I54< Mr Buchanan's Report on the Bridge of Suspension 
bridge, I would make, with an improved apparatus, such trials 
as would decide beyond all question, the strength of the iron to 
be employed in the construction of the bridge in question, more 
especially as this could be done at a trifling expence. In the 
mean time, we may safely assume the strength at 8 tons to the 
inch, which is the lowest estimate I can find, and which is only 
one-third of what the iron really will bear without breaking, as 
it takes from 20 to 30 tons to tear it asunder. Suppose, then, 
that the bridge is strained with the utmost load of carriages and 
people that can ever belaid on it, I propose having such a thick- 
ness of iron from pillar to pillar to support this load, that no 
part of it will ever be stretched with a force of more than 8 tons 
to the inch ; and this it will endure, not only for the moment du- 
ring which this extraordinary strain is ever likely to take place, 
but although it were continued for any length of time ; so that 
any notion of danger from the utmost load to which it can ever 
be subjected seems totally out of the question ; and much less 
can any hazard be incurred from the every-day traffic on the 
bridge. The thickness of iron required for this purpose will be 
about 124 inches, or about 62 on each side of the bridge. There 
will in fact be less strain by 100 tons in the centre of the arch 
than at each point of suspension, and the section of iron will re- 
quire to be varied on this account, but 124 inches is the average ; 
-and could we stretch, therefore, one whole arched bar of iron, of 
this thickness, from pillar to pillar, and from thence to the 
ground, this would form the most perfect suspending arch. But 
this is impossible ; such a vast mass can only be formed by uni- 
ting small pieces together, and it is of importance to consider 
what is the most convenient size for these pieces. In the Tweed 
equal to the horizontal tension into the tangent of the deflection, hence 
y — T tang D, and T : y : : It : Tang D. Thus, we have dy : dx : : T : y , 
y 2 ' y 2 
and ydy — T dx, which, by integration, gives — = To?, and T = — , as be- 
fore. 
For further information on the subject of the catenary, I may refer to 
Professor Leslie’s Geometry of Curve Lines, and Elements of Natural Phi- 
losophy, where the strain in the lowest part of the catenary is expressed by a 
b 2 d . 
very simple, yet extremely accurate formula, viz. — + ^ , b denoting the 
span, and d the depression. 
