Angle

“ The solid angle subtended at a given point by a closed curve is measured by the area of a spherical surface whose centre is the given point and whose radius is unity, the outline of which is traced by the intersection of the radius vector with the sphere as it traces the closed curve. ”

“ A right line drawn from the vertex and turning about it in the plane of the angle, from the position of coincidence with one leg to that of coincidence with the other, is said to turn through the angle, and the angle is the greater as the quantity of turning is the greater. Again, since the line may turn from one position to the other in either of two ways, two angles are formed by two lines drawn from a point. ”

“ What is the angle of reflection?
The angle of reflection is the angle which is formed by the returning rays of light, and a line perpendicular to the reflecting surface. It is always equivalent to the angle of incidence. ”

“ A solid angle may be conceived two ways. First, for the aggregate of all the angles, which are made by the motion of a strait line, while one extreme point thereof remaining fixed, it is carried about any plain figure, in which the fixed point of the strait line is not contained. ”

“ The Visual Angle.—To examine objects carefully we usually bring them as close to the eye as possible, for the nearer to the eye the object is brought, the larger is the visual angle formed by it (see Fig. 387) , and the larger is its image upon the retina. The visual angle of an object is the angle at the eye lens between the rays that have come from the ends of the object. Consequently the more distant the object, the smaller is its visual angle. ”

“ Apollonius defined angle as "a contracting of a surface or a solid at one point under a broken line or surface." Another of the Greeks defined it as "a quantity, namely, a distance between the lines or surfaces containing it." Schotten [56] says that the definitions of angle generally fall into three groups:
a. An angle is the difference of direction between two lines that meet. This is no better than Euclid's, since "difference of direction" is as difficult to define as "inclination."
b. An angle is the amount of turning necessary to bring one side to the position of the other side. ”

“ An angle is commonly represented as a pure number, but it has not really this character; an angle can only be measured in terms of a unit of angle, just as a length is measured in terms of a unit of length. ”

“ Do you mean that the arms of the angle are rigidly connected, so that it cannot change its magnitude, or that they are merely hinged loosely together, as it were, so that it depends entirely on the relative motions of the two arms whether the angle changes its magnitude or not? ”

“ Any face angle of a trihedral angle is less than the sum, but greater than the difference, of the supplements of the other two face angles.
2. A solid angle cannot be formed of equal plane angles which are equal to the angles of a regular polygon of n sides, except in the case of n = 3, 4, or 5. ”

“ All circles, of whatever dimensions, are supposed to be divided into 360 equal parts, called degrees; the opening of an angle, being therefore a portion of a circle, must contain a certain number of degrees: the larger the angle the greater is the number of degrees, and two angles are said to be equal, when they contain an equal number of degrees. ”

“ Why is m n g an acute angle?
What is an acute angle?
Here the attention is directed first to this particular angle; then this is taken as an example of its kind, and the idea generalized by describing the class. See also page 29.
Why are the lines e f and g h said to be parallel?
When are lines said to be parallel?
Many of the questions are intended to test the vividness of the pupil’s conception. See page 29.
Also page 78. If the circumference were divided into 360 equal parts, would each arc be large or small?
Many of the questions are intended to test the attention of the pupil. ”

“ Do the angles Blue, Yellow, Red, take up all the space on the line?
89Do the angles Blue, Yellow, Green, Red, take up all the space on the line?
Is there room between any two of the angles to put in another angle?
Then are not the angles Blue, Yellow, Green, Red, equal to all the space on the line a b?
Note.—The word space, as here used, means angular space; and it is indispensable that the teacher impress this fact upon the learner.
By means of former lessons, the pupil has learned positively, that an angle is the difference between the directions of two lines ”

“ If the angle Red were smaller, and the angle Green larger, would the adjacent angles still be equal to two right angles?
Then,—
Any two adjacent angles are equal to two right angles.
If we draw the straight line c d where the edges of the square corners come together, what kind of angles will a c d, d c b, be? ”

