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. Proof The proof is a consequence of the more widely known inequalities , and . Proof of the first inequality 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 inequality 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 also Aristarchus of Samos Eratosthenes Posidonius Notes and references External links Proof of the First Inequality Proof of the Second Inequality Trigonometry Inequalities