Book: Theoria motuum planetarum et cometarum

Overview
Leonhard Euler's Theoria motuum planetarum et cometarum (1744) presents a systematic, mathematically rigorous treatment of the motions of planets and comets under Newtonian gravitation. The text assembles analytic techniques to derive orbital elements, to propagate orbits forward in time, and to account for deviations from ideal two-body motion. Emphasis falls on practical methods that turn observational data into predictive orbits, marrying geometric insight with differential calculus to make celestial mechanics more calculable and accessible to working astronomers.

Mathematical foundations
Euler frames orbital motion using the language of calculus and analytic geometry, exploiting the inverse-square law to reduce motion to conic sections modified by perturbing influences. He develops expansions and transformations that simplify the relations between physical forces and the instantaneous elements of an orbit, treating the planet or comet as subject to both central attraction and external disturbances. Rather than only relying on purely geometric constructions, Euler uses series and algebraic manipulation to produce approximate solutions that converge to the true motion, thereby turning difficult integrals and transcendental equations into workable computational recipes.

Techniques for planets and comets
The treatise distinguishes between nearly Keplerian planetary orbits and the often highly eccentric paths of comets, tailoring techniques to each regime. For planets, Euler addresses how mutual planetary perturbations alter the elements slowly over time and introduces methods to compute secular and periodic corrections to mean motion and orbital orientation. For comets, whose high eccentricities and short observational arcs pose special difficulties, he gives procedures to extract reliable elements from limited data and to propagate such elements through perihelion passage. Across both cases, the focus remains on producing accurate tables and formulas that can be applied directly to observational records and ephemeris construction.

Computational approach and examples
Practical computation is central: Euler supplies worked examples and stepwise manipulations that show how to go from observed positions and times to orbital parameters, then how to use those parameters to predict future positions. He demonstrates how to linearize certain relations for iterative refinement, how to isolate dominant terms in perturbative expansions, and how to estimate the size of neglected terms to judge approximation quality. These procedural aspects make the treatise not only a theoretical statement of celestial dynamics but a manual for astronomers needing usable, finite computations.

Scientific context and influence
Appearing in the mid-18th century, the work builds on Newton's foundational principles while anticipating developments by Lagrange, Laplace, and later celestial mechanicians. By formalizing analytic techniques and emphasizing computational practicality, Euler's treatise helped bridge theoretical mechanics and observational astronomy. Its methods improved the preparation of ephemerides and the interpretation of cometary returns, and its analytic style influenced later expositions of perturbation theory. The combination of rigorous derivation and attention to numerical procedure marks it as a significant step toward the mature mathematical theory of planetary motion that followed.