CHAPTER LXXIIII.

USE AND PRACTICE OF THE FOURTH TABLE, IN THE FIRST EXAMPLE.

_The_ +use+.

Section 379. To shew in Feet, and Tenths, what is the Expansion of Air on each thousand Feet, from 1000 to 9000 Feet, _with_ each Degree of Temperature from 1 to 100 Degrees, on Farenheit’s Scale.

_The_ +practice+.

_The 12th Step applied in the first Example._

380. For the Expansion of Air with 11 Degrees of Heat on 4016.8 Feet, look in the fourth Table, _with_ 11 in the left Hand vertical Column of Temperature, and (first) _on_ 4000 Feet, along the upper Line: the Place of Meeting gives the Expansion of the Air, _with_ 11 Degrees _on_ 4000 Feet: viz. 106.92.[126]

Next; look _with_ 11 Degrees, and (as there is a Cypher only in the Place of Hundreds) _on_ 10, (viz. of the 16 Feet) call the 10, a 1000; the Place of Meeting, or Answer is 26.73:

Thirdly; _with_ 11, _on_ 6, (viz. of the 16,) calling it 6000; the Answer is 160.38:

Fourthly; _with_ 11, _on_ 8, (viz. the .8,) and the Answer is 213.84.

381. Having added the respective Expansions together, thus;

_with_ 11°, _on_ 4016.8 Feet. Tenths. } 4000 = 106.92 106.92 } 10 = 26.73 .2673 _with_ 11° } 6 = 160.38 .16038 _on_ } .8 = 213.84 .021384 } —————————— } Expansion 107.369064;

(See Page 306.)

THE FOURTH TABLE,

+shewing the expansion with HEAT, from 1 to 100 degrees, on each THOUSAND FEET in the atmosphere, from 1000 to 9000 feet.+

