Greek Numerals - Hellenistic Zero

Hellenistic Zero

Hellenistic astronomers extended alphabetic Greek numerals into a sexagesimal positional numbering system by limiting each position to a maximum value of 50 + 9 and including a special symbol for zero, which was also used alone like our modern zero, more than as a simple placeholder. However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.) — they were not used for the integral part of a number. This system was probably adapted from Babylonian numerals by Hipparchus c. 140 BC. It was then used by Ptolemy (c. 140), Theon (c. 380), and Theon's daughter Hypatia (died 415).

In Ptolemy's table of chords, the first fairly extensive trigonometric table, there were 360 rows, portions of which looked as follows:

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

Each number in the first column, labeled περιφεριῶν, is the number of degrees of arc on a circle. Each number in the second column, labeled ενθειῶν, is the length of the corresponding chord of the circle, when the diameter is 120. Thus πδ represents an 84° arc, and the ∠' after it means one-half, so that πδ∠' means 84.5°. In the next column we see π μα γ, meaning 80 + 41/60 + 3/602. That is the length of the chord corresponding to an arc of 84.5° when the diameter of the circle is 120. The next column, labeled ὲξηκοστῶν, for "sixtieths", is the number to be added to the chord length for each 1° increase in the arc, over the span of the next 12°. Thus that last column was used for linear interpolation.

The Greek sexagesimal place holder or zero symbol changed over time. The symbol used on papyri during the second century was a very small circle with an overbar several diameters long, terminated or not at both ends in various ways. Later, the overbar shortened to only one diameter, similar to our modern o macron (ō) which was still being used in late medieval Arabic manuscripts whenever alphabetic numerals were used. But the overbar was omitted in Byzantine manuscripts, leaving a bare ο (omicron). This gradual change from an invented symbol to ο does not support the hypothesis that the latter was the initial of ουδέν meaning "nothing".

Some of Ptolemy's true zeros appeared in the first line of each of his eclipse tables, where they were a measure of the angular separation between the center of the Moon and either the center of the Sun (for solar eclipses) or the center of Earth's shadow (for lunar eclipses). All of these zeros took the form 0 | 0 0, where Ptolemy actually used three of the symbols described in the previous paragraph. The vertical bar (|) indicates that the integral part on the left was in a separate column labeled in the headings of his tables as digits (of five arc-minutes each), whereas the fractional part was in the next column labeled minute of immersion, meaning sixtieths (and thirty-six-hundredths) of a digit.

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