April 12, 1877] 



NATURE 



517 



same shape all along, it satisfies Leibnitz's definition of a 

 straight line, and it is, in fact, a geodesic line of the 

 surface. 



Hence we have this second proposition — that all points 

 in the surface opposite to a given point lie in a straight 

 line. 



From the method of its construction, this straight line 

 is farther from the given point than any other line in the 

 surface. Travelling from the given point as a centre, in 

 whatever direction we might set out, we should, after 

 completing half our journey, arrive at this farthest straight 

 hne, we should cross it at right angles, and we should 

 then keep getting nearer and nearer to our starting-point, 

 until we finally reached it from the opposite side. 



Each separate point in the surface, moreover, has a 

 separate farthest line. For if any two points be taken, 

 the points opposite to them on the straight line which 

 joins them will be distinct. Hence their farthest lines will 

 cut this joining line in two separate points. They must, 

 therefore, be two separate lines, for the same straight line 

 cannot cut another straight line in two separate points. 

 In a similar manner it may be shown that each straight 

 line in the surface has a separate farthest point. Hence 

 there exists a reciprocal relation between the points and 

 straight lines of the surface, a relation which we may ex- 

 press by saying that every point in the surface has a polar, 

 and that every straight line in the surface has a pole. It 

 is then easy to show that when a point is made to move 

 along a straight line its polar will turn about a point, and 

 that when a straight line is made to turn about a point, 

 its pole will move along a straight line. 



It is interesting to compare these propositions with the 

 corresponding ones in spherical geometry. There, too, 

 each point has a farthest geodesic line ; that is to say, a 

 geodesic line which is farther from it than any other 

 geodesic line on the sphere. But each geodesic line has 

 two farthest points or poles, instead of having only one. 

 Hence there is not that perfect reciprocity of relationship 

 between points and geodesic lines which exists in -the 

 surface we have been examining ; and this is one of the 

 many ways in which the s{>here shows itself to be inferior 

 to that surface in simplicity. 



The most astounding fact I have elicited in connection 

 with this surface is one which comes out in the theory of 

 the circle. Defining a circle as the locus of points equi- 

 distant from a given point, we shall find that it assumes 

 a very extraordinary shape when its radius is at all 

 nearly equal to half the entire length of a straight line. 

 For let us again figure to ourselves a number of straight 

 lines radiating from a point. Let / be the total length 

 of each straight line. Then the supposition we have to 

 make is that the radius of our circle shall be nearly equal 



to — . Let us suppose it equal to — — in, where m is 



2 2 



small as compared with /. Each of the radiating lines 

 will cut the circle in two points, and each of these points 



1 I 



will be at a distance from O equal to — — in or — -}- m, 



2 2 

 according as the distance is measured in the one direction 

 or the other. And their distance from each other will be 

 equal to 2 m, that is to say, it will be comparatively small. 

 ~ It each point on the polar of o will be at a distance 



O equal to _. Hence each point on the circle will 

 2 

 It a distance from this polar equal to in. Moreover, 

 point at a distance of m from the polar will be a point 



i>the circle, because it will be at a distance oi — - m 



f ' 2 



O. But the locus of points at a distance of m from 



straight line, A B, will consist of two branches, C D 



' E F, one on either side of A u, and at the same dis- 



^ce from it along their whole length. It is true that 



se branches form^ in reality a , single continuous line. 



A point travelling along from C to D, and further in the 

 same direction, would ultimately appear at E, travel along 

 to F, and then, after a further journey, reappear at the 

 point c. But this does not alter the fact that when a 

 small portion only of this line is contemplated, it pre- 

 sents the appearance of two straight lines, each of them 

 parallel to, and equidistant from, A B. 



In the limiting case, where the radius becomes equal 

 / 

 to _, c D and E F both of them coincide with A B. The 

 2 



circle merges into a straight line, and becomes, in fact, 

 the polar of its own centre. It is not, indeed, quite accu- 

 rate to say that it merges into a straight line, for it re- 

 duces itself rather to two coincident straight lines, and its 

 equation in co-ordinate geometry would be one of the 

 second degree. 



In regard to the surface here treated of, it is easy to see 

 that, as with the sphere, the smaller the portion of it we 

 bring under our consideration, the more nearly its 

 properties approach to those of the plane. Indeed, if we 

 consider an area that is very small as compared with the 

 total area of the surface, its properties will not differ sen- 

 sibly from those of the plane. And on this ground it has 

 been argued that the universe may in reality be of finite 

 extent, and that each of its geodesic lines may return into 

 itself, provided only that its total magnitude be very great 

 as compared with any magnitude which we can bring 

 under our observation. 



In conclusion, I cannot do better than quote the passage 

 in which Prof. Clifford explains what must be the consti- 

 tution of space if this hypothesis should be true. " In 

 this case," he says, " the universe, as known, is again a 

 valid conception, for the extent of space is a finite number 

 of cubic miles. And this comes about in a curious way. 

 If you were to start in any directiori whatever and move 

 in that direction in a perfect straight line according to 

 the definition of Leibnitz, after travelling a most pro- 

 digious distance, to which the parallactic unit — 200,000 

 times the diameter of the earth's orbit — would be only a 

 few steps you would arrive at — this place. Only, if you 

 had started upwards, you would appear from below. Now 

 one of two things would be true. Either when you had 

 got halfway on your journey you came to a place that is 

 opposite to this, and which you must have gone through, 

 whatever direction you started in, or else all paths 

 you could have taken diverge entirely from each other 

 till they meet again at this place. In the former case 

 every two straight lines in a plane meet in two points, in 

 the latter they meet only in one. Upon this supposition 

 of a positive curvature the whole of geometry is far more 

 complete and interesting ; the principle of duality, instead 

 of half breaking down over metric relations, applies to all 

 propositions without exception. In fact I do not mind 

 confessing that I personally have often found relief from 

 the dreary infinities of homaloidal space in the consoling 

 hope that, after all, this other may be the true state of 

 things." F. W. Frankland 



HYDROGRAPHY OF WEST CENTRAL AFRICA 



W 



R. STANLEY'S second letter in last Thursday's 

 Telegraph contains important information on the 

 district between Tanganyika and the Albert and Victoria 

 Nyanza — information complementary to that given in his 

 former letters, which we embodied in a map, vol. xiv. 

 p. 374. He has, in fact, discovered another " source " of 

 the Nile, and one evidently of great length and volume — 

 the Kagera — which he has gallantly named the Alexandra 

 Nile. This river issues from a large lake, Akanyaru or 

 Alexandra Nyanza, in two branches and flows north, 

 uniting under 1° S. lat., and flowing east to the Victoria 

 Nyanza. Mr. Stanley was only able to see the Alexandra 

 Nyan?a from a distance, but it is evidently of consider- 



