JuNE 26, 1902] 
political leaders undergo a complete change. If the 
awakening does not come soon, the task of making up 
for the years of forced inaction will be almost impossible 
to accomplish. 
From what has been said it will be seen that no definite 
hope can be given of an immediately beneficial result 
accruing from the deputation. It serves, however, to 
bring the matter somewhat prominently to the notice of 
the Government and of the general public, and if it accom- 
plishes nothing more it will in this have paved the way 
for future reform. It is desirable that the public should 
be educated to know the advantages which electrical 
engineers are ready and anxious to confer upon them, and 
why it is that these have not yet been bestowed. Thirteen 
years ago Prof. Ayrton, in his oft-quoted Sheffield 
address, predicted that a time was coming when the 
Sheffield grinder would work amidst beautiful surround- 
ings, deriving the power he needed from a_ small 
electrically driven motor. The time is now ripe for the 
realisation of that prophecy ; in some few places, indeed, 
it is already begun, but for its free and rapid develop- 
ment there are many abuses and much restrictive legisla- 
tion which must be removed. For electrical distribution 
the days of the small station supplying a limited area are 
numbered, and with them the days of effective municipal 
control. So also with electric traction ; wide-spreading 
tramways connecting town with town and running 
far out into the country districts are needed to bring 
about decentralisation and to help to solve the pressing 
problem of overcrowding. We can hardly expect the 
municipalities to effect these changes; the arbitrary 
boundaries of the areas they control have no reference 
to the suitability of these areas as units for electrical 
distribution, and their interests are, moreover, to a 
certain degree directly opposed to decentralisation. 
Thus, quite apart from any considerations of the purity 
of the management or efficiency of municipal electrical 
undertakings, it will be seen that there is good reason in 
many cases for looking for better results to the nation 
from company working. In the train of developments 
such as would follow the removal of restrictive legis- 
lation, we may hope to find the improvement of our 
position as manufacturers of electrical machinery. The 
country lacks neither opportunities nor electrical 
engineers capable of making use of them. We may 
therefore reasonably look to the development of electrical 
undertakings to confer a double benefit upon the 
country ; directly, by increasing the comfort and health 
of the people, and by facilitating commercial work of 
all kinds; and indirectly by increasing the number 
and size of electrical factories, and so contributing to 
the wealth and prosperity of the nation and helping it in 
the struggle with foreign competitors. 
REPORT ON THE TEACHING OF GEOMETRY. 
HE immediate result of Prof. Perry’s Glasgow 
address has been the appointment of two com- 
mittees, the work of which is now near to completion. The 
British Association committee has, we believe, concerned 
itself with the more general aspects of the problem. The 
committee of the Mathematical Association, largely com- 
posed of schoolmasters, is formulating a set of detailed 
recommendations, of which the geometry section was 
published in the May number of the Mathematical 
Gazette (George Bell and Sons). 
The Mathematical Association committee contains 
delegates from the chief public schools within easy reach 
of London ; it has, therefore, something of a representative 
character. Its recommendations are very definite ; as 
the editor of the Gazefte remarks, “it is very desirable 
that mathematical masters and others should fully avail 
themselves of this opportunity of placing on record their 
NO. 1704, VOL. 66] 
NATURE 
20) 
views as to the proposed changes.” The secretary of 
the committee, Mr. A. W. Siddons, Harrow School, 
Middlesex, will be glad to receive criticisms of the 
report. . 
The study of formal geometry is to be preceded by a 
substantial introductory course, in which the subject- 
matter of geometry is to be treated experimentally and 
inductively. The pupil is to be carefully trained in the 
use of simple mathematical instruments ; he is to be 
allowed to convince himself of the truth of geometrical 
theorems by numerical measurements and calculations. 
In this way he will make his first acquaintance with the 
main facts of geometry. When he has thus gained 
familiarity with the subject-matter, he will be in a position 
to apply the machinery of logic to his knowledge ; he will 
be able to enter, with his eyes open, upon the task of 
consolidating into a consistent whole the facts he knows. 
Throughout his whole course he is to treat problems of 
construction in a practical way ; he is not to be content 
with describing how the thing is done, he is to do it. 
Passing to the formal study of geometry, Euclid, or 
rather a skeleton Euclid, is to be retained as a frame- 
work. Large omissions are recommended, but the 
logical order is to stand. 
Theorems are cut loose from the limitations of con- 
struction by the admission of “hypothetical constructions.” 
For example, the fons asimorum may be proved by 
bisecting the vertical angle, and thus dividing the 
isosceles triangle into two triangles that can be shown 
to be congruent by Prop. 4. For it is obvious that an 
angle has a bisector, even though the method of con- 
structing it with ruler and compass may appear later in 
Euclid ; the bisector might be found equally well by 
folding the triangle in two. 
Constructions are to be taken out of the formal course 
and treated in whatever order seems advisable. It is 
clearly absurd to keep to Euclid’s order of constructions 
unless we are confined to the use of his instruments, an 
ungraduated ruler and a pair of compasses that cannot 
be trusted to transfer a distance. 
The following order is recommended in teaching the 
theorems of the first three books :—Book i., Book 11. to 
32 inclusive, Book i1., Book ili. 35 to the end. 
The course is to be lightened by the omission of a 
considerable number of dull and obvious propositions, 
such propositions being found more especially in Book 
ii. Definitions are not to be taught ez d/oc at the 
beginning of each book, but are to make their appearance 
only when needed. 
It is suggested that two locus propositions should be 
added to Book i.—the locus of points equidistant from 
two points, and the locus of points equidistant from two 
lines. This will enable the pupil to inscribe and 
circumscribe circles to triangles at an early stage. 
Playfair’s axiom is preferred to Euclid’s ; and illustra- 
tion by rotation is recommended in dealing with angles 
connected with parallel lines, triangles and polygons. 
After Book i. we are to pass to Book iii., which by the 
omission of Props. 2, 4, 5, 6, 10, II, 12, 13, 18, 19, 23, 
24 is reduced to very modest dimensions. ‘To cover the 
ground of the omitted propositions there is to be a pre- 
liminary discussion of the symmetry of the circle about 
a diameter, which can be managed experimentally by 
folding the circle and pricking holes round the semi- 
circumference. 
The “limit” definition of the tangent is allowed ; 
and Euclid’s three propositions 16, 18, 19 are condensed 
into one—‘‘ The tangent at any point of a circle, and the 
radius to the point of contact are at right angles to one 
another.” 
Book ii. is to be illustrated by algebra ; and in order 
to simplify the geometrical proofs a rectangle is to be 
defined as a parallelogram with one of its angles a right 
angle. The use of the signs + and — is sanctioned. 
