Epitome of Ptolemy's Almagest

A complex geometric diagram situated in the left margin. It depicts arcs of circles, likely representing a portion of a celestial or terrestrial sphere. Several straight lines radiate from a point labeled 'h' (representing the center of the sphere) to points on the arcs labeled 'a', 'd', 'b', 'e', 'g', and 'z'. Intersecting lines create points labeled 'l', 'k', and 't'. The arcs are nested, with one larger arc extending from 'g' to 'b' and smaller segments within. Two lines 's' and another arc segment are visible at the very top right of the diagram. The labels use lowercase letters in a medieval/Renaissance script.

...also in the surface of triangle $a.d.g$. Therefore, it is necessary that they lie in the common intersection of these surfaces, which by the third [proposition] of the eleventh [book] A reference to Euclid's Elements, Book XI, Proposition 3: "If two planes intersect one another, their common section is a straight line." is established to be a straight line. Thus, from the endpoints of the two lines $a.t$ and $a.g$, two other lines $t.l$ and $g.d$ are reflected, intersecting each other at $k$. Therefore, by the fifteenth [proposition] of this [book], the ratio of $g.l$ to $l.a$ is composed of two ratios: namely, the ratio of $g.k$ to $k.d$ and the ratio of $d.t$ to $t.a$. Now, the ratio of $g.l$ to $l.a$, according to the tenth [proposition] of this [book], is as the ratio of the chord of the double arcoriginal: "chorda dupli". This is an archaic trigonometric function. In modern terms, the chord of a double arc $2α$ is equal to $2 \sin(α)$. $g.e$ to the chord of the double arc $e.a$. And the ratio of $g.k$ to $k.d$, by the same [proposition], is as the chord of the double arc $g.z$ to the chord of the double arc $z.d$. Likewise, by the twelfth [proposition] of this [book] and by inverse proportionality, the ratio of $d.t$ to $t.a$ is as the chord of the double arc $d.b$ to the chord of the double arc $b.a$. Wherefore, it must be that the ratio of the chord of the double arc $g.e$ to the chord of the double arc $e.a$ is composed of two: namely, the ratio of the chord of the double arc $g.z$ to the chord of the double arc $z.d$, and the ratio of the chord of the double arc $d.b$ to the chord of the double arc $b.a$. This was what was to be proven.

Proposition xvi.

Likewise, the ratio of the chord of the double arc of one of the descending arcs to the chord of the double arc of its upper part is composed of two ratios: one of which is the ratio of the chord of the double arc of the reflected coterminal arc of this descending arc to the chord of the double arc of its upper part; the other is the ratio of the chord of the double arc of the lower part of the other reflected arc to the chord of the double arc of that entire reflected arc.

¶ Let the arcs be as in the preceding figure. I say that the ratio of the chord of the double arc of arc $g.a$ to the chord of the double arc $a.e$ is composed of two: namely, the ratio of the chord of the double arc of arc $g.d$ to the chord of the double arc $d.z$, and the ratio of the chord of the double arc $z.b$ to the chord of the double arc $b.e$. For let $h$ be the center of the sphere, from which the radii $h.a, h.d, h.b$ are drawn to meet with the continued chords $g.e, g.z, z.t$ at the points $l, k, t$. It will be established that these three will be in one straight line: because they are in two plane surfaces: namely, of the circle $b.d.a$ and of triangle $z.e.g$. Wherefore it is established by the third [proposition] of the eleventh [book] that they intersect one another in a straight line. Thus you have it: that from the endpoints of the two lines $l.t$ and $l.g$, two others, $t.e$ and $g.k$, are reflected, intersecting each other in $z$. Therefore, by the eighth [proposition] of this [book], the ratio of $g.l$ to $l.e$ is composed of two: namely, $g.k$ to $k.z$ and $z.t$ to $t.e$. But by the twelfth [proposition] of this [book], it is clear that these ratios are as the chord of the double arc $g.a$ to the chord of the double arc $a.e$. Likewise, the chord of the double arc $g.d$ to the chord of the double arc $d.z$, and the chord of the double arc $z.b$ to the chord of the double arc $b.e$. Thus the proposition is established.

Proposition xvii.

To determine the distance between the two tropics by means of a skillfully made instrument.

¶ You shall arrange a quadrant of a circle upon the meridian line and a plane surface perpendicular to the horizonThe 'horizon' refers to the celestial horizon, the great circle on the celestial sphere whose plane is perpendicular to the observer's vertical line.; let this be $a.b$ upon the center $c$, so that $c.a$ is in the surface of the horizon and of the meridian circle. Let $b.c$ be the part of the axis passing through our zenithThe point in the sky directly above the observer. and its nadirThe point in the sky directly below the observer, opposite the zenith.. From this, you will fit a ruleroriginal: "regula". In astronomy, this is often called an alidade—a pointer used for measuring angles. $c.d$, which rotates upon the center $c$, having two sightsoriginal: "pinnulae". These are small plates with holes or slits used to align the instrument with a celestial body. with holes equally distant from the straight line $c.d$. You shall observe around the winter solsticeThe time of year when the sun reaches its minimum noon altitude. at midday, with the sun's ray [passing through] both holes of the sights...