136 PROCEEDINGS OF THE AMERICAN ACADEMY 



this then to be established, we have a very simple, easy, and direct 

 method of comparing, one with another, the forces of cannon-balls of 

 every possible weight, or rather inertia, as measured by their weight, 

 moving with any velocity that may be impressed upon them. In 

 making this comparison between the forces of the balls used, espe- 

 cially in the most powerful of the vaunted American and English 

 cannon of modern construction, it will be necessary first to establish 

 some standard which may be used as the unit of measure, with which 

 the others are severally compared and tried. To do this I shall take 

 the ball of the old 32-pounder under a velocity of 1,G00 feet a second, 

 this being a mass and velocity with which all artillerists are familiar, 

 being that produced by eight pounds of powder, the full charge of this 

 gun ; and I shall use the force possessed by this ball under these con- 

 ditions as the standard, or unit of the standard, by which the force 

 of any others may be compared or measured. 



It is very desirable that the unit of every standard of measure should 

 be taken from some simple and familiar object, of the quantity of which 

 we can not only form a conception, but with which we have a familiar 

 sensible acquaintance. Our common standards of weight and measure 

 have been thus derived, the grain of wheat forming the unit of one, 

 and the human arm and foot that of the other. In the more complex 

 instance of the power of the steam-engine, the strength of the horse 

 furnishes the basis of the standard of measure, and this again is defined 

 in a certain number of pounds raised one foot high against the opposing 

 force of gravitation. It will be at once perceived that, although a 

 32-pound shot moving with a velocity of 1,600 feet a second may 

 form some image capable of being grasped by the conception, yet we 

 must utterly fail to form a distinct idea of the quantity represented by 

 32 multiplied by the square of 1,600, or of the algebraic symbols mv 2 , 

 representing the product of a mass by the square of a velocity. It 

 will be seen, however, that we may bring the proposed standard of 

 comparison out of this dark envelope, by changing the factor of the 

 velocity into a physical equivalent taken in a vertical line a certain 

 number of feet high. Thus, instead of saying that the force of a 32- 

 pound shot under a velocity of 1,600 feet a second is numerically 

 represented by 32x2,560,000, we may substitute for this last factor 

 the height to which the shot would rise if it were pointed directly 

 upwards in vacuo, so that it should be freed from every atmospheric 

 and other resistance except that of gravitation alone. To do this we 



