Optics

BOOK ONE

...surface contained between them and between the lines joining the extremities of those lines, which was to be proposed.

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II.

A line drawn from a point on one of two parallel lines in the same surface must necessarily meet the other, if it be of indefinite length.

A vertical geometric diagram on the right margin showing parallel lines A, B and C, D. A line segments starts at point B and intersects line C-D at an angle, with a point E marked below.

Let there be two parallel lines, $ab$ and $cd$, one of which, namely $ab$, let the line $be$ cut at point $b$. I say that the line $be$ will also cut the line $cd$. For since the line $cd$ is assumed to be of indefinite length, let $be$ be drawn out toward it; if it meets $cd$, the proposition is held. If it should not meet it, it is clear by the definition of parallel lines that the line $be$ is parallel to the line $cd$; and because the lines $ab$ and $be$ are both parallel to the line $cd$, then by the 30th [proposition] of the first [book of Euclid], the line $eb$ will be parallel to the line $ab$. But this is clearly [false] from the hypothesis, since they meet, as at point $b$. Therefore the line $be$ is not parallel to the line $cd$; therefore the line $be$ necessarily meets the line $cd$, which was to be proposed.

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III.

Given three lines, to find a fourth proportional to any one of them according to the ratio of the other two.

Three horizontal line segments of varying lengths labeled A-B, C-D, and E-F.

Let the three given lines be $ab, cd, ef$, for one of which, as $ab$, a fourth proportional must be found according to the ratio of the other two, which are $cd$ and $ef$. Therefore, let two lines equal to the two lines $cd$ and $ef$ be cut off from one continuous line, which shall be $aef$ by the 3rd [proposition] of the first [book of Euclid]; and let the third given line, namely $ab$, be joined at an angle to that line $aef$ at point $a$. From the common point distinguishing the two cut-off lines, which is point $e$, let the line $eb$ be drawn to the extremity of the third of the given lines, which is $ab$; and from point $f$, let a line $fg$ be drawn parallel to the line $eb$ by the 31st [proposition] of the first [book of Euclid]. Then let the line $ab$ be extended in a continuous straight line until it cuts the line $fg$ (it will moreover cut it by the preceding [proposition]); let the point of meeting be $g$. I say that by the 2nd [proposition] of the sixth [book of Euclid], the ratio of line $ab$ to line $bg$ is the same as that of the given line $ae$ to the given line $ef$. Likewise, it can be demonstrated concerning any of the others in respect to the remaining two; therefore the proposition is clear.

A geometric diagram showing a triangle with vertex A. Two parallel horizontal lines intersect the sides: E-B (higher) and F-G (lower, forming the base).

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IIII.

When an addition of equal lines is made to two unequal lines of a known ratio, the ratio of the greater to the lesser is diminished.

Let there be two unequal lines $ab$ and $cd$ of a known ratio, and let line $ab$ be greater than line $cd$. Let line $be$ be added to $ab$, and line $df$ to $cd$, and let lines $be$ and $df$ be equal. I say that the ratio of line $ae$ to line $cf$ is less than the ratio of line $ab$ to line $cd$. For since three lines $ab, cd,$ and $be$ are given, a line proportional to line $be$ according to the ratio of lines $ab$ and $cd$ is found by the preceding [proposition], which shall be $dg$. Since, therefore, line $ab$ is greater than line $cd$, it is clear that line $be$ is greater than line $dg$. Therefore, line $df$ is also greater than line $dg$. Let line $dg$ be cut from line $df$ by the 3rd [proposition] of the first [book of Euclid]. Because, therefore, the ratio of line $ab$ to line $cd$ is as the ratio of line $be$ to line $dg$, by the 13th [proposition] of the fifth [book of Euclid] the ratio of the whole line $ae$ to the total line $cg$ will be as the ratio of line $ab$ to line $cd$. But by the 8th [proposition] of the fifth [book of Euclid], the ratio of line $ae$ to the greater line $cf$ is less than to the smaller line $cg$. Therefore, the ratio of line $ab$ to line $cd$ is greater than the ratio of line $ae$ to line $cf$, and this is what was to be proposed.

Two parallel horizontal lines divided into segments. The top line consists of segments A-B and B-E. The bottom line consists of segments C-D, D-G, and G-F.

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V.

When the ratio of the first to the second is as the third to the fourth, inversely the ratio of the second to the first will be as the fourth to the third.

Four horizontal line segments of different lengths arranged in two columns, labeled A, C (top) and B, D (bottom).

For let $a$ be the first, $b$ the second, $c$ the third, and $d$ the fourth; and let the ratio of $a$ to $b$ be as $c$ to $d$. I say that inversely the ratio of $b$ to $a$ will be as $d$ to $c$. For since the ratio of $a$ to $b$ is as $c$ to $d$, by the 16th [proposition] of the fifth [book of Euclid], alternately—