Book: Dioptrica

Overview
Leonhard Euler's Dioptrica (1769) is a systematic, mathematical treatment of refraction and the optical behavior of lenses. It brings together geometric optics and the analytic tools of calculus to derive precise relations for how light bends at interfaces and how combinations of refracting surfaces form images. The work sets out general principles for lens design, explores the limits of simple spherical surfaces, and addresses the practical optical problems of aberration that limit image quality.

Core content and methods
The book begins from the basic laws governing refraction and proceeds to express these laws in analytic form. Euler employs differential equations and geometry to describe rays as they pass through media of different refractive indices, treating refracting surfaces as curves or surfaces of revolution that can be described and optimized mathematically. Key ideas include the derivation of the shapes that will send rays from one point to another after refraction and the formulation of conditions that must be satisfied for stigmatic imaging, where all rays from a point converge to a single image point.

Solutions and optical surfaces
Euler examines the family of curves that can produce perfect focusing under refraction and shows how Cartesian ovals arise naturally as general solutions to the refraction problem between two homogeneous media. He analyzes why ordinary spherical lenses cannot, in general, produce perfect stigmatic images and he derives more general aspheric shapes that correct for the errors that spheres introduce. Much of the work turns on setting up and solving differential relations that link surface curvature, refractive index, and the desired ray behavior, giving designers explicit mathematical blueprints for theoretically perfect surfaces.

Aberration and chromatic effects
A central concern of Dioptrica is optical aberration. Euler distinguishes spherical aberration, produced by geometry, from chromatic dispersion, the wavelength-dependent bending of light. He develops theoretical criteria for minimizing or eliminating certain aberrations and discusses combining multiple elements with differing refractive properties to lessen chromatic effects. While practical glassmaking and mounting limited immediate technological application of some of his mathematically ideal surfaces, the text supplies the conceptual foundation for later achromatic lens solutions and for systematic approaches to correcting image defects.

Applications and influence
Although highly theoretical, Dioptrica was written with instrumentation in mind: Newtonian and refracting telescopes, microscopes, and other optical devices are treated as examples where the theory applies. Euler's mathematization of optical design helped shift the field from empirical tinkering toward design guided by analysis. The work influenced subsequent generations of opticians and mathematicians who sought to realize the aspheric and compound solutions Euler described and to reconcile mathematical ideals with the constraints of manufacturing and glass chemistry.

Legacy
Dioptrica stands as a major 18th-century contribution to physical optics, notable for transferring the power of calculus to the problems of lens design and image formation. Its clear articulation of the limits of simple lens shapes and its provision of general equations for refracting surfaces furnished a durable theoretical framework. The book's blend of rigorous analysis and practical concern helped prepare optics for the systematic engineering advances of the 19th century and remains an important landmark in the history of optical science.