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The Ariaus endeavour'd to have put him into that of Samofata, but could not effect, it: In lieu thereof, the Emperor Vatens r -ttored him to Cystous.
EUNOMIOEUPSYCHIANS, a Seft of Hereticks of the IV th Centuty. See Heresy.
The Eunomioeupfychians, mention'd by Nicephorus, L. XII. C. 30. are the 'fame with thofe call'd Eutychians by Sozomen, L.VII. C. 17. The Author whereof, according to Sozomen, was an Eunomian, named Eutychus, and not Eupfychms, as Nicephorus has it : And yet this latter Writer only copies Sozomen in this Paffage ; fo that it is paft doubt, they both fpeak of the fame Seer. But on whofe Side the Error lies, is not eafy to decide : Valefius durft not undertake to fhew it ; but contented himfelf to mark the Difference in his Notes on Sozomen, as Fa. Fron- ton has done on Nicephorus.
EUKUCH, Eunuchus, a Term applied, in the general, to all who have not the Faculty of Generating, either thro' Imbecility, or Frigidity 5 but particularly to fuch as have been cajirated, or have lofl: the Parts necefTary thereto. See Castration.
In England, France, &c. Eunuchs are never made but on Occafion of fome Difeafes, which render fuch an Opera- tion neceilary ; but in Italy, they make Eunuchs, for the Sake of preferving the Voice ; and, in the Eaft, they make Eunuchs to be Guards, or Attendants on their Women.
Great Numbers of Children, from one to three Years of Age, are yearly caifrated in Italy, to fupply the Opera's and Theaters, not only of Italy, but other Parts of Eu- rope, with Singers : Tho' 'tis not one in three, that after having loft their Virility, have a good Voice for a Recompence.
"Tavemier allures us, that in the Kingdom of Soutan in the Eaft-Indies, there are every Year made Twenty Thoufand Eunuchs, and fold thence into other Countries.
The Seraglio's of the Eaftem Emperors, are chiefly ferv'd, and guarded by Eunuchs. And yet we have very good Teflimonies, that the rich Eunuchs in 'Ferfia and other Countries, keep Seraglio's for their own Ufe.
By an Arret of the grand Chamber of 'Paris in 1*65, it is adjudged, that an Eunuch could not matry, not even with the Confent of the Woman and all the Parties on both Sides.
Claudian has a very fevere Satyr againft. the Eunuch Eutropius, who had been Elected Conful of Rome. He reprefents him as an old Woman, drefs'd up in the Ho- nours of the Confulate. •
The Story of Origen is notorious: That learned and pious Father, upon a too literal Interpretation of that Paffage in St. Matthew, C. XIX. ver. iz. where mention is made of, Eunuchs Jo born from their Mothers Womb ; — Eunuchs ivho -were made fo of Men; — And Eunuchs -who made themfelves Eunuchs for the Kingdom of Heaven : Caftrated himfelf.
In the Councel of Nice, thofe were condemned, who, out of an indifcreet Zeal and to guard themfelves from fenfual Pleafurcs, Ihouid make 'emfelves Eunuchs : Such as thus mutilated their Bodies, were excluded from Holy Orders 5 witnefs Leontius Bifhop of Autioch, who was depofed for having praclifed this Cruelty on himfelf. And the Bifhop of Alexandria excommunicated two Monks, who had follow'd his Example on Pretence of fecuring 'emfelves from the impetuous Motions of Concupifcence. Several of the Emperors made very fevere Prohibitions againft the making of Eunuchs, or caflrating one's felf.
The Word is derived from the Greek, iw*%Q; Eunuch ; form'd of \mh I^m, Zetli curam habet, q. d. Guardian, or Keeper of the Bed.
In the III d Centuty, there was even a Seel of Hereticks form'd, call'd Eunuchs, Eumtchi ; as having the Folly or Madnefs, to caflrate not only thofe of their own Perfua- fion, but even all they could lay Hands on. They took their Rife from the Example of Origen, who, upon a Mifunderftanding of our Saviour's Words in St. Matthew, made himfelf an Eunuch, by cutting off the Part, as fome fay; or, as others, particularly S. Epiphanius, Ha:ref. 58. by the Ufe of certain Medicines. Thefe Hereticks were alfo call'd Valefians. See Valesians.
EVOLVENT, in Geometry, a Term fome Writers ufe for the Curve refulting from the Evolution of a Curve ; in Contra-diftinction to the Evolute, which is the firft Curve fuppofed to be open'd, or Evolved. See Evo- lute.
The Evolute always both touches and cuts the Evol- vent at the fame Time : The Reafon is, that it has two of its infinitely fmall Sides in common with the Evol- vent, or rather exactly placed on two equal Sides thereof. One of 'em withinfide that of the Evolvent, i. e. on the concave Side thereof; and the other, on the convex Side of its correfpondent Side: So that the Evolute touches
the Evolvent in two Points; whence, inflead of beino Tangent, it is faid to Ofculate the Evolvent, and hence it is alfo call'd Ofculator, and Circulus Ofculator See
OSCULUM.
