[ er ] 
equilibrio, is in certain cafes greater than that of a femicircular 
arch of equal fpan, yet a given altitude of folid building ereéted 
on the Gothic will in fome cafes, but not in all, produce a greater 
aberration from a perfet equilibrium than in the femicircular. 
Let BNED, BHD, be the Gothic and femicircular arches, whofe 
common fpan is BD; let GR reprefent indifferently the ex- 
tradoffo of either, Poi which let fall the perpendicular GM on 
the fpan, meeting the curves in N, 2; and let a be the altitude 
of the folid building erected on each. If the Gothic arch were 
in perfect equilibrium, the height above N would be to a, as EK3 
“to NM:; and the height above m in the circle would be to the 
fame a as HK? tom M3 (Art. 13.).. Now if an ellipfe were defcribed 
with B D as a minor axis, and with E K asa femitranfverfe, it would 
fall without the Gothic arch, and meet the perpendicular GM 
fomewhere between N and G, for the radius of the circle ofcu- 
lating the ellipfe in B is equal to aie (Ham. Con. Prop. 17. L. 5.) 
that is, equal to the greater abfciffa of the circle B NE, and there- 
fore greater than the radius of the Gothic arch; and the ofculating 
circle falls without the ellipfe (Ham.Con. Prop. 15.L.5.); but fince the 
ofculating circle touches the ellipfe fo intimately, that no circle can 
pafs between it and the ellipfe, the Gothic arch muft fall within 
it; therefore the ratio of EK to NM is greater than the ratio of 
EK to mM, that is, than the ratio of HK tonM (Prop. 30. 
L. 1. Ham. Con.) ; therefore the height of the folid wall over N, 
in the Gothic arch, muft be greater than over z, in the circle; 
and confequently, when the line bounding the wall. is not the 
extradoffo of equilibrium but an horizontal line, there will be a 
M greater 
Fig. 4. 
