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EXAMPLE. Let the lower Barometer be supposed to stand at 29.68 Inches, the upper at 25.28 Inches:

The Log. of 29.6814724639 taken as whole
Log. of 25.2814027771

=

Numbers.

Their Diff. by 1000 696.868=Grofs-Height [of the Hill, in Fathoms.

N. B. What is here called the Grofs-Height of the Hill is the true Height, when the Thermometer ftands at 40° both at Top and Bottom of the Hill; but, if not, it will require two Corrections, as will be explained in the following Rules.

130. Secondly, With the Difference of the Temperatures of the Quickfilver, (as fhewn by the Thermometers in the Cases of your portable Barometers,) take out of Table I. from the Column of Fathoms, the Equation correfponding thereto, which call the Equation of the Mercury: But, if the Thermometer, in the bigber Station, gives a warmer Air than in the lower Station, then we must add this Equation to, otherwise fubtract it from, the Grofs-Height, and the Sum, or Remainder, will be the Approximate-Height.

EXAMPLE. Let the Grofs-Height of a Hill, found by the last Article, be 696.868, and let us fuppofe the Thermometer, at the Bottom of the Hill, ftood at +57, and, at the Top of the Hill, at +43, their Difference is 14.

Against 10, in the 3d Column of Table I. is 4.518

4

1.807

The Sum is the Equation of the Mercury 6.325

From the Grofs-Height of the Hill 696.868
Subtract the Equation of Mercury

6.325

Gives the Approximate-Height 690.543

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131. Thirdly, Add together the Temperatures of the Air, fhewn by the Thermometers (in open Air) under their proper Signs, and call Half the Sum the Temperature of the Air, being the mean State of the Air. With this Temperature, from Table II. find Part of the Equation for the Air thus: Look at Top for the Number of Decades (or Tens) next lefs than the Temperature of the Air; and from that Column collect (by repeated Entries, if neceffary) the Corrections for the Units, Tens, and Hundreds, of Fathoms, in the Approximate-Height (And, if you thould want for Thousands of Fathoms, it is only removing the Decimal-Point in Hundreds of Fathoms, by the Rules of Decimals:) Find the Sum of these feveral Parts of the Correction, and call it N.

To find that Part of this Equation, called n, which corresponds to the Degrees in the Temperature of the Air, which are over and above the Decades, obferve, that the Correction for 1 Fathom, at the feveral Degrees, is here annexed.

Degrees. Equation for 1 Fathom.

002

I

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018 020

.004.007.009.011.013.015.018

Therefore, to find n, we have only to multiply the Equation for 1 Fathom, taken from this little Table, by the Number of Fathoms in the ApproximateHeight, and the Product will give n.

If the Sign of the Correction, N, found by Table II. is, add the Sum N+ to the ApproximateHeight; but, if the Sign of Correction, N, is --, fubtract their Difference, viz. N-n, from the Approximate-Height, for the correct Height of the Object, which was required."

EXAMPLE. Let the Approximate-Height be, as found by the laft Article, 690.543, and the Height of Mercury in the Thermometers, out of Doors, 42 and 57; required, the true Height of the Hill. Thermometers

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The Temperature of Air 49 40°+9 d.

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In Table II. in Column 40+, the Equation is o; therefore No. In the little Table, in this Art. the Equation for Fathom, correfponding to 1 Degree, is .002; and therefore to a deg.

I

2

.001

Under 9 d.

.020

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x by Number of Fathoms, 690, gives 14.49 n. the whole Equation N+0+14.49

Fathoms.

To the approximate Height
Add Equation for the Air N+n

14.49

Fathoms.

690.543

14.490

The Sum is the required Height = 705.033

132. EXAMPLE 2. Let the Heights of the Mercury in the Barometers and Thermometers be as under:

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Now, to find N, we have given the Temperature of the Air 33°=30°+3°.

By Table II. 30d. upon 300 Fathoms give 6.683

Upon 90
Upon 6

2.005

0.134

Sum is N 8.822

To find n. By the little Table in Art. 131, under 3 Degrees, we have for 1 Fathom .007, this xd by 396 gives n=2.772. Now, N being' and ʼn always, the Equation is the Difference of these Numbers, (viz.) 8.822 — -2.772=6.050.

From the approximate Height 396.301 Fath.

Subtract Equation for Air N-n=

6.050

Gives the correct Height in Fathoms 390.251

133. EXAMPLE 3. Let the Heights of the Barometers be 30.0 and 29.9 Inches, and the Heights of the Thermometers each +40 of Fahrenheit's Scale. 30.0 14771212 29.914756712

Log. of
Log. of

Their Diff. by 1000 = 14500 Fathoms; this, multiplied by 6, gives 87 Feet; the Height anfwering to of an Inch Fall of the Mercury, in a mean State of the Air; viz. when the Thermometer stands at 40 d. and the Barometer at 30 Inches.

T

134.

134. TABLE I.

TABLE II.

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1 2 3 4 5 78

Deg.

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3

9

10

1807 1.807

20

2259 2.259 30

2711 2.711 40

73163 3.163

3614 3.614

9 4066 4.066 10 4518 4.518 20 9036 9.036

30 1355413.554 40 18072 18.072 -50 22590 22.590 60-27108 27.108 70 3162631.626

40+

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0.022 0.044 0.067 0.089 0.044 0.089 0.134 0.178 0.067 0.134 0.200 0.267 0.089 0.178 0.267 0.3561

111 0.223 0.334 0.445 0.134 0.267 0.401 0.535 0.156 0.312 0.468 0.624 0.178 0.356 0.535 0.713 0.200 0.401 0.601 0.802 0.223 0.445 0.668 0.891 0.445 0.891 1.337 1.782 0.668 1.337 2.005 2.673 0.891 1.782 2.673 3.564 1.114 2.228 3.342 4.455 1.337 2.673 4.010 5.347 1.559 3.119 4.678 6.238 1.7823.564 5-347 7.129 2.005 4.010 6.015 8.020 2.228 4.455 6.683 8.911 4.455 8.91113.367 17.822 6.68313-36720.050 26.733 8.91117.822 26.733 35.645 500 o 11.13922.27833.41744.556 o 13.367 26.733 40.100 53.467 0 15.594 31.18946.783 62.378 o 17.82235.645 53.467 71.289 9001 0 20.05040.10060.150 80.200

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135. L

BOOK III. CHA P. II.

Longimetry.

ONGIMETRY fhews how to apply the Doctrine of Triangles to find the Diftances of Objects.

Prop. 1. Standing on an Object of a known Height, to find its Distance from another Object.

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