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Leonhard Euler

(1707 - 1783)

The origin of quantitative arterial mechanics, like so much of quantitative mechanics in general, lies in the work of Leonhard Euler (1707-1783). His influence on mechanics and mathematics is unparalleled and far too extensive to be summarised here. As an indication of his influence on modern mathematics, Euler introduced the notation e for the base of natural logs, i for the square root of -1, &pi for the ratio of the circumference and the diameter of a circle, &Sigma for summation, &Delta for a finite difference and &part for partial derivatives.

After studying mathematics at the University of Basel, Euler took up a position at the St. Petersburg Academy of Sciences in 1727, two years after its foundation by Catherine I, the wife of Peter the Great. His original appointment was in the application of mathematics and physics to physiology, but was soon transferred to the mathematics-physics section at the urging of Daniel Bernoulli (the originator of the Bernoulli equation in fluid mechanics). In 1741 he moved to the Academy of Science in Berlin, founded by Frederick the Great, where he became the director of mathematics. In 1766 Euler returned to the St. Petersburg Academy where he was to remain until his death. In Russia his health, never very robust, continued to worsen and he became totally blind in 1771. With the help of colleagues and his remarkable memory, Euler continued to be very productive until his death in 1783.

Apart from his many contributions to mechanics in general, Euler contributed specifically to arterial mechanics with his essay entitled 'Principia pro motu snaguinis per arterias determinando' (Principles for determining the motion of the blood through the arteries) written in 1775. It was submitted to the Academy of Dijon which had set up a prize of 30 Louis d'or on the subject. Euler's contribution shared the prize with 2 other submissions but, for some reason, he did not hear form the Academy and wrote to a friend that he regretted that he had not kept a copy of the work. C. Truesdell, who edited the 1950 collection of Commentationes Mechanicae: ad theoriam corporum fluidorum pertinentes volume of the Leonhardi Euleri Opera Omnia project, instigated a search of the Dijon archives which came up with a fragment of Euler's letter. This was published as a fragment with the first 14 sections missing. In fact, the entire document had been recovered and catalogued. I am indebted to my colleague Natalya Kizilova for the following information about the convoluted history of Euler's work on arteries: (See her very informative essay On Leonard Euler and His Interest in Blood Flow Through the Arteries by Natalya Kizilova)

I was also surprised that the first paragraphs of the Euler's msnuscript on the pulse waves in arteries were found and published in 1962, a hundred years after the publication of the last paragraphs. The problem is after the death of Leonard Euler all the papers scattered on the tables, chairs and shelves in his study had been gathered and joined in an arbitrary order in three giant volumes of unpublished materials. The volumes had been kept in an archive of the St-Petersburg Academy of Science for many years. In 1955 Prof. Gleb Mikhajlov, a well-known expert in the history of science, read all the pages in French and Latin, attributed separate pages to different articles and rejoined the volumes into a correct order. Later all the papers were published in the Opera Omnia. The first 14 paragraphs were published in the Opera Omnia in 1962, so it is strange the event remained unknown for most researchers.

Section 14 of Euler's letter gives the 1-D equations derived from I. mass and II. momentum conservation in an elastic tube.

Because the work was largely unknown, these equations were only rediscovered more than 100 years after Euler first derived them with a large number of variants being proposed and studied in the interim. The excerpt has a remarkably modern appearance, being much easier to read and follow than, for example, Newton's work. This is partly because Euler invented and used the modern notation of calculus in his work and partly because he wrote so clearly. Even readers who do not know Latin, like myself, can understand the equations with little effort. Apart from a slightly different ordering of terms, they are identical to those written in the analytical section of these pages. [conservation equations]

In the excerpt, v is the average velocity across a cross-section of the tube, s is the cross-sectional area, p is the average pressure (measured as the pressure head) in a cross-section, 2g is the acceleration due to gravity*, t is time and z is the axial distance along the tube. The third equation in the excerpt is a constitutive equation relating p to s. This constitutive relationship is not derived but assumed as a reasonable choice.

Euler tried and failed to solve these equations by resolving them into a single differential equation which could be integrated. He had used this approach successfully in a number of other problems and his disappointment in not finding the solution is poignantly expressed in the final paragraph of the work.

My Latin is very rudimentary, but it is not difficult to understand insuperabiles difficultates and the suggestion (very crudely translated) If God wanted us to understand flow in the arteries, he would not have made the equations so difficult. In fact, the solution to these equations was found a century later by Bernard Riemann [to be linked].

*Footnote: The term 2g in equation II poses some difficulty for the modern reader. Euler defines g as the 'distance through which heavy things will fall in one second'. Since this distance is one half times the acceleration due to gravity, 2g is the acceleration due to gravity, which is confusing because of the now universal use of g for that quantity. It should also be noted that Euler is defining p as the pressure head so that it has units of distance.

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