VOL. XLIV.] PHILOSOPHICAL TRANSACTIONS. 223 



they are to the sentiment of L., may deserve examination. But one thing is 

 particularly deserving of notice in these notations, viz. a as ev, which is to be de- 

 monstrated: whence it seems that L. not even then, after l6 years, had not 

 found out a demonstration of the supposition formerly put off to Bernoulli ; viz, 

 that an action performing any thing in a single time, is double of an action per- 

 forming the same thing in a double time : since an action performing any thing 

 in a single time, does it with twice the velocity of an action performing the same 

 thing in a double time. But how W. demonstrates this, we shall examine pre 

 sently. 



Definitions. Def. 1. — I call that, with Leibnitz, vis viva, or merely vis 

 or force, which adheres to local motion. 2. A pure force, is that which is not 

 resisted in acting by any contrary force. Hence such a pure force remains un^- 

 varied in the whole time of action, and is not in the least exhausted by the effect it 

 produces. 3. A pure action, is that which is exercised by a pure moving force. 

 Such as the action of a moveable carried with an equable motion in an unresist- 

 ing medium. 4. A uniform action, is that which is double in a double time, 

 triple in a triple, &c. or in general which is as the time ; or such an action as 

 obtains in equable motion. 5. The effect of a moving force beyond the confiict, 

 is the translation of a moveable through a space. 



Axioms. Axiom 1. — If two or more equal moveables be moved with equal 

 celerity, their force is the same. 2. The same action is performed by the same 

 force in the same time. That a greater action is performed by the same force in 

 a longer time than in a shorter, and that a greater action is performed in the same 

 time by a greater force than by a less, no one doubts. Therefore the quantity of 

 an action depends on the quantity of force and time. So that if the forces be 

 equal, and the time the same, the action also must be the same. 3. If the same 

 moveable be transferred through the same space, the effect is the same. 



Theorems. Theor. 1 . — When unequal bodies are moved with the same ve- 

 locity, the forces are as the masses. 2. Uniform actions, performed in the same 

 time, are to each other as their forces. 3. Uniform actions, performed with 

 equal forces, are to each other as the times in which they are performed. 4. 

 Uniform actions are in the ratio compounded of the times and forces. 5. Un 

 equal forces perform the same action in times reciprocally proportional to each, 

 other. 6. When two equal moveables are carried through unequal spaces, the 

 effects are as the spaces. 7- When two moveables are carried through the same 

 space, the effects are as the masses. 8. When two moveables are carried 

 through any spaces, the effects are in the ratio compounded of the masses and 

 spaces, g. In equable motions, the effects are in a ratio compounded of the 

 masses, velocities, and times. The demonstrations of these g theorems, about 



