BRITISH ASSOCIATION, SEPTEMBER 1883. 
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bination (the distributive, commutative, and associative laws) but which are such that 
for any one of them, say x, we have x — x 2 — 0, this equation not implying (as in ordinary 
algebra it would do) either x=0 or else x—1. In the latter part of the work relating 
to the Theory of Probabilities, there is a difficulty in making out the precise meaning 
of the symbols; and the remarkable theory there developed has, it seems to me, passed 
out of notice, without having been properly discussed. A paper by the same author, 
“ Of Propositions numerically definite ” (Gainb. Phil. Trans. 1869), is also on the border 
land of logic and mathematics. It would be out of place to consider other systems 
of mathematical logic, but I will just mention that Mr C. S. Peirce in his “Algebra of 
Logic,” American Math. Journal, vol. III., establishes a notation for relative terms, and 
that these present themselves in connexion with the systems of units of the linear 
associative algebra. 
Connected with logic, but primarily mathematical and of the highest importance, 
we have Schubert’s Abziihlende Geometrie (1878). The general question is, How many 
curves or other figures are there which satisfy given conditions ? for example, How 
many conics are there which touch each of five given conics ? The class of questions 
in regard to the conic was first considered by Chasles, and we have his beautiful 
theory of the characteristics g, v, of the conics which satisfy four given conditions; 
questions relating to cubics and quartics were afterwards considered by Maillard and 
Zeuthen; and in the work just referred to the theory has become a very wide one. 
The noticeable point is that the symbols used by Schubert are in the first instance, 
not numbers, but mere logical symbols: for example, a letter g denotes the condition 
that a line shall cut a given line; g 2 that it shall cut each of two given lines ; and so 
in other cases; and these logical symbols are combined together by algebraical laws: 
they first acquire a numerical signification when the number of conditions becomes equal 
to the number of parameters upon which the figure in question depends. 
In all that I have last said in regard to theories outside of ordinary mathematics, I 
have been still speaking on the text of the vast extent of modern mathematics. In 
conclusion I would say that mathematics have steadily advanced from the time of the 
Greek geometers. Nothing is lost or wasted ; the achievements of Euclid, Archimedes, 
and Apollonius are as admirable now as they were in their own days. Descartes’ method 
of coordinates is a possession for ever. But mathematics have never been cultivated 
inore zealously and diligently, or with greater success, than in this century—in the last 
half of it, or at the present time : the advances made have been enormous, the actual 
field is boundless, the future full of hope. In regard to pure mathematics we may 
most confidently say :— 
Yet I doubt not through the ages one increasing purpose runs, 
And the thoughts of men are widened with the process of the suns. 
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