14. We are now prepared to solve the following problem-having any arc given to make another equal to itBut it will be proper first to remark that the instrument used in making arcs and measuring them is called a compass or more generally compasses. Being very common we shall not describe it. If the arc B C (fig. 6) be given, F 6 and you wish to make another as D F equal to it, you first describe an indefinite arc D F with the same radius as that of B C, because they must belong to the same circle. Then take the chord B C in the compasses, and placing one foot in D move the other round till it cuts D F in F. The arc D F will be equal to B C because their chords are equal (13).

15. It often becomes necessary to compare an arc with an entire circumference or with another arc of the same circumference. For this purpose every circumference is supposed to be divided into 360 equal arcs called degrees and marked thus (0). For instance 600 is read 60 degrees. As all circumferences whether great or small, are divided into the same number of parts, it follows that a degree, which is thus made the unit of arcs, is not a fixed value, but varies for every different circle. It merely expresses the ratio of an arc, namely, to the whole circumference of which it is a part, and not to any other. As we sometimes have occasion for an unit less than a degree, each degree is divided into 60 equal parts called minutes and marked thus ('). Again each minute is divided into 60 equal parts called seconds and marked thus ("). When extreme minuteness is required the division is sometimes continued to thirds and fourths, marked thus (""), (""). As a quarter of a circumference, or, as it is generally called, a quadrant contains 90°, and as small numbers are more convenient than larger ones, it is usual in practice to refer all arcs to a quadrant, instead of an entire circumference. Thus considering a quadrant as unity, we say that a degree is a minute and a second

5400'

Angles and their Measure.

1

324000

16. Two lines A B, A C (fig. 7) which meet each F 7 other, must form an opening B A C of greater or less exThe point

tent. This opening B A C is called an angle.

of meeting A. is called the vertex of the angle, and the straight lines A B, A C are called sides. The best way to obtain a definite idea of an angle is to suppose the line A B at first to coincide with A C, and then to turn about the fixed point A in the manner of a radius (11) till it reaches its present position. Then we define an angle to be-the quantity by which a straight line, turning about one of its points, has departed from coincidence with another straight line. To designate an angle, we make use of three letters as BA C, that at the vertex being in the middle. This is necessary when there are several angles F 5 formed at the same vertex, as at A (fig. 5). But if there F 7 be only one as at A (fig. 7), a single letter is sufficient to designate it.

17. The question now arises, how are angles to be measured and compared? It is evident that their magnitude does not depend at all upon the length of their sides, for F 7 the angle A (fig. 7) is the same, according to the definition, whether we consider A D, A B, or A B produced, as the moving side. Now the measure of an angle must be some known magnitude which increases and diminishes simultaneously with the angle itself. Where shall we find such a magnitude? We answer the definition itself suggests one. For while the line A B moves as a radius about the fixed point A, every point in the line A B describes an arc of a circle; and since the arcs and the angle are formed by one and the same motion, beginning, increasing, and ending simultaneously, we have in the arcs thus formed, every property included in the idea of a measure. Accordingly we say that-angles are measured and compared by means of the arcs described from their vertices and centres—. If for example we wish to make an angle equal to a given angle, it is only necessary to make the arc which measures it equal to that which measures the given angle. But here it is to be observed that the two arcs must be described with the same radius; otherwise we could not make them equal by making their chords equal (14), nor would the degrees by which the arcs are measured, have the same value (15). The student will now understand why the value of angles as well as arcs is expressed in degrees, minutes, &c.

18. For the sake of illustrating what has now been said, we shall solve the following problem-to make an angle

Draw F 8

equal to a given angle. Let the given angle be A (fig.
8), and let the vertex of the required angle be D.
the straight line D F indefinitely.

Then with the centre A and any convenient radius, describe the arc B C. Again with the centre D and the same radius (17), describe the arc F E, which is to be made equal to B C by the method shown before (14). We have now two points D and E, through which the remaining side of the required angle is to pass. These determine its position (6). Draw D E, and the angle D will be equal to A, because the arcs which measure them are equal by construction. But this problem, as well as many others of a similar nature, is more readily solved in practice, by means of a small metallic semicircle called a protractor, which is accurately graduated, that is divided into degrees, and which is usually found in cases of mathematical instruments. If, for example, we wish to make an angle D (fig. 8) equal to 40°, we apply F 8 the diameter of the protractor to the straight line D F, so as to make the notch marking the centre fall exactly at the point D intended for the vertex; then we have only to seek the number 40, mark the point, and draw the other side through the vertex and this point.

