Philosophical Discoveries.
334. The many curious and interesting Conjectures which Mons. de Luc
(before the Invention of Balloons) throws out, in the Course of 4 large
Volumes, on the Subject and Qualities of the Atmosphere; may now be
determined by actual Trial.
335. The Abbée Bertholon wrote in 1784: and has particularly mentioned
the following Points, as capable of ample Investigation, and.
Discussion.
Sect. 336. Art. 1. _The Temperature of the Air at different Heights._
Which will determine whether the Atmosphere be _practically Navigable_,
at all Times and Places.
336. 2. _The dissolvent Power of the Air by Means of an Atmometer for
Evaporation._
Probably the Height may be determined, to which Clouds commonly ascend
in order to find the proper horizontal Level, in which Balloons can
move with the greatest Ease, Safety, and Expedition.
336. 3. _Variations of the Barometer._
This will ascertain the exact Height, without Mensuration.
336. 4. _The_ +densities+ _at different Heights_.
A principal Object in de Luc’s abstruse and scientific Researches: not
only useful but necessary to determine the Laws of Refraction; without
which, Astronomy, and consequently +navigation+, must remain defective.
336. 5. _The different Effects of Tastes and Odors, at different
Heights_: _Experiments on Plants and Animals: also of_ +sound+.[111]
These may produce new and salutary Effects on the human Body: and
determine how far a Change from hot, putrid, and impure, to cool pure
Air, impregnated with the invigorating aërial Acid, may contribute,
without the Aid of Drugs, to the Recovery of the Sick, and Invalid: or
promote Longevity.
336. 6. _The Direction and Velocity of the Wind._
The different Currents and their different Heights, the Limitation of
each Stratum of Wind, together with their different Temperatures at
the same Time, will point out the proper Paths for the Balloon to move
in, at all Times, and _possibly_ without the Necessity of accurate
Direction: the Mode of Ascent and Descent being _already_ known, and
proper Instructions given for a secure Landing.
336. 7. _Electricity of the Air_, +meteors+.
This may lead to the Birth Place of Lightening, and Methods how to
avoid its Effects in the Air. Tho’ it be already known, that little
Danger is to be apprehended, on Account of the mutual Repellency
between the electric Fluid, inflammable Gass, and oiled Silk.
The Irides, the Coronaes, Haloes, and other Phenomena of Colours: the
Generation and Solution of which may be investigated on the Spot.
336. 8. _Geography may become a new Science._
336. 9. _Use of the Balloon for Signals in the calm Air, above
Molestation; above Winds still blowing below: to discover the Positions
of an Army, or Navy._[112]
336. 10. _To throw principal Men into a Town: and convey others out of
it._
336. 11. With the Montgolfier Balloon, to try Experiments on Light, and
Fire: to transport great Weights: raise them out of the Water: draw up
Piles, raise Trees, Vessels, &c.
336. 12. The Parashute to secure a Man from too precipitate a Fall, is
to be 5 Yards in Diameter, when extended: the Man,—weighing 140 Pounds,
and the Parashute weighing 10 Pounds, with a Surface of 150 square
Feet,—woud, in that Case, feel no greater Shock than if he had fallen
from the Height of six Feet.
336. 13. _Compass and its Variations: also the different Branches in
Astronomy._
His Hints on the Direction of the Machine are ingenious.
337. 1. Wheels furnished with Wings.
337. 2. Imitations of the Form and Motions of Fish.[113]
337. 3. Vessels to condense Air, as the Bladders of Fish.
337. 4. Wind-Guns, Wind-Fountains.
337. 5. Elopile and Vapour Steam.
337. 6. Contrary Currents at different Heights: Proof of.
337. 7. New Hints for Balloons to be raised by Steam.
337. 8. Mons. Gouan’s Invention to go +three hundred miles a day in a
calm+.
338. The general Use to which Balloons seem capable of being applied,
with the Assistance of propulsive Machinery, in the Calm which exists
above the Level of a +contrary+ Wind; is that of a common Vehicle, not
subject to the Inconvenience of Roads and Inns, between distant Places
and Countries, for Passengers, properly accommodated in a Boat-shaped
covered Car, furnished with Provisions, and occasional Siberian
Cloathing: the Car to be surrounded with, and resting on Bladders, one
_fourth blown_, and having each a few Drops of Water within, to keep
them moist and elastic;—to prevent an _accidental_ Shock in alighting
on Land; and from sinking, if on Water.
Such a Conveyance (the Balloon being once made _Air-tight_, and
furnished with an _Air-Bottle_ to ascend and descend _without Loss_
of Gass) is ready at all Seasons and Times: both Night and Day: for,
as the Aironauts will enjoy continual Sunshine without a Cloud, from
his Rising to his Setting: so, during the Night, the Light of the
+stars+, always intercepted in its Passage to the Earth by Clouds or
thick Vapours, will be greatly augmented, when above both: besides
the probable Increase of Light _reflected_ from the upper Fields
of white Clouds shone on continually by the different Planets and
Constellations: all which will afford an Illumination equal, if not
greater, than that of a cloudless frosty Night, when the Ground is
covered with Snow.
And such Light will be sufficient to read or write by: also to examine
the _Barometer_,[114] in order to know the _Height_ and Level of
the Balloon above the Surface of the Earth; and the +compass+ for
Direction.
If Aironauts propose to ascend by Night, and in the Moon’s Quarters;
observing likewise the Precautions already given; it may be proper also
to consult and take with them the Ephèmeris, in order to know the Time
when the Moon rises, and also when she is at the highest, i.e. in the
South, or has remained about half her Time above the Horizon.
The plainest Points, on which not only the Success of an Excursion, but
the Lives of Aironauts may depend, are too frequently neglected, as
unimportant and trivial.
CHAPTER LXVII.
THE PROCESS OF INFLATION.
[Sidenote: Process of Inflation on the Day of Ascent, viz. on Thursday
the 8th Sept. 1785.]
Sect. 339. Art. 1. Three cylindric wooden Vessels were sunk more than
half their Depth into the Ground: two of them, each, 5 Feet Diameter,
and 5 Feet high: the third, 8 Feet in Diameter, and 8 Feet high.
An oblong Hole, 4 Inches by 3, was made in each Vessel: and each Hole
was furnished with a solid wooden Plug (made tapering) 6 Inches in
Length: throu’ these the Vitriol was poured.
Besides which, there was an oblong Opening in each Vessel, large enough
to admit a Workman, to distribute the Iron equally over the Bottom, and
to pour in Buckets of Water: which Openings were well stopped, as soon
as the Iron and Water were poured in.
As the vitriolic Acid is _corrosive_, burning the Skin or Cloaths; the
following Precautions were taken.
An occasional moveable Tub was provided, 3 Feet high, and 3 wide: in
the Center of whose Bottom was an oblong Aperture, equal to that in
each of the Vessels: a corresponding Tin Tube, 6 Inches long, and
narrowing to the Bottom, was nailed by its Border on the Inside of the
occasional Tub; so as to go easily into any of the oblong Holes.
