1. Do neural computations in premotor circuits mirror the physical properties of the systems they control? In 1987, Tweed and Vilis showed that oculomotor theories where a neural integrator converts eye angular velocity commands into position commands cannot be correct, because angular position is not the integral of angular velocity. Recently Schnabolk and Raphan proposed that an angular velocity integrator is nevertheless used to generate tonic commands in the oculomotor system. Here we test the Schnabolk-Raphan (S-R) model against Tweed and Vilis's quaternion (Q) model of the velocity to position transformation. 2. The S-R model predicts large (up to 7 degrees) transient (approximately 700 ms) deviations ("blips") in torsional eye position during attempted horizontal and vertical saccades. The Q model predicts no blips. Search coil recordings of saccades by 7 normal human subjects showed no large blips. 3. For approximately 200 saccades by each subject, we plotted the area under the torsional blip versus the product of saccade eccentricity and magnitude. According to the S-R model, this graph should form a straight line with slope 1.00. According to the Q model, the slope should be zero. Measured slopes averaged 0.016 (range -0.073 to 0.061) for saccade targets at 20 degrees eccentricity and 0.040 (range 0.004-0.076) for targets at 40 degrees. 4. No parameter change can significantly improve the S-R model, but lowering one parameter eradicates the tiny inaccuracy in the Q model. We show that the fundamental reason for the S-R model's failure is its use of a commutative controller to steer a noncommutative plant.