May 16, 1912] 
NATORE 
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ancient temples. Finally, Sir Robert Hadfield’s 
account of ‘‘Sinhalese Iron and Steel of Ancient 
Origin ” throws an interesting light on the materials 
and methods in Ceylon many centuries ago, particu- 
laraly in the production of steel tools and implements. 
We hope to give a separate abstract of this paper in a 
later issue. 
The exceptionally full and interesting programme 
of the meeting is completed by a series of valuable 
papers dealing more directly with steel manufacture, 
including an historical survey of forty years’ progress 
of the industry by the president (Mr. Arthur Cooper) 
in his address, an interesting paper on steam engines 
for driving rolling-mills by Mr. J. W. Hall, and an 
account of the Nathusius electric steel furnace by its 
originator, with several other contributions of a 
similar character. Altogether the institute is to be 
congratulated upon a singularly successful meeting, 
which revives the traditions of the best days of its 
history. 
M. POINCARE’S LECTURES AT THE 
UNIVERSITY OF LONDON. 
1.—May 3.—The Logic of the Infinite——Some 
years ago, M. Poincaré said, he had = pub- 
lished a certain number of articles upon the 
subject, which had involved him in a veritable polemic. 
He would not attempt to renew the arguments that 
had been used on either side, or to bring forward any 
fresh arguments, as he believed that the divergence 
of the two schools was irreducible. It arose from an 
essential difference of mentality; he would therefore 
accept it as an experimental fact, and would endeavour 
to account for this divergence. For the first school, 
whom, for the sake of convenience, he would call 
Pragmatists, the infinite was derived from the finite ; 
for the second, the Cantorians, the infinite pre- 
existed, and the finite was only a small piece of the 
infinite. From another point of view, to use the 
language of the scholastics, the Pragmatists were 
extensionists, while the Cantorians were comprehen- 
sionists. This appeared in the nature of the defini- 
tions used by the two schools. For the first a 
definition consisted in the addition of one new object, 
expressed in terms of the aggregate of known objects; 
for the second a definition was a fresh subdivision of 
the aggregate of all objects known and unknown. 
The Pragmatists were idealists, and for them an 
object did not exist until it had been thought. The 
Cantorians were realists for whom the existence of 
objects was independent of a thinking subject. For 
them the infinite was independent of man or any 
thinking being; it was pre-existent and was discovered 
by man. 
IIl.—May 4.—Time and Space.—The conception of 
space arose from our muscular sense. When we saw 
an object we knew the movements necessary to attain 
it. The idea of space, then, was the association 
between certain sensations and certain movements. 
To the whole of space the principle of relativity 
applied, that is to say, we had no means of perceiving 
a transportation, a magnification, or a deformation 
of the universe, provided that in the transformation 
all objects were subject to the same law. Space, in 
fact, was “soft and without rigidity.” We appre- 
ciated the relations between objects in space by means 
of our instruments of measurement, of which our 
body was one, and the science of geometry was a 
study of these instruments. But the instruments 
were not perfect, and therefore we replaced them 
by a series of ideal instruments for the purposes of 
our geometry, which thus depended upon an aggre- 
NO. 2220, VOL. 89] . 
gate of conventions approximating to the actual laws, 
but simpler. The principle of relativity also applied 
to time; if all actions were retarded uniformly we 
had no means of perceiving it. 
A revolution had recently been brought about by the 
researches of modern physicists, especially those of 
Lorenz. Formerly the action of one body upon 
another was supposed to be instantaneous. But if we 
supposed that such an action was transmitted through 
the intervening space at a finite speed, the question 
of priority of action became very difficult. Formerly 
we had considered an action a to be anterior to a de- 
pendent action 8, when @ could be regarded as the 
cause of 8. But in the new mechanics, if 8 occurred 
too soon it might happen that « could not be regarded 
as the cause of 8, nor B as the cause of a. It might 
be necessary at this stage to abandon our former 
mechanical conventions and to adopt new ones. 
IlI.—May 10.—Arithmetical Invariants.—If{ an 
algebraic form, in two variables, say, F(x, y), was 
subjected to the transformation 
x—>ax+ By, y—>yx+by, where ai-By=1. . (1) 
there were certain functions of the coefficients of 
F which remained unchanged. These were algebraic 
invariants. Suppose now that a, B, 7, 6, and also the 
coefficients of F(x, y) were restricted to be whole 
numbers, positive or negative, F(x, y) would possess 
the same invariants as before, but it would also 
possess others which were termed arithmetical 
mvariants. 
The simplest form was F(x, y)=ax+by. This form 
possessed no algebraic invariants. Some arithmetical 
invariants, however, could be obtained which were 
related to the Weierstrassian elliptic functions, the 
thetafuchsian functions, and the functions of Jacobi. 
In the case of quadratic forms it was necessary to 
distinguish between the definite and the indefinite. 
The definite quadratic form might be reduced for this 
purpose to a pair of linear forms, but for an in- 
definite form invariants could only be found if we 
took certain subgroups of the group of transforma- 
tions considered instead of the group itself. 
IV.—May 11.—The Theory of Radiation.—Planck 
had enunciated some ideas, which, if they were 
accepted, would bring about in the science of physics 
the most profound revolution that had occurred since 
the time of Newton. We owed to Newton the prin- 
ciple that the laws of nature could be expressed in 
the form of differential equations. According to 
Planck, phenomena satisfy not differential, but finite 
difference equations. 
By the method of statistics applied to a very great 
number of separate molecules we arrived at one of 
the fundamental theorems of thermodynamics, that of 
Maxwell on the equipartition of mean kinetic energy. 
Upon the same basis we arrived at Wien’s law of 
radiation and Rayleigh’s law. The last was con- 
sistent with the theorem of Maxwell, but it was not 
justified by experiment. 
Planck supposed that there existed in incandescent 
bodies a very great number of resonators, each corre- 
sponding to a certain wave-length of light; these re- 
sonators could only acauire or emit energy by a definite 
increment: a quantum or atom of energy. Planck 
obtained in this way a law of radiation, which was 
justified by experiment, but which was not consistent 
with Maxwell’s theorem. M. Poincaré found that if, 
instead of considering the action of light upon a 
molecule, we applied the ideas of Planck to the action 
of a molecule upon light, we should be forced to 
conclude that diffusion took place with a certain 
retardation, and this was certainly not true. Thus 
the hypothesis of Planck was unsatisfactory, and no, 
solution to the problem was at present in sight. 
