The publication amongst Torricelli's Opera geometrica (Florence, 1644) of a tract on the properties of the cycloid involved him in a controversy with G.

The vermiform body is covered with cycloid imbricating scales, devoid of osteoderms. Limbs and even their arches are absent, excepting a pair of flaps which represent the hind-limbs in the males.

He there shows that the cycloid was investigated by Carolus Bovillus about r 500, and by Cardinal Cusanus (Nicolaus de Cusa) as early as 1451.

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No mention of the cycloid has been found in writings prior to the 15th century.

The method by which the cycloid is generated shows that it consists of an infinite number of cusps placed along the fixed line and separated by a constant distance equal to the circumference of the rolling circle.

Evangelista Torricelli, in the first regular dissertation on the cycloid (De dimensione cycloidis, an appendix to his De dimensione parabolae, 1644), states that his friend and tutor Galileo discovered the curve about 1599.

The mechanical properties of the cycloid were investigated by Christiaan Huygens, who proved the curve to be tautochronous.

It may be noticed that if the scales of x and be properly adjusted, the curve of positions in the present problem is the portion of a cycloid extending from a vertex to a cusp.

Among other early writers on the cycloid were Phillippe de Lahire (1640-1718) and Francois Nicole (1683-1758).

The cycloid was a famous curve in those days; it had been discussed by Galileo, Descartes, Fermat, Roberval and Torricelli, who had in turn exhausted their skill upon it.