5i6 



NATURE 



[April 12, 1877 



is no longer true. Geodesic lines diverging from a point 

 do not continue to diverge for ever. They meet again 

 and inclose a space. The first question which presents 

 itself is with reference to the situation of the point towards 

 which they ultimately converge. In the case of a spherical 

 surface they will converge towards a point which is sepa- 

 rated from the starting-point by half the length of a 

 geodesic line. And this is the only case we are able to con- 

 ceive. The surface of a sphere is the only doubly extended 

 manifoldness of uniform positive curvature which geometry 

 recognises, and it is the only one which we can figure to 

 ourselves in thought. It is not, however, the simplest of 

 such manifoldnesses. To obtain the simplest case we must 

 suppose that the point towards which two geodesic lines 

 converge is separated from their starting-point, not by 

 half, but by the entire length of a geodesic line ; or, what 

 amounts to the same thing, that it coincides with the 

 starting-point. It is true that we are utterly unable to 

 figure to ourselves a surface in which two geodesic lines 

 shall have only one point of intersection, and shall yet 

 inclose a space. But we are perfectly at liberty to reason 

 about such a surface, because there is nothing self-contra- 

 dictory in the definition of it, and because, therefore, the 

 analytical conception of it is perfectly valid. It is the 

 simplest continuous manifoldness of two dimensions, and 

 of finite extent, and those few properties of it which I 

 have worked out appear to me to be very beautiful. In 

 order to make my observations m.ore intelligible, I shall 

 for the future speak of it as a surface, and its geodesic 

 lines I shall speak of as straight lines. I have the highest 

 authority for using this nomenclature, and though it will 

 impart to my theorems a very paradoxical sound, it is 

 calculated, I think, to give a juster idea of their meaning, 

 than if I were to use the more accurate, but less familiar 

 terms. 



Assuming, then, as the fundamental properties of our 

 surface, that every straight line is of finite extent (in other 

 words, that a point moving along it will arrive at the 

 position from which it started after travelling a finite 

 distance), and that two straight lines cannot have two 

 points in common, the first corollary I propose to estab- 

 lish is that all straight lines in the surface are of equal 

 extent. 



Let A, B, be two straight lines in the given surface. If 

 possible, let A be greater than B. From A cut off a portion 

 equal to B. Let P, Q, be the extreme points of this seg- 

 ment, and let R be any point in B. Apply the line A to 

 the line B in such a manner that the point p falls on the 

 point R, then, since in a surface of uniform curvature 

 equal lengths of geodesic lines may be made to coincide, 

 the segment PQ will coincide with the entire straight line 

 B. Hence Q will fall upon R. But P coincides with R, 

 and P and Q do not coincide with one another, since PQ is 

 less than the entire straight line A ; therefore Q cannot 

 coincide with R. Hence A cannot be greater than B. 



The straight lines here spoken of are, of course, not 

 terminated straight lines. What the proposition asserts 

 is that the entire length of all straight lines in the given 

 surface is the same. The corresponding proposition in 

 spherical geometry is that all great circles of a given 

 sphere are equal. 



There are a great many other analogies between the 

 imaginary surface here treated of and the surface of a 

 sphere. Its straight lines, though they are like the straight 

 hnes of a plane in the circumstance that any two of them 

 have only one point of intersection, are in many other 

 respects analogous to great circles. In any of its straight 

 lines, for instance, each point has a corresponding point 

 which is opposite to it, and farther from it than any other 

 point in the line. For if by setting out from a point and 

 travelling a finite distance in a particular direction we 

 get back to the starting-point, there must be a point half 

 way on our journey which is farther from the starting- 

 point than any other point in the line, and which may 



very appropriately be called its opposite point. It is an 

 obvious corollary that the distance between any two 

 points will be the same as the distance between their 

 opposite points. 



Let us now consider the case of a number of straight 

 lines radiating from a centre. In each of them there will 

 be a point which is opposite to that centre. And it will 

 be a separate point for every separate straight line. For 

 no two straight lines can have two points in common, 

 and since these radiating lines have a common centre of 

 radiation, they can have no other point in common. 

 Hence, if we suppose one of these lines to rotate about 

 the centre, the point opposite to the centre will describe 

 a continuous line, and one which finally returns into itself. 

 It is the locus of all points in the surface opposite to the 

 centre of radiation. What now is the character of this 

 locus ? In the first place it is a line which is of the same 

 shape all along, and of which all equal segments there- 

 fore can be made to coincide. For any two positions of 



the rotating line which contain a given angle may be 

 placed upon any other two positions which contain an 

 equal angle. Then, since the length of all straight lines 

 in the surface is the same, the opposite points will coin- 

 cide, and by parity of reasoning all intermediate points 

 of the locus. But, in the second place, the locus is also 

 of the same shape on both sides. For each point in it 

 may be approached from the centre of radiation in two^ 

 different ways, and it is at the same distance from ths " 

 centre, whether it be approached in the one way or tl 

 other. Any particular segment, in fact, of the locus hj| 

 its extreme points joined to the centre of radiation 

 lines which are of equal length, and which in clue 

 an equal angle— lines, therefore, which may be madi 

 to coincide. Since this is the case for any segmen| 

 whatever, and for every subdivision of a segment, 

 the points of a segment will still remain on it if th^ 

 segment be turned round and applied to itself. Henc 

 the locus is of the same shape, whether viewed from tl 

 one side or from the other. But since it is also of thS 



