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It is demonftrated by Euclid, that all the Angles in the fame Segment are equal to one another ; that is, any Angle E H G, is equal to any Angle E F G in the fame Segment E F G.
The Angle at the Periphery, or in the Segment, is comprehended between two Chords AB and BD, and flands on the Arch A B. See Chord, lie.
The Mcafure of an Angle without the Periphery G, (fig. 95.) is the Difference between half the Concave Arch LM, whereon it ftands, and half the Convex Arch, NO, inter- cepted between its Legs. . • . , . -
Angle in a Semi-circle, is an Angle in a Segment ot a Circle whofe Bafe is a Diameter thereof. SccSegmfnt.
It is demonftrated by Euclid, that the Angle in a Semi- circle is a right one ; in a Segment greater than a Semi- circle, is Jets than a right one ; and in a Segment lels than a Semi-circle, greater than a right one.
Since an Angle in a Semi-circle Hands on a Semi-circle, its Meafure is a Quadrant of a Circle ; and therefore is a right Angle. . .
Ang li flf rfe Cra«"e, is m Angle whofe Vertex is in the Centre of a Circle, and its Legs terminated in the Peri- phery thereof Such is the Angle CAB. See Centre.
The Angle at the Centre is comprehended between two Radii, and its Meafure is the Arch BC. See Radius, f£c.
Euclid demonitrates that the Angle at the Centre, BAC, is double of the Angle BDC, Handing on the fame Arch BC. And hence, half of the Arch AD, is the Mea- fure of the Angle at the Periphery.
Hence alfo, two or more Angles HLI, and HMI (tig. 97. j {landing on the fame Arch HI, or on equal Arches, are equal. , , r , T
Angle without the Centre, HIK, is that whofe Vertex K. is not in the Centre, bur its Legs HK and IK are terminated in the Periphery. _
The Mcafure of an Angle without the Centre, is half ot the Arches HI and LM, whereon it and its Vertical K do
A»cn ofContaB, is that made by the Arch of a Circle
and a Tangent in the Point of Contaa. Such is the
Angle HLM, (fig. «'.; -Set Contact.
The AWc of Contaa, in a Circle, is proved by Euclid to be lefs than any right lined Angle: But from hence it does not follow, that the Angle of Contaa is of no Quan- tity as fbme have imagined.— Si* Mac Ntwtoit fhews, that if the Curve HAE, fig. 91- be a cubick Parabola, the Angle. of Contaa, where the Ordinate DF is in the fubtriple Ratio of the Abfciffe AD, the Angle BAF contained under the Tangent AB in its Vertex, and the Curve, is infinitely grea- ter than the circular Angle of Contaa BAC ; and that it other Parabola's of highter kinds be defenbed to the fame Axis and Vertex, whofe Abfciffcs AD are as the Ordinares DP" DI S , DF', U-c. you will have a Series of Angles ot Contaa going on infinitely, of which any one is infinite- ly greater than that next before it.
Angle cf a Segment, is that made by a Chord with a Tangent, in the Point of Contaa.— Such is the Angle MLH. See Segment. .
It is demonftrated by Euclid, that the Angle MLC is equal to any Angle MflL in the alternate Segment MsL.
For the EffeBs, ^Properties, Relations, &c of Angles, when combined into triangles, Qiadrangles, and polygonaus Figures, fee Triangle, Quadrangle, Sqjjare, Pa- rallelogram, Polygon, Fi&ure, i£c.
Angles are agiin divided into Tlane, Spherical, and
" '-Plane Angles are thofe we have hitherto beenfpeak- ing of; which are defined by the Inclination of two Lines ink Plane, meeting in a Point. See Plane.
Spherical Angle is the Inclination of the Planes of two great Circles of the Sphere. See Circle and Sphere.
The Meafure of a Spherical Angle, is the Arch of a sreat Circle at righr Angles to the Planes of the great Circles forming the Angle, intercepted between them.
For the 'Properties of Spherical Angles, fee Spherical Angle. . ,
Solid Angle is the mutual Inclination ot more than two Planes, or plane Angles, meeting in a Point, and not con- tain'd in the fame Plane.
For the Meafure, 'Properties, &c, cffilid Angles, iee
We alfo'meet with other lefs ufual forts of Angles among fome Geometricians ; as,
Horned Angle, Anguhts Cornntus, that made by a right Line, whether a Tangent or Secant, with the Periphery of a Circle. — . , . . , c ,, .
Lunula;- Angle, Angulus Zuuulans, is that form d by the Intcrfeaion of two Curve Lines j the one Concave, and the other Convex. See Lone.
Ciffoid Angle, Angulus Cijfoidei, is the inner Ang\ t made by two Spherical Convex Lines interfefling each other'. See Cissoides.
