ANGLE IN A SEMICIRCLE

ANGLE IN A SEMICIRCLE, is an Angle in a Segment of a Circle, whose Base is a Diameter thereof. See SEGMENT.
It is demonstrated by Euclid, that the Angle in a Semicircle is a right one; in a Segment greater than a Semicircle, is less than a right one; and in a Segment less than a Semicircle, greater than a right one.
Since an Angle in a Semicircle stands on a Semicircle; its Measure is a Quadrant of a Circle; and therefore is a right angle.
Angle at the Centre, is an angle whose Vertex is in the Centre of a Circle, and its Legs terminated in the Periphery thereof—Such is the Angle CAB. See CENTRE.
The Angle at the Centre is comprehended between two Radii, and its Measure is the Arch BC. See RADIUS, &c.
Euclid demonstrates that the Angle at the Centre, BAC, is double of the Angle BDC, standing on the same Arch BC. And hence, half of the Arch AD, is the Measure of the Angle at the Periphery.
Hence also, two or more Angles HLI, and HMI (fig. 97.) standing on the same Arch HI, or on equal Arches, are equal.
Angle without the Centre, HIK, is that whose Vertex K is not in the Centre, but its Legs HK and IK are terminated in the Periphery. The Measure of an Angle without the Centre, is half of the Arches HI and LM, whereon it and its Vertical K do stand.
Angle of Contact, is that made by the Arch of a Circle and a Tangent in the Point of Contact—Such is the angle HLM, (fig. 43.) See CONTACT.
The Angle of Contact, in a Circle, is proved by Euclid to be less than any right-lined Angle: But from hence it does not follow, that the Angle of Contact is of no Quantity, as some have imagined.—Sir Isaac Newton shows, that if the Curve HAE, fig.97. be a cubic Parabola, the Angle of Contact, where the Ordinate DF is in the subtiple Ratio of the Abscissa AD, the Angle BAF contained under the Tangent AB in its Vertex, and the Curve, is infinitely greater than the circular Angle of Contact BAC; and that if other Parabolas of higher kinds be described to the same Axis and Vertex, whose Abscissas AD are as the Ordinates DF, DE', DF", &c. you will have a Series of Angles of Contact going on infinitely, of which any one is infinitely greater than that next before it.
Angle of a Segment, is that made by a Chord with a Tangent, in the Point of Contact—Such is the Angle MLH. See SEGMENT.
It is demonstrated by Euclid, that the Angle MLC, is equal to any Angle MAL in the alternate Segment MaL.
For the Effects, Properties, Relations, &c. of Angles, when combined into Triangles, Quadrangles, and polygonous Figures, see TRIANGLE, QUADRANGLE, SQUARE, PARALLELOGRAM, POLYGON, FIGURE, &c.



Angles are again divided into Plane, Spherical, and Solid.
Plane Angles are those we have hitherto been speaking of; which are defined by the Inclination of two Lines in a 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 Measure of a Spherical Angle, is the Arch of a great Circle at right angles to the Planes of the great Circles forming the Angle, intercepted between them.
For the Properties of Spherical Angles, see SPHERICAL ANGLE.

Solid Angle is the mutual Inclination of more than two Planes, or plane Angles, meeting in a Point, and not contained in the same Plane.
For the Measure, Properties, &c. of solid Angles, see SOLID ANGLE.

We also meet with other less usual sorts of Angles among some Geometricians; as,Horned Angle, Angulus Cornutus, that made by a right Line, whether a Tangent or Secant, with the Periphery of a Circle —-—— Lunular Angle, Augulus Lunularis, is that formed by the Intersection of two Curve Lines; the one Concave, and the other Convex. See LUNE.

Cissoid Angle, Augulus Cissoides, is the inner angle made by two Spherical Convex Lines intersecting. See CISSOID.

Sistrum Angle, Augulus Sistroïdes, is that in the figure of a Sistrum. See SISTRUM.

Pelecoid Angle, Augulus Pelecoides, is that in the figure of a Hatchet. See PELECOID.

Angle, in Trigonometry, See TRIANGLE and TRIGONOMETRY.

For the Sines, Tangents, and Secants of Angles, see SINE, TANGENT, and SECANT.

Angle, in Mechanics, Angle of Direction is that comprehended between the Lines of Direction of two conspiring Forces. See DIRECTION.

Angle of Elevation, is that comprehended between the Line of Direction of a Projectile, and a horizontal Line. Such is the Angle ARB, (Tab. Mechanics, fig. 47.) comprehended between the Line of Direction of the Projectile AR, and the horizontal Line AB.
See ELEVATION and PROJECTILE.

Angles of Incidence, is that made by the Line of direction of an impinging Body, in the Point of Contact. Such is the Angle DCA, (fig. 63.) See INCIDENCE.

Angle of Reflection, is that made by the Line of Direction of the reflected Body, in the Point of Contact from which it rebounds. Such is the Angle ECF. See REFLECTION.

