Book of Various Mathematical and Physical Speculations

—I say, to multiply the number to which the difference of the two sought [numbers] must be equal by itself, and to this square join the proposed number, which must be equal to the product of the sought numbers, and from this sum extract the square root, which, when joined to half of the first proposed number, will give the greater of the two numbers; and by subtracting half of the first number from the same root, the lesser of the two sought numbers [will be given].

For example, if 12 were proposed, to which the difference of one number from the other should be equal, and then 64 were proposed, to which the product of the multiplication of the two sought numbers should together be equal. Half of the first number multiplied by itself would produce the square 36, which, when the second [number], namely 64, is joined to it, the whole would be 100; from which, having extracted the square root 10, and having joined to it the number six, the half of the first number, and having subtracted the same half, 6, from the same [root], there results 16 for the greater number and 4 for the lesser.

The demonstration of this matter is as follows. Let a.o. be the known difference of the two unknown numbers a.o. and a.e., whose given or known product is a.i.. Let us now consider e.i., the half of e.o., the given difference; and from the composite a.i. let a square a.x. be imagined, in which is drawn i.n. parallel to the side a.l., and as far removed from a.l. itself as x.i. is from s.e., whence l.e. will be the square of e.i., namely, of the half of the given difference e.o.. And the rectangle t.n. will be equal to the rectangle n.c., as anyone may consider for himself; whence it follows that the gnomon e.r.t. is equal to the product a.i. and is therefore known. If this gnomon is joined to the square e.r., known from the known root e.i. (as half of the total given difference a.o.), we shall have the known total square a.x., and thus its known root a.l., and consequently all the remaining things. This demonstration is the same as [Proposition] 6 of the second [book] or 8 of the ninth of Euclid.

Nevertheless, from the manner and reasons brought forward in the preceding theorem, one may conclude this very thing.

Geometric diagram of a divided square labeled with points a, o, e, s, t, r, n, i, l, c, x to illustrate a mathematical proof.

THEOREM XLI.

WHY those who, given some proposed number, are to find two numbers differing from each other, the sum of whose squares is equal to another proposed number, correctly multiply the first proposed number by itself, which square they subtract from the second number, and take half of the residue, which will be the product of the multiplication of the two numbers with each other; in the rest, they follow the order of the preceding theorem.

For example, if 12 were proposed as the number to which the difference of the two sought numbers must be equal, and moreover 272 were proposed, to which the sum of the squares of the two sought numbers must be equal: it would indeed be necessary to multiply the first number, namely 12, by itself, whose square in this place would be 144, and to subtract this from the second number; 128 would remain. Then, having taken half of this number, namely 64, I say, [as] the product of the two sought numbers: with this 64, and afterwards with the first proposed number twelve, we would follow the order of the preceding theorem.

In order to demonstrate this, let us consider the figure below, similar to the figure of the twenty-ninth theorem, in which the sought numbers are signified by the two directly joined lines q.g. and g.p.. Their squares will be a.c. and g.s., the sum of which is proposed, and therefore also known. Let the difference of the two numbers first proposed be q.i., and its square m.c., which is known from its root q.i.. Wherefore the gnomon e.n.m., together with the smaller square g.s., will be known, which sum is equal to double [the rectangle] g.r., the product of the given numbers. Therefore g.r. itself will also be known; now, if we apply the demonstration of the preceding theorem to the remaining parts, we shall achieve the objective.

Geometric diagram showing nested squares and rectangles within a larger square, labeled with points a, c, u, m, o, i, h, n, f, r, g, q, l, d, p.

THEOREM XLII.

EVEN still, we could achieve this same thing by another method, without consulting the fortieth theorem. For, having subtracted the square of the difference—of the first, I say, proposed number—from the sum of the two squares, namely from the second proposed number, the residue should be added to the aforesaid second number, and from this sum the square root must be taken, which will be the sum of the two numbers; by subtracting the first number from this [root], there will remain double the lesser sought number, to whose half, by adding the first proposed number, or by subtracting the found lesser number from the last-found root, the greater number which is sought will be given.

For example, when 128 remained: if we join this with the second number, namely 272, they will give 400, whose root will be 20; from which number, having subtracted the first proposed [number], namely 12, 8 will remain, whose half will be 4; by subtracting this from 20, or by joining 12 [to it], the greater number will arise.

The contemplation of this matter is opened by the preceding figure. For the residue of the subtraction of the square m.c. from the sum of the two squares a.c. and g.s. provides a number equal to the two supplements q.n. and n.u., from [Proposition] 8 of the second [book] of Euclid. This [residue], being joined to the two squares (whose sum was the second proposed [number]), brings forth knowledge of the square q.u. and its root q.p., from which, having subtracted the first given number, namely q.i., there remains i.p., whose half, namely g.p., is the lesser number which is sought; but the residue of the whole, g.q., is the greater.

THEOREM XLIII.

WHY those who wish to find two numbers whose sum will be equal to some proposed number, and the sum of the squares [is] greater than their product by the quantity of another proposed number, correctly multiply half of the first given number by itself, which square they subtract from the second given number, and they take the square root of the third part of the residue, which they join to half of the first number; from which is given the greater number of the two sought [numbers], which being subtracted from the whole first [number], the lesser will remain.

For example, the number 20 being proposed, to which the sum of the two sought numbers is to be equal, and the second number 208 being given, which must always be greater...