August 15, 1878] 



NATURE 



409 



a^pregate, and observing the tints which they scatter laterally 

 when clustered in the form of actinic clouds.^ The small bodies 

 with which experimental science has recently come into contact 

 are not confined to gaseous molecules, but comprise also complete 

 organisms ; and the same philosopher has made a profound study 

 of the momentous influence exerted by these minute organisms in 

 the economy of life.^ And if, in view of their specific effects, 

 whether deleterious or other, on human life, any qualitative 

 classification, or quantitative estimate be ever possible, it 

 seems that it must be effected by some such method as that 

 indicated above. 



Again, to enumerate a few more instances of the measure- 

 ment of minute quantities, there are the average distances of 

 molecules from one another in various gases and at various 

 pressures ; the length of their free path, or range open for their 

 motion without coming into collision ; there are movements 

 causing the pressures and differences of pressure under which 

 Mr. Crookes' radiometers execute their wonderful revolutions.^ 

 There are the excursions of the air while transmitting notes 

 of high pitch, which through the researches of Lord Rayleigh 

 appear to be of a diminutiveness altogether unexpected.'* There 

 a re the molecular actions brought into play in the remarkable 

 experiments by Dr. Ker,^whohas succeeded, where even Faraday 

 failed, in effecting a visible rotation of the plane of polarisation 

 of light in its passage through electrified dielectrics, and on its 

 reflection at the surface of a magnet. To take one more 

 instance, which must be present to the minds of us all, there are 

 the infinitesimal ripples of the vibrating plate in Mr. Graham 

 Bell's most marvellous invention. Of the nodes and ventral 

 segments in the plate of the telephone which actually converts 

 sound into electricity and electricity into sound, we can at 

 present form no conception. All that can now be said is that 

 the most perfect specimens of Chladni's sand figures on a 

 vibrating plate, or of Kundt's lycopodium heaps in a musical 

 tube,^ or even Mr, Sedley Taylor's more delicate vortices in the 

 films of the phoneidoscope,^ are rough and sketchy compared 

 with these. For notwithstanding the fact that in the movements 

 of the telephone- plate we have actually in oiu- hand the solution 

 of that old world-problem the construction of a speaking- 

 machine, yet the characters in which that solution is expressed 

 are too small for our powers of decipherment. In movements 

 such as these we seem to lose sight of the distinction, or perhaps 

 we have unconsciously passed the boundary between massive and 

 molecular motion.^ 



Through the phonograph ^ we have not only a transformation, 

 but a permanent and tangible record of the mechanism of speech. 

 But the differences upon which articulation (apart from loudness, 

 pitch, and quality) depends, appear from the experiments of 

 Fleeming Jenkin and of others, to be of microscopic size. The 

 microphone affords another instance ^^ of the unexpected value 

 of minute variations, — in this case of electric currents ; and it is 

 remarkable that the gist of the instrament seems to lie in 

 obtaining and perfecting that which electricians have hitherto 

 most scrupulously avoided, viz., loose contact. 



Once more, Mr. De La Rue has brought fonvard as one of 

 the results ^^ derived from his stupendous battery of 10,000 cells, 

 strong evidence for supposing that a voltaic discharge, even when 

 apparently continuous, may still be an intermittent phenomenon ; 

 but all that is known of the period of such intermittence is, that 

 it must recur at exceedingly short intervals. And in connection 

 with this subject, it may be added that, whatever be the ultimate 

 explanation of tbe strange stratification which the voltaic dis- 

 charge undergoes in rarefied gases, it is clear that the alternate 

 disposition of light and darkness must be dependent on some 

 periodic distribution in space or sequence in time m hich can at 

 present be dealt with only in a very general way. In the ex- 

 hausted column we have a vehicle for electricity not constant 

 like an ordinary conductor, but itself modified by the passage of 



1 Pkzl. Trans, of the Royal Society, 1870, p. 333; and 1876, p. 27. 



2 Phil. Tratts. 1877, p, 149. 



3 " On Attraction and Repulsion Resulting from Radiation," PA//. Trans. 

 1S74, p. 501 ; 1875, p. 519; 1876, p. 325. 



4 Philosoj>hical Magazine, April, 1878. 



i Philosophical Magazine, 1875, vol. ii. pp. 337, 446; 1S77, vol. i. 

 p. 321; 1878, vol. i. p. 161. 



* Poggendorff's Annalen, torn. xxxv. p. 337. 



