Book of Various Mathematical and Physical Speculations

This very thing can also be performed by another method: namely, by joining the sum of k. b., b. d., and b. t. to another rectangle equal to b. d., which let be b. c.; from which the whole square of line d. c. will be known, and thus consequently we shall know its root d. c., by means of which and the product d. b. we shall know d. p. and p. k., as will appear from the forty-fifth theorem of this book.

Michael Stifel, in the eleventh chapter of the third book, proposes a problem of this kind, which he himself, however, solves by way of algebra.

A geometric diagram showing a sequence of rectangles and line segments labeled with letters t, x, k, b, d, p, c along the horizontal base and z, u, n, w marking vertical segments.

THEOREM XXXVIII.

WHY for those who wish to find two numbers whose product is equal to some proposed number, and the difference of their squares is equal to another proposed number, it is correct that they multiply half of the second proposed number by itself—to which number indeed the difference of the squares must be equal; furthermore, let them add to this square the square of the first proposed number, to which the product of the sought numbers is to be equal; then let [one] join the square root of this sum to half of the second proposed number—to that, I say, to which the difference of the squares must be equal—from which the greater square arises, from which, when the second number is subtracted, the lesser square remains.

For example, if there were proposed in the first place the number 8, to which the product of the sought numbers is to be equal, and then there were proposed the number 12, to which, by subtracting the lesser from the greater, the difference of the squares of each sought number must be equal: it is necessary to multiply half of this last number, 12, by itself, and it will be 36, the square of the half; whence we should collect into a sum the square of the first number 8, which would be 64, which with 36 would make 100; the root of this hundred, namely 10, being collected into a sum with half of the second number, namely 6, would give the greater square, namely 16; from which, when the second number, namely 12, is subtracted, the lesser square, 4, would remain.

For the sake of this speculation, let the greater unknown square be signified by line q. g., and the lesser likewise unknown square by line g. i.; therefore q. i., their difference, will remain known as a given, together also with b. i. and q. b., its halves; then let there be imagined a square y. g. upon b. g. and a rectangular parallelogram g. r. designated, and thus also a gnomon u. g. t. as proposed in the sixth [proposition] of the second [book] of Euclid, from which the square b. i., namely u. t., will be known; but the gnomon is equal to rectangle g. r. from the aforesaid, or from the 8th after the 16th [proposition] of the ninth [book],

Geometric construction for a mathematical proof, showing a horizontal base line q-b-i and various squares and a gnomon built upon it, with points labeled q, b, i, g, y, u, t, s, and r.

of the ninth [book], and this rectangle g. r. is the square of the first proposed number from the 19th theorem of this book, and thus it will be known. Together also the gnomon u. g. t. will be known, wherefore the whole square g. y. and its root b. g. will be manifest; to which, when the given q. b. is joined, the greater square q. g. will be known, from which b. g., when the given b. i. is subtracted, the lesser square i. g. will be known, and consequently their roots also will be known.

THEOREM XXXIX.

THIS SAME THING can also be determined by another method, setting aside the way of the ancients: namely, by the first and second proposed numbers being multiplied by themselves, and the square of the first being quadrupled; which sum being joined with the square of the second number, and the square root being extracted from this other sum, from which root the second number is subtracted, and half of the remainder is taken, which will be the lesser square; which being subtracted from the root joined last, the greater square will remain.

For example, if the number 8 were proposed, to which the product of the two sought numbers is to be equal, and the same 12 were proposed, to which the difference of the squares of the two numbers must be equal: I command that the first number, namely 8, be multiplied by itself, from which will arise 64 as the number of its square, which I wish to be quadrupled, and the product will be 256; which I judge should be joined with the square of the second proposed number, namely 144, and the sum will be 400. From this the root will be taken, namely 20, and from this the second number 12 will be subtracted, and half of the remainder, namely 4, for the lesser square; which, being collected into a sum with 12, will give the greater square 16.

For the sake of this speculation, let the greater square be signified by line q. g., and the lesser by g. p. Moreover, upon the whole q. p. let there be erected the whole square d. p., divided like square f. g. of the twenty-seventh theorem of this book (the same would happen if the square were divided in the manner of the eighth [proposition] of the second [book] of Euclid), which division indeed is the way of the four products of q. g. into g. p., of which one is g. r., which will be known from the 19th theorem since it is the square of the first proposed number, from which those four will be known. Now indeed, if we consider q. p. cut at point t. so that q. t. is equal to p. g., the difference t. g. will be given as known, like the root of square e. o., since from what was presupposed r. n. is equal to q. g. and r. e. [is equal to] g. p., from which also q. t.; thus likewise e. n. will be equal to t. g. Therefore, the square e. o. of t. g. itself being collected with the quadruple of g. r., the square d. p. of q. p. itself will be known; wherefore q. p. will be known, from which number, when the known difference of the squares t. g. is subtracted, the known aggregate of p. g. and q. t. will remain. Wherefore, in consequence, half of the aggregate, namely g. p., will be known as the lesser of the two squares; to which g. t. being joined, or p. g. being subtracted from p. q., the greater square q. g. will remain known.

A geometric diagram of a large square labeled d, s, n, p at the corners, divided internally into several rectangles and squares, with points q, t, g, r, o, e used as labels for various parts of the construction.

THEOREM XL.

WHY for those who wish to find two such numbers that the greater of them exceeds the lesser by a proposed number, and the product of one into the other is made equal to another proposed number, it is most advisable that half of the first proposed number...