Non-Euclidean geometry that are both perpendicular to a third line: Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century. The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid's work "Elements" was written. In the "Elements", Euclid began with a limited number of assumptions (23 definitions, five common notions, and five postulates) and sought to prove all the other results (propositions) in the work. The most notorious of the postulates is often
Ptolemy II of Telmessos of Antipater. Ptolemy II was born and raised in Telmessos in Lycia at an unknown date during his father’s reign of Telmessos. His father Lysimachus ruled as the Ptolemaic Client King of Telmessos from February 240 BC until his death in 206 BC. Little is known on his early life prior to succeeding his father. When his father died in 206 BC, Ptolemy II became the fourth and final ruler from the Lysimachid dynasty, which is also known as the Ptolemaic/Lysimachid dynasty in Lycia to rule the city. Ptolemy II also, was the third and final Ptolemaic Client King of