ANGLE
, Angulus, in Geometry, the opening or mutual inclination of two lines, or two planes, or three planes, meeting in a point called the vertex or angular point. Such as the angle formed by, or between, the two lines AB and AC, at the vertex or angular point A.—Also the two lines AB and AC, are called the legs or the sides of the angle.
Angles are sometimes denoted, or named, by the single letter placed at the angular point, as the angle A; and sometimes by three letters, placing always that of the vertex in the middle. The former method is used when only one angle has the same vertex; and the latter method it is necessary to use when several angles have the same vertex, to distinguish them from one another.
The measure of an angle, by which its quantity or magnitude is expressed, is an arch, as DE described from the centre A, with any radius at pleasure, and contained between its legs AB and AC.—Hence angles are compared and distinguished by the ratio of the arcs which subtend them, to the whole circumference of the circle; or by the number of degrees contained in the arc DE by which they are measured, to 360, the number of degrees in the whole circumference of the circle. And thus an angle is said to be of so many degrees, viz, as are contained in the arc DE.
Hence it matters not, with what radius the arc is described, by which an angle is measured, when great or small, as AD, or Ad, or any other: for the arcs DE, de, being similar, have the same ratio as their respective radii or circumferences, and therefore they contain the same number of degrees.—Hence it follows, that the quantity or magnitude of the angle remains still the same, though the legs be ever so much increased or diminished.—And thus, in similar figures, the like or corresponding angles are equal.
The taking or measuring of angles, is an operation of great use and extent in surveying, navigation, geography, astronomy, &c. And the instruments chiefly used for this purpose, are quadrants, sextants, octants, theodolites, circumferentors, &c. Mr. Hadley invented an excellent instrument for taking the larger sort of angles, where much accuracy is required, or where the motion of the object, or any circumstance causing an unsteadiness in the common instruments, renders the observations difficult, or uncertain. And Mr. Dollond contrived an instrument for measuring small angles. See Hadley's Quadrant, Micrometer, and the Philos. Trans. Numbers 420, 425, and vol. 48.
1. On paper. Apply the centre of a protractor to the vertex A of the angle, so that the radius may coincide with one leg, as AB; then the degree on the arch that is cut by the other leg AC, will give the measure of the angle required.
Or thus, by a line of chords. Take off the chord of 60 with a pair of compasses; and with that radius, from the centre A, describe an arc as DE. Then take this arc DE between the compasses, and apply the extent to the scale of chords, which will give the degrees in the angle as before.
M. De Lagny gave, in several memoirs of the Royal Academy of Sciences, a new method of measuring angles, which he called Goniometry. The method consists in measuring, with a pair of compasses, the are which subtends the proposed angle, not by applying its extent to a pre-constructed scale, like chords, but in the following manner: From the angular point as a centre, with a pretty large radius, describe a circle, producing one leg of the angle backwards to cut off a semicircle; then search out what part of the semicircle the arc is which measures the given angle, in this manner; viz, take the extent of this arc with a very fine pair of compasses, and apply it several times to the arc of the semicircle, to find how often it is contained, with a small part remaining over; in the same manner take the extent of this small part, and apply it to the first arc, to find how often it is contained in it; and what remains this 2d time, apply in like manner to the first remainder; then the 3d remainder apply to the 2d, and so on, always counting how often the last remainder is contained in the next foregoing, till nothing remain, or till the remainder is insensible, and too small to be measured: Then, beginning at the last, and returning backwards, make a series of fractions of which the numerators are always 1, and the denominators are the number of times each remainder is contained in its next remainder, with the fractional part more, as derived from the following remainder; then the last fraction, thus obtained, will shew what part the given angle is of 180° or the semicircle; and being turned into degrees &c, will be the measure of the angle, and nearer, it is asserted, than it can be obtained by any other means; whether it be measuring, or calculating by trigonometrical tables.— Thus, if it be required to measure the angle GFH: With a large radius describe the semicircle GHI, meeting the leg FG produced in I; then take the extent of the arc GH in the compasses, and applying it from G upon the semicircle, suppose it contains 4 times to the point 4, and the part 4 I over; take 4 I and apply it from H to 1, so that HG contains 4 I once, and 1 G over; also apply this remainder to the former 4 I, and it contains 5 times, from 4 to 5, and 5 I over; | and lastly the remainder 5 I is just two times contained in the former remainder 1 G or 12, without any remaindor. Here then, the series of quotients, or numbers of times contained, are 4, 1, 5, 2; therefore, beginning at the last, the first fraction is 1/2, or the last remainder is half the preceding one; and the 2d fraction is 1/(5 1/2) or 2/11; the 3d is 1/(1 2/11) or 11/13; and the fourth is 1/(4 11/13) or 13/63; that is, the arc GH is 13/63 of a semicircle, or the angle GFH is 13/63 of two right angles, or of 180°, which is equivalent to 37 1/7 degrees, or 37° 8′ 34″ 2/7.
