20 G. H. KNIBBS. 



In this brief and very imperfect sketch of the world's advance 

 in mathematical knowledge, it has been possible all through to 

 mention but few of those who have taken part in that great move- 

 ment. The achievements of many, whose names are unmentioned, 

 are such as the most brilliant might well envy. What has been 

 said will, however, have served its purpose if it has, even faintly 

 helped us to realize with what success the realm of exact know- 

 ledge has been exploited, and under what obligations we have 

 been placed by the devoted labours and splendid genius of the 

 mathematician. The world which he explores, and in which his 

 discoveries are made, is the world of mind : the depths he sounds 

 are the depths of human consciousness : the forms of truth which 

 he perceives are the structures of that imponderable world not 

 seen by the eye, but by the soul, for the relations and laws dis- 

 covered are conceptional not physical. The elements of the 

 mathematician's world are those ideas, which it is the high 

 function of intellect to project on to the world of sense in order 

 to render it intelligible. This truth, as old as Plato, perhaps as 

 the foundation of things, is that which lends its chief lustre to the 

 " queen of sciences." It has been naively said that the truths of 

 mathematics flow merely from its definitions : true, they are con- 

 sequent upon its definitions, but that very fact asserts the reach 

 and the structure of exact human thought, in respect of which 

 mathematics attains to a unique position, both as regards generality 

 and certainty. We may be allowed one illustration serving to 

 shew how transcendently the power of abstract mathematical 

 thinking surpassed all possibilities of physical verification, for 

 exact knowledge is never attained by physical experiment or 

 physical measurement. When once the law of a geometrical curve, 

 surface, or solid, is denned, it is in general possible to ascertain 

 by the processes of pure mathematics its length, area, or volume, 

 to any degree of precision one might wish. In 1854 Richter 1 

 carried the calculation of the ratio subsisting between the diameter 

 and circumference of a circle to 500 places of figures true to the 



1 Grunert's Archiv. xxv., p. 472. 



