The action of a paddle wheel in the water resembles in this respect that of the wheel of a carriage running along the ground: that point in the radius of the paddle of which the rotative speed is just equal to the velocity of the vessel will describe a cycloid; points nearer the centre, prolate cycloids, and points further from the centre, curtate cycloids.

A.When the wheel of a coach rolls along the ground, any point of its periphery describes in the air a curve which is termed a cycloid; any point within the periphery traces a prolate or protracted cycloid, and any point exterior to the periphery traces a curtate or contracted cycloidthe prolate cycloid partaking more of the nature of a straight line, and the curtate cycloid more of the nature of a circle.

Such is the cycloid, first conceived by Galileo, and a stumbling-block and cause of contention among geometers long after he had left it, together with his system of the universe, undetermined.