If it is imagined to underline a point on a circumference and to make the circle roll down on a plane surface the cycloid is that curve described by the point . This curve is repeated always in the same way and, if they are took into consideration two consequentially moments in which the point is in the lower point (that is when it touches the plane surface), the distance between them is always the same and equal to the circumference. The area drawn by the cycloid and its chord results the triple of the area of the originating circle., however Galileo could just intuish this propriety, but wasn't able to demonstre it probably because of some exoerimental mistakes. Inbì 1640 he wrote , with regard to the cycloid: “It was more than fifty years that the idea of describing that curve had come into my head and I admired it because of its beautyful curvature in order to use it for bridges. I tried several times to demonstre any propriety of the space between it and its chord that I studied long, and it seemed to me that it could be the triple of the area of the originating circle, but it wasn't so, even if the difference is small.”
Another peculiarity of the cycloid is that it is the brachistochrone (from greek: “brachistos” shorter and “chronos” time) that is the curve wich join two points in the shorter time if they are the initial and the final position, not vertically lined up, of a body which is falling in frictionless conditions.