Epitome of Ptolemy's Almagest (1496) - Page 19

First [Book]
31

Proposition 9

If from the endpoints of two lines descending from some angle, two lines intersecting each other are reflected The term "reflexe" here refers to transversal lines that cross between the two main legs of the angle. back upon the descending lines: the ratio of the descending line to its upper part will be composed of two ratios: one of which is from the endpoint of this reflected line to its part above the intersection; the other is the ratio of the part below the intersection of the other reflected line to the whole of that same reflected line.

A geometric diagram showing a triangle with vertices a, b, g. Two lines, b-e and g-d, are drawn from the base vertices to the opposite sides, intersecting at point z. An auxiliary line e-h is drawn parallel to g-d.

¶ As two lines $a.b$ and $a.g$ descend from angle $a$; and from their endpoints $b$ and $g$, two lines are reflected back across the descending lines, which are $b.e$ and $g.d$, intersecting each other at $z$. I say that the ratio of $g.a$ to $a.e$ is composed of two: namely, the ratio of $g.d$ to $d.z$ and the ratio of $z.b$ to $b.e$. ¶ For, by the 31st [proposition] of the first [book of Euclid], let $e.h$ be drawn parallel to $g.d$. By the 29th of the first, angle $d.g.a$ will become equal to angle $h.e.a$, and angle $g.d.a$ will be equal to angle $e.h.a$; and angle $a$ is common to both triangles. Therefore, by the fourth [proposition] of the sixth [book], the ratio of $g.a$ to $a.e$ will be as $g.d$ to $e.h$. Between $g.d$ and $e.h$, I shall place $d.z$ as a middle term; and thus the ratio of $g.d$ to $e.h$ becomes composed of two: namely $g.d$ to $d.z$ and $d.z$ to $e.h$. But by the 29th of the first and the fourth of the sixth, $d.z$ to $e.h$ is as $z.b$ to $b.e$. Therefore, $g.d$ to $e.h$ is composed of two, namely $g.d$ to $d.z$ and $z.b$ to $b.e$. Wherefore the ratio of $g.a$ to $a.e$ is also composed of two: namely, $g.d$ to $d.z$ and $z.b$ to $b.e$, which was the intent.

Proposition 10

Likewise, the ratio of the parts of the lower descending line to the upper will be composed of two: one of which is the ratio of the parts from the endpoint of this lower reflected line to the upper; the other is the ratio of the lower part of the other descending line to that whole same descending line.

A geometric diagram similar to the one above, depicting a triangle a-b-g with intersecting internal lines b-e and g-d. An auxiliary line a-h is extended parallel to line e-b. The ratio of two triangles, in which one angle of one is equal to an angle of the other, the sides corresponding to the equal angles are proportional. 7. 6. original: "proportio duoz triāg[u]l[orum] quoz an[gulus] / un̄ alteri sit eq[u]ales lat[er]a equos a[ngulos] / resp[ici]entia sūt p[ro]p[orti]onalia 7. 6." This gloss refers to Euclid's Elements, Book VI, Proposition 7, regarding similar triangles.

¶ Let the descending and reflected lines be as before. I say that the ratio of $g.e$ to $e.a$ is composed of two: namely, the ratio of $g.z$ to $z.d$ and the ratio of $d.b$ to $b.a$. ¶ For, by the 31st of the first, let $a.h$ be drawn parallel to $e.b$, which the continued line $g.d$ meets at $h$. As before, triangles $a.h.d$ and $b.z.d$ become equiangular. Moreover, in triangle $g.a.h$, the line $e.z$ cuts two sides while being parallel to the third. Therefore, by the second [proposition] of the sixth [book], $e.g$ to $e.a$ is as $g.z$ to $z.h$. But between $g.z$ and $z.h$, let us place $z.d$ as a middle term. Therefore, the ratio of $g.z$ to $z.h$ will be composed of two: namely, $g.z$ to $d.z$ and $d.z$ to $z.h$. However, by the fourth of the sixth, combined with the converse of the proportions, $z.d$ to $z.h$ is as $d.b$ to $b.a$. Wherefore the ratio of $g.z$ to $z.h$ is composed of two: namely $g.z$ to $z.d$ and $d.b$ to $b.a$. It is clear, therefore, that the ratio of $g.e$ to $e.a$ is composed of the two: $g.z$ to $z.d$ and $d.b$ to $b.a$, which is the intent.

Proposition 11

With two continuous arcs Arcs that share an endpoint. taken in a semicircle, the radius drawn to their common endpoint will divide the chord of the arc composed from them according to the ratio of the chord of the double of one arc to the chord of the double of the other.

¶ In a semicircle let there be two arcs $a.b$ and $b.g$: of which the sum... original: "rū aggregati" - the text breaks off here at the bottom of the page.

At the bottom of the page there are several rough hand-drawn ink scribbles, including zig-zag lines resembling mountains or water waves, and some unintelligible cursive practice strokes.