wategs” 
ADDRESS. 37 
a given line ; g* that it shall cut each of two given lines; and so in other 
cases ; and these logical symbols are combined together by algebraical 
Jaws: 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 co- 
ordinates is a possession forever. But mathematics have never been culti- 
vated more zealously and diligently, or with greater success, than in this 
century—ain 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. 
