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27 28 29 30

PROBLEM IX.

To meafure heights and distances by the geometrical square.

When the plane is horizontal, the instrument is to be fupa 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 sights; then turn the index, till through the fights upon it, you see any accessible object B; then place the instrument 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, &t. approach towards or retire from the object, till it subtends 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 fide 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 inove with a celerity conflantly encreasing in such a manner as would carry it through

a 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 fecond, 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(729(27 seconds.

4

112

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

r

PROBLEM IX.

To measure beights and disances by the geometricel Square.

When the plane is horizontal, the instrument 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 measuItd, is seen through the fixed sights; then turn the dex, till, through the lights upon it, you see any accessible object B; then place the instrument 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 side of the square, as in

.
the point E, then, from similar triangles, ES : SB :: as BA :
AG; lut 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 acceslible, may be obtained by the quadrant only. For example, If you wanted the height of a house, tree, &c. approach towards or retire from the object, till it subtends an angle of 45° ; then shall the height of the object be equal to its horizontal diftance. Euclid, I. 6.

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

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

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CONTAINING,

I. A TABLE OF THE LOGARITHMS OF NUMBERS

FROM I TO 10000.

II. A TABLE OF LOGARITHMIC SINES, TAN.

GENTS, SECANTS, AND VERSED SINES,
TO EVERY DEGREE AND MINUTE OF THE QUA-
DRANT

III. A TABLE OF LOGARITHMIC SINES, TAN

GENTS, AND SECANTS, TO EVERY POINT,
HALF POINT, AND QUARTER POINT OF THE
COMPASS.

A TABLE of the LOGARITHMS of NUMBERS from 1 to 10000.

¡N. Leg. No. Log. No. Log No. Log. (No. Log.

1.C.ocovo 211.32222 471.612781 61 1.78533 811.90848 20.301b3 221.34242 42 1.62325 621.79239 821.91381 30.47712 23 1.36173) 43 1.63347 63 1.79934 831.91908 40.60206 241.38021 44 1.64345 641.80618 841.92428 50.69897 25 1.39794 451.65321 651.81291 851.92942 60.77815 261.41497 461.00276 66 1.81954 801.93450 7 0.84510 27 1.43136 471.67220 67 1.82607 87 1.93952 80.90309 28/1.44786 481.68124 68 1.83251 88 1.94448 90.95424 29 1.4624049 1.69020 69 1.83885 891.94939 101.00000 301.47712 501.69897 701.84510_901.95424 II 1.04139 311.49136 511.707571 711.85120 9111.95904 12 1.07918 321.gogis 52 1.71600 721.85733 921.96379 131.11394 331.51851 53.1.72428 731.86332 931.96848 141.14613 341.53148 541.73239 741.86923 94 1.973:3) 151.1?609351.54407 5511.74036 75 1.87500 951.97772 16 1.20412 3611.55636 56 1.74819 761.87081 9011.y227 171.23045 37 1.56820 571.75587| 77 1.886499, 1.98677 181.25527 38|1.57979] 581.76 343 786.892041 9811.99123 191.27875 391.59101 59 1.77085 794.89763 991.99563 201.3010; 4011.6020 | 601.77814) 801.00.102|102.00000

I

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