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Conon served as court astronomer to Ptolemaeus III in Alexandria. He was a lifelong friend of Archimedes and the two exchanged mathematical ideas. Pappus states that the curve now known as the spiral of Archimedes was discovered by Conon although it was much used by Archimedes.
Conon's work on conic sections became a basis for the fourth book of Conics of Apollonius of Perga.
Conon's major works (now all lost) included the seven books of De astrologia which included solar eclipse observations, and Pros Thrasydaion which studied the points of intersection of two conics.
Reference (One book/article)
References elsewhere in this archive:
Tell me about Archimedes' spiral
There is a Crater Conon on the moon. There is also a Rima Conon. You can see a list of lunar features named after mathematicians.