Mathematical Bridge

Mathematical BridgeThe bridge was designed in 1748 by William Etheridge (1709–76), and was built in 1749 by James Essex the Younger (1722–84). It has subsequently been repaired in 1866 and rebuilt to the same design in 1905.

The red-brick building seen on the right is the President’s Lodge (ca. 1460), the oldest building on the river at Cambridge.

For those who have fallen prey to the baseless stories told by unscrupulous guides to gullible tourists, one should point out that Isaac Newton died in 1727, and therefore was unlikely to have had anything to do with this bridge. Other baseless stories are that Etheridge had been a student or a Fellow, and/or had visited China, or that the bridge is of chinese design. Anyone who believes that students or Fellows could have dismantled the bridge (and then failed to re-assemble it, as the myth runs) cannot have a serious grasp on reality, given the size and weight of the wooden members of the bridge. The joints of the present bridge are fastened by coach-bolts and nuts. Earlier versions of the bridge used iron pins or coach-screws at the joints, driven in from the outer elevation. Only a pedant could claim that the bridge was originally built without nails.

The bridge has also been mis-attributed to John Michell: John Michell may perhaps be better known as the builder of the mathematical bridge across the Cam, at Queen’s College, Cambridge [English Mechanic and World of Science, No. 325, 1871 June 16, p. 310] in a letter from the pseudonymous Khoda Bux, later identified as Andrew Thomas Turton Peterson, great-grandson of Michell. The letter relied on oral tradition within the family, and some of its statements have been questioned by later scholars [for instance: Memoir of John Michell, by Sir Archibald Geikie, 1918, pp. 14–19; The Reverend John Michell: A Letter from his Great-grandson, by Eric Hutton, in The Antiquarian Astronomer, No. 3, 2006, pp. 65–8]. Michell was an undergraduate at Queens’ from 1742, and appointed a Fellow in 1749, the year after Etheridge’s model of the bridge had been paid for.

History of the Bridge at Queens’

The erection of the bridge occurred at the same time as the building of brick walls along the river banks and alterations in the layout of the Grove. It can therefore be difficult in reading the college records to distinguish one project from another. The following entries in the Magnum Journale appear to be relevant to the Bridge or riverside works:

    £. s. d.
1748 Oct 6 Mr Etheridge for the Design & Model of the Bridge 21. 00. 00
1749 Mar 25 Paid for removeing the Mud from the Bridge 01. 02. 6
1749 Jul 1 Paid for five Men for digging the River Bank 01. 11. 6
1749 Jul 8 Paid for six Men for digging the River Bank 02. 01. 6
1749 Sep 30 Paid for work in levelling the River Bank 0. 07. 6
" Two Men levelling the Bank at the Bridge 0. 07. 0
" 300 Green Turf for the River Bank 0. 18. 0
" Paid Wm Christmas for the Use of a Boat 00. 03. 0
" Mr Groom for the Use of a Boat 03. 00. 0
" Mr Tucke, the Bricklayer’s Bill 114. 16. 8
" Mr Bottomley, the Mason’s Bill 151. 12. 9
" Mr Essex’s Bill for the new Bridge 160. 00. 0
1750 Sep 30 Cook ... a Supper on finishing the Bridge to Mr Essex’s Men  0. 17. 9

It was the practice to account for the bills of tradesmen (including the College’s own cook) in a single lump sum on the last day of the academic year, so nothing much can be deduced about the timing of works accounted for on September 30th, other than that they occurred sometime in the previous twelve months. The supper that was accounted for in September 1750 might have occurred as early as October 1749. As Essex was paid in the year ending September 1749, it is reasonable to assume that all the works were executed during the summer of 1749. Willis & Clark appear to have been in error in stating that the bridge was finished in September, 1750. [The Architectural History of the University of Cambridge by Willis & Clark, Vol. 2, p. 56]

Bridge modelThe college possesses an old model of the bridge, and we assume that this is Etheridge’s model of 1748 (although we have no proof of this). It is to be noted that even the model has screw-eyes at the joints. The model is at 1/16th scale.

