4o8 Edward Livingston Youmans. 



and with a varying application to universal phenomena, 

 they are an indispensable key to universal science, and 

 their basis is, therefore, a broad and solid utility. But the 

 devotees of tradition are not satisfied with this ; they make 

 extravagant claims for mathematics, on the ground of the 

 discipline they afford, and then usurp for them an edu- 

 cational predominance to which they are not entitled. In 

 their subordinate place they are invaluable ; as a too en- 

 grossing subject of study, injurious. Mathematics are 

 suited to form habits of continuous attention by dealing 

 with trains of proof, to help the imagination steadily to 

 grasp abstract relations, and to familiarize the mind with a 

 system of necessary truth. But they do not afford a com- 

 plete exercise of the reasoning powers. They begin with 

 axioms, self-evident truths, established principles, and pro- 

 ceed to their conclusions along a track each step of which 

 is an intuitive certainty. But it so happens that in our 

 mental dealings with the experiences of life, the first, the 

 most important, and most difficult thing is to get the data 

 or premises from which to reason. The primary question 

 is, What are the facts, the pertinent facts, and all the facts, 

 which bear upon the inquiry ? This is the supreme step ; 

 for, until this is done, reasoning is futile, and it may be 

 added that, when this is done, the formation of conclusions 

 is a comparatively simple process. Now mathematical 

 training cannot help to this important preliminary work ; 

 it leaves its cultivator to the blind acceptance or blind re- 

 jection of his premises. Those, therefore, who have ex- 

 clusively pursued these studies, so as to form mathematical 

 habits of thinking, have no preparation for the practical 

 emergencies of thought, where contingencies are to be 

 taken into account, where probable evidence is to be 

 weighed, and conclusions from imperfect knowledge are to 

 be formed and acted upon. The pure mathematician is 

 therefore liable to a one-sided and erratic judgment of 