A Bottle of Vitriol being brought in its Basket by two Men, and made
to rest on the Top of one of the fermenting Vessels; a third Assistant
held the occasional Tub in his Hands, with the Plug-Staff fastened in
the Aperture of the Tin Tube; and the Instant a fourth Person opened
the Hole in the fermenting Vessel; the Assistant placed the Tin Tube in
the Hole, keeping the Plug tight, to prevent the Escape of Gass.
The Bottle of Vitriol was then immediately poured into the occasional
Tub: and the Bottle being removed, the Plug-Staff was taken out, and
the Vitriol suffered to run into the fermenting Vessel: the Assistant
watching for the Instant when the Vitriol was run out, in order to
_force in_ the Plug-Staff again, and prevent the Escape of Gass: after
which, the Tub was rinced with a few Quarts of Water, let also into the
Vessel.
The same Tub was then removed: the oblong Hole in the fermenting
Vessel instantly covered; and, by driving down the solid wooden Plug,
continued _Air-tight_; by Means of moist Clay, and a little Water,
kept purposely on the Tops of each Vessel, to discover by the Bubbles,
whether Gass escaped.
[Sidenote: 20 Hundred Weight of Iron-Turnings.]
In these Vessels, early on the Morning of the Inflation, were
distributed 20 Hundred Weight, at 120lb. Averdupoise to the Hundred,
consisting of cast Iron-Filings, and of a Mixture of Cannon-Borings.
The Borings were bright and fresh when thrown into the Water: and any
Bits of Wood that swam, were skimmed off.
_Rusty_ Iron emits Gass, that is heavier than common Air, and therefore
is improper.
[Sidenote: 16 Bottles of Vitriol.]
At the same Time, 16 Bottles of concentrated vitriolic Acid, or as it
is improperly called Oil of Vitriol, were brought in their Packages
near the Place, to be ready for Use: each Bottle at an Average
containing 112 Pounds Averdupoise, of Vitriol: each full Bottle and
Package together weighing from 136 to 148 Pounds.
[Sidenote: 4 Pints of Water to a Pound Averdupoise of Acid.]
339. 2. To the Iron in each Vessel, was then poured a Quantity of
Water, which was measured in the Proportion of about 4 to 1: i.e. 4
_Pints_ of Water to one _Pound_, of the vitriolic Acid.
The Height of Water and Iron in each Vessel, being then gaged, was
about 14 Inches.
In a Line with the two smaller Vessels, and between them, was fixed
another wooden Vessel or Cistern, filled with Water.
[Sidenote: Improvements suggested.]
(N. B. Fresh Water ought to have flowed continually into it, and to
have run over the Top of the Cistern: for the same Quantity being
once saturated, can no longer absorb the alcaline and fixed Air to be
separated from the Gass before the latter enters the Balloon.)
In the Cistern was fixed a Stage, consisting of 4 long Feet, (reaching
to the Bottom of the Cistern,) nailed at their upper Ends to the Inside
of an inverted Tub or Funnel, so placed over the Center of the Cistern,
that 3 Inches of the lower Part of the Rim of the Funnel were under
the Surface of the Cistern-Water: the Funnel was _cylindric_, 3 Feet
across, and 2 Feet high.
An Open was cut, 1 Foot Diameter, in the Bottom of the inverted Funnel:
on the Circumference of which was nailed a Tin-Cylinder or common
Conductor, 2 Feet high: and at a _certain_ Angle, as most convenient,
was soldered a cylindric Arm, of equal Diameter, and 1 Foot long;
having a Lip, Ring or Rim, on its outward circular Edge.
Round this Rim was fastened a varnished Linen Tube, of equal Diameter
with the Cylinder.
At a small Distance, about a Yard from the Cistern, stood a slender
Stillage, 3 Feet high; on which was supported a detached Tin-Cylinder
or Connecter, 1 Foot long and 1 Foot Diameter, made with a Rim at
each End: in the Center of whose lower Side was soldered, at right
Angles, another Tin-Cylinder or Evacuatory, 6 Inches long and 6 wide:
its Use is to let out any Water, that the Heat of the Mixture might
cause to boil and rise up out of the fermenting Vessels: and thus be
_evacuated_, without entering the Balloon: or, if condensed in the
Balloon, might run out by the same Orifice.
The opposite End of the varnished Linen Tube was fastened round one End
of the detached Cylinder on the Stillage: and round the other, was tyed
the Neck or Bottom-Opening of the Balloon.
Each of the 2 smaller fermenting Vessels was furnished with a cylindric
Tin-Tube; each Tube 4 Inches and a half Diameter, nailed on the Outside
of a circular Opening in the Top or Head of each Vessel; communicating
by additional rectangular Bends under the Funnel and Water in the
Cistern: the great fermenting Vessel had 2 Tubes, each 4 Inches and a
half Diameter; communicating with the Funnel.
[Sidenote: Improvements suggested.]
340. The Process woud have been more complete, if the fermenting
Vessels had been sunk till their Tops were even with the Ground: and
plaistered round their Outsides with soft moist Clay, six Inches thick,
to keep them Air-tight.
Also, if the common Conductor had been only 1 Foot high: its horizontal
or rectangular Arm only 6 Inches long: the Linen Trunk but 3 Feet,
joining the Connecter on the Stillage 1 Foot high, to communicate with
the Neck of the Balloon; which Neck shoud be 3 Yards in Length, and its
circular Opening 1 Foot, at least in Diameter.
CHAPTER LXVIII.
[Sidenote: Inflation began about X. in the Morning.]
Section 341. The Process of inflating the Balloon began about X. in the
Morning, by pouring 4 Bottles of Vitriol, immediately one after the
other, into the occasional Tub, properly placed over one of the smaller
fermenting Vessels: the Tub being instantly rinced with a few Quarts of
Water, which was suffered to fall into the same Vessel.
The oblong Hole was left purposely open for a Minute, till the strong
Smell of the Gass was perceived above the Orifice: i.e. till the Gass
had pressed out all the common Air that remained floating over the
Surface of the Mixture in the fermenting Vessel: which Smell being
_plainly perceived_, the _solid_ Plug was immediately _driven_ down.
And presently the Gass was known to press forward with an elastic
Force throu’ the Tin Conductor, by the Motion it communicated to the
Surface of the Water in the Cistern: thence upwards throu’ the common
Conductor: at its Departure from both of which throu’ the Linen Trunk,
and Neck into the Balloon, the Gass makes a guggling obtuse Sound by
quick Intervals according to the Quantity of Gass protruded.
And as the Intervals encreased, a Judgment was formed, that the
Operation began to be less vigorous: and consequently that it became
necessary, either to renew it by an Addition of more Vitriol and Water
in the same Vessel, or to set the other small Vessel in Fermentation,
the latter of which Mr. Lunardi preferred: this happened about half an
Hour after the Vitriol was poured into the first Vessel.