The earliest printed description of the bridge appeared in 1753:

The Bridge from the Cloister to the Stable, &c. which was wholly rebuilt A.D. 1746; may without Flattery, be esteemed one of the most curious pieces of Carpentery of this kind in England, it contains in length upwards of 50 Foot, being of one Arch, composed of Timbers curiously joined together, and supported on Abutments of Rustic Stone-work, between which is a Passage for the Cam, 40 Foot in the Clear, and of such Heigth[sic], that the Waters in a common Flood, cannot reach the lowest Timbers thereof.
[The History of the University of Cambridge by Edmund Carter, 1753, p. 186]

Carter was not renowned for his accuracy: the incorrect date of 1746 was reproduced in many subsequent histories and guides.

The earliest discovered use of the term Mathematical Bridge in relation to the bridge at Queens’ was in A Guide through the University of Cambridge, of 1803, as a footnote to a description of the bridge on page 68: Usually known by the name of the Mathematical Bridge. A different publication, Cantabrigia Depicta, by Harraden & Sons, made a similar comment in 1809. Some care is required in interpretation, as the title Mathematical Bridge was also being used in the same period for the Garret Hostel Bridge (see below), and either bridge might have been confused with the other.

The bridge is 50 feet 8 inches (15·44 metres) long. The angle between two adjacent radials (except the ones on the abutments) is one 32nd of a revolution. The arch is composed of tangents to a circle of radius 32 feet. [Queens’ Bridge, by Gunnar de Vries, 1992, p. 8]

The design of the Queens’ bridge resembles Etheridge’s much greater Old Walton Bridge, a three-arch bridge over the Thames, built 1748–50 (see below).

William Etheridge was one of a very long family line of carpenters called variously Edrich, Edriche, Eteridge, or Etheridge from Stradbroke and Fressingfield in Suffolk. His birth is not recorded, but his baptism took place on 3rd January 1708/09 at St Margaret’s parish church, South Elmham, about 5 miles north-east of Fressingfield. His career as a master carpenter first comes to light in 1738–49 when he worked under James King in the building of the first bridge to cross the Thames at Westminster, first as King’s foreman, then replacing him after King’s death in 1744. He was credited as the inventor of a battering ram to assist in the striking of the centres, and an underwater saw to cut off piles underwater. From 1747–50 he worked on the Walton Bridge, and in 1748 produced the design and model for the Queens’ bridge. From 1751–56 he worked as surveyor for the construction of Ramsgate Harbour in Kent. In 1761 he was consulted on the state of St Olave’s bridge at Herringfleet, Suffolk. He died aged 67 on 3rd October 1776 in Westminster and was buried on 15th October 1776 at Fressingfield. [Information in this paragraph is derived mainly from Etheridge’s family descendants, private communication].

The Queens’ Bridge is the earliest recorded built work of James Essex the Younger. He later constructed the Essex Building 1756–60, which was intended to be part of a new building along the complete riverfront, replacing the 1460s part of the President’s Lodge, but which was never completed. He is buried in St Botolph’s, church of the parish in which Queens’ lies. His father James Essex had also been a carpenter, doing work in many colleges, including the panelling of the Hall at Queens’.

Decay of, and repairs to, the Queens’ bridge

By the mid-19th century, the original bridge of 1749 was developing problems from the decay of certain timber elements. Judging by the photographic evidence, the main problems were the cross-beams under the bridge which connected together the lower ends of opposing radials from each side rib, and on which the load of the pedestrian decking was carried out to the side ribs. These cross-beams were horizontal, and presumably prone to water not draining away effectively, or draining into the joints with the radials of the side ribs. There was no redundancy: if a cross-beam failed, the decking would sag. We can track the development of these problems from old photographs.

