Parallel (geometry) as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate. Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius' definition as well as its modification by the philosopher Aganis. At the end of the nineteenth century, in England, Euclid's Elements was still the standard textbook in secondary schools. The traditional treatment of geometry was being
Ptolemy Philadelphus (son of Cleopatra) Ptolemy Philadelphus (son of Cleopatra) Ptolemy Philadelphus (, "Ptolemy the brother-loving", August/September 36 BC – 29 BC) was a Ptolemaic prince and was the youngest and fourth child of Greek Ptolemaic Queen Cleopatra VII of Egypt, and her third with Roman Triumvir Mark Antony. Ptolemy was of Greek and Roman heritage. He was born in Antioch, Syria (now a part of modern Turkey). Ptolemy was named after the original Ptolemy II Philadelphus (the second Pharaoh of the Ptolemaic dynasty) and Cleopatra's intention was recreating the former Ptolemaic Kingdom. In late 34 BC, at the Donations of Alexandria, Ptolemy was made