On August 28, 1742 in the letter to Christian Goldbach, Euler wrote that in March he sent his manuscript on fluid motion in elastic tubes to participate in the contest announced by Dijon Academy of Sciences, but he did not get any response. The topic of the contest was to reveal the differences in the flow rates of fluids moving through rigid and deformable tubes ("Déterminer la différence des vitesses d'un liquide qui passe par des tuyaux inflexibles et de celui qui passe par des tuyaux élastiques"), and the prize of 30 Louis d'ors was proposed for a winner. Euler was disappointed that he hadn't made a copy for himself and now the manuscript could be irretrievably lost. According to the rules of the contest the titles of the manuscripts and information about the authors including their motto (secret code) had been kept in separate sealed envelopes. In 1742 it was Daché who won the contest, so he had to be informed on of his prize while the files of other participants had to be deleted. That is why Euler did not receive any notification from Dijon. Nevertheless the manuscripts would have been stored at the Academy and Mr. M.P.Gras, conservator of the Public Library of Dijon in the 1950s checked the archives and did not find Euler's manuscript. Up to now it is not known whether it had been received or not. In fact, in the letter to Goldbach Euler mentioned two manuscripts, one of them had been sent to Dijon and another to the contest announced by Paris Academy of Sciences on "the nature of magnets". The last manuscript had been safely received by the Academy and even won the contest together with the manuscript by Dufour and a joint manuscript by Daniel and Johann II Bernoulli, so the prize was divided between the three winners. In that way there is a chance some day the Dijon manuscript will also be found in an archive or a library collection.
In 1733, after the departure of his friend D.Bernulli, Euler got a position in the Department of Mathematics of Petersburg Academy of Sciences. In 1741, because of an unfavourable political situation in Russia, he moved to Berlin and returned back to Petersburg in 1766 where he remained until his death in 4783. In 1755 he published his manuscript dedicated to basic principles of equilibrium and motion of liquids and gases. Maybe his interest in blood flow in the arteries expressed in his Dijon manuscript inspired him to the deep study of hydromechanics. The word "hydrodynamics" was introduced by Daniel Bernoulli in his book of the same name published in 1738. In "Hydrodynamics" the mass conservation law was formulated based on the "hypothesis of perpendicular layers". In accordance with the hypothesis, the liquid flow was imaginarily separated into thin layers, perpendicular to the fluid velocity vector, and it was assumed that all the particles of the same layer moved with equal speeds. The approach was applicable to 1d flows only.
Johann Bernoulli (Daniel's farther) thought his son had based his formulation "on indirect grounds" and tried to develop a more general method for mathematical description of fluid motion. In 1743 he published the memoir "Hydraulics, now first opened and grounded on exceptionally mechanical basis", in which he used a "direct method, which is based only on the dynamic principles which nobody disputes". Actually in the memoir hydrodynamics in the modern meaning of the term, not hydraulics, was developed. The general principle by J. Bernoulli was to write down Newton's equations of motion for an infinitesimal small volume of the fluid whereby the problem is reduced to an ordinary differential equation. J. Bernoulli managed to apply his general principle to 1d flows as his son did but he failed to derive the general equations of fluid motion. Euler had an opportunity to see J. Bernoulli's book before it had been published and as he wrote in 1738 in one of his letters to J. Bernoulli, "you illuminate the question by the brightest light, because before this object was hidden from me in a thick haze and the indirect method only could help to grasp anything clear". Most likely, the general dynamic principle developed by J. Bernoulli stimulated Euler to his studies on fluid mechanics and then on the mechanics of blood circulation.
In the summer of 1775 Pallas proposed a topic "To explain the blood function" ("Explicare sanguificationem") for the following year contest of the Petersburg Academy of Sciences. Euler did not participate personally in the contest (by that time he was blind) but he prepared his manuscript on the blood flow in the arteries. Euler's results had been reported by his student and secretary of the Academy of Sciences, Nicklaus Fuss, at several meetings of the Petersburg Academy during the winter 1775/76. In some literature it is incorrectly stated that Euler himself presented his results at the meetings. The reported results were posthumously published in 1862 as a manuscript entitled "Principia pro motu sanguinis per arterias determinando" and in fact contained only the second part (§§16-43) of the original manuscript. The first part (§§1-15) was lost among numerous sheets of papers scattered on the tables and shelves in Euler's study after his death. After the death of Nicklaus Fuss, the sheets had been bound into three huge volumes. A hundred years later a Russian expert in the history of mechanics and Euler's heritage professor, Gleb Mikhajlov, identified the first paragraphs of the manuscript whose pages were distributed in the three volumes. On his request the volumes were unbound and the entire manuscript was restored. All the 43 paragraphs were published in 1979 in the next volume of Euler's Opera omnia, ser.2, though the title of the manuscript did not correspond to the topic of the volume ("Commentationes mechanicae ad theoriam machinarum pertinentes"). Probably, this is the reason that the publication of the rediscovered original Euler's manuscript was practically unnoticed and why the first (incomplete) publication is quoted and contained in the numerous papers published in connection with the 300 anniversary of L.Euler as well as in the web archives of the Euler's manuscripts.
1. Calinger R. Frederick the Great and the Berlin Academy of Science (1740-1766). // Annals of Science. Vol.24. 1968. P. 239-249.
2. Calinger R. Leonhard Euler: The First St. Petersburg Years (1727-1741). // Historia Mathematica. V. 23. 1996. P. 121-166.
3. Cerny L.C. Leonhardi Euleri's "Prineipia pro motu sanguinis per arterias determinando" // J. Biol. Phys. V. 2. 1974. P. 41-56.
4. Euler L. Principes généraux du mouvement des fluides // Mémoires de l'Académie royale des sciences et belles lettres, Berlin, 1757. T. 11 (1755). P. 274-315 = LEOO II, 12. P. 54-91
5. Euler L. Principia pro motu sanguinis per arterias determinando.§§16-43. Opera Postuma. 2, 1862. ?. 814-823.
6. Euler L. Principia pro motu sanguinis per arterias determinando.§§1-43. // Opera omnia. 2. Commentationes mechanicae ad theoriam machinarum pertinentes. 1979. P.178-193.
7. Truesdell C.A. Leonhardi Euleri, Commentationes Mechanica ad Theoriam Corporum Fluidorum Pertinentes, Volumen Posterius Ser. Secunda XIII, (Lausannae, Orell Fussli Turici, 1955) LXXVII. 8. Vischer D. Daniel Bernoulli and Leonard Euler, the advent of hydromechanics // Hydraulics and Hydraulic Research: A Historical Review. Garbrecht G. (ed.). Rotterdam-Boston, 1987. ?. 145-156.