Great Kabbalistic Art — Hartmann Schopper, 1564-1569

...the terms are similarly varied. For if the comparison is of a Larger number with a Smaller, then the elicited difference must be placed absolutely as it is, without any exception. But if, conversely, the comparison to be made is of a Smaller number with a Larger, then after the subtraction is made, you shall take away one unit from their difference; and thus corrected, you shall place it under its own sum, as with the other. For example: if the difference were to be elicited between 9 and 1, then because it is a larger with a smaller—for the larger number occupies the first place, namely 9—the difference is 8 without any defect, which you will give where required. But if it contains the opposite place, such as 1 and 9; then because the comparison is of a Smaller with a Larger, for the reason already stated, you shall take away one unit from their difference of 8, and 7 will remain as the true Number of the difference, which you place underneath them.

It should further be noted that the numbers or differences of one sum are not to be mixed with the Numbers or differences of another, whether it be the following or the preceding; for in order to best obtain each sum, it is necessary to be decorated with its own difference.

The second line of this operation will be very easily composed from the terms; for it is formed simply by the sums of the two differences, and subsequently that sum will receive the total: for example, if any sum here or there were 9 1 9, its Difference Numbers constituting the first line here are 8 and 7 according to the said Rules, whose sum is 15. This will compose the second line; and since this sum exceeds Ten original: "Denarium", you place the whole division beneath them, as seen done here:
9 1 9 — Sum

It must be observed, however, that if the difference of any sum forming the first line consists of a single Number, then without exception from the sum of one Number, or from the sum of two (for in either case, if for the sake of one Number)—then I say, because to produce this second line there, we can by no means collect a sum of differences since only a single difference exists there—one must operate no differently than to repeat the same difference intact, and thus also in this case complete the second line.

The third line which follows is likewise easily finished, if you have understood whatever was said above regarding the first line. For it, therefore, you will compare the first number of the Second line with the second Number of the same line, and you will place their elicited differences in the middle underneath; and thus you will constitute the number for the third line. Proceeding in this way, you will compare the second of the same second line with the third: then the third with the fourth, continuing in the same way until you have consumed the entire second line. Beware, however, that you do not forget the Rules given above concerning comparison, lest you treat a smaller with a Larger incorrectly, which I have commanded to be observed in every line and operation.

But, since I said above that the first number should be compared with the second...

...and the second with the third; yet I want you to understand that the superior sums are to be divided according to the line, if by chance a Double number is recognized, as happens when the sum exceeds Ten, as was said above. (For if the sum were noted by some simple number, and the following sum, with which it is necessary to make a comparison, consisted of a double number, then you shall note that double number itself as if it were of one, placing another smaller one below it; as will be more clearly shown by the example below).

The final line, at last, is thus the most difficult of the others, not because the combinations and differences of numbers are made differently, but because it requires greater attention. (For it is necessary to extract the difference, which is elicited by comparing to form this line, from three known numbers, so that a fourth unknown number may be had, for which line act as follows).

Compare the first Number of this Second line with the first number of the third line, eliciting the difference in the usual manner as we taught above for the first and third lines; and you will note the resulting difference either in your mind, or at least (to avoid mental errors) beneath the number of the third line. Then you will compare the same first number of the line, that is, with the second number, placing the elicited difference near the other. After both differences are taken—either kept in the mind or noted below as I said—you will compare them to each other, and you will have as a result the unknown fourth proportional original: "quartum proportionalem" arising from the three known proportional notes, as you will perceive better in the following example. But the numbers, or differences, which you kept under the third line, and from which I brought forth the fourth unknown, are left as superfluous and not necessary in the work, keeping only the fourth arising from the three known numbers, with which you will complete the fourth and final line. And this is the reason why I said above that you should hold them in your mind, or only note them down, rather to avoid mental errors than to constitute the line. But take the example drawn from the petition prescribed above, so that you may perceive this first operation more clearly.

A numerical calculation diagram following the Kabbalistic method described. It starts with the sums of the words in the query: 41, 510, 634, 540, 510, and 509. Below these are derived "Differences" forming the First Line (5, 6, 4, 9, 6, 8). These are summed to create the Second Line (11, 10, 13, 15, 14). Differences between these form the Third Line (0, 2, 3, 1). The process converges through a series of intermediate subtractions (1, 15, 3, 4, 7, 5, 5 and 0, 2, 1, 2) to reach a final central point.