CYCLOID

, or Trochoid, a mechanical or transcendental curve, which is thus generated: Suppose a wheel, or a circle, AE, to roll along a straight line AB, beginning at the point A, and ending at B, where it has completed just one revolution, thereby measuring out a right line AB exactly equal to the circumference of the generating circle AE, whilst a nail or point A in the circumference of the wheel, or circle, traces out or describes a curvilineal path ADB; then this curve ADB is the cycloid, or trochoid.

Schooten, in his Commentary on Des Cartes, says that Des Cartes first conceived the notion of this elegant curve, and after him it was first published by Father Mersenne, in the year 1615. But Torricelli, in the Appendix de Dimensione Cycloidis, at the end of his treatise De Dimensione Parabolæ, published 1644, says that this curve was considered and named a cycloid, by his predecessors, and particularly by Galileo about 45 years before, i. e. about 1599. And Dr. Wallis shews that it is of a much older standing, having been known to Bovilli about the year 1500, and even considered by cardinal Cusanus much earlier, viz, before the year 1451. Philos. Trans. Abr. vol. 1, pa. 116. It would seem however that Torricelli's was the first regular treatise on the Cycloid; though several particular properties of it might be known prior to his work. He first shewed, that the cycloidal space is equal to triple the generating circle, (though Pascal contends that Roberval shewed this): also that the solid generated by the rotation of that space about its base, is to the circumscribing cylinder, as 5 to 8: about the tangent parallel to the base, as 7 to 8: about the tangent parallel to the axis, as 3 to 4: &c.

Honoratus Fabri, in his Synopsis Geom. has a short treatise on the cycloid, containing demonstrations of the above, and many other theorems concerning the centres of gravity of the cycloidal space, &c; which he says he found out before the year 1658.

From the preface to Dr. Wallis's treatise on the eycloid we learn, that, in the year 1658, M. Pascal publicly proposed at Paris, under the name of D'Ettonville, the two following problems as a challenge, to be solved by the mathematicians of Europe, with a reward of 20 pistoles for the solution: viz, to find the area of any segment of the cycloid, cut off by a right line parallel to the base; also the content of the solid generated by the rotation of the same about the axis, and about the base of that segment. This challenge set the Doctor upon writing that treatise upon the cycloid, which is a much better and compleater piece than had been given before upon this curve. He here gives the curve surfaces of the solids generated by the rotation of the cycloidal space about its axis, and about its base, with determinations of the centres of gravity, &c. He here asserts too, that Sir Christopher Wren, in 1658, was the first who found out a right line equal to the curve of the cycloid; and Mr. Huygens, in his Herolog. Oscillat. says that he himself was the first inventor of the segment of a cycloidal space, cut off by a right line parallel to the base at the distance of 1/4 the axis of the curve from the centre, being equal to a rectilinear space, viz, to a regular hexagon inscribed in the generating circle; the demonstration of which may be seen in Wallis's treatise.

Several other authors have spoken or treated of the cycloid: as Pascal, in his treatise, under the name of D'Ettonville: Schooten in his Commentary on Des Cartes's Geometry, near the end of the 2d book; M. Reinau, in his Analyse Démontrée, tom. 2, pa. 595: also Newton, Leibnitz, de la Loubere, Roberval, Des Cartes, Wren, Fabri, the Bernoulli's, De la Hire, Cotes, &c, &c.

Properties of the Cycloid.—The circle AE, by whose revolution the cycloid is traced out, is called the generating circle; the line AB, which is equal to the circumference of the circle, is the base of the cycloid; and the perpendicular DC on the middle of the base, is its axis. The properties of the cycloid are among the most beautiful and useful of all curve lines: some of the most remarkable of which are as follow:

1. The circular arc DG = the line GH parallel to AB.

2. The semicircumf. DGC = the semibase AC.

3. The arc DH = double the chord DG.

4. The arc DA = double the diam. DC.

5. The tang. HI is parallel to the chord DG.

6. The space ADBA = triple the circle AE or CGD &c.

7. The space ADGCA = the same circle AE, &c.

8. A body falls through any arc KL of a cycloid reversed, in the same time, whether that arc be great or small; that is, from any point L, to the lowest point K, which is the vertex reversed: and that time | is to the time of falling perpendicularly through the axis MK, as the semicircumference of a circle is to its diameter, or as 3.1416 to 2. And hence it follows that, if a pendulum be made to vibrate in the arc LKN of a cycloid, all the vibrations will be performed in the same time.

9. The evolute of a cycloid, is another equal cycloid. So that if two equal semicycloids OP, OQ, be joined at O, so that OM be = MK the diameter of the generating circle, and the string of a pendulum hung up at O, having its length = OK or = the curve OP; then, by plying the string round the curve OP, to which it is equal, and then the ball let go, it will describe, and vibrate in the other cycloid PKQ.

10. The cycloid is the curve of swiftest descent: or a heavy body will fall from one given point to another, by the way of the arc of a cycloid passing through those two points, in a less time, than by any other rout. See the Works of James and John Bernoulli for many other curious properties concerning the descents in cycloids, &c.

Cycloids are also either curtate or prolate.

, Curtate, or contracted, is the path described by some point without the circle, while the circumference rolls along a straight line; and a

, Prolate, or Inflected, is in like manner the path of some point taken within the generating circle.

Thus, if, while the circle rolls along the line AB, the point R be taken without the circle, it will describe or trace out the curtate or contracted cycloid RST; but the point being taken within the circle, it will describe the prolate or inflected cycloid RVW.

These two curves were both noticed by Torricelli and Schooten, and more fully treated of by Wallis, in his Treatise on the Cycloid, printed at Oxford in 1659; where he shews that these have properties similar to the first or primary cycloid; only the last of these is a curve having a point of inflection, and the other crossing itself, and forming a node.

By continuing the motion of the wheel, or circle, so as to describe a right line equal to the generating circumference several times repeated, there will be produced as many repetitions of the cycloids, which so united together will appear as in these figures following: Curtate Cycloid. Common Cycloid. Inflected Cycloid.