NATURE 
THURSDAY, DECEMBER 3, 1903. 
THE REFORMATION OF THE TEACHING OF 
GEOMETRY. 
Elementary Geometry, Practical and Theoretical. By 
C. Godfrey, M.A., Winchester College, and A. W. 
Siddons, M.A., Harrow. Pp. xi+355.  (Cam- 
bridge: University Press, 1903.) Price 3s. 6d. 
A New Geometry for Schools. By S. Barnard, M.A., 
Rugby, and J. M. Child, B.A., Technical College, 
Derby. Pp. xxvit+514. (London: Macmillan and 
Co., Ltd., 1903.) Price 4s. 6d. 
pie years ago the writer of this review, when 
publishing a small book on geometry for the 
use of pupils of eight or nine years of age, was assured 
by many friends that the attempt to get rid of Euclid’s 
order and language was a hopeless one, and that, even 
if it were successful, the foundations of all logical 
thought in England would be destroyed. Against a 
strong conservatism it seemed vain to point out that 
the great developments of modern geometry were made 
by European mathematicians who were not brought 
up on Euclid. The question had been tackled before, 
but with no success. When, however, the British 
Association and the Mathematical Association 
appointed committees to grapple with the matter in 
earnest, the victory of reform was assured. 
former body thought it wise to lay down generalities, | 
while the latter went into such minute details as to 
the course to be pursued by teachers, and the proposi- 
tions which should be included in school instruction, 
that it has been possible to embody its recommend- 
ations in definite systematic treatises, such as the two 
excellent works the titles of which are quoted above. 
That some recognised order of deduction must be 
established is a fact which is forced upon anyone who 
has to perform the part of an examiner, more 
especially for the public service, and the fact that there 
is a close agreement, not only in method, but in order, 
between the two works before us shows that the diffi- 
culty of dethroning Euclid is quite imaginary. 
Each of these books is a vigorous protest against 
the extraordinary contention which we have sometimes 
heard, that ‘‘ you must make bad figures in geometry 
so that the logical faculty of the pupil shall receive no 
assistance from them.’’? Rule, compass, set square, 
and protractor are now the tools with which the young 
pupil begins his acquaintance with this subject; and 
we venture to say that, under the new system, the 
typical schoolboy will change his attitude of repug- 
nance to “that beastly Euclid’’; the subject will 
The | 
actually become popular. 
The work of Messrs. Godfrey and Siddons begins | 
with fifty-nine pages of ‘‘ experimental geometry,’ in | 
which the pupil is taught to draw various figures by 
the use of scales, compasses, &c. There is no formal 
list of definitions; the definitions are given as they 
are required. 
Messrs. Barnard and Child open with a list of 
definitions, each of which, however, is illustrated by 
a good clear figure, and then follows part ii. of the | 
NO. 1779, VOL. 69] 
97 
book, which is ‘ practical,’’ and occupies 224 pages. 
This is occupied wholly by constructions, and many of 
these constructions are to be taken in conjunction with 
corresponding theorems, to which the pupil is duly 
referred in part iii. of the book, which is ‘‘ theoretical.”’ 
It must not be supposed, however, that part ii. is 
merely constructive—that is, that the pupil is directed 
to perform certain operations without understanding 
the reason. The constructions are, almost invariably, 
accompanied by a justifying proof, and the whole 
collection seems to be exhaustive. Here the nature of 
an envelope is also explained, and a few examples of 
the drawing of envelopes are given. 
Under the head of constructions we find also the 
definitions of the trigonometrical functions, and some 
constructions founded thereon, so far as one angle is 
concerned. There is also a section dealing with the 
displacement of a lamina in its own plane, and the 
nature of the instantaneous centre of rotation. It is 
needless to say that the plotting of figures on squared 
paper and the measurement of areas thereby occupy 
a fair space in this section. The principles of folding 
and superposition, also, are largely employed as a 
means of proof. There is no doubt that in this work 
of Messrs. Barnard and Child the teacher will find 
every requisite for the modern teaching of geometry, 
including a very large number of illustrative examples. 
The collection of all construction propositions into one 
large section by themselves is the main difference 
between the two works before us. 
In the work of Messrs. Godfrey and Siddons the 
constructions appropriate to each branch of the subject 
form a special section in that branch; thus construc- 
tions relating solely to triangles are talken together in 
the part of the book dealing with congruent triangles, 
those relating to circles in the part dealing with 
circles. 
There is a remarkable similarity of procedure in the 
theoretical or deductive portions of both works. Each 
begins with the discussion of angles at a point, then 
follow the treatment of parallel lines, angles of a 
triangle and external angles of a polygon, congruent 
triangles, inequalities (i.e. of sides and angles of a 
triangle) and parallelograms, closely followed in each 
work by the discussion of areas. 
Messrs. Godfrey and Siddons adopt the invariable 
plan of accompanying each proposition with a large 
series of examples and applications. Sometimes we 
come across a well chosen practical example calculated 
to enlist the interest and sympathy of the young pupil 
—such as the application of a simple case of congruent 
triangles to the finding of the breadth of a river, of 
which a figure is given. The work is a charming one, 
marked by great simplicity. 
Squared paper and the plotting of coordinates find 
also ample space in this book. Geometry is supposed 
to have arisen from the necessities of land-surveyors, 
but any such mundane connection has been so ‘long 
severed that we find ourselves astonished when we 
actually see (p. 180) an irregular figure plotted and its 
| area estimated by a process of give and take—and this 
in the midst of some of Euclid’s propositions, too! 
Truly the times have altered rapidly—a still further 
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