Book of Various Mathematical and Physical Speculations

To demonstrate this, let the first of all numbers? be represented by the line .a. b., which let us imagine divided at point .c. into the sought parts, from which it is presupposed that the two lines .a. c. and .c. b. are two squares, which in the other figure are represented by .d. and .e.; moreover, let the product of the roots be known as .f. since it is given, the square of which will be equal to the product of the squares .d. [and] .e. by each other, namely .b. c. times .a. c., from the 19th theorem of this [book]. Let this be, for example, .x. and thus known; having done this, following the doctrine of the 45th theorem of this book, we shall achieve what was proposed.

A square diagram divided into four quadrants. The top-left quadrant is labeled 'd', the top-right 'f', the bottom-left 'f', and the bottom-right 'e'. Points on the outer edges are labeled 'a', 'c', and 'b'.

THEOREM LX.

WHY the product of the difference of two roots times their sum is always the difference of the squares of the same roots?

For example, if we take any two numbers as roots, such as 3 and 5, the difference of which is 2; certainly, if we multiply this difference by the sum of the roots, namely 8, the number 16 will be given, which product is the difference of their squares, namely between 9 and 25.

To investigate this, let two roots be represented in the line .n. i., of which one is .n. e. and the other .e. i.; and let their difference be .n. t., from which .t. e. will be equal to .e. i. Hence, having considered the whole square .d. i., and therein with the diameter .d. i., and having drawn a parallel to the side .n. d. from point .e., and another from point .t., and from point .o. a third [parallel] to .n. i. itself, and from point .a. a fourth .x. a. e. parallel to .o. itself, we shall find that .b. n. is the product of the difference .n. t. into the sum of the roots .n. i. And since .d. o. and .a. o. are the squares of the aforesaid roots, .b. c. will be equal to .n. u., since each of these products is equal to .x. u., from which the gnomon .e. d. n. will be equal to the product .b. n., which we desired to know.

A geometric diagram showing a large square 'd i' divided by various parallel and diagonal lines. Labels include n, t, e, i, o, a, x, b, c, d, and u. A shaded L-shaped section (gnomon) is visible.

THEOREM LXI.

WHY, when about to divide some proposed number into two such parts that the difference of the square roots is equal to another proposed number (the square of which, however, does not exceed the square of half the first), they correctly multiply the second number by itself, and subtract the product from the first number, and again square half of the remainder, and subtract this square from the square of half the first, and thus, with the square root of the remainder joined to half the first, the larger part is given, which, when subtracted from the half itself, the smaller part is left.

For example, with the number 20 proposed to be divided as was set forth, namely so that the difference of the square roots of the said parts is equal to two, we shall multiply this two by itself, the square of which, 4, we shall subtract from the first number, 20;

subtract, and the number 16 will remain, the half of which, namely 8, we shall multiply by itself, and the number 64 will be given; when this has been subtracted from the square of half the first, namely from 100, and from the remainder, 36, the square root, namely 6, joined to ten (the half of the first), will give 16 as the larger part, and subtracted from ten, the smaller part.

For the sake of this investigation, let the first proposed number be represented by line .x. y., divided as desired at point .c., so that .x. r. is the product of .x. c. times .c. y.; likewise let .q. p. be the sum of the square roots, namely .q. g. of .x. c. and .g. p. of .c. y. Then upon .q. p. let a square .q. u. be constructed and divided by that method by which we divided in the 41st or 29th theorem, in which square indeed we shall perceive the square of .q. r. of the given difference, and the squares .x. c. and .c. y. placed therein; and so also the ratio by which we know the product .g. r. (as just now in the 29th theorem), the square of which .g. r., from the 19th theorem, will be equal to the product .x. r., and therefore also known; and accordingly, since we know .x. y., we shall follow the method; by the 45th theorem we shall know that this is correctly performed not only by the method brought forward in the 41st theorem, but also by this other method.

A square diagram labeled with letters u, p, c, y, q, t. It shows internal divisions including a diagonal line from q to u and various horizontal/vertical line segments.

THEOREM LXII.

WHY, when about to divide a proposed number into two such parts that the difference of their square roots is equal to another proposed number (the square of which, however, is not greater than the square of the half of the first proposed number itself), they also correctly subtract the square of half the second number from half the first, and multiply the root of the remainder by the second [number], and subtract the product from half the first, so that the remainder is the smaller part sought, and that other remainder of the whole is the larger part.

For example, if the number 50 were proposed to be divided into the two aforesaid parts, and the other [number] were 6, the square of half the second number would be 9; this being subtracted from half the first, 16 would remain, the root of which, 4, being multiplied by the whole second [number], namely 6, would produce 24; which product being subtracted from half the first, namely 25, the smaller part will be given as 1, while the larger will be the remainder of 50, which is 49; moreover, the roots will be 1 and 7, differing from one another by the number six.

To know this, let two numbers be represented by lines, the first by line .b., the second by line .c.; moreover, let the two parts of .b. be noted by two squares .q. and .l. d., and their roots by lines .a. g. and .g. d.; furthermore, let the difference be equal to .c. itself and known as .a. h., from which .h. g. [is] equal—

A geometric diagram consisting of a larger rectangle with labels q, e, f, r, t, s, u. Below this, there are separate line segments labeled a, h, g, d and k, b, c with multiple markers and dashed connecting lines.