December i, 1904] 



NA TURE 



In 1870 he succeeded in solving a question which was 

 then attracting much attention — the question of the unique- 

 ness of the representation of a function by Fourier's series. 

 The extension of the result to cases in which the convergence 

 of the series fails, at an infinite number of suitably dis- 

 tributed points, led him to construct a theory of irrational 

 numbers, which has since become classical. From the same 

 starting point he developed, in a series of masterly memoirs, 

 an entirely new branch of mathematics — the theory of sets 

 of points. 



Having established the fundamental distinction between 

 those aggregates which can be counted and those which 

 cannot. Cantor showed that the aggregates of all rational 

 numbers and of all algebraic numbers belong to the former 

 class, and that the arithmetic continuum belongs to the 

 latter class, and further, that the continuum of any number 

 of dimensions can be represented point for point by the 

 linear continuum. Proceeding with these researches he in- 

 troduced and developed his theory of " transfinite " ordinal 

 and cardinal numbers, thus creating an arithmetic of the 

 Infinite. His later abstract theory of the order-types of 

 aggregates, in connection with which he has given a purely 

 ordinal theory of the arithmetic continuum, has opened up 

 a field of research of the greatest interest and importance. 



Hughes Medal. 

 The Hughes Medal is awarded to Sir Joseph Wilson 

 Swan, F.R.S., for his invention of the incandescent electric 

 lamp, and his other inventions and improvements in the 

 practical applications of electricity. Not as directly in- 

 cluded in the award, his inventions in dry-plate photography, 

 which have so much increased our powers of experimental 

 investigation. 



NOTES. 



The council of the Royal Society of Edinburgh at its 

 recent meeting decided to award Sir James Dewar, F.R.S., 

 the Gunning Victoria Jubilee prize for 1900-4 for his re- 

 searches on the liquefaction of gases extending over the 

 last quarter of a century, and on the chemical and physical 

 properties of substances at low temperatures. 



The Times reports that a telegram by wireless telegraphy 

 has been transmitted by Mr. Marconi from the Marconi 

 Company's station at Poldhu, Cornwall, to a station belong- 

 ing to the Italian Government at Ancona, Italy. The 

 distance between Poldhu and Ancona, about 1000 miles, is 

 almost entirely overland, and in order to reach their destin- 

 ation the ether waves had to pass over nearly the whole 

 of France and a considerable part of Italy, including some 

 of the highest mountains of the Alps. 



The will of the late Dr. Frank McClean, F.R.S., in- 

 cludes the following bequests : — 5000!. to the University of 

 Cambridge to be expended in improving the instrumental 

 equipment of the Newall Observatory, 5000/. to the Uni- 

 versity of Birmingham (in addition to his previous sub- 

 scription) to be applied in the department of physical science, 

 2000/. to the Royal Society, 2000L to the Royal Institution, 

 2O0oi. to the Royal Astronomical Society, and to the Uni- 

 versity of Cambridge for presentation to the FitzwiUiam 

 Museum all the testator's illuminated or other manuscripts 

 and early printed books, and all objects of mediEeval or 

 «arly art which the director of the museum may select as 

 being of permanent interest to the museum. 



In a recent letter to the Times Prof. T. Clifford Allbutt 

 ■directs attention to the paramount importance of consider- 

 ing the question of diet in all schemes of physical education. 

 It is important that there should be no hasty legislation in 

 this matter, especially in view of the important researches 

 which are now approaching completion. Prof. Allbutt gives 

 In his letter a brief account of the results at which Prof. 

 Atwater, of Middletown, Connecticut, and Prof. Chittenden, 

 of Yale University, have arrived. Prof. Atwater has 

 NO. 183I, VOL. 71] 



measured accurately, upon healthy persons in uniform 

 circumstances, the intake of food, and the output of waste 

 and work, and has endeavoured to determine the modes and 

 rates of conversion of foods into bodily and mental energy. 

 Much of this expenditure of energy is upon an excess of food 

 taken beyond the needs of the individual. Such excess (or 

 not more than 4 per cent, of it) does not escape mechanically 

 and cheaply from the body, but is absorbed, distributed, 

 and excreted ; to this process no little energy is diverted. 

 In this useless effort energy is chiefly wasted by the nitro- 

 genous foods. Excessive starches and sugars are burned 

 off in the lungs almost directly, and at far less cost. Prof. 

 Atwater teaches that the ordinary man eats too much, and 

 in so doing wastes energy which he might have used to 

 profit. Prof. Chittenden comes to a like conclusion by some- 

 what different methods. He will publish shortly tables to 

 show how, on a closer adjustment of kinds and quantities 

 of food to the useful work required, not only is this much 

 work still sustained, but, by release of energy ordinarily 

 dissipated in the demolition of food excess, the sum of work 

 put out is prodigiously increased, in some cases even by so 

 much as 60 per cent, or 70 per cent. It is clear enough 

 already that one of the chief factors of physical well-being 

 is to know what to eat, and what quantity of it results in the 

 production of the maximum of useful energy. Until this 

 is known with more exactitude than is common to-day, 

 systems of physical education must be tentative and im- 

 perfectly conceived. 



Prof. S. Newcomb has been elected corresponding 

 member of the Berlin Academy of Sciences. 



Prof. Fehr contributes to V Enseignement mathimatique 

 for November 15 a list of the principal exhibits of models 

 and books at the mathematical congress last August. Among 

 the publishing firms exhibiting books, Germany was repre- 

 sented by six, Austria by two, France by four, Italy by five, 

 Switzerland, Belgium and Denmark each by one. This is 

 exclusive of books exhibited by societies and individuals, 

 under which category we find the solitary British exhibit, by 

 the Royal Irish Academy. Among the exhibitors of models 

 our country was represented by Prof. Greenhill. 



The Belgian Government has decided upon the construc- 

 tion of a turbine steamer for its Channel fleet. Gradually 

 the 19-knot steamers on this international service will be 

 replaced by new turbine boats, with a speed of 23 knots, so 

 that eventuallv even the slowest mail boats under the 

 Belgian flag will have a speed of 21J knots, or 24 miles 

 an hour. The steamer which will inaugurate this departure 

 in the progress of the service is at the present moment on 

 the stocks at Hoboken, near Antwerp, and it will shortly 

 be launched. Until quite recently, all steamships in the 

 Channel and Irish Sea services were of the paddle-wheel 

 type, a class admirably adapted for these comparatively 

 short journeys. Drawing little water, they were able to 

 enter any of the shallow harbours, and, at the same time, 

 were capable of developing a speed altogether out of pro- 

 portion to their draught. Since the introduction of turbines 

 the diminution of the diameter of the propeller and of the 

 weight of the engines has been rendered possible, so that 

 what was until lately considered a mechanical impossibility, 

 namely, to construct a steamer drawing only 91 feet and 

 developing 12,000 indicated horse-power, may now be taken 

 as a problem solved. The new Dover-Ostend mail boat 

 will be a triple-screw steamer driven by Parsons' marine 

 steam turbines. There will be three turbines — a high- 

 pressure one in the centre, receiving the steam direct from 

 the boilers, and a low-pressure one on each side, driven by 



