Optics

...alternately the ratio of $b$ to $a$ is as $d$ to $c$, namely of the second to the first as of the fourth to the third, which is what was proposed.

VI.

When there is a ratio of four quantities, that of the first to the second being greater than that of the third to the fourth, inversely the ratio of the second to the first will be less than that of the fourth to the third.

Geometric diagram consisting of three horizontal line segments. The top line is labeled C at the left end and A at the right. The second line is labeled D at the left and B at the right. The third, shorter line is labeled E in the center.

Let the ratio of line $a$ to line $b$ be greater than the ratio of line $c$ to line $d$. I say that inversely the ratio of line $b$ to line $a$ will be less than the ratio of line $d$ to line $c$. For thus, by the third [proposition] of this [book], let the ratio of line $e$ to line $b$ be the same as the ratio of line $c$ to line $d$; because, therefore, the ratio of line $a$ to line $b$ is greater than the ratio of line $c$ to line $d$, it is clear from the hypothesis that the ratio of line $e$ to line $b$ is less than the ratio of line $a$ to line $b$; therefore, by the 10th [proposition] of the fifth [book of Euclid], line $a$ is greater than line $e$; and because the ratio of line $e$ to line $b$ is as the ratio of line $c$ to line $d$, by the preceding [proposition] the ratio of line $b$ to line $e$ will be the same as that of line $d$ to line $c$. Moreover, by the 8th of the fifth, the ratio of line $b$ to line $a$ is less than to line $e$; therefore the ratio of line $b$ to line $a$ is less than the ratio of line $d$ to line $c$, which is what was proposed.

VII.

If, of four proportional quantities, the first be greater than the second, and the third greater than the fourth, then by conversion the ratio of the first to its excess over the second will be the same as that of the third to its excess over the fourth.

Geometric diagram consisting of two long horizontal line segments. The top line is labeled AA at the left end, B in the middle, and C at the right end. The bottom line is labeled DD at the left end, E in the middle, and F at the right end.

Let there be four proportional lines: $ac$ the first, $bc$ the second, $df$ the third, and $ef$ the fourth. And let line $ac$ be greater than line $bc$, and line $df$ greater than line $ef$; let line $ac$ also exceed line $bc$ by line $ab$, and line $df$ exceed line $ef$ by line $de$. I say that the ratio of line $ac$ to line $ab$ will be the same as that of line $df$ to line $de$; for since the ratio of line $ac$ to line $bc$ is as the ratio of line $df$ to line $ef$, therefore by the 16th of the fifth, alternately, the ratio of line $ac$ to line $df$ is as the ratio of line $bc$ to line $ef$; therefore by the 19th of the fifth, the ratio of line $ab$ to line $de$ will be as the ratio of line $ac$ to line $df$; therefore by the 4th [proposition] of this [book], the ratio of line $ac$ to line $ab$ will be as the ratio of line $df$ to line $de$, which is what was proposed.

VIII.

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

Geometric diagram showing two pairs of horizontal line segments. The top pair is labeled A and C. The bottom pair is labeled B and D.

Let there be four lines $a, b, c, d$, and let $a$, the first, be greater than $b$, the second, and let $c$, the third, be greater than $d$, the fourth. I say that the ratio of line $a$ to line $d$ is greater than the ratio of line $b$ to line $c$; for since line $c$ is greater than line $d$, it is clear from the hypothesis by the 8th of the fifth that the ratio of line $a$ to line $d$ is greater than to line $c$; but the ratio of line $b$ to line $c$ is less than the ratio of line $a$ to line $c$ by the same 8th of the fifth, since, as was previously stated, line $a$ is greater than line $b$; and since whatever is greater than the greater is greater than the lesser, it is clear that the ratio of line $a$ to line $d$ is greater than the ratio of line $b$ to line $c$; therefore the proposition is clear.

IX.

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

Geometric diagram showing two pairs of horizontal line segments. The top pair consists of a longer line A and a shorter line C. The bottom pair consists of a shorter line B and a longer line D.

Let there be four lines, $a$ the first, $b$ the second, $c$ the third, $d$ the fourth, and let $a$ be greater than $c$, and let $b$ be less than $d$. I say that the ratio of $a$ to $b$ is greater than that of $c$ to $d$; for since line $a$ is greater than line $c$, it is clear by the 8th of the fifth that the ratio of line $a$ to line $b$ is greater than the ratio of line $c$ to line $b$, but because...