( 74o )
tofNova; to 5 Zires or Limes of Genoa, 4 Zires and 10 Soldi of Lucca, 8 Zires of Sergama, 3 Z;'ra and 1 5 &£# of Savoy ; top Carlms of Naples, and as many of Sicily-, 21 Groats ana three Fifths of Venice, 24 of Neumbcurg ; to 372 Maravedes of Spain, to 5oo i?^s of 'Portugal, to 4 •/«-«! and 1 5 Swffi of Malta, to 120 Afpers of Constantinople, to a Tiemi-hongre of Gold of Hungary, to 2 Florins of Zrcge, 3 of Strasbourg and 20 of Raconis, to 90 Gf-M« or Greek's of 'Poland, and 24 of "Berlin, to 80 Aforfo" of Copper of Sweden, to jo Grities or Gr//i of Copper of Mufcovy, and laftly to 4ifo-.s of 'Denmark.
Par, in Anatomy. See Pair.
Par Vagum,or the fourth Pair,is a very notable Conjugation ofNerves,of the Medulla oblongata ; thuscall'd fromth'cir wide !M£!/e Diftribution throughout the feveral Parts of rhe Body.
See the Origin, Courfe, Diftribution, £5f& of the ?Ptf/* vagum under Nerve.
Par, aTerm of Nobility. See Peer.
PARABLE, a Fable, or Allegorical Inftruction founded on fomething real, or apparent in Nature or Hiftory ; from which fome Moral is drawn, by comparing it with fomeother Thing, wherein the People are more immediately concerned.
Such are thofe Parables of the Ten Virgins, of Hives and Lazarus, of the Prodigal Son, Sec. in the New Tefiament. St. Matthew fays our Saviour never fpoke to the People but by 'Parables.
~B.de Coloniacalls theParable,a Rational Fable. SeeFABLE.
The Word is form'd from the Greek iraey.Ca.Mtiv to compare. Whence Arif.otle defines it a Similitude drawn from Form to Form. Cicero calls it a Collation, others a Simile.
In the New Teftament it is ufed varioufiy. In Luke IV. 13. for a Proverb or Adage. lnMattb.XV. 15. for aThing darkly and figuratively exprefl'd. In Heb. IX. 9. &c. for a 'type. In Luke XIV. y. Hie for a fpecial InfiriiBion. Matth. XXIV. 32. fora Similitude or Companion. The Piebre-wsczW it ~YUD from "WU to predominate, to aflimilate ; whence the Proverbs of Solomon are call'd "1^7^ Parables or Proverbs.
Aquinas defines Parable a fimilitudinary Difcourfe ; or a Speech which fays one thing and means another. Gtaffms more accurately defines it a Simile wherein a fictitious Thing is related as real, and compared with fome Spiritual Thing, or accommodated to fignify it.
Some make Parable differ from Fable ; Grotius and others will have them the fame. Kircher derives the Ufe of Parables from the Egyptians.
PARABOLA, in Geometry, a Figure arifing from the Se- ction of a Cone, when cut by a Plane parallel to one of its Sides. See Section
From the fame Point of a Curve, therefore, only one Parabola can be drawn: All the other Sections within thofe Parallels being Ellipfes j and all without, Hyperbola's. See Cone.
Wolfius defines the Parabola to be a Curve wherein the Square of the Semi-ordinate is equal to the Rectangle of the Abfcifle, and a given right Line call'd the Parameter of the Axis or La- ms rttimn.
Hence, a Parabola is a Curve of the firft Order; and as the AbfcifTes increafe, the Semi-ordinates increafe likevvife ; confequently the Curve never returns into itfelf. Hence alfo the Abfcifle is a third Proportional to the Parameter and Se- mi-ordinate ; and the Parameter a third Proportional to the Abfcifle and Semi-ordinate ; and the Semi-ordinate a mean Proportional between the Parameter and Abfcifle.
"To describe a Parabola. The Parameter A B Tab. Conicks. Fig. 8. being given; continue it to C, and fromB let fall a Per- pendicular, toN. From Centres taken at Pleafure, with the Compafles open to A, defcribe Arches cutting the right Line B Vin I, II, III, IV, V, He And the right Line B C in 1, 2, 3, 4, 5, lie Then will B 1, B 2, B 3, B4, B 5, lie be Ab- fciffes, B I, B II, B III, B IV, B V, tic Semi-ordinates. Wherefore if the Lines B 1, B 2, B 3, \}§c. be transferr'd from the Line B C to that B N, and in the Points 1, 2, 3, 4, &c t Perpendiculars be raifed, 1 1 = B I, 2 II = B II, 3 III - B III, igc. The Curve pafling thro' the Points I, II, III, He. is a Pa- rabola ; and P N its Axis.
Every Point of the Parabola may alfo be determined geo- metrically. E.gr. If it isinquired whether thePointM be in the ■parabola or not ? From M to BN let fall a Perpendicular M P. And let P N be equal to the Parameter A B ; upon B N de- fcribe a Semicircle. For if that pafs thro' M, the Point M is in the Parabola.
In a Parabola the Diftance of the Focus from the Vertex is to the Parameter in a fubquadruple Ratio : And the Square of the Semi-ordinate is quadruple the Rectangle of the Di- ftance of the Focus from the Vertex, into the Abfcifle.