+-----------------------------------------------------------------------+ |+degrees of the THERMOMETER, from 1 to 50, on farenheit’s scale.+ | | +------+------+------+------+-------+-------+-------+-------+-------+ | | 1000 | 2000 | 3000 | 4000 | 5000 | 6000 | 7000 | 8000 | 9000 | | +------+------+------+------+-------+-------+-------+-------+-------+ | 1| 2.43| 4.86| 7.29| 9.72| 12.15| 14.58| 17.01| 19.44| 21.87| | 2| 4.86| 9.72| 14.58| 19.44| 24.30| 29.16| 34.02| 38.88| 43.74| | 3| 7.29| 14.58| 21.87| 29.16| 36.45| 43.74| 51.03| 58.32| 65.61| | 4| 9.72| 19.44| 29.16| 38.88| 48.60| 58.32| 68.04| 77.76| 87.48| | 5| 12.15| 24.30| 36.45| 48.60| 60.75| 72.90| 85.05| 97.20| 109.35| | 6| 14.58| 29.16| 43.74| 58.32| 72.90| 87.49| 102.06| 116.64| 131.22| | 7| 17.01| 34.02| 51.03| 68.04| 85.05| 102.06| 119.07| 136.08| 153.09| | 8| 19.44| 38.88| 58.32| 77.76| 97.20| 116.64| 136.08| 155.52| 174.96| | 9| 21.87| 43.74| 65.61| 87.48| 109.35| 131.22| 153.09| 174.96| 196.83| | 10| 24.30| 48.60| 72.90| 97.20| 121.50| 145.80| 170.10| 194.40| 218.70| | 11| 26.73| 53.46| 80.19|106.92| 133.65| 160.38| 187.11| 213.84| 240.57| | 12| 29.16| 58.32| 87.48|116.64| 145.80| 174.96| 204.12| 233.28| 262.44| | 13| 31.59| 63.18| 94.77|126.36| 157.95| 189.54| 221.13| 252.72| 284.31| | 14| 34.02| 68.04|102.06|136.08| 170.10| 204.12| 238.14| 272.16| 306.18| | 15| 36.45| 72.90|109.35|145.80| 182.25| 218.70| 255.15| 291.60| 328.05| | 16| 38.88| 77.76|116.64|155.52| 194.40| 233.28| 272.16| 311.04| 349.92| | 17| 41.31| 82.62|123.93|165.24| 206.55| 247.86| 289.17| 330.48| 371.79| | 18| 43.74| 87.48|131.22|174.96| 218.70| 262.44| 306.18| 349.92| 393.66| | 19| 46.17| 92.34|138.51|184.68| 230.85| 277.02| 323.19| 369.36| 415.53| | 20| 48.60| 97.20|145.80|194.40| 243.00| 291.60| 340.20| 388.80| 437.40| | 21| 51.03|102.06|153.09|204.12| 255.15| 306.18| 357.21| 408.24| 459.27| | 22| 53.46|106.92|160.38|213.84| 267.30| 320.76| 374.22| 427.68| 481.14| | 23| 55.89|111.78|167.67|223.56| 279.45| 335.34| 391.23| 447.12| 503.01| | 24| 58.32|116.64|174.96|233.28| 291.60| 349.92| 408.24| 466.56| 524.88| | 25| 60.75|121.50|182.25|243.00| 303.75| 364.50| 425.25| 486.00| 546.75| | 26| 63.18|126.36|189.54|252.72| 315.90| 379.08| 442.26| 505.44| 568.62| | 27| 65.61|131.22|196.83|262.44| 328.05| 393.66| 459.27| 524.88| 590.49| | 28| 68.04|136.08|204.12|272.16| 340.20| 408.24| 476.28| 544.32| 612.36| | 29| 70.47|140.94|211.41|281.88| 352.35| 422.82| 493.29| 563.76| 634.23| | 30| 72.90|145.80|218.70|291.60| 364.50| 437.40| 510.30| 583.20| 656.10| | 31| 75.33|150.66|225.99|301.32| 376.65| 451.98| 527.31| 602.64| 677.97| | 32| 77.76|155.52|233.28|311.04| 388.80| 466.56| 544.32| 622.08| 699.84| | 33| 80.19|160.38|240.57|320.76| 400.95| 481.14| 561.33| 641.52| 721.71| | 34| 82.62|165.24|247.86|330.48| 413.10| 495.72| 578.34| 660.96| 743.58| | 35| 85.05|170.10|255.15|340.20| 425.25| 510.30| 595.35| 680.40| 765.45| | 36| 87.48|174.96|262.44|349.92| 437.40| 524.88| 612.36| 699.84| 787.32| | 37| 89.91|179.82|269.73|359.64| 449.55| 539.46| 629.37| 719.28| 809.19| | 38| 92.34|184.68|277.02|369.36| 461.70| 554.04| 646.38| 738.72| 831.06| | 39| 94.77|189.54|284.31|379.08| 473.85| 568.62| 663.39| 758.16| 852.93| | 40| 97.20|194.40|291.60|388.80| 486.00| 583.20| 680.40| 777.60| 874.80| | 41| 99.63|199.26|298.89|398.52| 498.15| 597.78| 697.41| 797.04| 896.67| | 42|102.06|204.12|306.18|408.24| 510.30| 612.36| 714.42| 816.48| 918.54| | 43|104.49|208.98|313.47|417.96| 522.45| 626.94| 731.43| 835.92| 940.41| | 44|106.92|213.84|320.76|427.68| 534.60| 641.52| 748.44| 855.36| 962.28| | 45|109.35|218.70|328.05|437.40| 546.75| 656.10| 765.45| 874.80| 984.15| | 46|111.78|223.56|335.34|447.12| 558.90| 670.68| 782.46| 894.24|1006.02| | 47|114.21|228.42|342.63|456.84| 571.05| 685.26| 799.47| 913.68|1027.89| | 48|116.64|233.28|349.92|466.56| 583.20| 699.84| 816.48| 933.12|1049.76| | 49|119.07|238.14|357.21|476.28| 