Leonhard Euler (1707–1783) published two lunar theories in 1753 and 1772. He also published lunar tables in 1745, 1746, and—anonymously—in 1750. There are notebook records, unpublished manuscripts, and manuscript fragments by Euler reflecting the development of his lunar theories between about 1725 until about 1752. These documents might be used to reconstruct Euler’s theory on which he based his calculations of those lunar tables and to analyze the development of his lunar theories within this time span. The results of this (...) analysis will be published here in three parts representing three stages of Euler’s research on this topic: First approaches (about 1725–1730), developing the methods (about 1730–1744), and the breakthrough (about 1744–1752). In this part, I analyze Euler’s manuscripts and, predominantly, Euler’s records of his first two notebooks written between 1725 and 1730. I found that his early theoretical approach is coined by his development of analytical (rational) mechanics of punctiform bodies moved by central forces. He tried to describe the Moon’s motion in terms of two simultaneously acting centripetal forces, Huygens’ centrifugal theorem, and associated osculating radii. (shrink)

Up to now only three lunar tables by Leonhard Euler (1707–1783), published in 1745, 1746, and 1772, were known. For a long time, however, it was assumed that the first two of these tables were identical. The author compared these tables with each other and proved the contrary. This fact also transpires from an examination of their history, which was reconstructed using Euler’s correspondence. In addition, evidence has been found in Euler’s voluminous scientific correspondence and in contemporary publications of the (...) eighteenth century that, between 1742 and 1750, Euler published additional lunar tables anonymously. The author proved Euler’s authorship of at least five lunar tables containing some 100 pages, which were neither recorded in the list of his works nor published in the Leonhardi Euleri Opera Omnia. (shrink)

The analysis of unpublished manuscripts and of the published textbook on mechanics written between about 1730 and 1744 by Euler reveals the invention, application, and establishment of important physical and mathematical principles and procedures. They became central ingredients of an “embryonic” lunar theory that he developed in 1744/1745. The increasing use of equations of motion, although still parametrized by length, became a standard procedure. The principle of the transference of forces was established to set up such equations. Trigonometric series expansions (...) together with the method of undetermined coefficients were introduced to solve these equations approximatively. These insights constitute the milestones achieved in this phase of research, which thus may be characterized as “developing the methods”. The documents reveal the problems Euler was confronted with when setting up the equations of motion. They show why and where he was forced to introduce trigonometric functions and their series expansions into lunar theory. Furthermore, they prove Euler’s early recognition and formulation of the variability of the orbital elements by differential equations, which he previously anticipated with the concept of the osculating ellipse. One may conclude that by 1744 almost all components needed for a technically mature and successful lunar theory were available to Euler. (shrink)