CYCLOID, in geometry, a curve of the transcendental kind, called also the tro choid. It is generated in the following manner : if the circle C D H (Plate III. Miscel. fig. 15.) roll on the given straight line A B, so that all the'parts of the cir cumference be applied to it one after another, the point C that touched the line A B in A, by a motion thus compounded of a circular and rectilinear motion, will describe the curve A C E B, called the cycloid, the properties of which are these: 1. If on the axis E F be described the generating circle E G F, meeting the ordi nate C K in G, the ordinate will be equal to the sum of the arc E G and its right sine G K ; that is, C K will be equal to E G+ G K. 2. The line C H parallel to the chord E G is a tangent to the cycloid in C. 3. The arc of the cycloid E L is double of the chord E M, of the corre sponding arc of the generating circle E M F: hence the semi-cycloid E L B is equal to twice the diameter of the gene rating circle E F; and the whole cycloid AC E B is quadruple of the diameter E F. 4. If E R be parallel to the base A B, and C R parallel to the axis of the cycloid E F, the space E C R,bounded by the arc ofthe cycloid E C, and the lines E R and R C, shall be equal to the circle area F. G K hence it follows, W A T, perpendicular to the base A B, meet E R in T, the space E T ACE will be equal to the semi-circle E G F: and since A F is equal to the semi circumference EGF,the rectangle EFAT, being the rectangle of the diameter and semi-circumference, will be equal to four times the semi-circle E G F ; and there fore the area. E C A FE will be equal to three times the area of the generating semi-circle E G F. Again, if you draw the
line E A, the area intercepted betwixt the cycloid E C A, and the straight line E A, will be equar to the semi-circle E G F; for the area E CAF E is equal to three times E G F, and the triangle E A F =-_ AF xi B F, the rectangle of the semi circle and radius, and consequently equal to 2 E G F; therefore their difference, the area E C A E, is equal to E G F. 5. Take E b = O K, draw b Z parallel to the base, meeting the generating circle in X, and the cycloid in Z, and join C Z, F X; then shall the area C Z EC be equal to the sum of the triangles G F K and b F X. Hence an infinite number of segments of the, cy cloid may be assigned, that are perfectly quadrable. • For example, if the ordinate C K be supposed to cut the axis in the middle of the radius 0 E, then K and b coincide ; and the area E C K becomes in that case equal to the triangle G K F, and E b Z be comes equal to F b X, and these triangles themselves become equal.
This is the curve on which the doc trine of pendulums and time-measuring instruments in a great measure depend ; Mr. Huygens having demonstrated, that from whatever point or height a heavy body oscillating on a fixed centre begins to descend, while it continues to move in a cycloid, the time of its falls or oscilla tions will be equal to each other. It is likewise demonstrable, that it is the curve of quickest descent, i e. a body falling in it from any given point above, to another not exactly under it, will come to this point in a less time than in any other curve passing through those two points.