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Explanation of an Optical Deception in the Appearance of the Spokes 

 of a Wheel seen through vertical Apertures. By P. M. Roget, M.D. 

 F.R.S. Read December 9, 1824. [Phil. Trans. 1825, p. 131.] 



The optical deception which Dr. Roget describes, takes place when 

 a carriage-wheel, rolling along the ground, is viewed through the in- 

 tervals of a series of vertical bars, such as those of a palisade, or of a 

 Venetian window-blind. In these circumstances, all the spokes ap- 

 pear curved, those which are situated vertically excepted, the degree 

 of curvature increasing as their position recedes from the vertical, 

 and being a maximum for the horizontal spokes. The convexity of 

 these curved images is always turned downwards, and the direction 

 of their curvature is the same whichever way the wheel be moving. 



To determine the influence of variations of circumstances on these 

 phenomena, appeared to Dr. Roget the proper mode of obtaining a 

 clue to their explanation. It results from his experiments, that a 

 certain degree of velocity in the wheel is necessary to produce the 

 deception in question ; and if this be communicated to it gradually, 

 the principal effect of curvature is observed to come on suddenly, but 

 the degree of bending is independent of the velocity of the wheel, and 

 each image appears, during the moment it is viewed, motionless. The 

 number of spokes makes nodifference in their degree of curvature. The 

 deception is favoured by the diminution of the interval between the 

 bars, by throwing a strong light on the wheel, and by every circum- 

 stance which tends to draw attention from the bars and fix it on the 

 wheel. The number of curved images was found to depend on the 

 ratio of the angles subtended at the eye by the intervals between the 

 bars, and those between the extremities of the spokes ; being greater 

 as this ratio was less. If the bars were inclined to the horizon, the 

 phenomena were those of a wheel revolving with a less velocity in a 

 direction perpendicular to their length. Finally, the combination of 

 a progressive with a rotatory motion (whether produced by a real 

 motion of the wheel, the bars, or the spectator,) was found to be 

 essential to the production of the phenomena. 



From all these circumstances, the author is led to refer the ex- 

 planation of the deception in question, to the principle that an im- 

 pression made by a pencil of rays on the retina, if sufficiently vivid, 

 will remain for a certain time after the cause has ceased. He then 

 explains at length the application of this principle to the case in 

 question, and shows that the apparent form of each spoke will be 

 a curve formed by the continual intersection of the revolving and 

 advancing radius, with the immoveable interval between the bars, 

 referred, not to the plane in which the bars lie, for then it would be 

 a straight line, but by an effort of attention, to the plane of the 

 wheel. The general form of these curves he refers to the class of 

 Quadratrices, and the most remarkable among them is that first 

 discovered by Dinostrates, and known by his name. By varying a 

 certain parameter in the equation of these curves, other forms, hav- 

 ing infinite branches and points of contrary flexure, arise ; and these 