7 Royal Society's Proceedings, 1878. 



8 The papers on the telephone are too numerous to specify. 



5 See various papers in Nature, and elsewhere, during the last twelve 

 months. 



■0 Royal Society's Proceedings, May 9, 1878. 



^' Phil. Trails, vol. clxix. pp. 55 and 155, and other papers catalogued in 

 the Appendix to Part II. of the Memoir. 



the discharge, and perhaps subject to laws differing materially 

 from those which it obeys at atmospheric pressure. It may also 

 be that some of the features accompanying stratification form a 

 magnified image of phenomena belonging to disniptive dis- 

 charges in general ; and that consequently so far from expecting 

 among the known facts of the latter any clue to an explanation 

 of the former, we must hope idtimately to find in the former an 

 elucidation of what is at present obscure in the latter. A prudent 

 philosopher usually avoids hazarding any forecast of the practical 

 application of a piurely scientific research. But it would t^eem 

 that the configuration of these striae might some day prove a very 

 delicate means of estimating low pressures, and perhaps also for 

 effecting some electrical measurements. 



Now, it is a curious fact that almost the only small quantities 

 of which we have as yet any actual measurements are the wave- 

 lengths of light ; and that all others, excepting so far as they 

 can be deduced from these, await futiu-e determination. In the 

 mean time, when unable to approach these small quantities indi- 

 vidually, the method to which we are obliged to have recourse 

 is, as indicated above, that of averages, whereby, disregarding 

 the circumstances of each particular case, we calculate the average 

 size, the average velocity, the average direction, &c., of a large 

 number of instances.^ But although this method is based upon 

 experience, and leads to results which may be accepted as sub- 

 stantially true ; although it may be applicable to any finite 

 interval of time, or over any finite area of space (that is, for all 

 practical purposes of life) there is no evidence to show that it is 

 so when the dimensions of interval or of area are indefinitely 

 diminished. The truth is that the simplicity of nature which we 

 at present grasp is really the result of infinite complexity ; and 

 that below the uniformity there underlies a diversity whose depths 

 we have not yet probed, and whose secret places are still beyond 

 our reach. 



The present is not an occasion for multiplying illustrations, 

 but I can hardly omit a passing allusion to one all-important 

 instance of the application of the statistical method. Without 

 its aid social life, or the History of Life and Death, could not 

 be conceived at all, or only in the most superficial manner. 

 Without it we could never attain to any clear ideas of the con- 

 dition of the poor, we could never hope for any solid ameliora- 

 tion of their condition or prospects. Without its aid, sanitary 

 measures, and even medicine, would be powerless. Withoutit, 

 the politician and the philanthropist would alike be wandering 

 over a trackless desert. 



Mathematical Methods and other Subjects. 



" It is, however, not so much from the side of science at large 

 as from that of mathematics itself, that I desire to speak. I 

 wish from the latter point of view to indicate connections 

 between mathematics and other subjects, to prove that hers is 

 not after all such a far-off region, nor so undecipherable an 

 alphabet, and to show that even at unlikely spots we may trace 

 under-currents of thought which, having issued from a common 

 source, fertilise alike the mathematical and the non-mathematical 

 world. 



Having this in view, I propose to make the subject of special 

 remark some processes peculiar to modern mathematics ; and, . 

 partly with the object of incidentally removing some current 

 misapprehensions, I have selected for examination three methods • 

 in respect of which mathematicians are often thought to have 

 exceeded all reasonable limits of 'speculation, and to have- 

 adopted for unknown purposes an imknown tongue. 



And it will be my endeavom: to show not only that in these 

 very cases our science has not outstepped its own legitimate 

 range, but that even art and literatm-e have unconsciously em- 

 ployed methods similar in principle. The three methods m 

 question are first, that of imaginary quantities ; secondly that of 

 manifold space, and thirdly, that of 'geometry not according to 

 Euclid, 



Imaginaries. 



First It is objected that, abandoning the more cautious methods 

 of ancient mathematicians, we have admitted into our formulae 

 quantities which by our own showing, and even in our own 

 nomenclature, are imaginary or impossible ; nay, more, that out 

 of them we have formed a variety of new algebras to which 

 there is no counterpart whatever in reality, but from which we 

 claim to arrive at possible and certain results. 



On this head it is in Dublin, if anywhere, that I may be per- 



' See Maxwell " On Heat," chap. xxiL 



