Optics (1551) - Page 21 - Source Library

But since, by hypothesis, line $b$ is less than line $d$, it is clear by the same 8th [proposition] of this fifth [book], that the ratio of line $c$ to line $b$ is greater than to line $d$; therefore, the ratio of the first line $a$ to the second line $b$ is greater than the third line $c$ to the fourth line $d$, and this is what was proposed.

X.

If, of four quantities, the ratio of the first to the second be greater than that of the third to the fourth, the ratio of the first to the third will be, by permutation, greater than that of the second to the fourth.

Let there be four lines $a, b, c, d$, and let the ratio of $a$ to $b$ be greater than $c$ to $d$. I say that, by permutation, the ratio of line $a$ to line $c$ will be greater than the ratio of line $b$ to line $d$. For let the ratio of line $e$ to line $b$ be as the ratio of line $c$ to line $d$, by the 3rd [proposition] of this [book]; therefore, by the hypothesis and by the 10th of the fifth [book], line $e$ will be less than line $a$. Therefore, by the 8th of the fifth, the ratio of line $a$ to line $c$ is greater than the ratio of line $e$ to line $c$. But by the premises and by the 16th of the fifth, the ratio of line $a$ to line $c$ is as the ratio of line $b$ to line $d$; it is evident, therefore, that the ratio of line $a$ to line $c$ is greater than the ratio of line $b$ to line $d$, which is what was proposed.

A geometric diagram showing four horizontal lines of varying lengths labeled A, B, C, and D on the right. Below them is a fifth line labeled E.

XI.

When, of four quantities, the ratio of the first to the second is greater than that of the third to the fourth, the ratio of the first and second to the second will be, by composition, greater than that of the third and fourth to the fourth.

Let there be 4 lines $a, b, c, d$, and let the ratio of $a$ to $b$ be greater than $c$ to $d$. I say that the ratio of the whole line $ab$ to line $b$ will be greater than the ratio of the whole line $cd$ to line $d$. For let the ratio of line $e$ to line $b$ be as the ratio of line $c$ to line $d$, by the 3rd [proposition] of this [book]; therefore, by the hypothesis, the ratio of line $a$ to line $b$ is greater than the ratio of line $e$ to line $b$. Therefore, by the 10th of the fifth, line $a$ is greater than line $e$. The whole line $ab$, therefore, is greater than the whole line $eb$. Thus, by the 8th of the fifth, the ratio of the whole line $ab$ to line $b$ is greater than the ratio of the whole line $eb$ to line $b$. Truly, by the 18th of the fifth, the ratio of line $eb$ to line $b$ is as the ratio of line $cd$ to line $d$; for by the premises, the ratio of line $e$ to line $b$ is as the ratio of line $c$ to line $d$. There is, therefore, a greater ratio of line $ab$ to line $b$ than of line $cd$ to line $d$, which is what was proposed.

A geometric diagram similar to the first, showing four lines labeled A, B, C, D and a fifth line labeled E below.

XII.

If, of four quantities, the ratio of the first and second to the second be greater than that of the third and fourth to the fourth, the ratio of the first to the second will be, by division, greater than that of the third to the fourth.

Let the ratio of the whole line $ab$ to its part, line $b$, be greater than the whole line $cd$ to its part $d$. I say that by division the ratio of line $a$ to line $b$ will be greater than the ratio of line $c$ to line $d$. For let the ratio of line $eb$ to line $b$ be as line $cd$ to line $d$, by the 3rd [proposition] of this [book]. Therefore, by the hypothesis, the ratio of line $ab$ to line $b$ is greater than the ratio of line $eb$ to that same line $b$. Thus, by the 10th of the fifth, line $ab$ will be greater than line $eb$. Therefore, subtracting the common line $b$ from both sides, line $a$ remains greater than line $e$. Thus, by the 8th of the fifth, the ratio of line $a$ to line $b$ is greater than that of line $e$ to the same line $b$. But by the premises, the ratio of line $eb$ to line $b$ is as line $cd$ to line $d$; therefore, by the 17th of the fifth, the ratio of line $e$ to line $b$ is as line $c$ to line $d$. There will be, therefore, a greater ratio of line $a$ to line $b$ than of line $c$ to line $d$, and this is what was proposed.

A geometric diagram showing two long horizontal lines. The top line is divided into segments labeled A, B, and E. The bottom line is divided into two segments labeled C and D.

XIII.

For any three quantities arranged in any order whatever, where there is some ratio of the mean to each of the extremes, the ratio of the first to the third will be composed of the ratio of the first to the second and of the second to the third; from which it is evident that the ratio of the extremes to one another is always composed...