PROBLEM IX. Ta meafure heights and diftances by the geometrical square. When the plane is horizontal, the inftrument is to be sup ported and placed horizontally at any point A, and it is to be turned till the remote point F, whose distance is to be measured, is feen through the fixed fights; then turn the index, till through the fights upon it, you see any acceffible object B ; then place the inftrument at the point B, directing the fixed fights to the first station A, and the moveable ones to the point F; and if the index cut the reclined fide of the square, as in the point E, then, from fimilar triangles, ES: SB:: as BA : AG; but if the index cut the right fide of the fquare K, it will be BR: RK:: BA: AF. In either of these cafes, the distance required may be found by the rule of three*. Perpendicular heights, when acceffible, may be obtained by the quadrant only. For example: If you wanted the height of a house, tree, &t. approach towards or retire from the object, till it fubtends an angle of 45°; then fhall the height of the object be equal to its horizontal diftance. Euclid, I. 6. A fimilar obfervation may be made of the other inftruments ufed for heights and distances; but this, and many more, will daily occur in practice. * The fide DE is called the right fide, E the reclined fide. TABLES. The velocity acquired at the end of any given time may be found thus. Suppofe a body begins to move with a celerity conflantly encreasing in fuch a manner as would carry it through 16 feet in one fecond, at the end of this space it will have acquired fuch a degree of velocity as would carry it 32 feet in the next fecond, though it should then receive no new impulfe from the cause by which its motion had been accelerated. But as the fame accelerating caufe continues conftantly to act, it will move 16 feet farther the next fecond, confequently it will have run 64 feet, and acquire fuch velocity as would, in the fame time, carry it over double the space. And fo on. EXAMPLE I. How far will a body fall in 6 feconds? 62=36 36x16 576 feet. EXAMPLE II. In what time will a body descend through 11664 feet? Required the last acquired velocity, when a body has fallen 8 feconds of time. 32 the additional velocity per fecond. 8 the time. 256 the laft acquired velocity is 256 feet per fecond. EXAMPLE EXAMPLE IV. If a body move at the rate of 1376 feet per fecond, How far muft it fall to acquire that velocity? In the following Table, the column titled T denotes the feconds of time from 1" to 60"; S the spaces paffed over in any fecond of time. The third column gives the heights from which a body would fall at the end of any given time, from " to 60"; and column 4th denotes the laft acquired velocity at the end of any given time. Thus, at the end of 22 feconds, the body has fallen from the height of 7744 feet, and moves with a velocity of 704 feet per fecond. TABLE 81 26806 1312 83 28224 1344 85295841376 $730976 1408 89 32400 1440 46 91 33856 1472 47 93 353441504 48 95 36864 1536 49 97 38416 1568 50 99 40000 1600! 51101 41616 |1632| 52 103 43264 (1664) 53 105 44944 1696 54 107 46656 1728 55 109 48400 1760 56111 50176 |1792| 57 113 51984 1824 58 115 53824 1856 59 117 55696 1888 60 119 57600 1920 41 42 43 44 45 |