L Wag 
therefore the re@angle EKx AK= ae Xadx ; whofe fluxion 
ait ita? XN —axx 
That is, the rectangle E K x AK isa maximum, when CK=3CD: 
or, becaufe C D isconftant, the ratio of the ftrength of a femicircular 
=o. Whence x*+3a~ =5a?, and x=Faz. 
= x 
q@i—x7 
arch to a Gothic arch of the fame fpan is greateft when the fub- 
tenfe of half the Gothic arch is equal to its fpan. In this cafe the 
ftrength of the femicircular arch is to the ftrength of the Gothic 
as 1299 to 1000, or 13 to Io, nearly. 
Ir the radius of the Gothic arch be three-fourths of the fpan, 
in which cafe it is called the fharp arch of the fourth point, the 
ftrength of a femicircle will be to the ftrength of a Gothic arch of 
the fame fpan as 1257 to rooo. And if the radius be two-thirds 
of the fpan, in. which cafe it is called the fharp arch of the third’ 
point, the ftrength of the femicircle will be to the ftrength of the 
Gothic arch of equal fpan as 1210 to 1000. 
17. Because the rectangle E K x AK exceeds the fquare of the 
femidiameter when the point K bife€&ts C D, and vanifhes when K 
arrives at D, there muft be fome intermediate pofition, in which 
the reétangle EK x AK is equal to the fquare of radius. To 
find the magnitude of K D in this cafe, let C K=7, and. Cp=2 
as before; then EK =/a?—x*, and AK=a+»; therefore 
a+-xxXVa@—x?=a", and x*+2ax3-2a3x=0; and if a=1, 
x?+2x?=2. Of this equation two roots are impoflible: The 
third, 
