Euler and the fiasco at Sanssouci

What has happened then in the Park at Sanssouci? A history of the constructions at Potsdam, authored by the King’s last architect, provides us with detailed information about these events. Frederic the Great wished that, besides a variety of fountains dispersed all over the Park, a 100 feet high fountain jet springs up in front of the castle—in a Park which could be fed on water only from the slowly flowing Havel about a mile away. The plan foresaw to lead the water from there via a canal to a pumping station in the Park from where it should be pumped through a pipeline into an elevated reservoir. The pump should be driven by a windmill. The reservoir was at a distance of about 3,000 feet away from the pumping station at the, a hill behind the castle. The height difference of about 150 feet between the reservoir and the level of the Park would be sufficient to provide the required pressure for the fountain jets.

The water-art project started in 1748, shortly after the constructions of the castle at Sanssouci were completed. At first everything proceeded according to the plan. The canal from the Havel to the Park, the reservoir up on the Höneberg, the windmill and the pump were accomplished within about a year. The tubes for the pipeline between the pump and the reservoir were made like wooden barrels, held together by iron bands. But when water was pumped up to the reservoir, the pipeline burst at the lower end long before the water arrived at the reservoir. Subsequently, the barrel-like tubes were replaced by entire drilled out trunks of trees. But these new tubes also burst. Now it was evident that wooden tubes would not withstand the high water pressure. A new pipeline was constructed from metal tubes. Although this pipeline did not burst, the flow rate to the reservoir was inefficient because the tubes had a too narrow inner diameter. The flawed pipeline constructions went on for years until 1756, when the Seven Years War caused a longer interruption. After the war the project was resumed for a short time, but finally the King lost patience and ordered that it was abandoned.

The chronicler of the failed water-art project at Sanssouci had made himself painful experiences with constructions under Frederic the Great. He reported other instances in which the King’s extravagance collided with his stinginess when it came to the costs. He named all practitioners who were involved with the realization of the water-art project and criticised them for their bungling. If Euler would have been responsible for the failure, as the King stated in his letter to Voltaire, this would have certainly entered the construction report in some detail. But Euler is not even mentioned.

Fortunately we do not have to speculate about Euler’s involvement. Both his scientific work and his correspondence with the King and with Maupertuis, the president of the Berlin Academy, is available in a multi-volume edition. An evaluation of these sources leads to the conclusion that Euler was unjustly blamed for the fiasco at Sanssouci. To the contrary, his analysis founded the hydraulic theory on nonstationary pipe flow. When the water is set in motion, the pressure within the tube is bound to rise. This pressure increases with the amount of water in the pipeline, and it occurs even when there is no height difference. Euler calculated this dynamic pressure increase which escaped traditional hydrostatics, and he derived recommendations concerning the power of the pump and the dimensions of the pipeline. He even warned explicitly that the project was bound to fail if the bungling in the Park proceeded. But his advice was completely ignored.

Compared to the long duration of the Sanssouci project, Euler was involved only very shortly. Less than a month passed between his first letter, addressed to Maupertuis on 21 September 1749 about the Sanssouci affair, until his comprehensive analysis issued to the King on 17 October 1749. This explains why his name is missing in the account of the constructions which extended over more than a decade. But Euler’s analysis concerned the crucial problem of the entire affair. He demanded that the thickness of the walls of the lead tubes be determined from experiments. He assumed that such tubes would be used for the pipeline after the barrel-like tubes burst—but the practitioners in the Park continued unabated with wooden tubes. With regard to the critical question of the pressure Euler wrote: "I have made calculations about the first trials at which the wooden tubes burst, as soon as the water reached a height of 70 feet. I find that the tubes actually sustained a pressure corresponding to a 300 feet high water column. This is a certain indication that the machine is still far from its perfection." Concerning the dimensions of the pipeline he suggested "that at all costs one has to use larger tubes." This advice, too, was ignored.

Euler’s pipe flow theory which resulted from his involvement with the "machine de Sanssouci" was a prelude to his formulation of the general equations of motion for ideal (i. e. frictionless) fluids. It was left to the 19th century to extend these "Euler equations" to the "Navier-Stokes equations" which take friction into account and form the basis of modern fluid dynamics. Euler’s theory, of course, would not serve as a blueprint for modern hydraulic engineering because friction modifies the outcome of the calculations according to the chosen design and materials. But this cannot be used as an excuse for the bungling at Sanssouci. To account properly for friction remained a problem until the 20th century.