595.35| 714.42| 833.49| 952.56|1071.63| | 50|121.50|243.00|364.50|486.00| 607.50| 729.00| 850.5 | 972.00|1093.5 | | 51|123.93|247.86|371.79|495.72| 619.65| 743.58| 867.51| 991.44|1115.37| | 52|126.36|252.72|379.08|505.44| 631.80| 758.16| 884.52|1010.88|1137.24| | 53|128.79|257.58|386.37|515.16| 643.95| 772.74| 901.53|1030.32|1159.11| | 54|131.22|262.44|393.66|524.88| 656.10| 787.32| 918.54|1049.76|1180.98| | 55|133.65|267.30|400.95|534.60| 668.25| 801.90| 935.55|1069.20|1202.85| | 56|136.08|272.16|408.24|544.32| 680.40| 816.48| 952.56|1088.64|1224.72| | 57|138.51|277.02|415.53|554.04| 692.55| 831.06| 969.57|1108.08|1246.59| | 58|140.94|281.88|422.82|563.76| 704.70| 845.64| 986.58|1127.52|1268.46| | 59|143.37|286.74|430.11|573.48| 716.85| 860.22|1003.59|1146.96|1290.33| | 60|145.80|291.60|437.40|583.20| 729.00| 874.80|1020.60|1166.40|1312.20| | 61|148.23|296.46|444.69|592.92| 741.15| 889.38|1037.61|1185.84|1334.07| | 62|150.66|301.32|451.98|602.64| 743.30| 903.96|1054.62|1205.28|1355.94| | 63|153.09|306.18|459.27|612.36| 755.45| 918.54|1071.63|1224.72|1377.81| | 64|155.52|311.04|466.56|622.08| 767.60| 933.12|1088.64|1244.16|1399.68| | 65|157.95|315.90|473.85|631.80| 779.75| 947.70|1105.65|1263.60|1421.55| | 66|160.38|320.76|481.14|641.52| 791.90| 962.28|1122.66|1283.04|1443.42| | 67|162.81|325.62|488.43|651.24| 814.05| 976.86|1139.67|1302.48|1465.29| | 68|165.24|330.48|495.72|660.96| 826.20| 991.44|1156.68|1321.92|1487.16| | 69|167.67|335.34|503.01|670.68| 838.35|1006.02|1173.69|1341.36|1509.03| | 70|170.10|340.20|510.30|680.40| 850.50|1020.60|1190.70|1360.80|1530.90| | 71|172.53|345.06|517.59|690.12| 862.65|1035.18|1207.71|1380.24|1552.77| | 72|174.96|349.92|524.88|699.84| 874.80|1049.76|1224.72|1399.68|1574.64| | 73|177.39|354.78|532.17|709.56| 886.95|1064.34|1241.73|1419.12|1596.51| | 74|179.82|359.64|539.46|719.28| 899.10|1078.92|1258.74|1438.56|1618.38| | 75|182.25|364.50|546.75|729.00| 911.25|1093.50|1275.75|1458.00|1640.25| | 76|184.68|369.36|554.04|738.72| 923.40|1108.08|1292.76|1477.44|1662.12| | 77|187.11|374.22|561.33|748.44| 935.55|1122.66|1309.77|1496.88|1683.99| | 78|189.54|379.08|568.62|758.16| 947.70|1137.24|1326.78|1516.32|1705.86| | 79|191.97|383.94|575.91|767.88| 959.85|1151.82|1343.79|1535.76|1727.73| | 80|194.40|388.80|583.20|777.60| 972.00|1166.40|1360.80|1555.20|1749.60| | 81|196.83|393.66|590.49|787.32| 984.15|1180.98|1377.81|1574.64|1771.47| | 82|199.26|398.52|597.78|797.04| 996.30|1195.56|1394.82|1594.08|1793.34| | 83|201.69|403.38|605.07|806.76|1008.45|1210.14|1411.83|1613.52|1815.21| | 84|204.12|408.24|612.36|816.48|1020.60|1224.72|1428.84|1632.96|1837.08| | 85|206.55|413.10|619.65|826.20|1032.75|1239.30|1445.85|1652.40|1858.95| | 86|208.98|417.96|626.94|835.92|1044.90|1253.88|1462.86|1671.84|1880.82| | 87|211.41|422.82|634.23|845.64|1057.05|1268.46|1479.87|1691.28|1902.69| | 88|213.84|427.68|641.52|855.36|1069.20|1283.04|1496.88|1710.72|1924.56| | 89|216.27|432.54|648.81|865.08|1081.35|1297.62|1513.89|1730.16|1946.43| | 90|218.70|437.40|656.10|874.80|1093.50|1312.20|1530.90|1749.60|1968.30| | 91|221.13|442.26|663.39|884.52|1105.65|1326.78|1547.91|1769.04|1990.17| | 92|223.56|447.12|670.68|894.24|1117.80|1341.36|1564.92|1788.48|2012.04| | 93|225.99|451.98|677.97|903.96|1129.95|1355.94|1581.93|1807.92|2033.91| | 94|228.42|456.84|685.26|913.68|1142.10|1370.52|1598.94|1827.36|2055.78| | 95|230.85|461.70|692.55|923.40|1154.25|1385.10|1615.95|1846.80|2077.65| | 96|233.28|466.56|699.84|933.12|1166.40|1399.68|1632.96|1866.24|2099.52| | 97|235.71|471.42|707.13|942.84|1178.55|1414.26|1649.97|1885.68|2121.39| | 98|238.14|476.28|714.42|952.56|1190.70|1428.84|1666.98|1905.12|2143.26| | 99|240.57|481.14|721.71|962.28|1212.85|1443.42|1683.99|1924.56|2165.13| |100|243.00|486.00|729.00|972.00|1215.00|1458.00|1701.00|1944.00|2187.00| +---+------+------+------+------+-------+-------+-------+-------+-------+

