Airopaidia

by Thomas Baldwin

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.
with 11° on		4000	=	106.92	106.92
		10	=	26.73	.2673
		6	=	160.38	.16038
		.8	=	213.84	.021384
					—————
			Expansion		107.369064;

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.

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]

1st. Step, in Section 353.

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

2d. Step, in Section 354.

Below. Barometer, Inches 29, .4 Tenths.

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

3d. Step, in Section 355.

Above. Barometer, Inches 25, .19 Tenths.

Attached Thermometer 46°, Air Thermometer, 29°​1⁄2.

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

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:

5th Step, in Section 364.

6th Step, in Section 366.

with 4° on 25.	= .0101
with 4° on .19	= .0000076
—————————
25.2|
Upper Barometer, Inches 25, .2 Tenths.
Lower Barometer, 29, .4

End of the first Stage.

7th Step, in Section 368.

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

8th Step, in Section 371.

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

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.

11th Step, in Section 376.

Detached Air-Thermometer, above,	29​1⁄2
Ditto below,	45°
	——
Whole Heat	2)84​1⁄2
Half Heat or mean Temperature	43​1⁄4
Deduct Standard	31​1⁄4
	———
Moiety above Standard	11°

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.