45" 



NATURE 



[October 5, igu 



LETTERS TO THE EDITOR. 



[The Editor does not hold liimsclf responsible for opinions 

 expressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers oj, rejected 

 manuscripts intended for this or any oilier part of Nature. 

 .Yo notice is taken of anonymous communications.] 



Non-Euclidean Geometry. 



Mr. Frankland (Nature, September 7) has raised the 

 old problem of Bertrand's proof of the parallel-axiom by 

 a consideration of infinite areas. This is perhaps the 

 most subtle and the most specious of all the attempted 

 proofs, and this character it owes to the fact that a process 

 of reasoning which is sound for finite magnitudes is ex- 

 tended to a field which is beyond our powers of compre- 

 hension — the field of infinity. The fallacy which underlies 

 Bertrand's proof becomes more apparent in Legendre's 

 simpler device ("Elements de Geometrie," 12 £d., 

 Note ii.). A straight line divides a plane in which it lies 

 into two congruent parts — this, of course, has no real 

 meaning, since we are dealing with infinite areas, but 

 such is the argument — and two rays from a point enclose 

 an (infinite) area which is less than half the whole plane. 

 Hence, if two intersecting lines are both parallel to the 

 same straight line, the area of half the plane can be 

 enclosed within an area which is less than half the plane. 



This is the same sort of paradox as the well-known one 

 by which the part is made to appear equal to, or even 

 greater than, the whole. The even numbers 2, 4, 6, . . . 

 form a part of the aggregate of integral numbers 

 1, 2, 3, . . ., but a (1, 1) correspondence can be estab- 

 lished between them, viz. to 211 in the part corresponds n 

 in the whole aggregate, and to n in the whole corresponds 

 2ii in the part. Hence the part is equal to the whole. 

 And, again, a (2, 1) correspondence can be established 

 between the part and the whole, viz. to 411 in the part 

 corresponds n in the whole, while the numbers of the 

 form 4»+2 have no correspondent. Thus the part is 

 greater than the whole. 



Mr. Frankland's comparison of the areas of a circle 

 and a regular inscribed polygon is not quite fair to the 

 polygon. The area of a regular N-gon, as its radius 

 tends to infinity, tends to a finite limit, ?rfe 2 (N — 2), which, 

 of course, tends to infinity as N is increased. The area 

 of a circle is qrrk 2 smhrrhk, which also tends to infinity 

 as r is increased. The first he calls a linear infinity, and 

 the second an exponential infinity, and certainly 

 tends to infinity with «, if n is any finite number. But 

 what is the relation between r and N? If we take the 

 expression for the area of a regular N-gon inscribed in a 

 circle of radius r, and then let N increase, we get a 

 limit 47rfc 2 sinh 2 r/2fr, which is the expression for the area 

 of the circle. Again, if in the regular N-gon with infinite 



radius we inscribe a circle, its area is 27rfc"(coseL- N - 1) 



and this always bears a finite ratio to the area of the 

 N-gon ; it is thus an infinity of the same order, if N is 

 increased indefinitely, and the N-gon, the inscribed circle, 

 and the circumscribed circle all tend to the same geo- 

 limit — the absolute. 

 I he fact that the cuspidal edge of the surface of rota- 

 tion of the tractrix forms a line of discontinuity in this 

 representation, and that none of the types of surfaces "I 

 constant negative curvature exactly images the hyperbolic 

 plane in the properties belonging to analysis sinis, appears 

 to be no objection to hyperbolic geometry. An 1 

 similar difficulty occurs in the representation of elliptic 

 try, since there is no continuous surface of constant 

 v- curvature on which two geodesies have i"i! one 

 point of intersection. Geometry has become entirel) 1 

 er of postulation ; but, at ihe same time, it is of 

 interest to observe that the non-Euclid 



of being truly repi 1 v. ithin :> restricted 



Euclid iii space. 



I ). M. Y. SoMMERVIl I I ■ 



The University, St. Andrews, September 30. 



NO. 2l88, VOL. 87] 



Elements of Comet 1911/. 

 From M. Ouenisset's observation of September 23, and 

 ray own of September 20 and 30, 1 obtain lie following, 

 approximate elements : — 



T =1911 November 12-67 



w=I23 



a = 35- 306' 



i = 102 i9 - 3 



i -'11111. 



The comet is now 

 ing the sun. and then' is no reason to expect much in- 

 crease in brilliancy. The only point of interest is that 

 when at tie descending node on December id it will be 

 about half a million miles outside the earth's orbit. The 

 difference of the heliocentric longitudes of the earth and 

 comet will, however, be 132°, so that no near approach is 

 possible. J. B. Dale. 



Craigness, New Maiden, Surrey, October 3. 



Rainfall in the Summer of 1911 and of 1912. 



Has Mr. MacDowall the courage to apply his own 

 experience, to which he refers in Nature of September 2S, 

 to "supply long-range forecasts of months, seasons, &c"? 

 Will he publish in advance a forecast for the winter 

 1911-12 or for the spring and summer of 1912, such as 

 he considers could have been done for the summer of 

 1911? Or is it only after the event that he can discover 

 what points in the past have to be considered and in what 

 grouping they have to be compared in order to yield an 

 a posteriori " forecast "? Hugh Robert Mill. 



62 Camden Square, London, N.W., October 2. 



Miniature Rainbows. 



When returning one day in August of last year from 

 the Fame Islands to Berwick in a pleasure steamer, I was 

 standing in the bow of the boat, and was much struck by 

 the display of a permanent rainbow in the spray that was 

 thrown up. The rainbow was inverted, the result, pre- 

 sumably, of my position above it. The sea was very 

 rough, and thus the spray was constant. 



Edward A. Martin. 



2S5 Holmesdale Road, South Norwood, S.I'.. 



THE STONE AGES OF SOUTH AFRICA. 1 



THE papers in this volume are a very full and 

 important addition to the work already published 

 by Mr. J. P. Johnson; but it is doubtful whether they 

 bring us any nearer to a solution of one of the most 

 interesting' questions connected with archaeological or 

 palajontological discoveries in South or Central Africa 

 — namely, the approximate age to which the existence 

 of man can be traced back in South Africa, East 

 Africa, the Congo basin, West Africa, and the Sudan. 

 Though Dr. Peringuey would seem, from one or two 

 phrases, to lean to the theory of a very ancient dale 

 for the human colonisation of tropical Africa, he has 

 to admit repeatedly that so far no cogent evidence 

 has been produced in the shape of geological features 

 associated with the finds of human remains or imple- 

 ments to indicate, as positively as is the case in 

 Europe and Asia, the period in the earth's history 

 with which such remains are to be associated. 



As our knowledge advances towards perfection, as 

 we become better and better able to read that new 

 Bible, the book of the Earth itself, we may have to 

 revise our estimate of the ages of the hitherto dis- 

 covered prehistoric, palaeolithic, and eolithic human 

 1. mains in Europe and Asia. Still, there can be little 



1 Anna Ktri ra Mu eum : vol. viii., part 1, containing the 



Stone Ages of Si uth Africa, &c. Bj Dr. I . Pennguey, with furthi 



tributions by Mr. A. L. Du Toil and Dr. !•'. C. Shrubsall. (London: 



Pri Ii e I'ri es of the South African Museum by West, Newman 



1 



