An arch embodies strength from compression; tensile stress is avoided and stone is incompressible. Structural stability is a function of geometry rather than the nature of the material. The forces of gravity are directed sideways and downwards through the masonry voussoirs towards the ground. A freestanding arch, with no weight on it, has gravity force lines in the shape of a catenary: a complex hyperbolic function represented by the shape of an inverted hanging chain. In fact, catenaries are a family of curves, just as a hanging chain may be configured to be loose or tight. A catenary physically equalises the stress at every point- in its whole and in every part of it - i.e. every subsection of a catenary is also a catenary. An inverted catenary, forming a stand-alone arch without loading, will comply with Leibnitz's principles of least-action; it will be self supporting, requiring neither buttress nor abutment. It would seem sensible, therefore, to build arches in a general catenary shape.
However, to comply with the principles of limit analysis, it is only necessary that a single catenary line can be found that lies wholly within the masonry. If such a line exists, then the structure, before collapse, will find it and remain safe. If such a line cannot be found then tension can be expected and stress fractures will be followed by collapse of the arch. For safety, it is usually proposed that this line should be within the middle third of the stonework. Clearly, the wider the voussoirs of a circular arch, the easier it is for a catenary line to fit within it. A range of different arch shapes is therefore possible, some of them more optimal than others, but if a pure catenary shape were chosen, the voussoirs could be very slim indeed without compromising safety.
All this applies mathematically and geometrically to a theoretical arch. However, a real masonry bridge is different because it has weight and substance- the additional burden of dead-load: decking, parapets, spandrels and infill. This weight on the arch alters the shape of the force-lines a little; they change their profile to approach that of a parabola, which is different mathematically as well as having a little less spread. The heavier the whole structure, the more parabolic the force lines become. Expressed simply, weighting an arch, especially on the haunches, brings the force lines in a little. Compare above and left.
Today, live-loads such as heavy lorries or locomotives apply point pressures with quite different force line profiles; however, in 1750 the maximum live load was dwarfed by the weight of the masonry itself.
Of course, unlike a simple arch, a bridge has more material to contain the force lines, but in practice the lines of thrust are best contained by solid quality, anchored masonry- voussoirs, abutments or buttressing, perhaps ideally in ashlar. In summary, most old masonry bridges have shapes which are less than optimal for containing the forces of gravity; consequently, they must have either wide voussoirs or solid abutments on both sides.