Fig. 5. 
8 
[ 84 | 
a2. Ir an ellipfe be deferibed on the fpan- of a femicircular 
arch, its ftrength, when in equilibrio, will ‘be‘to‘that ‘of a femi- 
circular arch’* reciprocally sas the radii of curvature at the key- 
ftones 
* Hence the ftrength of an elliptic arch whofe leffer axis is perpendicular to the 
horizon, and of'a portion of a circle of ‘the fame’ fpan’and ‘altitude may be eafily 
compared. For the radius of the circle pafling through .the points B E D is 
egual to epee te aol 
2) EK 
KD? ares : ; : 
equal to. EK? therefore the ftrength of the fegment of the circle is to the! ftrength 
of the femi-elliptic arch’ as 2K D* to EK? +4 KD}, that is, always in a ratio 
of greater inequality. If to this we add, that the aberration from a perfect equi- 
librium ‘in the femi-ellipfe, arifing from the herizontab termination of the building 
erected ‘on it, is very great, but of no great confequence in‘ any fegment ofa 
circle which contains lefs than 120°, (fee Hutton’s Efiay on Bridges,) per- 
haps it may feem reafonable to conclude, that portions of circles are in all cafes 
preferable to femi-elliptic arches, or thofe curves of many centres, which of late 
spare become fo fafhionable in the conftruétion of bridges. In the celebrated bridge 
the Sein, at Neuilly near Paris, confifting of five arches, each 120 feet opening, 
and rifing but 30 feet, the radius of curvature at the crown of the arch is 150 feet, 
and therefore is to the radius of a circular arch of the fame fpan, and rifing only 
to the fame height, as 2 to 1; and confequently the relative ftrength of thefe arches, 
without taking into account the enormous error from a true equilibrium in the 
» and the radius of the circle ofculating the ellipfe in D is 
former, is in the reciprocal ratio of thefe numbers. And the ftrength of a femi- 
ellipfe of the fame {pan and altitude is to the ftrength of the curve ufed in the bridge 
of Neuilly as § to 4. A 
The radius of curvature of a circle ofculating the vulgar cycloid at its vertex is 
equal to twice the diameter of the generating circle; and the radius of a fegment of 
a circle of the fame opening and height is equal to the fum of the fquares of half 
the circumference and diameter applied to twice the diameter; therefore a cycloid 
is weaker than a circular arch of the fame fpan and height in the ratio of 11 to 16 
nearly. ‘To'this we may add, that the accurate, conftruction of cycloidal arches, in 
ftone, is almoft impracticable, and therefore a confiderable deduction fhould be 
made from their ftrength, on account of their imperfect exceution. For thefe rea- 
fons arches of this kind are not introduced into practice; though Drummond, in 
his trayels, tells us, that thofe of the bridge of the Holy Trinity, over the Arno, at 
Florence, are cycloidal. And indeed what is related of that bridge is confiftent with 
theory, for on account of its extreme weaknefs, it is found neceffary to prevent! 
wagons and other heavy carriages from paffing over it. 
