CHAPTER III.

CIRCLES.

Construction of Circles.-We have now to consider the wellknown figure represented below (Fig. 35), called a circle. Before attempting to define it, however, let us

con

sider some of the ways in which it may be formed.

1. Take a fine thread, fix one end of it to a drawing-pin pressed into the paper on a drawingboard, and make at the other end a loop through which the point of a pencil may pass. Move the pencil round the pin, always keeping the thread stretched and the pencil-point touching

FIG. 35

B

the paper. The line thus traced out encloses a circle, and, on this account, is very often itself called a circle; but, in order to distinguish between the space enclosed and the enclosing line, the latter is termed the circumference. A part of the circumference is termed an arc.

2. Instead of the drawing-pin and pencil the two legs of a pair of compasses may be used. The fine steel point, P (Fig. 36), of the instrument must be held in one position on the paper while the marking-point is moved round it, the distance between the two points being kept constantly the same.

3. For the construction of larger circles, a rod with sliding points is employed, the two points being fixed at the required distance by screws. Fig. 37 represents an instrument of this description.

FIG. 37.

FIG. 36.

4. It is often convenient to be able to make a circle on paper without the use of instruments. This may be done by placing the hand, as in the figure (Fig. 38), touching the paper only by the nail of the little finger at o, then turning the

sheet of paper round with the other hand. The point of the pencil being

FIG. 38.

at a fixed distance from the centre, o, will describe a circle.

These different methods of construction serve to bring out the defining characteristics of the circle; namely, that:

I. It is a plane figure.

2. It is Lounded by one line, termed the circumference.

3. All points in the cir

cumference are at the same distance from a fixed point termed the centre. Equality of Circumferences.-The equality of two circumferences may be tested by super-position. With the same radius describe two circles, and place one upon the other so that the centre of the first is on the centre of the second. The two circles having the same centre, and all points in their two circumferences being at the same distance from the centre, it is evident that the two circumferences must coincide. Hence, when the radii of two circles are equal, the circumferences are equal.

To illustrate

From equal circumferences we may cut off equal arcs. this fact, suppose we have a carefully-made box with a circular lid (Fig. 39). If the lid fits the box exactly, the outer circumference of the lid and the inner circumference of the box touch one another at all points; they are therefore equal.

Mark upon the circumference of the box and the lid in Fig. 39, two points, a and b, cutting off an arc; then turn the lid round; the points a and b will come to a'b', and the arcs ab and a'b' marked upon the same circle will be equal, for one can be placed above the other, so as to coincide with it. Take away the lid, and the equal arcs will be on two circles which have the same radius but different centres. We cannot in this manner apply two circles having different radii to one another; so that we cannot take into consideration equal arcs in these circles. We may, however, treat of arcs which are the same part of the circumference in two circles of a different radius.

FIG. 39

If the points a b be joined to the centre we shall have two angles at the centre standing on equal arcs. When the arc a' b' is placed on the arc ab these two angles will coincide. Angles at the centre on equal arcs are equal.

B

C

Circumferences having different Radii.-Let us now consider twc circumferences having the same centre but different radii. Let AB, BC, CD be equal arcs on the outer circumference. Draw the lines OA, OB, OC from the centre O, and let them cut the inner circumference in the points A', A B', C', D', and it is evident that the angles at O will be equal, and also that the arcs A'B', B'C', C'D' will be equal. Whatever part A B is of its circumference that same part is A'B' of its circumference. If for instance, twelve times A B make the outer circumference, twelve times A'B' make the inner circumference. Again, the amount of turning which must be given to the radius A O to carry it to BO will be the same part of a whole revolution as the arc A B or A'B' is of a whole circumference. This fact affords a useful method of measuring angles.

FIG. 40.

Measurement of Angles.-We have seen that the magnitude of an angle is measured by the amount of rotation required to carry one of the sides to the other; hence we now see that we measure the magnitude of an angle when we find what part of a circumference, having the vertex of the angle as centre, is intercepted by the sides of the angle. Certain fractions of a circumference have special names:The 360th part of a circumference is termed a Degree. The 60th part of a Degree is termed a Minute.

The 60th part of a Minute is termed a Second.

The signs °, ', ", stand respectively for degrees, minutes, and seconds; thus, 2 degrees 5 minutes 6 seconds is written, 2° 5′ 6′′.

It may be asked why these particular numbers are chosen for the division of a circumference. The answer is, that the numbers are particularly convenient, on account of the number of parts into which they may be divided; thus, 60 contains the following factors: 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. No smaller number contains so many factors. The following numbers of degrees are contained exactly in one circumference: 1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360.

Since the angle and the arc intercepted by its sides are measured by the same numbers, the size of the angle may be also expressed in degrees.

For measuring arcs and angles in degrees, and for drawing angles when the magnitudes are given in degrees, an instrument termed a protractor or a graduated semicircle is frequently used.

The protractor is a thin semicircle of brass or transparent horn, the

FIG. 41.

circumference of which is generally divided into degrees, and (where the size permits) half-degrees. In order to find how many degrees there are in a given arc of a circle, A B (Fig. 42), its extremities are joined to the centre by the two radii O A and OB; the graduated

semicircle is then placed so that its centre coincides with the centre of the circle, and that the radius marking o° rests upon one of the

M

FIG. 42.

FIG. 43.

radii, O A, of the circle; the second radius, O B, produced if necessary, now marks upon the scale of the protractor the size A B of the arc in degrees and parts of a degree. By a similar process we may mark upon a circumference an arc of a given number of degrees.

The protractor is used in the same manner for measuring an angle. To test the correctness of a protractor, it is sufficient to use different parts of it for the measurement of the same angle, then, of course, the indication ought to be the same. Symmetry. Draw with ink upon a sheet

M

of paper any line, curved or straight; fold the sheet in two, so that the crease passes through the line drawn. The wet ink will print

The

on the other side of the crease another line similar to the first. crease in the paper forms a straight line, which is called the symmetrical axis, or line of symmetry of the whole figure; and the two parts of the figure are said to be symmetrical with regard to this axis. It is evident, therefore, that the part of the figure which is on one side of the axis is exactly equal to the part on the other side; for when the figure is folded about the axis the one part coincides with the other (Fig. 43).

Symmetrical Points.-Let M be a point in a symmetrical figure, and let M' be the corresponding point on the other side of the axis. Let the straight line M M' cut the axis in C. Now, in order that M' may fall on M when the figure is folded about the axis, it is evident that CM' must fall on CM; hence the adjacent angles at C must be equal to one another—that is, they must be right angles. Again, that M' may fall on M, C M' must be equal to CM. Hence, two points are symmetrical about an axis when they are equidistant from it and the straight line joining them is perpendicular to the axis.

Straight Lines Symmetrical about an Axis.-Two straight lines coincide throughout when the extremities of the one coincide with the extremities of the other. Hence the straight line joining two points is symmetrical to the straight line joining the corresponding points. If one of the lines meets the axis it is evident that the line which is symmetrical with it also meets the axis in the same point (Fig. 45). Since the angles which two such lines make with the axis coincide when the figure is folded about the axis, it is evident that the axis bisects the angle between two such symmetrical lines.

小

FIG. 45

FIG. 44

Arithmetical Questions.

1. How many seconds in 17°?

2. How many seconds in 19° 17'?

3. How many degrees in 5690 minutes ?

4. How many degrees in an arc which is a fifth part of the circumference?

5. How many times is an arc of 5° 37′ 30′′ contained in its circum. ference?

6. What part of the circumference is an arc of 48′′ ?

7. What fraction of an arc of 7° 5′ 50′′ is an arc of 5° 4 10"?