This section is from "The American Cyclopaedia", by George Ripley And Charles A. Dana. Also available from Amazon: The New American Cyclopędia. 16 volumes complete..

**Angle**, a portion of space between two lines or between two or more surfaces intersecting each other. Geometry distinguishes four kinds of angles: plane, spherical, dihedral, and polyhedral. 1. Plane angles. When two lines are situated in the same plane and not parallel to each other, they intersect at some point, and around this point of intersection they form four plane angles; the point of intersection is called the vertex, and the lines the sides of the angles. If all the four angles thus formed are equal, they arc called right angles, and the lines are said to be perpendicular to one another; when not equal, those smaller than a right angle are called acute, and those larger obtuse angles. Angles are measured by degrees, which are nothing but angles so small that 360 of them are situated around one point, and therefore 90 in a right angle. For practical measurement of angles the circumference of a circle is divided into 360 equal parts (see fig. 1), and its centre laid on the vertex of the angle, in which case the parts of the circumference between the sides of the angle will indicate the number of degrees contained in the same. Each degree is again divided into 60 parts called minutes, and each minute into 60 seconds.

The whole circumference of the circle is therefore subdivided into 1,296,000 seconds, which is about the limit of accuracy of astronomers in measuring angles at the firmament. When angles have curved sides (as represented in fig. 2), tangents are drawn to the curves at the vertex, and the angle these tangents make with one another is measured. 2. Spherical angles. Under this name is designated the space included between two arcs of great circles, drawn on a sphere. A D and B D, fig. 2, form together a spherical angle, which, if the plane B O E D is perpendicular to the plane A O 0 D, is a spherical right angle: the intersections of the meridians with the equator of the earth are such right angles, while the intersections of the meridians at the poles form a number of acute spherical angles. The angles which the astronomers measure in their celestial triangles are all spherical angles. 3. Dihedral angles are formed by the intersection of two planes. The planes ABCD and A B F E, fig. 3, form a dihedral angle; the line of intersection, A B, is called the edge, and the planes are called the faces.

Such angles are measured by the plane angle formed when passing a plane perpendicular through the edge, or, what is the same, drawing two lines OT and ST from the same point in the edge A B, perpendicular to the same, and one in each plane; the arc S T is in that case the measure of the dihedral angle. 4. Polyhedral angles are the spaces included between three or more planes which intersect at one point. Thus O, fig. 4, is the vertex of a trihedral, and O, fig. 5, the vertex of a tetrahedral angle, respectively bounded by three and four faces. As an arc of a circle is used for measuring plane and dihedral angles, so a portion of the surface of a sphere, of which the centre is at the vertex, is used to measure polyhedral angles. - Angle of Total Reflection. When a ray of light falls on a polished surface separating a transparent denser medium from a similar rarer one, it will be reflected and refracted, that is, split up into two rays; one of which will be thrown back, and the other will pass on and be diverted more or less from its course. Such a splitting up of a ray of light always takes place when it passes from a rarer into a denser medium.

But when the light passes from a denser into a rarer medium, for instance, from glass into air, this will not be the case under all inclinations of the ray. When the angle of incidence is not very acute, no refraction, but total reflection, will take place. Let A B O represent a cross section of a glass prism; then the ray D R will be split up, being reflected to R E and refracted to R R, because the angle of incidence, D R Q, is very acute, the ray F T, however, making with the perpendicular T P a less acute angle. As F T P is only reflected in the direction T G, and not refracted at all, it cannot pass out of the prism at T, and this constitutes there a case of total reflection. The minimum number of degrees required for such a case is calculated according to a law discovered by Descartes, which is that "the sines of the angles of incidence and refraction bear a fixed relation to one another, different for each substance." When the calculation gives for the sine of the angle of refraction a quantity greater than 1, it gives a sine which cannot exist, which indicates that no refraction can exist in this case, and that consequently all the light is reflected.

The smallest angle of incidence with which this takes place, or the angle of total reflection, differs according to the relative power of refraction of the two transparent media. For light passing into air, it is when coming from water 48° 30'; from crown glass, 42°; from flint glass, 38°; and from diamond, 24°. This is one of the reasons of the special brilliant lustre of the last-named substance. - For other special applications of the term (angle of incidence, of least deviation, of polarization, of repose), see Mechanics, Polar-iztion, and Spectrum.

Fig. 1. - Plane Angles.

Fig. 2. - Spherical Angles.

Fig. 3. - Dihedral Angles.

Fig. 4. - Trihedral Angle.

Fig. 5. - Tetrahedral Angle.

Fig. 6. - Angle of Total Reflection.

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