There is one, and but one Ofculator, to each Point of the Evolvent- but to the fame Point there are an Infr. nity of other Circles, which only touch, and don't Ofcu- late. The Ofculator and the Evolute make no Angle in the Place where they touch and cut : Nor can any Curve Line be drawn between; as there may betwixt a Tangent and a Curve. See Angle of Contact.
EVOLUTE, Evoluta, in the higher Geometry', a Curve, firft propofed by Mr. Huygens ; and fince, much ftudied by the later Mathematicians. See Curve.
The Evolute is a Curve, fuppofed to be evolved, or open'd ; and which in opening, defcribes other Curves.
To conceive its Origin and Formation the better ; fup- pofe a flexible Thread, wound exactly over the Convexity of any Curve, as A B C G, (I"ab. Geometry Fig. 20.) and fuppofe the Thread fix'd in G, and every where elfe at Liberty, to A. Now, beginning to unwind the Thread from the Point, and continuing it to G, and throughout keeping it tight on the Curve Surface A B C G ; when the Thread is become quite ftraight, and is only a Tan- gent, F G, to the Curve in the Point G ; 'tis evident the Extremity A, in its Progrefs to G, has defcribed another Curve Line A D E F.
Here, the firft Curve A B C G is call'd the Evolute - Each of its Tangents B D, C E, iSc. comprehended be- tween it, and the Curve AD E F, is call'd a Radius of the Evolute ; or Radius Ofculi, or Radius Ofculator of die Curve A D E F, in each Point D, E, (Sc. And the Circles whereoftheOfculatorsBD, C E, (Sc. are Radii, are call'd Circuit Ofculatores of the Curve A D E F, in D, a, iSc. And laftly, the new Curve, refulting from the Evolution of the firft Curve, begun in A, is call'd the Curve of Evolution, or Curve defcribed by the Evolution.
the Radius of the Evolute, then, is die Part of the Thread comprized between any Point where it is a Tan- gent to the Evolute, and the correfpondent Point, where it terminates on the new Curve. Which Appellation Radius is the more proper, as one may actually confider this Part of the Thread in every Step it takes, as if it defcribed an Arch of an infinitely fmall Circle, making a Part of the new Curve, which thus confifts of an infinite Number of fuch Arches, all defcribed from different Cen- tres and different Radii.
Every Curve, therefore, may be conceiv'd as form'd by the Evolution of another. And we are to find that whole Evolution that form'd it, which amounts to the finding of the Radius of the Evoluta in any Point. For, as it is always a Tangent to the generating Curve, it is properly no more than one of its infinitely fmall Parts, or Sides prolonged ; and all its Sides, whofe Petition is determined of Courfe, is no more than the generating Curve it felf.
The fame Thread is alfo called Radius Curvedinis or Radius Ofculi, by Reafon a Circle defcribed hereby from the Centre C, is faid to Ofculate or kif's it; as both touching and cutting at the fame Time, /. e. touchin" both the infide and the out. See Osculation.
Hence 1°. The Evolute BCF, (Fig. 21.) is the Place of all the Centres of the Circle that Ofculate the Curve defcribed by the Evolution AMI. 1'. When the Point B, falls on A, the Radius of the Evolute M C, is equal to the Arch B C ; or to the Aggregate of A B, and the Arch B C. 5°. Since the Element of the Arch M m in the Curve defcribed by Evolution, is an Arch of a Circle defcribed by the Radius CM; the Radius of the Evolute C M is perpendicular to the Curve A I. 4°. Since the Radius of the Evolute M C, is always a Tangent to the Evolute BCF; Curves of Evolution may be ' defcribed thro' innumerable Points, if only Tangents be produced in the feveral Points of the Evolute, till they become equal to their correfpondent Arches.
The finding of the Radii of Evolutes, is a Thing of great Importance in the higher Speculations of Geometry; and even, fometimes, is of Ufe in Practice, as the In- ventor of the whole Theory, Huygens, has iliewn in ap- plying it to the Pendulum. Horol. Ofcill. Part III. — The Doftrine of the Ofcula of Evolutes, is owing to M. Leibnitz ; who firft fhewed the Ufe of Evolutes in the meafuring of Curves.
To find the Radius of the Evolute in the divers Kinds of Curves, with Equations to the Evolutes. See Wolf. Elem. Math. Tom. I. p. 524> & c . feqaent. Or the Infinim. fetites of Monf l e Marquis de /' fflfital.
Since, the Radius of an Evolute is equal to an Arch of an Evolute, or exceeds it by fome given Quantity ; all the Arches of Evolutes may be rectified geometrically, whofe Radii may be exhibited by geometrical Con- ftrucf ions ; whence we fee why an Arch°of a Cycloid is
double
