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**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

- ,

- and

- .

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

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

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

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