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.