Mathematical Bridge early 1850sThe first is a detail from a very early photograph of the bridge, and possibly the earliest surviving photograph of any Cambridge scene, taken by Thomas Craddock in the early 1850s while he was an undergraduate at Queens’ 1850–56. One of the cross-beams is shown to have been reinforced with an extra layer of timber underneath. Note that, in this period, the pedestrian decking was stepped (just like the model), rather than sloping. Those interested in demolishing myths should note that there are clearly visible fastenings at the joints of the radials and tangents on the outer side facing the camera, but that those fastenings are not visible on the inside elevation of the far side rib.

Mathematical Bridge 1857We have confirmation in the next photograph, probably taken by Robert Cade, around 1857. The same cross-beam is shown with the same additional layer.

Mathematical Bridge, mid-1860sBy the time of this next photograph (one half of a 3D pair from the early to mid-1860s), the problems with the cross-beams had spread to two others, and the side ribs appear to be distorted and leaning inwards. The bridge has metal hoops strapped to pairs of radials: it is not clear whether those hoops are intended to support a canopy, or whether they are an emergency fix to prop up the side ribs.

The bridge was repaired in 1866:

Agreed that the Bills for repairing the Bridge over the Cam, amounting to £348-7 having been received, that £148-7 be deducted from revenue for this year and £100 from revenue for each of the two next succeeding years. [Conclusion Book, 1867 January 10th]

Mathematical Bridge, 1870sIn this photograph, ascribed to the 1870s, and definitely after the 1866 repairs, we can see that the cross-beams under the bridge have returned to their normal size and shape, with no extra layers strapped underneath. This photograph has sufficient detail to show that the side ribs appear to be unchanged: nothing appears to be new: the 1866 repairs appear to have been confined to the cross-beams under the bridge and the decking, which changed to be sloped rather than stepped, making it possible today for wheeled trolleys to be used over the bridge. Another change was the addition of metal clips on the top rail, to enable metal hoops to be slotted in when required. It can be seen that various timbers in the bridge have lead capping, especially near joints: presumably to keep rainwater out of the joints.

In 1905 the bridge was completely rebuilt to the same design, by the local builder William Sindall, in teak instead of oak. Once again, the cost of the new bridge was partly buried in the costs of a larger project of works to the riverside buildings, overseen by architect Thackeray H. Turner (1853–1937), of Balfour & Turner. The only invoice attributable solely to the new bridge was the final one of £207 6s 6d, but it seems unlikely that a bridge could have been built in 1905 for such a small sum. In this version of the bridge, the joints were fastened by coach-bolts passing right through each joint, the nuts being on the outer elevations, and the bolt heads being on the internal elevations of the side arches, and therefore visible to people passing over the bridge. In the pre-1905 bridge, the fastenings at the joints had not been visible on the inside elevation of the side ribs: this might have been either (a) because the fastenings had been coach-screws driven in from the outer elevation, not penetrating to the inside elevation, or (b) because the fastening had been rebated on the inside elevation and covered with a wood shim. In the 1905 bridge, fastenings are visible at the cross-over between tangents (intermediate between radials), where no fastenings had been visible before 1905. One may speculate that perhaps the sight of bolt heads where none had been seen before might have given rise to the myth of the failed re-assembly. This myth has not been found in print earlier than the 20th century.

History of the design

Etheridge had previously been foreman to James King, master carpenter during the building of the first Westminster Bridge 1737–50. It was reported that Mr K. was, without education, a man of ingenuity and considerable self-taught mechanical knowledge [Gentleman’s Magazine, Vol. 65, Pt 2, July 1795, p. 618]. It is to King that the interesting system of trussing may be attributed: a curious wooden Superstructure, of the Invention and Design of … Mr. James King. … the Model of this Superstructure was … shown and explained to a great many Persons of the first Rank in the Kingdom, and to many others, and generally approved of, as the most curious of its Kind… [A Description of Westminster Bridge, by Charles Labelye, 1751, p. 7]. The main members of each rib are set at tangents to the circle describing the underside arch of the bridge. 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 itself 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.

Westminster BridgeJames 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.

Wooden centre for stonework of Westminster Bridge 1741, drawn by Simon MillerIn 1741, construction of a Westminster Bridge in stone commenced, and James King was appointed again 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 wooden bridge. This design permitted shipping to pass under the arches while they were being erected.

