396 REPORT—1883. 
In close connection with these methods stand descriptive geometry and 
geometrical drawing, which teach how to represent figures on a plane or other 
surface. These have been treated as arts unknown at English universities, and 
relegated to the drawing office. Instead of this they ought to be an essential 
and integral part of the teaching of geometry in connection with the purely 
geometrical methods, 
As far as the progress of science is concerned, this neglect of pure geometry 
in England has been of little consequence—perhaps it has rather been a gain. For 
science itself it is often an advantage that a centre of learning becomes one-sided, 
neglects many parts in order to concentrate all its energy on some particular points, 
and make rapid progress in the directions in which these lie. At present, when 
mathematics flourishes as never before, when almost every nation, however small, 
has its eminent mathematician, there are so many such centres that what is neglected 
at one place is pretty surely taken up and advanced at another. But what may 
suffer, if one side of a science is not cultivated in a country, is the industry which 
would have gained by its applications. 
In considering the teaching of any mathematical or other scientific subject, we 
cannot at the present time neglect the wants of the ever-increasing class of men 
who require what has been called technical education. Among these, the large 
number who want mathematics at all require geometry much more than algebra 
and analysis, and geometry as applied to drawing and mensuration. 
This want has been supplied by the numerous science classes spread over the 
country, with their head-quarters at the Science and Art Department at South 
Kensington, whose examinations—now, however, put in competition with those of 
the City and Guilds of London Institute, and others—have pretty much guided 
and regulated the teaching. A great deal of good has thus been done, but there is 
still much room for improvement. The teaching of geometry especially, as judged 
by the text-books which have come before me, is somewhat deplorable. And this 
is so, principally, because the spirit of Euclid and the methods of the ancient 
Egyptians and Greeks, rather than the fundamentally different ideas and methods 
of modern geometry, still rule supreme; though the latter have had their origin 
partly in technical wants. 
In what is called Geometrical Drawing, or Practical Geometry, for instance, 
there are first given a number of elementary constructions—such as drawing parallels 
and perpendiculars, or bisecting the distance between two given points. They are 
solved by aid of those instruments only which Euclid knew—viz. the pair of com- 
passes for drawing circles, and the straight-edge for drawing straight lines. But 
there is no draughtsman who would not, as a matter of course, use set squares for 
the former problem, and solve the latter by trial rather than by construction, Then 
again there come constructions like the division of the circumference of a circle into 
seven parts, which cannot be solved accurately, but which is very easily solved by 
trial. Instead of that, a construction is given which takes much more time, and is 
by no means more accurate. For, after all, our lines drawn on the paper are not 
without thickness, so that, for this reason alone, every part of the construction is 
affected by some small error; and it is absurd to employ a construction, though 
theoretically true for ideal figures as conceived in our mind, in preference to a much 
simpler one which, within our practical limits, is equally, or perhaps more, 
correct. 
This is very much like the manner in which I found problems on decimal fractions 
treated by the candidates for the Matriculation Examination at the London Uni- 
versity, and which reflected little credit on the manner in which the important subject 
of decimals is handled at our schools. It is so characteristic that I may be excused 
for giving it here. The problem, for instance, being to give the product of two 
decimal fractions, exact to, say, four decimals, each of the factors having the same 
number of places. This was almost regularly performed as follows. First, the 
decimals are converted into vulgar fractions, then these are duly multiplied, nume- 
rator by numerator, and denominator by denominator, and then the resulting fraction 
is again converted to a decimal, with as many places as it may yield, and, lastly, of 
these the first four are taken and put down, duly marked Answer. Or a candidate, 
