The Pechot Memorandum

As a young man, Prosper Péchot studied at the prestigious Ecole Polytechnique in Paris. Those who passed the entrance examination were already graduates of prestigious institutions with the equivalent of a degree in mathematics. He would have enjoyed the company of such luminaries as Lionel Penrose, Laurent Bartoldi and André Henriques if they had been around at the time. Unfortunately, he died in 1928, but here follows my little flight of fancy. While at Staff College, 1880 to 1882, he was given the standard puzzle of the day, how to regain the Lost Provinces of Alsace-Lorraine. These were conceded to the new German Reich in 1871 at the conclusion of the Franco-Prussian War. The city of Metz was considered the key to a successful campaign. It was well-defended with the finest fortifications of the period, massive structures proof against siege guns of the time. Prosper Péchot reasoned that if the present guns weren’t big enough, larger ones would have to be used. Such guns existed and had indeed been used. During the lamentable war, the French had brought naval guns up the Seine and positioned them around Paris. The Prussians never actually took Paris. It had surrendered in January 1871 after a grisly siege when cold, hunger and disease had each taken their toll.

If you want an idea of the area defended by these guns and how such guns were moved into position, look at a map of modern Paris. Although now surrounded by urban sprawl, its most distinctive feature is the péripherique road which encircles it. This was built over the defences of 1870. Though warfare is no longer actually waged, a drive can be a somewhat unnerving experience for a visitor. The map has other interesting features. The main motorway intersections are still called ‘Ports’ recalling the time they were defensive gates on the road to Orleans etc. It is pierced by a number of obliging rivers, twice by the Seine and once each by tributaries such as the Marne. It’s not too much of a stretch to call it the Isle de France! There was, however, all the difference in the world between transporting naval guns by ship and dragging them – and their heavy ammunition – overland to a convenient position within range of an enemy fortification. He envisaged guns whose barrels alone had a mass of 34 tonnes and a total daily passage of freight of over 2000 tonnes. Péchot had the solution and he had seen it at work, well, sort of. In the 1870s, Paul Decauville the French industrialist had developed a portable railway for use on farms and quarries. There is quite a difference between a barrow-load of sugar-beet and a large gun, but Péchot had resolve, imagination and a grasp of mathematics. The Péchot system was born.

He seized his pen and turned a College exercise into Memorandum which, hardly out of Staff College he sent straight to the top, to the Ministre de Guerre – ministry of Defence we might say. It was a snappy document. With a small preamble, in the paragraph entitled Description, he explained the purpose of his system. ‘A means must be found to carry components weighing up to 34 tonnes and a significant mass of smaller freight.’ Hoping that his readers shared his enthusiasm and would read on in excitement, ‘Trials indicate that a portable railway could do this.’ Trials had proved that an adapted version of the existing Decauville railway was suitable. Unfortunately, as it turned out, his readers did not share his enthusiasm and it was a long time before the système Péchot was adopted. His claims were justified, both by mathematical theory and experiments in the field. Paul Decauville helped with the latter though, sadly, in the end his efforts did not receive due recognition. We come back to the mathematics. The Decauville system was ideal for transporting carts carrying 0.5 of tonne, a relatively easy ‘push’ for a human or 1 tonne, well within the capabilities of a horse.

Decauville prefabricated rail looked like an adapted ladder with some rungs missing; such rail was quite adequate for the force with which each axle of these carts pressed down. If, however, such wagons were overloaded, the rail would distort. To ensure a better match, a/ the rail could be strengthened b/ the number of axles could be increased. What about c/ a judicious increase of each? We can imagine a ‘sweet spot’ where the two lines intersected on a graph. Firstly Péchot found the theoretical ‘sweet spot’ and then he experimented at Decauville’s factory. As we have seen, the patron helped, with specially made track and wagons. Péchot devised ‘a track using rail of 9.5 kilogrammes per metre.’ He supplied a drawing of a 5m prefabricated track panel supported by 8 improved sleepers to be coupled with Decauville-style fishplates. These could resist a force of 3 ½ tonnes ie support a mass of 3 ½ tonnes.

Secondly, in order to take serious loads, he proposed the use of bogies instead of simple wagons. Although not unknown, the bogie was an innovation in this context. If each load were supported supported by a suitable number of axles, load-bearing was vastly increased. Two axles (ie four wheels) could safely convey a mass of 5 tonnes, three axles for 9 tonnes. By using pivoting links, such axles could be multiplied, if not infinitely, many times at least. See picture above. There was another issue, that of the track gauge. At one extreme, was the prefabricated track used by Decauville and others. A gauge of 40 cm was convenient for transport over ploughed fields and inside factory settings whereas national French railways used Standard Gauge. Broader gauges such as those of the British Great Western Railway and the railways of Imperial Russia also existed. Theory suggested that 60cm gauge offered ‘the best of both.’ A rival system dreamed up by the Génie (Engineers) branch of the French army came to grief wherever their system arrived at the gate of a fort – too narrow. A railway of 60cm gauge could tolerate a turning circle of 20 metres – track panels of this radius were available. There is a theoretical minimum which was later explored by Lionel Penrose. His son, Roger, created mazes for railways; the train had to get from A to B without making any excessively sharp turns. The illustration shows ‘allowed’ and ‘forbidden’ routes on a railway.Forbidden are marked in red. Péchot had to devise many mazes of the sort shown below right and if Penrose puzzles had existed in those days he would have enjoyed them.

There was another design point where Péchot’s strong mathematics helped. He realised that his system needed a smooth transition between curved and straight track. He appreciated the Euler spiral. This puts into mathematical language what the eye can see. Its curvature of 1/r where r is the radius of the best-fitting circle at a given point, is proportional to the distance along that line. A graph shows this. At the origin, there is no curve – it’s a straight line. At the two ends of the spiral, the curve is infinitely tight. To get your curved and straight lines to meet up smoothly – otherwise your train will fall off the track – just calculate the correct value of r.

Sadly, Péchot-style track was used on an industrial scale during the first world war. This picture shows a British War Department railway in action in 1917.

As Hannah Fry would say, ‘mathematicians aren’t the ones who find it easy. They just enjoy how hard it is.’ Prosper Péchot would smile. Further Reading Roy Link: Albums of the First World War RAM Publishing Roger Penrose: Railway Mazes from ‘A lifetime of puzzles’ Ed, Demaine et al, Wellesley 2008 Professor Ian Stewart: ‘Casebook of Mathematical Mysteries’ Profile Books 2014 Sarah Wright: ‘Tracks To The Trenches: Colonel Péchot’ Birse Press 2014