*“Did Isambard Kingdom Brunel calculate how strong his bridges would be; and if so, how?”*

The quick answer is that Brunel was indeed able to work out the strength of his bridges and other structures, and he did it in much the same way as engineers do today – but without the tremendous help that computers and other digital tools now offer.

This seems a remarkable feat for anyone to achieve in the 1800s; so let us dig a little deeper and find out what mathematical tools were available to Brunel to help him design his structures (not just bridges, but things like roof trusses, too). Let us also ask whether Brunel even had the skill and knowledge to use these tools in his work.

**Brunel had a brainy dad**

Taking the last question first, it turns out that Brunel was uniquely qualified among his contemporaries to take a mathematical approach to engineering design. For even though most British engineers of the time were not good at maths, he was lucky to be born into an educated family. The son of Sir Marc Brunel, his father was a Frenchman who learned his engineering skills in pre-revolutionary France before fleeing the French Revolution for America and finally settling in Britain. By the late 1700s, when Marc Brunel was learning his craft, the French were providing university-level training for bright young engineers. This education included a thorough grounding in the latest scientific and mathematical ideas. By contrast, British universities did not start offering engineering degrees until the 1870s. So Marc Brunel knew his maths, and as Brunel learnt his engineering as Marc’s apprentice, Marc ensured that his son knew his maths, too. It is interesting to note that a lot of Brunel’s well-known British contemporaries tended to be ignorant even of simple calculus, and placed a great deal of reliance on rules-of-thumb and practical experience when producing designs.

**How much did 19th Century engineers know anyway? A brief history.**

Surprisingly, the mathematical knowledge of structures in Brunel’s day was quite good: By 1800, mathematicians had produced a useful ‘toolkit’ of results for engineers – giving them access to good equations for working out the strength of beams and columns, the stability of arches as well as the behaviour of suspension bridge chains.

It is generally believed that Galileo was the first scientist to attempt a mathematical analysis of structures. In his *Dialogue concerning the Two New Science*s, published in about 1638, Galileo presents the first ever attempt at a theory for the strength of beams. He was wrong, and predicted that beams should be a lot weaker than they actually are, but it was a start.

Then in the 1670s Robert Hooke, a distinguished architect as well as a leading member of the Royal Society, published the first serious attempt at a mathematical analysis of arches. But the maths of buildings (‘structural mechanics’) only really took off after the invention of calculus by Isaac Newton and Liebnitz in the 1680s. Using this powerful new mathematical technique, Daniel Bernouilli and Leonhard Euler had, by 1750, developed the basic theory of loaded beams that is still used by engineers today.

**‘Beam me up, Berni’**

The mathematical equation used by Bernouilli and Euler in their beam theory calculates how much a beam will move when a force pushes on it (like a train moving over a bridge, for example). The beauty of their formula lies in its versatility – you can just plug in the corresponding values of pretty much anything that behaves like a simple beam, do the workings, and crank out the result. In fact, the most difficult part for engineers is to decide which parts of a structure are ‘beams’ (fixed, straight objects) that can successfully be used with the formula. For the parts of a building that are not ‘beams’, a scaled=down model or computer programme is used to figure out their expected behaviour.

**No computer? No problem**

Brunel, of course, would not have had access to any fancy engineering software. So when algebra was a no-go – for example, in calculating ‘statically indeterminate’ structures like bowstring girders – he would have had to make do by building small models to test his schemes: first designing a to-scale prototype of the proposed bridge, and then testing it with different weights to prove that the full-size structure would be strong enough. The equivalent of Brunel’s rough workings on projects falling under this category may be found in his notebooks in the Brunel archive at the SS Great Britain, Bristol – handily reachable by train!

**Bridging the final gap**

Even with a good formula, it was still not enough to just have the maths: good results also require good data about the properties of materials – how strong they are, how much they bend, etc. – and this was sorely lacking in the early Victorian period. A prudent engineer, like Brunel’s great friend Robert Stephenson, would never have placed their entire trust in the numbers alone. When, in the mid-1840s, Stephenson was planning his ‘tubular’ bridges to carry the Chester and Holyhead railway across the Menai Strait and the River Conwy, he asked William Fairbairn and Eaton Hodgkinson, both engineers with stronger mathematical backgrounds than his own, to help in the design calculations for the immense wrought-iron tubular girders his design demanded. Fairbairn and Hodgkinson could not agree upon whether the tubes would be strong enough, so Stephenson had a large-scale model of one of the tubes – about 75 feet long – built and tested, in order to validate the design and settle the argument! Brunel therefore would not only have required mathematical acumen, but also a fair amount of artistic creativity and lateral thinking in planning and testing models to ascertain that his final designs would hold up. No pressure, then!

Answer by Richard Ellam (member of the Newcomen International Society for the History of Engineering and Technology)

**Footnote:** Brunel’s actual calculations are hard to come by. His calculations are spread across various notebooks for any given project, and in some cases may have been done ‘on the fly’ and then thrown out with the envelope they were on the back of. Also, archivists are much better at preserving engineer’s drawings than they are at preserving the working notes that lead to their creation. Sorry maths fans!

Image Sources: Thames Tunnel walk by Emily Webber, on Flickr, Isambard Kingdom Brunel by Leo Reynolds, on Flickr and Brunel Bridge by jonno85uk, on Flickr