FEliRUARV 25, 1892J 



NATURE 



407 



experimental truths? But we do not experiment on straight 

 lines or ideal circles ; only material objects can be dealt with. 

 On what would depend, then, the experiments which would 

 serve to found a geometry ? The answer is easy. 



We have seen above that one argues constantly as if 

 geometrical figures behaved like solids. That which geometry 

 would birrow from experience is therefore the properties of 

 these bodies. 



But a difficulty exists, and it cannot be overcome. If 

 geometry were an experimental science, it would not be an exact 

 science — it would be liable to a continual revision. What do I 

 say? It would from today be convicted of error, since we 

 know that a rigorously invariable solid does not exist. 



Geometrical axioms, therefore, are neither synthetic a priori 

 conclusions nor experimental facts. 



They are conventions : our choice, amongst all possible con- 

 ventions, is guided by experimental facts ; but it remains free, 

 and is only limited by the necessity of avoiding all contradiction. 

 It is thus that the postulates can remain rigorously true, even 

 when the experimental laws which have determined their 

 adoption are only approximate. 



In other words, axioms of geometry (I do not speak of those of 

 arithmetic) are only definitions in disguise. 



This being so, what ought one to think of this question : Is 

 the Euclidian geometry true ? 



The question is nonsense. 



One might just as well ask whether the metric system is true 

 and the old measures false ; whether Cartesian co-ordinates 

 are true and polar co-ordinates false ; whether one geometry 

 cannot be more true than another — it can only be more con- 

 venient. 



Now, Euclidian geometry is, and will remain, the most 

 convenient : — 



(i) Because it is the simplest ; and it is not so simply on 

 account of our habits of thought, or any kind of direct intuition 

 which we may have of Euclidian space ; it is the most simple 

 in itself in the same way as a polynomial of the first order is 

 simpler than one of the second. 



(2) Because it agrees sufficiently well with the properties of 

 natural solids, those bodies which come nearer to our mem- 

 bers and our eye, and with which we make our instruments of 

 measurement. 



Geometry and Astronomy. — The above question has also been 

 stated in another way. If the geometry of Lowatchewski is 

 true, the parallax of a very distant star would be finite ; if that 

 of Riemann be true, it would be negative. Here we have 

 results which seem subject to experience, and it has been hoped 

 that astronomical observations would have been able to decide 

 between the three geometries. 



But what one calls a straight line in astronomy is simply the 

 trajectory of a ray of light. If then, as is impossible, we had 

 discovered negative parallaxes, or shown that all parallaxes 

 are greater up to a certain limit, we should have the choice 

 between two conclusions : — 



VVe could renounce Euclidian geometry, or modify the laws of 

 optics, and admit that light is not propagated strictly in straight 

 lines. 



It is useless to add that everyone would regard the latter 

 solution as the more advantageous, 



Euclidian geometry, then, has nothing to fear from new 

 experiments. 



Let me be pardoned for stating a little paradox in con- 

 clusion : — 



The beings which had minds like ours, and who had the 

 sanve senses as we have, but who had not received any previous 

 education, might receive conventionally from an exterior world 

 choices of impressions such that they would be led to construct 

 a geometry different from that of Euclid, and to localize the 

 phenomena of this exterior world in a non-Euclidian space, or 

 even in a space of four dimensions. 



For us, whose education has been formed by our real world, 

 if we were suddenly transported in this new one, we should not 

 have any difficulty in referring the phenomena to our Euclidian 

 space. 



j- Anyone who should dedicate his life to it could, perhaps, •■ 

 'eventually imagine the fourth dimension. 



I fear that in the last few lines I have not been very clear. I 

 can only be so by introducing new developments ; but I have 

 already been too long, and those whom these explanations might 

 interest have read their Helmholtz. 



NO. I165, VOL. 45] 



Desiring to be brief, I have affirmed more than I have proved : 

 the reader must pardon me for this. So much has been written 

 on this subject, so many different opinions have been put 

 forward, that the discussion of them would fill a volume. 



