Book of Various Mathematical and Physical Speculations

...that the root is the square root of the product of $l.e.$ into $e.p.$, which product is the square of the solids. Let us likewise imagine two squares, $l.e.$ and $e.p.$, to be equally solids of such depth as the linear unit of the roots $m.e.$ and $e.q.$ requires. These two solids shall be produced from a surface into a unit, and let them be called $l.x.$ and $x.p.$; having done this, let us imagine the solid $a.g.$ as the product of the cube $l.b.$ into the square $e.p.$ Whence, from the eighteenth or nineteenth [proposition] of the seventh [book of Euclid], the ratio of $a.g.$ to $c.g.$ will be the same as that of $l.b.$ to the solid $l.x.$ But from the 25th of the eleventh and the 1st of the sixth, $a.K.$ is to the linear unit $K.c.$ as $a.g.$ is to $c.g.$; and from the same, $b.e.$ is to the linear unit $e.x.$ as $l.b.$ is to the solid square $l.x.$ Therefore, $b.e.$ will be to the linear unit $e.x.$ (namely $K.c.$) as $a.K.$ is to the same $K.c.$ Whence, from the ninth of the fifth, $a.K.$ will be equal to $e.b.$, and consequently equal to $m.e.$ Now, however, let $u.g.$ be the product of the cube $l.b.$ into the cube $o.p.$, as was said above. From this it will be clear, from either of the two propositions—the eighteenth or the nineteenth of the seventh—that the ratio of $u.g.$ to $a.g.$ will be the same as that of $o.p.$ to the solid square $x.p.$ Wherefore, from the ratios last mentioned, the ratio of $u.K.$ to $a.K.$ will be the same as that of $o.e.$ to the linear unit $e.x.$ But from the stated eighteenth and nineteenth of the seventh, the number $m.q.$ is to the superficial number $m.e.$ (which is produced from the linear $m.e.$ into the linear unit of $e.q.$) as the number $q.e.$ is to its own unit. But since the number $a.K.$ is equal to the number $m.e.$, as has been proved, it will therefore be, from the eleventh and ninth of the fifth, that the number $u.K.$ is equal to the number $m.q.$ But $f.g.$ is likewise equal to the number $m.q.$ from the preceding theorem; whence $K.u.$ will likewise be equal to $f.g.$ Thus it follows that $u.g.$ is a cube, and $f.g.$ is its root, equal to the number $m.q.$, which was sought.

A perspective drawing of two geometric structures. On the left, a rectangular prism with vertical and horizontal lines, labeled with the letters l, b, f, m, x, q, and p at various vertices and intersections. On the right, a cube labeled with the letters a, u, k, b, f, and g, with projection lines connecting it to the structure on the left.

THEOREM XXI.

Moreover, so that the whole infinity of powers may be known universally, that is, that the root of the product of two similar powers is the product of the two roots of the same powers.

Let us take, for example, two unknown square roots $q.p.$ and $g.K.$, which someone wishing to multiply by each other is compelled to multiply their known squares $n$ and $i$, the product of which is the square $m$, the root of which is $b.d.$, which I say is equal

to the product of $q.p.$ into $g.x.$, which [product] is $o$. For it is evident that the ratio of $o$ to $q.p.$ is the same as the ratio of $g.x.$ to its linear unit, from the eighteenth or nineteenth of the seventh. Now let this linear unit be $r$, the superficial unit of which is $u$, namely the unit multiplied by itself as many times as the proposed power allows, even though in the present example a square power is taken. Therefore, from the same eighteenth or nineteenth propositions, $m$ is to $n$ as $i$ is to $u$. We know, furthermore, that the ratio of $m$ to $n$ (because in the proposed example they are squares) is double the ratio of $b.d.$ to $q.p.$, and that of $i$ to $u$ is likewise double the ratio of $g.x.$ to $r$. Now, it has already been said that $m$ is to $n$ as $i$ is to $u$. Therefore, $b.d.$ will be to $q.p.$ as $g.x.$ is to $r$, since the whole is to the whole as the part is to the part, provided they are similar. But it was proved above that $o$ is to $q.p.$ as $g.x.$ is to $r$. Therefore, $o$ will be to $q.p.$ as $b.d.$ is to $q.p.$; whence $o$ will be equal to $b.d.$ This same thing will apply to other powers, only the ratios of $m, n$ to the ratio $b.d. : q.p.$ being changed, as well as the ratios of the two powers $i, u$ to the ratio of their roots $g.x., r$.

A diagram showing five squares of different sizes. They are labeled with the letters m, n, o, i, and u. Dotted lines indicate proportional relationships between the corners of the squares.

THEOREM XXII.

The ancients teach that if any number is divided into two unequal parts, and the whole divided by one of the parts, and the other part is divided by the same [part], the difference of the results will always be unity; which is indeed most true.

For let the proposed number $b.d.$ be given, divided into two unequal parts $b.c.$ and $c.d.$ First, let the whole $b.d.$ be divided by $c.d.$, from which let $c.o.$ arise, and let the unit be signified by $p.i.o.$ Then let the part $b.c.$ be divided by the same $c.d.$, and let the result be $a$. Truly, from the definition of division, the ratio of $b.d.$ to $c.o.$ will be the same as that of $c.d.$ to $i.o.$, and so $b.c.$ to $a$ is as $c.d.$ to $i.o.$ From the 19th of the fifth, however, $b.c.$ is to $e.i.$ as $b.d.$ is to $c.o.$ But $b.d.$ is to $c.o.$ as $c.d.$ is to $i.o.$, that is, as $b.c.$ to $a$. Therefore, from the 11th of the fifth, $b.c.$ will be to $e.i.$ as it is to $a$; from which, by the 9th of the aforesaid [book], $a$ will be equal to $e.i.$ But $e.i.$ is less than $c.o.$ by [the unit] $i.o.$ Wherefore it follows that the proposition is true. This same thing can be defined in very few words if we say that such a divisor goes into the divisible part one time less than it does into the whole, since the other part is that by which the whole is made complete.

A geometric diagram consisting of three horizontal line segments labeled with letters b, e, f, c, d, and a to illustrate numerical divisions.

THEOREM XXIII.

We shall be able to contemplate this same thing by another reasoning.

For let the total number be signified by $a.e.$, divided into two parts $a.u.$ and $u.e.$ Moreover, let the divisor of the whole be $u.e.$, and also of the other part $a.u.$ Let the result of the whole be $a.c.$, and let the result of the part be $a.n.$; then let the difference $n.c.$ be the unit.

A horizontal line segment marked with points h, n, o, u, e, and i, accompanied by vertical tick marks above some points.