Book: Mechanica

Overview
Mechanica, published by Leonhard Euler in 1736 under the Latin title "Mechanica sive motus scientia analytice exposita," is a sweeping mathematical treatment that recasts the science of motion in analytic terms. Euler replaces much of the classical synthetic geometry used by his predecessors with a systematic application of differential and integral calculus, turning qualitative geometric arguments into precise algebraic and differential relations. The book sets out general principles for describing the motion of particles and rigid bodies and draws the foundations for modern dynamics.
Mechanica treats motion as a subject for rigorous mathematical analysis rather than rhetorical or purely geometric argument. Problems that had been addressed by geometric construction are reformulated so that they become problems of solving ordinary differential equations, subject to initial conditions and algebraic constraints.

Analytical method
Euler insists on coordinates and algebraic expressions for positions, velocities, and accelerations. He expresses forces and their effects by means of components and differential relations, showing how the language of calculus naturally encodes both magnitude and direction. This move allows a uniform treatment of diverse problems, from single particles moving under central forces to systems of interconnected masses and constraints.
The approach emphasizes deriving equations of motion directly from kinematic relations and force laws. Where earlier writers used geometric loci and constructions, Euler writes and manipulates differential equations, applies limiting processes, and interprets solutions in mechanical terms. The clarity of this analytic method makes it possible to handle more complex configurations and to follow the dependence of motion on initial conditions.

Rigid body dynamics
A central achievement of Mechanica is the systematic analysis of rigid bodies in motion. Euler formulates the kinematics of rotation and translation together, introducing the idea of resolving motion into components about fixed and moving axes. He develops relations for the motion of the center of mass and for rotation about that center, distinguishing translational and rotational effects with precision.
Euler derives the fundamental equations that govern the rotation of a rigid body under the action of external moments and forces. He identifies principal axes and relates the distribution of mass, what later becomes the moment of inertia, to resistance to angular acceleration. These formulations provide a clear pathway from the geometry of a body to quantitative predictions of its rotational behavior.

Key results and examples
Mechanica contains worked examples that illustrate the power of the analytic framework: the motion of bodies under central forces, the behavior of pendula, the response of systems subject to constraints, and the change of motion under impulsive forces. Euler translates many classical mechanical problems into differential equations and solves them or reduces them to solvable forms, thereby demonstrating unified methods that apply across problem types.
Throughout, emphasis falls on expressing physical quantities in algebraic form and on deriving conservation relations and integrals of motion when they exist. The methods spotlight how symmetry and the nature of applied forces simplify differential equations and lead to general solution techniques.

Legacy and influence
Mechanica marks a decisive step in the transformation of mechanics into a modern mathematical science. Euler's analytic presentation influenced contemporaries and later giants of mathematical physics, offering tools and concepts that were absorbed and extended by Lagrange, d'Alembert, Laplace, and others. The equations and procedures introduced in Mechanica remain recognizable in modern treatments of rigid-body dynamics, continuum mechanics, and celestial mechanics.
Beyond specific formulas, the book's lasting contribution is methodological: it established calculus-based analysis as the natural language of dynamics, encouraging subsequent generations to seek general principles and unified formulations rather than isolated geometric constructions. Mechanica therefore stands as one of the foundational texts that shaped the form and practice of mathematical physics.