May 23, 1982] 
NATORE 
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side of the bevel gears at those parts which were in 
mesh at the moment the error occurred, and the error 
was thereby reduced to about eight-tenths of a second. 
The sun was photographed on 148 days, and 
showed spots on 88 days. In future, the ** Nautical 
Almanac” publications are to be stored and dis- 
tributed by the Naval Observatory librarian. 
EPHEMERIS FOR BORRELLY’s COMET, 1911e.—To No. 
4572 of the Astronomische Nachrichten, M. Schau- 
masse contributes an ephemeris for comet 1grre, 
which is at present about a degree north of 36 Lyncis, 
and is travelling in the direction of 8 Leonis Minoris. 
This comet is extremely faint, but an observation by 
M. Schaumasse, with the Nice equatorial coudé on 
April 19, showed that the error of the ephemeris was 
only —3s., 0’. 
THE TEACHING OF MATHEMATICS.} 
The Content of the School Course in Mathematics. 
yay SYSTEM of education designed on broad lines 
to prepare pupils for some particular occupation 
is not only the best training for that particular 
occupation, but it is better as a “ general education "’ 
than a system which has been designed simply as a 
general education, and not as a preparation for any 
particular calling. For a boy willingly undertakes 
work which clearly leads up to the solution of a real 
and interesting problem, even if that problem is one 
that belongs to his neighbour’s after-life and not to 
his own. But the course designed for “ general 
education" tends to become a ‘mental discipline "’ 
lacking in interest, and such discipline deadens the 
mind and makes the boy a machine. 
In Papers Nos. 15 and 16 of this series, Mr. 
Carson and Mr. Durell advocate the inclusion in a 
school course of certain methods of great beauty, 
which to a few boys will be a source of delight. But 
the authors of those papers have no criterion of the 
suitability of these subjects beyond their own love of 
them. To a certain point that is a true criterion; 
what has given pleasure to one person has a good 
chance of giving pleasure to another; and all the sub- 
jects which they advocate deserve a place in a system 
of recreations for the mathematician’s leisure hour. 
But to determine which of these methods and subjects 
are to be thrust upon every boy of an ordinary degree 
of mathematical ability, some better criterion is 
necessary. I do not say that I would exclude any of 
these methods, but only that they have not yet been 
judged on a suitable criterion. 
That suitable criterion must be a consideration of 
the needs in after-life of certain groups of boys. In 
many cases mathematics is a form of technical know- 
ledge required for the after-career, e.g., for the 
careers of engineer, mathematical schoolmaster, pro- 
fessor of mathematics. In such cases the content of 
the subject will be determined by a wide interpreta- 
tion of the requirements of the career, the treatment 
of the subject being of the broadest and_ every 
problem viewed from many points of view. The boy 
to whom mathematics is merely a part of his general 
1 Papers on the Teaching of Mathematics in the United Kingdom, 
published by the Board of Education :— 
(12) ‘* Mathematics with relation to Engineering Work in Schools.” By 
T. S. Usherwood. (1912.) Price 2d. 
(13) “The Teaching of Arithmetic in Secondary Schools.” By G. W. 
Palmer. (1912). Price 2}d. 
(x4) “ Examinations for Mathematical Scholarships.” By Dr. F. S. 
Macaulay and W. J. Greenstreet. (1912). Price 3d. 
(15) “The Educational Value of Geometry.” By G. St. L. Carson. 
(1912.)_ Price 14d. 
(16) **A School Course in Advanced Geometry.” 
(t912.) Price 14d. 
(17) ‘* Mathematics at Osborne and Dartmouth.” 
C. 1. Ashford. (xg12.)_ Price 24. 
Earlier papers were noticed in NaTURE of March 14. 
NO.2221, VOL. 89| 
By C. V. Durell. 
By J. W. Mercer and 
| classification also held by Dr. 
education will, so far as he goes, study along with the 
technical group with which he has most in common. 
It is not necessary that each boy’s future career 
should be planned in advance; all boys, technical, 
semi-technical, and non-technical, will study together 
for a time; then gradually the non-technical. boys 
will drop out, and the remainder will bifurcate 
according to their varying intellectual powers and 
their varying technical needs. 
These are the views to which observation, experi- 
ment and reflection are leading students of educa- 
tion. Many a doubter will be converted by a study 
of Mr. Mercer’s admirable account of the teaching 
at the naval colleges (Paper 17). It is a document 
which every mathematical master should have by him. 
Some small portions of the course are special to the 
requirements of the Navy, but the course as a whole 
makes an excellent starting point from which to lay 
out a scheme for any school. c 
In Paper No. 12 Mr. Usherwood provides further 
evidence in favour of our principles. The close 
correlation of mathematics with engineering has 
given his boys a breadth of mathematical knowledge 
and a real grasp such as would have been incredible 
a generation ago. Mr. Usherwood justifies his pro- 
cedure by quoting Mr. Branford’s classification of the 
impulses which urge towards mathematical study, a 
c Nunn. Of these 
impulses, the utilitarian is the chief one at the school 
stage, and every central truth should be made to arise 
in response to some demand arising from a practical 
problem. Mr. Usherwood holds that manual as well 
as mental dexterity should be involved in the practical 
problem from which an investigation sets out, and he 
petitions for a greater place in the curriculum for 
suitable manual training. 
Further support to the principles enunciated above 
is given by Mr. Palmer’s historical account of the 
teaching of arithmetic. It is an excellent account 
of the changes which have been made in the last 
quarter-century. A generation ago ‘‘ general educa- 
| tion”? was the cry, and if any method had a ‘‘bread- 
and-butter” value that was sufficient reason for its 
exclusion. The course consequently contained such 
monstrosities as ‘‘true discount.’’ The true criterion 
has now been adopted; in part, half unconsciously. 
More conscious application of the criterion will in 
time recognise that most fractions should be dealt 
with in decimal form, and will greatly reduce the 
time spent on vulgar fractions, greatest common 
factor, and least common multiple. We learn from 
Mr. Palmer how far removed the school treatment of 
stocks and shares is from business practice. The 
whole subject seems to us unsuited to the school. 
The difficulty lies in the realisation of the cireum- 
stances of the problem; the circumstances are far 
removed from a boy’s experience, and the explanation 
of them profits him nothing. The circumstances 
once realised, the arithmetic is child’s play. 
The Methods of Mathematical Study. 
The various methods of mathematical investigation 
have been added one by one at various times to our 
available stock c! tools. On the historical principle 
that the development of the individual should copy 
the history of the race, it is appropriate that these 
various tools should be put in the pupil’s hand in 
the order of their discovery. It is, however, the 
practice to follow the development of the race too 
closely, and to discuss by the more primitive method 
all the problems for which our ancestors used it, 
regardless of the fact that a later method is a more 
| Suitable weapon with which to attack many of these 
problems. Such exactness of recapitulation cannot be 
justified; it is the haphazard result of the successive 