2. On the ground. Place a surveying instrument with its centre over the angular point to be measured, turning the instrument about till 0, the beginning of its arch, fall in the line or direction of one leg of the angle; then turn the index about to the direction of the other leg, and it will cut off from the arch the degrees answering to the given angle.
To plot or lay down any given angle, either on paper or on the ground, may be performed in the same manner; and the method is farther explained under the articles Plotting and Protracting, and under the names of the several instruments.
To bisect a given angle, as suppose the angle LKM. From the centre K, with any radius, describe the arc LM; then with the centres L and M, describe two arcs intersecting in N; and draw the line KN, which will bisect the given angle LKM, dividing it into the two equal angles LKN, MKN.
To trisect an angle, see Trisection.
Pappus, in his Mathematical Collections, book 4, treats of angular sections, but particularly and more largely, of trisections. He also treats of any section in general, in the 36th and following propositions.
Angles are of various kinds and denominations. With regard to the form of their legs, they are divided into rectilinear, curvilinear, and mixed.
Rectilinear, or right-lined Angle, is that whose legs are both right lines; as the foregoing angle CAB.
Curvilinear Angle, is that whose legs are both of them curves.
Mixt, or mixtilinear Angle, is that of which one leg is a right line, and the other a curve.
With regard to their magnitude, angles are again divided into right and oblique, acute and obtuse.
Right Angle, is that which is formed by one line perpendicular to another; or that which is subtended by a quadrant of a circle. As the angle BAC.—Therefore the measure of a right angle is a quadrant of a circle, or 90°; and consequently all right angles are equal to each other.
Oblique Angle, is a common name for any angle that is not a right one; and it is either acute or obtuse.
Acute Angle, is that which is less than a righ<*> angle, or less than 90 degrees; as the angle BAD And
Obtuse Angle, is greater than a right angle, or whos<*> measure exceeds 90 degrees; as the angle BAE.
With regard to their situation in respect of eac<*> other, angles are distinguished into contiguous, adjacen<*> vertical, opposite, and alternate.
Contiguous Angles, are such as have the same vertex<*> and one leg common to both. As the angles BAD CAD, which have AD common.
Adjacent Angles, are those of which a leg of th<*> one produced forms a leg of the other: as the angle<*> GFH and IFH, which have the legs IF and FG in <*> straight line.—Hence adjacent angles are supplement<*> to each other, making together 180 degrees. An<*> therefore if one of these be given, the other will b<*> known by subtracting the given one from 180 degrees. Which property is useful in surveying, to find the quantity of an inaccessible angle; viz, measure its adjacent accessible one, and subtract this from 180 degrees.
Vertical or opposite Angles, are such as have their legs mutually continuations of each other; as the two angles a and b, or c and d.—The property of these is, that the vertical or opposite angles are always equal to each other, viz, [angle] a = [angle] b, and [angle] c = [angle] d. And hence the quantity of an inaccessible angle of a field, &c, may be found, by measuring its accessible opposite angle.
