VOL. LXIX.] PHILOSOPHlCAI> TRANSACTIONS. 499 



bridges, moles, bulwarks, pyramids, columns, edifices, the great magnitude and 

 solidity of which we admire, and dare hardly to imitate. All these works 

 require very strong and solid foundations. If the place be marshy or loose 

 earth, piles must be driven in by the force of machines, to place the grated 

 frame on; then more piles with projecting tops, and their intervals filled with 

 large stones, flints, coarse sand, and mortar. 



The form of the ancient pile engine is not well known. But several are 

 given by the moderns, as Leopold, Desaguliers, and Belidor. Among all these, 

 that appears to be the best which was invented by Vauloue, described by 

 Desaguliers, and used at piling the foundation at Westminster Bridge. Its chief 

 properties are, that the ram or weight be raised with the fewest men; that it 

 fall freely from its greatest height; and that, having fallen, it is presently laid 

 hold of by the forceps, and so raised again. By which means, in the shortest 

 time, and with the fewest men, the most piles can be driven to the greatest 

 depth. 



Belidor has given some theory as to the effect of this machine, but it appears 

 to be founded on an erroneous principle: he deduces it from the laws of the 

 collision of bodies, considering the pile and the falling weight as two striking 

 bodies. But who does not perceive that the rules of collision suppose a free 

 motion and a non-resisting medium ? It cannot therefore be applied in the case 

 given, where a very great resistance is opposed to the pile by the ground. 

 We shall now endeavour to explain another theory of this machine. 



The problem amounts to this, that we may consider as two machines, a 

 certain weight falling from a certain height, and the head of the pile on which 

 it falls. Now, calling the falling weights w and w, the altitudes descended a 

 and a; the masses of the piles m and m; their surfaces within the earth s 

 and s; and the depths of the same d and d. The percussion of the falling 

 weight is to be estimated by the product of the mass or weight into the square 

 of the velocity, or by the mass into the altitude descended, since this altitude is 

 as the square of the velocity. But the effects are as the whole forces of their 

 causes. Therefore, if we apply the resistance equally both to the ground and to 

 the weights or masses of the piles, the depths, to which the piles are driven by 

 every stroke, will be in a ratio composed of the direct ratio of the weights and 

 of the altitudes; or </ : d = a X 't' : a X w. 



If we assume the cohesion of the ground as uniform and homogeneous, the 

 resistance will increase in the simple ratio of the rubbing surface only. 

 And if now we make the descending weights equal, with w = u>; also the alti- 

 tudes equal, a = a; it appears that the effects of the percussions, and hence the 

 depths driven to, will decrease as the superficies and weights or masses of the 

 piles increase. Hence, under the above data, the depths will be in a ratio com- 



3 s 2 



