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PROBLEM IX.

To meafure heights and diftances by the geometrical square.

When the plane is horizontal, the inftrument is to be fupported and placed horizontally at any point A, and it is to be turned till the remote point F, whofe diftance is to be measured, is seen through the fixed fights; then turn the index, till through the fights upon it, you fee any acceffible object B; then place the inftrument at the point B, directing the fixed fights to the firft ftation A, and the moveable ones to the point F; and if the index cut the reclined fide of the fquare, as in the point E, then, from fimilar triangles, ES: SB:: as BA : AG; but if the index cut the right fide of the square 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 accessible, 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 diftances; but this, and many more, will daily occur in practice.

* The fide DE is called the right fide, E the reclined side.

TABLES.

The velocity acquired at the end of any given time may be found thus. Suppofe a body begins to move with a celerity conftantly encreasing in such 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 fhould 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

36× 16=576 feet.

EXAMPLE II.

In what time will a body descend through 11664 feet? 16)11.664(729(27 feconds.

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Required the laft 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?

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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 1" 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 second.

TABLE

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PROBLEM IX.

To meafure heights and diftances by the geometrical squarė.

When the plane is horizontal, the inftrument is to be fupported 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 seen through the fixed fights; then turn the index, till through the fights upon it, you fee 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 square K, it will be BR: RK:: BA: AF. In either of these cases, the distance required may be found by the rule of three *.

Perpendicular heights, when accessible, 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.

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