Writings of Charles S. Peirce: A Chronological Edition, Volume 8: 1890–1892

6. The Non-Euclidean Geometry Made Easy

25 6 The Non-Euclidean Geometry Made Easy late Spring 1890 Houghton Library We have an a priori or natural idea of space, which by some kind of evolution has come to be very closely in accord with observations. But we find in regard to our natural ideas, in general, that while they do accord in some measure with fact, they by no means do so to such a point that we can dispense with correcting them by comparison with observations. Given a line CD and a point O. Our natural (Euclidean) notion is that 1 st there is a line AB through O in the plane OCD which will not meet CD at any finite distance from O. 2 nd that if any line AB or AB through O in the plane OCD be inclined by any finite angle, however small, to AB, it will meet CD at some finite distance from O. Is this natural notion exactly true? A. This is not certain. B. We have no probable reason to believe it so. C. We never can have positive evidence lending it any degree of likelihood. It may be disproved in the future. D. It may be true, perhaps. But since the chance of this is as 1: or S, the logical presumption is, and must ever remain, that it is not true. C D A A A B B B O     Writings of C. S. Peirce 1890–1892 26 E. If there is some influence in evolution tending to adapt the mind to nature, it would probably not be completed yet. And we find other natural ideas require correction. Why not this, too? Thus, there is some reason to think this natural idea is not exact. F. I have a theory which fits all the facts as far as I can compare them, which would explain how the natural notion came to be so closely approximate as it is, and how space came to have the properties we find it has. According to this theory, this natural notion would not be exact. To give room for the non-Euclidean geometry, it is sufficient to admit the first of these propositions. Either the first or the second of the two natural propositions on page 25 may be denied, giving two corresponding kinds of non-Euclidean geometry. Though neither of these is quite so easy as ordinary geometry, they can be made intelligible. For this purpose, it will suffice to consider plane geometry. The plane in which the figures lie must be regarded in perspective. Let ABCD be this plane, which I call the natural plane, 1 seen edgewise . Let S be the eye, or point of view, or centre of projection. From every point of the natural plane, rays, or straight lines proceed to the point of view, and are continued beyond it if necessary. If three points in the natural plane lie in one straight line, the rays from them through the point of view will lie in one plane. Let AB be the plane of the delineation or picture seen edgewise. It cuts all the rays through S in 1. Merely because so called by writers on Perspective. Nothing to do with the “natural assumptions” of page 25. S A B B C D D I A     6. Non-Euclidean Geometry Made Easy, 1890 27 points, and so many of these rays as lie in one plane it cuts in a straight line; for the intersection of two planes is a straight line. The points in which this plane cuts those rays are the perspective delineations of the natural points, i.e. the corresponding points in the natural plane. We extend this to cases in which the point of view is between the natural and the delineated points. It is readily seen that the delineation of a point is a point, and that to every point in the picture corresponds a point in the natural plane. And to a straight line in the natural plane corresponds a straight line in the picture. For the first straight line and the point of view lie in one plane, and the intersection of this plane by the plane of the picture is a straight line. All this is just as true for the non-Euclidean as for the Euclidean geometry. But now let us consider the parts of the natural plane...