Angle (geometry)

“ Curiously enough the generalized, or trigonometrical, definition of an angle is found not under the word angle at all, but from a cross reference to "Mathematical angle." It is only at this place that the essential quality of an angle as a magnitude is given.
The word congruent, for some unknown reason is not given the meaning applied to it in foreign and recent American text-books on elementary geometry.
Under "Parallel lines" we find "lying evenly everywhere in the same direction, however far extended; in all parts equally distant." This is said to be the Euclid idea of the term! ”

“ Now, the most elementary geometry teaches us that every triangle having these angles will have the lengths of its sides in the same invariable proportions one to another whatever may be the size of the triangle itself, and it has been found convenient to calculate these proportions once for all, not merely for angle 20°, but for every angle up to 90°. ”

“ It should also be observed that in the degraded circumstances of ordinary geometry it would be impracticable to employ the same constant c for the purpose of both linear and angular measurement.
It is easy to see that the definition of a right angle in the generalized sense is embodied in the statement that “if two corresponding legs of an harmonic pencil touch the fundamental conic then the two other legs are at right angles.” We also see that all the perpendiculars to a given line pass through a point, i.e., the pole of the given line ”

“ 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 ”

“ 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. ”

“ Similarly, if you draw the circle FGI from the center C, the amount of degrees in arc FG from circle FGI is the same as the degrees in vertex ABC according to arc DE from circle DEH. In the same manner, whatever the distance of the arc that lies between the two lines, it remains in proportion to the circumference, and the angle is the same, regardless whether the length of the lines are in cubits, handbreaths, or finger measurements.
Section 10 A right angle is one that contains 90 degrees and whose edges are perpendicular, not leaning towards either side. ”

“ 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.] ”

“ Any angle is easily trisected by means of certain higher curves, such as the conchoid of Nicomedes (ca. 180 B.C.) , the quadratrix of Hippias of Elis (ca. 420 B.C.) , or the spiral of Archimedes (ca. 250 B.C.) . But since this problem, stated algebraically, requires the solution of a cubic equation, and this involves, geometrically, finding three points, we cannot solve the problem by means of straight lines and circles alone. In other words, the trisection of any angle, by the use of the straightedge and compasses alone, is impossible. Special angles may however be trisected. ”

“ 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. ”

“ Teachers will naturally see that just as we may measure the plane angle by taking the ratio of an arc to the whole circle, and of a dihedral angle by taking the ratio of that part of the cylindric surface that is cut out by the planes to the whole surface, so we may measure a polyhedral angle by taking the ratio of the spherical polygon to the whole spherical surface. ”

“ Therefore, if we could easily measure ∠O and could measure the distance x, we could find the height of a [Pg 236] building y. In trigonometry we use a transit for measuring angles, but it is easy to measure them with sufficient accuracy for illustrative purposes by placing an ordinary paper protractor upon something level, so that the center comes at the edge, and then sighting along a ruler held against it, so as to find the angle of elevation of a building. ”

“ Connect the ends of the two lines by a straight line.
What kind of an angle is that opposite the last line drawn?
Are any two of its sides equal?
Write its two names under it.
Draw a horizontal straight line of any length.
At one end draw a vertical line of equal length.
Complete the triangle, and write two names inside.
Draw a right-angled triangle whose base is of any length, and its perpendicular twice as long.
Draw a right-angled triangle whose base is three times as long as any short measure, and its perpendicular five times as long as the same measure or unit.
54
Diagram 20. ”

“ 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. ”

“ Neither may a surface, of 2. dimensions, be bounded with [23] one right line: Nor a body, of three dimensions, bee bounded with two, at lest beeing plaine surfaces.
4. The shankes of an angle are the bounds compreding the angle.
Scèle or Crura, the Shankes, Legges, H. are the bounds insisting or standing upon the base of the angle, which in the Isosceles only or Equicrurall triangle are so named of Euclide, otherwise he nameth them Latera, sides. So in the examples aforesaid, ea. and ei. are the shankes of the superficiary angle e ”

“ The decimal system measurement of angles has never come into ordinary use. In analytical trigonometry the circular measure of an angle is employed. In this system the unit angle is the angle subtended at the centre of a circle by an arc equal in length to the radius. ”

“ 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. ”

“ What we have to investigate is the value of ; or, in other words, if the angle varies, we have to find the relation between the increment of the sine and the increment of the angle, both increments being indefinitely small in themselves. Examine Fig. 43, wherein, if the radius of the circle is unity, the height of is the sine, and is the angle. Now, if is supposed to increase by the addition to it of the small angle –an element of angle–the height of , the sine, will be increased by a small element . ”

“ Now the values of the sine, cosine, and tangent (functions of the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining them [Pg 235] by actual measurement, using the protractor and squared paper. Some of those needed for such angles as a pupil in geometry is likely to use are as follows:
It will of course be understood that the values are correct only to the nearest thousandth. ”

“ Then letting fall on the axis of the ellipsis the ordinate PR, from the proportion of the diameters of the ellipsis, the ordinate RQ of the circumscribed circle AQB will be given; which ordinate is the sine of the angle AOQ supposing AO to be the radius, and also cuts the ellipsis in P. It will be sufficient if that angle is found by a rude calculus in numbers near the truth. Suppose we also know the angle proportional to the time, that is, which is to four right angles, as the time in which the body described the arc Ap, to the time of one revolution in the ellipsis. ”

“ If, in order to find the measure of the angles of these rhombs, different angles be placed on them till their sides apparently coincide, an angle of 100°, or nearly so, appears to agree very well with the obtuse angles, so that the rhombs have apparently 100° and 80° for the measures of their angles. ”

“ The angle which one plane makes with another is called a dihedral angle.
iii. A solid angle is that which is made by more than two plane angles, in different planes, meeting in a point.
iv. The point is called the vertex of the solid angle.
v. If a solid angle be composed of three plane angles it is called a trihedral angle; if of four, a tetrahedral angle; and if of more than four, a polyhedral angle.
PROP. I.—Theorem.
One part (AB) of a right line cannot be in a plane (X) , and another part (BC) not in it.
Dem.—Since AB is in the plane X, it can be produced in it [Bk. I. Post. ii.] ”