To defcribe a Parabola by a continued Motion. Affuming a right Line for an Axis,letfAFig. 9.— A F=i a. In A fix a Ruler D B cutting the Axis/D at right Angles. To the Extremity of another Ruler E C, faften a Thread fix'd at its other Ex- treme in the FccusV which is to be —A D+ A F. If then a Style or Point be fix'd to the Ruler E C, and the Ruler be carried firft to the Right then to the Left, according to the Direction of the other D B ; the Style will mark out a Parabola : For F M will be conftantly —EM — Pfzzx-\- \a, and confe- cruenfly the Point M is in a Parabola.
properties of tie Parabola.
The Squares of the Semi-ordinates are to each other as the AbfcifTes; and the Semi-ordinates, themfeives, in a fubtri- plicate Ratio of the AbfcifTes.
The Rectangle of the Sum of the two Semi-ordinates into their Difference, is equal to the Rectangle of the Parameter into the Difference of the Abfciflcs : The Parameter therefore is ro the Sum of the two Semi-ordinates, as their Difference to the Difference of the AbfcifTes.
In zParabola the Rectangle of the Semi-ordinate into the Abfcifle, is to the Square of the Abfcifle, as the Parameter to the Semi-ordinate.
In a Parabola the Square of the Parameter is to the Square of one Semi-ordinate, as the Square of the other Semi-ordinate to the Rectangle of the Abfcifl'es.
In a Parabola the Subtangent is double the Abfcifle, and the Subternal fabduplc the Parameter.
Quadrature of the Parabola. See Quadrature.
Rectification of the Parabola. See Rectification.
Centre of Gravity of a parabola. See Centre of Gra- vity.
Centre of Ofcillation of the Parabola. See Oscillation.
P arabola'so/ 'the MgherKinds are Algebraic Curves.defin'd by «•"— ~ x=y m . E.gr. bya' x—y>, a' x—y*, a* x=y'\ a'x^=y*, Hie. See Curve.
Some call thefe Paraboloides: particularly, if a' x~y'$ they call it a Cubical Paraboloid, if a' x = y*, Hie. They call i t a Siquadratical Paraboloid, or a Surdefolidai Paraboloid. And in refpect of thefe, the Parabola of the firft Kind, above explained, they call the Apollonian or Quadratic Parabola.
Thofe Curves are likewifc ufed to be referred to Parabola's wherein a x m — ' =yLA2-bot (talk). as E. gr. ax'=y\ ax'=iy* 9 which fome call Semi-parabola's. They are all comprehended under one common Equation a m x"y r , which alfo extends re* other Curves, v. g. to thofe wherein a' x' =y* a" x" — <y' a , x t =:y.
Since in Parabola's of the higher Kinds, ym t= a™ — i x 5 If any other Semi-ordinate becalled v, the Abfcifle correfpond- ing to 2-,will be v~n - rf ra — 1 3 confequently ym : vta : . *jm — 1 x
- a m - ' z. That is a? : z. 'Tis a common Property, therefore 3
of thefe Para.bola's,that the Powers of the Ordinates are in the Ratio of the AbfcifTes.
But in Semi-parabola's yn •. vm - : a x m — * : : x m — ' : z m — '.■ Or the Powers of the Semi-ordinates are as the Powers of the Abfciflcs, one Degree lower. JE gr. In Cubical Semi-para-- bola's, the Cubes of the Ordinates^' and v' are as the Squares of the AbfcifTes x ' and a*.
Apollonian Parabola, is the Common, or Quadratic Para- bola or Parabola of the firft Kind ; fo called by way of Diftin- ctionfrom Parabola's of the higher Kinds : Which fee.
Quadratic Parabola, is the fame with the ^ " Which fee.
PARABOLAN, PARABOLANUS, among the Antients, was a Sort of Gladiator; called alfo ConfeBor. SecCoN-
The Name was given them from the Greek ■&«/&&>.& f 3«'aas> to throw, precipitate; in regard they threw themfeives on Danger and Death.
ParabolanIs alfo ufed in Church-Hiftory, for a Set of People, efpecially in Alexandria, who devoted themfeives to the Service of Churches, and Hofpitals. The Parabolans were not allowed to withdraw themfeives from their Function, which was the Service of the Sick. They made a Kind of Friary, amounting fometimes to fioo Perfons ; depending on the Bifhop.
The Defign of their Inftitution was, that the difeafed, e- fpecially thofe infected with the Plague, might not be without Atttendance.
PARABOLIC Space, the Space ox Area contained between any entire Ordinate as V V Tab. Conicks Fig. 8. and the Curve of the incumbent Parabola.
The Parabolic Space is to the Rectangle of the Semi-ordi- nate into the Abfcifle, as 2 to 3 ; to a Triangle inferibed on the Ordinate as a Bafe, the Parabolic Space is as 4 to 3.'
Every Parabolical and Paraboloidical Space is to the Rect- angle of the Semi-ordinate into the Abfcifle as r x y : (m -(- f) to x y, that is, as r to m -f-r.
Segment of a Paraklic Space, is that Space included be- tween two Ordinates. See Segment.
Quadrature of a Parabolical Segment. See Quadrature.
Parabolic Pyramidoid, a foiid Figure, generated by fup- pofing all the Squares of the ordinate Applicates in the Para- bola, fo placed, as that the Axis fhall pafs thro' all their Cen- tres at Right Angles; in which Cafe the Aggregate of the Planes will be arithmetically proportional.
The Solidity hereof is had by multiplying the Bafe, by half the Altitude; the Reafon whereof is obvious : for the compo- nent Planes being a Series of Arithmetical Proportionals begin- ning from o, their Sum will be equal to the Extremes multi- ply'd by half the Number of Terms, that is, in the prefent Cafe, equal to the Bafe multiply'd by half the Height.