Aristarchus's inequality


Aristarchus's 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

Ptolemy used the first of these inequalities while constructing his table of chords.[1]


The proof is a consequence of the more known inequalities


Proof of the first inequalityEdit

Using these inequalities we can first prove that


We first note that the inequality is equivalent to


which itself can be rewritten as


We now want show that


The second inequality is simply  . The first one is true because


Proof of the second inequalityEdit

Now we want to show the second inequality, i.e. that:


We first note that due to the initial inequalities we have that:


Consequently, using that   in the previous equation (replacing   by  ) we obtain:


We conclude that


See alsoEdit

Notes and referencesEdit

  1. ^ Toomer, G. J. (1998), Ptolemy's Almagest, Princeton University Press, p. 54, ISBN 0-691-00260-6

External linksEdit

  • Leibowitz, Gerald M. "Hellenistic Astronomers and the Origins of Trigonometry" (PDF).
  • Proof of the First Inequality
  • Proof of the Second Inequality