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.

${\begin{array}{|rlr|rlr|rlr|}\hline \alpha &\mathrm {alpha} &1&\iota &\mathrm {iota} &10&\varrho &\mathrm {rho} &100\\\beta &\mathrm {beta} &2&\kappa &\mathrm {kappa} &20&&&\\\gamma &\mathrm {gamma} &3&\lambda &\mathrm {lambda} &30&&&\\\delta &\mathrm {delta} &4&\mu &\mathrm {mu} &40&&&\\\varepsilon &\mathrm {epsilon} &5&\nu &\mathrm {nu} &50&&&\\\mathrm {\stigma} &\mathrm {stigma\ (archaic)} &6&\xi &\mathrm {xi} &60&&&\\\zeta &\mathrm {zeta} &7&\mathrm {o} &\mathrm {omicron} &70&&&\\\eta &\mathrm {eta} &8&\pi &\mathrm {pi} &80&&&\\\vartheta &\mathrm {theta} &9&\mathrm {\koppa} &\mathrm {koppa\ (archaic)} &90&&&\\\hline \end{array}}$

Thus, for example, an arc of 143+1⁄2° 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° through 7+1⁄2°, looks like this:

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

Later in the table, one can see the base-10 nature of the integer part of the arc. Thus an arc of 85° 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.

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