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 xλατις, 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

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 formulas, 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.

ANGLE of Refraction now generally means the angle which a ray of light, refracted by any medium, makes with a perpendicular to that point on the surface of which it was incident; but has sometimes been understood of the angle which it makes with the surface of the refracting medium itself.

Trisection of. The attempts of the Greek mathematicians to Double a Cube, and to Trisect an Angle, were their first steps beyond the limits of elementary geometry. They soon perceived that such problems cannot be solved by any combination of mere straight lines or circles. To this conclusion they were led directly by the application of geometrical analysis, a beautiful instrument of discovery which Plato had recently invented or improved. Their investigations pointed at some curves of a higher order than the circle, and opened to them a wide and interesting field of research.

The analysis of the trisection of an angle, conducted in two different ways, terminates in the construction of the conchoid, a complex curve which was first proposed by Nicomedes. As the subject is very curious, and throws great light on the theory of angular magnitude, we shall here not only give both the ancient methods of investigation, but subjoin a third which is due to the sagacity of Newton.

1. Let it be required to trisect the angle BAC or the arc BC. Suppose the thing already done, and the angle BAD to be the third part of the given angle. From the point C draw CE parallel to AD, meeting the extended diameter in E, and cutting the circumference of the circle in the point F; join FD and FA. It is obvious that the angle BAD is the half of DAC, the remaining part of the whole angle, and therefore equal to the angle DFC at the circumference. But AD and EC being parallel, the angle BAD is equal to AEF, which is hence equal to DFC; and consequently FD is parallel to EB. Wherefore the arc BD is equal to GF, and the angle BAD equal to GAF. The angle AEF is thus equal to EAF, and hence the side EF is equal to AF, the radius of the circle. To solve the problem, therefore, it would be requisite to inflect from C a straight line CFE, such, that the portion FE, intercepted between the circumference and the diameter, or its extension, should be equal to the radius of the circle. The radius AD, drawn parallel to this inflected line CE, would cut off an angle BAD, which is the third part of the given angle BAC.

But elementary geometry will not in general furnish the means of inflecting CE, according to the required conditions. This must be done either tentatively, that is, by repeated trials, or by the application of a curve, so constituted that every straight line drawn from the pole C to the directrix BG shall have the portion EF, intercepted by the curve, equal to AB. This curve is, from its general shape or resemblance to a conch or shell, named the conchoid; it consists of two branches, one above the directrix called the interior conchoid, and the other below it called the exterior conchoid. The conchoid being described, will, by its intersection with the circumference of the circle, give the point F, and consequently the position of the trisecting line AD'. But such a complex curve must cut the circumference in more points than one, and consequently the problem of angular trisection, viewed in its generality, admits of several answers. In fact, there are always three distinct positions of the inflected line CE, which will fulfil the conditions of the problem.

It is curious to examine these different positions of the inflected line. Draw AD' parallel to the second position CE', and join DF, AD', and AF. Because AF is equal to EF, the angle EAF is equal to AEF; and, consequently, the exterior angle AFC is the double of either of these. But CAF being an isosceles triangle, AFC is equal to ACF, which again is equal to the alternate angle CAD'; wherefore CAD' is the double of the angle AEF; and being likewise the double of CFD' at the circumference, the angles CFD' and AEF' are equal, and, consequently, FD' and EA parallel. Now the angle CAD' being double of EAF' or DAG, and the angle CAD double of DAB, the arc DCD' is double of the arcs DG and DB, which serve to complete the semicircumference; wherefore this arc DCD' is two third parts of the semicircumference, or one third of the whole circumference.

In the third position AD'' of the trisecting line, draw CF' parallel to it, and join AF' and DF''. The isosceles triangles D'AF' and AFE' have equal vertical angles; and, consequently, the angles at their base are likewise equal; wherefore AF'D'' is equal to the alternate angle F'AE', and the chord D'F'' parallel to the diameter BG. But the reverse angle CAD'' standing on the arc CDD'' is double of the angle CFD' at the circumference, and therefore double of BEF' or of BAD'; and the angle CAD being by the first construction likewise double of BAD, the reverse angle CAD'', together with CAD, must be double of BAD' and BAD, or the arc CD'GD'' is double of DBD'', which completes the circumference. Hence the arc DD''D' is two thirds of the circumference.

