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But here, so that the entire work of this Great Art may be seen more clearly, it must be noted that in the previous Chapter on petitions, you have seen how the Reader might have erred. The name or word count should neither be less than a specific amount, nor exceed the twenty-eighth word, nor should there be two requests, as can be seen from the observations of the Response. For if one makes it less than five words, the work easily falls into lightness. I do not doubt that the four previously prescribed conditions were asserted, since in my judgment it is almost impossible to make a petition in such a brief, circumscribed method without a defect in the remaining conditions. Then, when the matter stands thus, you would be least able to pursue the following operations that occur; because no one will doubt that at least five Roots Radices base numerical values derived from the petition are necessary for this lengthy multiplication, as you will see in its proper place.

On the other hand, if it exceeded the twenty-eighth word, he denies it here, not as a suggestion, but at least as a Multiplication or some foreign form. For what purpose was this labor performed? This opinion, although it appeared false, whenever you constructed a quantity larger than said amount without the stain of the aforementioned errors, then soon audacity would have wandered. Then, I hope, tell me how much one has sweated in this Chaos of Sums, Roots, Vocabularies, and other operations to lead these to their end without error? Although one might be able to penetrate all these difficulties with diligence, one can never (whenever the petition exceeds the twenty-eighth word) complete the Algebraic division original: "Algebraicam divisionem", nor use proportional numbers, which would then change their own nature; as in the second part of the Demonstration of our Art, where I have chosen to point all these things out here, so that whoever composes it may speculate more cautiously.

But now that the construction of the Rhomboids is finished, let us see in what form the numbers are to be arranged in the cells:

And first, having received the number, or the Differences of the second line, place the first difference of the said line in the first cell of the left Rhomboid, which is a whole cell, marked with the letter A in the following sections. Next, inscribe the second difference of this line in the second cell, marked with the letter B. Note the third difference of the same line in the third cell, marked with the letter C, and operate thus in the rest. Thus, the work of the left Rhomboid is adorned, and its cells are filled with the numbers of the second and third lines, beginning from the first difference of the said second line up to its last, proceeding then from the first of the third line to the end of the said line.

Otherwise, in placing the numbers in the cells of the Rhomboid, the following order must be observed. Specifically: the first and second cells should be filled with numbers. The third, however, should lack its number. The fourth, fifth, seventh, ninth, tenth, twelfth, fourteenth, fifteenth, and seventeenth describe a number. Conversely, the third, sixth, eighth, eleventh, thirteenth, and sixteenth should be empty; and understand the same for the rest; so that

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the first and last cell of each order should hold a number, as you will better perceive by the attached example, taken from the first query shown above.

A schematic diagram titled "Example of the Left Rhomboid" consists of a square grid containing letters arranged in a diamond configuration. Above the grid is the letter A. The grid itself is divided into 12 cells arranged in three columns and four rows. Row 1: B, C, (empty). Row 2: D, E, F. Row 3: G, H, I. Row 4: (empty), K, L. Below the grid is the letter M. Small dots appear within the cells for B, D, E, H, I, and L.

Two things, however, remain to be noted here. First, if any difference to be found in any cell were affected by a double number, you shall not divide it, but you will note the double for that one.

Second, to complete the cells of the same left Rhomboid which you see remain empty; for either two, or three, or a few will remain; because the Numbers, or the differences of the second and third lines, cannot for the most part complete an entire Rhomboid. I say that to adorn these also with numbers, take the closer Numbers of the higher order, which are noted on the right and left sides; and the difference elicited from them will be the number noted for this empty cell. As in the Rhomboid above, the last cell of it remains empty due to a lack of numbers to be placed: therefore, taking the difference between 4 and 7, which are closer to it in the higher order, both from the left and from the right; I set it as 3; and I write 3 in the aforementioned empty cell. Thus, one must operate in a similar way for the others which are read as empty.

However, the reader must note here that if that cell which remains empty were one of those that constitute the second half of the Rhomboid; for example, if it were as marked with a Cross in the above-mentioned scheme, then because there would not be closer numbers in the higher order, only one would be read, which is 13. Then to obtain the number for that cell, take the number of the first cell of the same Rhomboid, which is marked there as A, and compared with the said 13, you will elicit from them the difference, which you will note there for the true number.

Indeed, it is necessary to complete the Right Rhomboid with a similar construction, with the numbers of the last line of the same first operation placed in it, in that quantity and the usual number placed after this last line, specifically, which corresponds to the last sum of the words. But, because these numbers in any line clearly passed through the middle of this Rhomboid; indeed, they do not even occupy all the last cells of this half for the most part; for sometimes two, three, or more cells are vacant to complete the first half, therefore before anything is said about the second half of the same Rhomboid, it is first necessary