650 SCIENTIFIC THOUGHT. 



the highest rank, who for some time had lived apart 

 in the secluded regions of sublime analysis, descended 

 again into the region of elementary science, both pure 

 and applied, where they speedily remodelled the entire 

 mode of teaching. England possessed very early a writer 

 of great eminence who represented this tendency, and 

 whose merits were only partially recognised in his day 

 Augustus de Morgan. 

 is. It will now be necessary to -explain more definitely 



Extension of 



conception what is meant by the extension of our conception of 



of number. 



number and quantity through the introduction of com- 

 plex numbers or complex quantities. This extension 

 first forced itself on analysts in the theory of equations, 

 then in the algebraical treatment of trigonometrical 

 quantities i.e., in the measurement of angles, or, as 

 it is now called, of direction in geometry. The first 

 extension of the conception of number lay in the intro- 

 duction of negative numbers. These admitted of com- 

 paratively easy representation arithmetically by counting 

 backward as well as forward from a given datum ; 

 practically in the conception of negative possessions, 

 such as debts, geometrically by the two opposite direc- 

 tions of any line in space. In algebra, where the simple 

 operations on quantities are usually preserved in the 

 result and not lost in the simple numerical value of 

 the result as in arithmetic, compound quantities were 

 looked upon as generated by the processes of addition, 

 resulting in the binomial (of which the polynomial was 

 an easy extension), and further by the multiplication 

 with each other of different binomials or polynomials, 

 through which process expressions of higher order or 



