Book of Various Mathematical and Physical Speculations

THEOREM V.

By another way also, the cause of the aforesaid effect can become known through speculation, for the sake of which the following figure $e. o. a. u. n.$ is formed in such a manner that $a. e.$ is the linear number of wholes, and $o. e.$ is the product of the numerator of the fractions themselves and the wholes; from which $a. o.$ will be two-thirds, for example, of $a. i.$ or $a. u.$, each of which lines is established as equal to the linear unit. Let the parallelogram surface $u. n.$ be constructed equal in magnitude to the surface $o. c.$, from which $u. n.$ will be a surface known to us. Likewise, the quantity of the parts of $a. u.$ will be known, which in the proposed example we have said to be of three parts. Therefore, from the rule of three, we shall say: if $u. a.$ gives $a. e.$, without doubt $o. a.$ will give the linear number $a. n.$; which rule is drawn from the 15th [proposition] of the sixth [book] on continuous [quantities], and from the 20th of the seventh [book] on discrete [quantities]. Rightly, therefore, are the numerators of the fractions multiplied with the wholes, and the product divided by the denominator of the fractions.

A geometric diagram consisting of a grid-like rectangle. The vertices and various points along the segments are labeled with lowercase letters: n, i, a, o, u, c, e. The diagram represents a multiplication area corresponding to the theorem's proof.

THEOREM VI.

Likewise, it can be known by another speculation that this is done rightly; for multiplying these two-thirds by ten, we must consider the quantity of two-thirds produced ten times, from which arise 20 thirds, since individual units are then taken for two thirds; but since any whole contains three fragments, therefore, by the ratio of dividing [to see] how many times three enters into twenty, we shall immediately know that which we desired.
The same would happen if the wholes were divided into that same species of fractions; which having been done, these would have to be multiplied with the proposed numerator, and the product divided by the square of the denominator.
The speculation of this matter is thus: Let line $a. e.$ consist of five whole numbers, each of which is equal to $a. u.$ or $a. i.$, and let $a. o.$ be two-thirds of a linear whole unit. Let us now imagine these five wholes to be divided into their linear fragments, which in the proposed example will be 15. Now, with 15 multiplied by the proposed [parts], namely $a. o.$, there will arise a product $o. e.$ of thirty superficial fragments, of which nine fall into each superficial whole in this example; and when we have noted how many times nine enters into thirty, we shall attain the proposition.

A smaller rectangular grid labeled with letters n, i, a, o, u, e, showing a subdivision of units into fractional parts.

---

THEOREM VII.

When those who are about to multiply whole numbers and fractions with whole numbers and fractions must reduce the wholes to the species of the fractions by collecting them with the fractions: they then multiply these last numerators with each other and divide the product by the product of the denominators.
As (for the sake of example) if we wish to multiply one and two-thirds by two and three-fourths, all shall be reduced into fractions, from which on one side there would be five-thirds to be multiplied with eleven-fourths on the other; which being done, there will arise a product of fifty-five fractions, which being divided by the product of three and four, namely by twelve, four wholes will be produced with seven-twelfths fractions of one whole.
Let the subsequent figure be given in which line $a. i.$ is equal to line $u. a.$, each of which is considered as a whole number: let $a. i.$ be imagined to be worth four in the present example, and $a. u.$ three: let there then be given line $a. o.$ equivalent to one whole with two-thirds, and $a. e.$ equivalent to two wholes and three-fourths. Now if these two lines be reduced into their fractions, and $a. o.$ be multiplied with $a. e.$ (as appears in the following figure), there will arise a product $o. e.$ of fifty-five superficial fractions, of which a superficial whole is worth twelve, namely $u. i.$, as is manifest to everyone; from which, for one seeking by means of division how many times twelve enters into fifty-five, the result sought will occur without error.

A rectangular grid labeled with n, u, o, i, a, e. The grid visualizes the product of mixed numbers by showing how fractional subdivisions combine to form the total area.

THEOREM VIII.

The very same thing would happen if the fractions were reduced to one and the same denomination, and afterwards multiplied together, and we divided the product by the square of the common denominator.
For example, let the same five-thirds and eleven-fourths be multiplied together; if these are reduced to one and the same denomination, the numerator five of the one will be multiplied with the denominator four of the other, and the numerator eleven of the second with the denominator three of the first. From which, on one side there would be 20, on the other 33, as numerators of one common denominator, which would be the product of three and four, namely twelve, as is clear from the old rule. Now if 20 be multiplied by 33, there will be given 660 fractions, of which a whole will be the square of twelve, namely 144; with which 660 divided by 144, there will be produced four wholes and seven-twelfths.
For the sake of which matter, let line $a. i.$ and $c. i.$ in the figure below be equal to $a. u.$ for a linear whole; let $a. i.$ be divided into four parts, and $a. u.$ into three, and let line $a. e.$ be of eleven such parts as $a. i.$ is of four, and $a. o.$ be five even as $a. u.$ is of three. Then with $a. o.$ and $a. i.$ multiplied, there will arise a product $o. i.$ of twenty superficial parts. When multiplied...

A grid labeled with n, u, o, i, a, e, illustrating the mathematical demonstration of Theorem VIII with units and subdivisions representing a common denominator.