NATURE 



[November 2, 191 1 



Non-Euclidean Geometry. 



As many mathematicians give very little thought to the 

 theory of sets, it is perhaps worth while dwelling for a 

 moment on Dr. Sommerville's possibly misleading re- 

 marks in Nature of October 5. He, quite correctly, points 

 out the one-one correspondence between the aggregates of 

 integral numbers i, 2, 3, &c. (n), and even numbers 2, 4, 

 6, &c. Can). Thus the part appears equivalent to the whole. 

 This statement loses the character of a paradox to all 

 who will bear in mind that the notion of " the part cannot 

 equal the whole " has its origin in the contemplation of 

 finite quantities. 



Again, the sets of numbers of the form 4n and 4n+2 

 constitute aggregates of the same type ; they are equivalent 

 to each other as well as to n itself and 2n. I fail to see 

 that Dr. Sommerville shows the part to be larger than 

 the whole. An aggregate A would be larger than A' if 

 A contained a set equivalent to A', while A' does not 

 contain a set equivalent to A. 



Harold M. Sadow-Pittard. 



SS. Caledonia, October 14. 



In drawing a parallel between Legendre's proof and the 

 paradox concerning infinite aggregates, I had not thought 

 it necessary to point out the fallacy, which consists, as in 

 Legendre's proof, of transferring to infinities notions which 

 are derived from a study of finite magnitudes. 



Of the two propositions, a part is (i) equal to, and (2) 

 greater than, the whole ; the one is just as much a paradox 

 as the other until the meaning of the terms equal, greater, 

 and less has been extended and modified for infinite aggre- 

 gates ; and the proofs which I gave are equally in accord- 

 ance with notions derived from finite aggregates. Neither 

 of them was intended as a valid proof, though the first 

 happens to be in agreement with the usual extension of 

 the meaning of equivalent. D. M. Y. Sommerville. 



The University, St. Andrews, October 23. 



Dew-ponds and the Dry Season, 



With reference to the remark in Nature of October 26 

 (P- 559)' " paid a visit to the Chanctonbury Dew-pond 

 about the end of last August, and was surprised at its 

 flourishing condition. 



The water-level was, of course, much below the normal, 

 and the surrounding water plants were much trodden under 

 by sheep and cattle. There still remained, however, a good 

 fringe, and the usual pond plants seemed to be in a satis- 

 factory state. I could not get to the water's edge owing 

 to the moist and boggy nature of the ground, usually sub- 

 merged. 



Some other dew-ponds met with in the course of a few- 

 days' walk west of Chanctonbury were quite dry, so far 

 as I remember. J. P. Clatworthy. 



University College, Reading, October 30. 



CHARLES DARWIN'S EARLIEST DOUBTS 

 CONCERNING THE IMMUTABILITY OF 

 SPECIES. 



TN view of the great revolution in scientific thought 

 •■■ which was inaugurated by the publication of the 

 ''Origin of Species," the story of the evolution of 

 ideas in the mind of its author must always have a 

 deep fascination for the student of the history of 

 science — and the question of the nature of the initial 

 stage of that evolution is one especially worthy of 

 attention. 



In his autobiography, Charles Darwin has declared 

 his belief that, before leaving England for the memor- 

 able voyage in the Beagle, he was quite indifferent to 

 any speculations upon the subject of evolution — and 

 this in spite of his admiration for his grandfather's 

 '' Zoonomia " as a literary production.' Now concern- 

 ing the exact period in his life when Darwin ceased 

 to feel this indifference, and had his interest aroused 

 by that "mystery of mysteries "—to the solution of 



1 " Life and Letters of Charles Darwin," vol. i., p. 38. 

 NO. 2192, VOL. 88] 



which his whole after-life was to be devoted — thti 

 have been very marked differences of opinion. 



Huxley stated his conviction to be that no reali\ 

 important fruits 6f the observations made during the 

 voyage of the Beagle could have been gathered by 

 the ardent but untrained young naturalist until after 

 he reached England, and had the opportunity of con- 

 sulting specialists concerning the specimens which had 

 been sent home by him from time to time.* But, on 

 the other hand, Dr. Francis Darwin and Prof. Seward 

 maintain that during the voyage, and especialh 

 towards its close, when the Galapagos Islands w- 

 visited, Darwin's observations and his meditatiui 

 upon them had already begun to bear fruit, and had 

 led him to lose his absolute faith in the immutability 

 of species.' I am myself convinced, as the result - 

 a careful consideration of letters written at the tin. 

 that very early indeed in the course of the voyage 

 certain observations and reflections had given rise in 

 Darwin's mind to serious misgivings concerning th« 

 fixity of species, although, writing nearly fifty year- 

 afterwards, he dismissed them lightly as nothing 

 more than "vague douhts.'" 



It was this statement in Darwin's corresponden' 

 to which Huxley appears to have attached very gn . 

 importance. It occurs in a letter to Dr. Zacharia- 

 and is as follows : — 



"When I was on board the Beagle I believed in th- 

 permanency of species, but as far as I can rememb' 

 vague doubts occasionally flitted across my mind.'' ' 



It should be borne in mind, however, that as th' 

 lines were written as late as 1877, to one of his ver 

 numerous casual correspondents, we may not improl - 

 ably infer that Darwin penned them somewhat hasti!\ 

 and without any deep thought or reflection concernir 

 the interpretation that might be put upon them 

 published. For it must be remembered that nearl\ 

 twenty years before this he had written and printed 

 the following : — 



"When on board H.M.S. Beagle as naturalist, I 

 was much struck with certain facts in the distribution 

 of the inhabitants of South America, and in the 

 geological relations of the present to the past inliabi- 

 tants of the continent." ' 



Seeing that these words form the first sentence < 

 the introduction to the "Origin of Species," and mi: 

 therefore have expressed the result of very deliberao 

 thought and consideration, that they would certainly 

 have been frequently scanned by the author befc 

 publication, and that they are repeated without chan^ 

 or qualification in every succeeding edition of t;.. 

 book, it is undoubtedly only fair to attach far greater 

 weight to them than to a sentence hastily indited to 

 a casual correspondent so many years afterwards. 



In that marvel of candid introspection, the "Auto- 

 biography," Darwin wrote in 1876 as follows : — 



" During the voyage of the Beagle I had been deeply 

 impressed by discovering in the Pampean formation 

 great fossil animals covered with armour like that o^ 

 the existing armadilloes." * 



And this he enumerates as the first, though not tr. 

 most important, of the observations which turned his 

 thoughts in the direction of evolution during the 

 vovage. When writing to Haeckel in 1S64 he says : — 



"/ shall never forget my astonishment when I dug 

 out a gigantic piece of armour like that of the arma- 

 dilloes." ^ 



Owing to a singular blunder, for which Darwin 

 was in no way responsible, I shall have to point out that 



2 " Collected Essays," vol. ii., p. 271. 



•* "More Letters of Charles Darwin," vol. i 

 " Foundations of the Origin of "Species." p. xv. 



* " More Letters of Charles Darwin," vol. i., p. 

 and succeeding passages are our own. 



5 " OriEin of Species, "p. i. 6 "Life and Letters," vol. i., p. 82 



7 " History- of Creation," vol. i., p. 134. 



PP- 37-39- ?ee also 

 67. The italics in this 