Siflroid Angle, Angiihs Siflroides, is that in Figure of a Siilrum. Sec Sistrum.
- Petecoid Angle, Angulus 'Pelecoides, is that in figure f
a Hatchet. See Pelecoides.
Angle, in Trigonometry. See Triangle and Tat-
GONOMETRY.
For the Sines, tangents, and Secants cf Angles, ft e Sine, Tangent, and Secant.
Angle, in Mechanicks Angle of Diretlion, i s
that comprehended between the Lines of Direaion of two confpiring Forces. See Direction.
Angle of Elevation, is that comprehended between the Line of Direaion of a Project ile, and a horizontal Line — Such is the Angle ARB, (Tab. Mechanics, fig.47.) com- prehended between the Line of Direaion of the Projectile AR, and the horizontal Line AB. See Elevation and Projectile.
Angle of Incidence, it that made by the Line of Di.
reaion of an impinging Body, in the Point of Contaa .
Such istheyfeg/eDCA, (fig. S3.) See Incidence.
Angle of Reflexion, is that made by the Line of Di- reaion of the reflected Body, in the Point of Contaa from
which it rebounds, Such is the Angle ECF. See Re-
flexion.
Angle, in Optics Vifual or Optic Angle, is the
Angle included between two Rays drawn from the rwo ex- treme points of an Objea, to the Centre of rhe Pupil
Such is rhe Angle ABC, (Tab. Optics, fig.ffj.) compre- hended between the Rays AB, and BC. See Visual Angle.
ObjeBs feen under rhe fame, or an equal Angle, appear equal. See Magnitude and Vision.
Angle of the Interval, of two places, is the Angle fub- tended by two Lines direaed from rhe Eye to thofe places.
Angle of Incidence, in Catoptrics, is the lefler Angle made by an incident Ray of Light, with the Plane of a Speculum ; or, if the Speculum be concave or convex,with
a Tangent in the point of Incidence Such is the Angle
ABD (fig. 2<r.) See Ray and Mirrour.
Every incident Ray, AB, makes two Angles, the one a- cute, ABD, the other obtufe, ABE ; tho fometimes both right The leffer of fuch Angles is the Angle of Inci- dence. See Incidence.
Angle of Incidence, in Dioptrics, is the Angle ABI, (fig.5<r.) made by an incident Ray, AB, with a Lens or orher refrafling Surface, HI. See Lens, £Jc.
Angle of Inclination, is the Angle ABD, contained between an incident Ray, AB, and the Axis of Incidence, DB. See Axis, iSc.
Angle of Reflection, 7 in Catoptrics. See Reflec-
RefleSed Angle, 5 tion.
Angle cf RefraBion,~) in Dioptrics. See Refrac-
Refralied Angle, S tion.
Angle, in Aftronomy Angle of Commutation. Sec
Commutation.
Angle of Elongation, or, Ancle at the Earth. See Elongation.
(ParallaSic Angle. See Parallactic Angle.
Angle at the Sun, or the Inclination, is the Angle RSP, (Tab. Astronomy, fig. 25.) under which the Diflanceof a Planet P, from the Ecliptic PR, is feen from the Sun. See Inclination.
Ancle of the Eajl. See Nonasesimal.
Angle of Obliquity, of the Ecliptic. See Obliquity and Ecliptic.
the Angle of Inclination of the Axis of the Earth, to the Axis of the Ecliptic, is 23°, 30' ; and remains inviola- lably the fame in all points of the Earth's annual Crbit. By means of this Inclination, fuch Inhabitants of the Earth as live beyond 45 of Latitude, have mote of the Sun's Heat, taking all the Year round 5 and thofe who live with- in 45', have lefs of his Heat, than if the Earrh always moved in the Equinoaial. See Heat, iSc.
Angle of Longitude, is the Angle which the Circle of a Star's Longitude makes with the Meridian, at the Pole of the Ecliptic. See Longitude.
Angle cf right Afcenflon, is the Angle which the Circle of a Star's right Afcenflon makes with the Meridian at ths Pole of the World. See Right Afcenflon.
Angle, in Navigation Angle of the Rhumb, ° r
Loxodromic Angle. See Rhumb and Loxodromy.
Angles, in Fortification, are undcrftood of thofe formed by the feveral Lines ufed in Fortifying. See Fortifi- cation, Fortifying, S£c.
Angle of, or at the Center, is the Angle formed at the Center of the Polygon, by two Semi-diameters drawn thi- ther from rhe rwo neareft Extremities of thc Polygon. See Polygon Such is the Angle CKF (Tab. Fortificat-
fi *'° Angle