W. J. L. 



SOCIETIES AND ACADEMIES. 

 London. 



I Royal Society, February 11.— "The Rdle played by Sugar 



I in the Animal Economy : Preliminary Note on the Behaviour of 



I Sugar in Blood." By Vaughan Harley, M.D. 



This communication was to show that the causes why the 

 whole amount of added sugar can seldom be recovered from blood 

 are threefold. Firstly, the imperfections in the as yet known 

 methods of analysis. Secondly, the different ways in which the 

 albumens of the blood behave themselves while coagulating; some 

 coagulating in the form of firm clots, which retain the saccharine 

 matter in their interstices, rendering it impossible to extract all 

 the sugar from them by washing ; others separating as loose, 

 flocculent curds, from which the sugar can be regained with 

 comparative facility. While, thirdly, as bacteria were distinctly 

 ascertained to have nothing to do in the matter, and yet the loss 

 of the sugar added to the blood is in every instance distinctly pro- 

 gressive—according to the period of time the sugar is left in 

 contact with the blood before the analysis is begun — Dr. 

 Vaughan Harley considered himself justified in saying that there 



I must exist in the normal blood itself a sugar-transforming agent. 

 This he described as an enzyme ; but refrained from going 

 into any further particulars regarding it until his researches upon 

 the subject are more advanced. 



He gave tables of the results of his experiments, and compared 

 them with those recently published by Schenk, Rohmann, and 

 Seegen ; showing that while the percentages of the sugar 

 regained by the first observer ranged from 20 to 55 per cent., and 

 those recovered by the two last experimenters fluctuated between 

 80 and 96 per cent., in his three different series of experiments, 

 where different methods of analysis were employed, the percent- 

 ages of the added sugar regained ranged respectively between 85 

 and 100 ; 92-9 and 99*3 ; and 947 and 99*9 per cent. 



Mathematical Society, February 11.— Prof. Greenhill, 

 F.R.S., President, in the chair. — The following communica- 

 tions were made : — On the logical foundations of applied 

 mathematical sciences, by Mr. Dixon. He maintained the 

 importance of distinguishing in all sciences between what is 

 dependent on verbal conventions and what is not. He thus 

 distinguished between that part of the meaning of a term which 

 is laid down as its definition, and the part which remains to be 

 discovered as a consequence of the definition. So also sciences 

 might be divided into purely symbolic sciences, which being 

 based on definitions alone conveyed no real information ; sub- 

 jective sciences, which deal with concepts and objective 

 sciences, which deal with actual things. He then stated the 

 conditions under which a set of assertions might be arbitrarily 

 laid down as the definition of a term ; and applied these condi- 

 tions to show that Newton's three laws of motion could be 

 regarded as a definition of the term force, that if this was 

 done there could no longer be any discussion as to whether or 

 not force alone is sufficient to account for the movements of 

 matter. The anomaly that we are apparently able to determine 

 directions absolutely, though we can determine positions only 

 relatively, was explained, and a formal proof of all the 

 elementary theorems of mechanics, including the principle of 

 virtual work, might be deduced. — Note on the inadmissibility 

 of the usual reasoning by which it appears that the limiting 

 value of the ratio of two infinite functions is the same as the 

 ratio of their first derived, with instances in which the result 

 obtained by it is erroneous, by Mr. Culverwell. — On Saint 

 Venant's theory of the torsion of prisms, by Mr. A. B. 

 Bass-!, F,R.S. 



Dublin 

 Royal Society, January 20,— Prof. W. N. Hartley, F.R.S., 

 in the chair. — Reports on the zoological collections made by 

 Prof. Haddon in Torres Straits, 1888-89 : the Hydrocorallinae, 

 by S. J. Hickson. The specimens described are a female stock 

 of Stylaster gracilis, Distichopora violacea, and Millepora 

 Murrayi. Some of the smaller colonies of Distichopora are 

 bright orange in colour, others vandyke brown, and the larger 

 ones are deep purple with pale yellowish tips. The author 



