# 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, A*mol*
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 :: MP^{2} : AP^{2}. 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 LOA*ol* &c, of the cissoid, is equal to triple the
generating circle AOB*o*A.

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.