342. After the second half Hour, eight Bottles were poured, by four at
a Time, into the great Vessel.
And at one o’Clock, the Balloon, without any farther Trouble was
beautifully inflated.
No Iron Rods were used to stir up the Borings or Filings at the Bottom
of the Vessels: the Vitriol being found so heavy as to penetrate them
as fast as the Iron, contiguous to the Vitriol, had parted with its
Gass.
At each of the two former Inflations, a similar Accident happened which
may be imputed to the same Cause.
343. During the first Inflation, the solid oblong wooden Plug fell into
one of the fermenting Vessels: the hot Vapour, forcibly issuing from
the Orifice, was condensed in the Form of a _white_ Smoke; which being
mistaken by the Company, a Cry was immediately heard of Fire, Fire:
on which the Workmen retreated. Mr. Lunardi incautiously thrust his
Arm into the Orifice to extract the Plug: at the same Time being much
burnt, and failing in the Attempt; the Gass continued to escape, till a
new Plug was prepared.
344. During the second Inflation, one of the Plugs being driven too
forcibly; it was with Difficulty extricated, by the Strokes of a Hammer
against the Sides of it, which tended at the same Time to displace the
Boards forming the Top or Head of the Vessel: and, a little afterwards,
occasioned it to burst, unexpectedly +inwards+,[115] rendering the
Vessel useless for the Purpose of Inflation.
_Observation._ Therefore instead of the solid oblong wooden Plug, a
circular Hole, 4 Inches Diameter shoud be drilled in each Vessel: and a
corresponding solid wooden Plug 8 Inches long, 5 Diameter at the upper
Part, and tapering to near 3 at the Bottom, shoud be prepared by the
Turner.
In the upper Part of the Solid shoud be turned an inside Screw, to
which an outside Screw of the circular Plug-Staff, made of Oak, Ash, or
other heavy Wood, 4 Feet long, and 4 Inches Diameter, shoud be adapted:
the Worm of the Screw to be 5 Inches long.
A wooden Peg of Ash, about a Quarter of an Inch Diameter, may be put
throu’ a Hole near the Top of the Staff, as a Handle.
A Lever of such a Length and Weight will probably answer every
Intention, as no sudden Blows will be required to _fasten or extract
it_.
The occasional Tub, Tube, Plug, and Staff, shoud be fashioned after
this Model.
345. _The Price of the Iron and Vitriol for Inflation._
2000lb. of Iron Filings or Borings[116] delivered
on the Spot, at 6s. a Hundred, £. 6 0 0
16 Bottles of Vitriol, at an Average
38s. a Bottle 30 8 0
Concomitant Expences, 3 12 0
———————
£. Total 40 0 0
_Observation 1._ A great Saving might be made by conducting the Process
in a different Manner.
The Author making two Journies to Manchester, purposely to observe the
Process by Mr. Sadler; found that his Balloon was inflated in two Hours
each Time; by Means only of the two smaller _identical_ fermenting
Vessels which Mr. Lunardi afterwards purchased; but the Levity procured
by the former, tho’ he also expended 16 Bottles, was by no Means so
great as that gained with the Assistance of the great Vessel.
It has likewise been remarked by the Author, who has made several
Experiments to this End, that the Vessels always continued in
Fermentation and Ebullition, with a _quick Pulsation_, for at least 24,
and commonly during 48 Hours, after the Inflation was completed.
And, that not more than the Depth of _half an Inch_ of Filings had been
_calcined_ during the Operation: the rest being perfectly _bright_, and
untouched by the Acid.
_Observation. 2._ If therefore one Inch in Depth of Filings, be spread
over the Bottom of each of the _smaller_ Vessels only; the proper
Quantity of Water poured in; and _not more_ than two Bottles of Acid
used at once, in each Vessel; also, as soon as the Fermentation begins
to decline; other two Bottles, and a proportionable Supply of Water be
added; if suffered to work double, triple, or quadruple the Time;—the
Inflation will be as great, if not greater, for Instance, in six Hours
with eight Bottles, and two small _Tubs_, as it woud in three Hours,
with 16 Bottles, in the _same Vessels_.
The small conducting Tin Tubes ought instead of four and a half, to be
nine Inches Diameter: by which Means there will be no violent Pressure
of Gass to endanger the Bursting of the Vessels: particularly if the
Gass is not suffered to descend; but, on the contrary, according to
Instructions already given, either to rise, or move, in an horizontal
Direction, past the Evàcuatory, into the Balloon.
346. The Workmen may begin the Operation at twelve at Night, or at six
in the Morning: and the Time previously fixed for the Exhibition, may
be eight or ten Hours after the Operation has commenced.
The Necessity of a Current of fresh Water, throu’ a Pipe of at least
half Inch Bore, the larger the better, to supply the overflowing
Cistern, cannot be too much _insisted_ on: as the Levity of the Gass
almost wholly depends upon so trivial a Circumstance, as that of having
a plentiful Supply of _cold fresh_ and _soft_ Water.
347. _Observation 3._ Supposing the Balloon +air-tight+, near half the
Expence is thus saved in the Inflation.
Besides the greater Probability of +calm+ Weather for the Inflation,
if completed before X. in the Morning, more Time is given to remedy
Accidents, and rectify Mistakes: the Warmth of the Air likewise
encreases.
But above all; if an upper Current carry the Balloon to Sea, the
Aironaut may, (as before mentioned) drop into the Sea-Breeze, which
will waft him safe back till IV. in the Afternoon, or even later.
CHAPTER LXIX.
MENSURATION OF HEIGHTS.
[Sidenote: Rules for calculating Heights by Means of the Barometer and
Thermometers.]
Section 348. Rules for calculating the Height of Mountains, when
applied to those elevated Stations in the Atmosphere _attainable_ only
by Means of the Balloon, will henceforward become more useful, and
be more frequently practised: as the Lives of Aironauts _may_ depend
on a Knowledge of their _Height_ above the Earth; which, not being
determinable by _Sight_, in _all Weathers_, or at all Times, must be
referred to the _Barometer_ and _Thermometers_, they carry up with them.
De Luc, Horseley, Maskelyne, Shuckburgh, and Roy, have each written
+ably+ on the Subject, in the _Transactions_: tho’ few have either
Leisure or Inclination to follow them.
Sir George Shuckburgh has made successful Attempts to smooth the Way,
by Examples and Tables, yet is still too concise for actual Learners,
and the Generality of those who will have Spirit enough to go before
the Calculators in exploring the Atmosphere; but cannot dedicate
sufficient Leisure to overtake them in their Studies.
Each may therefore assist the other.
349. Whoever is at the Trouble of comparing the Observations made by
Shuckburgh, with the Directions here given, will find that the latter
contains the _Essentials_ of the former, with this material Difference,
that the Investigation moves here by Steps, which are all pointed out
to the Learner; and not by Strides.
