IN §4.9 it was mentioned that within the domain of geometrical optics the departure of the path of light from the predictions of the Gaussian theory may be studied either with the help of ray-tracing or by means of algebraic analysis. In the latter treatment, which forms the subject matter of this chapter, terms which involve off-axis distances in powers higher than the second in the expansion of the characteristic functions are retained. These terms represent geometrical aberrations.

The discovery of photography in 1839 by Daguerre (1789–1851) was chiefly responsible for early attempts to extend the Gaussian theory. Practical optics, which until then was mainly concerned with the construction of telescope objectives, was confronted with the new task of producing objectives with large apertures and large fields. J. Petzval, a Hungarian mathematician, attacked with considerable success the related problem of supplementing the Gaussian formulae by terms involving higher powers of the angles of inclination of rays with the axis. Unfortunately, Petzval's extensive manuscript on the subject was destroyed by thieves; what is known about this work comes chiefly from semipopular reports. Petzval demonstrated the practical value of his calculations by constructing in about 1840 his well-known portrait lens [shown in Fig. 6.3(b)] which proved greatly superior to any then in existence. The earliest systematic treatment of geometrical aberrations which was published in full is due to Seidel, who took into account all the terms of the third order in a general centred system of spherical surfaces. Since then, his analysis has been extended and simplified by many writers.