Optics (1551) - Page 22 - Source Library

from the proportion of the means to each other and to the extremes themselves.

Let there be three lines outside the scale, which are $a, b, g$, of which the first, $a$, is greater than the mean, $b$, and $b$ is greater than the third, $g$; and let the ratio of $b$ to both extremes be known. I say that the ratio of line $a$ to the third line $g$ is composed of the ratio of line $a$ to line $b$, and of the ratio of line $b$ to line $g$. For since the ratio of line $a$ to line $b$ is known, let the quantity $d$ be the denomination of 그 ratio; and similarly, since the ratio of line $b$ to line $g$ is known, let the denomination of that ratio be the quantity $e$, and let the quantity $z$ be the denomination of the ratio of line $a$ to line $g$. I say that $z$ is produced from the multiplication of $e$ into $d$. For since, by definition, line $a$ is produced from the multiplication of $z$, the denomination of the ratio of line $a$ to line $g$, into

A set of five horizontal lines of varying lengths. The top three are labeled A, B, and G in decreasing order of length. Below them are two lines labeled D and E. A small triangular mark is positioned between lines D and E.

the line $g$ itself (which is less than $a$), and similarly, line $a$ is produced from the multiplication of $d$ into line $b$. Therefore, $z$ is proposed as the first, $d$ as the second, line $b$ as the third, and line $g$ as the fourth; since, therefore, that which is produced from the multiplication of the first into the fourth is equal to that which is produced from the multiplication of the second into the third, it is evident by the 15th of the sixth [book] that the ratio of the first to the second is as the third to the fourth. Therefore, the ratio of $z$ to $d$ is as line $b$ to line $g$. Thus, the denomination of the ratio of $z$ to $d$ is, by the same supposition, the same as the denomination of the ratio of line $b$ to line $g$. But the denomination of the ratio of line $b$ to line $g$ is the quantity $e$; therefore, the denomination of the ratio of $z$ to $d$ is the same $e$. Therefore, $z$ is produced from the multiplication of $e$ into $d$. Since, therefore, the denomination of the ratio of line $a$ to line $g$, which is $z$, is produced from the multiplication of the denomination of the ratio of line $a$ to line $b$ into the denomination of the ratio of line $b$ to line $g$, it is evident by definition that the ratio of the first line $a$ to the third line $g$ is composed of the ratio of the first line $a$ to the second line $b$, and of the ratio of the second line $b$ to the third line $g$, which is what was first proposed. In the same manner, it can also easily be demonstrated for any number of means placed between any two extremes; for the ratio of the extremes to each other is always composed of all the ratios of the means to each other. And the extremes themselves similarly by the way of division; if it happens that the mean is greater than either of the extremes, the proposition is then clear.

XIIII.

If a straight line falls upon two straight lines and makes the alternate angles unequal, or the two interior [angles] less than two right angles, or an exterior [angle] unequal to the interior [angle], it is necessary that those lines meet on the side of the lesser angles, but on the other side it is impossible; and if the lines meet, it is necessary that the said angles relate to one another in any of the proposed ways.

A geometric diagram showing two horizontal lines, AB and CD. A transversal line EF intersects both, slanted from top-left to bottom-right. Points are labeled E at the top of the transversal, AA at the left end of the top line, CC at the left end of the bottom line, B at the right end of the top line, D at the right end of the bottom line, and F at the bottom of the transversal.

Let there be two lines $ab$ and $cd$, which the line $ef$ makes according to what is proposed. I say that lines $ab$ and $cd$ will meet. For if they do not meet, it is clear that they are parallel; therefore, by the 29th of the first [book], the contrary of the hypothesis follows, which is inconsistent. Therefore they meet. Moreover, it is necessary that they meet on the side of the lesser angles; for if they should meet on the side of the greater angles, it would follow that the exterior angle of the triangle would be made less than the interior angle, and this is against the 16th and 32nd of the first [book]. And because they meet on the side of the lesser angles by the preceding propositions, if by concession they should meet on the side of the greater angles, it would follow that straight lines enclose a surface, which is impossible. It is therefore impossible that they meet on the side of the greater angles, which is the first [part] proposed. But also if it is given that those lines meet, it is necessary that the angles relate to one another in any of the proposed ways by the 32nd of the first [book]. Therefore everything that is proposed is clear, the hypothesis always being preserved.

XV.

When, with lines intersecting each other between two parallel lines from whose endpoints they are produced, the parts of the same line are equal to each other on each side of the intersection, it is necessary that the lines between which the intersection occurs be equal.