Each Step is self evident: and, by carrying Conviction to the Mind, is
just what the Mind itself woud make use of, in the Attainment of any
_distant_ Truth.
To do every Justice to Sir George, the Merit of whose Performance wants
no Eulogium; his three Precepts are copied; tho’ rather as a Memorandum
for those who understand the Methods; than as plain Directions for such
as are yet to learn them.
It will be found likewise, that the first, second, and fourth
Tables are greatly enlarged: being calculated for those _extreme_
Temperatures, and Heights, which the Balloon _only_ can attempt to
reach: and the fourth Table, for greater Dispatch in computing the
Expansion of the Air.
The Foundation and Construction of each Table, is also methodically
traced and elucidated.
CHAPTER LXX.
METHODS TO ASCERTAIN THE TRUE HEIGHT.
Section 350. Methods to be pursued on taking and comparing Heights, in
order to ascertain the true Height of any Station in the Atmosphere, by
the Barometer and Thermometers.
For this Purpose it is necessary, 1st, to provide a Barometer, (whose
Bulb or Cistern is _large_ enough to contain all the Quicksilver in the
Tube;)—into the Frame of which, a Thermometer, on _Farenheit’s_ Scale,
is to be fixed or _attached_.
The Use of the _attached Thermometer_ is to point out the Temperature
of the Barometer.
2d. A second or _detached Thermometer_ is also to be provided.[117]
This is to be hung in the Shade at the Distance of a Yard (or two) from
the other:—to shew the _general_ Temperature of the Air at the same
Time and Place: and may be called the _Air Thermometer_.
A proper Person, on the Ground, having a good Watch, with Pen Ink and
Paper at Hand, is to attend the Instruments _below_ every ten Minutes,
(or at any other _preconcerted_ Intervals of Time,) putting down,
1st. The Time of each Observation.
2d. The Point at which the Quicksilver stands in the Barometer.
3d. The Degree of Temperature of the _attached_ Thermometer.
4th, and lastly, the Degree of Temperature of the _detached_ or
_Air_-Thermometer.
This Employment is to be carefully attended to; during the Time, that
_similar_ Observations, by _preconcerted_ Agreement, are making, with
three other _similar_ Instruments, on the Top of the Mountain, or any
elevated Station in the Atmosphere, by Means of the _Balloon_; and to
be written with a _red Lead_ Pencil, in a Patent Asses Skin Pocket Book.
[Sidenote: The Instruments to be compared on Return from the Mountain,
or upper Station.]
Each single Observation, made with one Set of Instruments _below_, is
to be compared with each single corresponding Observation, made with
the other Set _above_.
And two Observations are said to _correspond_, when both are made
_nearly_ at the _same_ Time, the one _below_, and the other _above_.
351. Take Shuckburgh’s first Example, (Ph. Tr. for 1777, 2d Part, Page
577.) viz.
“Let the Point at which the Quicksilver stands in the Barometer, on
the Ground, be 29 Inches 4 tenths: the attached Thermometer 50 Degrees
of Temperature, and the Air Thermometer, or general Temperature of the
Air 45°: at the same Time, that at the Top of the Mountain, or other
elevated Station in the Atmosphere, the Barometer stands at 25 Inches
19 Tenths, the attached Thermometer at 46°, and the Air Thermometer at
39° and ½: required the upper Height in English Feet.”
[Sidenote: Rules for the Work: and Practice of the first Example.]
352. The Work is divided into three Stages.
The End proposed in this first Stage is to bring the colder Barometer,
to the same _Expansion_ or _Temperature_ with the _other_.
353. 1st. Step. First, write down the Observation made on the Ground,
or at the Bottom of the Mountain, thus:
+Below.+ Barometer, 29 Inches 4 Tenths. attached Thermometer, 50
Degrees. Air Thermometer, 45°.
354. 2d. Step. Secondly, write down the Observation made at the Top of
the Mountain, or upper Station in the Atmosphere, thus:
+Above.+ Barometer, 25 Inches, .19 Tenths. attached Thermometer, 46°.
Air Thermometer, 29½.
355. 3d Step. Subtract the _colder attached_ Thermometer, from the
other attached Thermometer, thus: 46 colder from 50 warmer, and there
remains 4° warmer, viz. the Number of Degrees of Temperature to which
the _colder_ Barometer must be _expanded_, before it becomes equal in
Temperature to the _warmer_ Barometer: each Barometer being always
supposed _equal_ in Temperature with its _attached_ Thermometer.
356. 4th Step. Give the _colder_ Barometer the same _Temperature_ with
the warmer: or, which amounts to the same, give the _colder_ Barometer
that _Expansion_ which is communicated by the Addition of 4 Degrees of
Temperature.
Both Barometers will then have the same _Temperature_, or _Expansion_,
viz. an Expansion equal to the warmer Barometer.
This is to be done by referring to the first Table, for the Application
of which there are separate Instructions: see the Explanation of the
first Table.[118]
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.
CHAPTER LXXII.
Section 365. Having therefore understood the Foundation, Construction,
and Use of the first Table; in the present Case, having also added the
decimal Parts of an Inch just found, for the Expansion,—to the Inches
and Tenths, expressing the colder Barometer; which will then have the
_same Expansion_, or _Temperature_ with the warmer, thus;
Inches.
25.19 _colder_ Barometer:
.0101 Expansion _on_ the same, in Parts of an
Inch with 4° of Temperature, (rejecting
all but the first Decimal as too
———————— minute,)
25.2|001 added; this Sum will express the
Point at which the Quicksilver in the colder Barometer
woud stand, when equally expanded, i.e.
in the same Temperature, with the warmer.
366. 6th Step. Place both Barometers, now of _equal_ Temperature with
the warmer, together, first, the _upper_ Barometer; and under it the
_lower_, thus: Inches 25. 2 Tenths.
29. 4
END OF THE FIRST STAGE.
* * * * *
367. The Ends proposed in the _second Stage_ of the Work, (the colder
Barometer being _now_ brought to the same Expansion or Temperature
with the warmer,) are two: First, to find, (by the Application of the
second Table) the Heights, in Feet and Tenths, in the Atmosphere,
corresponding to the Points at which the Quicksilver stands in both
Barometers, which have now the same Temperature, viz. that of the
warmer equal to 50°: on a Supposition that they were both exposed to
the Temperature of 31°.24, on Farenheit’s Scale, which is about the
Standard or freezing Point, for which sole Purpose the 2d Table is
calculated.
N. B. The _Second Stage_ includes two Steps only, viz. the 7th and
8th.
368. 7th Step. The Barometers being placed in one View, as before
directed, thus:
Upper Barometer, Inches 25 .2 Tenths.
Lower Barometer, Inches 29 .4; find, with the Temperature of 31°.24,
the corresponding Heights in the Atmosphere.
This is to be done by referring to the 2d Table, for the Application
of which there are separate Instructions: See the Explanation of the
second Table.[120]
CHAPTER LXXIII.
USE AND PRACTICE OF THE SECOND TABLE IN THE FIRST EXAMPLE.
