SOUE News Issue 4

John Wallis and the Roof of the Sheldonian Theatre - Structural Engineering in Oxford in the 17th Century

Paul H Taylor
[Originally published in a Keble journal]

John Wallis was the Savilian Professor of Geometry in Oxford for just over 50 years around the second half of the 17th century. He was a most remarkable figure - probably the leading English mathematician of that era apart from Newton. His appointment to the Savilian Chair may have come about as a reward for deciphering intercepted Royalist dispatches during the Civil War. After the Restoration this work apparently continued. His Royal Society biography describes him as "decipherer to William III", making him an early predecessor of GCHQ. Like his modern day counterparts, he also complained about his role becoming widely known - governmental leaks were a problem then as now!

Whatever the reason for his arrival in Oxford, Wallis became a leading mathematician and corresponded with scholars throughout Europe including Fermat, most famous for his Last Theorem only proved in 1995. Wallis worked extensively in geometry and calculus, publicising both his own work and also that of others including Pell and Newton, each notoriously reluctant to publish anything. In much of his mathematical work, one can detect his interest in patterns of both symbols and numbers - with echoes of his skill as a decipherer. This interest in patterns is obvious in his single foray into structural engineering.

Wallis tackled a practical problem - how to provide a flat structure such as a roof to span a large square open space, using only wooden beams much shorter than the required span and only supported around the edges. He devised a repeating pattern of short but interlocking beams, which could be extended to span any sized space. Each beam within the body of the pattern is supported at its ends by other beams, and provides end supports for two other beams. All these internal beams are identical. The whole pattern is only supported around the periphery. The edge beams are shorter and each rest on an external support only at their outside end. Individually, these edge beams are also supported by the internal array at their inner ends. They themselves support the array at their centres. Overall, this is a clever but somewhat confusing arrangement of short interleaved structural elements, making up a repeating pattern. A 1 m square model of the pattern of beams is shown below.

Model of Sheldonian roof

To the modern engineer, what is perhaps most remarkable about Wallis's structure is that it requires no glue or screws at the joints - in modern engineering parlance, only simple vertical forces are transmitted at the contact points between the beams. The structurally awkward problem of transmitting bending and twisting moments through connections is eliminated by clever design. Wallis worked out the mathematics of how such a structure carries load - what a modern engineer would recognise as structural analysis. This required the solution of a set of 25x25 simultaneous equations with a repeating pattern, the mathematics reflecting the geometry of the array. These he solved exactly by hand, an impressive achievement as the forces transmitted from each beam to the next involve ratios such as 3088694/340167. This was as far as Wallis got with his structural analysis. However, these results are essential for deciding what cross-section would be required for each beam. Given the technical knowledge of the middle 17th century, could Wallis have estimated the required size of each beam for a given structural load?

In 1638 Galileo presented a discussion of the length of a cantilever beam able to support its own weight. This combined the theory of structures with the strength of materials, a combination still reflected in engineering teaching today. Although not correct in the details, Galileo's results reflect enough of the actual physics to allow the use of scale-models to correctly size the beams in a practical application of Wallis's design. So, Wallis could have sized his beams for an actual building. Could he also have solved the more difficult problem of predicting the deflection of his structure under load? The answer this time is no. This requires the linear elastic theory of beam bending - a problem not solved until many years after Wallis's time.

There is an intriguing further connection between Wallis as the Savilian Professor, his structural design and the visitors' Oxford of today. Christopher Wren came to Oxford as a student in 1649, the same year as Wallis was appointed Savilian Professor, so presumably Wallis taught him. Certainly they knew each other and were both members of the group whose meetings triggered the foundation of the Royal Society. Wren is most famous as an architect but he was also a distinguished mathematician. Perhaps, even more than Wallis, he should also be regarded as a structural engineer, as is clear from his ingenious design for the magnificent dome of St. Paul's Cathedral. One of his more modest achievements was the design of the Sheldonian Theatre. Here he aimed for a flat ceiling without any internal supporting columns. Such columns would have interfered with the use of the building as a venue for dancing!

There is documentary evidence in Wren's family papers that he had seen and admired Wallis's design and that he seriously considered it for the roof of the Sheldonian Theatre. Unfortunately, it was never actually built: Wren designed a type of composite roof truss that was used instead, although there is perhaps an echo of Wallis's overlapping beams in some of the lead work in the windows.

At this point, I should perhaps explain the origin of my interest in John Wallis. Robin Wilson, a colleague of mine at Keble, has recently co-authored a book on the history of mathematics in Oxford1 which contains a drawing of Wallis's design. It struck me that a structural analysis of the design using modern methods would make an interesting and unusual final year undergraduate project. This project also involved constructing probably only the third model of Wallis's structure ever made. Wallis himself instructed carpenters to make two small models, the second one he presented to King Charles II, "who was well pleased with it".

Structural analysis of Wallis's structure highlights some interesting subtleties in the way it carries loads. With the simple joints between the beams only transmitting vertical forces, there is a smooth distribution of supporting reactions around the periphery. In contrast, had the beams been rigidly connected to transmit bending and torsion moments as well, the distribution of edge reactions would be quite different, with the largest edge reaction being almost doubled and the edge force for the beam closest to the each corner being inverted. This somewhat counter-intuitive behaviour is related to significant torsion or twisting of the beams, which only occurs with rigid connections at the internal joints. For a given load, Wallis's structure will deflect more than an orthodox rigidly connected array of beams - known as a grillage to modern engineers. However, the smooth distribution of edge reactions would be much easier to provide on the top of brick or stone walls - with advantages for the design of the rest of the structure. Perhaps, his design is cleverer than even Wallis realised, and Wren did miss an opportunity with the roof structure of the Sheldonian Theatre.

In conclusion, modern engineers can greatly admire the work of the famous Oxford mathematician John Wallis. He produced a beautiful and sophisticated structural design, with engineering calculations, approximately 250 years before the University recognised engineering as an academic discipline.

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