Book of Various Mathematical and Physical Speculations

[multipli-]cato $a. e.$ by $a. u.$, the product $u. e.$ of thirty-three parts will be given. In addition, the square $u. i.$ will consist of twelve parts of the same ratio as the other two products, which square $u. i.$ is the superficial unit and the common denominator of the two products. Now if for the present we consider the line $c. d.$ to be of thirty-three equal parts, and $c. t.$ of twelve similar ones, and $c. f.$ of twenty, and $c. n.$ of twelve; by multiplying $c. d.$ by $c. f.$, the surface $f. d.$ of 660 superficial fractions will be given, whose whole superficial unit will be the square $n. t.$ of 144 parts, of which kind the parts of $f. d.$ are 660. Therefore, by dividing $f. d.$ by $n. t.$, the proposition will follow, for the reason that the proportion of the product $f. d.$ to $n. t.$ will be the same as that of the product of $a. e.$ and $a. o.$ to $u. i.$ For the proportion of $c. d.$ to $c. t.$ is the same as $a. e.$ to $a. i.$, and $c. f.$ to $c. n.$ as $a. o.$ to $a. u.$, from the first [proposition] of the sixth [book] or the 18th of the seventh. But as $f. d.$ is to that which results from $f. c.$ into $c. t.$, so is $c. d.$ to $c. t.$; and as that which results from $f. c.$ into $c. t.$ is to $n. t.$, so is $f. c.$ to $c. n.$ from the said proportions. Therefore, by equal proportionality, proceeding in the same way in figure $o. a. e.$, $f. d.$ will be to $n. t.$ as $o. e.$ is to $u. i.$ Furthermore, from those things which have been considered thus far concerning the multiplication of fractions, the reason is clearly perceived why the product is always less than the individual factors, since the products are superficial, while the factors are always linear—setting aside corporeal products, all of which are reduced to superficial ones.

A rectangular geometric diagram containing a grid of squares. The vertical axis on the left is marked with points a, i, e, c, and d at the bottom. The horizontal axis is marked with u at the top left, o in the middle, and n at the top right of a sub-section. The bottom is marked with t and f. The diagram illustrates numerical proportions described in the text.

THEOREM IX.

IN THE division of fractions itself, it must be observed that the denominating numbers must always be equal to one another, that is, of one species; but if they are not equal, it is necessary to make them equal by means of multiplying the denominators by each other, from which a product arises of such a kind that it is suited to hold the parts of the fractions which were desired.

For example, if seven-eighths were proposed to be divided by three-fourths, the rule of the ancients prescribes that they be reduced to only one denomination. Therefore they multiply the denominators by each other, from which, in the matter proposed, there arises a common denominator of thirty-two parts, whose two numerators are twenty-four and twenty-eight, produced from the multiplication of one numerator by the denominator of the other; from which twenty-four are given as three-fourths of thirty-two, and twenty-eight as seven-eighths of the uniform small parts, just as can be known by the aid of the first of the sixth or the eighteenth of the seventh in the figure written below.

Let there be—

Therefore, let line $a. i.$ be divided into eight parts, and [let line] $a. u.$, equal to it in length, [be divided] into four; and let the product of one by the other be $u. i.$, of thirty-two superficial small parts, similar and equal to one another. Then likewise let $a. e.$ be of seven parts of line $a. i.$, and $a. o.$ of three parts of $a. u.$ Then the product of $a. e.$ into $a. u.$ will be $u. e.$ of twenty-eight superficial small parts, and the product of $a. o.$ into $a. i.$ will be $o. i.$ of twenty-four given superficial parts of the same nature as the thirty-two parts of the whole common denominator. Whence, by dividing the numerator twenty-eight by the numerator twenty-four, there will be given one and a sixth part of that one.

A square grid diagram consisting of 8 vertical columns and 4 horizontal rows, forming 32 small squares. The vertices are labeled u, c, a on the left vertical edge and d, e, i along the bottom edge. Point o is marked on the top edge.

THEOREM X.

TO PART or divide one number by another number is also, in a certain way, to find such a part of the divisible number with respect to the whole divisible number as unity is in the divisor with respect to the whole divisor; a part, I say, of the divisible number having itself toward the whole divisible number just as unity [has itself] toward the whole divisor, which we likewise perform by the rule of three, saying: if such a dividing number gives unity, what will the divisible number give? Just as may be observed from the 15th [proposition] of the sixth [book] or the 20th of the seventh. Therefore, whenever we divide a smaller number by a larger, that which results is always a fraction.

For example, if we were to consider a line $a. c.$ divided into eight equal parts, of which one—namely, a unit—would be $a. i.$, and we wished to divide it into nine parts and to know how great its ninth part is; it would be manifest that the ninth part of $a. c.$ itself will be less than $a. i.$ itself, since $a. i.$ must be diminished from its integrity in the same proportion as $a. c.$ is found to be less than a line of nine equal parts [each equal] to the individual [unit] $a. i.$

In order that this may become clearly known to anyone, it will also be possible to see it in this way. Let line $n. c.$ be ninefold to $a. i.$ and parallel to $a. e.$; there is no doubt that $a. c.$ will be larger than $a. e.$ itself. Now if their extremities are joined by two intermediate lines $n. a.$ and $c. e.$, which meet together at point $o$ (which is very easy to prove), there will certainly be given two similar triangles, $a. o. e.$ and $n. o. c.$ Then let $n. t.$ be one of the parts of $n. c.$ itself, which $n. t.$ will be equal to $a. i.$ by the presupposition. Then let $o. t.$ be drawn, which intersects $a. i.$ at point $x$. I say that $a. x.$ will be as much smaller than $a. i.$ as $a. e.$ is smaller than $n. c$; for there can be no doubt that the proportions of $n. t.$ to $n. c.$ and $a. x.$ to $a. e.$ [are the same]...

A geometric diagram showing a large right-angled triangle with vertex o and base n-i. The base is divided by points n, c, e, and i, with tick marks indicating equal units. A vertical line connects o to n. A horizontal segment a-x is drawn within the triangle parallel to the base. The line o-i forms the hypotenuse. Additional construction lines connect o to points on the base, demonstrating similar triangles and proportional division.