Arches of the Bridge of the Holy Trimly. 5 



when ip denotes the angle of the curve with its ordinate, and 

 m ■=. 2.302585 to find (p by approximation, and thence the 



constant quantity = — • some doubt may be enter- 



tained, whether it was designed to be a catenary, for in five 

 cases by calculation, the longest constant quantity was 2.59, 

 and shortest 2.46. 



It may be observed that if the curve had been an arc of a 

 ])arabola or of a catenary, curves likely to be adopted at the 

 time of the erection of this bridge, from the inquiries then com- 

 menced respecting them, that the ordinate in the middle of the 

 arch, as well as the absciss, would have been whole numbers, 

 and the same conclusion will be come to, if it had been a scheme 

 as Ferroni supposes. But in adopting an ellipse for the curve, 

 it may be presumed that the axes and the span would have been 

 integers, and the height fractional, dependent on them. 



In trying whether the curve be elliptical, we must assume one 

 of the axes. It is a reasonable presumption to conclude from 

 the obtuseness of the angle at the intersection of the arcs, that 

 the semi-conjugate axis was taken, the next greatest whole 



B. S. D. 



number to 7 16 6, the ordinate in the middle of the arch, that 



is, eight braccia. Letx =: the absciss, y = the corresponding 



ordinate, c = the serai-conjugate, and t =: the semi-transverse. 



c x f "I 



Then by the common formula t = —^\ c + V (c- -y") = ^'^ 



nearly. Hence we derive an ellipse whose semi-conjugate axis 

 is j5j = ith of the transverse axis *. 



If now we take t = 32 braccia, and c = 8 braccia, and then 



by the common formula y = - V (3xt — x^) we may obtain 



ordinates by calculation to compare with Salvetti's. 



When X ^ 2, then y by calculation, = 2 15 9, by measurement, 2 19 2, 



5 4 5 10, ... 4 7 I, 



9 5 11 2, ... 5 11 4, 



10 5 16 1, ... 5 15 10, 



* Ferruui lays great stress upon AiuuiauuU's prcposties$Ton for diiiicDsiou:^ 

 of which S is an aliqiiut part. 



