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NA TURE 



\AugMst 8, 1889 



have bad in the process has made me more alive to these 

 dangers than even he is. I need only say here that I have not 

 yet come across a case where I should feel warranted in stating 

 that a break existed on the evidence of one curve only where 

 the break depended on differentiation for being clearly visible. 

 In my own work I have never considered any breaks as being 

 niore than "suggested " unless they were shown by at least two 

 liifferent properties of the substance under examination ; the 

 majority of the breaks which I insist on are shown by more than 

 two, in some cases by as many as seven different properties. 



As to the examination of the curves by means of empirical 

 formulae, nothing of the sort has been done, and it is difficult to 

 understand how Prof. Lodge, even though he speaks under cor- 

 rection, should have so misunderstood the methods adopted. 

 If Mendeleeff's paper may have been open to misinterpretation, 

 Crompton's certainly was not, for he gives in a tabular form the 

 results of the direct differentiation of the experimental numbers 

 themselves ; an abstract only of my own paper has as yet 

 appeared, and I have not got it by me to refer to, but I do not 

 think that the terms "formula" or "equation" occurred 

 throughout it. The impossibility that seems to exist of getting 

 either chemists or physicists to understand that the method of 

 examining curves which we have employed does not involve the 

 use of any equation at all is indeed extraordinary. My own 

 opinion on the use of equations will be best illustrated by the 

 following extract from my paper : — "It is necessary to say a 

 word at starting to correct a'l erroneous opinion which is pre- 

 valent as to the method of examining curves which I have 

 adopted. ... It is imagined by many that this method con- 

 sists in fitting sundry equations to the curves, and, on the 

 strength of their concordance with these equations, to conclude 

 that they are continuous or otherwise. Now, it is quite true 

 that if a curve differentiates into a straight line after a certain 

 number of differentiations, an equation of a certain form must 

 represent that curve, and if it yields several straight lines there 

 must be as many different equations applicable to different parts 

 of it ; but it is one thing to find equations empirically, and 

 prove (?) their truth by a display of those most fallacious of 

 arguments known as tables of ' found ' and ' calculated ' 

 values, and another thing to apply an ordinary process of 

 mathematical analysis to the curves, letting them speak for 

 themselves, and tell us whether they are continuous or not. 

 On the former of these methods 1 would place absolutely 

 no reliance, and so far have I been from making use of 

 it, that I have not found the equation for any single curve 

 here depicted, and have purposely avoided finding any. 

 The mathematical argument on which this work depends is, 

 that a curve, if it be continuous, will on differentiation give 

 either a straight line or another continuous curve, whereas, if it 

 be not continuous, but be made up of different curves, will yield 

 on differentiation a series of straight lines or curves. This, I 

 think, is an incontestable fact." 



That the majority of chemists are not mathematicians I 

 willingly admit ; this painful fact is shown only too clearly by 

 their blind acceptance as gospel truth of everything which is 

 "proved" mathematically. But Prof Lodge must do us the 

 justice to admit that we have occasionally some glimmers of 

 common-sense, glimmers which would be inconsistent with our 

 assuming that a certain curve was a parabola, and then being 

 pleased, or even surprised, that it behaved after the manner of 

 parabolas. 



However much I may envy the powers of a mathematician, 

 and however firmly I may believe that chemical facts will 

 eventually be translated into mathematical expressions, I feel 

 that at the present day the introduction of mathematical formulae 

 into chemistry almost invariably involves the exclusion of com- 

 mon sense. It is curious that Prof. Lodge's letter should have 

 been immediately followed by an article on chemical affinity, 

 which, I think, will be found to give a striking illustration of 

 this dictum. What may be termed the x and y theory of 

 chemical action, studied on paper by Guldberg and Waage, and 

 followed up in the laboratory by Ostwald, has led unfortunate 

 chemists into a labyrinth of cumbrous mathematical expressions 

 for erroneous facts, where the common-sense of BerthoUet would 

 have given them a simple explanation of all the true facts of the 

 case (see Trans. Chem. Soc, 1889, 26). 



