OF MECHANICS AND GEOMETRY. 275 



be the original force reduced in the proportion of FO : PP' or of 1 : cos t) The question remaining 

 would be, whether a force could properly be represented by the distance between two impossible 

 planes, a question which might perhaps be answered satisfactorily and without much difficulty if 

 we consider tiiat a finite line taken in the direction of a force will represent the two fundamental 

 properties of force, namely, magnitude and direction. But if it appear in this way that PF 

 represents the effective part of the force acting on P it will be seen that in like manner P O repre- 

 sents the ineffective or destroyed part. And therefore the result of the artifice of constraining 

 the particle P would be that when a force I acts on a particle, which is constrained by a 

 plane inclined to the direction of the force at an angle Q, the force is equivalent to a force 

 /cos 9 which is effective and a force / sin which is destroyed, or a force/ cos in the direction 

 in which motion is possible and a force / sin 6 in the impossible plane. And this is exactly what 

 would result from considering force under the light of the formula 



le^ ^~' = / cos e + v^^l I sin 0. 



13. This symbolical representation*, though depending on refined principles, is nevertheless, 

 I apprehend, valuable in the discussion of the question before us, because it is generally admitted 

 as a complete method of geometrical representation, and those who study the question must 

 perceive that its complete character is founded on something much deeper than a mere symbolical 

 artifice, inasmuch as it expresses the equivalence between a line, considered in its direction and 

 maa;nitude, and the two rectangular projections of that line. Now it has been the intention of 

 what immediately precedes to point out the corresponding necessary connexion between a force 

 and its resolved parts, and the perfect applicability of the same symbolical method to the two 

 cases tends, it is presumed, to strengthen the characteristic view of this paper, viz., the essential 

 identity of the Geometrical and Mechanical Sciences, considered as developements of the same 

 combined fundamental ideas. 



l-l. The preceding remarks have been wholly devoted to the consideration of force as acting 

 on a single particle ; it was my intention to have attempted a discussion of the case of a system of 

 forces acting on a rigid body, and to have shewn how the science of Mechanics diverges from that 

 of Geometry, by the introduction of this new idea oi Rigidity; but perhaps what has been already 

 said will be sufficient to put in a clear light the fundamental views which it is my desire to 

 explain : my belief is, that these views contain the shadows at least of important truth, and 

 that they will be seen to do so by any one who will devote attention to the subject. The 

 great question is, what are the fundamental ideas of Elementary Mechanics, and what of Geometry ? 

 Are they the same, or are they cognate, or are they altogether distinct .'' If the last, then 

 the resemblance between certain demonstrations and propositions in the two sciences is a curious 

 and unexplained fact ; but if the second or the first, then the explanation is obvious. And if 

 the relation of the two sciences be such as I have represented it, then it seems to me to be most 

 important that it should be recognized, and that for more reasons than one ; first, this view 

 connects two streams of truth, usually I believe considered distinct, and traces them to one 

 fountain head, and this is an important simplification, in the same sense and for the same 

 reason tliat it is an important simplification to trace two phenomena to the same physical cause ; 

 but, again, the foundation of geometrical truth is a matter of less question in general than that 

 of mechanical ; it is I suppose universally allowed that the propositions in pure Geometry are 

 an they are, because they could not be otherwise, that they are necessary truths in every sense 

 in which truths can l)0 necessary, but there is not, I apprehend, such clearness of thought prevalent 

 respecting mechanical truth, it is difficult to make out from the ordinary books on the subject 



See Note (C). 



N MS 



