Book of Various Mathematical and Physical Speculations

...from the square of the half, as will be known from the speculation of this type of work, to which must be equated the difference between the sum of the squares of the two numbers being sought, together with the product of their roots. Half of the number 20 would have to be multiplied by itself, and the square subtracted from 108, so that 108 would remain; of which 108, the square root of the third part would be 6. If this were joined to half of 20, namely 10, the greater sought number, 16, would be given; which, being subtracted from 20, 4 would be given.

For the sake of this speculation, let the first given number be signified by line g. h., in which the greater unknown number is g. h. [sic], but the lesser is b. h.; let their squares be y. t. and b. l. in the greatest square g. p.; then let the product of g. b. and b. h. be g. c.; and let two diameters, q. h. and g. p., be imagined, divided in the middle at point o., through which two lines are drawn, f. d. and r. m., parallel to the sides of the greatest square. These will divide the said square into four equal squares, each of which will be equal to the square g. f. of half the given g. h. itself; wherefore each of them will be known. Again, let us imagine s. x. through c. parallel to g. r., as far distant from g. r. as y. l. is found to be distant from g. h.. Likewise, let z. i. a. be imagined through point i. parallel to d. p.; wherefore a. t. will be equal to f. c., and y. x. equal to f. c., and y. s. equal to b. l.. Thus, with a. t., y. x., and b. l.—equal to the product y. b.—subtracted from the two squares mentioned above, there will remain known k. d. and a. c. x., as being equal to the second given number. But k. d. is the square of the known half g. f.; therefore the remainder a. c. x. will be known, as well as each of the three individual parts, namely the squares o. i., o. c., and o. e., and the root b. f. or f. s. of each; which, when joined to the half g. f. and again subtracted from the same, we shall attain the proposed goal.

A complex geometric diagram consisting of a large square divided into smaller rectangles and squares by internal lines and two diagonals. Labels for points include letters q, d, t, p along the top; g, k, y, s along the left; g, z, s, f, b along the bottom; p, t, m, n, b along the right; and internal points such as x, o, i, c, a, k, e.

THEOREM XLIV.

WHY, if anyone should desire to divide a proposed number into two such parts that the square of the greater exceeds the square of the lesser by the quantity of another proposed number, he will correctly first multiply the first number by itself, and from this subtract the second number, and divide the remainder by double the first; from which the resulting value will be the lesser part of the first, which, being subtracted from that first, will produce the greater part.

For example, if 20 were proposed, to be divided into two such parts that the square of the greater exceeds the square of the lesser by a number equal to 240, it will be necessary to multiply the first number (which, when squared, will be 400) by itself, and from this square subtract the second number, namely 240; then 160 will remain. These being divided by 40, a number double the first, four will be given for the lesser number; but from the remaining 20, with four subtracted, there will be 16 for the greater number.

In order that we may consider this exactly, let the first proposed number be signified by line q. h., to be divided into two parts q. p. and p. h. such as we seek. Afterward, let the square q. e. be erected, divided by the diameter f. h., and with p. o. t. and a. o. c. drawn parallel to the sides of the square, the imaginary squares c. t. and p. a. of the two unknown parts q. p. and p. h. will be given. To these let us imagine the square u. n. equal to square p. a., extracted from the

greater square c. t.; wherefore the remainder of the square c. p. will be known. Since this known quantity is given in the second place, let us imagine it to be subtracted from the whole known square q. e., from which the sum of the two supplements q. o. and o. e. will be known, together with the squares u. n. and p. a.; namely, double q. a.. When this is divided by double q. h., or the simple q. a. by the simple q. h., a. h. will be given, namely p. h., the lesser sought number.

A geometric diagram showing a square (f, t, e, c) with a diagonal line from top-left 'f' to bottom-right 'a'. An internal rectangle or square is defined by points u, n. Labels include f, t, e, u, n, c, q, p, b, a.

THEOREM XLV.

WHY those wishing to divide a proposed number into two such parts that the product of one by the other is equal to another proposed number, correctly multiply half of the first given number by itself, from which square they subtract the second given number, and take the square root of the remainder; which root, being joined to one half of the first number, the greater part is given, but being subtracted from the other half, it will reveal the lesser.

For example, if the number to be partitioned were 34, and the other number were 64, to which the product of one part by the other ought to be equal. We would multiply half of the first number by itself, whose square would be 289; from which, with the second number, namely 64, subtracted, 225 would remain; whose square root, namely 15, joined to 17 (half of 34), will produce 32 for the greater part, and being subtracted from 17, there would remain 2, the lesser part, I say.

For the sake of this speculation, let the first proposed number be signified by line a. d., whose half c. d. will be known, as well as its square c. f.. With this divided by the diameter c. d. [sic], let the unknown parts of a. d. be supposed to be a. b. and b. d.; and from point b. let the line b. h. g. be drawn parallel to d. f., and m. h. x. parallel to d. a., with a figure constructed similar to the fifth figure of the second [book] of Euclid; wherefore the gnomon l. d. g. will be given, equal to the product b. x. and therefore known. When this is subtracted from the square c. f., there will remain the square g. l., whose root being equal to c. b., joined to a. c. and subtracted from c. d., the sought parts a. b. and b. d. will be given.

A geometric diagram showing a subdivided rectangle illustrating a gnomon construction related to Euclid's geometry. Points are labeled k, g, f, m, l, a, c, b, d.

THEOREM XLVI.

WHY, when three numbers are proposed, of which the first is to be divided into two such parts that, when mutually divided, and the second number being divided by the sum of the resulting values, the final result is equal to the third of the proposed numbers. It is most advisable to divide the second number by the third, from which the result is the sum of the results from the two parts mutually divided; which sum, if one wishes to distinguish [the parts], he can correctly do by means of the operation of the preceding theorem, taking a superficial unit for the second number. With the results afterward distinguished, we shall correctly operate, in my judgment, by the rule of three (which was omitted by the ancients). If we should say: