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Though an example of a cycloid is not given inthe tables of bridges, the ratio of the versed sine tothe span in the arches surbaisees most frequentlyadopted by the French and other architects, ap-proaches very nearly to that of the cycloid, viz.As the diameter to the circumference of a circle, or1 to 3.1416; and mathematicians have very properlyrecommended this curve in preference to the segmentof a circle and to the ellipse “ surbaissee au tiers,”and its simplicity of construction, and the facility offinding the tangents (whence the joints of the vous-soirs) warrants practically the recommendation.

Assuming the incumbrances on bridges to be aconstant quantity, the radius of the circle of curvatureat the vertex to be a measure of the strength of theform of the arch, the height of the key-stone to bedeterminable from the ability of the material to resistcompression ; and, that ordinary stone may be taken,after the proper allowances have been made to obtaina practical strength, of a constant strength in respectto compression ; marble and granite, also, of a con-stant strength, but of greater strength, in that respect,than common stone; and brick of a constant strength,but of less strength than common stone; and so inrespect to woods, and in respect to iron ; then in thecolumn of the specific height of the arch at the vertex,an approximating ratio may be had of the science dis-played, of the quantity of materials used, and of thecost of the several bridges. For example, compar-ing the bridge of Neuilly , by Perronet, with thatof Alcantara, over the Tagus, near Lisbon , Per-ronet’s science will be to that of the architect of thebridge of Alcantara as 8.32 to 1., and the quantityof materials in the Alcantara bridge, and the cost ofit will also be in the same ratio in respect to thebridge of Neuilly , viz. as 8.32 to 1.

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