Ptolemy's table of chords

How Ptolemy computed chords

Chapter 10 of Book I of the Almagest presents geometric theorems used for computing chords. Ptolemy used geometric reasoning based on Proposition 10 of Book XIII of Euclid's Elements to find the chords of 72° and 36°. That Proposition states that if an equilateral pentagon is inscribed in a circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the hexagon and the decagon inscribed in the same circle.

He used Ptolemy's theorem on quadrilaterals inscribed in a circle to derive formulas for the chord of a half-arc, the chord of the sum of two arcs, and the chord of a difference of two arcs. The theorem states that for a quadrilateral inscribed in a circle, the product of the lengths of the diagonals equals the sum of the products of the two pairs of lengths of opposite sides. The derivations of trigonometric identities rely on a cyclic quadrilateral in which one side is a diameter of the circle.

To find the chords of arcs of 1° and ⁠1/2⁠° he used approximations based on Aristarchus's inequality. The inequality states that for arcs α and β, if 0 < β < α < 90°, then

${\displaystyle \frac{\sin \alpha }{\sin \beta }<\frac{\alpha }{\beta }<\frac{\tan \alpha }{\tan \beta }.}$

Ptolemy showed that for arcs of 1° and ⁠1/2⁠°, the approximations correctly give the first two sexagesimal places after the integer part.

Accuracy

Gerald J. Toomer in his translation of the Almagest gives seven entries where some manuscripts have scribal errors, changing one "digit" (one letter, see below). Glenn Elert has made a comparison between Ptolemy's values and the true values (120 times the sine of half the angle) and has found that the root mean square error is 0.000136. But much of this is simply due to rounding off to the nearest 1/3600, since this equals 0.0002777... There are nevertheless many entries where the last "digit" is off by 1 (too high or too low) from the best rounded value. Ptolemy's values are often too high by 1 in the last place, and more so towards the higher angles. The largest errors are about 0.0004, which still corresponds to an error of only 1 in the last sexagesimal digit.

The numeral system and the appearance of the untranslated table

Lengths of arcs of the circle, in degrees, and the integer parts of chord lengths, were expressed in a base 10 numeral system that used 21 of the letters of the Greek alphabet with the meanings given in the following table, and a symbol, "∠′", that means ⁠1/2⁠ and a raised circle "○" that fills a blank space (effectively representing zero). Three of the letters, labeled "archaic" in the table below, had not been in use in the Greek language for some centuries before the Almagest was written, but were still in use as numerals and musical notes.

${\displaystyle \begin{array}{|ccccccccc|}\hline \alpha & \mathrm{a}\mathrm{l}\mathrm{p}\mathrm{h}\mathrm{a}& 1& \iota & \mathrm{i}\mathrm{o}\mathrm{t}\mathrm{a}& 10& \rho & \mathrm{r}\mathrm{h}\mathrm{o}& 100\\ \beta & \mathrm{b}\mathrm{e}\mathrm{t}\mathrm{a}& 2& \kappa & \mathrm{k}\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{a}& 20& \sigma & \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{m}\mathrm{a}& 200\\ \gamma & \mathrm{g}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}& 3& \lambda & \mathrm{l}\mathrm{a}\mathrm{m}\mathrm{b}\mathrm{d}\mathrm{a}& 30& \tau & \mathrm{t}\mathrm{a}\mathrm{u}& 300\\ \delta & \mathrm{d}\mathrm{e}\mathrm{l}\mathrm{t}\mathrm{a}& 4& \mu & \mathrm{m}\mathrm{u}& 40& \upsilon & \mathrm{u}\mathrm{p}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{o}\mathrm{n}& 400\\ \epsilon & \mathrm{e}\mathrm{p}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{o}\mathrm{n}& 5& \nu & \mathrm{n}\mathrm{u}& 50& \phi & \mathrm{p}\mathrm{h}\mathrm{i}& 500\\ ϛ& \mathrm{s}\mathrm{t}\mathrm{i}\mathrm{g}\mathrm{m}\mathrm{a}(\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{i}\mathrm{c})& 6& \xi & \mathrm{x}\mathrm{i}& 60& \chi & \mathrm{c}\mathrm{h}\mathrm{i}& 600\\ \zeta & \mathrm{z}\mathrm{e}\mathrm{t}\mathrm{a}& 7& o& \mathrm{o}\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{n}& 70& \psi & \mathrm{p}\mathrm{s}\mathrm{i}& 700\\ \eta & \mathrm{e}\mathrm{t}\mathrm{a}& 8& \pi & \mathrm{p}\mathrm{i}& 80& \omega & \mathrm{o}\mathrm{m}\mathrm{e}\mathrm{g}\mathrm{a}& 800\\ \theta & \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{a}& 9& ϟ& \mathrm{k}\mathrm{o}\mathrm{p}\mathrm{p}\mathrm{a}(\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{i}\mathrm{c})& 90& ϡ& \mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{i}(\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{i}\mathrm{c})& 900\\ \hline\end{array}}$

Thus, for example, an arc of ⁠143+1/2⁠° is expressed as ρμγ∠′. (As the table only reaches 180°, the Greek numerals for 200 and above are not used.)

The fractional parts of chord lengths required great accuracy, and were given in sexagesimal notation in two columns in the table: The first column gives an integer multiple of ⁠1/60⁠, in the range 0–59, the second an integer multiple of ⁠1/60²⁠ = ⁠1/3600⁠, also in the range 0–59.

