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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
1st there is a line AB through O in the plane OCD which will not meet CD at any finite distance from O.
2nd that if any line A′B′ or A″B″ 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 , the logical presumption is, and must ever remain, that it is not true.
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 A′B′ be the plane of the delineation or picture seen edgewise. It cuts all the rays through S in 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 which lie at an infinite distance and see how they look in perspective.
First suppose the natural, Euclidean, or parabolic geometry to be true. Then all the rays through S from infinitely distant parts of the plane, themselves lie in one plane. For let SI′ be such a ray, then if SI′ be turned about S the least bit out of the plane parallel to the natural plane it will cut the latter at a finite distance.
These rays through S from the infinitely distant parts of the natural plane, since they lie in one plane, will cut the plane of the picture in a straight line, called the vanishing line of the natural plane.
The delineations of any two parallel lines will cross one another on this vanishing line.
Next, suppose that the 2nd natural proposition on p. 25 is false and that SI′ may be tilted through a finite angle without cutting the natural plane at a finite distance.
Then, I will state, what there is no difficulty in proving, that the delineation of the infinitely distant parts of the natural plane will occupy a space on the picture bounded by a conic section.
The picture looks exactly as before, only that the REAL DISTANCES of certain parts which on the first assumption were finite, now become infinite.
The following two figures show the straight line which on the first assumption alone represents parts really infinitely distant, as well as the conic which on the second assumption bounds the delineation of the really infinitely distant parts. With two lines parallel on the first assumption.
In this case, space is limited. But though limited, it is immeasurable. Imagine, for instance, that every kind of unit of linear measure should shrink up as it was removed from a fixed centre, and perhaps differently according as it was radially or peripherally placed, so that it never could in any finite number of repetitions get beyond a certain spherical surface. Then that surface would be at an infinite distance and no moving body could ever traverse it, for the distance moved over in a unit of time would be a unit of distance. But unless this linear unit were to shrink according to a peculiar law, the result would be that different parts of space would be unlike. That is to say, for example, if we were to draw the plane figure ABCD, by means of given lengths AB, AC, AD, BC, we should find the resulting length of CD to be different in different parts of space. Now, geometers assume, perhaps with little reason, that the fact that rigid bodies move about readily in space without change of proportions, shows that all the parts of space are alike.
With this condition, I will state, what is again readily proved, what the law of the representation of equal distances and equal angles is in this second or hyperbolic geometry. For the distances. Let the conic be the “absolute,” or the perspective representation boundary of infinitely distant regions. Let AB be a unit of distance. Through AB draw the line
IJ, let I′J′ be any other line. Draw straight lines through their intersections II′ and JJ′ with the absolute. Let O be the point where these lines intersect; from O draw rays to A and B; then the distance A′B′ cut off by these rays on the line I′J′ is equal to AB.
The rule for angles is this. Let the conic be the absolute. Let ACB be a given angle. From C draw CI and CJ tangent to the absolute. Take any other vertex C′ and draw tangents C′I and C′J to the absolute. Through I the intersection of CI and C′I and J the intersection of CJ and C′J draw the straight line IJ cutting AC and BC in A and B. Then AC′B = ACB.
The third kind of geometry, which denies the 1st natural assumption, supposes space to be unlimited but finite, that is measurable. It is as if the linear unit so expanded in departing from a centre as to enable us to pass through what is naturally supposed to be at an infinite distance.
1. Merely because so called by writers on Perspective. Nothing to do with the “natural assumptions” of page 25.
