Book of Various Mathematical and Physical Speculations

with .12. and a half will be .155. which will be equal to the sum of the squares of the two parts, namely .100. with .55.

To demonstrate this, let the proposed divisible number be represented by line .a. d. and the sum of the roots, taken in our manner, by line .e. h. of which the first and greater is .e. f. the second .f. g. and the third .g. h. Let us also imagine line .a. d. to be divided in the same ratio as .e. h. for it will be clear from the method of the preceding theorem that each of the parts of .a. d. will relate to its whole just as the individual parts of .e. h. relate to their own whole. I say this so that we may understand that we speak correctly: If .e. h. is to .a. d. then .e. f. will give .a. b. and so for the rest. Wherefore, by permutation, .a. b. will be to .b. c. as .e. f. is to .f. g. and I say the same for the rest. Therefore, from the 18th [proposition] of the sixth [book] or the 11th of the eighth, the ratio of the square of .a. b. to the square of .b. c. will be the same as that of the square of .e. f. to the square of .f. g. for the wholes are equal since their similar parts are equal to one another. I say the same of the ratio of the square of .a. b. namely that it is to .c. d. as the square of .e. f. is to the square of .g. h. From which, by the 24th of the fifth, the ratio of the square of .a. b. to the sum of the squares of the two parts .b. c. and .c. d. will be as that of the square of .e. f. to the sum of the squares of .f. g. and .g. h. But the square of .e. f. is equal to the sum of the squares of .f. g. and .g. h. therefore the square of .a. b. will also be related in this way, namely being equal to the squares of .b. c. and .c. d. You shall say the very same regarding other powers, and you shall use the 21st theorem of this book.

A geometric diagram consisting of a horizontal line divided into three main segments. Markers 'a', 'b', 'c', 'd' are placed above the line and 'e', 'f', 'g', 'h' are placed below the line, aligned with the division points. Vertical dotted lines connect some of these points.

THEOREM LVII.

A SIMILAR problem is also proposed by the ancients as indeterminate, which is of this sort:
Whether any number can be divided into three such parts that the square of one is equal to the sum of the squares of the other two parts together with the product of one into the other.
For example, if the number .30. is proposed to be divided as already stated, any other number must be found which is nevertheless the sum of three roots so related that the square of one is equal to the sum of the squares of the two parts together with the product of one into the other; let us take the one that first occurs, such as .30. which is the sum of the numbers .6. 10. 14. parts so related that the square of .14. itself is equal to the sum of the squares of the other parts together with the product of one into the other; and let us apply the rule of three, and say, if .30. is worth .30. what will .14. be worth? namely .14. with a third part. We shall effect the same for the other parts, of which one will be .16. with two thirds, and the other .10. without fractions; from which the square of the first will be .544. with .4. ninths, of the second .277. with seven ninths, of the third .100. and the product of the second into the third .166. with .6. ninths; which product, collected with the squares of the second and third, will be .544. with .4. ninths.
The investigation of this matter is the same as that of the preceding theorem, until you know that the ratio of the square of .a. b. to the sum of the squares of .b. c. and .c. d. is the same as that of the square of .e. f. to the sum of the squares of .f. g. and .g. h. But since here we do not give the square of .e. f. as equal to the sum of the squares of .f. g. and .g. h. but greater by the product of .g. h. into .f. g. (or what is the same, vice versa), the subsequent figures must be considered, of which .i. is the square of .a. b. .l. is the square of .e. f. .x. is the square of .b. c. .y. is the square of .f. g. .p. is the square of .c. d. .q. is the square of .g. h. let .k. be the product of .b. c. into .c. d. and let .m. be the product of .f.

g. into .g. h. Now, from the investigation of the preceding theorem, the ratio of .n. t. to .o. u. will be the same as that of .n. s. to .o. r. wherefore the product .k. by definition will be similar to the product .m. since both are rectangles; whence the ratio of .k. to .m. will be double the ratio of .n. t. to .o. u. from the 18th of the sixth. Therefore, the ratio of .k. to .m. will be equal to the ratio of .x. to .y. and .p. to .q. and .i. to .l. and by permutation it will be thus: .k. to .i. as .m. to .l. But it has been proved that .x. p. is to .i. as .y. q. is to .l. Wherefore, from the same 24th of the fifth, .x. p. k. will be to .i. as .y. q. m. is to .l. But .y. q. m. is equal to .l. Therefore .x. p. k. will likewise be equal to .i.

A complex geometric diagram composed of several square and rectangular figures.
Top left: A simple square labeled 'i'.
Top right: A larger composite rectangle divided into four segments labeled 'x', 'n' (upper row), and 'K', 'P' (lower row). A vertical line extends downwards labeled 's'.
Bottom left: A square similar to 'i', labeled with a script 'f' (possibly 'l').
Bottom right: A composite rectangle divided into segments labeled 'u', 'm', 'q', with a numeric value '112' or symbol '11z' in the lower left section.

THEOREM LVIII.

THE ancients proposed another problem as well, though not defined: namely, whether any number can be divided into .4. such parts that the sum of the squares of two parts is double the sum of the squares of the remaining two.
Indeed, the execution and investigation of this will not be difficult, since it is the same as that which was brought forward in the two immediately preceding theorems, taking indeed the sum of any roots so related as was stated. For example, .44. whose parts will be .16. 12. 14. 2. then we shall proceed by the rule of three, saying: If .44. is worth the proposed number, what is the greater part .16.? namely, it will be worth the greater part of the proposed number corresponding to .16. I say the same for the others.
Furthermore, the investigation is the same as with the above.

THEOREM LIX.

WHY [the method for] dividing a proposed number into two such parts that the product of the square roots of the parts themselves is equal to another proposed number, provided however that its square is not greater than the square of the half of the first proposed number. He correctly multiplies the second proposed number by itself, and subtracts the same from the square of the half of the first, and subtracts the square root of the remainder from the half of the first itself, from which the smaller sought part is given; which being joined to the half itself, the larger part is obtained.
For example, if the number .20. is proposed to be divided in the proposed manner into two such parts that the product of the roots is equal to (for instance) .8. We shall multiply the half of the first number by itself, the square of which will be .100. from which we shall subtract the square of the second number, namely .64. and there will remain .36. its square root being joined to .10. (I mean the half of the first proposed number), the number .16. the greater part, will be given, and being subtracted from the half, the smaller part will be given, namely .4.