CUR
C Y S
Baulk Curvature is ufed for the curvature of a line, gll the parts of which are not fituated in the fame plane. See the next article.
CURVE (Cyd. and Sup'l.) — The theory of curves is a very considerable branch of the mathematical fciences, Thofe who are curious of advancing beyond the knowledge of the circle and the conic- fections, and to confidcr geometrical curves of a higher nature, and in a general view, will do well to finely Mr. Cramer's Introduction a LAnalyfe des li- gnes Courbes Algebriqua, printed at Geneva, 4. 1750, which the learned and ingenious author compofed for the ufe of beginners. We have an elegant pofthumous work of Mr. Mac Launn's, printed at the end of his Algebra, and in- tituled De Liwarv.m geometricarum proprictatibus gcneralibus. The fame author at a very early age gave a remarkable fpe- cimen of his genius and knowledge in his Gecmetria Organica, and carried thele (peculations farther afterwards, as may be feen in the theorems he has given us in the Philofophical Tranfadtions. See Dr. Martin's Abridg. vol. vui. p. 62, feq. Curves may be organically defcribed by the rotation of andes, in the manner mentioned in the Cyclopedia, which is Sir Ifaac Newton's invention.
But there is another general method of defcribing curves by the rotation of rulers or ftrait lines, inftead of angles. Thus, if inftead of angles we ufe three rulers, D Q,, C N, S P, (fig. IV.) which arefuppofed to revolve about the poles D,C, S, and to cut one another always in three Points N, Q, and P ; if any two of thefe interfech'ons as N and Q, be carried along the given ftrait lines A E, E B, the third interferon P will defcribe a conic-fcction. See Mac Laurins Algebra, p. 34.6. feq.
Fig. IV.
If you aflume any number of poles whatfoever, and make rulers revolve about each of them, and all the interfeclions but one, be carried along given right lines, that one fhall never defcribe a line of a higher nature than a conic- fecti on. And if inftead of rulers you fubftitute given angles which move on the fame poles, the curve defcribed will (till be no more than a conic- fechon.
But by carrying one of the interfections neceflary in the de- fcription over a conic-fection, lines of higher orders may be defcribed. Mac Laurin, ib. p. 351.
The Rev. Mr. Brakenridge has given us a general method of defcribing curves by the interfe&ion of right lines moving about points in a given plane. See Phil. Tranf. N°. 436. Dr. Martin's, Abridg. vol. viii. p. 58. feq. But the demon- ftrations are not yet extant, excepting the particular cafes
demonftratcd in his Exercitatio Gecmetrica de Curvarum <&- jcripiione, Land. 1733, 4 .
Curves may alfo be defcribed by the projection, or fhadows of other curves. Thus the projection, or fliadow of the circle upon different planes, will form the reft of the lines Of the fecond order, or conic-fections. This is evident, be- caufe the rays of light proceeding from a point out of the plane of a circle, and falling upon the circumference of that circle form a cone, which being cut by the plane upon which the fliadow of the circle is projected, the different conic- fections will be formed according to the pofition of the inter- fering plane.
In like manner the projections or fhadows of lines of the third order, will form other lines of the third order ; and pro- jections or fhadows of lines of the fourth order, will form lines of the fourth order, csV.
And as the circle, by the projection of its fhadow, forms the conic- fections, fo the five diverging parabolas among the lines of the third order, will, by their fhadows, form and exhibit all the reft of the lines of that order. See Newton, enumerat. lin. tertij. ordin. published by Mr. Jo?ies, 171 1. This hint of Sir Ifaac Newton has been lately purfued and iliuftrated with great elegance by Mr. Murdoch, in his trea- tife entitled, Newtoni Genefis curvarum per Umbras, feu per- fpeiTiViE univtrfalis elementa. Land. I 746, S\ By an accurate enumeration of thefe projections, Mr. Mur- doch finds, that the number of fpecies of the lines of the third order amount to feventy-eight in all. Curve of a double curvature, or Curve having a double curva- ture, is ufed for a curve, all the parts of which do not lie in the fame plane, that is, fuch as cannot be defcribed on the fame plane.
The curves commonly treated of in geometry, are fuppofed to be defcribed, or to have all their points placed in the fame plane ; but if a curve be fuppofed to be defcribed on a curve furface, in fuch a manner that all the points of that curve cannot lie or be fituated in one and the fame plane, then will the curve fo defcribed have a double curvature. Monfieur Clairaut has publifhed an ingenious treatife on curves of a double curvature. See his Recherches fur les Courbes, a double Courbure, ax Paris, 4". 1731. Mr. Euler has alfo treated this fubjedt in the Appendix to his Analyfis infinitorum, Vol. II. p. 323, feq. CUSTARD-^//,?, a name ufed by fome for the guanabanus, a
fpecies of emona. See the article Anona, Suppl. CUTTLE-fJb, the Englifh name of the fepia of authors, called alfo the ink-fijb. See the articles Sepia and Ink-M. Append. CYANEUS, in zoology, the name of a fpecies of coluber. See
the article Coluser, Append. CYANUS. This is made a diftinct genus of plants by Tour- nefort, but comprehended under that of centauria by Linnseus. See the article Centauria, Append. CYCLIDIA, in zoology, the name ufed by Dr. Hill for a ge- nus of animalcules of a roundifh or elliptic figure, and with- out any vilible limbs or tail. Seethe article Animalcule, Suppl. and Append. CYMBIUM, in natural hiftory, a name given to the gondola-
fliell. See Gondola. CYNOCRAMBE, in botany, a name given to a fpecies of
dog's mercury. See the article Mercurialis, Append. CYNOCTONUM. See Aconite, Cycl. CYPRESS (Cycl. and Suppl.) — The timber of the cyprefs-tree is good for making chefts, mufical inftruments, and other utenfils. It never cleaves, and is extremely hard and durable ; its bitter juice refilling worms and putrefaction. Thus, we are told, that the gates of St. Peter's church at Rome, made of cyprefs-zvood, had lafted had fix hundred years as frefh as new, when pope Eugenio ordered gates of brafs in their ftead. Some will have it, that the wood, gophir, of which Noah's ark was made, was cyprefs; which Plato preferred to brafs itfelf, for writing his laws on. Build. Diet, in voc. Summcr-CypRESS, a name ufed by fome for the chenopodium
of authors. See the article Chenopodium, Append. CYST1CAPNOS, in botany, a name ufed by Boerhaave for thefumaria, or fumitory. See Fumaria, Suppl,
D.
