122 THE PRINCIPLES OF SCIENCE. 



On the Fictitious Mean or Average Result. 



Although the average when employed in its proper 

 sense of a fictitious mean, represents no really existing 

 quantity, it is yet of the highest scientific importance, as 

 enabling us to conceive in a single result a multitude 

 of complex details. It enables us to make a hypothetical 

 simplification of a problem, and avoid complexity without 

 committing error. Thus the aggregate weight of a body is 

 the sum of the weights of the indefinitely small particles, 

 each acting at a different place, so that the simplest 

 mechanical problem concerning a body really resolves itself, 

 strictly speaking, into an infinite number of distinct pro 

 blems. We owe to Archimedes the first introduction of 

 the beautiful idea that one point might be discovered in 

 a gravitating body such that the weight of all the par 

 ticles might be regarded as concentrated in that point, 

 and yet the behaviour of the whole body would be exactly 

 represented by the behaviour of this heavy point. This 

 Centre of Gravity may be within the body, as in the 

 case of a sphere, or it may be in empty space, as in 

 the case of a ring. Any two bodies, whether connected 

 or separate, may be conceived as having a centre of 

 gravity ; that of the sun and earth, for instance, lying 

 within the sun and only 267 miles from its centre. 



Although we most commonly use the notion of a centre 

 or average point with regard to gravity, the same notion 

 is applicable to many other cases. Terrestrial gravity 

 is only one case of approximately parallel forces, so that 

 the centre of gravity is but a special case of the more 

 general Centre of Parallel Forces. Wherever a number 

 of forces of whatever amount act in parallel lines, it 

 is possible to discover a point at which the algebraic 

 sum of the forces may be imagined to act with exactly 

 the same effect. Water in a cistern presses against the 



