634 THE PRINCIPLES OF SCIENCE. [CHAP. 



found to be 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 

 generalisation 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 tri- 

 angle of velocities, the triangle of forces, the triangle ot 

 couples, and perhaps other cognate theorems, are reduced 

 by analogy to one simple theorem, which amounts 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. 1 In the system 

 of quaternions of the late Sir W. E. 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. 



It ought to be added that to the discovery of analogy 

 between the forms of mathematical and logical expressions, 

 we owe the greatest advance in logical science. Boole 

 based his extension of logical processes upon the notion 

 that logic is an algebra of two quantities o and I . His 

 profound genius for symbolic investigation led him to per- 

 ceive by analogy that there must exist a general system of 

 logical deduction, of which the old logicians had seized 

 only a few fragments. Mistaken as he was in placing 

 algebra as a higher science than logic, no one can deny that 

 the development of the more complex and dependent 

 science had advanced far beyond that of the simpler science, 

 and that Boole, in drawing attention to the connection, 

 made one of the most important discoveries in the history 

 of science. As Descartes had wedded algebra and geo- 



1 See Goodwin, Cambridge Philosophical Transactions (1845), vol. 

 riii. p. 269. O'Brien, "On Symbolical Statics," Philosophical 

 Magazine, 4th Series, vol. i. pp. 491, &c. See also Professor Clerk 

 Maxwell's delightful Manual of Elementary Science, called Matter 

 and Motion, published by the Society for Promoting Christian 

 Knowledge. In this admirable little work some of the most advanced 

 results of mechanical and physical science are explained according to 

 the method of quaternions, but with hardly any use of algebraic 

 symbols. 



