236 Researches in the Theory and Calculus of Operations. 



or product l^xl^; from which result, relieving the burthen 

 1 xl 



by division — there remains l t as the relative measure 

 of a unit of operation; and finally eliminating the element 

 of space, i =1 gives the rational or numerical unit. 



Thus a first multiplication of 1 M by 1^ gives 1, = 1, the 

 first power of a geometrical unit; a second multiplication 

 carries 1,, from the 1 to the point 2, giving 1? = 2; 



a third multiplication gives 1?= 3 ; and 

 a fourth " gives If = 4, etc. 



Dismissing now 1 M and taking for multiplicand the unit 

 line 01 (fig. 30) regarded as a rigid physical line equal to the 

 unit of mass, and for multiplier the perpendicular unit line 

 01/, the operation l„xl, = 01x01/ shall express the transfer 

 of the former line parallel to itself, from the position 01 to 

 that of l'l", by uniform movement; thus covering the area 

 Oll'l" equalling the square constructed in 01=1^; that is, 

 by this method of interpretation, 1J=011'1". If then 04 (fig. 

 31) be multiplicand, successive multiplications by l x =01 

 (perpendicularly) will produce the respective areas 4, 8, 12, 

 16, as products of geometrical factors. 



Pursuing this method of interpretation, let the radius of 

 the circle (fig. 32) OA=rl, be multiplicand, and the mul- 

 tiplier 01== l x : the product becomes l^x r lj=r 1 2 = area O 

 A12 by movement of OA parallel to itself into the position 

 1 2. This shows at the same time that the area of the triangle 

 OA2 = Jr l 2 , and conterminously that the area of the sector 

 O A* equals half the product of the radius by the arc A f y 

 or area OA 2 = |r 2 dl 2 , & standing for the arc A*. If now the 

 radius be fixed at the centre O, it must necessarily describe 

 the sector OAt under the application of the multiplier l x , 

 instead of the including parallelogram. By a continuance 

 of this rotatory movement, the radius will describe the entire 

 area of the circle <7rr 2 l 2 , in a certain time tl t equal to the 

 sum of the partial times enumerated in describing the partial 



