E Saad 
greater deviation from a perfect balance in the Gothic arch. 
But the higher the Gothic arch the lefs will be the aberration ; 
and it may, at length, be fo far diminifhed as to become lefs 
than the error in the femicircle, for the ordinates EK, NM, 
vary in the fubduplicate ratio of their greater abfciffe,; and 
thefe abfcifiz, their difference MK being given, approach to- 
wards equality, as the radius of the circle or height of the arch 
increafes; but the ratio of HK to 2M is the limit of their 
variable ratio; fo that though the height of the wall to be 
erected over the correfponding points in the Gothic and femicircular 
arch, in order to render them curves of equilibration, continually 
approach to equality, and can be brought nearer to an actual 
equality, by elevating the point of the Gothic arch, than by any 
affignable difference, yet the height of the building to be erected 
on the points of the Gothic arch will always be greater than in the 
femicircle. Now asE 4, the difference between the ordinates N M, 
EK, is always greater than Ez, the difference between the 
ordinates mM, EK, and thefe are always in the conftant ratio 
of xM to HK, it follows, that by producing the ordinates 
mM, EK, their difference may be made greater than any 
affignable quantity; and at the fame time the difference be- 
tween the heights of the building to be erected on the points 
N, 2, of the Gothic and femicircular arch, may be made lefs 
than any aflignable quantity. It is evident therefore, that the 
defe& of equilibrium over the point N, in the Gothic arch, may 
be made lefs than the defe& over the correfponding point a of 
the femicircle, arifing from the horizontal termination of the building 
erected on them. Thus let the difference of the heights of the 
wall 