Angle, in Optics, Visual or Optic Angle, is the Angle included between two Rays drawn from the two extreme points of an Object, to the Centre of the Pupil. Such is the Angle ABC, (Tab. Optics, fig. 69.) comprehended between the Rays AB, and BC. See VISUAL ANGLE.

Objects seen under the same, or an equal angle, appear equal. See MAGNITUDE and VISION.

Angles of the Interval, of two places, is the Angle subtended by two Lines directed from the Eye to those places.

Angle of Incidence, in Catoptrics, is the lesser 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. 26.) See RAY and MIRROR.

Every incident Ray, AB, makes two Angles, the one acute, ABD, the other obtuse, ABE; though sometimes both right. The lesser of such angles is the Angle of Incidence. See INCIDENCE.

Angle of Incidence, in Dioptrics, is the Angle ABI (fig.56.) made by an incident Ray, AB, with a Lens or other refracting Surface, HI. See LENS, &c.

Angle of Inclination, is the Angle ABD, contained between an incident Ray, AB, and the Axis of Incidence DB. See AXIS, &c.

Angle of Reflection, in Catoptrics. See REFLECTION.

Reflected Angle, in Catoptrics. See REFLECTION.

Angle of Refraction, in Dioptrics. See REFRACTION.

Refracted Angle, in Dioptrics. See REFRACTION.

Angle, in Astronomy, Angle of Commutation. See COMMUTATION.

Angle of Elongation, or, ANGLE at the Earth. See ELONGATION.

Parallactic Angle. See PARALLACTIC ANGLE.

Angle at the Sun, or the Inclination, is the Angle RSP, (Tab. Astronomy, fig.25.) under which the Distance of a Planet P, from the Ecliptic PR, is seen from the Sun.
See INCLINATION.

Angles of the East. See NONAGESIMAL.

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 inviolably the same in all points of the Earth’s annual Orbit.

By means of this Inclination, such Inhabitants of the Earth as live beyond 45° of Latitude, have more of the Sun’s Heat, taking all the Year round; and those who live within 45°, have less of his Heat, than if the Earth always moved in the Equinoctial. See HEAT, &c.

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 of right Ascension, is the Angle which the Circle of a Star’s right Ascension makes with the Meridian at the Pole of the World. See RIGHT ASCENSION.

Angle, in Navigation Angle of the Rhumb, or Loxodromic ANGLE. See RHUMB and LOXODROMY.

Angles, in Fortification, are understood of those formed by the several Lines used in Fortifying. See FORTIFICATION, FORTIFYING, &c.

Angle of, or at the Center, is the Angle formed at the Center of the Polygon, by two Semi-diameters drawn thither from the two nearest Extremities of the Polygon.
See POLYGON—Such is the Angle CKF (Tab. Fortification fig.1.) Angle of the Circumference, is the mixed Angle made by the Arch drawn from one Gorge to the other. See GORGE.

Angle of the Counterscarp, is that made by the two Sides of the Counterscarp, meeting before the middle of the Curtain. See COUNTERSCARP.

Angle of the Curtain, or of the Flank, is that made by, or contained between, the Curtain and the Flank; such is the Angle BAB. See CURTAIN.

Angle of the Complement of the Line of Defence, is the Angle arising from the Intersection of the two Complements one with another. See COMPLEMENT.

Diminished Angle, is the Angle which is made by the meeting of the exterior Side of the Polygon, with the Face of the Bastion—Such is the Angle BCR.

Angle of the Polygon, or of the exterior Figure, is the Angle FCN, formed at the Point of the Bastion C, by the meeting of the two outermost Sides or Bases of the Polygon, FC and CN.

Angle of the Epaule, or Shoulder, is that formed by the Flank and the Face of the Bastion. Such is the Angle ABC. See EPAULE.

Angle of the Interior Figure, is the Angle GHM, made in H, the Center of the Bastion, by the meeting of the innermost Sides of the Figure GH and HM.

Flanking Angle outward, or Angle of the Tenaille, is that made by the two rasant Lines of Defence, i.e., the two Faces of the Bastion when prolonged. See TENAILLE.

Angle flanking inward, is the Angle CIH, made by the flanking Line with the Curtain.

Angle flanked, by some called the Angle of the Bastion, is the Angle BCS, made by the two Faces of the Bastion, BC, CS; being the outermost part of the Bastion, and that most exposed to the Enemy’s Batteries, and therefore by some called the Point of the Bastion. See BASTION.

Angles of the Triangle, in Fortification, is half the Angle of the Polygon.

Angle forming the Flank, is that consisting of one Flank, and one Demi-gorge.

Angle forming the Face, is that composed of one Flank and one Face.

Angle of the Moat, is that made before the Curtain, where it is intersected. See MOAT.

Re-entrant Angle, is that whose Vertex is turned inwards, towards the Place. See RE-ENTRANT.

Salient Angle, is that which advances its Point towards the Field. See SALIENT.

Angle of the Tenaille, or the outward flanking Angle, called also the Angle of the Moat, or the dead Angle, is made by the two Lines rasant in the Faces of the two Bastions, extended till they meet in an Angle towards the Curtain—This always carries its Point in towards the Work.

ANGLE IN A SEMICIRCLE