ANGLE. Without entering into the science of Geometry, which does not come within the scope of the present work, a few practical de tails concerning angles may be useful.
An angle is the opening of two lines ; rectilinear, of two straight lines ; curvilinear, of two curves ; mixtilincar, of a straight line and a curve. But in truth angle always means rectilinear angle; and when a curve enters, its tangent is the straight line which is used in determining the angle.
Aright angle. is half the opening of a straight line and its continuation : an acute angle is that which is less than aright angle; an obtuse angle is that which lies between one and two right angles. Complemental angles are two which together make a right angle ; supplemen tal angles are two which together make two right angles. When lines meet and make a pair of angles, the one less than two right angles is called salient ; the other, greater than two right angles, is called re-entrant or re-entering. For salient and re-entering (which are borrowed from fortification) direct and retrylected have sometimes been used.
The angles which two lines make with the same part of a third, on opposite sides of it, are called alternate. Two lines which cross one another make two pair of vertically oppo site angles. The angles made by adjacent sides of a figure are called internal ; those made by any sides with adjacent sides pro duced are external. When the angular point is the centre or on the circumference of a circle, the angle is said to be at the centre, or at the circumference. Beginners often confound the angle with the angular point.
A spherical angle is made by two -circles (usually great circles) of a sphere. When the circles meet at the pole of the equator, and one of them is the meridian, the angle is an horary or hour-angle ; and when neither is the meri dian, the angle is frequently called horary. The angle of position of a star is that made by the circles drawn to it from the poles of the equator and ecliptic. The angle of eleva tion is the angle made by a line drawn from the eye to any object with the horizontal line which is in the same vertical plane as the first line ; but when the object is below the horizon, the term is angle of depression. When lines are drawn from two points to a third, these two points, and also the line joining them, are said to subtend the angle which is made at the third point. The angle which two objects
subtend at the eye is their angle of elongation. The angle of the vertical is a name given to the angle which a line drawn to the spectator's zenith makes with the radius of the earth pro duced: it is taken as nothing when the earth is supposed to be a perfect sphere.
When one line falls upon another, the angle of incidence is the acute angle which the inci dent line makes with the perpendicular to the other. When the incident line is thrown off again on the same side as that from which it came, the acute angle made with the perpen dicular is called the angle of reflexly» ; when on the opposite side, the angle of refraction. These terms are nearly confined to optics.
A dihedral angle is the opening made by two Planes : it is measured by a rectilinear angle, namely, that made by two lines drawn in the two planes perpendicular to their common in tersection. A solid angle is said to exist when three or more straight lines, not in the same plane, meet at a point.
In measuring angles, a circle is considered as the curved boundary of four right angles. This is divided into 360 degrees (°), each of which is divided into 60 minutes ('), and each minute into 60 seconds ("). Formerly, the second was divided into 60 equal parts called thirds, and so on ; but it is now customary to use the tenths, hundredths, &c., of seconds. The present table therefore stands thus : A whole revolution A right angle = 90°= 5400`= 324000"1 Degrees. Minutes. Seconds.
1 = 00 = 3600 1 = 60 In the attempt to effect a universal change of weights and measures which followed the French Revolution, the circle was divided into 400 degrees, each degree into 100 minutes, each minute into 100 seconds, and so on. This innovation obtained only a partial introduction, and is now entirely abandoned. When used, it is customary in this country to distinguish the French degrees by the name of grades, and to denote one grade by lsr. The convenience of this method, from its close affinity with the decimal system, is certainly great : for exam ple, grades and decimals of grades, such as 12e' 1329 are converted into grades, minutes, and seconds, by mere separation of the figures : thus, 13' 29".
lgr• is or 54' or 3240" 1' „ 0 •009 „ 0'•51 „ 32"•1 1" „ 0 •00009 „ „ 0"•324