Epitome of Ptolemy's Almagest

...for by the sixth [proposition], $g.h$ is half the chord of the double arcThe "chord of the double arc" was the standard way to express trigonometric values before the modern sine function was adopted. Numerically, the chord of a double arc $2α$ is equal to $2 \sin(α)$. of $a.g$, and $b.z$ is half the chord of the double arc $a.b$. Therefore, by the 17th [proposition] of the fifth [book], the proportion of $g.e$ to $e.b$ is as the proportion of the chord of the double arc $a.g$ to the chord of the double arc $a.b$, which was what was to be proposed.

Proposition 14.

A geometric diagram in the right margin depicting a circle with center d. Points a, b, and g lie on the circumference. A line segment from d passes through midpoint z to point g. Chords are drawn between a and g, a and b, and b and g. A radius is drawn from d to b. This illustrates the relationship between arcs and their chords as described in the text.

Given one part of an arc—with the lines drawn out as already stated—and the division being known: because the proportion of the chord of the double arc of the whole to the chord of the double arc of its part (which the drawn lines include) is known, the arc included by those lines shall also be known.
¶ Let the proportion of one arc $b.g$ of the arc $a.g$ be known, and let the proportion of the chord of the double arc $a.g$ to the chord of the double arc $a.b$ be given. I say that arc $a.b$ becomes known. For let the perpendicular line $d.z$ be drawn from the center $d$ to $b.g$, such that $z$ is the midpoint of $g$ original: "equalis z.g.". Therefore, since the whole chord $b.g$ is given (because its arc is known), $b.z$ will be known. And by the last [proposition] of the sixth [book], the angle $b.d.z$ subtends half of arc $b.g$, and is therefore known. But $b.d$ is known because it is the radiusoriginal: "semidiameter". Therefore, by the penultimate [proposition] of the first [book]This refers to the Pythagorean Theorem (Euclid, Elements, Book I, Prop. 47)., $d.z$ will become known. Likewise, because the proportion of the chord of the double arc $a.g$ to the chord of the double arc $a.b$ is given—but by the preceding [proposition], $e.a$ is as $g.e$ is to $e.b$—and since $g.b$ is known by disjunct proportionA method of manipulating ratios found in Euclid's Book V. and the 15th [proposition] of the sixth [book], $e.b$ will be known. Therefore, the whole $e.z$ is known. From the known values of $e.z$ and $d.z$, by the penultimate [proposition] of the first [book], $e.d$ will be known. Thus, for the right-angled triangleoriginal: "orthogonij" $e.d.z$, with the lengths of the sides known by the method mentioned in the previous sections, all the angles will become known. Thus, angle $a.d.z$ is known; from which, by subtracting the already known angle $b.d.z$, there remains angle $a.d.b$, the quantity of which is the arc $a.b$ that was sought.

Proposition 15.

A complex geometric diagram in the right margin illustrating spherical trigonometry. It shows several intersecting circular arcs on the surface of a sphere, forming a network of spherical triangles and segments. Points are labeled a, b, g, d, e, z, h, l, k, t. The diagram visualizes the proportional relationships between the chords of doubled arcs on a sphere.

If on the surface of a sphere there are four arcs of great circlesA great circle is a circle on the surface of a sphere that has the same circumference as the sphere and shares its center, such as the equator on Earth., of which none is greater than a semicircle: two indeed descending from one angle, and the remaining two reflected alternately from the ends of the former and intersecting each other: then the proportion of the chord of the double arc of the lower part of one descender to the chord of the double arc of its upper part is composed of two [proportions]. One of these is the proportion of the chord of the double arc of the lower part of the line reflected from the end of that descender to the chord of the double arc of its upper part; the other is the proportion of the chord of the double arc of the lower part of the other descender to the chord of the double arc of the whole of this descender.
¶ Let there be four arcs of great circles on the surface of a sphere, and let each of them be less than a semicircle. Let the two descending from angle $a$ be $a.b$ and $a.g$; and let the two reflected upon them from their ends be $b.e$ and $g.d$, intersecting each other at $z$. I say that the proportion of the chord of the double arc $g.e$ to the chord of the double arc $e.a$ is composed of two proportions: one of which is the chord of the double arc $g.z$ to the chord of the double arc $z.d$, and the other is the proportion of the chord of the double arc $d.b$ to the chord of the double arc $b.a$. For let us place the center of the sphere at $h$, from which the radii $h.b$, $h.z$, and $h.e$ are drawn to points $b$, $z$, and $e$. And let the chord $a.d$, extended as far as needed, meet the radius $h$ (similarly extended) at point $t$. Likewise, let the chords $g.a$ and $g.d$ intersect the radii $h.e$ and $h.z$ at points $l$ and $k$. It is necessary that the three points $l$, $k$, and $t$ be in one straight line; for they are on the surface of the circle $b.z.e$, they are