Harpenden, July 22. Spenxer Pickering. 



P.S. — Since writing the above I have obtained the most 

 absolute justification of my method of differentiation which could 

 possibly be obtained. 1 have isolated in the solid crystalline 



form a new hydrate of sulphuricfacid, the existence of which I 

 had predicted from an examination of the density and heat 

 results of solutions of the acid. A few further details on the 

 subject will, I believe, be foitnd in the last issue of the Chemical 

 News. S. P. 



Ilfracombe, August 4. 



PHOTOGRAPHIC STAR-GAUGING. 



T"* HE mere equal-surface counting of the stars visible 

 -*• with the same instrument in ditTerent sections of the 

 sky gives results open to misinterpretation. Admirable 

 in itself, the method fails because it encounters what we 

 may call "systematic errors" in the distribution of the 

 stars. With incidental anomalies it is fully competent to 

 deal ; they should, on a large average, be mutually com- 

 pensatory ; but it breaks down before the clustering 

 tendency which pervades, more or less markedly, the 

 entire sidereal system. Not only are certain parts of 

 space more crowded than others, but the crowded parts 

 are related according to an obvious plan. They do not 

 occur casually. Their effect is then heightened, instead 

 of being eliminated, by multiplied observations. 



The present resources of science, however, seem to 

 ofifer the means of discriminating, to some extent, be- 

 tween real crowding and the simple extent of star-strewn 

 space. Although the total number of the stars visible in 

 each case with the same telescope might be precisely the 

 same, their relative numbers, counted by magnitudes, 

 would in all probability be very different. In a stratum, 

 supposing the distribution of the stars equable, and their 

 size uniform, their numbers should be nearly quadrupled 

 at each descent of a magnitude. This of course is an 

 ideal law of progression which we cannot expect to find 

 anywhere ^strictly obeyed ; but even approximate con- 

 formity to it must be held to indicate with tolerable 

 certainty that the lessening ranks of the stars are, on the 

 whole, at distances from us corresponding with their 

 light. Now it is approximately conformed to by the 

 stellar multitude down to about 8"9 magnitude over the 

 general expanse of the sky, as well as over the zone of 

 the Milky Way. But in that zone, stars of the ninth and 

 higher magnitudes very much exceed their due numerical 

 proportions ; in other words, they are physically, no less 

 than optically, condensed. 



From these circumstances two very important infer- 

 ences may be derived : first, that the lower margin of the 

 galactic aggregations lies at a distance from us corre- 

 sponding roughly to the mean distance of a ninth magni- 

 tude star, costing light some fourteen hundred years of 

 travel ; next, that the aggregated objects are average 

 stars, neither larger nor smaller than those in our nearer 

 neighbourhood. Both conclusions seem inevitable should 

 the facts turn out, on closer investigation, to be as above 

 stated. A regular increase in the numbers of the suc- 

 cessive photometric orders of stars, tallying with the in- 

 creased cubical contents of the successive spheres of 

 which the radii are the theoretical mean distances of 

 those same orders, affords strong, if not demonstrative, 

 evidence of a corresponding real penetration of space. ^ 

 And since the sequence continues unbroken down just to 

 the ninth magnitude, we see that the galactic condensa- 

 tions of ninth magnitude stars cannot be situated nearer 

 to us than their brightness would lead us to suppose — 

 cannot, in other words, be stars on a lower than the_ 

 ordinary level of lustre. 



It is tolerably certain, however, that the denser star 

 clouds of the Milky Way lie far beyond ninth magnij 

 tude distance. The ground for this assertion is not thtf 

 apparent minuteness of their components, but the singulaj 

 fact, adverted to by Argelander, that, in the divider 

 Milky Way, running from Cygnus to the Centaur, the 



' The idea of determining distance by distribution seem'; to have presente^ 

 itself to Dr. Gould in 1874. See American Journal 0/ Science, vol. viii 



