The last published work of SACCHERI was a recital of his endeavors at demonstrating the parallel-postulate. This received the "Imprimatur" of the Inquisition, July 13, 1733; the Provincial Company of Jesus took possession of the book for perusal on August 16, 1733; but unfortunately within two months after it had been reviewed by these authorities, SACCHERI passed away. All efforts which had been made prior to the work of SACCHERI were based upon the assumption that there must be an equivalent postulate which, if it could be demonstrated, would lead to a direct, positive proof of EUCLID'S proposition. Although these and all other attempts at reaching such a proof have signally failed and although it may correctly be said that the entire history of demonstrations aiming at the solution of the famous postulate has been one long series of utter failures, it can be asserted with equal certitude that it has proven to be one of the most fruitful problems in the history of mathematical thought. For out of these failures has been built a superstructure of analytical investigations which surpasses the most sanguine expectations of those who had labored and failed. In 1766 JOHN LAMBERT (1728-1777) wrote a paper upon the _Theory of Parallels_ dated Sept. 5, 1766, first published in 1786, from the papers left by F. BERNOULLI, which contained the following assertions:[2] 1. The parallel-axiom needs proof, since it does not hold for geometry on the surface of the sphere. 2. In order to make intuitive a geometry in which the triangle's sum is less than two right angles, we need an "imaginary" sphere (the pseudosphere). 3. In a space in which the triangle's sum is different from two right angles there is an absolute measure (a natural unit for length). At this time IMMANUEL KANT (1724-1804), the noted German metaphysician, was in the midst of his philosophical labors. And it is believed that it was he who first suggested the idea of different _spaces_. Below is given a statement taken from his _Prolegomena_[3] which corroborates this view. "That complete space (which is itself no longer the boundary of another space) has three dimensions, and that space in general cannot have more, is based on the proposition that not more than three lines can intersect at right angles in one point.... That we can require a line to be drawn to infinity, a series of changes to be continued (for example, _spaces_ passed through by motion) in indefinitum, presupposes a representation of space and time which can only attach to intuition." [2] Vide _New York Mathematical Society Bulletin_, Vol. III, 1893-4, p. 79, G. B. HALSTEAD on _Lambert's Non-Euclidean Geometry_. [3] _Prolegomena_, KANT, p. 37, Trans. by J. P. MAHAFFY and J. H. BERNARD.