any one of them be assumed the others are readily proved. The theorem that the sum of the three angles of a triangle is equal to two right angles belong to this set. Ptolemy (Claudius Ptolemæus, second century A. D.) seems to have been the first to publish an attempted proof of this postulate of Euclid. Almost all mathematicians down to the beginning of the nineteenth century have given more or less attention to this question, and the account of their efforts to prove the postulate forms one of the most interesting chapters in the history of mathematics. Cajori, in his 'History of Elementary Mathematics' says, page 270: "They all fail, either because an equivalent assumption is implicitly or explicitly made, or because the reasoning is otherwise fallacious. On this slippery ground good and bad mathematicians alike have fallen. We are told that the great Lagrange, noticing that the formulas of spherical trigonometry are not dependent upon the parallel-postulate, hoped to frame a proof on this fact. Toward the close of his life he wrote a paper on parallel lines and began to read it before the Academy, but suddenly stopped and said: 'Il faut que j'y songe encore' (I must think it over again); he put the paper in his pocket and never afterwards publicly recurred to it."
About the time to which I have referred, the end of the eighteenth and beginning of the nineteenth century, the idea began to force itself upon mathematicians that perhaps there was more in the question than appeared on the surface. It was one of the many instances which have occurred in all branches of human knowledge where some truth of fundamental importance has begun to force itself simultaneously on a number of minds. We leave the significance of this aspect of the question to the psychologists. Another curious fact to be noted in connection with the writings which have finally shown us the true meaning, of the parallel-postulate is that either they attracted little or no general attention when they first appeared, or else they remained unpublished. The names of Lobatchewsky and the Bolyais have been made immortal by their writings on this subject, but it was not until long after they were published that their vast importance was recognized. The inimitable Gauss wrote on the same subject, but left his work unpublished, and Cajori (ibid., p. 274) mentions two writers of much earlier date who anticipated in part the theories of Lobatchewsky and the Bolyais. These are Geronimo Saccheri (1667-1733), a Jesuit father of Milan, and Johann Heinrich Lambert (1728-1777), of Mühlhausen, Alsace.
Lobatchewsky (Nicholaus Ivanovitch Lobatchewsky, 1793-1856) conceived the brilliant idea of cutting loose from the parallel-postulate altogether and succeeded in building up a system of geometry without its aid. The result is startling to one who has been taught to look upon Ihe facts of geometry (that is, of the Euclidean geometry) as incon-