Alternate Angles, are those made on the opposite sides of a line cutting two parallel lines; so, the angles e and f, or g and h, are alternates. And these are always equal to each other; viz, the [angle] c = [angle] f, or [angle] g = [angle] h.
External Angles, are the angles of a figure made without it, by producing its sides outwards; as the angles i, k, l, m. All the external angles of any rightlined figure, taken together, are equal to 4 right angles; and the external angle of a triangle is equal to both the internal opposite ones taken together; also any external angle of a trapezium inseribed in a circle, is equal to the internal opposite angle.
Internal Angles, are the angles within any figure, made by the sides of it; as the angles n, o, p, q.—In any right-lined figure, an internal angle as n, and its adjacent external angle k, together make two right angles, or 180 degrees; and all the internal angles n, o, p, q, | taken together, make twice as many right angles, wanting 4 right angles; also any two opposite internal angles of a trapezium inscribed in a circle, taken together, make two right angles, or 180 degrees.
Homologous, or like Angles, are such angles in two figures, as retain the same order from the first, in both figures.
Angle out of the centre, as G, is one whose vertex is not in the centre of the circle.—And its measure is half the sum (a+b)/2 of the arcs intercepted by its legs when it is within the circle, or half the difference (a-b)/2 when it is without.
Angle at the centre, is an angle whose vertex is in the centre; as the angle AFC, formed by two radii AF, FC, and measured by the arc ADC.—An angle at the centre, as AFC, is always double of the angle ABC at the circumference, standing upon the same arc ADC; and all angles at the centre are equal that stand upon the same or equal arcs: also all angles at the centre, are proportional to the arcs they stand upon; and so also are all angles at the circumference.
Angle at the circumference, is an angle whose vertex is somewhere in the circumference of a circle; as the angle ABC.
Angle in a segment, is an angle whose legs meet the extremities of the base of the segment, and its vertex is anywhere in its arch; as the angle B is in the segment ABC, or standing upon the supplemental segment ADC; and is comprehended between two chords AB and BC.—An angle at the circumference is measured by half the arc ADC upon which it stands; and all the angles ABC, AEC, in the same segment, are equal to each other.
Angle in a semicircle is an angle at the circumference contained in a semicircle, or standing upon a semicircle, or on a diameter.—An angle in a semicircle, is always a right angle; in a greater segment, the angle is less, and in a less segment the angle is greater than a right angle.
Angle of a segment, is that made by a chord with a tangent, at the point of contact. So IHK is the angle of the less segment IMH, and IHL, the angle of the greater segment INH.—And the measure of each of these angles, is half the alternate or supplemental segment, or equal to the angle in it; viz, the [angle] IHK = [angle] INH, and the [angle] IHL = [angle] IMH.
Angle of a semicircle, is the angle which the diameter of a circle makes with the circumference. And Euclid demonstrates that this is less than a right angle, but greater than any acute angle.
Angle of contact, is that made by a curve line and a tangent to it, at the point of contact; as the angle IHK. It is proved by Euclid, that the angle of contact between a right line and a circle, is less than any right-lined angle whatever; though it does not therefore follow that it is of no magnitude or quantity. This has been the subject of great disputes amongst geometricians, in which Peletarius, Clavius, Taquet, Wallis, &c, bore a considerable share; Peletarius and Wallis contending that it is no angle at all, against Clavius, who rightly asserts that it is not absolutely nothing in itself, but only of no magnitude in comparison with a right-lined angle, being a quantity of a different kind or nature; like as a line in respect to a surface, or a surface in respect to a solid, &c. And since his time, it has been proved by Sir I. Newton, and others, that angles of contact can be compared to each other, though not to right-lined angles, and what are the proportions which they bear to each other. Thus, the circular angles of contact IHK, IHL, are to each other in the reciprocal subduplicate ratio of the diameters HM, HN. And hence the circular angle of contact may be divided, by describing intermediate circles, into any number of parts, and in any proportion. And if, instead of circles, the curves be parabolas, and the point of contact H the common vertex of their axes; the angles of contact would then be reciprocally in the subduplicate ratio of their parameters. But in such elliptical and hyperbolical angles of contact, these will be reciprocally in the subduplicate of the ratio compounded of the ratios of the parameters, and the transverse axes. Moreover, if TOQ be a common parabola, to the axis OP, and tangent VOW, and whose equation is , or x=y2, where x is the absciss OP, and y the ordinate PQ, the parameter being 1<*> and if OR, OS, &c, be other parabolas to the same axis, tangent, and parameter, their ordinate y being PR, or PS, &c, and their equations x=y3, x=y4, x=y5, &c: then the series of angles of contact will be in succession infinitely greater than each other, viz, the angle of contact WOQ infinitely greater than WOR, and this infinitely greater than WOS, and so on infinitely.
