Book of Various Mathematical and Physical Speculations

will give twelve: namely, they will give eighteen, which is the sought number. Then, however, we perform nothing other than seeking a number to which twelve has the same ratio as two has to three. So also, if someone should ask of what number two-thirds are three-fifths, he will say: if three give five, what will two-thirds give? Namely, they will give one whole with a ninth fraction. This will be, therefore, to seek a number to which two-thirds has the same ratio as three to five, which is manifest of itself.
By the same reasoning, he who wished to know of what number two-sevenths were eight wholes and two-fifths would say: if two give seven, what will eight wholes and two-fifths give? Namely, they will give 29 wholes and two-fifths, the sought number. Thus also, he who wished to convert a fractional number into another fraction would accomplish it by the rule of three.
For example, if eleven-thirteenths of one whole were proposed, the whole being divided into 13 parts, and we desired to know how many parts of the whole eleven-thirteenths would be if the whole were divided into 4 parts, we would say: if 13 give 11, what will 4 give? Namely, they will give three fourths with five-thirteenths of one fourth; this indeed is nothing other than seeking a number to which the whole divided into 4 parts has the same ratio as the same whole divided into thirteen has to eleven-thirteenths. Furthermore, this rule is adapted to many other things as well.
For these things were not said without purpose, but so that everyone might see that the cause of similar operations written by practitioners concerning fractional numbers all draw their origin from that divine rule of three, as we shall also see in the following.

THEOREM XI.

Why the product of the quotient and the divisor is always equal to the divisible number, understand as follows.
Let the divisible number be b, the quotient c, the divisor d, and the unit of the divisor e. Since therefore, as was said in the preceding theorem, the ratio of b to c is the same as that of d to e, it is clearly understood from the 20th of the seventh [book of Euclid] that the product of b and e is equal to the product of c and d.

A simple geometric diagram consisting of two parallel horizontal lines. The top line has point 'b' on the left and 'c' on the right. The bottom line has point 'd' on the left and 'e' on the right.

THEOREM XII.

The same thing may be considered by another reasoning.
Let the divisible number be represented by line n.e., the divisor by line a.e., the quotient by line u.e., and the unit of the divisor by line o.e., which we conceive as a linear unit. To these, let the product of u.e. and a.e. be the surface u.a. I say that the surface u.a. is composed of as many superficial units as the line n.e. consists of linear units. For from those things we have noted regarding the method of dividing, it is established that the ratio of n.e. to u.e. is the same as that of a.e. to o.e. But from the first of the sixth or the 18th of the seventh, the total product u.a. has the same ratio to the partial product u.o. as a.e. has to o.e. Wherefore, u.a. will have the same ratio to u.o. as n.e. has to u.e. But u.e. and u.o. do not differ in number, since they are of one and the same species (provided, however, that the number u.o. is superficial and u.e. linear). Therefore, from the ninth of the fifth, the number u.a. will be equal to the number n.e.

THEOREM XIII.

Why, when dividing a divisible number by the quotient, does the divisor arise?
Let the rectangle o.e. written below be the divisible number, which is produced both from a.o. into a.e. and from a.e. into a.o. Wherefore, if a.o. were the divisor, a.e. will be the quotient; but if a.e. were the divisor, a.o. will be the quotient.

A rectangular diagram labeled 'o' at the top left corner and 'a' at the bottom left. The bottom right corner is labeled 'e'. A point 'c' is placed to the right of 'e'.

THEOREM XIIII.

This same thing may also be observed in another way.
Let line a denote the divisible number, line o the first quotient, line e the first divisor, and line u the second quotient—that is, when o is taken as the divisor. Now from the indicated definition of division in the ninth theorem of this book, the ratio of a to o will be given just as e is given to the unit signified by line i; and by permutation, a is to e as o is to i. But a has the same ratio to u as o has to i from the same definition of division; therefore a will have the same ratio to u as a to e, whence u will be equal to e from the ninth of the fifth.

A set of horizontal line segments labeled with letters representing mathematical proportions. The top group shows segments 'a' and 'o' over 'u' and 'i'. Below is a segment 'u' over 'c'.

THEOREM XV.

Whence it arises that he who wishes to know of what number four-fifths parts are two-thirds, or anything similar, would act most wisely if he reduced them to one and the same denomination.
Just as in the proposed example, since the common denominator is fifteen, of which two-thirds are ten and four-fifths are twelve, the common denominator 15 must be multiplied by four-fifths—namely twelve—and the product divided by two-thirds—that is ten—from which eighteen, the sought number, arises.
As for the reduction of the numerators to one and the same denominator, it is done for the purpose that we may use the rule of three, which requires only three known terms, so that a fourth depending on the aforesaid may be found, since those two relationships can be encompassed in three terms. But as for the multiplication of the common denominator by the numerator of the unknown denominator and the division of the product by the known numerator, these are nothing other than finding a fourth term, so proportioned to the third as the second is to the first.
For example, let a denote the numerator of the known denominator, which is signified by line o; and let e be the numerator of the unknown denominator, denoted by line u; or rather, the known [denominator] o is four-fifths. Now if we multiply o by e and divide the product by a, u will be given, having the same ratio to e as o to a, from the 20th of the seventh.

A complex proportional diagram with horizontal lines. The first section has points 'h', 'g', 'r' with the number '15' below it, and 't' with '10'. The second section has 'u', 'e' with '18' below it, and 'd', 'f' with '12'.