Arithmetical Questions. 1. How many standard yards are equal in links? 2. How many links will be equal in length to length to 2761 3. Express a length measuring 387 53 chains in yards, feet, Ans. 8525 yds. I ft. 1176 in. and inches. 4. How many links are equal in length to 17776 inches, and how many to 9 miles ? Ans. 2244 50; 72000. 5. An angle, A B C, less than a right-angle, is turned about a side, A B, then about A C, then about A B again, and so on until A B exactly takes its first position. It passes this position three times in the process: find what part of a right-angle is A B C. Ans. 3. CHAPTER III. CIRCLES. Circumferences.-Use of compasses. 43. If we place one of the points of a pair of compasses upon a perfectly even board, and the compasses be then turned round this point, the other being properly pressed upon the board, a line is drawn which we call a circumference or circle (fig. 38). The distance between the two points of the compass is called the radius of the circle, the fixed point being called the centre. It is easily seen that the straight lines which join different points in the circumference with the centre are all of the same length; that is, all radii in the same circle are equal. Fig. 38. Two circumferences having the same radius are equal. 44. With the compass opened to the same extent, describe a new circumference. If the two centres coincide the circumferences will also coincide; for every point in each will be the same distance from the common centre; in other words, two circumferences having the same radius are equal. The two ends of a cask or of a round box, the two wheels of a carriage mounted upon the same axle-tree, are circles having the same radius. B Fig. 39. To draw a circumference, when the radius and the position of the centre are known. 45. Take a pair of compasses, and separate the points by a distance equal to the given radius. Place one point of the compasses at the given centre, and with the other describe the circumference. Compasses of different kinds and different dimensions are employed. For drawing circles on paper, one leg of the compasses is replaced by a drawing pen or pencil (fig. 39). An instrument called a sliding-guage is used by wheelwrights and others for striking out large circles (fig. 40). This instrument consists of a ruler having a fixed needle at one of its extremities, and a movable one at the other. Gardeners use a cord of convenient length, with a ring at each end; one is placed over a stake fixed in the centre of the circle, the other over Fig. 40. a sharp piece of iron or wood, which is held upright in the hand, and with which the circle is described. In order to draw a circle without using the compass, the hand is placed as in the figure (fig. 41), and held motionless, touch ing the paper only by the nail at o; the sheet of paper is then turned round by the other hand, so that the point of the pencil being at a fixed distance from the centre o, describes the circle. Fig. 41. 46. If a circle described upon a sheet of paper be cut out by a thin blade, the circumference of the part taken away, is equal to the corresponding circumference of the part left. If the disc be made to turn in the hole, its edge everywhere touches that of the paper; the same takes place when the disc is fixed, and the paper is turned round it, and therefore the circles are equal. Thus the inner circle on the lid of a box is equal to the outer circle of the box itself (fig. 42). Arcs. 47. A part of the circumference is called an arc. The ends of the boards which form the bottom of a cask are arcs of the circle (fig. 43). Arcs are frequently employed in architecture and ornamental work (fig. 52). Fig. 42. Mark upon the circumference of the box and the lid in (fig. 43) two points, A and O, cutting off an arc; then turn the lid round; the points A and B will come to A' B', and the arcs A B and A' B' marked upon the same circle will be equal. They can, as you see, be placed one above the other. Take away the lid, you have equal arcs of 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. Division of the circumference into 360 equal arcs. Degrees, minutes, seconds. 48. Compare the dial of a watch with that of a clock. They are divided in the same manner; that is to say, they are both divided into twelve equal arcs, each of these being further divided into five smaller arcs. When the small hand has traversed one of the large divisions on either dial, an hour has elapsed; while the large hand traverses one of the small divisions, a minute passes away. We say then that the dials have the same graduation. Suppose we have a circumference sufficiently large to be divided easily into 360 equal parts, each part is called a degree; again, let the degree be divided into sixty equal parts, each part is called a minute; and in the same way, the sixtieth part of a minute is called a second. So that a circumference = 360 degrees = 360 × 60 minutes = 360 × 60 × 60 seconds. If we take an arc upon this circumference, it will contain a certain number of degrees, minutes, and seconds: for example,-43 degrees, 10 minutes, 16 seconds, which is written 43° 10' 16". Consider now an arc of 43° 10' 16" upon another circumference. C Take two circumferences having the same centre, but radii of different lengths (fig. 44). If, after having divided one of them into equal parts, A B C D etc., the points of division are B joined to the centre, we shall A obtain upon the inner circumference the same number of divisions, all equal to one another. If A D is an arc of 30° upon one, the lines O D, OA will intercept an arc of 30° upon the other. Fig. 44 We see, therefore, that when two circles have the same centre, the arcs intercepted by two radii are the same portion of the circumference, and contain, therefore, the same number of degrees. Two arcs which contain the same number of degrees, minutes, and seconds, are equal if the circles are of the same radius. 49. For example, in two equal circles, two arcs of 60° will be equal; for each is the sixth part of the circumference. An arc of a certain number of degrees being given, to find how many times it is contained in the circumference. 50. The whole circumference contains 1,296,000 seconds; hence, if we divide this by the number of seconds in the given arc, the quotient will show how many times the circumference contains the given arc. Take, for example, an arc of 2° 48′ 45", or 10,125"; divide 1,296,000 by 10,125; the division gives the quotient 128′′ and no remainder. And so an arc of 2° 48′ 45′′ would be contained 128 times in the circumference. |