above it, keep the other side fast, open till the same edge touches E, and by it mark the point e, on the produced line. A line drawn from D to e, will take in as much as it leaves out. Thus have you the triangle c De, equal to the irregular polygon A B C D E F G H. If when the ruler's edge be applied to the points A and C, the point B falls under the ruler, hold that side next the said points fast, and draw back the other to any convenient distance ; then hold this last side fast, and draw back the former edge to B, and by it mark b, on the produced line ; and thus a parallel may be drawn to any point under the ruler, as well as if it were above it. It is best to keep the point of your protracting pin in the last point in the extended line, till you lay the edge of the ruler from it to the next station, or you may mistake one point for another. This may also be performed with a scale, or ruler, which has a thin sloped edge, called a fiducial, edge ; and a fine pointed pair of compasses. Thus, Lay that edge on the points A and C, take the distance from the point B to the edge of the scale, so that it may only touch it, in the same manner as you take the perpendicular of a triangle ; carry that distance down by the edge of the scale parallel to it, to b; and there describe an arc on the point b; and if it just touches the ruler's edge, the point bis in the true place of the extended line. Lay then the fiducial edge of the scale from b to D, and take a distance from C, that will just touch the edge of the scale; carry that distance along the edge, till the point which was in C, cuts the produced line in C; keep that foot in c, and describe an arc, and if it just touches the ruler's edge, the point c is in the true place of the extended line. Draw a line from c to D, and it will take in and leave out equally : in like manner the other side of the figure may be balanced by the line e D. Let the point of your compasses be kept to the last point of the extended line, till you lay your scale from it to the next station, to prevent mistakes from the number of points. That the triangle c D e, is equal to the rightlined figure ABCDEFGH, will be evident from problems 18. 19. geom. for thereby, if a line were drawn from b to C, it will give and take equally, and then the figure b C D E F G H, will be equal to the map. Thus the figure is lessened by one side, and by the next balance line will lessen it by two, and so on, and will give and take equally. In the same manner an equality will arise on the other side. The area of the triangle is easily obtained, as before, and thus you have the area of the map. It is best to extend one of the shortest lines of the polygon, because if a very long line be produced, the triangle will have one angle very obtuse, and consequently the other two very acute ; in which case it will not be easy to determine exactly the length of the longest side, or the points where the balancing lines cut the extended one. This method will be found very useful and ready in small enclosures, as well as very exact; it may be also used in large ones, but great care must be taken of the points on the extended line, which will be crowded, as well as of not missing a station. PROB. XVII. A man with its area being given, and its scale omitted to be • either drarn or mentioned ; to find the scale. Cast up the , it will be As the area found Is to the square of the scale by which you cast up, :: The given area of the map To the square of the scale by which it was laid down. The square root of which will give the scale. EXAMPLE. A map whose area is 126A. BR. 16P, being given ; and the scale omitted to be either drawn or mentioned; to find the scale. Suppose this map was cast up by a scale of 20 perches to an inch, and the content thereby produced be 31A. QR. 34P, As the area found, 31A, 2R, 34P.5074P. up, that is to 20 x 20=400, : : The given area of the map 126A, 3R. 16P. =20296P. To the square of the scale by which it was laid down. 5074: 400:: 20296 : 1600 the square of the required scale. Root. 1600(40 16 8(00 Answer. The map was laid down by a scale of 40 perches to an inch. PROB. XVIII. How to find the true content of a survey, though it be taken by a chain that is too long or too short. Let the map be constructed and its area found as if the chain were of the true length. And it will be, As the square of the true chain EXAMPLE. If a survey be taken with a chain which is 3 inches too long; or with one whose length is 42 · feet 3 inches, and the map thereof be found to con tain 920A. 2R. 20P. Required the true content. As the square of 42F. oln.=the square of 504 = inches = 254016. Is to the content of the map 920A, IR. 20P.= 147260P. = |