merous geometers throughout the centuries. Hundreds tried it, and failed. As in squaring the circle, some claimed to have accomplished it; but against each one all the rest decided.
It now seems queer that no one during all this time systematically developed the results obtainable when this postulate is denied, is negatived, is thrown overboard. Euclid's method, the reductio ad absurdum, would have led them on to this if only it had ever entered their heads to suspect a plural to space. But the perfect originality of this step required genius, and has given a permanent rank in the history of science to two names of which otherwise we should probably never have heard, Bolyai and Lobatchewsky. Their publication of a non-Euclidean geometry gave the entire question a totally new aspect, and from that moment everything previously printed on the subject became antiquated; everything else became moribund, and the world of geometries was dualized into Euclid and non-Euclid. Like Columbus, they discovered and opened a new continent, into which for the last forty years geometers have been swarming, rewarded by many gold-mines. On non-Euclidean spaces and the kindred subject, hyperspaces, I have given in the "American Journal of Mathematics" a list of about one hundred and eighty publications since 1844. In dividing spaces with reference to the parallel-postulate, those in which through one point outside of a straight line can be drawn more than one parallel to that line are called hyperbolic spaces; that space in which through the point we can draw one and only one parallel is called parabolic; those spaces in which we can draw no parallel straight lines are called elliptic. In hyperbolic spaces the sum of the three angles of any triangle is less than two right angles, in parabolic equal to, in elliptic greater than, two right angles. Elliptic spaces are positively curved spaces, hyperbolic are negatively curved spaces, while the parabolic has no curvature, is a flat or homaloidal space.
This pluralization of the idea of space is independent of dimensionality and came synthetically. But about the same time came analytically a plural having reference to dimensions. Our perceptions, intuitions, imagings, are confined to a flat space of three dimensions, and this gives us a strong prejudice in favor of the belief that our bodies and the stars are also confined in a tridimensional homaloid. But this is simply a question of fact in the domain of physical experimentation.
How this belief might be negatived is easily illustrated. In 1872 Clifford said before the British Association: "Suppose that three points are taken in space, distant from one another as far as the sun from a Centauri, and that the shortest distances between these points are drawn so as to form a triangle. And suppose the angles of this triangle to be very accurately measured and added together: this can at present be done so accurately that the error shall certainly be less than one minute, less therefore than the five-thousandth part of a right
