Book of Various Mathematical and Physical Speculations

...tributed, as u. x. is to n. x. from the first proposition of the sixth book, or the 18th or 19th of the seventh; wherefore, from the 11th of the fifth, o. x. will be to e. x. as s. x. is to unity; but as s. x. is to unity, so also is o. x. to l. m. Hence, from the aforementioned 11th, o. x. will be to m. as the same o. x. is to e. x.; and so from the 9th of the aforementioned fifth, m. will be equal to o. x.

T H E O R E M XXXI.

WHY, given any proposed number divided into two unequal parts, if it is further divided by each of those parts, the quotients added together produce as much as they do when multiplied.
For example, let ten be the proposed number, divided into two and eight; the quotients will be five and one and a quarter, which added together will be 6 and a quarter in linear measure, and if multiplied together, they will likewise be 6 and a quarter in superficial measure.
For the sake of this demonstration, let the total number be signified by line q. p., its two parts by the greater x. and the lesser u., and unity itself by t. Let the quotient from the division of q. p. by x. be q. i., and the quotient of the same q. p. by u. be q. f. Wherefore, from the definition of division, q. p. is to q. i. as x. is to t., and q. p. is to q. f. as u. is to t., which is to say q. f. is to q. p. as t. is to u. Hence, from the equality of proportions, q. f. will be to q. i. as x. is to u., and vice versa. Furthermore, let unity be signified on line q. p. by line q. o. This done, let us say: if q. p. is to q. i. as x. is to q. o., then by permutation, q. p. will be to x. as q. i. is to q. o., which is to say u. is to x. as i. q. f. is to q. f. (for x. and u. are the integral parts of the whole q. p., and the ratio of x. and u. to x. is as i. q. f. is to q. f. from the 18th of the fifth). Wherefore i. q. f. will be to q. f. as q. i. is to unity, q. o., from the 11th of the fifth. Then let q. i. be added to q. f., and let q. i. be multiplied by q. f., and let the product of this multiplication be x. f., which I shall prove to be equal to the sum of f. q. and q. i. For let line q. x. be cut at point s. such that q. s. is equal to q. o., and let the product be marked s. f. Wherefore the proportion of the quantity x. f. to s. f. will be the same as q. x. is to q. s. from the first of the sixth, or the 18th or 19th of the seventh, that is, as q. i. is to q. o., and from the 11th of the fifth (as was said) as i. q. f. is to q. f. But the superficial number s. f. is as great as the linear q. f.; wherefore from the 9th of the fifth the number x. f. (superficial) will be as great as the number f. q. i. (linear), which was the thing proposed.

A mathematical diagram consisting of horizontal line segments. A main segment is labeled with points q, o, i, f, p. Below this, smaller segments are labeled s, x, a, k, and u, with numerical indicators like 5 and 4 and vertical connecting lines.

T H E O R E M XXXII.

WHY, when any number is divided into two unequal parts, and is then divided by those individual parts, the sum of the two quotients is always greater by two than the sum of the quotients arising from the division of one part by the other.
For example, if the number 24 were proposed, which was divided

into two unequal parts, namely 20 and 4; certainly, 24 being divided by the individual parts, one quotient would be six whole units, and the other one and a fifth part, whose sum would be seven whole units with a fifth part. Then, with one part divided by the other, one quotient would be five whole units and the other one fifth only, whose sum would be five whole units and one fifth part, which is less than the first sum of the other two quotients by the number two.
For the sake of this consideration, let the proposed number be signified by line q. p., its two parts by lines q. x. and x. p.; then let q. f. be the quotient from the division of the whole q. p. by x. p., and q. i. be the quotient from the division of the same q. p. by q. x. Furthermore, let h. m. be the quotient from the division of q. x. by x. p., and h. x. the quotient from the division of q. x. by q. x. It is evident, therefore, from the 22nd theorem of this book, that the quotient h. m. is less than the quotient q. f. by one unit, and the quotient h. x. is less than the quotient q. i. by another unit. Therefore f. q. i. will be greater than m. h. x. by the number two, which was the thing proposed.

A line diagram showing a horizontal line segment q-x-f-p with markers above and below, and another segment h-m located beneath it.

T H E O R E M XXXIII.

ANY number is the mean proportional between its square and unity.
For let the proposed number be given, signified by line a. u., its square be u. n., the linear unity be i. a., and the superficial unity be o. It will be evident from the 18th of the sixth or the 11th of the eighth that the proportion of u. n. to o. will be the double of the proportion of u. a. to i. a. But i. a. and o. are the same thing in kind, namely, a. i. is as great as the unit o. Therefore the proportion of the number u. n. to u. a. will be equal to the proportion of u. a. to i. a. Wherefore the number u. a. will be the mean proportional between the number u. n. and unity.

A geometric diagram showing a 3x3 square grid labeled 'n' at the top left corner and 'u' at the bottom right. To its left is a small unit square labeled 'o' and '1'. Below the grid are horizontal line segments labeled 'u-n', 'u-q', and 'i-a'.

T H E O R E M XXXIIII.

THIS same thing which we have said may be observed by another reasoning.
Let the proposed number again be signified by a. u., its square by u. n., the linear unity by a. i., and let the product of a. u. into a. i. be bounded, and let it be n. i. Wherefore n. i. will consist of a superficial number equal to the linear number a. u.; and from the first of the sixth or the 18th or 19th of the seventh, the proportion of u. n. to i. n. will be the same as a. u. to a. i. But the number a. u. is the same in kind as the number n. i. Therefore it is the mean proportional between u. n. and unity.