 27 from 34, and write the remainder, 7, above it, in the same way as is usually done in common subtraction.

Mathematical scratch notation for cube root extraction. The number 34,258,630,921 is shown with dots under the 4, 8, 0, and 1. The number 7 is written above the 34. Below the 34 is the number 27 with a horizontal strike-through. To the right is the root digit 3 following a parenthesis.

5. Triple the root just found, and place this triple under the digit next preceding the digit marked by a dot. If there are more digits of this triple, they should be placed in order toward the left.

6. Prepare the divisor in this way: multiply the quotient by the triple, note the product one position further removed toward the left than you started the triple, and in a lower position, so that there are now two distinct numbers, one of which we shall call the Triple, and the other the Divisor. If you divide the number written above by this divisor, you will have the second digit of the root in the quotient.

7. Multiply everything that was in the quotient by the Triple, then multiply the product again by the digit of the quotient found by division. Add to this product the cube of the same number, in such an order that the last digit of that cube toward the right is not placed immediately under the last digit of the superior product, but is set back at an interval of one digit toward the right.

8. From the aggregate of all numbers disposed in this order, subtract from the upper numbers (if it can be done), and write the remainder (if there is any) above. If not, the quotient must be reduced until the aggregate found in the said manner can be subtracted from the higher number, the same divisor and triple always remaining. As in the example above: triple the root 3, and it becomes 9, which you write under 5. Then multiply 3 by 9; the result is 27, which you place lower than the Triple, and one position thereafter toward the left, namely under 72. Now divide 72 by 27; you will have the quotient 2, to be added to the previous 3, so that the whole quotient becomes 32. Multiply this by the triple; 9 becomes the product 288. Multiply this again by the number just found, 2; you will have the second product, 576. Finally, add to this the cube of the number just found, 2, namely 8. The aggregate will be made from the numbers disposed in the order as it appears here: 5 7 6 8, which, subtracted from the upper number 7 2 5 8, leaves 1 4 9 0 as a remainder.