Thus in Heiberg's edition of the Almagest with the table of chords on pages 48–63, the beginning of the table, corresponding to arcs from ⁠1/2⁠° to ⁠7+1/2⁠°, looks like this:

${\displaystyle \begin{array}{ccc}\pi \epsilon \rho \iota \phi \epsilon \rho \epsilon \iota \tilde{\omega }\nu & \epsilon \stackrel{\text{'}}{\upsilon }\theta \epsilon \iota \tilde{\omega }\nu & \stackrel{\text{‘}}{\epsilon }\xi \eta \kappa o\sigma \tau \tilde{\omega }\nu \\ \begin{array}{|l|}\hline {\mathrm{\angle }}^{\prime }\\ \alpha \\ \alpha {\mathrm{\angle }}^{\prime }\\ \beta \\ \beta {\mathrm{\angle }}^{\prime }\\ \gamma \\ \gamma {\mathrm{\angle }}^{\prime }\\ \delta \\ \delta {\mathrm{\angle }}^{\prime }\\ \epsilon \\ \epsilon {\mathrm{\angle }}^{\prime }\\ ϛ\\ ϛ{\mathrm{\angle }}^{\prime }\\ \zeta \\ \zeta {\mathrm{\angle }}^{\prime }\\ \hline\end{array}& \begin{array}{|rrr|}\hline \circ & \lambda \alpha & \kappa \epsilon \\ \alpha & \beta & \nu \\ \alpha & \lambda \delta & \iota \epsilon \\ \beta & \epsilon & \mu \\ \beta & \lambda \zeta & \delta \\ \gamma & \eta & \kappa \eta \\ \gamma & \lambda \theta & \nu \beta \\ \delta & \iota \alpha & \iota ϛ\\ \delta & \mu \beta & \mu \\ \epsilon & \iota \delta & \delta \\ \epsilon & \mu \epsilon & \kappa \zeta \\ ϛ& \iota ϛ& \mu \theta \\ ϛ& \mu \eta & \iota \alpha \\ \zeta & \iota \theta & \lambda \gamma \\ \zeta & \nu & \nu \delta \\ \hline\end{array}& \begin{array}{|rrrr|}\hline \circ & \alpha & \beta & \nu \\ \circ & \alpha & \beta & \nu \\ \circ & \alpha & \beta & \nu \\ \circ & \alpha & \beta & \nu \\ \circ & \alpha & \beta & \mu \eta \\ \circ & \alpha & \beta & \mu \eta \\ \circ & \alpha & \beta & \mu \eta \\ \circ & \alpha & \beta & \mu \zeta \\ \circ & \alpha & \beta & \mu \zeta \\ \circ & \alpha & \beta & \mu ϛ\\ \circ & \alpha & \beta & \mu \epsilon \\ \circ & \alpha & \beta & \mu \delta \\ \circ & \alpha & \beta & \mu \gamma \\ \circ & \alpha & \beta & \mu \beta \\ \circ & \alpha & \beta & \mu \alpha \\ \hline\end{array}\end{array}}$

Later in the table, one can see the base-10 nature of the numerals expressing the integer parts of the arc and the chord length. Thus an arc of 85° is written as πε (π for 80 and ε for 5) and not broken down into 60 + 25. The corresponding chord length is 81 plus a fractional part. The integer part begins with πα, likewise not broken into 60 + 21. But the fractional part, $\frac{4}{60}+\frac{15}{{60}^2}$, is written as δ, for 4, in the ⁠1/60⁠ column, followed by ιε, for 15, in the ⁠1/60²⁠ column.

${\displaystyle \begin{array}{ccc}\pi \epsilon \rho \iota \phi \epsilon \rho \epsilon \iota \tilde{\omega }\nu & \epsilon \stackrel{\text{'}}{\upsilon }\theta \epsilon \iota \tilde{\omega }\nu & \stackrel{\text{‘}}{\epsilon }\xi \eta \kappa o\sigma \tau \tilde{\omega }\nu \\ \begin{array}{|l|}\hline \pi \delta {\mathrm{\angle }}^{\prime }\\ \pi \epsilon \\ \pi \epsilon {\mathrm{\angle }}^{\prime }\\ \pi ϛ\\ \pi ϛ{\mathrm{\angle }}^{\prime }\\ \pi \zeta \\ \hline\end{array}& \begin{array}{|rrr|}\hline \pi & \mu \alpha & \gamma \\ \pi \alpha & \delta & \iota \epsilon \\ \pi \alpha & \kappa \zeta & \kappa \beta \\ \pi \alpha & \nu & \kappa \delta \\ \pi \beta & \iota \gamma & \iota \theta \\ \pi \beta & \lambda ϛ& \theta \\ \hline\end{array}& \begin{array}{|rrrr|}\hline \circ & \circ & \mu ϛ& \kappa \epsilon \\ \circ & \circ & \mu ϛ& \iota \delta \\ \circ & \circ & \mu ϛ& \gamma \\ \circ & \circ & \mu \epsilon & \nu \beta \\ \circ & \circ & \mu \epsilon & \mu \\ \circ & \circ & \mu \epsilon & \kappa \theta \\ \hline\end{array}\end{array}}$

The table has 45 lines on each of eight pages, for a total of 360 lines.