It thus appears that the construction of the problem assumes three different aspects, and that the trisecting lines, to which a close analogy conduits us, mutually divide the whole circuit into equal portions. These results are perfectly conformable with the theory of angular magnitude. For if A denote the arc BC, and C the whole circumference, this arc will be generally expressed by A, A+C, A+2C, &c.; consequently, the third part will be expressed by $\frac{1}{3}A$, $\frac{1}{3}A + \frac{1}{3}C$, $\frac{1}{3}A + \frac{2}{3}C$, &c., which evidently correspond to BD, BD' and BD''. But any farther extension of this progression only brings the trisecting line back into its former positions.

To solve completely, therefore, the problem of the trisection of an angle; from the pole C, on either side of the directrix BG, with a measure equal to the radius of the circle, describe the exterior and the interior or nodated conchoid; draw CF, CF', and CF'' to the three points of intersection with the circumference, and the radii AD, AD', and AD'' parallel to these will mark the triple section of the angle BAC.

It may be perceived that the exterior branch of the conchoid cuts the under semicircle in another point besides F. This occurs in the extension of the radius CA, or where the diameter passing through C, the extremity of the original arc, meets the opposite circumference; the portion of the inflected line, intercepted below BG by the conchoid, being evidently equal to the radius. The fourth intersection, however, affords no real solution, but only exhibits the amount of repeated division, as completing the arc itself.

2. But another analysis leads to a similar result. Let the angle BAD, as before, be the third part of BAC; draw BC perpendicular to AB, and CE parallel to it, meeting AD produced in E. The right angle DCE would be contained in a semicircle having DE for its diameter; join C with the centre H, and the triangle CHE being isosceles, the exterior angle CHA is double of CEH, or of the alternate angle BAD, and therefore equal to the remaining portion CAD of the divided angle BAC. Whence the triangle ACH is isosceles, and the side CA equal to HC, or the diameter DE must be double of AC.

The construction of the problem is thus reduced to the drawing from the vertex of the given angle a straight line ADE, such, that the part DE, intercepted between the perpendiculars BC and CE, shall be equal to the double of AC. This can only be done by describing a conchoid from the pole A to the directrix BC, and with the double of AC as the measure; the intersection of the curve with the perpendicular CE will determine the position of the trisecting line ADE. The exterior branch of the conchoid will cut the perpendicular in the point E, and the interior or nodated branch will meet and cross it at the two points E' and E''. The radiating lines AE, AE', and AE'', or its extension Ae'', will indicate the complete trisection of the angle BAC. These lines will be found, as in the first construction, to make angles with each other that are equal to the thirds of an entire circuit. It may be worth while to examine the several cases. In the second position D'AE' of the trisecting line, draw CH' to the middle point. Because the triangle E'H'C is isosceles, its exterior angle CH'A is double of CETH', or of the angle BAD'; but H'CA being also an isosceles triangle, CH'A is equal to CAH', and consequently double of BAD'. Add CAD, which is double of BAD, and the compound angle DAE' is double of DAD', which would complete two right angles; whence DAE' is two thirds of two right angles, or one third of a whole circuit.

In the third position, AE'D' of the trisecting line, or rather its extension Ae", draw CH" to bisect ED'. The triangles CHPD' and ACH" are then isosceles, and consequently the angle CAD" or CH"A is double of CD'A; but CAE is likewise double of CEA, and therefore the combined angle D'AE is double of the angles CD'A and CEA. Now this angle D'AE, together with the two angles CD'A and CEA, is evidently equal to the exterior angle DC'E or a right angle. Whence D'AE is two thirds of a right angle, or one third of two right angles, and therefore the adjacent angle DAE" is two thirds of two right angles, or one third of a whole circuit.

