Aristarchus' inequality

Aristarchus' inequality (after the Greek astronomer and mathematician Aristarchus of Samos; c. 310 – c. 230 BCE) is a law of trigonometry which states that if α and β are acute angles (i.e. between 0 and a right angle) and β < α then

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

Ptolemy used the first of these inequalities while constructing his table of chords.

External links

• Hellenistic Astronomers and the Origins of Trigonometry, by Professor Gerald M. Leibowitz