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PROBLEM IX.

To meafure heights and distances by the geometrical square.

When the plane is horizontal, the instrument is to be supa 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 seen through the fixed lights; then turn the index, till through the fights upon it, you see any accessible object B; then place the inftrument at the point B, directing the fixed Lights to the first station A, and the moveable ones to the point F; and if the index cut the reclined side of the square, as in the point E, then, from similar triangles, ES : SB :: as BA : AG; but if the index cut the right side 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, &to approach towards or retire from the object, till it fubtends an angle of 45° ; then shall the height of the object be equal to its horizontal distance. Euclid, I. 6.

A similar obfervation may be made of the other instruments used for heights and distances; but this, and many more, will daily occur in practice.

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

TABLES.

The vclocity acquired at the end of any given time may be found thus. Suppose a body begins to move with a celerity constantly encreasing in such a manner as would carry it through 16 feet in one second, at the end of this space it will have acquired such a degree of velocity as would carry it 32 feet in the next second, though it should then receive no new impulse from the cause by which its motion had been accelerated. But as the same accelerating cause continues constantly to act, it will move 16 feet farther the next fecond, consequently it will have run 64 feet, and acquire such velocity as would, in the same time, carry it over double the space. And so on.

EXAMPLE I.
How far will a body fall in 6 seconds ?

62=36
36*16=576 feet.

EXAMPLE II.

In what time will a body descend through 11664 feet?
16)11.664(72927 seconds.

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46 47)329
32 329

144 144

EXAMPLE III.

Required the last acquired velocity, when a body has fallen 8 seconds of time.

32 the additional velocity per second.
8 the time.

256 the last acquired velocity is 256 feet per second.

EXAMPLE

EXAMPLE IV.

If a body move at the rate of 1376 feet per second, How far muft it fall to acquire that velocity?

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In the following Table, the column titled T denotes the leconds of time from 1" to 60" ; S the spaces passed 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 last acquired velocity at the end of any given time. Thus, at the end of 22 seconds, the body has fallen from the height of 7744 feet, and moves with a velocity of 704 feet per second.

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TABLE

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