December 9, 1915J 



NATURE 



397 



I- thereby determine an area, and we express this 



lortly by saying: "Area is derived from length." 



I his is all we mean by the conventional term " deriva- 



;i)n," and in stating the dimensions of a derived 



aantity we do not make use of any hypotheses. 



Now there is no known process by which, having- 



mailable only standards of mass, length, and time, 



.<■ can fix and reproduce any temperature such as the 



e point. To do that we require something more — 



r instance, that the mass shall be a mass of some 



particular substance having other properties than mere 



inertia, some one of which may serve as a fourth 



standard. There is no uncertainty in answering the 



question referred to at the beginning of this section ; 



whatever Maxwell's demons might do, we cannot 



derive temperature from any three purely mechanical 



magnitudes. There can therefore be no doubt of the 



validity of Lord Rayleigh's deduction. 



(3) Though the question suggested by M, Riabouch- 

 insky's note is thus answered immediately by an appeal 

 tt) facts, it may not be amiss to add a few words for 

 those who have fallen into the habit of setting propor- 

 tionality constants equal to unity and then forgetting 

 all about them. 



If we accept the molecular theory, the information it 

 affords on the subject now in hand is that the numerical 

 value of any temperature, on Kelvin's scale, is propor- 

 tional to the mean molecular kinetic energy of an ideal 

 gas which is at that temperature. We may describe 

 this relation by writing 6/0o = T/To, in which T and 

 To are the molecular kinetic energies at the tempera- 

 tures Q and 6„ respectively. Both members are pure 

 ratios, and it is obvious that the equation does not 

 furnish any dimensional relation between B and T; 

 and yet this equation embodies all the knowledge 

 which the molecular theory affords on the matter under 

 discussion. To say that the molecular theory authorises 

 us to "define" temperature as the mean kinetic 

 energy of the molecules, would be quite on a par with 

 saying that a peach may be defined as a shilling 

 because the number of peaches we can buy is propor- 

 tional to the number of shillings we spend upon them, 

 and, in some states of the market, not only propor- 

 tional but equal. On our ordinary scale, an interval 

 of time is proportional to the angle through which the 

 arth rotates during that interval ; but no one thinks 

 I saying that we may define time as angle, or 

 ui assigning to time the dimensions of angle. Propor- 

 tionality of numerical values does not imply qualitative 

 identity. 



As Lord Rayleigh remarks : — " It would indeed be 

 a paradox if the further knowledge of the nature of 

 heat afforded by molecular theory put us in a worse 

 position than before in dealing with a particular 

 problem." In reality, the worse position in which 

 M. Riabouchinsky suggests that we place ourselves, 

 would be due not to utilising further knowledge but 

 to ignoring what we already have. 



(4) Cases do occur, though the foregoing is not one 

 of them, in which it seems doubtful, at first sight, 

 how many independent units we ought to use. Such 

 a doubt may arise when we ask ourselves if we ought 

 not to use the law of gravitation to eliminate one of 



ur three mechanical units, or the constancy of the 

 leed of light to derive time from length. The dis- 

 ussion of this subject, which involves the question 

 how we are to interpret "universal constants," must 

 be postponed to a future occasion, but the following 

 hint may be given of the conclusion to which such a 

 discussion will lead. 



Suppose that we have n independent simultaneous 

 equations, involving n + fe quantities, and that we re- 

 duce them to a single equation. Each equation repre- 

 sents a single known fact, and when a given equation 

 has been used once, there is nothing further to be 



NO. 2406. VOL. 96] 



gained by using it again ; for only a formal and not 

 a real change in the result can be thus produced. If 

 one pf the quantities is known to be constant, it may 

 be removed from the list of variables before starting 

 the reduction. But as regards the final result, it is 

 immaterial whether the constancy of a particular quan- 

 tity is recognised explicitly at the start or not until 

 the end ; the conclusion to be drawn regarding the 

 quantities which do vary is the same in either case. 



If, for example, the phenomenon under considera- 

 tion involves the operation of the law of gravitation, 

 as in Lord Rayleigh's problem of the vibration of 

 liquid globe (Nature, March i8), one of the facts of 

 the problem is expressed by the equation f = ymm/r-. 

 We may treat the gravitation constant 7 as one of the 

 physical quantities involved in the problem, and use 

 this equation to find its dimensions f7] = [m-'Pf-^] ; 

 or we may treat 7 as a pure number and use the 

 equation to eliminate one fundamental unit by setting 

 [m-'Ff-2] = fi] ; but we cannot do both. The final 

 result Is in either case that given by Lord Rayleigh. 



E. Buckingham. 



Washington, November 23. 



Grime's Graves Flint Mines. 



Prehistoric archaeologists will be grateful for the 

 excellent account given In N.ature of November 18 of 

 the report recently published by the Prehistoric Society 

 of East Anglia on the excavations conducted in 1914 

 at Grime's Graves, Norfolk. It is evident that your 

 reviewer regards the flint implements found at this 

 site as referable to the Neolithic period, and while this 

 view may possibly be correct, the present writer is of 

 the opinion that a close and dispassionate study of the 

 specimens recovered, and of the exhaustive report pre- 

 pared by Mr. Reginald A. Smith, will not tend to 

 foster any feeling of certainty on this point. 



The question of the age of the flint implements 

 found at Grime's Graves is of great importance, and 

 can only be fully and adequately dealt with by experts 

 in prehistoric archaeology. The contributor of the article 

 in Nature is evidently a geologist, and I venture to 

 enter a protest against his taking an authoritative part 

 in the discussion on a technical subject altogether 

 outside the realm of geology. Unfortunately, It does 

 not seem to be generally recognised that the study of 

 flint implements is of a highly complex and dh'ricult 

 nature, requiring as much, if not more, detailed know- 

 ledge than is required in many otTier sciences. The 

 geologist would object, and rightly so, to a prehlstorian 

 giving an authoritative opinion upon a question of 

 geology; the archaeologist simply asks for a like 

 immunity from inexpert criticism of his particular sub- 

 ject. Your reviewer has every right to give an opinion 

 on the geological problems presented by the excava- 

 tions at Grime's Graves, and there can be little doubt 

 but that his opinions must carry weight. But 

 the flint implements present a problem that can only 

 be discussed with any profit by experts in prehistoric 

 archaeology. J. Reid Moir. 



12 St. Edmund's Road, Ipswich. 



I I inferred from the report of the "experts in pre- 

 I historic archaeology," that if the various flint imple- 

 ments met with at Grime's Graves had been found 

 { separately in different localities, they would have been 

 I referred " authoritatively " to several successive periods 

 I of human culture. To aid them in dealing with this 

 ! strange admixture of supposedly distinct industries, 

 ' I merely pointed out that the geological evidence, so 

 I far as discovered, is perfectly harmonious and con- 

 I elusive, showing that the deposits cannot be older than 

 I the Neolithic period. A. S. W. 