382. The decimal Points in the Answer must be changed, thus:

1. For the Place of _Thousands_ in the Question, (viz. 4000,) the Answer must remain, viz. 106.92, as in the Table, which is calculated for the Place of _Thousands_.

2. For the Place of _Hundreds_, in the Question, (viz. which in the present Case was a Cypher;) if there had been a Figure or Figures in the Place of hundreds; then the decimal Point in the Answer must have been removed over _one_ Figure or Place to the left.

3. For the Place of _Tens_, in the Question, (viz. 10 Feet,) the decimal Point in the Answer, must be removed over _two_ Figures, or Places, to the left.

4. For the Place of _Units_, in the Question, (viz. 6) the decimal Point in the Answer, must be removed over _three_ Figures, or Places, to the left.

5. For the Place of a _Decimal_, in the Question, (viz. .8) the decimal Point, in the Answer, must be removed over _four_ Figures, or Places to the left, by adding a Cypher: and for the Place of each further Decimal in the Question;—_one_ Place more in the Answer, by the further occasional Addition of a Cypher, thus: _on_

Feet 4000, the Ans. 106.92 is still 106.92 10 26.73 becomes .2673 6 160.38 .16038 .8 213.84 .021384 —————————— 107.369064

383. Which Sum, by rejecting all but the first Decimal, in the Answer, is Feet 107.3 Tenths equal to the Expansion of Air, _with_ 11° of Heat, _on_ 4016.8 Feet, the Height of the upper Barometer, with the Temperature of 31°.24, according to the 2d Table.

END OF THE LAST STAGE.

* * * * *

[Sidenote: +Rule+ copied.]

384. The +rule+ underneath, consisting of 3 Precepts _only_, is laid down by Sir George Shuckburgh, in the Transactions for 1777, Page 574, in order to ascertain the Height of Mountains, &c. (See Section 349).[127]

[Sidenote: 1st. Step, in Section 353.]

385. Recapitulation for each Step of the Work, in the first Example; referring to the Sections.

[Sidenote: 2d. Step, in Section 354.]

Below. Barometer, Inches 29, .4 Tenths.

Attached Thermometer, 50 Degrees, Air-Thermometer 45°.

[Sidenote: 3d. Step, in Section 355.]

Above. Barometer, Inches 25, .19 Tenths.

Attached Thermometer 46°, Air Thermometer, 29°½.

From 50° subtract 46 —— and there remains 4 Degrees of Temperature to be added to the colder Barometer.

[Sidenote: _4th Step, in Section 356._]

By Means of the first Table, find the Expansion of the _colder_ Barometer, with Degrees of Heat, viz. 4° on Inches 25, .19, _gradually_, thus:

[Sidenote: 5th Step, in Section 364.]

with 4° on 25. = .0101 with 4° on .19 = .0000076 ———————————————— 25.2|

[Sidenote: 6th Step, in Section 366.]

Upper Barometer, Inches 25, .2 Tenths. Lower Barometer, 29, .4

_End of the first Stage._

[Sidenote: _7th Step, in Section 368._]

By Means of the 2d Table, find the corresponding Heights in the Air, at 31°. 24.

[Sidenote: 8th Step, in Section 371.]

25, .2 Answer 6225.0 29, .4 2208.0 —————— The Remainder is 4016.8 Height in Feet, &c.

[Sidenote: _9th and 10th Steps, in Section 373._]

The 3d Table, or Table for _Heights_ in the Atmosphere corresponding to the _Tenth_ of an Inch _on_ the Barometer, including the 9th and 10th Steps, is useless in this first Example.

_End of the Second Stage._

[Sidenote: 11th Step, in Section 376.]

Detached Air-Thermometer, _above_, 29½ Ditto _below_, 45° ——— Whole Heat 2)84½ Half Heat or mean Temperature 43¼ Deduct Standard 31¼ ——— Moiety above Standard 11°

[Sidenote: _12th step, in Section 377._]

By Means of the 4th Table, find the Expansion of Air, with 11° on 4106.8 Feet viz. 107.3 which added to the same Height gives 4124.1 for the true Height, in English Feet, of the _Mountain_, or _upper Station_, sought.

_End of the last Stage._