Epitome of Ptolemy's Almagest

Book [1]
31

A geometric diagram of a semicircle. Point d is the center on a diameter. A chord ag is drawn, intersected at point e by the radius bd. Perpendicular lines az and gh are dropped from points a and g to the line of the radius bd. Point b is on the arc of the semicircle.

Let the semi-diameter $b.d$ cut the chord $a.g$ at point $e$. I say that the ratio of $a.e$ to $e.g$ is as the ratio of the chord of double the arc $a.b$ to the chord of double the arc $b.g$. In ancient trigonometry, the "chord of the double arc" is equivalent to twice the sine of the single arc. These proofs were used to build the "Table of Chords," the precursor to modern trigonometric tables. ¶ For let $a.z$ and $g.h$ be perpendiculars dropped upon $d.b$. By the fourth [proposition] of the sixth [book of Euclid's Elements], the ratio of $a.e$ to $e.g$ will be as the ratio of $a.z$ to $g.h$. But by the third [proposition] of the third [book], $a.z$ is half the chord of the double arc $a.b$, and $g.h$ is half the chord of the double arc $b.g$. Therefore, by the 15th [proposition] of the fifth [book], the ratio of $a.e$ to $e.g$ is as the ratio of the chord of double the arc $a.b$ to the chord of double the arc $b.g$, which was to be shown. original: "quod fuit ostendendum"

Proposition 12

A geometric diagram of a semicircle with center d. A chord ag is drawn. A perpendicular dz is dropped from the center d to the midpoint z of the chord ag. A radius db intersects the chord ag at point e. Points a, b, and g are on the circumference.

If a known arc in a semicircle is divided into two: and the ratio of the chord of the double of one to the chord of the double of the other is given: each of those arcs which it divides will be known. ¶ Because the whole arc $a.b.g$ is known, its chord $a.g$ will be given from the table of chords. And because the ratio of the chord of the double arc $a.b$ to the chord of the double arc $b.g$ is given—but this is, by the preceding [proposition], as $a.e$ to $e.g$—therefore the ratio of $a.e$ to $e.g$ is given. And since the whole $a.g$ is given: by the rule of joined proportionality joined proportionality: a method of solving ratios where if a:b and the total a+b are known, the individual parts can be found. and the 15th of the sixth [book], both of the two segments $a.e$ and $e.g$ will be revealed. ¶ Now, let the perpendicular $d.z$ be drawn from the center $d$ to $a.g$. By the third [proposition] of the third [book], $a.z$ will be equal to $z.g$; therefore $e.z$, the difference of the half of $a.g$ over $a.e$, will be known. But since the triangle $a.d.z$ is right-angled original: "orthogonius" and subtends half of the arc $a.g$, it is therefore known. And since the angle $z$ in triangle $a.d.z$ is a right angle: by the 32nd [proposition] of the first [book], the angle $d.a.z$ will be known, because angle $z.a.d$ and angle $a.d.z$ together make one right angle. Therefore, since triangle $a.d.z$ is right-angled and has known angles, its sides will be known through the table of chords; or, by the penultimate [proposition] of the first [book] Refers to the Pythagorean Theorem., $z.d$ will be known from $a.z$ and $a.d$. Likewise, by the same penultimate [proposition] of the first, $e.d$ will be known from the known lengths $e.z$ and $d.z$. Thus, in the right-angled triangle $e.d.z$, with known sides in parts where $a.d$ is 60 The author uses the sexagesimal (base-60) system common in medieval astronomy, where the radius of a circle is divided into 60 parts., the sides will be known by the 15th [proposition] of the first in parts where $d.e$ is 120. From this, its angles will be known by the table of chords, just as the three angles of a right-angled triangle correspond to the whole circle circumscribed around it—that is, where a right angle is 180 degrees. Therefore, its angles will be known when a right angle is 90 degrees; thus angle $z.d.e$ will be known. But $a.d.z$ was known previously; therefore angle $a.d.e$ will be known, the quantity of which is arc $a.b$, which was sought.

Proposition 13

A geometric diagram showing a full circle with center d. A diameter lda is extended to point e. From point g on the circle, a line gbe is drawn passing through point b on the circle and meeting the extended diameter at e. Perpendicular lines gz and bh are dropped from points g and b to the diameter line.

If a line is drawn from one endpoint of an arc smaller than a semicircle, passing away from the center and cutting the arc until it meets the diameter joined through the other endpoint of the same arc: the ratio of the line passing beyond the center to its part outside the circle will be as the ratio of the chord of double the whole arc to the chord of double the part which the extended lines include.
¶ Let there be a circle $a.b.g$ upon center $d$, in which the diameter $l.d.a$ passes through the endpoint of arc $a.g$ to $e$; and let another line passing away from the center from the other endpoint of the arc be $g.b.e$, cutting the arc at $b$ and meeting the extended diameter at $e$. I say that the ratio of $g.e$ to $e.b$ is as the ratio of the chord of double the arc $a.g$ to the chord of double the arc $a.b$. ¶ From points $b$ and $g$, let the perpendiculars $b.z$ and $g.h$ descend upon $l.e$. Therefore, by the 28th [proposition] of the first [book], triangles $g.h.e$ and $b.z.e$ will be equiangular. Wherefore, by the fourth [proposition] of the sixth [book], $g.e$ is to $e.b$ as $g.h$ is to $b.z$. But by the ninth [proposition] of the third and last...