3. The simplest and most elegant solution of the trisection of an arc was indicated by Pappus, and is given in Castillon's Commentary on Newton's Universal Arithmetic. The problem is there reduced to the combination of the circle with a certain kind of hyperbola. But the general property of the directrix, which belongs to all the conic sections, or the lines of the second order, affords the readiest mode of investigation. Let the arc BD be the third part of BDC. Complete the circle, and draw the chords BD, BC, and CD. The arc CD is evidently double of BD, and therefore the angle CBD is double of BCD. Bisect the angle CBD by the straight line BH, let fall the extended perpendicular, and draw the parallel DK. The triangle CHB is evidently isosceles, and HI bisects the base CB. But the triangle CBD having its vertical angle at B bisected, the side CB is to BD as the segment CH of the base to HD; that is, since the triangles CHI and DHK are similar, as CI to KD; therefore, CB being the double of CI, BD is likewise double of DK. The ratio of the distances BD and DK is thus given, while the point B is given, and the straight line IH given in position. Whence, from the theory of lines of the

Second Order, the locus of the point of section D is an hyperbola, of which B is a focus, and IH a directrix, with the determining ratio of two to one. Let this construction be made, and the arc CDB is trisected in D. For since BD is, from the property of the curve, double of DK, it is evident that BC is to CI as BD to DK; and the triangles CIH and KDH being similar, and CI to CH as DK to DH, it follows that BC is to CH as BD to DH, or alternately BC is to BD as CH to DH. Therefore the vertical angle CBD is bisected by BH, or the angle CBD is double of CBH or of BCD, and consequently the arc CD is double of BD, or BD itself is the third part of the whole arc CDB.

But the opposite branch of the hyperbola, which passes through C, also comes into play; and the intersection of these two branches with the circle assigns three different positions of the point D, separated from each other by intervals equal to the third of the whole circumference. Thus, in the second position D', produce the perpendicular DK' to the opposite circumference L; and since BD' is double of D'K', it must be equal to the chord DL, and consequently the arc BDCD' is equal to LD'MD'. Wherefore the double of BDCD', together with the interval BL or CD', is equal to the whole circumference; that is, the double of DCD', with the double of BD and CD', is equal to the whole circumference; and since the double of BD is DC, the triple of the arc DCD' must complete the circumference. In the third position D", produce the perpendicular D"K" as before; the double of this, or the chord MD", is hence equal to D'B, and the arc BLD" equal to DM; consequently the double of BLD", with the compound arc BDCM, completes the circumference; but D'M being parallel to BC, the arc BLD" is equal to CDM, and therefore three times the arc BLD", with the arc BDC, or the triple of BD, will fill up the circumference, or the arc DBD" is a third part of it.

The trisection of innumerable arcs described on the same chord is rendered very conspicuous by combining the separate branches of two hyperbolas that have the determining ratio of two to one, and their foci situate in the extremities of the given line. Thus, let the chord BC be trisected at the points N and O, and from B and C, as distinct foci, and in the determining ratio of two to one, describe branches of independent hyperbolas. All the arcs erected on BC are each of them divided by those curves into three equal portions. These arcs, as they flatten, approach to the trisected chord CNOB on the one hand; and as they become enlarged, they constantly tend, on the other, to the complete circumference which the asymptotes of the hyperbola, making angles on each side of the axis equal to two thirds of a right angle, would themselves trisect.

It may be observed in general, that the section of an arc or angle admits of as many different answers as the number of divisions proposed. Thus, the quadrisection would give four distinct results, and a quintisection would involve no fewer than five separate products. Nay, the bisection itself of an arc, though within the limits of the most elementary geometry, yet brings out a double result. Thus, the arc CDB is bisected by the perpendicular or diameter DID', which not only gives BD for the half of that arc, but also BDCD', or the same half-arc augmented by a semicircumference.

These conclusions agree with the results derived from the general theory of equations. The expression for the sine of a multiple arc is always an equation of corresponding dimensions, which therefore admits of as many distinct roots as the index contains units.

The ancient geometers were only acquainted with the original division of the circumference into two, three, or five equal portions. The only subdivisions were obtained from the differences of those arcs or their continued bisection. But the very ingenious Professor Gauss has discovered a series of more complex regular polygons, which may be inscribed in a circle by elementary geometry. The expression $2^n + 1$, when a prime number, will represent the sides of the figure: for the general equation of a cosine of the section can be decomposed into quadratics of the simplest kind, which can be constructed by the repeated application of circles and straight lines. A polygon of 17 sides is the first that occurs after the pentagon; and then follow the polygons of 257, 65537, &c.