A Brief  History of Lens Design 2GRL`.1  
Since about 1960, the way lenses are designed has changed profoundly  as a result of  the introduction of  electronic digital computers and numerical optimiz-ing methods. Nevertheless, many of  the older techniques remain valid. The lens designer still encounters terminology and methods that were developed even in previous centuries. Furthermore, the new methods often  have a strong classical heritage. Thus, it is appropriate to examine, at least briefly,  a history of  how the techniques of  lens design have evolved. Ky:y1\K1^K  
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A.2.1 Two Approaches to Optical Design @ ={Hx$zL  
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The equations describing the aberrations of  a lens are very nonlinear func-tions of  the lens constructional parameters (surface  curvatures, thicknesses, glass indices and dispersions, etc.). Boundary conditions and other constraints further complicate the situation. Thus, there are only a few  optical systems whose con-figurations  can be derived mathematically in an exact closed-form  solution, and these are all very simple. Examples are the classical reflecting  telescopes. _o,Mji|  
This predicament has produced two separate and quite different  approaches to the practical task of  designing lenses. These are the analytical approach and the numerical approach. Historically the analytical dominated at first,  but the numer-ical now prevails. kF,_o/Jc  
Neither approach is sufficient  unto itself.  A lens designed analytically using aberration theory requires a numerical ray trace to evaluate its actual perfor-mance. In addition, an analytically designed lens can often  benefit  significantly from  a final  numerical optimization. Conversely, a lens designed numerically cannot be properly understood and evaluated without the insight provided by ab-erration theory. W.67};',  
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A.2.2 Analytical Design Methods R#