_The_ +use+.
Section 369. To find the Heights, in Feet and Tenths, in the
Atmosphere, corresponding to the Points at which the Quicksilver
stands in both Barometers, which have _now_ the _same Temperature_,
viz. that of the _warmer_ Barometer, on a Supposition that they were
both exposed to the Standard-Temperature of 31°.24, on Farenheit’s
Scale.
_The_ +practice+.
_The 7th Step applied in the first Example._
370. Look at the first Column, in the _second_ Table, for
25.2, and the Answer is 6225.0 in the second Column; and for
29.4, and the Answer is 2208.2. The Answers are the Heights, in Feet
and Tenths, in the Atmosphere, at which the Quicksilver stands in both
Barometers, with the Temperature of 31°.24: corresponding to their
respective Points, for which _sole_ Purpose this Table is calculated.
371. 8th Step. Having placed the Barometers and their corresponding
Heights in the Atmosphere, shewn by the second Table, at one View:
subtract the lesser from the whole Height, and there will remain,
+secondly+; (see Section 367) the greater Height, viz. the Height
corresponding to the Barometer in the elevated Station, above the
Height corresponding to the Barometer, on the Ground, (both being at
the Temperature of 31°.24) thus:
Feet.
Inches 25.2 correspond to 6225.0
Inches 29.4 correspond to 2208.2; subtract:
——————
and the Remainder is 4016.8
viz. a Number in Feet and Tenths corresponding to the Height of the
upper above the lower Barometer, both being in the Temperature of
31°.34.
(See Page 295.)
THE SECOND TABLE.
The 1st Column shews the Quicksilver in the barometric Tube
standing at each Inch from 1 to 10, and at each Tenth from 10 to
32 Inches.
The 2d Column shews the Height of the barometric Tube, above
the _imaginary Level at 32 Inches_,—_with_ the Temperature of
31.24;—in Feet and Tenths, answering to Inches and Tenths of the
Barometer in the first Column.
The 3d Column shews the Height in Feet and Tenths, answering to a
Tenth of an Inch on the Barometer, being the +difference+ between
each two adjoining Heights in the 2d Column.
+-----+---------+-----------+
|Inch.| Feet. |Difference.|
| | | |
| 1 | 90309.0 | |
| | | 18061.8 |
| 2 | 72247.2 | |
| | | 10565.4 |
| 3 | 61681.8 | |
| | | 7496.4 |
| 4 | 54185.4 | |
| | | 5814.6 |
| 5 | 48370.8 | |
| | | 4750.9 |
| 6 | 43619.9 | |
| | | 4016.8 |
| 7 | 39603.1 | |
| | | 3479.5 |
| 8 | 36123.6 | |
| | | 3069.2 |
| 9 | 33054.4 | |
| | | 2745.4 |
|10.0 | 30309.0 | |
| | | 259.6 |
| .1 | 30049.4 | |
| | | 256.4 |
| .2 | 29793.0 | |
| | | 254.3 |
| .3 | 29538.7 | |
| | | 251.8 |
| .4 | 29286.9 | |
| | | 249.3 |
| .5 | 29037.6 | |
| | | 247.0 |
| .6 | 28790.6 | |
| | | 244.7 |
| .7 | 28545.9 | |
| | | 242.4 |
| .8 | 28303.5 | |
| | | 240.2 |
| .9 | 28063.3 | |
| | | 237.9 |
|11.0 | 27825.4 | |
| | | 235.8 |
| .1 | 27589.6 | |
| | | 233.7 |
| .2 | 27355.9 | |
| | | 231.6 |
| .3 | 27124.3 | |
| | | 229.6 |
| .4 | 26894.7 | |
| | | 227.6 |
| .5 | 26667.1 | |
| | | 225.6 |
| .6 | 26441.5 | |
| | | 223.7 |
| .7 | 26217.8 | |
| | | 221.7 |
| .8 | 25996.1 | |
| | | 220.0 |
| .9 | 25776.1 | |
| | | 218.0 |
|12.0 | 25558.1 | |
| | | 216.3 |
|12.1 | 25341.8 | |
| | | 214.4 |
| .2 | 25127.4 | |
| | | 212.7 |
| .3 | 24914.7 | |
| | | 211.0 |
| .4 | 24703.7 | |
| | | 209.3 |
| .5 | 24494.4 | |
| | | 207.7 |
| .6 | 24286.7 | |
| | | 206.0 |
| .7 | 24080.7 | |
| | | 204.3 |
| .8 | 23876.4 | |
| | | 202.8 |
| .9 | 23673.6 | |
| | | 201.2 |
|13.0 | 23472.4 | |
| | | 199.7 |
| .1 | 23272.7 | |
| | | 198.2 |
| .2 | 23074.5 | |
| | | 196.6 |
| .3 | 22877.9 | |
| | | 195.2 |
| .4 | 22682.7 | |
| | | 193.7 |
| .5 | 22489.0 | |
| | | 192.4 |
| .6 | 22296.6 | |
| | | 191.0 |
| .7 | 22105.6 | |
| | | 189.4 |
| .8 | 21916.2 | |
| | | 188.1 |
| .9 | 21728.1 | |
| | | 186.8 |
|14.0 | 21541.3 | |
| | | 185.5 |
| .1 | 21355.8 | |
| | | 184.1 |
| .2 | 21171.7 | |
| | | 182.9 |
| .3 | 20988.8 | |
| | | 181.6 |
| .4 | 20807.2 | |
| | | 180.3 |
| .5 | 20626.9 | |
| | | 179.0 |
| .6 | 20447.9 | |
| | | 178.0 |
| .7 | 20269.9 | |
| | | 176.7 |
| .8 | 20093.2 | |
| | | 175.4 |
| .9 | 19917.8 | |
| | | 174.3 |
|15.0 | 19743.5 | |
| | | 173.1 |
|15.1 | 19570.4 | |
| | | 172.0 |
| .2 | 19398.4 | |
| | | 170.9 |
| .3 | 19227.5 | |
| | | 169.8 |
| .4 | 19057.7 | |
| | | 168.6 |
| .5 | 18889.1 | |
| | | 167.6 |
| .6 | 18721.5 | |
| | | 166.5 |
| .7 | 18555.0 | |
| | | 165.4 |
| .8 | 18389.6 | |
| | | 164.1 |
| .9 | 18225.5 | |
| | | 163.7 |
|16.0 | 18061.8 | |
| | | 162.4 |
| .1 | 17899.4 | |
| | | 161.3 |
