# C W Oseen's plenary: ICM 1936

C W Oseen was a plenary speaker at the International Congress of Mathematicians in Oslo in 1936. He delivered his lecture, Probleme der geometrischen Optik, on the morning of Thursday 16 July 1936 with Ernst LindelĂ¶f as chair of the session. We give below an English version of the beginning of his lecture.

Probleme der geometrischen Optik

Dear attendees! The importance of geometric optics for both mathematics and physics, both experimental and theoretical, is well known to all of you. If it were necessary to convince a doubting person, it would suffice to give one name, the name William Rowan Hamilton. With his geometric-optical work, Hamilton was led to the discovery of the so-called contact transformations. On the other hand, Hamilton has created a bridge between Newtonian mechanics and optics through the equations named after him. On this bridge, SchrĂ¶dinger found the basic idea of wave mechanics in our day.

But geometric optics are not only important for mathematics and physics. There is another science that works with light rays, ophthalmology. We had an excellent representative of this science in Sweden. I believe that many of you have heard the name Allvar Gullstrand. I was his colleague at the university in Uppsala for eighteen years. When I speak of geometric-optical problems here, the reason for this is my preoccupation with Gullstrand's life's work.

Allvar Gullstrand was a doctor by profession and by training. He was a mathematician and an inventor by nature. He acquired his mathematical knowledge himself. He was a self-taught mathematician.

If someone does mathematics on his own in our day, he obviously runs the risk of having to search for results on long thorny paths that can be achieved without any effort. I want to show by example that Gullstrand did not escape this danger. One of his most well-known theorems states that optical imaging does not come about point by point, but that on any surface element of the object space there are two single-parameter groups of mappable curves, which correspond to two image curves in the image space that are located on different surface elements. With Gullstrand this theorem is a corollary of a so-called Fundamental equation, which it takes long and tedious calculations to derive. The essential content of the theorem can be clearly derived. A point of the object space and the corresponding focal surface in the image space determine a contact transformation between the object space and the image space. A point P of a surface element F of the object space can be assigned the curve, C, on a corresponding surface of the image space, in which this surface is intersected by the focal surface that corresponds to the point P. The different directions through P correspond to the different line elements of C. If you now ensure that only one element of the image surface is illuminated, this means that you have made a choice among the different directions through P. The point P is thus assigned a specific direction on F. If it is possible to arrange the diaphragms in such a way that this takes place for every point of the surface element F, the geometrically and optically possible one has been achieved in relation to the illustration. At least that was the view of Gullstrand. The mapping of the two surface elements onto each other does not come about point by point. It is a mapping of certain mappable lines on F to the corresponding image lines of the corresponding surface element in the image space. The two groups of which Gullstrand speaks come from the two sheets of the focal surface.

It is to be hoped that in Gullstrand's great work there will be many theorems for which a clear derivation is possible. But it is certain that the simplification of his work on a large scale, which is necessary if this work is not forgotten, must consist in replacing his analytical methods with other, more expedient, but still analytical methods. Gullstrand himself did not value geometry other than analytical.

After this introduction I turn to the problem of geometric optics, to which the first period in Gullstrand's scientific life (1890-1904) was devoted. The problem can be said as follows: what different types of light beams are possible according to the laws of geometric optics? Gullstrand was led to this problem by a contradiction between the ophthalmological experiences and the geometrical optics of his youth. According to this geometric optic, the astigmatic light beam should always have two focal lines. They should appear ophthalmologically in the ability of the eye to adapt. The two accommodation limits should be characterised by the incidence of a focal line on the retina. After the discovery of the nature of accommodation, this doctrine was no longer tenable.

How had one been led to this doctrine by the two focal lines?

The basis of any theory of geometrical-optical light beams must be Malus's theorem. As you know, he says u. a. from the fact that a light beam that started from a luminous point after any number of refractions and reflections in an isotropic, homogeneous medium, is a normal congruence, in other words that the light rays are always perpendicular to a certain surface, the wave surface.

Last Updated April 2020