Have fun with strabismus!

The polar coordinate system and Listing's Law

The only coordinate system for describing eye rotations that does not have these flaws is the polar coordinate system. In a polar coordinate system the position of the eye is also determined by two angles: One angle defines the direction of eye movement out of the primary position and a second angle defines the angle of eye movement out of the primary position. In this coordinate system, all tertiary positions of gaze are reached by simple rotation about a single axis. This principle was invented by Professor J.B. Listing, a good friend of Professor Ruete, who had participated with Ruete in the Göttinger Studien with a booklet on entoptic phenomena and cataract (1845).

Ruete (1853) therefore called this principle Listing's Law: 'Aus der oben angegebenen normalen Stellung (Anfangsstellung, Primärstellung) des Auges wird das Auge in irgend eine andere, secundäre, durch die Cooperation der sechs Muskeln in der Weise versetzt, dass man sich diese Versetzung als das Resultat einer Drehung um eine bestimmte Drehungsaxe vorstellen kann, welche jederzeit, durch das Augencentrum gehend, auf der primären und secundären Richtung der optischen Axe zugleich senkrecht steht, so dass also jede secundäre Stellung des Auges zur primären in der Relation steht, vermöge welcher die auf die optische Axe projicirte Drehung=0 wird. Diesem Princip zufolge lässt sich aus der bekannten Lage der drei auf je zwei antagonistische Muskeln bezäglichen Drehungsaxen fär jede gegebene Secundärstellung des Auges der Wirkungsbetrag jedes Muskels, d.i. die Grösse seiner Verkärzung durch Rechnung bestimmen. Unter den vielfachen Consequenzen dieses Princips verdient die hervorgehoben zu werden, dass nämlich das Auge beim Uebergange aus einer secundären Stellung in eine andere eine ihrer Grösse nach bestimmbare Drehung um seine optische Axe erfährt, welche nur in dem besonderen Fall null ist, wenn die drei Richtungen der optischen Axe in der primären und in den beiden secundären Stellungen in einer Ebene liegen' (in short, all secondary and tertiary positions of gaze can be reached by rotation about a single axis that is perpendicular to the primary position of gaze and to the new position of gaze... Among the many consequences of this principle, one needs particular emphasis, namely, that the eye will rotate about its optical axis in eye movements from one tertiary to another tertiary position of gaze, this rotation being zero only when the two tertiary positions of gaze and the primary position of gaze are all in a single plane.)

Fig. 1. Model from Halle

The Ophthalmic Collection of the former Royal Netherlands Ophthalmic Hospital has a copper model that illustrates this principle beautifully (Fig. 1). The name of the place of manufacture, Halle, is engraved in the model. Halle is about 30 km from Leipzig, so it was probably Ruete who sent it to Donders.

In Ruete's first ophthalmotrope (1845) the model eye was suspended in gimbals, i.e. the model eye rotated in a ring that itself could rotate about an axis that was perpendicular to the first axis, this method of suspension having been invented by Cardano in the sixteenth century. An improved version of his ophthalmotrope, presented by him in Leipzig in 1857, no longer employed suspension with gimbals. The globe was simply pulled against a ring with screws by the 'muscles'.

Fig. 2. Ruetes second ophthalmotrope

It can now be seen why Ruete did not employ a gimbal suspension in a second version of his ophthalmotrope (1857): this kind of globe suspension will not bring the eye in a tertiary position that complies with Listing's Law: Pseudotorsion will occur in tertiary positions of gaze. Ruete wrote that the ring with screws supporting the model eye represented the 'fat pad behind the eye, the nutshell in which the eye was suspended', quite a modern concept for his time.

Fig. 3. Donders ophthalmotrope.

Donders (1870) later presented his own ophthalmotrope to illustrate Donders' Law. In this model he used gimbal suspension on purpose, to make the pseudotorsion visible. This ophthalmotrope was equipped with a camera obscura, to obtain an image of, for instance, the left upper hand corner of the wall in front of us. This image was to be compared with the retinal meridians, which were represented by 4 copper bars surrounding the camera obscura.