| .2 | 17738.1 | |
| | | 160.4 |
| .3 | 17577.7 | |
| | | 159.3 |
| .4 | 17418.4 | |
| | | 158.4 |
| .5 | 17260.0 | |
| | | 157.5 |
| .6 | 17102.5 | |
| | | 156.5 |
| .7 | 16946.0 | |
| | | 155.6 |
| .8 | 16790.4 | |
| | | 154.6 |
| .9 | 16635.8 | |
| | | 153.7 |
|17.0 | 16482.1 | |
| | | 152.9 |
| .1 | 16329.2 | |
| | | 151.9 |
| .2 | 16177.3 | |
| | | 151.1 |
| .3 | 16026.2 | |
| | | 150.2 |
| .4 | 15876.0 | |
| | | 149.3 |
| .5 | 15726.7 | |
| | | 148.5 |
| .6 | 15578.2 | |
| | | 147.6 |
| .7 | 15430.6 | |
| | | 146.8 |
| .8 | 15283.8 | |
| | | 146.0 |
| .9 | 15137.8 | |
| | | 145.2 |
|18.0 | 14992.6 | |
| | | 144.3 |
|18.1 | 14848.3 | |
| | | 143.6 |
| .2 | 14704.7 | |
| | | 142.8 |
| .3 | 14561.9 | |
| | | 142.0 |
| .4 | 14419.9 | |
| | | 141.2 |
| .5 | 14278.7 | |
| | | 140.5 |
| .6 | 14138.2 | |
| | | 139.7 |
| .7 | 13998.5 | |
| | | 139.0 |
| .8 | 13859.5 | |
| | | 138.2 |
| .9 | 13721.3 | |
| | | 137.5 |
|19.0 | 13583.8 | |
| | | 136.8 |
| .1 | 13447.0 | |
| | | 136.1 |
| .2 | 13310.9 | |
| | | 135.3 |
| .3 | 13175.6 | |
| | | 134.5 |
| .4 | 13041.1 | |
| | | 134.2 |
| .5 | 12906.9 | |
| | | 133.3 |
| .6 | 12773.6 | |
| | | 132.6 |
| .7 | 12641.0 | |
| | | 131.9 |
| .8 | 12509.1 | |
| | | 131.3 |
| .9 | 12377.8 | |
| | | 130.6 |
|20.0 | 12247.2 | |
| | | 130.0 |
| .1 | 12117.2 | |
| | | 129.3 |
| .2 | 11987.9 | |
| | | 128.7 |
| .3 | 11859.2 | |
| | | 128.0 |
| .4 | 11731.2 | |
| | | 127.4 |
| .5 | 11603.8 | |
| | | 126.8 |
| .6 | 11477.0 | |
| | | 126.2 |
| .7 | 11350.8 | |
| | | 125.6 |
| .8 | 11225.2 | |
| | | 125.0 |
| .9 | 11100.2 | |
| | | 124.4 |
|21.0 | 10975.8 | |
| | | 123.7 |
| .1 | 10852.1 | |
| | | 123.3 |
| .2 | 10728.8 | |
| | | 122.6 |
| .3 | 10606.2 | |
| | | 122.0 |
| .4 | 10484.2 | |
| | | 121.5 |
| .5 | 10362.7 | |
| | | 120.9 |
| .6 | 10241.8 | |
| | | 120.4 |
| .7 | 10121.4 | |
| | | 119.8 |
| .8 | 10001.6 | |
| | | 119.2 |
| .9 | 9882.4 | |
| | | 118.8 |
|22.0 | 9763.6 | |
| | | 118.1 |
|22.1 | 9645.5 | |
| | | 117.7 |
| .2 | 9527.8 | |
| | | 117.1 |
| .3 | 9410.7 | |
| | | 116.6 |
| .4 | 9294.1 | |
| | | 116.0 |
| .5 | 9178.1 | |
| | | 115.6 |
| .6 | 9062.5 | |
| | | 115.1 |
| .7 | 8947.4 | |
| | | 114.5 |
| .8 | 8832.9 | |
| | | 114.0 |
| .9 | 8718.9 | |
| | | 113.6 |
|23.0 | 8605.3 | |
| | | 113.0 |
| .1 | 8492.3 | |
| | | 112.6 |
| .2 | 8379.7 | |
| | | 112.1 |
| .3 | 8267.6 | |
| | | 111.6 |
| .4 | 8156.0 | |
| | | 111.1 |
| .5 | 8044.9 | |
| | | 110.6 |
| .6 | 7934.3 | |
| | | 110.2 |
| .7 | 7824.1 | |
| | | 109.7 |
| .8 | 7714.4 | |
| | | 109.3 |
| .9 | 7605.1 | |
| | | 108.8 |
|24.0 | 7496.3 | |
| | | 108.3 |
| .1 | 7388.0 | |
| | | 107.9 |
| .2 | 7280.1 | |
| | | 107.5 |
| .3 | 7172.6 | |
| | | 107.0 |
| .4 | 7065.6 | |
| | | 106.6 |
| .5 | 6959.0 | |
| | | 106.1 |
| .6 | 6852.9 | |
| | | 105.7 |
| .7 | 6747.2 | |
| | | 105.3 |
| .8 | 6641.9 | |
| | | 104.9 |
| .9 | 6537.0 | |
| | | 104.4 |
|25.0 | 6432.6 | |
| | | 104.0 |
| .1 | 6328.6 | |
| | | 103.6 |
| .2 | 6225.0 | |
| | | 103.2 |
| .3 | 6121.8 | |
| | | 102.8 |
| .4 | 6019.0 | |
| | | 102.4 |
| .5 | 5916.6 | |
| | | 102.0 |
| .6 | 5814.6 | |
| | | 101.6 |
| .7 | 5713.0 | |
| | | 101.2 |
| .8 | 5611.8 | |
| | | 100.8 |
| .9 | 5511.0 | |
| | | 100.6 |
|26.0 | 5410.4 | |
| | | 99.8 |
| .1 | 5310.6 | |
| | | 99.7 |
| .2 | 5210.9 | |
| | | 99.3 |
| .3 | 5111.6 | |
| | | 98.8 |
| .4 | 5012.8 | |
| | | 98.6 |
| .5 | 4914.2 | |
| | | 98.1 |
| .6 | 4816.1 | |
| | | 97.8 |
| .7 | 4718.3 | |
| | | 97.4 |
| .8 | 4620.9 | |
| | | 97.0 |
| .9 | 4523.9 | |
| | | 96.7 |
|27.0 | 4427.2 | |
| | | 96.4 |
| .1 | 4330.8 | |
| | | 95.9 |
| .2 | 4234.9 | |
| | | 95.7 |
| .3 | 4139.2 | |
| | | 95.2 |
| .4 | 4044.0 | |
| | | 95.0 |
| .5 | 3949.0 | |
| | | 94.5 |
| .6 | 3854.5 | |
| | | 94.3 |
| .7 | 3760.2 | |
| | | 93.9 |
| .8 | 3666.3 | |
| | | 93.6 |
| .9 | 3572.7 | |
| | | 93.2 |
|28.0 | 3479.5 | |
| | | 92.9 |
| .1 | 3386.6 | |
| | | 92.6 |
| .2 | 3294.0 | |
| | | 92.2 |
| .3 | 3201.8 | |
| | | 91.9 |
| .4 | 3109.9 | |
| | | 91.6 |
| .5 | 3018.3 | |
| | | 91.3 |
| .6 | 2927.0 | |
| | | 90.9 |
| .7 | 2836.1 | |
| | | 90.7 |
| .8 | 2745.4 | |
| | | 90.3 |
| .9 | 2655.1 | |
| | | 90.0 |
|29.0 | 2565.1 | |
| | | 89.7 |
| .1 | 2475.4 | |
| | | 89.4 |
| .2 | 2386.0 | |
| | | 89.1 |
| .3 | 2296.9 | |
| | | 88.7 |
| .4 | 2208.2 | |
| | | 88.5 |
| .5 | 2119.7 | |
| | | 88.2 |
| .6 | 2031.5 | |
| | | 87.9 |
| .7 | 1943.6 | |
| | | 87.6 |
| .8 | 1856.0 | |
| | | 87.3 |
| .9 | 1768.7 | |
| | | 87.0 |
|30.0 | 1681.7 | |
| | | 86.7 |
| .1 | 1595.0 | |
