Airopaidia

CHAPTER LXXI.

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

The use.

Section 357. TO find the Expansion of Quicksilver, and of the barometric Tube in which it is contained: or, in other Words, to find the Point to which the Quicksilver will rise in the Tube, (in Parts of an Inch) with a given additional Temperature, on Farenheit’s Scale.

The Question in the first Example is, (Ph. Tr. for 1777, Page 578;)

To find the Expansion that arises, with the Addition of 4 Degrees of Heat, on the colder Barometer resting at Inches 25 .19 Tenths, in order to give it an Expansion equal to that of another Barometer, 4 Degrees warmer than the former: the Temperature of each Barometer, being indicated by its respective attached Thermometer.

N. B. During the Application of the first Table, the Investigation moves forward two Steps only, viz. the 4th and 5th.

The 4th Step, applied in the first Example.

358. The Order to be observed in finding the Expansion of the Quicksilver, with 4 Degrees on Inches 25 .19 Tenths of the Barometer.

1st. Find the Expansion, With 4° on 25 Inches only.

Then in order to obtain with 4° on .19, begin

2d. With 4° on 1 Inch above 25 Inches, i. e. on the 26th Inch.

3d. With 4° on .1, i. e. one Tenth of an Inch above 25 Inches: and lastly,

4th. With 4° on .19, Tenths above 25 Inches.

The practice.

359. 1st. In the first Table, with 4 Degrees on the left Hand vertical Column, and with 25 Inches, along the upper Range; at the Point of Meeting, is the Answer .0101⁠[119] viz. the Expansion, or Rise of the Quicksilver standing at 25 Inches, and receiving an additional Heat of 4°: the Answer .0101 being the Expression for the ten thousand one hundredth Part of an Inch, (viz. in Height, by Expansion.)

360. Add this Number, .0101, Part of an Inch, or Rise by Expansion, to the Barometer resting at Inches 25, .19 Tenths, Units under Units, &c. thus: .0101.

361. 2d. Now, in order to obtain the Expansion with 4 Degrees, on .19 Tenths i. e. the nine hundred and tenth Part of an Inch of Quicksilver in the Tube (above 25 Inches,) it must be considered, where it ought to be found in the first Table.

Tenths of 1 Inch, above 25 Inches, it must be observed, are at some intermediate Point between 25 and 26 Inches; that is, above 25, yet not so high as 26, or more than 25, yet less than 26.

Therefore, to find the Expansion with 4 Degrees, on 1 Inch above 25, i. e. on the 26th Inch; look in the Table, first, with 4 Degrees on 25 Inches: then with 4 Degrees on 26 Inches. The respective Numbers are .0101 and .0105.

And by taking the Expansion with 4° on 25 Inches, from the Expansion, with 4° on 26 Inches, thus;

Expansion		.0101 on 25 Inches,
		.0105 on 26 Inches,
		——
The Remainder		.0004 is the Expansion with 4° on 1 Inch, above 25, i. e. on the 26th Inch.

362. 3d. To find the Expansion, with 4° on .1 above 25 Inches; add a Cypher and decimal Point to the former Answer, which then becomes .00004, viz. the Expansion, with 4° on one Tenth, above 25 Inches.

363. 4th. Lastly, to obtain the Expansion with 4°, on .19, above 25 Inches, say: If one Tenth of an Inch, above 25 Inches, gives this Expansion viz. .00004, what Expansion will nineteen Tenths above 25, give? answer .19 Tenths more; thus:

If .1:	.00004	::.19?
	.19
	———
	00036
	0004
	———
	.00076;	then, in order to have

as many decimal Places in the Product as are contained both in the Multiplicand and Multiplier, add a Cypher and Point to the left, and the Product becomes .0000076 which, being divided by .1, gives a Cypher less. viz. the Expansion with 4° on .19 above 25 Inches.

