388 
NATORE 


[ Sept. 14, 1871 

one. And I think it is admitted that because this obser- 
vational or practical geometry is wanting in our elemen- 
tary mathematical teaching, geometry is generally found 
so difficult, so inexplicably difficult, by boys. 
It does not suffice to give a child a box of geometrical 
solids, and let him handle them and learn their names, 
though this is not useless. Nor does it suffice to give him 
a ruler and a pair of compasses to play with ; and, in fact, 
the more we reflect on what is required to give an interest 
to the observations out of which familiarity with geo- 
metrical facts is to spring, the more inevitably, it seems to 
me, are we led to the conclusion that practical geometry is 
to be taught not #7 se, but by practical work, by interest- 
ing and varied applications of geometrical methods to 
measure and copy actually existing things. 
And this at once suggests that the elementary teaching 
of practical geometry should consist in the manipulation 
of measuring instruments, and the calculations based on 
these measurements, which lie at the foundation of sound 
scientific work. I believe that all such measurements and 
calculations and practical constructions as are within 
the range of a boy, might be profitably laid before him 
as his work in elementary geometry: and I believe that 
this kind of training would moreover be of the very 
highest value in preparing him for good experimental 
work, and for a sound appreciation of scientific methods 
and results. 
It will be advisable, however, to go into some degree of 
detail in order to explain my meaning to such as are not 
familiar with these parts of education ; and in doing so, 
I must confess that I have not tested these details through- 
out by actual experience. For we have at Rugby to deal 
with older boys than those whom I am now contemplating, 
and though we do something of the kind with our younger 
boys, yet it is not what I should choose if I had the 
control of boys’ education from an earlier age. Any one 
whowishesto see ouractual course of practical geometry can 
do so by ordering Kitchener’s “ Geometrical Note-book” 
from the publisher of NATURE. But it must be under- 
stood to be a stopgap, and not a complete work. (I trust 
the author will forgive me for saying so.) 
Let a boy be furnished with a ruler, a triangle (in plain 
wood), a pair of compasses, and a protractor. Let him 
have a hard pencil, and be taught how to sharpen it. 
First let him draw on card a decimetre scale, divided into 
centimetres, and in part into millimetres. This, of course, 
he must copy from some trustworthy scale. Insist on this, 
and on every part of his work being done with great care 
and perfect neatness, and therefore not in ink. 
These are his tools. He must proceed to measure some 
figures provided him for this purpose ; a few triangles, 
quadrilaterals, &c, in wood, or figures drawn on paper, 
are sufficient for this purpose. Every one of his measure- 
ments is neatly written ina suitable note-book, and the 
figure to which they apply is drawn (freehand) therein. 
The next thing to proceed to is the measurement of 
angles, and the expression of the result in degrees and 
minutes, with exercises suggested by Euclid, i. 32, and its 
corollaries, properties of the circle, &c., which are to be 
practically verified, the observed results being written 
down, and compared with the theoretical results. 
Then the lad may go on to the practical measurement 
of areas, beginning of course with a rectangle, which he 
divides into square centimetres and millimetres ; he goes 
through the practical proof that thearea of the triangle is 
half that of the rectangle of equal altitude on the same 
base ; he proves Euclid, i. 47, and iii. 35 ; he finds the 
areas of various polygons of which drawings or models 
are given him. 
Mensuration of solids is next approached, and here 
probably a few rules will have to be given, by which 
volumes are calculated from linear measurements. But 
in all cases the measurements must be made by the boy 
himself with his compasses and scale. Any one who 


pleases can show his pupils how to prove the relation 
between the volumes of pyramids and prisms by weighing 
models of suitable dimensions. The same method may 
easily be applied to determine approximately the area 
of acircle ; and in this, as in some other measurements, 
it will be well to require an estimate of the degree of 
approximation attained, and a mean to be taken of several 
measurements, 
If more applications are wanting, the use of co-ordi- 
nates to express position may be explained, and some 
examples may be given of their application in simple 
problems, such as to make a plan of a room or of a garden, 
the scale being specified ; and to copy a drawing, such 
as the sun with a group of spots. More advanced work 
to any amount will be offered by projections. The boy 
would be required to draw the projections of the various 
regular solids given to him, and perform the usual exer- 
cises of geometrical drawing. The construction by ruler 
and compasses of exact copies of triangles and other 
figures may be introduced almost anywhere, and a clear 
statement given of the different data from which a triangle © 
can be constructed. 
I wonder what “ A Father” and mathematical teachers 
say to these suggestions. It will be obvious that they do 
not aim at making a boy a rapid analyst, or an expert pro- 
blem solver ; but I hope it is equally obvious that they 
are really calculated to make a careful and exact worker, 
one who shall attach precise meaning to his words, and 
shall be capable of using his head and hands in combi- 
nation with one another in practical problems. The 
method is, moreover, applicable to a class, as well as to 
an individual pupil, and involves a very trifling expendi- 
ture on materials. 
When some such course of practical geometry has been 
gone through, a boy may begin any scientific or deductive 
geometry ; he had better read whatever book is read in the 
school to which he is going. A boy so prepared will find 
Euclid easy enough, but rather unaccountably indirect and 
clumsy ; but he may be fortunate enough to be going toa 
school which has adopted some better arranged text book. 
In a year or two there will be better modern text books 
than now exist. Whatever book he reads, he ought to 
work many examples, and do original work. It is not a 
bad plan to give him the enunciations alone, and let him 
discover the proofs as far as he can. Perhaps the best 
text book now existing is the “ Eléments de Géometrie,” 
par Ch. Briot. 
My remarks have run to considerable length, much 
greater than I intended, and I can apologise for it only 
on the ground that many teachers are thinking of the 
question handled in it, and that it is only by imparting 
our notions and our experience to one another that we 
shall improve our methods. I sincerely hope that ‘A 
Father’s” letter to you may elicit answers from teachers 
more experienced and successful than I am, 
J. M. WILSON 
ON FRESH DISCOVERIES OF PLATYCNEMIC 
MEN IN DENBIGHSHIRE 
7 the course of 1869 I had the good fortune to discover 
and explore a sepulchral cave at Perthi Chwareu, a 
farm about fourteen miles north of Corwen, and high 
up in the region of hills. It contained fifteen or six- 
teen skeletons, some of which were buried in a sitting pos- 
ture, of ages varying from infancy upwards, and associated 
with the broken bones of animals that had been eaten, 
which belonged to the dog, fox, badger, horned sheep, 
Celtic shorthorn (Los ongi/rons), roe, stag, horse, wild boar, 
and domestic hog. The solitary work of art left behind 
by man consisted of a flint flake, but there were also frag- 
ments of A/ya ¢runcata, and of mussel and cockle-shells. 
The cave had been evidently used as a place of habitation 
