118 SECTIONAL ADDRESSES 
Throughout a long period in the eighteenth and nineteenth centuries 
mathematicians and others applied themselves to finding the exact form 
of the line of thrust that would ensure equilibrium in a mass of masonry 
bridging a void. ‘The upper boundary of the mass was a horizontal 
surface representing the road surface and the lower one the intrados of 
the arch, shaped to conform to the line sought. The effect of hollow 
spaces over the haunches was investigated also. ‘The influence of a moving 
load was regarded as negligible in comparison with the weight of the 
masonry or could be allowed for by adding an extra layer of masonry over 
the upper surface. 
The shape of this arch of equilibrium was compared in great detail 
with those of the ellipse, cycloid, parabola, catenary and semi-circle or 
segment of a circle. Different writers strongly advocated one or other 
of these curves as being the true curve for an arch. The elaboration with 
which this was done seems remarkable, for many must have known that 
to build an arch to conform to a particular curve with the exactitude 
suggested is practically impossible. When the centering on which an 
arch is built is removed and the arch supports itself, the compression of 
the mortar in the joints and of the voussoir stones, allows the arch to drop 
an amount which is quite sufficient to alter the shape appreciably ; thus, 
the arches of Perronet’s famous bridge at Neuilly dropped on decentering 
enough to alter the radius curvature at the crown from 150 ft. to 244 ft., 
and if intended to be elliptical, it might have conformed actually more 
closely to a cycloid. 
Differences of opinion on the correct proportions of arches were very 
sharp. ‘Two writers of ability and experience at the beginning of the last 
century disagreed on important principles of design, for instance, Samuel 
Ware, Professor at Woolwich, in a pamphlet published in 1822 maintained 
that the thickness of an arch at the crown should be proportional to the 
radius of curvature. In opposition to this John Seaward in an equally 
learned paper on the subject published in 1824 argued that Ware was 
entirely wrong and that the span should be the governing factor and 
added :— 
“An ingenious writer in a late publication has strongly recommended 
that in the forming of an arch, the depth of the voussoirs should be made 
to bear a certain ratio to the radius of curvature at the crown, without any 
reference to the span of the arch: by which I presume it is intended that 
the depth of the voussoirs should bear some certain relation to the lateral 
pressure. Much as I admire the talents of the gentleman in question, 
I feel obliged, in this particular, to differ from him toto celo.’ 
i . if there be two arches of the same span, but the one having double 
the radius of curvature to the other, it is certain that the liability of the 
equilibrium being destroyed, would not be greater in the former than it 
would be in the latter: therefore on that account it is clear that the flat 
arch would not need a greater depth of voussoir ; although according to 
the doctrine held out it would have been necessary to make it double. 
Indeed, it is demonstrable that with the same depth of voussoirs the flat 
arch (provided the abutments are immovable) would be by far the strongest ; 
because, from the increased lateral pressure, it would require a much 
greater force to disarrange the parts and destroy the equilibrium.’ 
