306 
NATURE 
[May 23, 1912 
origins of the various methods; the physiologist from 
whom education has borrowed the historical prin- 
ciple says that ‘‘the history of the individual is a 
blurred recapitulation of the history of the race.” 
Too exact a recapitulation is wasteful of time and 
deadens the intellect. The recapitulation must be a 
blurred one; the barriers between the various 
branches of mathematics must be broken down, and 
the pupil given freedom to select for any problem 
whatever tool he finds most appropriate. 
In Paper 16 Mr. Durell drives home this principle. 
The freedom to treat a problem by Euclid’s method, 
by Descartes’, or by Monge’s, by the principle of 
duality or by that of continuity, gives to the pupil 
a breadth of view and to the subject a unity other- 
wise unattainable. It reduces the multitude of 
properties of geometrical figures to a small number 
of greater generalisations which the mind can carry 
without effort. And it effects a saving of time, which 
makes possible a much further advance in mathe- 
matics than is now customary. 
Mr. Durell rightly reduces to small compass the 
Euclidean treatment of conics, but he _ retains 
conics as the chief material to which the various 
methods are to be applied. His course might be 
further improved by the substitution in some cases 
of other material, such as an occasional higher 
algebraic curve, a transcendental curve, or a surface. 
The Postulates of Geometry. 
Mr. Carson (Paper 15) pleads for more system in 
the treatment of elementary geometry, in order that 
the pupil may gain a better grasp of the subject and 
have time to pursue his studies further. Mr. Carson 
would assume as postulates all the geometrical 
properties which can be looked upon as “intuitive,” 
and build a system of reasoned geometry upon these; 
a suggestion which deserves serious consideration. 
The elaboration of this idea must involve some pre- 
liminary discussion of the nature of intuition. Intui- 
tion varies greatly from individual to individual; that 
“things equal to the same thing are equal to one 
another ’’ is not an intuition to every child (see Bran- 
ford’s ‘‘ Principles of Mathematical Education ”’); and, 
on the other hand, to an occasional genius results 
are intuitive which involve prolonged investigation 
for the average mathematician.  Intuitions depend 
upon experience, and differ according to the experience 
of the individual. 
It will clearly be necessary to give precision to each 
particular property which is to be assumed as an 
intuition. One valuable method of giving such pre- 
cision is strangely repugnant to Mr. Carson, I mean 
that of numerical illustration. This method has real 
value, not only for these intuitions, but also for 
ensuring the comprehension of a property of which 
the proof is to follow. Nevertheless, when worked 
out Mr. Carson’s scheme would probably differ little 
from some courses now in use. 
Mr. Carson’s main thesis is that if the inclusion of. 
mathematics in the school curriculum is to be upheld, 
its study must be justified as an end in itself, and 
not by any consideration of utility. This view is 
best judged by the conclusions to which it leads him. 
One such conclusion is that the study is essential 
for girls as well as for boys; perhaps if Miss 
Burstall’s excellent discussion of that topic in a recent 
number of The Mathematical Gazette had been avail- 
able at the time when Mr. Carson wrote this paper, 
he might have modified his views. 
We have already referred to Mr. Carson’s criterion 
of the content of the mathematical course—‘ mathe- 
matics for its own sake.’"’ To most of us beauty is 
closely connected with utility; there are on the high 
road of progress just as many and as lovely views 
NO. 2221, VOL. 89] 
to be seen as in Mr. Carson’s bypaths. For many 
of us, also, the high road provides bread and butter 
along with beauty; at the present day the view is 
all too prevalent that real work and beauty are in- 
compatible. 
But really Mr. Carson is barely half in earnest. 
He is constantly falling into some utilitarian justifi- 
cation for his teaching, and then pulling himself up 
short. And the programme he sketches is excellent, 
chiefly because he keeps so close to the concrete and 
to utility. 
Examinations. 
In recent years there has been much discussion of 
the value of literary examinations, some holding 
them to be the only true criterion of a pupil’s ability, 
others holding them entirely harmful. The truth 
would appear to lie between these extremes. On 
the one hand, no literary examination can tell us 
much of the character of a boy, and there are sub- 
jects in which training is the great element, and 
knowledge so small an element that any attempt to 
examine would spoil the value of the subject. There 
are, on the other hand, many subjects in which ex- 
amination has 
conducted. 
An examining body cannot escape the responsibility 
of influencing schools, whether for good or ill. If 
the examiner is ignorant of the schools his influence 
will be bad; he must in some way be put in close 
touch with the school. He must also not be a mere 
hack, but have a fresh interest in the subject and 
some knowledge of educational principles. With 
that granted, there is ground for hope that his 
influence on the schools will be good. Another 
thing of much value is difficult to get, namely, 
the criticism of the business man who has no 
expert knowledge of the subject but a real know- 
ledge of the kind of boy he wants in his business. 
I remember Prof. Henrici’s modest account of his 
early mathematical development as teacher in a tech- 
nical college. The business committee wanted cer- 
tain things done which seemed impossible to the 
young professor with his academic views. But he 
agreed to try, and speedily he concluded that the 
business men had been perfectly right. 
Messrs. Macaulay and Greenstreet (Paper 14) dis- 
cuss the scholarship examinations on which the 
universities select entrance scholars. The discussion 
concerns Cambridge chiefly, and the authors make a 
strong case for their view that the universities are 
not sufficiently acquainted with the conditions of the 
schools, and that more weight should be attached to 
the opinions of the schoolmasters who prepare the 
boys for the examinations. The authors deserve all 
sympathy in their desire that pupils should not waste 
time in exploring bypaths and in the acquisition of 
excessive skill in manipulation, but should push on 
along the main road. Some of their suggestions, 
however, scarcely carry conviction. Consider, for 
instance, their disapproval of the graphical method 
in statics, a method of such value for giving a grasp 
of principle. Take, again, their view that a boy 
should sit still and watch his master draw algebraic 
graphs without drawing them himself. 
Davip BEVERIDGE Matr. 
BIOLOGICAL PAPERS FROM PRAGUE. 
ROF. HLAVA (Bull. Internat. Acad. Sci., 
Prague, xv. Ann.) has found, in the blood of 
children infected with measles, oval or rod-like 
bodies, which he regards as probably of protozoan 
nature. In a blood-smear from another infected child 
(who also exhibited severe anzemia due to the pre- 
sence of numerous whip-worms in the intestine), 
real value—provided it is properly © 
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