296 



THE POPULAR SCIENCE MONTHLY. 



Ielogram of Fig. 4 the diagonal A C is the sum of A B and B C; 

 or, since A J) is geometrically equivalent to B C, A C is the geomet- 

 rical sum of A B and A I>. 



Fig. 3. 



Fig. 4. 



All this is purely conventional. It simply amounts to this : that 

 we choose to call paths having the relations I have described equal or 

 added. But, though it is a convention, it is a convention with a good 

 reason. The rule for geometrical addition may be applied not only to 

 paths, but to any other things which can be represented by paths. 

 Now, as a path is determined by the varying direction and distance 

 of the point which moves over it from the starting-point, it follows 

 that anything which from its beginning to its end is determined by a 

 varying direction and a varying magnitude is capable of being repre- 

 sented by a line. Accordingly, velocities may be represented by lines, 

 for they have only directions and rates. The same thing is true of 

 accelerations, or changes of velocities. This is evident enough in the 

 case of velocities ; and it becomes evident for accelerations if we con- 

 sider that precisely what velocities are to positions namely, states 

 of change of them that accelerations are to velocities. 



The so-called " parallelogram of forces " is simply a rule for com- 

 pounding accelerations. The rule is, to represent the accelerations by 

 paths, and then to geometrically add the paths. The geometers, how- 

 ever, not only use the " parallelogram of forces " to compound differ- 

 ent accelerations, but also to resolve one acceleration into a sum of 

 several. Let A B (Fig. 5) be the path which represents a certain 



acceleration say, such a change in 

 the motion of a body that at the 

 end of one second the body will, 

 under the influence of that change, 

 be in a position different from what 

 it would have had if its motion 

 had continued unchanged such that 

 a path equivalent to A B would 

 lead from the latter position to the 

 former. This acceleration may be 

 considered as the sum of the accelerations represented by A C and 

 C B. It may also be considered as the sum of the very different ac- 

 celerations represented by A D and D B, where A B is almost the 

 opposite of A C. And it is clear that there is an immense variety of 



Fig. 5. 