| | | 86.4 |
| .2 | 1508.6 | |
| | | 86.2 |
| .3 | 1422.4 | |
| | | 85.8 |
| .4 | 1236.6 | |
| | | 85.6 |
| .5 | 1251.0 | |
| | | 85.3 |
| .6 | 1165.7 | |
| | | 85.0 |
| .7 | 1080.7 | |
| | | 84.7 |
| .8 | 996.0 | |
| | | 84.5 |
| .9 | 911.5 | |
| | | 84.2 |
|31.0 | 827.3 | |
| | | 83.9 |
| .1 | 743.4 | |
| | | 83.7 |
| .2 | 659.7 | |
| | | 83.4 |
| .3 | 576.3 | |
| | | 83.1 |
| .4 | 493.2 | |
| | | 82.8 |
|31.5 | 410.4 | |
| | | 82.6 |
| .6 | 327.8 | |
| | | 82.4 |
| .7 | 245.4 | |
| | | 82.0 |
| .8 | 163.4 | |
| | | 81.8 |
| .9 | 81.6 | |
| | | 81.6 |
|32.0 | 00.0 | |
+-----+---------+-----------+
372. _Now_ apply the third Table, or Table for Tenths, _if necessary_;
including two more Steps, viz. the 9th and 10th: which, being useless,
in the first Example, are, for the present, omitted.
373. An Explanation of the third Table, or Table for Tenths, is,
however, for the Sake of _Order_, _here_ subjoined.[122] (See Page
298.)
THE THIRD TABLE, OR TABLE FOR TENTHS:
Serving to compleat the 2d Table, on Expansion of the Barometer,
with the Temperature of 31°.24.
1. The upper horizontal Figures shew the Number of Parts into which
the Tenth of an Inch has been divided.
2. The Figures in the left vertical Column express the Height
in +feet+, (above the imaginary Level, at 32 Inches of the
Barometer,) or Expansion corresponding to a single Tenth of an
Inch of Quicksilver.
3. The +feet+ in the Place of Meeting are called +tenths+: thus, 90
Feet are 9 Tenths of 100 Feet.
+-------+-----------------------------------------------------+
|_Feet._| _Parts into which the Tenth of an Inch is divided._ |
| | ⅒ | ²⁄₁₀ | ³⁄₁₀| ⁴⁄₁₀ | ⁵⁄₁₀ | ⁶⁄₁₀ | ⁷⁄₁₀ | ⁸⁄₁₀ | ⁹⁄₁₀ |
+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| 81 | 8 | 16 | 24 | 32 | 40 | 49 | 57 | 65 | 73 |
| 82 | 8 | 16 | 25 | 33 | 41 | 49 | 57 | 66 | 74 |
| 83 | 8 | 17 | 25 | 33 | 41 | 50 | 58 | 66 | 75 |
| 84 | 8 | 17 | 25 | 34 | 42 | 50 | 59 | 67 | 76 |
| 85 | 8 | 17 | 25 | 34 | 42 | 51 | 59 | 68 | 76 |
| 86 | 9 | 17 | 26 | 34 | 43 | 52 | 60 | 69 | 77 |
| 87 | 9 | 17 | 26 | 35 | 43 | 52 | 61 | 70 | 78 |
| 88 | 9 | 18 | 26 | 35 | 44 | 53 | 62 | 70 | 79 |
| 89 | 9 | 18 | 27 | 36 | 44 | 53 | 62 | 71 | 80 |
| 90 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 |
| 91 | 9 | 18 | 27 | 36 | 45 | 55 | 64 | 73 | 82 |
| 92 | 9 | 18 | 28 | 37 | 46 | 55 | 64 | 74 | 83 |
| 93 | 9 | 19 | 28 | 37 | 46 | 56 | 65 | 74 | 84 |
| 94 | 9 | 19 | 28 | 38 | 47 | 56 | 66 | 75 | 85 |
| 95 | 9 | 19 | 28 | 38 | 47 | 57 | 66 | 76 | 85 |
| 96 | 10 | 19 | 29 | 38 | 48 | 58 | 67 | 77 | 86 |
| 97 | 10 | 19 | 29 | 39 | 48 | 58 | 68 | 78 | 87 |
| 98 | 10 | 20 | 29 | 39 | 49 | 59 | 69 | 78 | 88 |
| 99 | 10 | 20 | 30 | 40 | 49 | 59 | 69 | 79 | 89 |
| 100 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 |
| 101 | 10 | 20 | 30 | 40 | 50 | 61 | 71 | 81 | 91 |
| 102 | 10 | 20 | 31 | 41 | 51 | 61 | 71 | 82 | 92 |
| 103 | 10 | 21 | 31 | 41 | 51 | 62 | 72 | 82 | 93 |
| 104 | 10 | 21 | 31 | 42 | 52 | 62 | 73 | 83 | 94 |
| 105 | 10 | 21 | 31 | 42 | 52 | 63 | 73 | 84 | 94 |
| 106 | 11 | 21 | 32 | 42 | 53 | 64 | 74 | 85 | 95 |
| 107 | 11 | 21 | 32 | 43 | 53 | 64 | 75 | 86 | 96 |
| 108 | 11 | 22 | 32 | 43 | 54 | 65 | 76 | 86 | 97 |
| 109 | 11 | 22 | 33 | 44 | 54 | 65 | 76 | 87 | 98 |
| 110 | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 |
| 111 | 11 | 22 | 33 | 44 | 55 | 67 | 78 | 89 | 100 |
| 112 | 11 | 22 | 34 | 45 | 56 | 67 | 78 | 90 | 101 |
| 113 | 11 | 23 | 34 | 45 | 56 | 68 | 79 | 90 | 102 |
| 114 | 11 | 23 | 34 | 46 | 57 | 68 | 80 | 91 | 103 |
| 115 | 11 | 23 | 34 | 46 | 57 | 69 | 80 | 92 | 103 |
| 116 | 12 | 23 | 35 | 46 | 58 | 70 | 81 | 93 | 104 |
| 117 | 12 | 23 | 35 | 47 | 58 | 70 | 82 | 94 | 105 |
| 118 | 12 | 24 | 35 | 47 | 59 | 71 | 83 | 94 | 106 |
| 119 | 12 | 24 | 36 | 48 | 59 | 71 | 83 | 95 | 107 |
| 120 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 |
| 121 | 12 | 24 | 36 | 48 | 60 | 73 | 85 | 97 | 109 |
| 122 | 12 | 24 | 37 | 49 | 61 | 73 | 85 | 98 | 110 |
| 123 | 12 | 25 | 37 | 49 | 61 | 74 | 86 | 98 | 111 |
| 124 | 12 | 25 | 37 | 50 | 62 | 74 | 87 | 99 | 112 |
| 125 | 12 | 25 | 37 | 50 | 62 | 75 | 87 | 100 | 112 |
| 126 | 13 | 25 | 38 | 50 | 63 | 76 | 88 | 101 | 113 |
| 127 | 13 | 25 | 38 | 51 | 63 | 76 | 89 | 102 | 114 |
| 128 | 13 | 26 | 38 | 51 | 64 | 77 | 90 | 102 | 113 |
| 129 | 13 | 26 | 39 | 52 | 64 | 77 | 90 | 103 | 116 |
| 130 | 13 | 26 | 39 | 52 | 65 | 78 | 91 | 104 | 117 |
+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
END OF THE SECOND STAGE.
