Degree (angle)

“ 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. ”

“ 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°. ”

“ 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. ”

“ Any two interior opposite angles are equal to two right angles.
Let the straight line e f intersect the two parallel straight lines a b, c d, in the points m and n.
Then will any two interior opposite angles be equal to two right angles.
For the interior alternate angles Yellow, Red, are equal.
If to the angle Green we add the angle Yellow, we shall have the adjacent angles Green, Yellow.
If to the same angle Green we add the equal angle Red, we shall have the interior opposite angles Green, Red. ”

“ A few illustrations, as of an open book, the wall and floor of a room, and a swinging door, serve to make the concept clear, while a plane at right angles to the edge shows the measuring plane angle. So manifest is this [Pg 299] relationship between the dihedral angle and its measuring plane angle that some teachers omit the proposition that two dihedral angles have the same ratio as their plane angles.
Theorem. If two planes are perpendicular to each other, a straight line drawn in one of them perpendicular to their intersection is perpendicular to the other. ”

“ If we extend these lines until they intersect, we will have the required triangle, and can measure the two sides, which will be found to be about 12 inches and 8 inches long, and the third angle will measure just 26 degrees. It doesn’t make any difference on what scale we draw the triangle, whether it be miles, yards, feet, inches or fractions of an inch, the proportions will be the same. If the base line had been 6 half-inches, or 3 inches long, and the same angles were used, the other two lines would measure 12 half-inches, or six inches, and 8 half-inches, or 4 inches. ”

“ 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. ”

“ We may in fact take, as a varied curve, the arc AQ of a neighbouring stationary curve, the straight line QP drawn from Q to a point of the arc AB, and the arc PB of the stationary curve AB (fig. 6) . The sign of the variation is then the same as that of the function E (x, y, tan φ, tan ψ) , where (x, y) is the point P, ψ is the angle which the straight line QP makes with the axis of x, and φ is the angle which the tangent at P to the curve APB makes with the same axis. We thus arrive at a new necessary (but not sufficient) condition for the existence of an extremum of the integral ∫Fds, viz. ”

“ 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. ”

“ 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. ”

“ How many degrees in the arc f h? In e h?
If the arc f h were divided into three equal parts, how many degrees would there be in each?
How many degrees in a circumference?
In a semi-circumference?
How many more degrees in a large circumference than in a small one?
If the arc i f is 40°, what is the size of the angle f o i?
If the angle f o g is an angle of 130°, what is the size of the arc f i h g?
How many degrees in each of the adjacent angles f o h, h o e?
When two adjacent angles are equal to each other, what is each called?
How many degrees in a right angle?
82
Diagram 26. ”

“ 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 ”

“ 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 . ”

“ In the case of a quadrantal triangle, rule 1 above is used, and rule 2 restated: angle C (the angle opposite the side of 90°) is more than 90° when A and B are in the same quadrant, and less than 90° when A and B are in different quadrants. If the rule requires an angle of more than 90° and the solution produces an angle of less than 90°, subtract the solved angle from 180°. ”

“ 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 ”

“ 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. ”

“ If we divide a square into equal angles we take the nearer horizontal ones to be larger, so that we often take an angle of thirty degrees to be forty-five. Habit has much influence here. It will hardly be believed, and certainly is not consciously known, that in the letter S the upper curve has a definitely smaller radius than the lower one ”

“ With a square and a compass it is a comparatively easy matter accurately to step off 45 degrees, as it is the line C, midway between A and B, and the other degrees may be calculated from the line C and the cardinal lines A or B.
Degrees Without a Compass.—But in the absence of a compass and when you do not wish to step off a circle, you will in such case lay down the square, and mark off at the outer margin of the limbs two equal dimensions. Suppose we take 2 inches on each limb of the square. The angle thus formed by the angle square blade is 45 degrees. ”

“ Because, by the grant, the right line ae, is cut proportionally, therefore the Oblong of the secant and outter segment, is equall to the quadrate of the greater segment, to which by the grant, the base ei, is equall. Here therefore the angle aie is the double of the angle at a: because it is equall to the angles aio, and oai, which are equall betweene themselves. For by the 27 e xvj it is equall to the angle oai in the alterne segment. And the remainder aio, is equall to it selfe. Therefore also the angle aei, is equall to the same two angles, because it is equall to the angle aie. ”

“ As here, when the diameter is continued it is perpendicular, and maketh twice two angles, within and without, the outter equall betweene themselves, and inner equall betweene themselves: But the outer unequall to the inner: And the angle of a greater semicircle is greater, then the angle of a lesser. Neither is this affection any way reciprocall, That all equall angles should bee right angles. For oblique angles may bee equall betweene themselves: And an oblique angle may bee made equall to a right angle, as a Lunular to a rectilineall right angle, as was manifest, at the 6 e. ”

“ 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. ”

“ 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.] ”

“ On AC the diagonal of a square ABCD, a rhombus AEFC is described of the same area as the square, and having its acute angle at A: if AF be joined, shew that the angle BAC is, divided into three equal angles. ”

“ These are right angles if they are equal, and they will be equal if one line is perpendicular to the line it crosses. If the angle be less than a right angle, it is called an acute angle; if more, it is called an obtuse angle. Acute means sharp, and obtuse means blunt. ”

“ For if the cantle wherein the angle is made, be greater then the halfe circle, then is that angle a sharpe angle. And generally the greater the cãtle is, the lesser is the angle comprised in that cantle: and contrary waies, the lesser any cantle is, the greater is the angle that is made in it. Wherfore it must nedes folowe, that the angle made in a cantle lesse then a semicircle, must nedes be greater then a right angle. So the angle by B, beyng made at the right line A.B, and the righte line B.D, is a iuste righte angle, because it is made in a semicircle. ”

“ A radius drawn from the centre to the number 30 on the graduated semicircle, will form an angle of 30 degrees with the diameter of the semicircle. And so for any other number of degrees. It will be seen that any number of degrees less than 90 will make an acute angle, and more than 90 degrees will form an obtuse angle thus, in fig. 4, 30 degrees form an acute angle, and the remaining 150 degrees of the half circle form an obtuse angle. ”

“ The new height will be the sine of the new angle , or, stating it as an equation,
;
and subtracting from this the first equation gives
.
The quantity on the right-hand side is the difference between two sines, and books on trigonometry tell us how to work this out. For they tell us that if and are two different angles,
.
If, then, we put for one angle, and for the other, we may write
But if we regard as indefinitely small, then in the limit we may neglect by comparison with , and may also take as being the same as . ”