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 {{frac|143|1|2}}° is expressed as $\backslash varrho\backslash mu\backslash gamma\backslash angle\text{'}$.

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 [http://www.wilbourhall.org/pdfs/HeibergAlmagestComplete.pdf 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 {{frac|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\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° is written as $\backslash pi\backslash varepsilon$ ($\backslash pi$ for 80 and $\backslash varepsilon$ for 5) and not broken down into 60 + 25, and the corresponding chord length of 81 plus a fractional part begins with $\backslash pi\backslash alpha$, likewise not broken into 60 + 1. But the fractional part, 4/60 + 15/60², is written as $\backslash delta$, for 4, in the 1/60 column, followed by $\backslash iota\backslash varepsilon$, 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}