Etheridge took over the work at Westminster after King’s death in 1744.

Detail from: London: The River Thames looking towards Westminster from Lambeth, by Canaletto 1746Paintings 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 the arches, and scaffolding around some of the piers. The balustrade above the arches is not yet built.

London seen through an Arch of Westminster Bridge, Canaletto, 1746-7Another painting by Canaletto, 1747, 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 invented most of this picture.

Etheridge went on to use this system of trussing again in his designs for Walton and Queens’, both in 1749. In the 18th century, a generic description for designs of this sort was geometrical construction, from which one might speculate that the phrase Mathematical Bridge could be derived. [Arch bridges and their builders 1735–1835, by Ted Ruddock, p. 35]

Other bridges to this design

Old Walton Bridge 1750The old Walton Bridge was also designed by William Etheridge, built by a Mr White of Weybridge during 1748–50, and paid for by Samuel Dicker, later M.P. for Plymouth. This print of the Walton Bridge was published in the Gentleman’s Magazine of 1750, Supplement, pp. 588–9.

The Walton Bridge was much admired for its strength, contrivance and remarkably great arch [Gentleman’s Magazine, Vol. 20, Nov 1750, p. 488], and was even described as the most beautiful wooden arch in the world. [Descriptions to the plates of Thames Scenery, engraved by W.B. Cooke & G. Cooke, published by John Murray, 1818]

Old Walton Bridge, Canaletto, 1754It attracted Canaletto to paint it twice: this is a detail from the first, painted in 1754. The artist in the bottom left-hand corner is reputed to be Canaletto’s depiction of himself.

Old Walton Bridge, Canaletto, 1755This is a detail from the second Canaletto, of 1755. The Walton Bridge’s main span was 130 feet, with two side arches of 44 feet. Unfortunately, it decayed and lasted only until 1783.

One of the asserted merits of this design was described as: The happy construction of this bridge was such, that, being composed of timbers, tangent to a circle of an hundred feet in diameter, either of them falling into decay, might with ease be unscrewed, and with equal facility, receive a new substitute, without disturbing the adjoining timbers. [Descriptions to the plates of Thames Scenery, 1818] No report is known of such a surgical repair ever having been carried out on either the Walton or the Queens’ bridge, but it is plausible that Etheridge’s model of the bridge for Queens’ was designed to demonstrate that possibility.

Old Garret Hostel BridgeThere once was a bridge of similar design on the River Cam at the site of the present Garret Hostel Bridge, between Trinity Hall and Trinity College, built in 1769. In 1812: On the 2nd of July, Garret Hostel bridge broke down. It had been in a decayed state for a considerable time, and the passage over it had been stopped for several days previously. It was frequently called the mathematical bridge, and was erected in 1769, from a design by Mr. Essex. [Annals of Cambridge, C.H. Cooper, vol. IV, p. 503]. That was the same Mr Essex who had built the bridge at Queens’ in 1749.

This view was drawn by Bt Leach and engraved by R.H. Leach (1794–1851). The date of the view is commonly quoted as circa 1810, but the print might have been engraved much later.

Iffley Lock footbridgeThe footbridge at Iffley Lock, Oxford, is a scaled-down copy of the Queens’ bridge, built in 1924, by the Thames Conservancy, to the design of their Chief Engineer, G.J. Griffiths, M.I.C.E.

There was a footbridge at Winchester to the same design which lasted until 1976.

Further reading:

1886: The Architectural History of the University of Cambridge, by Robert Willis and John Willis Clark, Vol. 2, pp. 55–​6. (OCLC 6104300)

1979: Arch bridges and their builders 1735–1835, by Ted Ruddock. (ISBN 978-0-521-21816-0)

1992: Queens’ Bridge, by Gunnar de Vries, in Industrial Design 02: Zwei Forschungsberichte, ed. Lambert Rosenbusch, pp. 6–​18. (ISBN 978-3-931185-27-5)