Oct. 19, 1871 | 

object of the problem, the bisection of the angle, though 
the line H K will converge in common with the two given 
lines. We must further enter protest against the wagual7- 
fied proposal “to draw a straight line equal to the true 
length of the circumference of a circle” (Prob, 184) as 
misleading to the learner. But, any such defects notwith- 
standing, here is a most wonderful eighteenpenny book. 

LETTERS TO THE EDITOR 
[The Editor does not hold himself responsible for opinions expressed 
by his Correspondents. No notice is taken of anonymous 
communications, | 

Geometry at Oxford 
In the last number of Nature Mr. Proctor remarks that ‘‘no 
one who considers carefully the mathematical course at either 
University, can believe that it tends either to form geometricians 
or to foster geometrical taste.” 
With regard to Oxford, I think it is only fair that some quali- 
fication should be offered to this conclusion, In Cambridge, 
candidates for mathematical honours have to run their race in a 
course clearly marked out for them, and loss of place is naturally 
the result of individual vagaries. But in Oxford the order of 
merit is not carried further than distribution into classes, and I 
do not believe there is anything to prevent a skilful geometrician 
finding himself in the first class with those who put their trust 
most in analytical methods. 
I cannot pretend to much geometrical capacity, but I know 
something of Oxford mathematical teaching. Speaking for my- 
self, the fascinating lectures of the present Savilian Professor of 
Geometry will never cease to hold perhaps the most prominent 
place in my recollections of university work. It is quite true 
that I remember conversing with a college tutor who was rather 
doubtful about modern geometrical methods, and seemed disposed 
to look upon these lectures as ‘‘dangerous.” He was a great 
stickler, however, for ‘‘legitimacy,” thinking it wrong, for ex- 
ample, to import differential notation into analytical geometry ; 
but I do not think he had a large following amongst younger 
Oxford men. I certainly did not find, in reading with some of 
them, that geometry was at all in disfavour. I have often had 
neat geometrical solutions pointed out to me of problems where 
other methods proved cumbrous or uninteresting ; and conversely 
I have found geometrical short cuts were far from objected to. 
On the whole, the characteristic feature of the Oxford exami- 
nation system (most marked in the Natural Science School, but 
making itself felt in all the others) being to encourage a student 
after reaching a certain point in general reading to make himself 
strong in some particular branch of his subject, I believe special 
attention to geometrical methods would pay very well. 
Oct. 13 W. T. THISELTON DYER 
Elementary Geometry 
Your ‘correspondent, ‘‘A Father,” has in view a very 
desirable object—to teach a young child geometry—but I fear 
that he is likely to miss altogether the path by which it may be 
reached. His principle, that ‘‘a child must of necessity commit 
to memory much that he does not comprehend,” appears to me 
to be totally erroneous, and not entitled to be called a fact. To 
this time-hallowed principle it is due that a large proportion of 
all wno go to school learn nothing at all, while those more suc- 
cessful learn with little improvement of their faculties. It isa 
convenient principle which allows the title of teacher to be as- 
sumed by those who only hear lessons. Children labour under 
this difficulty that they learn only through language, which is to 
them a misty medium, particularly when the matter set before 
them is in any degree novel or abstruse, and no pains are taken to 
clear up the obscurity of new expressions. Children know 
nothing of abstraction, and learn to generalise from experience, 
not from words. Committing tomemory what is not understood 
is a disagreeable task ; begetting a hatred of learning, and causing 
many to believe that they want the special faculty required for 
the task set before them. ‘The art of teaching the young ought 
to be the art of enabling them to comprehend, and memory 
ought to be strengthened not by drudgery but by being founded 
on understanding and by the rational connection of ideas. 
Now geometry is the science of figure ; it theorises reality, and 
the truth of every proposition in it may be made apparent to the 
NATURE 

485 
senses. Double a piece of paper and cut out a triangle in dupli- 
cate. The two equal triangles thus formed, A and B, may be 
put together so as to form a parallelogram in three different 
ways. The child who makes this experiment will learn at once 
LES NESS EEG EB ie aa 
what is meant by a parallelogram, and he will perceive its pro- 
perties, viz., that its opposite sides and angles are equal ; that it 
is bisected by the diagonal, &c. But if he learns all this by rote, 
he acquires only a cloud of words, on which his mind never 
dwells. Propositions touching abstractions and generalisations 
can never be understood by the young without abundant illus- 
tration. When a geometrical truth is made apparent to the 
senses, when seen as a fact and fully understood, the language in 
which it is expressed having no longer a dim and flickering light, 
is easily leaned and remembered, and the learner listens with 
pleasure to the discussion of the why and wherefore. 
It is not enough for a child to learn by rote the definition of 
anangle, He ought to be shown howit is measured by a circle ; 
and by circles of different sizes. In short, he ought to be taught 
what words alone will not teach him, that an angle is only the 
divergence of two lines. Let us now come to the important 
theorem that the three angles of any triangle are equal to two 
right angles (Euclid i. 32). Cut a paper triangle, mark the 
angles, then separate them by dividing the triangle and place the 
three angles together.. They will lie together, filling one side of 
a right line, «nd thus be equal to 
two right angles. Let the learner 
test the theorem with triangles of 
every possible shape to convince 
himself of its generality, and then, 
fully understanding what it means, 
he will also understand the lan- 
guage in which it is proved. 
It is a mistake to decry the use 
of symbols. They enable us to E ; ; 
get rid of the wilderness of words, which form a great impedi- 
ment in mathematical reasoning. Ordinary language can never 
group complex relations for comparison so compactly as to bring 
them within the grasp of the understanding. When we would 
compare objects, we place them close together, side by side. 
But the features and lineaments of objects described in language 
are too widely scattered to be kept steadily in view. Itis easier 
to learn the use of symbols than to commit to memory what is not 
understood. Those who would learn mathematics without sym- 
bols can advance but a little way. yt 
Neither is there any good reason for rejecting the second book 
of Euclid, though it certainly may be much abridged. The rela- 
tions of whole and parts, sum and difference are easily exhibited, 
and an acquaintance with them is of great value to arithme- 
ticians, Let us take for example the following propositions : 
“© The squares (A and B) of any two lines (or numbers) are equal 
to double the rectangle under those lines (R and R, or the pro- 
duct in case of numbers) and the square of their difference D.” 








Now these figures being constructed, it will be found that when 
the two squares are placed together as in Fig. 2, the rectangles 
cover exactly the parts marked with diagonals, and the square of 
the difference the remainder. 
In numbers, the square of 5 = 25 
” ay =e) 
34 
Double product of 5 and 3 30 
Square of diff. 4 
34 