19. Angles are denominated, according to their magnitude, right angles, acute angles, and obtuse angles. When the moving line A B (fig. 9), has reached that position, in F 9 which the two adjacent angles B A C and BAD are equal to each other, these are called right angles. In this case A B is said to be perpendicular to C D; so that to say a line is perpendicular to another, and to say a line makes a right angle with another, are the same thing. If the moving line has not reached the position of A B, the angle is called an acute angle, as E A C. If it has passed beyond A B, the angle is called an obtuse angle, as FA C. In each case, one of the lines is said to be oblique with respect to the other. Thus A E and A F are oblique with respect to A C. The substance of what is said above may be expressed by the following definitions-A right angle is when one straight line meets another so as to make the two adjacent angles equal. An acute angle is less than a right angle—. An obtuse angle is greater than a right angle―.

20. It follows from the definition that a right angle has for its measure a quadrant or 90°-. For as the adjacent angles D A C, D A B (fig. 10) are equal, and as both are F 10

measured by a semicircumference or 180°, it follows that one of them, as D A C, must have for its measure half a semicircumference, that is a quadrant or 90°. Moreover -the sum of all the angles which can be formed about a given point are equal to four right angles- because they all have for their measure an entire circumference or 360°. Thus F10 all the angles formed about the point A (fig. 10) are equal to 4 right angles. Again-all the angles formed about a given point on one side of a line are equal to two right angles-for they have for their measure a semicircumference or 180°. Thus all the angles whether two or more, formed at the point A on one side of the line B C, are equal to two right angles.

F11

21. Since, from what has just been shown, the adjacent angles formed by one straight line meeting another BA I+I A C=2 right angles, B A I must be just as much greater than a right angle, as IA C is less. In this case, each is said to be a supplement of the other. Hence the supplement of an angle is what that angle wants of 2 right angles or 180°. Thus B A I is the supplement of I A C, and I AC is the supplement of B A I. When the sum of two angles, as D A F+F A B=a right angle, each is said to be a complement of the other. Hence-the complement of an angle is what that angle wants of a right angle or 90o.- Thus BA F is the complement of F A D, and F A D is the complement of B A F. As we have just seen that all right angles have the same measure, we say-all right angles are equal-. Then from the definition of supplements and complements, we say-equal angles have equal supplements and the converse-; also-equal angles have equal complements and the converse-.

22. When two straight lines as A B, C D (fig. 11) cross each other, the angles which are opposite to each other at the vertex are called vertical angles. Thus CE A and BED, also C E B and A E D are vertical angles. Then the following proposition-all vertical angles are equalmay be easily demonstrated. For A E C B C D because both have the same supplement CEB (21). Also CE B=A E D because they have the same supplement B E D. The same reasoning will apply to all cases, therefore all vertical angles are equal.

23. If a perpendicular be erected upon the middle of a straight line, every point in the perpendicular is equally disF12 tant from the extremities of that line Let A B (fig. 12)

be the line, D the middle of it, and C D the perpendicular. We are to prove that every point in C D is equally distant from A and B. In the first place D is equally distant by hypothesis, that is by the conditions of the proposition. Now take any other point at pleasure as C, and draw C A and C B. We say C A=C B. For let the figure C D A be folded upon C D B so that the folded edge shall coincide with C D. Then since the angle C D A=C D B, being right angles, the line D A will fall upon D B'; and since they are by hypothesis of the same length, the point A will fall upon B. Therefore, since C A and C B have the two points C and B common, they must coincide (6) and are equal. C then is equally distant from A and B; and since C was taken at pleasure, that is, any where in the perpendicular, the same is true of every other point, and the proposition is demonstrated.

24. The last proposition leads to several important practical results. Since it is the property of a perpendicuÎar drawn to the middle of a straight line, that all the points in it are equally distant from the extremities of that line, and since two points are sufficient to determine the position of a straight line, we conclude that a perpendicular may be drawn to the middle of a line, by finding two points equally distant from the extremities of that line. We propose then to solve the following problem-to erect a perpendicular at a given point in a straight line. Let A B (fig. 13) be the given line and C the given point. Place F 13 one foot of the compasses in C and fix two points A and B at equal distances from C. Then with A as a centre and any radius greater than A C, make an arc D, and with B. as a centre and the same radius, make another arc cutting the first in D. The point D thus fixed, is equally distant from A and B. The point C was made so at first. Therefore the line D C is the perpendicular required.

25. From a given point without a straight line, to let fall a perpendicular to that line. Let A (fig. 14) be the F 14 given point and B C the given line. With A as a centre and any radius greater than the shortest distance from A to the line B C, make an arc cutting B C in two points B and C. A is equally distant from these two points. Then find another point D, as in the preceding problem, which is equally distant from B and C. The straight line drawn through A and D (23) is the perpendicular required.