THE FIRST TABLE:
shewing the expansion with HEAT
on inches of the BAROMETER.
degrees of the THERMOMETER, from 1 to 40, on farenheit’s scale.
	9	10	11	12	13	14	15	16
1	.00091	.00102	.00112	.00122	.00132	.00142	.00152	.00162
2	.00182	.00204	.00224	.00244	.00264	.00284	.00304	.00324
3	.00273	.00306	.00336	.00366	.00396	.00426	.00456	.00486
4	.00364	.00408	.00448	.00488	.00528	.00568	.00608	.00648
5	.00455	.00510	.00560	.00610	.00660	.00710	.00760	.00810
6	.00546	.00612	.00672	.00732	.00792	.00852	.00912	.00972
7	.00637	.00714	.00784	.00854	.00924	.00994	.01064	.01134
8	.00728	.00816	.00896	.00976	.01056	.01136	.01216	.01296
9	.00819	.00918	.01008	.01098	.01188	.01278	.01368	.01458
10	.00910	.01020	.01120	.01220	.01320	.01420	.01520	.01620
11	.01001	.01122	.01232	.01342	.01452	.01562	.01672	.01782
12	.01092	.01224	.01344	.01464	.01584	.01704	.01824	.01944
13	.01183	.01326	.01456	.01586	.01716	.01846	.01976	.02106
14	.01274	.01428	.01568	.01708	.01848	.01988	.02128	.02268
15	.01365	.01530	.01680	.01830	.01980	.02130	.02280	.02430
16	.01456	.01632	.01792	.01952	.02112	.02272	.02432	.02592
17	.01547	.01734	.01904	.02074	.02244	.02414	.02584	.02754
18	.01638	.01836	.02016	.02196	.02376	.02556	.02736	.02916
19	.01729	.01938	.02128	.02318	.02508	.02698	.02888	.03078
20	.01820	.02040	.02240	.02440	.02640	.02840	.03040	.03240
21	.01911	.02142	.02352	.02562	.02772	.02982	.03192	.03402
22	.02002	.02244	.02464	.02684	.02904	.03124	.03344	.03564
23	.02093	.02346	.02576	.02806	.03036	.03266	.03496	.03726
24	.02184	.02448	.02688	.02928	.03168	.03408	.03648	.03888
25	.02275	.02550	.02800	.03050	.03300	.03550	.03800	.04050
26	.02366	.02652	.02912	.03172	.03432	.03692	.03952	.04212
27	.02457	.02754	.03024	.03294	.03564	.03834	.04104	.04374
28	.02548	.02856	.03136	.03416	.03696	.03976	.04256	.04536
29	.02639	.02958	.03248	.03538	.03828	.04118	.04408	.04698
30	.02730	.03060	.03360	.03660	.03960	.04260	.04560	.04860
31	.02821	.03162	.03472	.03782	.04092	.04402	.04712	.05022
32	.02912	.03264	.03584	.03904	.04224	.04544	.04864	.05184
33	.03003	.03366	.03696	.04026	.04356	.04686	.05016	.05346
34	.03094	.03468	.03808	.04148	.04488	.04828	.05168	.05508
35	.03185	.03570	.03920	.04270	.04620	.04970	.05320	.05670
36	.03276	.03672	.04032	.04392	.04752	.05112	.05472	.05832
37	.03367	.03774	.04144	.04514	.04884	.05254	.05624	.05994
38	.03458	.03876	.04256	.04636	.05016	.05396	.05776	.06156
39	.03549	.03978	.04368	.04758	.05148	.05538	.05928	.06318
40	.03640	.04080	.04480	.04880	.05280	.05680	.06080	.06480

THE FIRST TABLE CONTINUED:
shewing the expansion with HEAT
on inches of the BAROMETER.
degrees of the THERMOMETER, from 1 to 40, on farenheit’s scale.
	17	18	19	20	21	22	23	24
1	.00172	.00182	.00192	.00203	.00213	.00223	.00233	.00243
2	.00344	.00364	.00384	.00406	.00426	.00446	.00466	.00486
3	.00516	.00546	.00576	.00609	.00639	.00669	.00699	.00729
4	.00688	.00728	.00768	.00812	.00852	.00892	.00932	.00972
5	.00860	.00910	.00960	.01015	.01065	.01115	.01165	.01215
6	.01032	.01092	.01152	.01218	.01278	.01338	.01398	.01458
7	.01204	.01274	.01344	.01421	.01491	.01561	.01631	.01701
8	.01376	.01456	.01536	.01624	.01704	.01784	.01864	.01944
9	.01548	.01638	.01728	.01827	.01917	.02007	.02097	.02187
10	.01720	.01820	.01920	.02030	.02130	.02230	.02330	.02430
11	.01892	.02002	.02112	.02233	.02343	.02453	.02563	.02673
12	.02064	.02184	.02304	.02436	.02556	.02676	.02796	.02916
13	.02236	.02366	.02496	.02639	.02769	.02899	.03029	.03159
14	.02408	.02548	.02688	.02842	.02982	.03122	.03262	.03402
15	.02580	.02730	.02880	.03045	.03195	.03345	.03495	.03645
16	.02752	.02912	.03072	.03248	.03408	.03568	.03728	.03888
17	.02924	.03094	.03264	.03451	.03621	.03791	.03961	.04131
18	.03096	.03276	.03456	.03654	.03834	.04014	.04194	.04374
19	.03268	.03458	.03648	.03857	.04047	.04237	.04427	.04617
20	.03440	.03640	.03840	.04060	.04260	.04460	.04660	.04860
21	.03612	.03822	.04032	.04263	.04473	.04683	.04893	.05103
22	.03784	.04004	.04224	.04466	.04686	.04906	.05126	.05346
23	.03956	.04186	.04416	.04669	.04899	.05129	.05359	.05589
24	.04128	.04368	.04608	.04872	.05112	.05352	.05592	.05832
25	.04300	.04550	.04800	.05075	.05325	.05575	.05825	.06075
26	.04472	.04732	.04992	.05278	.05538	.05798	.06058	.06318
27	.04644	.04914	.05184	.05481	.05751	.06021	.06291	.06561
28	.04816	.05096	.05376	.05684	.05964	.06244	.06524	.06804
29	.04988	.05278	.05568	.05887	.06177	.06467	.06757	.07047
30	.05160	.05460	.05760	.06090	.06390	.06690	.06990	.07290
31	.05332	.05642	.05952	.06293	.06603	.06913	.07223	.07533
32	.05504	.05824	.06144	.06496	.06816	.07139	.07456	.07776
33	.05676	.06006	.06336	.06699	.07029	.07359	.07689	.08019
34	.05848	.06188	.06528	.06902	.07242	.07582	.07922	.08262
35	.06020	.06350	.06720	.07105	.07455	.07805	.08155	.08505
36	.06192	.06534	.06912	.07308	.07668	.08028	.08388	.08748
37	.06364	.06716	.07104	.07511	.07881	.08251	.08621	.08991
38	.06536	.06892	.07296	.07714	.08094	.08474	.08854	.09234
39	.06708	.07078	.07488	.07917	.08307	.08697	.09087	.09477
40	.06880	.07260	.07680	.08120	.08520	.08920	.09320	.09720

