C E N
(182)
CEP
of a Surface, or a Solid, by means of its Centre of Gravity.
The Doctrine is compris'd in the following Theorem, with its Corollaries.
Every Figure, whether Superficial or folid, generated by the Motion of a Line or a Figure, is equal to the Fac- tum of the generating Magnitude into the -it-ay of its Cen- tre of Gravity, or the Line which its Centre of Gravity defcrihes. See Centre of Gravity.
2)emonJl. Tor fuppofe the Weight of the whole gene- rating Magnitude collected in the Centre of Gravity ; the whole Weight produe'd by its Motion, will be equal to the Fatlum, of the Weight mov'd, into the Centre of Gra- vity. But when Lines and Figures are confider'd like ho- mogeneous heavy Bodies, their Weights are as their Bulks: and therefore, the Weight mov'd is the generating Magni- tude ; and the Weight produced, that generated. The Figure generated, therefore, is equal to the Fa Bum of the Magnitude, into the Way of its Centre of Gravity. Jg.E.Z).
Corol. 1. Since a Parallelogram A BCD (Tab. MecbanickSy Fig. z6i) is defcrib'd, if the right Line A B, pro- ceed according to the Direction of another AC, with a Motion {till parallel to it felf 5 and the Way of the Centre of Gravity h, is equal to the right Line F_ F, perpendi- cular toCD; that is, to the Altitude of the Parallelogram : Its Area is equal to the FaBam of the Bafc C D, or the defcribent Line into the Altitude E F. See Parallelo- gram.
Corol. II. In the fame manner it appears, that the Soli- dity of all Bodies, which proceed according to the Direc- tion of any right Line A C, is had by multiplying the de- ferring Plane by the Altitude. Sec Prism, and Cylinder.
Corol. III. Since a Circle is defcrib'd, if the Radius CL (Fig.^i.) revolve round a Centre C,and the Centre of Gravity of the Radius C L, be in the Middle F ; the Way of the Centre of Gravity is a Periphery of a Circle as, defcrib'd by a fubduple Radius: Confequently the Area of the Circle, is equal to the FaHum of the Radius CL, into the Peri- phery defcrib'd by the fubduple Radius C F. See Circle.
Corol. IV. If a Redanglc AB CD, (Tab. Mechauicks, Fig. a8.) revolve about its Axis AD 5 the Rectangle will defcribc a Cylinder, and the Side BC the Superficies of a Cylinder. But the Centre of Gravity of the right Line BC, is in the Middle F ; and the Centre of Gravity of the generating Plane in the Middle G, of the right Line EGF. The way of this latter, therefore, is the Periphery of a Circle defcrib'd by the Radius E G ; that of the for- mer, the Periphery of a Circle defcrib'd by the Radius EF. Wherefore, the Superficies of the Cylinder is thcFacJum of the Altitude B C, into the Periphery of a Circle defcrib'd by the Radius E F, or the Bafe. But the Solidity of the Cylinder, is the Fa&imi of the generating Rectangle A BC D, into the Periphery of a Circle defcrib'd by the Radius EG, which is fubduple of EF, or of the Semi- diameter of the Cylinder.
Suppofe, v. g. the Altitude of the defcribing Plane, and therefore of the Cylinder B C -~a $ the Semidiameter of the Bafe D C = r ; then will EG=~ r : And fuppofxng the Ratio of the Semidiamcter to the Periphery =1 : 7n, the Periphery defcrib'd by the Radius of tr=^\mar. therefore, multiplying Imr by the Area of the Rectangle AC-=ar$ the Solidity of the Cylinder will be =±mar 1 . But ~mar z =y r. m r the Area of the Circle defcrib'd by the Radius DG. 'Tis evident, therefore, the Cylinder is equal to the Fattum of the Bafc into the Altitude. See Cylinder.
Corol. V. In like manner, fince the Centre of Gravity of the right Line AB, (Tab. Mechauicks, Fig. 17.) is in the middle M, and the Surface of a Cone is defcrib'd, if the Triangle ABC revolve about its Axis j if PM=4 C ; the Superficies of the Cone will be equal to the Fac- tum of its Side A B, into the Periphery defcrib'd by the Radius PM; or the Subduple of the Semidiameter of the Bate B C.
Suppofe v.g. B C = r, A B — a ; the Ratio of the Radius to the Periphery 1 : m h then will PM=^-, and the Pe- riphery defcrib'd by this Radius =~mr. Therefore, multiplying | mr into the Side of the Cone A B, the Pro- duct is the Superficies of \ amr. But kamr\% alfo the Factum of a a and m r ; therefore, the Surface of the Cone is the Produa of the Periphery, into half the Side. See Cone.
Corol. VI. If the Triangle A C B (Tab. Mechauicks, Fig. ^9.) revolve about an Axis 5 it describes a Cone ; but "if CB divided into two M D, and the right Line AD be drawn, and AO = - AD ; the Centre of Gravity will be mO. The Solidity of the Cone, therefore, is equal to the FaBwn of the Triangle CAB, into the Periphery de- fcrib'd by the Radius PO ; but AD: AO:: BD -OP and A O = I A D and D B = -i C B. Therefore, 6 P = |DB=iQB.