And farther, between the angles of contact of any two of this kind, may other angles of contact be found ad infinitum, which shall infinitely exceed each other, | and yet the greatest of them be infinitely less than the smallest right-lined angle. So also x2=y3, x3=y4, x4=y5, &c, denote a series of curves, of which every succeeding one makes an angle with its tangent, infinitely greater than the preceding one; and the least of these, viz, that whose equation is x2=y3, or the semicubical parabola, is infinitely greater than any circular angle of contact.
Angles are again divided into plane, spherical, and solid.
Plane Angles, are all those above treated of; which are defined by the inclination of two lines in a plane, meeting in a point.
Spherical Angle, is an angle formed on the surface of a sphere by the intersection of two great circles; or, it is the inclination of the planes of the two great circles.
The measure of a spherical angle, is the arc of a great circle of the sphere, intercepted between the two planes which form the angle, and which cuts the said planes at right angles. For their properties, &c, see Sphere, Spherical, and Spherical Trigonometry.
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; like the angles or corners of solid bodies. For their measure, properties, &c, see Solid Angle.
Angles of other less usual kinds and denominations, are also to be found in some books of Geometry. As,
Horned Angle, angulus cornutus, that which is made by a right line, whether a tangent or secant, with the circumserence of a circle.
Lunular Angle, angulus lunularis, is that which is formed by the intersection of two curve lines, the one concave, and the other convex.
Cissoid Angle, angulus cissoides, the inner angle made by two spherical convex lines intersecting each other.
Sistroid Angle, angulus sistroides, is that which is in form of a sistrum.
Pelecoid Angle, angulus pelecoides, is that in form of a hatchet.
Angle, in Trigonometry. See Triangle, TRICONOMETRY, Sine, Tangent, &c.
Angle, in Mechanics.—Angle of Direction, is that which is comprehended between the lines of direction of two conspiring forces.
Angle of Elevation, is that which is comprehended between the line of direction and any plane upon which the projection is made, whether horizontal or oblique.
Angle of Incidence, is that made by the line of direction of an impinging body, at the point of impact. As the angle ABC.
Angle of Reflection, is that made by the line of direction of the reflected body, at the point of impact. As the angle DBE.
Instead of the angles of incidence and reflection being estimated from the plane on which the body impinges, sometimes the complements of these are understood, viz, as estimated from a perpendicular to the reflecting plane; as the two angles ABF and DBF.
Angle in Optics.—Visual or Optic Angle, is the angle included between the two rays drawn from the two extreme points of an object to the centre of the pupil of the eye: as the angle HGI. The apparent magnitude of objects is greater or less, according to the angle under which they appear.—Objects seen under the same or an equal angle, always appear equal—.The least visible angle, or least angle under which a body can be seen, according to Dr. Hook, is one minute; but Dr. Jurin shews, that at the time of his debate with Hevelius on this subject, the latter could probably discover a single star under so small an angle as 20″. But bodies are visible under smaller angles as they are more bright or luminous. Dr. Jurin states the grounds of this controversy, and discusses the question at large, in his Essay upon distinct and indistinct Vision, published in Smith's Optics, pa. 148, & seq.
