292 THE PRINCIPLES OF SCIENCE. 



existence like force demands much power of conception, 

 but it has a perfect concrete representative in a line, the 

 end of which may denote the point of application, and the 

 direction the line of action of the force, while the length 

 can be made arbitrarily to denote the amount of the force. 

 Nor does the analogy end here; for the moment of the 

 force about any point, or its product into the perpen- 

 dicular distance of its line of action from the point, is 

 found to be correctly represented by an area, namely 

 twice the area of the triangle contained between the 

 point and the ends of the line representing the force. 

 Of late years a great generalization has been effected ; 

 the Double Algebra of De Morgan is true not only of 

 space relations, but of forces, so that the triangle of forces 

 is reduced to a case of pure geometrical addition. Nay, 

 the triangle of lines, the triangle of velocities, the triangle 

 of forces, the triangle of couples, and perhaps other 

 cognate theorems, are reduced by analogy to one simple 

 theorem, which amounts merely to this, that there are 

 two ways of getting from one angular point of a triangle 

 to another, which ways, though different in length, are 

 identical in their final results f . In the wonderful system 

 of quaternions of the late Sir W. R. Hamilton, these 

 analogies are embodied and carried out in the most 

 general manner, so that whatever problem involves the 

 threefold dimensions of space, or relations analogous to 

 those of space, is treated by a symbolic method of the 

 most comprehensive simplicity. Since nearly all physical 

 problems do involve space relations, or those analogous 

 to them, it is difficult to imagine any limits to the work 

 which may be ultimately achieved by this calculus. 



It ought to be added that to the discovery of analogy 



f See Goodwin, 'Cambridge Philosophical Transactions' (1845), 

 vol. viii. p. 269. O'Brien, ' On Symbolical Statics/ Philosophical 

 Magazine, 4th Series, vol. i. pp. 491 &c. 



