566 JOURNAL OF SCIENCE, 
exact sciences, viz. geometry and algebra. These two, although 
among the oldest, are, as Professor Cayley very justly reminded the 
Association last year, perhaps the most progressive and promising of 
all the sciences. Great names of antiquity are associated with them, 
and in modern times an army of men of genius have aided their 
advance. Moreover, it cannot be said that this advance concerns the 
higher parts of these sciences alone. On the contrary, the discoveries. 
of Gauss, Lobatschewsky, and Reimann, and of Poncelet, Mobius, 
Steiner, Chasles, and Von Standt, in Geometry, and the labours of 
De Morgan, Hamilton and Grassman, not to mention many others in 
Algebra, have thrown a flood of light on the elements of both these 
subjects. What traces of all this do we find in our school books? 
To be sure antiquity is stamped upon our geometry, for we use the 
text-book of Euclid, which is some two thousand years old; but where 
can we point to the influence of modern progress in our geometrical 
teaching? For our teaching of algebra, I am afraid, we can claim 
neither the sanction of antiquity nor the light of modern times. 
Whether we look at the elementary, or at what is called the higher 
teaching of this subject, the result is unsatisfactory. With respect to 
the former, my experience justifies the criticism of Professor Henrici; 
and I have no doubt that the remedy he suggests would be effectual. 
In the higher teaching, which interests me most, I have to complain 
of the utter neglect of the all-important notion of algebraic form. I 
found, when I first tried to teach University students co-ordinate 
geometry, that I had to go back and teach them algebra over again. 
The fundamental idea of an integral function of a certain degree, 
having a certain form and so many coefficients, was to them as much 
an unknown quantity as the proverbial x. I found that their notion 
of higher algebra was the solution of harder and harder equations. 
The curious thing is that many examination candidates, who show 
great facility in reducing exceptional equations to quadratics, appear 
not to have the remotest idea beforehand of the number of solutions 
to be expected; and that they will very often produce for you by 
some fallacious mechanical process a solution which is none at all. 
In short, the logic of the subject, which, both educationally and | 
scientifically speaking, is the most important part of it, is wholly 
neglected. The whole training consists in example grinding. What 
should have been merely the help to attain the end has become the end 
itself. The result is that algebra, as we teach it, is neither an art nor 
a science, but an ill-digested farrago of rules, whose object is the 
solution of examination problems. 
“The history of this matter of problems, as they are called, 
illustrates in a singularly instructive way the weak point of our 
English system of education. They originated, I fancy, in the Cam- 
bridge Mathematical Tripos Examination, as a reaction against the 
abuses of cramming book-work, and they have spread into almost 
every branch of science teaching—witness test-tubing in chemistry. 
At first they may have been a good thing; at all events the tradition 
at Cambridge was strong in my day, that he that could work the 
most problems in three or two and a-half hours was the ablest man, 
and be he ever so ignorant of his subject and its width and breadth, 
could afford to despise those less gifted with this particular kind of 
superficial sharpness. But, in the end, came all to the same: we 
