[ 85 J 
ftones (by Art. 14.); that is, as the femidiameter of the ellipfe 
which is perpendicular to the horizon to the femidiameter of the 
circle, (Prop. 17. L. 5. Ham. Con.) or as the height of the 
ellipfe to the height of the femicircle; that is, if BmED bean 
elliptic arch, its ftrength will be to that of the femicircle BHD, as 
EK to KD; but the ftrength of the femicircle is to the ftrength 
of the Gothic arch BED as EK x AK to CD’; therefore, 
ex @quo, Sc. the ftrength of the elliptic arch is to ie aha of 
the Gothic arch of equal height and fpan as EK’ x AK, or 
AK*xKD to CD*x KD, or as AK* to CD’, or in the du- 
_plicate ratio of the fum of radius and the cofine of the Gothic 
arch to radius, that is, in a ratio of greater inequality. And 
fince KD is conftant, AK increafes in a higher proportion than 
CD; therefore in ‘the elliptic arch, the ftrength increafes in a 
higher proportion, as the altitude increafes, than in the Gothic. 
Thus in an elliptic and Gothic arch in which the heights and 
{pans are equal, and the fubtenfes of their halves equal to the 
fpan, the ftrength of the elliptic will be to that of the Gothic 
as 223 toro. And when the ftrength of the Gothic arch, is equal 
to that of a femicircle of equal {pan, the ftrength of an elliptic 
arch of equal height and fpan exceeds it nearly in the ratio of 5 
to 2 by Art. Uy 
23. (THE extradoffo G R of an elliptic arch B ED may be thus in- 
veftigated : through any point-N of the ellipfe draw the tangent N Mi 
mecting theaxisKE produced i in V; “from K the ¢ centre let fall the per- 
_pendicular K A on the tangent; and draw K G the femiconjugate of 
K N. 
Fig. 4. 
Fig. 2. 
Fig. 6. 
