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 IV Philopator Ptolemy IV Philopator Ptolemy IV Philopator (, "Ptolemaĩos Philopátōr" "Ptolemy, lover of his Father"; 245/4–204 BC), son of Ptolemy III and Berenice II, was the fourth Pharaoh of Ptolemaic Egypt from 221 to 204 BC. The decline of the Ptolemaic dynasty began under the reign of Ptolemy IV. Among the children of Ptolemy IV Philopator and his sister-wife Arsinoe III of Egypt was Ptolemy V Epiphanes, who married Cleopatra I Syra, daughter of Antiochus III the Great and Laodice III. Ptolemy IV's reign was inaugurated by the murder of his mother, and he was always under the dominion of favourites,