About twenty years before the printing of the work of NASIR-EDDIN, CHRISTOPH CLAVIUS (1574) deduced the axiom of parallels from the assumption that a line whose points are all equidistant from a straight line is itself straight. In his consideration of the parallel-postulate he is said to have regarded it as EUCLID'S XIIIth axiom. Later BOLYAI spoke of it as the XIth and later still, TODHUNTER treated it as the XIIth. Hence, there does not seem to have been any general unanimity of opinion as to the exact status of the parallel-postulate, and especially is this true in view of the uncertainty now known to have existed in EUCLID'S mind concerning it. GIROLAMO SACCHERI (1667-1733), a learned Jesuit, born at San Remo, came next upon the stage. And so important was his work that it will perpetuate the memory of his name in the history of mathematics. He was a teacher of grammar in the Jesuit _Collegio di Brera_ where TOMMASO CEVA, a brother of GIOVANNI, the well-known mathematician, was teacher of mathematics. His association with the CEVA brothers was especially beneficial to him. He made use of CEVA'S very ingenious methods in his first published book, 1693, entitled _Solutions of Six Geometrical Problems Proposed by Count Roger Ventimiglia_. A +--------------------------------+ B | | | | | | | | | | | | C +--------------------------------+ D FIG. 1. SACCHERI attacked the problem of parallels in quite a new way. Examining a quadrilateral, _ABCD_, in which the angles _A_ and _B_ are right angles and the sides _AC_ and _BD_ are equal, he determined to show that the angles _C_ and _D_ are equal. He also sought to prove that they are either right angles, obtus acute. He undertook to prove the falsity of the latter two propositions (that they are either obtuse or acute), leaving as the only possibility that they must be right angles. In doing so, he found that his assumptions led him into contradictions which he experienced difficulty in explaining. His labors in connection with the solution of the problems proposed by COUNT VENTIMIGLIA, including his work on the question of parallels, led directly into the field of metageometrical researches, and perhaps to him as to no other who had preceded him, or at least to him in a larger degree, belongs the credit for a continued renewal of interest in that series of investigations which resulted in the formulation of the non-Euclidean geometry.