COMPOSITION OR AGGREGATION OF FORCES IS A CONSEQUENCE. 295 



in terms of the abstract numbers C : A and B : A, we can have no relation between C, A, B, 

 except in the form C = Acp {B: A), independently of c. But when ^ = 0, C = A + B : 

 therefore this relation always exists. It must be observed that the force of this demonstration 

 depends upon our having introduced C : A and B : A, not from any argument upon the 

 necessity of expression by ratios, but from the very postulates themselves. 



On the same principles we find that, a, b, c being the distances of the tendencies A, B, 

 and the aggregate C, from a given point in the line of application, the ratio of c - a and c - b 



is determined by the ratio of A to B. Hence an equation of the form c =/{'„] "■ + f {—) b, 



the unknown function being subject to the condition ^f—j +/(-:) = !> expressing that 



when the two tendencies are applied at one point, their aggregate is also applied at that 

 point. 



Let any three tendencies P, Q, R be applied in one straight line at the distances w, y, z 

 from a given origin. Aggregate P + Q at the distance /{P : Q) a? +/{Q : P)y with R at 

 the distance z : the aggregate P + Q + ^ is then at the distance 



R 



/m-{/©-"4§-f/{^«) 



. ss. 



This is not disturbed by going through the same process in a different order of relation ; 

 hence the above is identical with 



/m!/(l)-/(l)-f/(.f«) 



x. 



Equate the coefficients of .v, write tv and « for P : ^ and Q : R, which gives 



tv 



Differentiation with respect to v gives, after elimination o{ ft, 





.(1 + 



{v + If 



= 0. 



may be a function of an angle : the very angle itself deter- 

 mines those numbers (ratios of lines) which we call sines and 

 cosines. Imagine a being incapable of conceiving angles in 

 the relations of whole and part: he may, if he can only con- 

 ceive lines in such relation, be forced to 4, or cos 60°, in its con- 

 nexion with the angle of an equilateral triangle. Let the differ- 

 ence of angles be to him merely the difference of individuals of 

 a species, as different men or different horses, yet he shall, by 

 measurement, ascertain on right-angled triangles that different 

 angles determine different numbers. Nor shall it be necessary to 

 think of a second angle, for the triangle may be determined by 

 drawing the shortest distance from a point in one side to the 

 other. Hence, though an angle be not a number till we settle a 



Vol. X. Part II. 



unit to which it may have ratio, yet a function of an angle may 

 be a number. Consequently, if c were an angle, the equation 



C = A<j> ( c, J I does not expel c, for it contains under it 



C=A(i>{^c, -^1, and i/zc may be a number varying with c. 



If any objection should arise in the case of a second angle 

 implicitly constructed, it may be removed by considering the 

 converse: a m«mier determines an angle, without reference to 

 any second angle, fllake that number the ratio of the arc 

 to the radius, and the angle is obtainable by means which 

 make no reference, even by tacit consequence, to any otiier 

 angle. 



38 



