Book of Various Mathematical and Physical Speculations

THEOREM XXXV.

Any number multiplied and divided by another is a mean proportional between the product of the multiplication and the result of the division.

For example, if 20 be multiplied by five and then divided by five, the product will be 100 and the result 4; between which numbers 20 is the mean proportional.

To observe this, let there be proposed a number to be multiplied and divided, which is signified by line u. e., and the multiplier and divisor by line a. u. Let the product of the multiplication be e. a., and the result from the division be o. e. Now let the result e. o. be multiplied by the dividing number e. i., let the product of which multiplication be e. i. Wherefore, the proportion of the number a. e. to the number e. i. will be the same as that of the number u. e. to the number e. o., from the first [proposition] of the sixth, or the 18th or 19th of the seventh [books of Euclid]. But since the number u. e., from the 11th theorem of the present book, is equal to the number e. i., it follows that what was proposed is true.

A small geometric diagram consisting of a horizontal line segment divided into parts marked 'a', 'u', 'e', and a vertical line segment marked 'o'. A rectangle is indicated above the line segments.

THEOREM XXXVI.

Why those who desire to multiply and divide a proposed number such that the product of the multiplication is as many times the result of the division as is sought, correctly take as the multiplier and divisor some number which is the square root of the denominator of the sought multiplicity.

For example, 20 is proposed to be multiplied and divided, such that the product of the multiplication is ninefold the result of the division—that is, that the result be a ninth part of such a product—wherefore they take the square root of nine itself, that is, of the denominator, namely three. Thus they multiply and divide the given 20, from which the product will be 60 and the result 6 with two-thirds; and the proposition follows.

For the sake of this speculation, let the proposed number be signified by line u. e., and the multiplier and divisor by line u. a. Let the product be e. a., the result e. o., and the square of a. u. be x. a. Therefore, the proportion of a. e. to e. o. will be the double [duplicate] of the proportion of a. e. to the number u. e., from the preceding theorem. Furthermore, let us conceive unity u. i. on the line u. a., and let the two products be terminated at e. i. and x. i. Wherefore, the proportion of a. e. to e. i. will be the same as that of a. e. to u. e., for the number e. i. (though superficial) is the same as the linear number u. e. But a. e. is to e. i. as a. u. is to u. i. from the first of the sixth or the 18th or 19th of the seventh (the same of which I say regarding a. x. to x. i.); wherefore the proportion of a. x. to x. i., that is x. u., will be equal to the proportion of a. e. to u. e. But it was proved in the thirty-third and thirty-fourth theorems that the proportion of the number a. x. to unity is the double [duplicate] of the proportion of the same number a. x. to u. x. It follows therefore, since halves are equal, that the wholes are also equal: that is, that the proportion of the number a. e. to the number e. o. is equal to the proportion of the number a. x. to unity. Thus it is correctly taken that the number a. u. be such that its square

A geometric diagram showing a horizontal line segment with labels a, u, i. Above the segment u-i is a rectangular area labeled x, with point e at the top right.

a. x. be as many times the unity as we desire the number a. e. to be a multiple of the number e. o.

THEOREM XXXVII.

Why those desiring to find two numbers, whose squares collected into a sum are equal to a proposed number, and the product of the same numbers multiplied by each other is equal to another proposed number, correctly take half of the first proposed number (to which the sum of the squares must be equal), and multiply this half by itself, and likewise multiply the other proposed number by itself; they then subtract this square from the first [square of the half], and join the square root of the remainder to the half of the first proposed number; from which sum they extract the square root, which will be the greater of the two sought numbers; then, by subtracting this square [of the result] from the first number and extracting the square root from the remainder, the smaller of the two sought numbers is given.

For example, if 34 were proposed for the first number, to which the sum of two squares must be equal, and the product of whose roots must be equal to the other number, for instance 15; the rule of the ancients bids the half of the first number to be multiplied by itself, the square of which half will be 289; from which if you subtract the square of the second number, namely 225, there will remain 64. And if you take the square root of this, namely 8, and join it to the half of the first number, namely 17, there will be given the greater of the two sought square numbers, 25. Then, this square being subtracted from the half [Wait: subtraction from the first proposed number], the smaller square will be given, namely 9; the roots of which, 5 and 3, would be those numbers which are sought.

For the sake of this speculation, let us conceive the first number, to which the sum of the squares must be equal, to be signified by line a. n. Then let us conceive the sought squares to be signified and joined in the manner written below, t. b. x. Furthermore, let the second proposed number be signified by the product d. b. Now nothing else remains but that we seek the quantities d. p. and b. p.

Therefore, since in line a. n. the number of the sum of the squares is given, the square of the half o. a. would be s. a., which will be known to us; let also a. u. be the number of the greater square, and u. n. of the smaller, and a. z. the product of one by the other; which number a. z. will indeed be equal to the square of the number d. b. from the 19th theorem of this book. Thus a. z. will be known, since its root d. b. is the second proposed number, which will be less than a. s. from the fifth of the second, or the seventh consequence after the 16th of the ninth of Euclid. Now, the quantity z. a. having been subtracted from the square a. s., the square r. x. will be known, whose root will be equal to o. u. from those things last adduced. Therefore we shall know o. u., which number joined to the known half o. a. will give the known square a. u., and thus u. n. will be known likewise, and consequently their roots.

A complex geometric diagram showing multiple labeled line segments and divided rectangles. A main horizontal line at the bottom is labeled a, z, d, p, K, o, u, n. Above it are rectangles labeled t, b, x, containing internal points labeled c and z. A shorter horizontal segment at the top right is labeled with points f, r, s.