84 



NATURE 



[April 4, 19 18 



paratively brief report, and does not depreciate 

 the efforts of the various contributors. 



As outstanding- features in the various reports, 

 the following- may be mentioned. Considerable 

 space is devoted to the Bragg method of investi- 

 gating crystals by means of X-rays, both by Dr. 

 Dawson (general and physical chemistry) and by 

 Mr. T. V. Barker (crystallography and minera- 

 logy). An interesting discussion on phosphor- 

 escence is included in Prof. E. C. C. Baly's report 

 (inorganic chemistry). Prof. J. C. Irvine con- 

 tributes a very readable account of the year's 

 researches on the aliphatic organic compounds, 

 whilst homocyclic compounds are dealt with by 

 Dr. F. L. Pyman, and heterocyclic compounds by 

 Dr. A. W. Stewart. More than half of Prof. 

 F. G. Hopkins's report (physiological chemistry) 

 is devoted to the important subjects of "The Alka- 

 line Reserve of the Body " and "Some Aspects of 

 Nutrition." Dr. E. J, Russell writes on the year's 

 agricultural chemistry in his customary lucid 

 manner and emphasises the value of the present 

 tfo-operation between farm and laboratory. 



E. H. 



LETTERS TO THE EDITOR. 

 [The Editor does not hold himself responsible for 

 opinions expressed by his correspondents. Neither 

 can he undertake to return, or to correspond with 

 the writers of, rejected manuscripts intended for 

 this or any other part of Nature. No notice is 

 taken of anonymous communications.] 



A Proof that any Transfinite Aggregate can be 

 Well-ordered. 



The following sketch of a proof which seems to me 

 to ibe not wholly unimportant is given here for certain 

 reasons of priority. I hope that this short account is 

 not unintelligible. 



Hartogs's (Math. Ann., Ixxvi., 19 15, 438-43) con- 

 siderations may be generalised without difficulty to an 

 investigation of the consequences of the existence of a 

 least ordinal number which is greater than the ordinal 

 types of a'l possible well-ordered series that can he 

 constructed out of a given aggregate M. This con- 

 sideration throws no light on whether or not any one of 

 these series actually exhausts M, unless we assume 

 that of two different cardinal numbers one is greater 

 than the other. Instead of using Hartogs's method, I 

 consider all those parts of M which can be well- 

 ordered, well-order them in all possible ways, so that 

 they form what may be called for shortness "chains 

 of M " (so that the same part in different orders gives 

 different "chains"), and imagine as put on one side 

 all chains which are "segments," in Cantor's sense, 

 of other chains of M. 



At this point we must Introduce a definition : Given 

 a chain (K) of M, let us say that a class K' of chains 

 of M is a " class of direct continuations of K " if each 

 member of K' has K as a segment, and also, if L 

 is any member of K' of type A, those members of K' 

 which are of type less than A are segments of L. 

 Such a class K' evidently defines one chain and not a 

 class of independent chains, such as Hartogs considers. 



Now, in the above process of imagining, we do in 

 fact have a remainder of chains which are not seg- 

 ments of others ; for, if not, all chains of M would be 



NO. 2527, VOL. lOl] 



segments of other chains of M, and then we could 

 show indirectly that for any such chain K, any ordinal 

 number 7, however great, and any class K' of direct 

 continuations of K, there is a segment of K' of type 7. 

 In fact, if there were not such a segment, there would 

 be at least one definite example of each of 7, K, and 

 K', such that no segment of K' is of type 7; and 

 thence we can easily show that not every chain of M 

 is a segment of others. But we can prove {Phil. Mag. 

 (6), vii., 1904, 61-75) that there is no series which 

 has segments of any ordinal number 7, however great. 

 Thus there is at least one chain of M which is not a 

 segment of some other. It is easy to prove that this 

 chain exhausts M, and that there is a least type of 

 those of chains that exhaust 'M. Thence, from the 

 fact that the cardinal number of M is an Aleph,- we 

 can deduce Hartogs's theorem, determine the form of 

 the limit that Hartogs was really trying to find, and 

 prove Zermelo's (Math. Ann., lix., 1904, 514-16; lxv.„ 

 1908, 107-28, 261-81) "principle of selection." 



Philip E. B. Jourdain. 

 The Bourne, Basingbourne Road, Fleet, Hants, 

 March 12. 



Future Supplies of Laboratory Apparatus and 

 Materials. 



I HAVE been looking at my list of apparatus and 

 materials which the chemical dealer tells me must 

 wait until the war is over before they can be obtained 

 from Germany. I regret to say the list is a formidable 

 one ; I had to add to it this week. Few in our genera- 

 tion will ever knowingly purchase goods made in Ger- 

 many if they can be obtained from other countries. 

 We feel that German goods must appear to be smeared 

 with the blood of our relatives and countrymen. I 

 take it that my position is much the same as obtains 

 with the heads of other laboratories in the country. 

 Surely, therefore, it is time our British manufacturers 

 realised that it is not much use tinkering with labora- 

 tory glass and porcelain ware, if the thousand-and-one 

 other forms of laboratory apparatus have to be pur- 

 chased in Germany after the war. It seems reasonable 

 to suppose that the orders for laboratory glass and 

 porcelain ware are bound ultimately to accompany the 

 orders for the other requisites. X. Y. Z. 



Long-range Guns. 



By a slip of the pen, double velocity was said, in- 

 my article in last week's Nature (p. 65), to give 

 double range, instead of fourfold. 



At that rate, an increase of the velocity of our gun 

 in 1887 would be required from 2400 to 6000 ft. per 

 sec. to make the range grow from 12 miles to 75. 



The rule is, of course, not exact except when air 

 resistance is not taken into account. The 12-mile 

 range would have been nearly trebled if it was not 

 for the resistance of the air. 



G. Greenhill. 



I Staple Inn, W.C.i, March 30. 



LONG-RANGE GUNS. 



THE appearance of a gun with a range of 

 something like seventy or eighty miles has 

 naturally aroused considerable interest, and the 

 question is often asked as to how such long" 

 rang-es are attained. The answer is that if the 

 shot is to travel far it must get outside the atmo- 



