OF MECHANICS AND GEOMETRY. 273 



Duchayla. Now this proof is certainly convincing; that is to say, it is not possible to point 

 oat any flaw in the steps of demonstration, but for persuading the intellect it seems to have no 

 kind of fitness. The proof is essentially artificial, and is based on a simple case of composition 

 of forces which seems very insufficient to suggest, as it is pretended that it docs, the result 

 sought. The character of the proof seems, if I may so express myself, to be that of cunning 

 rather than honest argument ; and yet I think that, however unsatisfactory the proof may appear 

 in this light, we must feel convinced that, supposing it accurate as we do, there must be a meaning 

 and principle about it at bottom, and that these are only smothered and obscured by the artificial 

 contrivances of the demonstration. This I think we shall find to be really the case if we 

 examine the proof in the light of the preceding observations. The first part of the proof seems 

 to involve very faintly the idea of force ; the only principle introduced being this — that a force 

 may be applied at any point in its direction ; and thus the distinctness of the proposition as a 

 mechanical one seems rather obscured, but this difficulty of course vanishes if this first part of 

 the proposition be what I should call a proposition in the science of Pure Direction ; the proof 

 involves the idea of force only indirectly, and this is exactly what ought to be the case if the 

 proposition be true of several things, of which force is one : it is equally true of velocity, for 

 example: force is an embodiment of the pure idea of direction, and therefore all theorems of 

 pure direction will belong to force, not singly, but to it in common with all other emliodinients 

 of the same idea. In fact, the first portion of Duchayla's proof appears to be simply this, given 

 two straight lines in position to ascertain the direction which will be determined by them. 



But direction is not the only idea involved in force : there is magnitude as well, and there- 

 fore there is a second portion of the proof we are considering, in which it is shewn that, allowing the 

 triangle of forces so far as direction is concerned, that part which regards magnitude necessarily 

 follows ; the extreme simplicity of this part of the proof shews how intimate the connexion must be 

 between the two parts of the proposition, a connexion which I think we should not have been led to 

 expect from anything occurring in the proof itself, for, although the fact that the direction of 

 the resultant of two equal forces will bisect the angle between them is taken as suggestive of 

 the general law of direction, there is not a shadow of a hint that in this simple case the law will hold 

 as respects magnitude : so that a very remarkable proposition is proved by a mere artifice without 

 apparently the least reason in the nature of things why we should anticipate the result. But if we 

 consider the proposition from the same point of view as that from which we regarded the question 

 of the resultant of two straight lines, we shall see that there is a necessary connexion between 

 the two propositions, I mean those respecting direction and magnitude ; for when we had two 

 lines AB, BC given, the resultant AC became at once known both in direction and magnitude ; the 

 two things were co-ordinate, in fact, as this word suggests, they were merely two new co-ordinates 

 of C which became known from the two given co-ordinates AB, and BC. 



10. On the whole, therefore, I would urge that the proposition which we call the triangle of 

 forces is a result of the combination of the pure ideas o{ direction and magnitude, and will therefore 

 be true in some sense of all concrete existences which are embodiments of these two ideas and 

 no other: and therefore I explain the fact of the unmechanical character of the proof we have been 

 considering by observing, that the proposition is more general than the merely mechanical one, 

 includes in fact the triangle of forces, the triangle of lines, the triangle of velocities, the triangle 

 of couples, and perliaps other cognate propositions. 



11. This subject will, I think, receive further elucidation as follows: 



If / represent any quantity in magnitude only ; then if the quantity depend on direction also, it 



will be necessary to nssign the direction in which / is to be measured ; but if this be done, it is 



possible to affect I l)y a symi)ol or sign of afl'ection, which shall indicate for itself the diiection 



ill which it is measured. This symbol it is well known is e"^^', which is such that if/ represent a 



Vol.. VIII. Paht III, N N 



