TRANSACTIONS OF SECTION L. 707 
these are framed with reference to axes; the mind becomes interested in the 
linear arrangement, and there follows finally the demonstration of the remark- 
able equivalence of line and equation. 
Comparison with the mere proof that a linear equation represents a straight 
line shows that current ideas regarding reform are on sound lines. The danger 
used to lie in teaching formal conclusions with little or no account of their 
growth. It would be small gain if we fell into the opposite danger of teaching 
to look without observing or to observe without inquiring. In trigonometry, 
for instance, the formula a = 2R sin A and its proof represent the last stage ; 
the preceding stage is the observation that we could draw and measure a chord 
of a circle if we knew the diameter of the circle and the angle subtended by the 
chord at the circumference; and this is preceded by beholding the beautiful 
scene of equal angles in the same segment of a circle. In descriptive geometry 
we must see the line given by its plan and elevation or the plane by its traces. 
Not till it is noted that from the traces of a plane a joiner could mechanically 
construct the scene, that the plane is there, can there be any observation of such 
a thing as the real angle between its traces, far less any inquiry into its deter- 
mination, or any sound understanding of what a text-book conveys by the head- 
ing, ‘Given the traces of a plane, to find the real angle between the traces,’ 
and the corresponding construction. The preliminary study of the calculus 
requires the survey of scenes where 5y/dz and Xydz are matters of observation 
and interest before the notions of a limit and dy/dx and fydx are reached. 
Otherwise we may know about dy/dx and fydz, but we do not know them, at 
any rate we do not know them in the sound sense of knowing the spirit through 
the letter. 
(ili) The Present Position of Mathematical Teaching. 
By Wiuu1am P. Miune, M.A., D.Sc. 
In passing in review the changes that have taken place in the teaching of 
mathematics throughout the country during the last ten or twenty years, one 
has to consider not only the instruction given to the rank-and-file of mathe- 
matical pupils, but also that given to those of higher endowments, seeing that 
for a nation’s educational efficiency it is essential to train carefully the average 
person and to train equally carefully the expert. It is found that at the present 
moment practically all the best pedagogic thought of the country is being 
directed towards rendering mathematics at once useful and educative to the 
pupil of average attainments, while very little is being done for those who are 
being definitely trained in the more advanced branches. It is here maintained 
that most of the methods discovered and applied with such conspicuous success 
to the teaching of elementary mathematics can be extended in scope and 
modified in application so as to improve the teaching of the higher mathematics, 
and bring a larger proportion of the more difficult branches of the subject within 
the powers of comprehension of the general body of the more advanced pupils 
than is at present possible. The published views of such distinguished teachers 
as Tait, Chrystal, and Hobson support this hypothesis. 
If we consider the teaching of the higher mathematics given in the secondary 
schools to scholarship candidates, we see that the following difficulties at once 
present themselves and militate against efficient teaching :— 
(1) No systematic effort has ever been made by secondary teachers carefully 
to scrutinise the schedule of knowledge required of scholarship candidates, and 
to discover what could be best omitted and what could advantageously be added. 
(2) The subject has left the stage when it is easily within the teacher’s 
grasp, without any private study. Owing to the length and stress of his official 
routine, very little time is left him to keep abreast of the latest developments 
of the subject as it leaves the hands of the great masters and investigators. 
(3) Owing to the labour of preparation and the small pecuniary returns, 
text-books covering the ground of the scholarship work are rarely published, 
and continue to be used when long out of date. 
(4) Detailed discussions as to the best methods of teaching such difficult 
subjects as Limits, Virtual Work, Homography and Involution, &c., have never 
been carried out. 
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