is equal to DEF; and universally-Two angles which have their sides parallel and directed the same way, are equal. 25. To measure a straight line, we apply to it a scule or rule, or some other straight line of known and standard measure, and thus ascertain its length. This corresponds with general practice in analogous cases; one quantity is usually measured by another quantity of the same kind. In measuring an angle, geometers have adopted a method somewhat different. To ascertain the magnitude of an angle, they measure the portion of a circular arc embraced by the two sides of the angle, the vertex of the angle being at the centre of the circle. 26. To obtain a clear idea of the magnitude of an angle, and the connexion which it has with a circular arc, let us suppose (fig. 13) that at first the two lines AC Fig. 13. and BD, coincide in the part BC, and that the part A, of the line AC, be raised, so that this line, departing from BC, may revolve about the point C; it becomes immediately inclined to BC; and this inclination increases as the arc described by the point a increases, and in the same degree; that is, for the same amount of angular motion in any part of the revolution of the line AC, the arc described by the point a will be the same; so that when two angles are equal, the arcs drawn with the same radius, from the vertices of the angles, as centres, will be also equal. We hence see how the magnitude of an angle may be designated by a circular arc. For this purpose, the ancients divided the circumference of the circle into 360 equal parts, called degrees; each degree into 60 minutes; and each minute into 60 seconds. And the magnitude of an angle they expressed by the degrees, (0) minutes (') and seconds (",) which express the value, or magnitude, of the arc comprehended between the sides, the vertex being at the centre of the circle; thus, an angle of 35 degrees, 27 minutes, 15 seconds; usually written 35° 27′ 15′′. 27. We have seen (15) that the sum of all the plane angles made at the same point, is equal to four right angles; and it is manifest that the sum of all the arcs, which they would embrace in a circle described from their common vertex as a centre, would be the entire circumference. A circumference then, or 360°, is the measure of four right angles. If, therefore, through the Fig. 14. centre of the circle (fig. 14) we draw two diameters, perpendicular to each other, we shall divide the circle into four equal parts, called quadrants. ABC is a quadrant ; the arc AB, is an arc of 90°, and the measure of the angle ACB, which is a right-angle. A right angle then is an angle of 90 degrees. If we divide the arc AB into 90 equal parts, and from C through the 55th division, draw the line CD, we shall have the angle ACD, an angle of 55°; and the angle BCD, an angle of 35°. The obtuse angle ACD is measured by the arc AD', greater than a quadrant. 28. Of the two acute angles DCA and DCB, each is called the complement of the other, because each is just what the other wants to make it a right-angle. The acute angle DCA, and the obtuse angle DCA', are called supplements of each other, because each being subtracted from the sum of two right-angles, or 180°, will give the other. Angles complemental to each other, are both acute. Of two angles supplemental to each other, if one be acute the other will be obtuse, and vice versá. Two right-angles are supplements of each other. 29. A method practised by surveyors for measuring angles in the field, is by means of an instrument called Fig. 15. a semicircle, (fig. 15). At the two extremities of the diameter AB, sights are fixed, through which you look directly along the diameter. To the centre C, a moveable index is attached, with sights at its extremities, D, E, so that the line DCE is a straight line. When an angle of a field is to be measured, this instrument is placed horizontally with its centre C at the vertex of the angle, or, in other words, exactly at the corner of the held. The instrument is so placed that, by looking through the sights A, B, we look along one side of the field; the index DE is then turned about the central point C, till the line of sight is directly along the other side of the field. The magnitude of the angle will be expressed by the number of degrees, minutes, &c. in the graduated arc BE, comprehended between the stationary and moveable diameters. 30. When these angles are to be transferred to paper, for the purpose of giving a plan of the ground; an instrument called a protractor, is used. This is usually a Fig. 16. semicircular piece of brass (fig. 16), graduated in de grees and parts; and the centre C accurately determined. When an angle is to be set off upon the plan, one of the lines embracing the angle is first drawn; the protractor is then placed with its centre at the point where the vertex of the angle is to be, and its semidiameter lying along this line. The number of degrees, &c, is then counted off upon the arc, or limb (as it is called) of the instrument, and the point carefully indicated; then through the centre and this point a line is drawn, and we have the angle required. 31. When the magnitude of an angle already constructed, is required, the protractor is placed with its centre at the vertex of the angle, and its diameter lying along one of the sides; the number of degrees &c. will be read upon the graduated limb, at the point where it is cut by the other side of the angle. 