ARITHMETICAL INSTRUMENTS. 23 



had learned to distinguish between discrete, or discontinuous, and 

 continuous quantity. All counting, properly so called, is of dis- 

 continuous quantity ; all measurement is of continuous quantity. 

 To use a simple illustration : if we are counting points or dots on 

 a line, we can say, " two dots and one dot make three dots ; " if 

 we are measuring inches we can equally say, " two inches and one 

 inch make three inches." But in the latter case we can, if we 

 please, pass by insensible degrees, and through every intermediate 

 gradation of magnitude, from two inches to three inches : in the 

 former case we can only pass 'abruptly from counting two dots to 

 counting three dots ; there is no such thing as half a dot, and no 

 intermediate stage is conceivable. 



But while this important distinction was clearly seen in very 

 ancient times, being indeed of a nature to commend itself specially 

 to the philosophical spirit of classical antiquity, there was not an 

 equally distinct apprehension of the truth that continuous quan- 

 tity, no less than discontinuous, appertains to the domain of 

 arithmetic. By whom the first dim perception, or by whom the 

 first vivid realisation, of this truth was attained, we have no 

 means of ascertaining with precision. It must have been gradu- 

 ally impressed on the minds of men by the growth of science. 

 It is, perhaps, hardly discernible in the writings of Plato and 

 Aristotle : it underlies, but is carefully excluded from, the fifth 

 book of the Elements of Euclid. It must have been present to 

 the mind of Archimedes when he measured the proportions to 

 one another of the sphere, the cylinder, and the cone ; it must 

 have forced itself on the notice of the Greek astronomers, whose 

 business it was to record numerically at discontinuous intervals 

 the phases of continuous phenomena ; and it became firmly esta- 

 blished as an axiomatic principle by the development of that 

 mode of arithmetic which is called algebra ; by the great inven- 

 tion of Descartes which reduced geometry to algebra ; and, last 

 of all, by the creation of those arithmetical methods which are 

 briefly described as the infinitesimal calculus. 



