394 REPORT—1883. 
equations containing 1, 2, 3, or more variables are employed. In such a division, 
Von Staudt’s system, without a vestige of algebra, would occupy the one end, and 
the purely algebraical theory of invariants with geometrical interpretation the 
other. 
There is, however, not only a difference in the amount of algebra used, but, if 
possible, a greater one in the manner in which the symbols are interpreted. And 
it is here that algebra has apparently the greater power. One algebraical theorem, 
by being read in different ways, by giving ever different meanings to the symbols, 
reveals a variety of geometrical and other theorems. We have in it the crystallised 
form, the very essence of the mathematical truth, but in the most abstract form 
conceivable. Now this most abstract form is the highest and the most perfect 
which mathematical truth as such can assume, and which it must assume before a 
theory is really complete in the eyes of a pure mathematician. It is only in this 
shape that it is ready to be turned fo account in any direction where it may be 
needed. 
In thus placing algebra on the highest pinnacle, the reasons will be apparent 
which will make many mathematicians, not to mention others, prefer the truths it 
reveals cast in a mould which connects them with concrete things rather than with 
abstract notions. In fact, to be thoroughly at home in the highest theories of pure 
algebra requires some of the genius of men like Cayley and Sylvester who have 
founded, and to a great extent built up, modern algebra. But even they constantly 
make use of geometry to assist them in their investigations, and no one could have 
expressed this more strongly than Professor Sylvester himself in his brilliant 
address delivered from this chair at the Exeter meeting of our Association. 
If this is so, surely every progress in the spread of the knowledge of pure 
geometry should be welcomed and encouraged ; but in England pure geometry is 
almost unknown excepting in the elements as contained in Euclid and in the old- 
fashioned geometrical conics. The modern methods of synthetic projective geo- 
metry as developed on the Continent have never become generally known here. 
The few men who have thoroughly made themselves acquainted with them, and 
who have preferred purely geometrical reasoning, have not belonged to Cambridge, 
and have thus stood somewhat outside the national system of training mathe- 
matical teachers. The late Professor Smith introduced these methods at Oxford, 
and there was some expectation that he would have written, if he had been spared, 
a text-book which might have done much to introduce the subject more widely. 
His principal mathematical work lay, however, in another direction. 
The one English mathematician whose mathematical thought is purely geo- 
metrical is Dr. Hirst, a pupil of Steiner, who in the position which he has just 
relinquished has been able to introduce, as the first, modern geometrical methods 
into 4 regular system of professional education, whilst showing at the same time by 
his original work what can be done with these methods. 
Other methematicians who have studied these methods—and I believe there are 
many—have made use of them by translating the geometrical into algebraical 
reasoning. 
Towards the early possibility of such a translation much was done by the 
labours of the late Mr. Spottiswoode, who years ago wrote the first connected 
treatise on the theory of determinants, and who up to the last few years employed 
some of his leisure hours in working out geometrical problems, the work consisting 
always of some beautiful piece of algebra. 
It is not often that our section has to mourn in one year the loss of two such 
men as Smith and Spottiswoode. 
It is easy to see how the neglect complained of has come to pass. In England 
when mathematics, after having lain dormant for about a century, began to revive, 
the first necessity was to become acquainted with the enormous amount of work 
meanwhile done on the Continent. This acquaintance was made through France, 
at that time nearly all the standard works being in the French language, which 
was at the same time the language best Inown to English students. The subjects 
principally taken up were the calculus and its application to mechanics. And I 
believe I am not far wrong when I say that the wonderful writings of Lagrange 
