# CISSOID

, is a curve line of the second order, in- | vented by Diocles for the purpose of sinding two continued mean proportionals between two other given lines. The generation or description of this curve is as follows:

On the extremity B of the di- ameter AB of the circle AOB, erect the indefinite perpendicular CBD, to which from the other extremity A draw several lines, cutting the circle in I, O, N, &c; and upon these lines set off the corresponding equal distances, viz, HM = AI, and FO = AO, and CL = AN, &c; then the curve line drawn through all the points M, O, L, &c, is the cissoid of Diocles, who was an ancient Greek geometrician.

This curve is, by Newton, reckoned among the defective hyperbolas, being the 42d species in his Enumeratio Linearum tertii ordinis. And in his appendix de Æquationum Constructione Lineari, at the end of his Arithmetica Universalis, he gives another elegant method of describing this curve by the continual motion of a square ruler. Other methods have also been devised by different authors for the same thing.

The Properties of the Cissoid are the following:

1. The curve has two insinite legs AMOL, Amol meeting in a cusp A, and tending continually towards the indefinite line CBD, which is their common asymptote.

2. The curve passes through O and o, points in the circle equally distant from A and B; or it bisects each semicircle.

3. Letting fall perpendiculars MP, IK from any corresponding points I, M; then is AP = BK, and AM = HI, because AI = MH.

4. AP : PB :: MP2 : AP2. So that, if the diameter AB be = a, the absciss AP = x, and the ordinate PM = y; then is ; which is the equation of the curve.

5. Sir Isaac Newton, in his last letter to M. Leibnitz, has shewn how to find a right line equal to one of the legs of this curve, by means of the hyperboia; but he suppressed the investigation, which however may be seen in his Fluxions.

6. The whole infinitely long cissoidal space, contained between the infinite asymptote BCD, and the curves LOAol &c, of the cissoid, is equal to triple the generating circle AOBoA.

See more of this curve in Dr. Wallis, vol. 1, pa. 545.

CIVIL Day. See Day.

Civil Month. See Month.

Civil Year, is the legal year, or annual account of time, which every government appoints to be used within its own dominions.

It is so called in contradistinction to the natural year, which is measured exactly by the revolution of the heavenly bodies.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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