NOTES. 



I. 



On the Measure of Forces. 



THE name of living forces has been applied to the forces of bodies 

 in motion, and that of dead forces to those, which, like a simple 

 pressure, suppose no actual motion in the operating cause. 



There was, for a considerable time, a difference of opinion 

 among mathematicians in regard to the measure of living forces, or 

 the forces of bodies in motion. Some affirmed that these forces 

 ought not to be measured by the product of the mass into the velo- 

 city, according to the rule that has been given, but by the product 

 of the mass into the square of the velocity. As this difference in 

 the estimate of forces may be deemed of great importance in me- 

 chanics, it will be proper to make a few remarks upon it. 



It is altogether unimportant whether we measure the force of 

 bodies in motion by the product of the mass into the velocity simply, 

 or by the product of the mass into the square of the velocity, pro- 

 vided that in the two cases we assign different significations to the 

 word/orce. When we assume as the measure of force, the product 

 of the mass into the square of the velocity, we mean by the term 

 force the number of obstacles which a moving body is capable of 

 overcoming ; it is certain that the number of obstacles which mov- 

 ing bodies of equal masses are capable of overcoming is proportional 

 to the squares of the velocities. For example, if the bodj r A has Fig. 255. 

 precisely the velocity necessary to close the spring j?CB, an equal 

 body M will require only double this velocity to close four springs, 

 each equal to ACB. For, in the first instant, for example, the body 

 M advancing with a velocity double that of .#, will close the four 

 springs, considered as one, twice as much as A would close its sin- 

 gle spring; each of the four springs then will be closed only half 

 as much as ACB, and consequently will have opposed during this 

 instant a resistance only half as great as that of ACB ; all the four 



