CHAPTER LXXVIII.

PRACTICE OF THE FOURTH EXAMPLE,[134] FOR MEASURING SMALL HEIGHTS.

[Sidenote: By this Example, _small_ Heights are easily measured.]

Section 419. Attached Therm. _below_, 71°.0 Attached Therm. _above_, 70 .5 ————— Subtract, and there remains .5 Tenths of a Degree of Heat to be added to the _colder_ Barometer (which in the present Case is the _upper_, but might possibly have been otherwise) by the 1st Table.

_First_, _with_ 0°.5 _on_ 29 Inches. To obtain which, begin _with_ 1°.0 _on_ 29 Inches = .002: _with_ 0°.1 above 1°, _on_ 29 = .0002: then _with_ 0°.5 above 1°, _on_ 29 = .001.

Prepare it for Addition to the _colder_ Barometer. colder Barometer 29.985 Expansion _with_ .5 above 1°, _on_ 29 .001 —————— 29.986

_Secondly_, _with_ .5 Tenths above 1°, _on_ .985 Tenths above 29 Inches. To obtain which, (having already found the Height from Expansion _with_ .5 above 1°, _on_ 29 Inches, to be .001;) since the Expansion on .985 Tenths above 29 Inches, is somewhere above 29, yet below 30 Inches; find the Expansion _with_ .5 above 1°, _on_ 30 Inches, thus:

first, _with_ 1°, _on_ 30 = .003 2d. _with_ 0°.1 above 1°, _on_ 30 = .0003 3d. _with_ 0°.5 above 1°, _on_ 30 = .0015

Subtract the Expansion _with_ .5 Tenths above 1°, _on_ 29 Inches, from the Expansion _with_ .5 Tenths above 1°, _on_ 30 Inches:

viz. _on_ 30 = .0015 _on_ 29 = .001 ————— The Answer is .0005, the Height from Expansion, _with_ .5 Tenths above 1°, _on_ 1 Inch above 29, i.e. on the 30th Inch: Then, if 1 Inch above 29 gives .0005; .1 gives .00005: and 985 ——————— multiplied 00025 as whole 00040 Numbers, 00045 ———————— give .0004|925 add the former Number 29.986 and, for the three remaining Decimals, _may_ be substituted 1 Decimal in the fourth Place 1 ——————— colder Barometer of equal Heat } 29.9865 with the _warmer_ }

420. _When the Quicksilver in each Barometer indicates the same Number of Inches_, differing _but_ one _or_ two Tenths _at the most; (which will frequently be the Case, in levelling flat Countries, or measuring small Heights;—instead of the usual Method, (to find the Height of each Barometer_ separately, _with the Standard-Heat, by the_ 2d Column _of the 2d Table, as in Section 411;)—it will be more convenient,_

1st. _To subtract the lower Barometer from the upper. Then,_

2dly. _By the_ 3d Column _of the same Table, find the_ +difference+, (_viz. of_ one _or_ two Tenths _at the most_) below _the Inches and_ nearest Tenth _of the_ lower _Barometer_.

_And_ lastly, _with that_ +difference+, _find by the_ 3d Table, _the Height at the Standard-Heat, corresponding to the remaining Decimals_ above _the_ upper _Barometer_.

421.(_1st._) From the lower Barom. viz. 30.082 Subtract the upper 29.9865 ——————— Remaining Decimals _above_ the upper .0955

_2d._ Find, by the 2d Table, the Height corresponding to the Inches, and _nearest_ Tenth _above_ and _below_ the Point at which the Quicksilver rests in the lower Barometer.

The Inches and _nearest_ Tenth is _above_ 30 Inches, correspond. to Feet 1681.7 and _below_ 30.1, corresponding to 1595.0 —————— 86.7 which is the +difference+ of .1 _below_ 30.1.

_Lastly._ Find, by the 3d Table, _with_ the +difference+, viz. 86 Feet, _on_ the remaining Decimals, for the Height, in Feet, corresponding to the Standard-Heat. viz. .09 77 = 77. Feet. .005 43 = 4.3 .0005 43 = .43 ————— Answer, Height in Feet 81.73 corresponding to .0955 above Inches 29.9865 Tenths of an Inch, of Quicksilver in the upper Barometer thus brought to the Standard-Heat.

422. Prepare for Expansion of Air from Excess above Standard-Heat, on the same Number of Feet: Detached Thermom. _above_ 76°. Detached Thermom. _below_ 68.0 ———— Whole Heat 144.0 Half Heat 72.0(0 adding a Cypher) Standard-Heat 31.24 ————— which deduct, and there remains 40.76: with which, by the 4th Table, find the Expansion of Air on Feet 81.73: _First, with_ 40°, _on_ 81.73, thus: _on_ 80. as 8000 – 777.6 = 7.776 1. as 1000 – 97.2 = .0972 .7 as 7000 – 680.4 = .06804 .03 as 3000 – 291.6 = .002916 ———————— 7.944156 _Second_, _with_ .76 _on_ 81.73, thus: _on_ 80. as 8000 – 1477.4 = .14774 1. as 1000 – 184.6 = .001846 .7 as 7000 – 1292.7 = .0012927 .03 as 3000 – 554.0 = .0000554 ———————— Expansion .1509341 add the former Expansion 7.944156 ————————— Sum of the Expansions, viz. } Height in Feet } 8.0950901 from Excess of Heat above Standard, _with_ 40°.76 _on_ 81.73, +added+ to the Height at the Standard-Heat, } in Feet } 81.73 gives, in Feet and Tenths, the true —————— Height of the Tarpeian Rock 89.8|2.