32. To construct an angle equal to a given angle without the protractor. Suppose ACB (fig. 17) to be Fig. 17. the given angle; and suppose at c, in the straight line cb, it is required to make the angle acb, equal to ACB. With a convenient radius CD, from C as a centre, describe the arc DE; and from c, as a centre with the same radius, describe the arc de; then from d as a centre with a radius equal to DE, describe another arc cutting the arc de, in the point c; and through the points c and e, draw the line c e a, and you have the angle a cb, equal to the angle ACB. It is evident that, in the angle ACB, if the sides were more inclined, that is, if the angle were increased, the distance of the points D and E would be increased; and if the angle were diminished this distance would be diminished; therefore, while the distance of these points is the same, and their distance from the vertex of the angle is not changed, the angle must be the same; but these conditions are the same in the two angles ACB, acb; they are therefore equal. Of Plane Figures. 33. A plane has been defined,—a surface, to which a straight line, being applied in every direction, will touch the surface in its whole extent. Plane figures are portions of plane surface bounded or enclosed by lines. Those bounded by straight lines, or right lines (as they are frequently called), are denominated rectilinear figures. Those bounded by curve lines are curvilinear figures. 34. Among rectilinear figures, as two straight lines cannot enclose a space, the simplest is that of three sides, called the triangle. Triangles are differently denominated, according to the different relations of their parts. When the three sides are of equal length it is called an Fig. 18. equilateral triangle (fig. 18). When two of its sides are equal, it is called an isosceles triangle. When no two sides are equal, the triangle is called scalene Fig. 19. (fig. 19). When it has one right angle, it is called a Fig. 20. right-angled triangle (fig. 20). In the right-angled triangle, the side opposite the right-angle is called the hypothenuse. 35. The first inquiry concerning the properties of triangles, respects the sum of the three angles; is it always Fig. 21. the same? Let us take the triangle ABC (fig. 21), and through the vertex A draw the straight line DE parallel to the base BC; we then have AB, a straight line meeting the two parallel lines DE and BC; the alternate-internal angles DAB and ABC, are equal; and AC being a straight line meeting the same parallels, the angles EAC and ACB, are equal, for the same reason; then the angle DAB being equal to the angle B, and the angle EAC, equal to the other angle C, the three angles DAB, BAC and CAE, are equal to the three angles of the triangle; but the sum of these is equal to two rightangles (14); therefore, the sum of the three angles of the triangle is equal to two right-angles. It is evident. that, however the sides and angles of the triangle may be changed, they will always admit of a straight line (as DE) being drawn through the vertex of one of the angles, and parallel to the opposite side; and that the two outer angles at this vertex must therefore be equal to the other angles of the triangle. We say, therefore, that, the sum of the three angles of every triangle, is equal to two right-angles, or 180°. 36. (1.) If we know two angles of a triangle, how can we find the remaining angle? (2.) How many obtuse angles can a triangle have? Why? (3.) How many right angles can any triangle have? Explain. (4.) If one angle of any triangle, be a right-angle, Fig. 21. what will be the sum of the other two? (5.) If one of the oblique angles of a right-angled triangle be given, how can we find the other? 37. If we produce the base of the triangle BC to F, the angle which it makes with AC on the outside is called the exterior angle; and because AC is a straight line meeting the two parallels DE and BF, this angle is equal to the angle DAC; but DAC is composed of the two angles DAB and BAC; and DAB is equal to the angle B; therefore this exterior angle ACF, is equal to the sum of the two angles ABC and BAC, of the triangle; these two angles with respect to the exterior angle, are called interior and opposite. We say then- The exterior angle, made by producing one of the sides of the triangle, is equal to the sum of the two interior and opposite angles. 38. Problem. The three sides of a triangle being given, to construct the triangle. Let the three given sides be the lines A, B, C, (fig. 22). Fig. 22. Draw the line DE equal to the given line A ; then from D as a centre with a radius equal to the given line B, describe an arc; and from E as a centre, with a radius equal to the other given line C, describe an arc cutting the other arc in F; draw DF and EF, and you have the triangle required. 39. It is plain that no different triangle can be formed with these three lines. The only different construction which the case admits, is to make the triangle on the lower side of the base, as the triangle D'E'F'; but this triangle is not really different from the first. To show this, turn the last triangle over by lifting up the part F' and making the whole turn about the base D'E'; then place it upon the first so that the point D' will be upon D, and E' upon E; this may be done, as each of the bases is equal to the given line A. The triangles will then coincide in all their parts, and must therefore be equal. The point D' being upon D, the point F' must be at the same distance from D, as the point F is, it must therefore be in the first described arc; and as E' coincides with E, F' must be at the same distance from E, that Fis; it must therefore be somewhere in the other arc which crosses the first in F; if F is in each of these two arcs, it can only be at their intersection, and there |