This term is, owing to the poverty of language, employed to signify very different things. In Plane Geometry, it means the opening or separation of two straight lines which meet in a point; but in Solid Geometry, it variously denotes the deviation of a straight line from a plane, the divergence of one plane from another at their line of junction, or even a cluster of plane angles terminating in a common summit. This diversified application of the same word is not likely, however, among mathematicians, to occasion any misconception. But it would be more perspicuous, and certainly more philosophical, to imitate the practice of naturalists in framing a set of cognate words to express the several transitions of meaning.

The word angle was drawn from common discourse into the vocabulary of science. Its primitive sense, in all the languages in which it can be traced, is merely a nook or corner; but it has acquired a more precise and extensive application in its transfer to geometry. In its simplest form, it now denotes generally the divergence or difference of direction between two concurring straight lines. Yet a learner still experiences some difficulty in seizing the correct idea of its nature, which has always baffled the attempts of authors to reduce to the terms of a strict definition. Apollonius, at once the most elegant and inventive of the Greek geometers, was satisfied with representing an angle as a collection of space about a point,—a description which is not only extremely loose, but which intimates quite a different conception. Euclid, the great compiler of the Elements, has defined an angle to be the ἀκολούθησις, or mutual inclination of two straight lines that meet. But, in strict language, this definition should apply only to the acute angle, in which one of the sides leans towards the other, and deflects from the perpendicular. Without an extension of the meaning of the term inclination, it will not include the obtuse angle, and far less comprehend angles in general; which, since they are capable of repeated additions, must evidently, as much as lines themselves, be susceptible of all degrees of magnitude.

It is indeed impossible, by any combination of words, to express completely and accurately the primary notions which form the ground of geometrical science. The more profitable task is to trace the process by which the mind, refining on external observation, comes to acquire such abstract ideas. We seem to get the idea of length, or of linear extension, by viewing progressive motion; and the enlarged conception of angles, or of angular magnitude, is easily attained, from the contemplation of revolving motion. In opening, for instance, the legs of a pair of compasses, we perceive that their difference in direction gradually increases, keeping pace with the turning at the joint. The quantity of this opening properly constitutes the measure of an angle; and an entire revolution, which brings the moving side of the angle back to its first position, furnishes a standard of reference. The bisected revolution marks the divergence of a directly opposite position, or that of two segments of a straight line at their point of separation; and the half of this, again, or the divergence of a line proceeding from the same point, and turned equally aside from both segments, is the right angle, which, therefore, being constant, serves to measure all the rest.

Suppose an inflexible straight line AB to turn from right to left, about the point or vertex A. It first comes to the position AC, then to AD, next to AE, and now returning it reaches AF, and lastly it gains its original site AB. The angles thus formed at the point A arise from the combination of successive openings.

The angle BAD is composed of BAC and CAD; the angle BAE, or that of direct opposition, is compounded of BAC, CAD, and DAE; and the entire circuit is made of the accumulated angle BAC, CAD, DAE, EAF, and FAB. This circuit, being quartered by the straight lines BE and DF, is divided at the vertex A into four right angles. By comparison, therefore, the angle BAC is acute, and CAE obtuse.

But the side AB can attain the direction AC either by moving onwards, or by turning backwards through the points F, E, and D. The angle compounded of the openings BAF, FAE, EAD, and DAC, may hence be termed appropriately the reverse of BAC. The defect of an angle from a right angle is called its complement, the defect from two right angles its supplement, and the defect from four right angles, or the entire circuit, might be conveniently named its explement. Thus, CAD is the complement of the angle BAC, CAE is its supplement, and the reverse angle BAC its explement.

If we consider attentively the formation of angles about a point, we shall be convinced that two concurring straight lines do not contain merely a single angle, but involve an indefinite multitude of angles; in short, that they comprehend all the revolutions and parts of a revolution by which the one line would successively attain the direction of the other. Hence AB will, after describing repeated revolutions, always return into the same position AC. Thus, if A represent the measure of an angle, and C that of a whole circuit, or four right angles; then the primary angle will include likewise $A + C$, $A + 2C$, $A + 3C$, $A + 4C$, continued for ever. Of those successive angles, $A$, $A + C$, $A + 2C$, $A + 3C$, $A + 4C$, &c., the sines, tangents, and secants are severally the same; and so are the versed sines, the cosines, cotangents, and cosecants. This extension of the doctrine of angles is of the greatest importance in the higher branches of geometry, in the application of trigonometrical formulæ, and in algebraical analysis.

Euclid, in the course of his reasoning, has frequent occasion to combine angles together; and yet he never ventures beyond the consideration of those angles which are less than two right angles. Had he composed his Elements after the science of trigonometry came to be cultivated, he could not have failed to take more enlarged views of angular magnitude. In consequence of his narrow conception of the constitution of angles, the Greek geometer is not a little cramped sometimes, and obliged to adopt a circuitous mode of demonstration. For instance, in the 20th prop. of his third book, that "the angles at the circumference are the halves of those at the centre standing on the same arc," he quite overlooks the case of obtuse angles at the circumference. But, in the annexed figure, the angle ABC is clearly the half of the reverse angle AOC at the centre, which is subtended by the large arc AEC. It hence follows that the obtuse angles ABC and ADC contained in the same segment must be equal, since they are both of them halves of the same reverse angle AOC. Yet, in demonstrating this very obvious corollary, Euclid is constrained to divide the obtuse angles into portions which are shown to be the halves of corresponding angles at the centre. For the same reason he finds it necessary to give a distinct demonstration of the celebrated proposition, that "the angle contained in a semicircle is a right angle." But this property ought likewise to be considered as a mere corollary; for if the radii OA and OC were supposed to extend in one straight line, and thus form the diameter of the circle, their angle AOC would become equal to two right angles, and consequently ABC, its half, would be one right angle. See Leslie's Geometry.

Angle of Incidence, in Optics, the angle which a ray of light makes with a perpendicular to that point of the surface of any medium on which it falls; though it is sometimes understood of the angle which it makes with the surface itself.