Book of Various Mathematical and Physical Speculations

[...n. c. to a. e.] are mutually equal, since each of them, from the similarity of the triangles, is equal to the proportion of o. n. to o. a. Therefore n. t., that is a. i., will be as much greater than a. x. as n. e. is greater than a. e.; whence just as a. e. consists of eight-ninths of n. c., so the part a. x. of a. e. will consist of eight-ninths of a. i.

From this the reason is evident why those about to divide a smaller number by a larger one place the smaller above the bar and the larger below, and a zero to the left.

It must be known, furthermore, that to divide a number by a number is to find the other side from which it is produced, it being always supposed that the divisible number is superficial and rectangular.

For example, if thirty were proposed to be divided by five, this division will be nothing other than the finding of the other number which, multiplied by five, would produce thirty rectangular surfaces; of this kind, indeed, is six, whose individual units will be superficial.

A geometric diagram of a right-angled triangle with vertex o at the top, a at the bottom left, and n at the bottom right. A horizontal line segment an forms the base. Points e and c are marked on the hypotenuse on, and points i and t are marked on the base an. Internal construction lines illustrate proportional relationships discussed in the text.

For the sake of this matter, let the rectangle a. e. above be of thirty superficial units, whose side e. n. is of five units; hence side a. n. will be of six units. Thus, in dividing the rectangle a. e., we shall do nothing else than find what the side a. n. is worth, which will be of six units. But if we divide by the side a. n., we shall seek the side e. n. of five units. From which, the proportion of the whole divisible number to the number that arises will be as the divider to the unit, from the first [proposition] of the sixth [book], or the 18th or 19th of the seventh; and alternately, the divisible will be to the divider as the number that arises is to the unit.

A rectangular grid of 5 rows and 6 columns, labeled with vertices a, i, n, e. It represents a product of 30 units (5 units by 6 units).

Therefore, to divide is nothing other than to find the side of a rectangle which, multiplied by the divider, contains the divisible number; from which the divisible number is superficial, but the divider and that which arises are linear numbers and the sides producing such a divisible number. For multiplying and dividing are opposed to each other; since, however, a surface is generated from the multiplication of linear sides, from the division of that surface thereafter the other side is found. Wherefore it is not surprising if that which results from a division (by way of fractions) is always greater than the divisible number.

For example, by dividing a half by a third part, there results one whole number with a half for the number that arises. Let, therefore, the superficial divisible half be b. e., whose whole is the square b. p. Let the linear divider indeed be the third b. n., whose linear whole is b. d. We must seek the side b. s., which with the side b. n. produces the rectangle n. s. equal to the proposed superficial half b. e. If this be done, from the 15th [proposition] of the sixth [book], or the 20th of the seventh, the proportion of b. n. to b. q. will be the same as q. c. to b. s. We shall say, therefore, if n. b. gives b. q., what will q. c. give? Certainly b. s. But n. b. is the linear third and b. q. the whole linear [unit], and b. s. the resulting linear [side]. And because the superficial half b. e. is produced from the linear half q. c. into the whole linear q. b., wherefore since n. s. is equal to b. e., and [the rectangle is] produced from the smaller b. n. and q. c., it is necessary that it be produced in the larger b. s. than q. b.; which q. b. is greater than q. c., which indeed q. c. is named just as b. e. [i.e., a half]. Wherefore it is not surprising if that which results from a division of numbers by fractions is greater than the divisible number.

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From this it is clearly evident that any division or partition arises from the Rule of Three, since individual dividers are equivalent to one whole and are taken in its place. For it is the same to divide one hundred by twenty as to observe the Rule of Three, saying: if twenty are equivalent to one, to what will one hundred be equivalent? This, moreover, will be easily perceived from the following figure, in which line a. b. signifies twenty, and a. o. the linear unit, and a. c. one hundred linear units; o. c. indeed one hundred superficial units, and a. d. five linear units, and d. b. one hundred superficial units; from which it is clearly perceived that just as to multiply is nothing other than to find the product from two proposed sides, so to divide is nothing other than, given one side, to find the other side of the proposed product.

Two rectangular diagrams demonstrating division and proportion. The left rectangle has vertices a (top left), o, d (top right), c (bottom left), and b (bottom right), with internal proportions marked 1 and 11. The right rectangle is labeled p, q, b, d with proportions 1 and 11 marked inside.

For whenever in reasoning we say "so much of a number," we immediately produce a surface, mediately through the unit in such a number; which number, before it is produced into the unit, must be conceived in the mind as if linear—as a line, I say, divided into so many linear particles, each continuous and equal to the proposed unit. But when the number has been produced as a superficial unit, it will be as if there were so many square units; if it were not so, no mention would be made of any fractions. From the same Rule of Three, the third theorem can be reduced to practice.

Wherefore, wishing to know what those parts are which are three-quarters of five-sixths themselves, we shall say: if four gives three, what will five-sixths give? They will give fifteen twenty-fourths, which fifteen are three-quarters of twenty itself. But twenty [is] five-sixths parts of twenty-four, as is known of itself.

From the same Rule of Three, such a question can be answered if we establish the aforesaid five-sixths to be the number of which three-quarters are sought, saying: if one whole gives three-quarters, what will five-sixths give? Wherefore, following the Rule of Three, fifteen twenty-fourths will be given. The same Rule of Three is valid so that one may know what part or parts of a proposed number any number might be.

For example, to one wishing to know what part or parts of twenty-four sixteen might be: twenty-four will be established as one whole, of which sixteen may be a part or parts. We shall say, therefore: if 24 gives sixteen, what will one give? Namely, sixteen twenty-fourths, which when they have been reduced to prime numbers, will be two-thirds. By the same reasoning, he who would wish to know what parts or part were three-quarters of eight-ninths would say: if eight-ninths gives three-quarters, what will one give? There will result twenty-seven thirty-seconds.

It serves equally for knowing the nature of the parts of a proposed number. For the sake of example, if someone asks of what number twelve are two-third parts. He will say: if two give three, what will [twelve] give...