Epitome of Ptolemy's Almagest

3. According to the eighth [proposition] of the fifth [book], the ratio of sector $e.d.t$ to sector $e.d.b$ will be greater than the ratio of triangle $e.d.z$ to sector $e.d.b$. But also, by the same [proposition], the ratio of triangle $e.d.z$ to sector $e.d.b$ is greater than the ratio of triangle $e.d.z$ to triangle $e.d.a$. Therefore, even more strongly original: "a fortiori", the ratio of sector $e.d.t$ to sector $e.d.b$ is greater than the ratio of triangle $e.z.d$ to triangle $e.d.a$.

But the ratio of one sector to another in the same circle, according to Archimedes' demonstrations in On the Measurement of the Circle, is as the ratio of one arc to the other. Furthermore, according to the last [proposition] of the sixth [book] Refers to Euclid's Elements, Book VI, Prop. 33, which states that in equal circles, angles have the same ratio as the arcs on which they stand., one arc is to another as the angle of one (which is at the center) is to the angle of the other. Likewise, the ratio of triangle $e.d.z$ to triangle $e.d.a$, according to the first [proposition] of the sixth [book], is as the line $z.e$ is to $e.a$. Therefore, joined together by the third of the added [propositions], the ratio of the combined angle $z.a.d$ to angle $e.d.a$ is greater than the ratio of $g.e$ to $e.a$.

By the last [proposition] of the sixth [book], however, the ratio of angle $g.d.b$ to angle $b.d.a$ is as arc $b.g$ is to arc $a.b$; and by the third [proposition] of the sixth [book], the ratio of $g.e$ to $e.a$ is as chord $b.g$ is to chord $a.b$. Therefore, the ratio of arc $b.g$ to arc $a.b$ is greater than the ratio of chord $b.g$ to chord $a.b$, which was the thing proposed [to be proved].

Proposition 7.

To reveal the chord of an arc of one degree without noticeable error.

Let arc $a.b$ be half a degree and a quarter of one This equals three-quarters of a degree, or 45 arc-minutes.. Its chord $a.b$, according to the preceding principles and following the discovery of Ptolemy Claudius Ptolemy (c. 100–170 AD), the Greco-Roman astronomer whose Almagest provided the foundation for this trigonometry., will be 47 minutes and 8 seconds. Likewise, if there is an arc $a.g$ of one degree, its chord is sought. Through the preceding [proposition], it is clear that the ratio of arc $a.g$ to arc $a.b$ is greater than the ratio of chord $a.g$ to chord $a.b$. But since arc $a.g$ contains arc $a.b$ plus its third part, it follows that chord $a.g$ contains chord $a.b$ and less than its third part.

Now, a third of chord $a.b$ is 15 minutes, 42 seconds, and two-thirds of a second. When these are added to the 47 minutes and 8 seconds [of the original chord], they make 1 degree, 2 minutes, 50 seconds, and two-thirds of a second. This value is therefore necessarily greater than the chord of one degree.

Next, let arc $a.b$ be one degree, and arc $a.g$ be a degree and a half ($1.5^\circ$). From his earlier work, Ptolemy found the chord of $a.g$ to be 1 degree, 34 minutes, and 15 seconds. From this, the chord $a.b$ is sought. By the preceding [proposition], the ratio of arc $a.g$ to arc $a.b$ is greater than the ratio of chord $a.g$ to chord $a.b$. But since arc $a.g$ contains arc $a.b$ and its half, then chord $a.g$ contains chord $a.b$ and less than its half.

Thus, if I subtract a third of arc $a.g$ (namely $b.g$) from arc $a.g$, arc $a.b$ remains. Therefore, if I also subtract a third of the chord of arc $a.g$ (namely 31 minutes and 25 seconds) from the total $a.g$, which is 1 degree, 34 minutes, and 15 seconds, there remains 1 degree, 2 minutes, and 50 seconds. This value must necessarily be less than the chord of an arc of one degree.

Therefore, the chord of an arc of one degree will be more than 1 degree, 2 minutes, and 50 seconds, and less than 1 degree, 2 minutes, 50 seconds, and two-thirds of a second. It was therefore appropriate to set the chord of an arc of one degree at 1 part, 2 minutes, and 50 seconds. From this, no noticeable error would follow in astronomical calculations, because of the small and imperceptible difference between the quantities within which it has now been concluded to stand. From the chord of an arc of one degree, following the teaching of the fourth [proposition] of this [book], the chord of half a degree will be established. From here, following the teachings of the preceding [propositions], you will complete the chords of all arcs increasing by half a degree.