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TRANSACTIONS OF SECTION A. 4.59 
England. The author held that the importance of the practical method was now 
being generally recognised, but that there was considerable danger of the logical 
method being neglected, and in this way some of the best minds in the country 
being lost to mathematics. 
The author explained in some detail how the idea of infinity naturally pre- 
sented itself at an early stage in a boy’s education, and how it might and should 
be elucidated. In this connection the importance of the concept of limiting point 
was referred to, and illustrated by the fact that this concept was involved in 
a precise definition of the centre of mass of a heterogeneous body. 
2. The Teaching of the Elements of Analysis. By C. O. Tuckry. 
How must the teaching of the elements of analysis be modified in view of the 
early stage at which the calculus is now reached ? 
The main defects (which we must hope to avoid) of the now usual course are— 
(i) The initial elaborate discussion of convergence, which ignores the lesson 
from history, that only as much of this theory should be given as is shown to be 
necessary by examples of false results deduced by plausible reasoning. 
(ii) The apparent lack of object in discussing infinite series. 
(iii) The abrupt introduction of the exponential function. 
(iv) The unnecessary difficulty of the proofs of the series for sin 2 and cos w. 
(v) The proceeding from difficult to easy by taking first the rigorous proofs of 
the expansions and later the easy method of obtaining the series by the calculus, 
their existence being assumed. 
The course advocated is based on the calculus; the necessity of commencing it 
before differentiating @, and other reasons, point to the convenience of beginning 
it when the differentiation of cos 2 and sin 2 is known. 
The numerical calculation of sin x, cos 2, and also log x is put forward as 
the object of the course, and it is explained that series are obtained for these 
functions. 
The series for (a+ a)", n integral, and =. being known to the student, the 
i avr 
calculus method of obtaining by repeated differentiation the coefficients of a series 
(assumed to exist) is illustrated by these known series. 
Then the binomial series for fractional and negative powers are worked out in 
the same way. 
Taking series for (a—a)~", say, and putting a>, shows necessity for dis- 
cussion of convergence, which is then taken in hand and proofs given that— 
(i) Su, where Lt “"*1 <1 is convergent (comparison with G.P. used) ; 
Un 
(ii) u,—u,+u,—...18 convergent if uy>u,>u,>... 
An example from partial fractions, taking 
tsa A B 
Te I= ee 
shows that the fact that no hitch occurs in finding the coefficients must not 
ye regarded as a proof of these series, and shows the necessity for rigorous proofs 
ater on. 
The series for sin w and cos a are obtained by the calculus method (existence 
assumed), and considerable practice is given with these series and the various 
forms of the binomial series before proceeding to the next stage, viz., the expo- 
nential function. 
The importance of a function obeying the law 2 =kx, the ‘compound interest’ 
Z 
law, is emphasised by instances—compound interest, and examples from physics 
