can be drawn parallel to a given straight line. It was felt, even by the earliest commentators on Euclid, that this postulate did not possess quite the same degree of self-evidence as was manifested by the others. It was necessary, they felt, to give a proof of this postulate; they attempted to improve on Euclid’s work in a number of minor ways, but it was the parallel axiom which they were most concerned to revise; the proof of this postulate should be contained, they thought, in the other postulates. The attempts to supply this proof were all fruitless, and the sixth century was reached with this nine-hundred-years-old disfigurement still persisting. For some time after the sixth century the world rested from Euclid’s parallel axiom; indeed, it rested from geometry altogether, and the old empirical outlook of the Egyptians, and even their formulæ, again became current. But the Greek culture penetrated to the Arabs, and with the Greek culture came the riddle of Euclid’s axiom. Again proofs were attempted; a famous attempt is that of Nasir Eddin, who flourished in the thirteenth century. In 1663 John Wallis made the important discovery that unless the parallel axiom be assumed, similar figures of different sizes are not possible, that is to say, that if we are to assume that _shape_ is independent of _size_, then we must assume Euclid’s parallel axiom. Many of these attempts brought out points of interest, but none of them were successful. In the year 1733, however, the whole research took on a new complexion with the publication of Girolamo Saccheri’s _Euclides ab omni naevo vindicatus_. The importance of this work consists in the fact that, although it was written to vindicate Euclid’s parallel axiom once for all, it contains the first real outline of a non-Euclidean geometry.