Book of Various Mathematical and Physical Speculations

11. will be given 110. Which product, being multiplied by 12, will give 1320. This result, divided by the first [number], namely 10, will give 132, a number equal to the product of the second into the third of the proposed numbers, that is, 132.
To investigate this, let the first number be represented by line .o.u., the second by .e.o., and the third by .e.a. Let the product of .o.u. into .o.e. be .o.i.; and let the solid product of this .o.i. by .e.a. be .i.c. Then let the product of .e.o. into .e.a. be .e.c. I say now that if the solid number .i.c. is divided by the first .o.u., the result will be equal to the number of the product .e.c. Wherefore, it must first be considered that since the product .i.c. arose from the multiplication of .o.i. into .e.a., the said .o.i. will enter into .i.c. as many times as unity is found in .e.a. By the same reasoning, .e.c. [enters] into .i.c. as many times as there are units in .o.u. Therefore, it follows that if .i.c. is divided by .o.u., the result is the solid .e.c., which is nevertheless equal to the surface product .e.c.

A geometric diagram of a rectangular parallelepiped with vertices and edges labeled a, c, e, i, o, u.

THEOREM LIII.

When dividing a proposed number into three parts so related that the product of the first into the second, multiplied by the third, provides a number equal to another proposed number: he rightly divides the second number by any other [part] smaller than the first, which divisor will be one of the three sought parts, and the result will be the product of one into the other of the remaining two, the sum of which will be known by subtracting the dividing number from the first given [number]; if anyone wishes to distinguish these parts, he shall use Theorem 45.
For example, the number 20 is proposed to be divided into three parts so related that the product of the first into the second, multiplied by the third, gives 90. Therefore, one part of 20 must be taken for the first, whatever it may be—for example, 2. Let the second number, namely 90, be divided by this; thus 45 will be given, which will be the product of the other parts with each other, the sum of which is 18. If you wish to distinguish this sum into the other two separate parts, you will use Theorem 45, so that you may achieve what you desire as quickly as possible; the parts, moreover, will be 3 and 15.
For the sake of this investigation, nothing else occurs except what was brought forward in the preceding theorem and the earlier Theorem 45.

THEOREM LIIII.

To divide a number into 3 parts of such a kind that the square of one is equal to the product of the remaining two with each other, is exactly the same as in Theorem 51. For he who takes any part of the proposed number (provided it is not greater than the third part), and divides the remainder into two such parts that the first part taken is the mean proportional, will achieve the proposal by the proof brought forward in Theorem 51.

THEOREM LV.

The same thing can be performed by another method different from that which we introduced in Theorem 51.

For let three continuously proportional numbers be taken, of whatever proportionality, which are collected into a sum; and afterward, by the Rule of Three, let us say: If this sum corresponds to the first number proposed to be divided into three parts, to what will one of the three parts of this sum correspond? I say the same for the remaining two parts.
For example, if the number 57 is proposed to be divided into three continuous proportional parts in a sesquialter ratio, we shall take three numbers distinguished in such proportionality, such as 4, 6, and 9, which, collected into a sum, will give the sum 19. And we shall say: if 19 gives 4, what will 57 give? Whence the result for one part will be 12. Then if we say: if 19 gives 6, what will 57 give? Namely, it will give 18. Finally, if 19 gives 9, what will 57 give? Namely 27. And thus 18 will be given, the square of which will equal the product of the remaining two parts with each other.
That we may know this, let the proposed number to be divided into any three parts be represented by line .a.d.; let the three numbers of the said proportionality be denoted by lines .c.f., .f.g., and .g.h. joined directly to one another. Let us likewise conceive the line .d.a. divided into three parts .a.b., .b.c., and .c.d. in the same proportionality as the others; then the ratio of .a.d. to any of its parts will be the same as that of .c.h. to its corresponding part in .a.d. For example, .a.b. corresponding to .c.f., .b.c. to .f.g., and .c.d. to .g.h. For I say that .a.d. will be to .c.d. as .c.h. is to .g.h. For since .a.b. is to .b.c. as .c.f. is to .f.g. by presupposition, by permutation .a.b. will be to .c.f. as .b.c. is to .f.g.; and by the same reasoning, .c.d. will be to .g.h. as .b.c. is to .f.g., and consequently as .a.b. is to .c.f. From which, by the 13th [proposition] of the fifth [book], the whole .a.d. will be to the whole .c.h. as .c.d. is to .g.h., or .b.c. to .f.g., or .a.b. to .c.f. Therefore, by permutation, the proposal will be manifest; and the product of .a.b. into .c.d. will be equal to the square of .b.c. by the 15th [proposition] of the sixth [book] or the 20th of the seventh.

Lines and labels illustrating geometrical proportions with points marked a, b, c, d and c, f, g, h.

THEOREM LVI.

The ancients also proposed another indeterminate problem, which shall be defined by me according to custom and reason; it is of this kind:
How a proposed number may be divided into three parts of such a kind that the square of one is equal to the sum of the squares of the remaining two parts.
To effect this, let us take three separate squares, one of which is equal to the remaining two, and let their roots be collected together into a sum; then we shall follow the Rule of Three, by the reasoning demonstrated in the preceding theorem, and we shall proceed rightly as there. What I say concerning squares, I assert also concerning cubes and any powers whatsoever.
For example, if the divisible number 30 is proposed to be divided into three such parts that the square of one is equal to the sum of the squares of the remaining two parts, we shall first take the roots of three squares, related in any way so that the greatest of them is equal to the sum [of the squares] of the other two—for example, 3, 4, and 5; namely 5, 4, and 3, which if they are collected into a sum, result in 12. Then from the Rule of Three we shall say: if 12 corresponds to 30, to what will the greater root 5 correspond? Namely, to 12 and a half.
Then if we say: if 12 is worth 30, what will the middle root 4 be worth? Namely, it will be worth 10. And the third, the lesser, 7 and a half. Thus the whole sum is 30, and the square...