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or taste, not by logic. Now, what is in the mind of 
those who desire a uniform sequence, whether agreed 
or imposed ? We do not know ; but it may be useful 
to consider the case for both kinds of sequence—that 
of logic and that of convenience. We will take the 
latter first. 
It would, no doubt, be convenient, as boys frequently 
move from school to school, if all followed the same 
general order—taking, for example, the circle before 
similarity or vice versa. But agreement on an open 
question like this is unlikely, for each of the equally 
admissible orders would find strong advocates, and 
teachers keenly interested in their work would not 
willingly surrender their liberty. 
The graver question, of course, is as to the logical 
sequence. But in fact, the current practice of schools 
has eliminated the question in this form; for the 
practice is now widespread (and the Assistant Masters’ 
Association’s Report will give it further currency) of 
beginning the formal study of geometry at a point 
where a sufficiently broad quasi-axiomatic basis has 
been established, namely, the conditions of congruency 
of triangles and the angle properties of parallel lines. 
This means in effect the abandonment, or at least 
the postponement, of most of Euclid’s propositions 
up to I. 32. Experience has shown that many of these 
individual propositions are not really grasped by the 
ordinary boy, and if these are omitted others become 
unnecessary, as they are mere links between the others. 
The advantage of the omission is that a boy can begin 
where the work is easy instead of where it is most 
difficult. 
Two questions of great importance emerge, however, 
and it is probably to these that those who are, quite 
justly, dissatisfied with the present state of things 
should address themselves. First, how can we recover 
anything that we have lost by departure from the strict 
traditional system ; and second, when, if at all, should 
boys be introduced to the initial difficulties which have 
been evaded ? 
As to the former, it is suggested that the proper 
guiding word is not “‘ sequence,”’ but “ interconnexion ”’ 
—that the idea required is not so much that of a 
single thread, as of a network of argument. It is an 
excellent practice to take a known proposition and 
trace its connexions backward. Thus the property 
of a cyclic quadrilateral depends on the relation 
between the angle at the centre and that at the 
circumference ; this, again, depends on two early pro- 
positions, namely, the exterior angle of a triangle is 
equal to the two interior angles, and the angles at the 
base of an isosceles triangle are equal ; the former de- 
pends on the angle properties of parallels, the latter 
on congruence. Following this process, wherever we 
NO. 2783, VOL. 111] 
NATURE 
{Marcu 3, 1923 
begin, we always get back to one or both of these 
fundamental principles. fe 
This illustration shows how grasp of sequence can 
be strengthened ; as illustration of interconnexion 
take Pythagoras’s proposition. It may be proved, 
as in Euclid, by use of parallelograms and congruent 
triangles ; or by variants, using parallelograms only, 
which, however, depend on congruent triangles ; but 
again it may be proved by the use of similar triangles — 
(Euclid VI. 8). But similar triangles rest on the 
angle properties of parallels and on Euclid VI. 2, or 
the equivalent proposition as to the segments made 
on transversals by parallels, and this, again, depends 
on congruence. Similarly, it seems unwise to neglect 
either of the proofs of Euclid III. 35, 36 (rectangles 
contained by segments of chords); the proof by 
similarity is the easier and shows the inwardness of — 
the proposition better ; Euclid’s proof brings out the 
important fact that the rectangle is. equal to k®—7?, 
the “power of the point.’ Illustrations might be 
multiplied ; but these will suffice to indicate what 
is meant, the habit of tracing connexions which gives 
mastery of the whole, and, it may be added, greatly 
increased power in what, after all, is the essential 
thing, the art of doing riders. 
The second question does not, perhaps, as yet 
admit of so definite an answer: when and how far 
should pupils be asked to face the initial difficulties— 
congruence, parallelism, and the link propositions — 
(e.g. inequalities) necessary for dealing with them? — 
A partial answer may be given with some confidence : 
not until they have mastered the rest of the work 
and have gained power in solving problems. Beyond 
this it is not safe to dogmatise, but if geometry is 
worth studying for its own sake, for its beauty and 
essential interest, and not merely as an exercise in — 
logic, it is quite possible, and, indeed, for most boys 
probable, that they will gain more by going on— 
by studying the ordinary developments not contained 
in Euclid, e.g. coaxal circles, pole and polar, inversion, 
etc., and geometrical conics, to say nothing of solid 
geometry—than by going back to examine first: 
principles. Still in sixth form work, possibly in ~ 
favourable circumstances in a fifth form, time 
might well be found for this; properly handled it 
would arouse great interest and would certainly be — 
well within the power of the boys—as it is not within 
that of a third form. It involves, above all, the 
parallels axiom and some consideration of the relation- 
ship between axioms and definitions ; in fact, it is 
quite as much a philosophic as a mathematical question. 
Its treatment would be rendered more effective by 
some knowledge of non-Euclidean geometry on the 
part of the teacher. 