“ Heighths, by the Perpendicular or Plumbe: And Angles, by the square.] Contrariwise, an Oblique angle it is, when the one shanke standeth so upon another, that it inclineth, or leaneth more to one side, then it doth to the other: And one angle on the one side, is greater then that on the other.
Therefore,
14. All straight-shanked right angles are equall.
[That is, they are alike, and agreeable, or they doe fill the same place; as here are aio. and eio. And yet againe on the contrary: All straight shanked equall angles, are not right-angles.] ”

“ ANGLE, a term employed in different senses, the most common and simplest use of it—to which all the rest may be referred, and by which they may be explained—being to indicate the inclination of two straight lines to each other. Suppose O F in the figure to turn about O from the position O B towards O C, like the opening out of the legs of a pair of compasses, the angle formed by O B and O F, which is distinguished as the angle B O F, measures the extent or degree of the turning or opening, being of course entirely inde pendent of the length of the lines. ”

“ So angulus solidus, a solid angle, in the common section of three surfaces at the least.
[But the learned B. Salignacus hath observed, that all angles doe not consist in the common section of the bounds, Because the touching of circles, either one another, or a rectilineal surface doth make an angle without any cutting of the bounds: And therefore he defineth it thus: Angulus est terminorum inter se invicem inclinantium concursus: An angle is the meeting of bounds, one leaning towards another.] So is aei. ”

“ Hence the ratio of the distance e h to the line h i being known, we shall be able to find, by means of a table of sines, the magnitude of the angle subtended at the eye by the object h i, which we shall find to contain only some minutes. ”

“ Thirdly, that the bases be equall: which three conditions if they doe meete, it must needes be that both the angles shall bee equall: but if one of them be wanting, of necessity againe they must be unequall.]
This shall hereafter be declared and made plaine by many and sundry practises: and therefore here we bring no example of it.
12. An angle is either right or oblique.
Thus much of the Affections of an angle; the division into his kindes followeth. An angle is either Right or Oblique: as afore, at the 4 e ij. a line was right or straight, and oblique or crooked. [29]
13. ”

“ In endeavouring, for example, to grasp the relation of the exterior angle to the two interior and opposite angles, the pupil may fail because he has not a clear knowledge of the equality of angles in connection with parallel lines. ”

“ If a right line making an angle with a right line given, and knit unto it with a base, be cut according to any rate assigned, a parallell to the base from the ends of the segments, shall cut the line given according to the rate assigned. ”

“ A line at a right angle to another is said to be perpendicular to it; thus, in Figures 39, 40, and 41, lines A are in each case perpendicular to line B, or line B is in each case perpendicular to line A.
A point is a position or location supposed to have no size, and in cases where necessary is indicated by a dot. ”

“ The perpendicularity of the one line, and the horizontality of the other, must not in any way be interfered with; in the sketch they should, in fact, meet and form a right angle, which should just be rounded off. ”

“ Suppose the inclined plane A B to turn upon a hinge at B, and to be depressed so that its angle of elevation shall be diminished, it is evident that as this angle decreases the height of the plane decreases, and its base increases. ”

“ Angle of Incidence.—Sixty degrees is safe for most solids, but for bright or polished surfaces, such as varnishes, polished metals, etc., the angle must be lessened as the degree of smoothness increases. ”

“ Because A B is a tangent, and A E a radius at the point of contact, the angle B A F is a right angle, and is measured by the semicircle A D F.
Because the angle F A D is at the circumference, it is measured by half the arc D F.
153Then the difference between the angles B A F and D A F, or B A D, must be measured by half the difference of the arcs A D F and D F, or A D;
That is, the angle B A D is measured by half the arc A D.
PROPOSITION XXIII. THEOREM. DEMONSTRATION. We wish to prove that
A tangent and chord parallel to it intercept equal arcs of the circumference. ”