L PR 
15: Let BD, confidered as variable, reprefent the fpan of Mig. 2. 
the Gothic arch BED, fpringing from the points B,D; and 
the invariable line-C D the radius of the circle by which the 
Gothic arch is defcribed. On BD defcribe the femicircle BHD; 
the ftrength of the femicircular arch BHD at H is to the 
ftrength of the femicircular arch AGD at G, as CD to KD, 
by Art. 1o.. But, fince the relative ftrength of arches is to be 
determined by comparing them in their weakeft points, (for if 
an arch fails in any one point, the whole falls to ruin) the 
ftrength of the femicircular arch AGD will be to the ftrength 
of the Gothic arch, as the ftrength of the femicircle at G to 
its ftrength at E, that is, as the greateft weights thefe points 
are able to bear when in equilibrium ; that is, reciprocally as 
the cube of the angle IE F contained by the vertical line TE and 
the tangent to the circle at E, or its equal EC K, that is, as the 
cube of EK to thecube of radius, Art. 14. Therefore, ex equo, &e. 
the ftrength of the femicircular arch BHD at H, is to the 
ftrength of the Gothic arch of the fame fpan, defcribed with the 
radius CD, as GCXEK? to KDxGC3, or as EKX AK 
to C D?. 
16. Since the re€tangle EKxAK _ vanifhes at both extre- 
mities when K arrives cither at A or D, there muft be fome 
intermediate ftate where it is a maximum; to find this 
ftate, let CD=a, and CK=x; EK=a@’~x"), and AK=a+.», 
therefore 
