* influences on the progress of mathematical ° 
} 
opinions, and it is therefore of great importance 
_ that a reviewer should be able to state his own 
4 
3 
b 
| 
THURSDAY, JANUARY 22, 1914. 
MATHEMATICIANS IN COUNCIL. 
Proceedings of the Fifth International Congress 
of Mathematicians. (Cambridge, August 22— 
28, 1912.) Edited by Prof. E. W. Hobson and 
Prof. A. E. H: Love. Vol. i., Part i., Report 
of the Congress. Part ii., Lectures: Com- 
munications (Section I.) Pp. 500. Vol. ii., 
Communications to Sections II-IV. Pp. 657. 
(Cambridge University Press, 1913.) Price 
30s. net, two vols.) _ ; 
REVIEW of theaeieautiflly printed pub- 
lications of the Cambridge Press neces- 
sarily constitutes in some measure a survey of 
the proceedings of the Fifth International Con- 
gress of Mathematicians. Although more than a 
_ year has elapsed since these meeting's were held at 
Cambridge, it may not yet be too late to form 
an opinion on the work that was then done, and 
science, and on the position of mathematics in 
Great Britain. These are subjects on which no 
_ two people can be expected to hold the same 
_ Views without prejudice to those held by other 
members present at the Congress, readers of 
the proceedings, or, indeed, anyone else. 
While the corresponding records for Heidelberz 
(1904) are contained in one volume of 756 pages, 
and for Rome (1go8) in three volumes, of which 
the first two contain 218 and 318 pages, the 
Cambridge volumes occupy 500 and 657 pages 
respectively. Nor was the attendance at the 
meeting less satisfactory. While Great Britain 
only contributed 2 per cent. of the members at 
Heidelberg and 4 per cent. at Rome, the attend- 
ance of 221 British members out of a total of 
574 at Cambridge compares favourably with 
Germany’s representation of 173 out of 336 at 
Heidelberg, and Italy’s 190 out of 535 at Rome. 
Turning next to the published papers, these 
reflect in no small degree the influences that have 
been making themselves felt in recent years in 
raising higher mathematics to the dignity of a 
science, and saving it from degenerating into mere 
cut-and-dried algebra. Even in that most difficult 
of all to popularise section—arithmetic, algebra, 
and analysis—the papers deal largely with analysis, 
and are not overloaded with formule, while a 
pleasing variety is introduced by descriptions of 
“mechanisms for solving equations, and cases 
where a sum of powers is equal to the same power 
‘of one number. A physicist who was exclusively 
NO. 2308, VOL. 92] 
NATURE , | 
75 
a physicist might find much to interest him in 
some of these papers. On the other hand, in 
| the geometry section, where one naturally ex- 
| pects to find results adapted to visualisation, there 
are very few papers in which the investigations 
are not expressed in symbolic form. The paper 
on rational right-angled triangles would have been 
better placed alongside of the one above referred 
to on sums of powers. Is it the fact that pure 
geometry is exhausting its resources in three 
dimensional space, and that it is becoming  in- 
creasingly difficult to find new subjects of investi-. 
gation which do not require the use of extended 
algebraic formula ? 
Coming now to applied mathematics, the most 
noticeable feature is that the papers presented 
contain no conspicuous reference to aéroplanes, 
and, indeed, judging from their general character, 
it seems almost, if not quite, certain that the aéro- 
plane has nowhere received mention in the pro- 
ceedings of this section. We have work sub- 
mitted on the old hackneyed “problem of three 
bodies,” performing motions which no living man 
will ever see realised experimentally, also theories 
of the ether and gravitation. Now the peculiar 
type of brain which is capable of investigating 
the hypothetical motions of three hypothetical 
bodies is just the intellect required to investigate 
the the general | character 
described by an aéroplane, and if it is necessary 
to assume a simplified law of air resistance, the 
conclusions will certainly afford some definite 
basis for a comparison of theory with experiment. 
As for the ether, this might well stand over when 
we know so little about the air, and with regard 
to gravitation, the fact that it may be propagated 
with finite velocity through space can scarcely give 
an aviator any hope of saving his life in the 
event of his aéroplane collapsing. The aviation 
problems awaiting solution at the time that this 
congress was heid—and after—would have amply 
sufficed to occupy the proceedings of a separate 
section. 
Possibly an appeal to Prof. Reissner might have 
elicited some contribution on this subject. With 
regard to workers in our own country, it seems 
not improbable either that Lord Kelvin died too 
soon, or that aéroplanes came too late. Lord 
Kelvin had an extraordinary power of command- 
ing both the attention and respect of the practical 
engineer and the interest of the mathematician, 
and had he been able to investigate the stability 
of aéroplanes, it is certain that the present dead- 
lock would never have arisen; on the contrary, 
mathematical proceedings would have been filled 
| with aéroplane papers, and aviators would be 
motions of most 
