Skull openings in 20th-century classification

Main article: Skull roof

The synapsid/sauropsid division supplemented, but was never as popular during the 20th century as a Linneaean approach splitting the reptiles into four subclasses based on the number and position of temporal fenestrae, openings in the sides of the skull behind the eyes. This classification was initiated by Henry Fairfield Osborn and elaborated and made popular by Romer's classic Vertebrate Paleontology.^[9][10] Those four subclasses were:

• Anapsida – no fenestrae - cotylosaurs and Chelonia (turtles and relatives)^{[note 2]}
• Synapsida – one low fenestra - pelycosaurs and therapsids (the 'mammal-like reptiles')
• Euryapsida – one high fenestra (above the postorbital and squamosal) - protorosaurs (small, early lizard-like reptiles) and the marine sauropterygians and ichthyosaurs, the latter called Parapsida in Osborn's work.
• Diapsida – two fenestrae - most reptiles, including lizards, snakes, crocodilians, dinosaurs and pterosaurs

The composition of Euryapsida was uncertain. Ichthyosaurs were at times considered to have arisen independently of the other euryapsids, and given the older name Parapsida. Parapsida was later discarded as a group for the most part (ichthyosaurs being classified as incertae sedis or with Euryapsida). However, the scheme with four (or three if Euryapsida is sunk into Diapsida) subclasses remained more or less universal for non-specialist work throughout the 20th century, and has only been challenged with the rising popularity of phylogenetic nomenclature.