DEMONSTRATION, AND NECESSARY TRUTHS. 335 



physical fact. They are names of the same objects, 

 but of those objects in two different states : though 

 they denote the same things, their connotation is 

 different. Three pebbles in two separate parcels, and 

 three pebbles in one parcel, do not make the same 

 impression on our senses ; and the assertion that the 

 very same pebbles may by an alteration of place and 

 arrangement be made to produce either the one set of 

 sensations or the other, though it is a very familiar 

 proposition, is not an identical one. It is a truth 

 known to us by early and constant experience : an 

 inductive truth ; and such truths are the foundation 

 of the science of Number. The fundamental truths of 

 that science all rest upon the evidence of sense ; they 

 are proved by showing to our eyes and our fingers 

 that any given number of objects, ten balls for 

 example, may by separation and re-arrangement 

 exhibit to our senses all the different sets of numbers 

 the sum of which is equal to ten. All the improved 

 methods of teaching arithmetic to children proceed 

 upon a knowledge of this fact. All who wish to 

 carry the child's mind along with them in learning 

 arithmetic ; all who (as Dr. Biber in his remarkable 

 Lectures on Education expresses it) wish to teach 

 numbers, and not mere ciphers now teach it through 

 the evidence of the senses, in the manner we have 

 described*. 



* See, for illustrations of various sorts, Professor LESLIE'S Philo- 

 sophy of Arithmetic; and see also two of the most efficient books 

 ever written for training the infant intellect, Mr. HORACE GRANT'S 

 Arithmetic for Young Children, and. his Second Stage of Arithmetic, 

 both published by the Society for the Diffusion of Useful Know- 

 ledge. 



" Number," says the reviewer of Mr. Whewell, already cited, 

 " we cannot help regarding as an abstraction, and consequently its 



