History of the Design
Etheridge had previously been foreman to James King, master carpenter during the building of the first Westminster Bridge 1737-1750. It is to King that the interesting system of trussing may be attributed. The main members of each rib are set at tangents to the circle describing the underside arch of the bridge. An engineering analysis will show that, in the arch itself, each member is in compression with little or no bending moment, an ideal application of wood as a structural material. Where the main members cross, the wood joint is designed to transmit the compressive stress from one member to the next, with a bolt serving to hold the joint together laterally, rather than carrying any stress. There are also radial members which both support the top rail and lock all the overlapping tangents into a rigid structure, by creating triangles out of quadrilaterals. The load bearing deck is attached to the bottom of the radials, close to the junction of two tangents, where the applied load can balance the resultant of the two compressive forces from the tangents.
James King used this system of tangent-and-radial trussing in his 1737 design for a wooden Westminster Bridge (seen right), but this was abandoned after the structure was damaged when the Thames froze over in the winter of 1739-40. This is the earliest known example of this style of tangent and radial design.
In 1741, construction of a Westminster Bridge in stone commenced, and James King was appointed to erect the wooden centres on which the stone arches would be laid. He used the same system of tangent and radial trussing for his wooden arched centres as he had employed in the earlier failed bridge. This design permitted shipping to pass under the arches while they were being erected.
Pictures by Canaletto show Westminster Bridge under construction with centres of this design under each arch. In the detail of 1746 shown right, wooden centres can be seen still in position under 5 arches, and scaffolding around some of the piers. The balustrade is not yet built, apart from the centre arch on the left.
Another painting by Canaletto purports to show a view of London through an arch of Westminster Bridge while the wooden centre was still in place. St Paul's can be seen on the right. But the artist has fallen into serious error: he has shown the wooden arch to have a semi-circular outline, when we know it was made from intersecting tangents. Also the perspective of the buildings seems to have been taken from roof-level rather than water-level. So I suspect that Canaletto never saw the underside of the wooden centres at all, and that this painting is more of a capriccio than an accurate landscape.
Etheridge took over the work at Westminster after King's death, and went on to use this system of trussing again in his designs for Walton and Queens', both in 1749.
The angle between two adjacent radials in the Queens' bridge (except the ones on the abutments) is one 32nd of a revolution: other examples of this design used different spacings. This is also, of course, the angle between adjacent tangents.
In the 18th century, the generic description for designs of this sort was "geometrical construction", from which one might speculate that the phrase "Mathematical Bridge" could be derived.
Other bridges to this design
As well as Etheridhge's Old Walton Bridge (see above), there was also once a bridge of similar design on the site of the present Garret Hostel Bridge, between Trinity Hall and Trinity College. This was also built by James Essex the Younger in 1769. It was called a Mathematical Bridge in Cooper's Annals of Cambridge
The footbridge at Iffley Lock, Oxford, is a scaled-down version of the Queens' bridge, built in 1924. There was a footbridge at Winchester to the same design which lasted until 1976.
Further reading
Arch Bridges and their Builders 1735-1835, by Ted Ruddock, Cambridge University Press, 1979 (out of print).