454 Prof. Forbes on the Form of the Gothic Pendent. 
sign. We accordingly find the peculiar figure of the pen- 
dent carried into the minuter depending ornaments for the 
sake of symmetry; though the scale is almost too small to re- 
quire the curve of equal strength to satisfy the eye. It is 
quite obvious too, that to reverse the case we have described, 
and to make masses of the form of pendents resting on their 
smaller bases to sustain weights, is equally repugnant to the 
principles of good architecture and good sense. 
In all cases the strength actually given to pendents enor- 
mously exceeds that requisite for their cohesion. It appears 
from the following simple analysis that the modulus or sub- 
tangent of the logarithmic curve, must, in order exactly to 
prevent rupture, be equal to twice the modulus of cohesion 
of the substance in feet. 
« Required the figure of a depending body which shall be 
just within the limit of cohesion at every part of its length.” 
Let s* represent the area of its section corresponding to any 
point z in a given vertical ascending line. Since the condi- 
tion infers that the increase of section shall be in a constant 
ratio to the increased volume of the solid, 
med seus? gia 
(a being a constant) ; and integrating 
xz =a.hyp.log s? +c. 
If we assume the body to bea solid of revolution, and like- 
wise that the variable radius 7 shall become equal to unity 
when w = 0, we shall have for the corrected integral 
xv = 2a.hyp. log r. 
Hence the contour of the pendent will be a logarithmic curve, 
whose subtangent = 2a. 
Now, since it {is required that the increment of cohering 
surface shall be just capable of supporting the increment of 
Ss aa 
Tgp? 8% equal to the mo- 
dulus of cohesion of the substance employed expressed in 
linear measure. Consequently the subtangent is equal to twice 
the modulus of cohesion, and for a self-supported body of 
uniform thickness, the measure of the one and the other would 
be the same. 
In the cases of white marble and Portland stone the moduli 
of cohesion have been stated at 1542 and 945 feet respec- 
tively. The subtangents would, therefore, be 3084 and 1890 
feet. We may thence calculate the logarithm of 2 upon those 
scales, or the vertical height in which the radius of the section 
doubles itself. This will be found to be 2138 feet in the case 
mass, we must have the quantity 
