V&L. XLIV.] PHILOSOPHICAL TRANSACTIONS. 225 



and the effect will be a page written. Or, if a workman whiten a wall, his 

 action will be the whitening of a wall, and the effect will be a v/all whitened, 

 Or, if a labourer dig a garden, his action is the digging of a garden, and the 

 efiect is a garden digged. 



Our Theor. 10. — Of equal actions, the effects are equal. 

 Let any vis viva a perform any action ; and let there be supposed any other vis 

 viva B. Now that the vis viva b may perform an action equal to that of the vis viva 

 A, it is necessary that the vis viva b should act exactly as much as the vis viva a 

 has acted. Therefore, after the completion of the action of b, as much will be 

 acted by the force b, as has been acted by the force a ; that is, the effect of the 

 vis viva b will be equal to the effect of the vis viva a, the actions of which were 

 equal. 



Our Theor. 11. — Actions are in proportion to the effects. For let the effect 

 e be produced by the action a. Therefore another effect e, equal to the first, 

 will, by theor. 10, be produced by another equal action a ; consequently the 

 effect twice e will be produced by the action twice a. In like manner 

 it appears that the effect thrice e must be produced by the action thrice a, 

 &c. And generally, that the effect we (= e) must be produced by the action na 

 (= a.) Therefore a : a :: e : e; that is, the actions are in the ratio of the effects. 

 Our Theor. 12. — Forces are in the ratio compounded of the masses and ve- 

 locities. 



For, by theor. 4, actions are in the ratio compounded of the times and forces. 

 And, by theor. 11, the actions are in the ratio of the effects. Therefore, the 

 effects are in the ratio compounded of the times and forces. But, by theor. 8, 

 effects are in the ratio compounded of the masses and spaces. Therefore the 

 ratio compounded of the times and forces, is equal to the ratio compounded of 

 the masses and spaces. Therefore the forces are in the ratio compounded of the 

 masses and spaces directly, and of the times reciprocally ; that is, in the ratio com- 

 pounded of the masses and velocities. 



Of Two Extraordinary Deers" Horns, found under- ground in different Parts of 

 Yorkshire. By Mr. Tho. Knowlton. N" 47g, p. 124. 



The head and horns here described, were found in a sand-bed, in the river 

 Rye, which runs into the Derwent, in the east-riding, belonging to Ralph Cra- 

 thorn, esq. They were discovered as he was fishing for salmon ; the net happen- 

 ing to hang on 1 or 2 of the antlers, he ordered to pull away ; by which some of 

 the antlers were broken off, and discovered it to be part of a deer's horn. At 

 length, with somt; difficulty, it was dug out pretty entire. Mr. Crathorn sup- 

 poses, that these wild moors were once inhabited with this kind of deer, not any 

 such now being known to be in this kingdom ; and supposes it is, at least, 7 or 



VOL. IX. Gg 



