Book of Various Mathematical and Physical Speculations

...if anyone should consider by what reasoning the fractional number $c. i.$ in its whole $d. b.$ is smaller than the fractional [number] $a. i.$ in its whole $a. b.$, or the fractional $a. c.$ in its whole $a. d.$, let him consider how the ratio of $c. i.$ to $d. b.$ is less than the ratio of $a. i.$ to $a. b.$ and of $a. c.$ to $a. d.$ by this reasoning. It is manifest from the first [proposition] of the sixth [book] concerning continuous quantity, or the 18th of the seventh [book] of Euclid concerning discrete [quantity], that the ratio of $d. i.$ itself to $d. b.$ is as $a. i.$ to $a. b.$; and since $c. i.$ is less than $d. i.$, like a part to its whole, the ratio of $c. i.$ to $d. b.$ will be less than the ratio of $d. i.$ to $d. b.$ from the 8th [proposition] of the fifth [book]; wherefore it will likewise be less than the ratio of $a. i.$ to $a. b.$ from the 12th [proposition] of the same [book]. At the same time also, the ratio of $c. i.$ to $d. b.$ will be less than $a. c.$ to $a. d.$ for the same reasons, with $c. b.$ as the mean. From which the reason is evident why fractions of different denominations are reduced to a single one. Also why it is permitted to break whole numbers into parts similar to fractions, all of which can easily be known from the following figure.

A rectangular diagram divided by a vertical line labeled i and a horizontal line labeled c. The vertices are labeled a (bottom-left), b (bottom-right), and d (top-left). The point i lies on the segment ab, and c lies on the segment ad.

THEOREMA II.

WHAT the reason is why those who wish to add fractional numbers of different denominations and reduce them to a sum, multiply one of the numerators by the denominator of the other, and afterwards the denominators with each other; the last product of which is the common denominator of the two previous products, which, being added into a sum, produce what was sought.

In which matter it must be known that the denominators are considered as parts of one and the same magnitude of continuous quantity—of lines, for example, $a. b.$ and $a. d.$ equal in length—of which let $a. b.$ be divided into four parts and $a. d.$ into three. Wherefore, if we wish to add two-thirds with three-quarters, we shall multiply $a. c.$, two-thirds, by $a. b.$ divided into 4 parts, and the product $c. b.$ of eight superficial parts will be produced; then, by multiplying $a. i.$, three-quarters, by $a. d.$ divided into 3 parts, $i. d.$ will be produced, equal to the first individual parts, [consisting] of nine superficial parts; then $a. b.$ divided into 4 parts being multiplied by $a. d.$ divided into 3, the square $d. b.$ will be produced in a continuous [quantity], divided into 12 parts, which will be the common whole for the individual products, the first of which was $c. b.$ Wherefore $c. b.$ is to the whole $d. b.$ as $a. c.$ is to $a. d.$ from the first of the sixth in continuous [quantities], or the 18th of the seventh in discrete quantities; and $d. i.$ to $d. b.$ as $a. i.$ to $a. b.$ from the same propositions. Then, the parts of the product $c. b.$ having been added with the parts of the product $i. d.$, it will be clearly perceived that a sum of this kind is composed of the parts of one whole common to each of them.

A rectangular grid of 12 squares (3 rows by 4 columns). The bottom edge is labeled with points a, i, and b. The left vertical edge is labeled with points a, c, and d. Vertex a is at the bottom-left corner.

THEOREMA III.

WHEN those about to find what sort of fraction any number is in respect to another, must multiply the numerators together and so also the denominators, from which the product of the numerators takes its name from the product of the denominators.

If you wish to know the cause of this, take $o. i.$ and $o. u.$ for the whole denominations, then $o. e.$ and $o. a.$ for the numerators; for example, let $o. i.$ be six, $o. u.$ four, $o. e.$ five, and $o. a.$ three. If you wish to know what are three-quarter parts of five-sixths, it is clear from practical rules that fifteen twenty-fourths arise. How that is done will be perceived from the figure written below; however, it is necessary to be mindful that any product is considered as a surface, but the factors as lines. In this figure, therefore, the product of the linear wholes is $u. i.$, a total of 24 parts, and $u. e.$ is the product total of 20. This will be to the total product $u. i.$ as $o. e.$ is to $o. i.$ from the first of the sixth or the 18th of the seventh; thus $u. e.$ will be five-sixth parts of $u. i.$, of which, in the proposed example, three-quarters are sought. If, therefore, $o. e.$ is multiplied by $o. a.$, the product $a. e.$ will arise, so proportioned to $u. e.$ as $o. a.$ is found to be to $o. u.$, from the aforesaid reasonings. Because if it is established that $o. a.$ is three-quarter parts of $u. o.$ itself, $a. e.$ will also be three-quarter parts of $u. e.$; but $u. e.$ is five-sixths of $u. i.$; from which it follows that a work of this kind is correct.

A rectangular grid of 24 squares (4 columns by 6 rows). Points are labeled: u (top-left), a (partway down the left edge), e (bottom-left corner), i (partway along the bottom edge), and v (bottom-right corner).

THEOREMA IIII.

WHEN those about to multiply fractions with whole numbers, rightly multiply the numerator of the fraction by the number of wholes, and divide the product by the denominator of the fraction, from which the sought number is gathered.

On account of which, let us conceive in the mind in the following figure the number of wholes as the line $a. e.$, which, for example, may be ten, each one of which is equal to $a. i.$; and let the product of $a. e.$ into $a. i.$ be thought of, and let it be $u. e.$, which indeed will be ten superficial units, $a. u.$ having first been established equal to $a. i.$; and let $a. o.$ be two-thirds of $a. u.$, the product of which two-thirds into the number $a. e.$ becomes $o. e.$ Likewise let $u. i.$ be a superficial unit just as $a. i.$ is a linear unit, which $u. i.$ the product $o. e.$ must regard; from which the superficial whole $u. i.$ will be as three, and the product $o. i.$ [sic] as two; and because any part out of the twenty of $o. e.$ itself is equal to a third part of the superficial unit $u. i.$; if we desire to know how many whole units are in the parts of $o. e.$, it is advisable to divide the same by the denominator $u. i.$ composed of three superficial parts; and since the line $u. a.$ as well as the surface $u. i.$ is divided into 3 equal parts, it is very fitting to know that a division of this kind of the number $o. e.$ is made by the number of $u. i.$ itself, not $u. a.$, for the aforesaid causes.

A horizontal diagram of a narrow subdivided rectangle. Letters are placed along the left vertical edge: u (top), o (middle), and a (bottom corner). Along the bottom horizontal edge, the points i and e are labeled.