5 20 THE POPULAR SCIENCE MONTHLY. 



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 Colum- 

 bus, 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, hyper- 

 spaces, 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 par- 

 allel 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 posi- 

 tively 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 dimension- 

 ality and came synthetically. But about the same time came ana- 

 lytically 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 bod- 

 ies and the stars are also confined in a tridimensional homaloid. But 

 this is simply a question of fact in the domain of physical experimen- 

 tation. 



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 



