32 Observations on the taking down and 



table of natural sines and tangents, v may be obtained *. Tang. 



t 



When n =s 6 feet 3 inches, d becomes 111 feet 6 inches, and 

 <p — 71° 23'. When n — 3 feet 10 inches, d becomes 103 feet 

 6 inches, and <p = 75° 7'. 



Whence in the first case the width of the grating at the level of 

 the foundation is 51 feet, exceeding by 8 feet, the width of the 

 grating in the drawing, but the width of the piers, at the level 

 where the line of water is shewn, and they rise vertically, becomes 

 22 feet, very nearly conformable to the dimensions of the drawing 

 taken from the scale ; and in the second case, the width of the 

 grating at that level becomes 41 feet, and the pier at the level of 

 the water 16 feet. 



The 51 feet, and the 41 feet, give the extreme points of strut- 

 ting; but the first would be strutted to a balance, if the grating were 

 28 feet wide, and the second 26 feet, which dimensions, in the 

 case of bearing piers, would generally be considered safe. In 

 Waterloo Bridge, the bearing piers are sufficiently thick for 

 abutment piers, and the width of the grating in the bearing 

 piers, conforms to what would be obtained from the above formulae, 

 so that it is probable, in that bridge Mr. Rennie used the geo- 

 metrical construction before referred to, or one producing the same 

 result. 



If the thickness at the vertex be taken at 6 feet 3 inches, and 

 the width of the vertex stone, at the intrados be taken 20 inches ; 



31 

 then the summering of that stone will be — of an inch, that is, 

 5 12 



the extrados of that stone will be 7 ^th of an inch wider than the 

 intrados, that stone approaching very near to a parallelopiped. If 

 the thickness be 3 feet 10 inches, then the difference between the 

 extrados and intrados of that stone will be T 4 „th of an inch. 



Partly to avoid the hazard of so little summering in the case 

 of a deficiency of strutting in the abutments ; the Gothic archi- 



* This operation is facilitated by Mr. Barlow's Table IV., New Math, 

 Tub. 1814. 



