Book of Various Mathematical and Physical Speculations

...we ask, if the sum of one of the said quotients with unity gives the first number, what will that same unity give? From which the proposition may arise.

For example, three numbers are proposed: the first, 20; the second, 34; the third, 8. Now we seek to divide the first, 20, into two parts which, when divided by each other, provide two quotients whose sum is such that when 34 is divided by it, there results a number equal to the third number, 8. To perform this, that rule bids that the second, 34, be divided by the third, 8, from which results 4 with one-fourth part; this result will be the sum of the quotients from the division of the two sought parts. If we wish to distinguish these parts, we shall follow the method of the preceding theorem, taking a surface unit for the second proposed number, as if we were saying: let 4 and one-fourth part be divided into two such parts that the product of one by the other is a surface unit. Certainly, by joining the fractions with the whole numbers for the fourth part, there would be seventeen linear fourth-parts of a unit. But since it is necessary, according to the preceding theorem, to multiply the half by itself, and the half would be 8 fourth-parts with an eighth, it will be more conveniently established as 34 eighths; the half of which, namely seventeen eighths, multiplied by itself, will be 289 sixty-fourths of one surface whole—since the surface whole, whose linear unit is divided into 8 parts, is 64, as can be gathered from the first theorem of this book. Now, with this surface unit (namely 64) subtracted from 289, 225 remains; the square root of which, namely 15, joined to the half of the said quotients (namely 17), will give the greater quotient, 32; and subtracted from the other half, it will give the lesser quotient, 2. That is, 32 eighths for the greater quotient, and two for the lesser—specifically, four wholes for the greater and a fourth part of one whole for the lesser. Now if, by the Rule of Three, we say: if 4 joined to one, namely 5, gives 20 (the first number), what will 4 wholes (I mean the greater quotient) give? They will certainly give 16, the greater part. Then if we say: if a fourth part joined to unity gives 20, what will that fourth part (that is, the lesser quotient) give? It will indeed give four, specifically the lesser part. This was certainly unknown to the ancients, who, having found the quotients, rested, not knowing how to use them to find the two parts of the first number.

For the sake of this speculation, let us grant that the first number is signified by line e.u., whose parts e.a. and a.u. are those which are sought; let the other number be signified by line b.d., and the third by line g.f. Let the quotient of the division of e.a. by a.u. be n.t., and the quotient of the division of a.u. by a.e. be t.o.; the sum will be n.t.o., while the unity is n.i. and o.i. Now if the number f.g. proposed thirdly must be produced from the division of the second by o.t.n., it is clear from the 13th theorem that if we divide b.d. by g.f., o.t.n. will be produced; which, when it has been found, must be the sum of the two quotients from the mutual division of the two numbers, namely a.e. by a.u. and a.u. by a.e. Furthermore, it is manifest from the 24th or 25th theorem that their product (multiplying the quotients by each other) will be a surface unit. Thus far, therefore, the whole o.n. is divided at point t. according to the doctrine of the preceding theorem, such that the product of o.t. into t.n. contains only a surface unit. This being done, if, as was said before, we consider n.t. to be the quotient from the division of e.a. by a.u. and t.o. the quotient from the division of a.u. by a.e., it will be clear from the definition of division that the proportion of a.e. to n.t. will be the same as that of a.u. to the unity n.i.; and the proportion of a.u. to o.t. the same as that of e.a. to

Geometric line diagram. A main line is labeled e, a, u. Below it is a line b, d. Below that a shorter line g, f. To the right is a vertical and horizontal line set labeled with points i, n, t, o, and u.

the unity o.i. And by permutation, e.a. is to a.u. as t.n. is to n.i.; and by composition, e.a.u. is to a.u. as t.n.i. is to n.i.; and by conversion, e.a.u. is to e.a. as t.n.i. is to t.n. Wherefore, from the 20th of the seventh [book of Euclid], we rightly use the Rule of Three. I say the same regarding the other part, although he who holds one will also have the other. It is no wonder, however, if a problem of this kind was not defined by the ancients, who did not know this last part.

THEOREM XLVII.

WHY, when two numbers are mutually divided, if their product is multiplied by the sum of the quotients, the final product will be equal to the sum of the squares of the two numbers.

For example, with 16 and 4 proposed and mutually divided, the sum of the quotients will be 4 wholes with a fourth part; when this sum is multiplied by the product of the first numbers, namely 64, it will give 272 surface wholes, which are equal to the sum of the squares of the two numbers.

To consider this, let two numbers be signified by the parts a.e. and e.i. on the line a.i.; let their product be e.d., and let the square of a.e. itself be e.f., while that of e.i. itself be e.q. Let the quotient from the division of e.i. by a.e. be o.u., and let the quotient of a.e. by e.i. be o.t.; let their sum be o.u.t. Then let the product e.d. be signified by the line u.n. joined at a right angle at point u., the extremity of o.u.t. itself; and let the product of u.o.t. into u.n. be n.r. Now we must prove that n.r. is equal to the sum of the two squares q.e.f. I prove this piece by piece, and I assert that the product o.n. is equal to the square q.e., and the product s.t. is equal to the square e.p. For it is clear from the 35th theorem that the number e.i. is a mean proportional between e.d. and o.u.; since the number e.i. is multiplied and divided by a.e. as presupposed, the product of which multiplication is d.e. (namely u.n.) and the quotient from the division is o.u. Wherefore, from the said theorem, e.i. is a mean proportional between u.n. and u.o. Therefore, the product o.n. is equal to the square e.q. from the 15th of the sixth [book] or the 20th of the seventh. I say the same of the product s.t., namely that it is equal to the square e.p., since the number a.e. is multiplied and divided by e.i.; the product of which multiplication is d.e. (namely o.s.) and the quotient from the division is o.t., between which a.e. is a mean proportional according to the 35th theorem. Wherefore, from the cited propositions, the product s.t. is equal to the square e.p. But the whole product n.t. is the sum of the two products o.n. and s.t. from the first [proposition] of the second book of Euclid. It follows, therefore, that what was said is true.

A geometric diagram showing line segments and squares. Line segments are labeled a, e, i. Squares and rectangles are constructed from these segments, with additional points labeled f, p, q, r, o, u, t, n, and s.

THEOREM XLVIII.

WHY, if one divides the larger of two numbers differing from each other by only unity by the smaller, and multiplies the larger by the quotient, the product will be equal to the sum of the larger itself with the same quotient.

For example: 10 being divided by 9 gives one with a ninth part; when this is multiplied by the larger number [sic], namely 10, the product 11 with a ninth part is given, which is just as much...