User:Michael Hardy/Greek.chord.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 letters of the Greek alphabet with the meanings given in the following table, and a symbol, "∠'", that means 1/2. Two of the letters, labeled "archaic" in this table, had not been in use in the Greek language for some centuries before the Almagest was written.

\begin{array}{|rlr|rlr|rlr|} \hline \alpha & \mathrm{alpha} & 1 & \iota & \mathrm{iota} & 10 & \varrho & \mathrm{rho} & 100 \\  \beta & \mathrm{beta} & 2 & \kappa & \mathrm{kappa} & 20 & & & \\  \gamma & \mathrm{gamma} & 3 & \lambda & \mathrm{lambda} & 30 & & & \\  \delta & \mathrm{delta} & 4 & \mu & \mathrm{mu} & 40 & & & \\  \varepsilon & \mathrm{epsilon} & 5 & \nu & \mathrm{nu} & 50 & & & \\  \stigma & \mathrm{stigma\ (archaic)} & 6 & \xi & \mathrm{xi} & 60 & & & \\  \zeta & \mathrm{zeta} & 7 & \omicron & \mathrm{omicron} & 70 & & & \\  \eta & \mathrm{eta} & 8 & \pi & \mathrm{pi} & 80 & & & \\  \vartheta & \mathrm{theta} & 9 & \koppa & \mathrm{koppa\ (archaic)} & 90 & & & \\  \hline \end{array} $$ Thus, for example, an arc of $143 1/2$&deg; is expressed as $$\varrho\mu\gamma\angle'$$.

The fractional parts of chord lengths required great accuracy, and were given in three columns in the table: the first giving 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, and the third an integer multiple of 1/603 = 1/21600, again 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&deg; through $7 1/2$&deg;, looks like this:



\begin{array}{ccc} \pi\varepsilon\varrho\iota\varphi\varepsilon\varrho\varepsilon\iota\tilde\omega\nu & \varepsilon\overset{\text{'}}\nu\vartheta\varepsilon\iota\tilde\omega\nu & \overset{\text{`}}\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\ \begin{array}{|l|} \hline \angle' \\ \alpha \\ \alpha\;\angle' \\  \hline\beta \\  \beta\;\angle' \\  \gamma \\  \hline\gamma\;\angle' \\  \delta \\  \delta\;\angle' \\  \hline\varepsilon \\  \varepsilon\;\angle' \\  \stigma \\  \hline\stigma\;\angle' \\  \zeta \\  \zeta\;\angle' \\  \hline \end{array} & \begin{array}{|r|r|r|} \hline\circ & \lambda\alpha & \kappa\varepsilon \\  \alpha & \beta & \nu \\  \alpha & \lambda\delta & \iota\varepsilon \\  \hline \beta & \varepsilon & \mu \\  \beta & \lambda\zeta & \delta \\  \gamma & \eta & \kappa\eta \\  \hline \gamma & \lambda\vartheta & \nu\beta \\  \delta & \iota\alpha & \iota\stigma \\  \delta & \mu\beta & \mu \\  \hline \varepsilon & \iota\delta & \delta \\  \varepsilon & \mu\varepsilon & \kappa\zeta \\  \stigma & \iota\stigma & \mu\vartheta \\  \hline \stigma & \mu\eta & \iota\alpha \\  \zeta & \iota\vartheta & \lambda\gamma \\  \zeta & \nu & \nu\delta \\  \hline \end{array} & \begin{array}{|r|r|r|r|} \hline \circ & \alpha & \beta & \nu \\  \circ & \alpha & \beta & \nu \\  \circ & \alpha & \beta & \nu \\  \hline \circ & \alpha & \beta & \nu \\  \circ & \alpha & \beta & \mu\eta \\  \circ & \alpha & \beta & \mu\eta \\  \hline\circ & \alpha & \beta & \mu\eta \\  \circ & \alpha & \beta & \mu\zeta \\  \circ & \alpha & \beta & \mu\zeta \\  \hline \circ & \alpha & \beta & \mu\stigma \\  \circ & \alpha & \beta & \mu\varepsilon \\  \circ & \alpha & \beta & \mu\delta \\  \hline \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 integer part of the arc. Thus an arc of 85&deg; is written as $$\pi\varepsilon$$ ($$\pi$$ for 80 and $$\varepsilon$$ for 5) and not broken down into 60 + 25, and the corresponding chord length of 81 plus a fractional part begins with $$\pi\alpha$$, likewise not broken into 60 + 1. But the fractional part, 4/60 + 15/602, is written as $$\delta$$, for 4, in the 1/60 column, followed by $$\iota\varepsilon$$, for 15, in the 1/602 column.

\begin{array}{ccc} \pi\varepsilon\varrho\iota\varphi\varepsilon\varrho\varepsilon\iota\tilde\omega\nu & \varepsilon\overset{\text{'}}\nu\vartheta\varepsilon\iota\tilde\omega\nu & \overset{\text{`}}\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\ \begin{array}{|l|} \hline \pi\delta\angle' \\ \pi\varepsilon \\  \pi\varepsilon\angle' \\  \hline  \pi\stigma \\  \pi\stigma\angle' \\  \pi\zeta \\  \hline \end{array} & \begin{array}{|r|r|r|} \hline \pi & \mu\alpha & \gamma \\  \pi\alpha & \delta & \iota\varepsilon \\  \pi\alpha & \kappa\zeta & \kappa\beta \\  \hline \pi\alpha & \nu & \kappa\delta \\  \pi\beta & \iota\gamma & \iota\vartheta \\  \pi\beta & \lambda\stigma & \vartheta \\  \hline \end{array} & \begin{array}{|r|r|r|r|} \hline \circ & \circ & \mu\stigma & \kappa\varepsilon \\  \circ & \circ & \mu\stigma & \iota\delta \\  \circ & \circ & \mu\stigma & \gamma \\  \hline \circ & \circ & \mu\varepsilon & \nu\beta \\  \circ & \circ & \mu\varepsilon & \mu \\  \circ & \circ & \mu\varepsilon & \kappa\vartheta \\  \hline \end{array} \end{array} $$