Great Kabbalistic Art — Hartmann Schopper, 1564-1569

Then, whatever remainder comes from the subtraction, divide it by the whole number of the first line of that sum of the operation, and the one matching the said sum: and the remainder of each is the Root of that sum and word, which Root you will place after the line of this second operation under the proper word, as you will see done below.

But in this extraction of Roots, beyond the general Rule now stated above, not a few things are to be noted.

First, if the last number, or the one of the fourth line of the first operation (of which we have spoken), which must be multiplied by the sum corresponding to it, should be a zero; you will consider it as if it were a unit. Specifically, in its multiplication it will not generate zeros, but will retain the same sum in its own state. As in the first example described above, the first sum of the word An is 41 and the number collecting it placed in the fourth line is 0. Wherefore, multiplying 0 by 41 according to the common method of Arithmetic, we would always have 0; but in this our work it will not generate zeros, but will leave the same sum, namely 41.

Second, it must be noted that the number dividing the subtraction, and which (as I said) is to be taken from the first line of the first operation; if it is a simple number, take the simple one for its divisor. If, however, it stands as a composite number, you will also place the whole composite for the divisor. For example, if the Divisor were 34, you will make the division for the whole number 34, and not first by 3 and then by 4.

Third, if a zero precedes this double Divisor, namely, placed on the left part of it; then you will consider that zero as if it were on the right part. As in the second example handed down above concerning the Surname of the Urbani Pontificis Pope Urban in its second sum, which is 645, for the number that is the divisor of that subtraction, there stands in the first line 02. This you will take as if it were 20, taking it as the divisor, extracting the Root by its means. If, however, you distinguish the Root thus elicited with some sign, whether with ±, or any other small mark, it signifies that the Root is not a direct one, but an inverted one. Thereafter it must be used in a different mode, as will be clear in the following pages.

Fourth, if you see the Divisor is either a Zero or a unit (for each is powerless to divide the remainder of the subtraction), then leaving the subtraction of that sum of the first operation, you will only divide the last line of this second work by the sum corresponding to it, keeping the Remainder for the Root. For example: In the same second operation concerning the Barberini Surname, in its fourth sum, a divisor of 9 is indeed lacking. Wherefore, without any subtraction, once the sum has been multiplied from the last line of the pyramid, the last quantity of this pyramidal line, which is 65.31000, I divide by the number of the sum of the same Word, which is 99, and in

the division I see there remains 213 units, which I set down in the place of the Root under the last line of the pyramid adhering to it, and marking it as superior with the sign ± already shown in the description.

Fifth, if in the number or quantity to be divided, the Divisor perfectly hides itself, namely, without any remainder of the quantity to be divided; then you will establish for the Root that same Divisor (number 8). In the example of the surname of Bonifacij Boniface, the sum composing the word of the fourth word is 460. This sum multiplied by the number of the fourth line corresponding to it, which is 4, you will see the product is 18400. The last line of the same corresponding pyramid has the number 100000000. From which line, according to the Rules, subtract the aforesaid multiplied sum, namely 18400, and you will see the Remainder of this subtraction is 100000060. This you divide by the true divisor, namely 16, which in the already mentioned remainder of subtraction will give for the true quotient the quantity 6250000, yet without any Remainder of division. Wherefore in such a case the same divisor is decently to be placed for the Root under the proper pyramidal lines.

Sixth, if the last pyramidal line of any word, from which according to the noted Rules it is necessary to subtract the multiplied sum adhering to it, should be a quantity smaller than the sum already multiplied, so that subtraction cannot be made; then you should proceed with a reversed subtraction, namely, by subtracting the quantity of the pyramidal line from the quantity of the multiplied sum, and whatever comes from the subtraction, you should write down for the true Root under the pyramidal line. But because you extracted this Root in a reversed order or method, you will therefore note ±. As in the first example, the sum forming this word An is 21, which multiplied by 2, the number of the fourth line corresponding to it, generates the same sum, and its last pyramidal line adhering to it has for its quantity one 1. To subtract 42 from this was a labor; afterward, in reversed order, from the same multiplied sum, namely 42, subtract 1, and 41 will remain, which you will replace for the Root under the last pyramidal line marked with ±: in such a case, all other operations of algebraic division are the same.

Seventh, if the sum of any word already multiplied by the number of the fourth line shall equal the quantity of the last pyramidal line responding to it, so that when the subtraction is done no remainder appears: then for the purpose of denoting the Root, observe what number would be the divisor of this subtraction: for you will place it for the Root at the end of the pyramid. As in the second example handed down above, let the fifth sum of the word