THE FIRST TABLE CONCLUDED:
shewing the expansion with HEAT
on inches of the BAROMETER.
degrees of the THERMOMETER, from 1 to 40, on farenheit’s scale.
	25	26	27	28	29	30	31	32
1	.00253	.00263	.00274	.00284	.00294	.00304	.00314	.00324
2	.00506	.00526	.00548	.00568	.00588	.00608	.00628	.00648
3	.00759	.00789	.00822	.00852	.00882	.00912	.00942	.00972
4	.01012	.01052	.01096	.01136	.01176	.01216	.01256	.01296
5	.01265	.01315	.01370	.01420	.01470	.01520	.01570	.01620
6	.01518	.01578	.01644	.01704	.01764	.01824	.01884	.01944
7	.01771	.01841	.01918	.01988	.02058	.02128	.02198	.02268
8	.02024	.02104	.02192	.02272	.02352	.02432	.02512	.0259?
9	.02277	.02367	.02466	.02556	.02646	.02736	.02826	.02916
10	.02530	.02630	.02740	.02840	.02940	.03040	.03140	.03240
11	.02783	.02893	.03014	.03124	.03234	.03344	.03454	.03564
12	.03036	.03156	.03288	.03408	.03528	.03648	.03768	.03888
13	.03289	.03419	.03562	.03692	.03822	.03952	.04082	.04212
14	.03542	.03682	.03836	.03976	.04116	.04256	.04396	.04536
15	.03795	.03945	.04110	.04260	.04410	.04560	.04710	.04860
16	.04048	.04208	.04384	.04544	.04704	.04864	.05024	.05184
17	.04301	.04471	.04658	.04828	.04998	.05168	.05338	.05508
18	.04554	.04734	.04932	.05112	.05292	.05472	.05652	.05832
19	.04807	.04997	.05206	.05396	.05586	.05776	.05966	.06156
20	.05060	.05260	.05480	.05680	.05880	.06080	.06280	.06480
21	.05313	.05523	.05754	.05964	.06174	.06384	.06594	.06804
22	.05566	.05786	.06028	.06248	.06468	.06688	.06908	.07128
23	.05819	.06049	.06302	.06532	.06762	.06992	.07222	.07452
24	.06072	.06312	.06576	.06816	.07056	.07296	.07536	.07776
25	.06325	.06575	.06850	.07100	.07350	.07600	.07850	.08100
26	.06578	.06838	.07124	.07384	.07644	.07904	.08164	.08424
27	.06831	.07101	.07398	.07668	.07938	.08208	.08478	.08748
28	.07084	.07364	.07672	.07952	.08232	.08512	.0879	.09072
29	.07337	.07627	.07946	.08236	.08526	.08816	.09106	.09396
30	.07590	.07890	.08220	.08520	.08820	.09120	.09420	.09720
31	.07843	.08153	.08494	.08804	.09114	.09424	.09734	.10044
32	.08096	.08416	.08768	.09088	.09408	.09728	.10048	.10368
33	.08349	.08679	.09042	.09372	.09702	.10032	.10362	.10692
34	.08602	.08942	.09316	.09656	.09996	.10336	.10676	.11016
35	.08855	.09205	.09590	.09940	.10290	.10640	.10990	.11340
36	.09108	.09468	.09864	.10224	.10584	.10944	.11314	.11664
37	.09361	.09731	.10138	.10508	.10878	.11248	.11618	.11988
38	.09614	.09994	.10412	.10792	.11172	.11552	.11932	.12312
39	.09867	.10257	.10686	.11076	.11466	.11866	.12246	.12636
40	.10120	.10520	.10960	.11360	.11760	.12160	.12560	.12960

The 5th Step, applied in the first Example.

364. Add this, to the former Expansion, thus:

Inches 25.19 Tenths
with 4° on 25	.0101 Expansion
with 4° on .19	.0000076 Expansion
——————
The Answer is 25.2\|001076,

viz. the Point at which the Quicksilver woud stand, in the coldest Barometer, when equally expanded, i. e. of the same Temperature with the warmer. Reject all but the first Decimal as too minute: this is seen by a Line drawn between the first and second Decimal.

Practice will shew how far to proceed, without computing the decimal Parts of an Inch, to more than 4 Places; but it is always more exact, to follow minutely the above Rules.