Suppofe v.g.CB~r, KB— a ; the Ratio of the Radius tothe Periphery = 1 : ?;;. Then will O P = ' r the Peri- phery defcrib'd by this Radius £«r$ the Triangle A CB
= 4r; an<* therefore, the Solidity of the Cone ^mr^kr = y amr 7 '. But £amr x =?r. mr. ~ a. Or, the 'Fac- tum of the Bafe of the Cone into the third Part of the Al- titude. See Triangle.
This elegant Theorem, which may be rank'd among the chief Inventions in Geometry of the laft Age, was ta- ken notice of long ago by 'Pappus ; but the Jefuit Guldi- nus was the firft who fet it in its full Light, and exhibited its Ufe in a variety of Examples. Several other Geome- ters, after Guldinus and 'Pappus, alfo us'd it in meaiurino Solids, and Surfaces generated by a Rotation round a fix'd Axis; efpecially before the late Invention of the Calculus Summatorius : and it may itill take place in fome Cafes, where the fummatory Calculus wou'd be more difficult. M. Leibnitz has obferv'd, that the Method will hold, tho the Axis or Centre be continually chang'd during the gene- rative Motion.
CENTRUM, in Geometry, Mcchanicks, &c. See Centre.
Centrum Pho;iicum, in Acouftics, is the Place where the Speaker Hands in the Polyfyllabical and articulate Echoes. See Echo.
Centrum Phonocampticum, is the Place, or ObjecF that returns the Voice in an Echo. See Echo.
Centrum 'fendinofum, in Anatomy, a Point, or Centre, wherein the Tails of the Mufcles of the Diaphragm meet. This Centre is perforated towards the right Side, for the Vena Cava ; towards the left backward : Its flefhy Part gives way to the Gula. The defcending Trunk of the great Artery, Thoracic Duel, and Vena j4zygos, pafs be- tween its two inferior ProcefTes. See Diaphragm.
CENTRY Box, a wooden Cell, or Lodge, made tofhel- ter the Certify, or Sentry, from the Injuries of the Weather.
In a Fortification, they are ufually plac'd on rhe flanked Angles of the Baitions, on thofe of the Shoulder, and fome- times in the middle of the Curtain.
CENTUMVIRI, among the Romans, a Court compos 'd of 100 Magitlrates, or Judges, appointed to decide Diffe- rences between the People. See Decemviri, Quindecemviri, &c.
CENTURION, Centurio, among the Romans, an Of- ficer in the Infantry, who commanded a Century, or a a hundred Men. See Century.
The Centurion of the firft Cohort of each Legion, was call'd Primipilus : he was not under the Command of any Tribune, as all the reft were 3 and had four Centuries un- der his Direction. He guarded the Standard, and the Ea- gle of the Legion. See Primipilus.
CENTURY, a thing divided, or rang'd into an hun- dred Parts.
At the Time when the Roman People were affembled for creating of Magistrates, eftablifhing of Laws, or deli- berating of publick Affairs, they were divided into Centu- ries -j and to the end their Suffrages might be more eafily collected, they voted by Centuries : This was done in the Campus Martins $ and thefe AfTeinblies were hence call'd Comitia Centuriata.
The Roman Cohorts were distributed into D;cus, commanded by Decurions ; and Centuries, by Centurions. Each Cohort confifled of fix Centuries, and a Legion of fixty. See Cohort.
Century, in Chronology, is the Space of 100 Years. Church Hiftory is computed chiefly by Centuries, com- mencing from our Saviour's Incarnation : In this Senfe, we fay, The firft Century 5 the Fathers of the fecond Century*, the Councils of the third Century, Sic. See Council, Fa- thers, &c.
Centuries of Madgehurg, a celebrated Ecclefiaftical Hiflory, divided into thirteen Centuries, containing thir- teen hundred Years, ending at 129S ; compil'd by feveral learned Proteftants of Magdeburg. The chief of the Cen- turiators, was Matthias Flacius Illyricus. 'Tis faid Saro- nius undertook his Annals, purely to oppofe the Madge- hurg Cent-uriators.
Century, or Centaury the leffer, in Medicine, an Herb, chiefly noted for a Stomachic, and reftoring decay'd Appetites : It is feldom prefcrib'd otherwife than by Infu- fion or Decoclion ; it is an Ingredient in the Venice Treacle 5 and, as a Stomachic, it enters as one of the In- gredients in the Officinal Bitter Draught ; and is for the fame Purpofe order'd in extemporaneous Prefcriptions, with others of the fame Nature, for the making of bitter Wines, &c.
A ftrong Decoftion hereof is faid to be of fervice, drank for fome time, in Obftruclions of the Menfes, and alfo to deflroy Worms; a Property attributed to molt' Bitters. It is often prefcrib'd externally in difcutient Fomentations. The Tops, or Flowers, are only us'd.
The greater Century is never prefcrib'd.
CEPHALIC, in Medicine, is apply'd to any thing be- longing to the Head, or its Parts. See Head. ' The \Vor d is form'd of the Greek M$ahe ) Caput.
1 Cbpha-
