Book of Various Mathematical and Physical Speculations

...it is allowed, however great the sum of the greater [part] is with the quotient.
For the sake of this investigation, let the greater number be represented by line .a.i. and the lesser by line .a.o., from which, by what was previously established, .o.i. will be unity. Now, let the quotient resulting from the division of .a.i. by .a.o. be .a.u.; let .e.a. be joined directly to .a.i., and let the product of .a.i. into .a.e. be .u.i. I shall prove that the surface number .u.i. is equal to the linear number .e.i.a.; for which it is necessary to remember that it was proved in the thirteenth theorem that if a divisible number is divided by the quotient, the resulting quotient will be the dividing number; wherefore .a.o. will be the quotient from the division of .a.i. by .a.u. And from the definition of division, .e.a. will be to .a.i. as .o.i. is to .a.o.; and by composition, as .e.i. is to .a.i., so is .i.a. to .o.a. Therefore, .a.i. will be the mean proportional between .e.i. and .a.o. But .a.i. is not only conceived here as divided by .e.a., from which the quotient .a.o. results, but it is also multiplied by the same .e.a., from which the number .u.i. is produced. Thus, from the 23rd theorem, .a.i. is the mean proportional between .u.i. and .a.o. Therefore, from the 1st [proposition] of the fifth book, the ratio of .u.i. to .a.i. will be the same as that of .e.i. to the same .a.i. Therefore, from the 9th [proposition] of the aforesaid [book], the number .u.i. will be equal to the number .e.i., which was the thing proposed.

A rectangular geometric diagram with vertices and points labeled. Top line: u, i, n. Bottom line: e, a, o. Vertical lines connect u-e, i-a, and n-o.

T H E O R E M X L I X.

The same thing can be considered by another method.
Let the line .u.a. be cut at point .a. such that .a.r. is equal to unity .o.i., and let the middle parallel .t.n. be terminated by the produced line .t.i., which will consist of a number, although a surface one, equal to the number .a.i.r., even though it is linear. Then let a parallel be drawn from point .o. to .a.u., and let it be terminated by the produced line .u.u.; from which two products will be given, .u.o. and .t.i., equal to each other by the 15th of the sixth book or the 20th of the seventh, since .a.i. is to .a.u. as .a.o. is to .a.r. But by permutation, .a.i. is to .a.o. as .a.u. is to .a.r.; and from the 1st of the sixth book or the 18th or 19th of the seventh, .u.i. is to .u.o. as .a.i. is to .a.o.; that is, .u.i. is to .t.i. by the help of the 15th of the fifth book. Now, from the definition of division, .e.a. is to .a.i. as .o.i. is to .o.a.; and by composition, .e.i. is to .a.i. as .i.a. is to .o.a. Therefore, from the aforesaid .u.i., .e.i. will be to .i.a. as .u.i. is to .t.i. But .t.i. consists of a number equal to .a.i.; wherefore, from the 9th of the fifth book, the number .u.i. will be equal to the number .e.i.

A geometric diagram showing a rectangle with labels u, t, n across the top and e, a, o along the bottom. Points i and r are marked on the vertical segments.

T H E O R E M L.

WHY [those] dividing a proposed number into two such parts, that the product of one by the other added to their difference equals some other number greater than the first: They correctly first subtract the first number from the second, but keep the remainder; then they always take away the number two from the first number, and keep the half, but multiply the other half by itself; and from this square they take out the saved number, and the square root of the remainder from the saved half, which final remainder is the sought lesser part of the proposed number.
For example, if the number 20 is proposed to be divided so that the product of one part by the other, added to the difference of the parts, is equal to the proposed

number, for instance, 92: the rule commands that the first number be subtracted from the second, namely 20 from 92, the remainder of which, i.e., 72, is kept; then it commands the number two to be subtracted from the first, thus in the proposed example 18 will remain; it then commands the half of this 18 to be multiplied by itself, and since this is 9, the number 81 is given; from which 81 the rule requires the first saved number, namely 72, to be subtracted, thus 9 will remain; then the square root of this 9 must be subtracted from the half of the 18 itself which was [determined] before the squaring, so 6 will be left over—that is, 9 with the square root taken out—which 6 will be the sought lesser part, and the greater part will be 14, the product of which, 84, joined with the difference of the parts [8], exactly provides 92.
The theoretical investigation of this matter is as follows. Let the first lesser number, which is proposed as divisible, be represented by line .q.g., and the greater [number] by line .x.; then let us imagine .q.g. divided, of which the greater part is .q.o. and the lesser .o.g., the difference being .q.p., from which .p.o. will be equal to .o.g.; let the product be .b.o. It is necessary, therefore, that .b.o. together with the difference .q.p. be equal to the second proposed number .x., which is known; wherefore the sum of the product .b.o. with the difference .q.p. will also be known, from which, if the first number .q.g. is subtracted, the remainder will be known. Now then, what will this remainder be? Let us observe by what reasoning .q.g. is to be subtracted from the sum of .b.o. and .q.p. In the first place, if we subtract from the said sum .q.p., which is part of .q.g., there will remain to be subtracted .p.g. from .b.o.—a part, I say, of .q.g. itself—which will happen whenever we imagine .q.o. diminished by two units and multiplied by .o.g., and the product becomes .b.e.; for since .o.g. enters into .b.o. as many times as there are units in .q.o. from the 1st of the sixth book or the 18th or 19th of the seventh, and .p.g. is to be subtracted from .b.o., which .p.g. is double .o.g., it will be evident that .o.e. is equal to .p.q.; there will remain, therefore, .b.e., the product of .q.e. into .o.i., [which is] known; and after taking those same two units from .q.g., there will remain .q.i., known to us, from which .e.i. will be equal to .e.e. Since, therefore, we know the product of .q.e. into .e.i. together with .q.i., if we wish to know the parts .q.e. and .e.i., we shall use the 45th theorem of this book, and we shall obtain the proposition; for we shall know .e.i., and consequently .o.g., which is its equal.

A line diagram used for geometric proof. A main horizontal line is labeled with points b, q, e, i, p, o, g. Above it, point x is indicated as a length, and vertical lines extend to create segments labeled with points like n.

T H E O R E M L I.

TO DIVIDE a number into two such parts which take another proposed number as a mean proportional—[that number being] less than half of the first—is nothing other than to find two parts of the first number which, when multiplied together, produce a number equal to the square of the second number, by the 16th of the sixth book or the 20th of the seventh, which thing was nevertheless investigated by us in the 45th theorem.

T H E O R E M L I I.

WHY, given any three numbers, if the product of the first into the second is multiplied by the third, and this solid product is divided by the first number, the quotient will be a number equal to the product of the second into the third.
For the sake of example, let these three numbers be proposed: 10, 11, 12. Let 10 be multiplied by 11.