OF UNDISTURBED MOTION. 71 



acting separately." It appears, however, to be more na- 

 tural and perspicuous to defer the consideration of force 

 until the simpler doctrine of motion has been separately 

 examined. 



227. Theorem. Any equable motions, 

 represented by the sides of a triangle or poly- 

 gon, supposed to take place in the same 

 moveable point, in directions parallel to those 

 sides, and in the order of going round the 

 figure, destroy each other, and the point 

 remains at rest. 



For two sides of the triangle, AB, BC, J^ 



are sides of the parallelogram ABCD, 

 therefore by the motions AB, BC, or AB, 

 AD, A would arrive at C, while by the mo- I) 

 tion CA it would be brought back to A in the same time ; 

 and all the motions being equable, it will always remain 

 in A : and, in the same manner, the proof may be ex- 

 tended to a figure with any greater number of sides. The 

 truth of the proposition will also appear by considering 

 several successive planes as moving on each other, and the 

 point A as moving in the last : or we may imagine each 

 motion to take place in succession for an equal small in- 

 terval of time ; then the point would describe a small po- 

 lygon similar to the original one, and would be found, at 

 the end of the whole of the small intervals, in its original 

 situation. 



Scholium. When the motions to be combined are 

 numerous and diversified, it is often convenient to resolve 

 each motion into three parts, reduced to the directions of 

 three given lines perpendicular to each other : and, in this 



