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 scale 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, (9) 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 the angle, or, in other words, exactly at the corner of the Fig. 16. field. The instrument is so placed that, by looking grees and parts; and the centre C accurately determin- 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. Fig. 17. Fig. 18. Fig. 19. Fig. 21. Those bounded by straight lines, or right lines (as they (4.) If one angle of any triangle, be a right-angle, what will be the sum of the other two 2. (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). 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 F is ; 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 Fig. 21. Fig. 22. |