Angle of the interval, of two places, is the angle subtended by two lines directed from the eye to those places.
Angle of incidence, or reflection, or refraction, &c. See the respective words Incidence, Reflection, Refraction, &c.
Angle in Astronomy.—Angle of Commutation. See Commutation.
Angle of elongation, or Angle at the Earth. See Elongation.
Parallactic Angle, or the parallax, is the angle made at the centre of a star, the sun, &c, by two lines drawn, the one to the centre of the earth, and the other to its surface. See Parallactic, and Parallai.
Angle of the position of the sun, of the sun's apparent semi-diameter, &c. See the respective words.
Angle at the sun, is the angle under which the distance of a planet from the ecliptic, is seen from the sun.
Angle of the East. See Nonagesimal.
Angle of obliquity, of the ecliptic, or the angle of inclination of the axis of the earth, to the axis of the ecliptic, is now nearly 23° 28′. See Obliquity, and Ecliptic.
Angle of longitude, is the angle which the circle of a star's longitude makes with the meridian, at the pole of the ecliptic.
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 equator.
Angle in Navigation. Angle of the rhumb, or loxodromic angle. See Rhumb and Loxodromic.
Angles, in Fortification, are understood of those formed by the several lines used in fortifying, or making a place defensible.
These are of two sorts; real and imaginary.—Real angles are those which actually exist and appear in the works. Such as the flanked angle, the angle of the epaule, angle of the flank, and the re-entering angle of the counterscarp. Imaginary, or occult angles, are those which are only subservient to the construction, and which exist no more after the fortification is drawn. Such as the angle of the centre, angle of the polygan, flanking angle, sallant angle of the counterscarp, &c.
Angle of, or at, the centre, is the angle formed at the centre of the polygon, by two radii drawn from the centre to two adjacent angles, and subtended by a side | of it, as the angle ACB. This is found by dividing 360 degrees by the number of sides in the regular polygon.
Angle of the Polygon, is the angle intercepted between two sides of the polygon; as DAB, or ABE. This is the supplement of the angle at the centre, and is therefore found by subtracting the angle C from 380 degrees.
Angle of the Triangle, is half the angle of the polygon; as CAB or CBA; and is therefore half the supplement of the angle C at the centre.
Angle of the Bastion, is the angle FAG made by the two faces of the bastion. And is otherwise called the flanked angle.
Diminisbed Angle, is the angle BAG made by the meeting of the exterior side of the polygon with the face AG of the bastion.
Angle of the curtin, or of the flank, is the angle GHI made between the curtin and the flank.
Angle of the epaule, or shoulder, is the angle AGH made by the flank and the face of the bastion.
Angle of the tenaille, or exterior flanking angle, is the angle AKB made by the two rasant lines of defence, or the faces of two bastions produced.
Angle of the counterscarp, is the angle made by the two sides of the counterscarp, meeting before the middle of the curtin.
Angle, flanking inward, is the angle made by the flanking line with the curtin.
Angle forming the flank, is that consisting of one flank and one demigorge.
Angle forming the face, is that composed of one flank and one face.
Angle of the moat, is that made before the curtin, where it is intersected.
Re-entering, or re-entrant Angle, is that whose vertex is turned inwards, towards the place; as H or I.
Saliant, or sortant Angle, is that turned outwards, advancing its point towards the field; as A or G.
Dead Angle, is a re-entering angle, which is not flanked or defended.
Angle of a wall, in Architecture, is the point or corner where the two sides or faces of a wall meet.
Angles, in Astrology, denote certain houses of a figure, or scheme of the heavens. So the horoscope of the first house, is termed the angle of the east.
ANGUINEAL Hyperbola, a name given by Sir I. Newton to four of his curves of the second order, viz, species 33, 34, 35, 36, expressed by the equation ; being hyperbolas of a serpentine figure. See Curves.