374. The Ends proposed in the third and last Stage of the Work, are,
first, to add the _general_ Temperatures of the Air, or detached
Air-Thermometers, at each Place of Observation _above_ and _below_,
into one Sum.
Secondly, to divide that Sum: each Moiety of which is called the _mean
Temperature_ of the Air.
Thirdly, to apply that Moiety to each Barometer, (both of which have
been already brought to the Standard-Temperature of 31°. 24;) in order
to prove whether the Moiety (or Quantity of Heat assigned to each
Barometer by the _general_ Temperature of the Air) _exceeded_, _fell
short of_, or equalled the Standard-Temperature of the Barometers, by
the 2d Table.
And fourthly, from the Moiety or mean Temperature of the Air, to find
the true Height of the upper Barometer: which Temperature resolves
itself into three Cases.
375. 1st. If the Moiety or mean Temperature of the Air is greater than
the Standard Temperature, viz. that to which the Barometers are now
brought; find the Expansion of Air corresponding to such _Excess_ of
Temperature by the fourth Table, which Height by Expansion, being added
to the Height already found in the 2d Table, shews the true Height,
viz. of the upper Barometer.
N. B. The 3d and last Stage includes two Steps only, viz, 11th and
12th.
376. 11th Step. The detached Air-Thermometer _above_ was — — 39½
Degrees.
The detached Air-Thermometer _below_ was — — 45
1st. Add them, for the whole
Heat. — — — 2)84½ Degrees.
2d. For _mean Temperature_ of the
Air-Thermometers, or a _Moiety_ of
the Heat, divide by 2. — 42¼
3d. Deduct the Standard-Temperature
of — — — 31¼
from either Moiety, and the Remainder ———
— — — 11
is the 11 Degrees of Heat, more than the Standard[123]
for each Barometer.
For 42°¼, and 42°¼, equal to 84°½, was the whole Height of the Air at
both Places of Observation in the upper and lower Stations; of which
whole Height the detached or Air-Thermometer _above_ received 39°½, and
the detached or Air-Thermometer _below_, received 45°.
377. 12th Step. Find the Height corresponding to the Expansion of Air,
with Excess of Heat or Temperature above the Standard-Temperature of
the Barometers: and add it (as in the first Example) to the Height of
the upper Barometer, corresponding to the Standard-Temperature already
found in the _second_ Table, and the Sum is the _true_ Height of the
upper Barometer.
This is to be done by referring to the 4th Table, shewing Expansion
of Air with Heat; for the Application of which there are separate
Instructions: see the Explanation of the 4th Table.[124]
378. The Expansion of Air, in the first Example, is found by the
4th Table to be Feet 107.3 Tenths _higher_ than the 4016.8, viz. the
Remainder from the 2d Table (Section 371); which Numbers added give
4124.1 Feet: viz. the true Height of the upper Station required.
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._
CHAPTER LXXV.
PRACTICE OF THE SECOND EXAMPLE:
_With a distinct View of the Work. (Ph. Tr. for 1777, Page 579.)_
Section 386. The Point at which the Quicksilver stood in the Tube of
the Barometer on the Mountain, or in the Car of the Balloon, being
Inches 24.178 Tenths; its _attached_ Thermometer, Degrees 57.2 Tenths,
and its Air-Thermometer 56°; while the Barometer on the Ground stood
at Inches 28, .1318 Tenths; its _attached_ Thermometer, Degrees 61,
.8 Tenths, and its Air-Thermometer 63°, .9; what is the Height of the
_upper_ Station?
[Sidenote: 1st. Step.]
387. 1st. Step. Set down the Observation on the Ground, thus:
+Below+, Barometer, Inches 28, .1318 Tenths,
_Attached_ Thermometer, Degrees 61, .8 Tenths.
_Air_-Thermometer, 63°, .9.
[Sidenote: 2d. Step.]
388. 2d. Step. Set down the Observation, on the Mountain, or _in the
Car_, thus:
+Above+, Barometer, Inches 24, .178 Tenths.
_Attached_ Thermometer, Degrees 57, .2 Tenths.
_Air_-Thermom. 56°.
[Sidenote: 3d. Step.]
389. 3d. Step. From the _warmer attached_ Thermometer, subtract the
colder, thus:
61°, .8
57, .2
———————
4, .6
390. 4th. Step. Give the _colder_ Barometer the same Expansion, viz.
4°, .6 with the warmer, by the _first_ Table.
CHAPTER LXXVI.
PRACTICE OF THE FIRST TABLE IN THE SECOND EXAMPLE.
_4th Step applied in the 2d Example._
[Sidenote: _4th Step applied._]
Section 391. The _Order_ to be observed in finding the Expansion _with_
4°.6, i.e. with 4 Degrees, .6 Tenths of Heat, on 24.178, i.e. 24
Inches, .178 Tenths of the coldest Barometer.
Find the Expansion required, thus:
_Case the 1st._
1st. Part. _With_ 4° _on_ 24 Inches.
2d. Part. _With_ 4° _on_ .178 Tenths of an Inch above 24 Inches.
_Case the 2d._
1st. Part. _With_ .6 Tenths of a Degree, _on_ 24 Inches.
2d. Part. _With_ .6 Tenths of a Degree, _on_ .178 Tenths above 24
Inches.
+specifically+, _thus_:
1st. Part of _Case the 1st._ To find the Expansion,
_With_ 4° _on_ 24 Inches.
2d. Part of _Case the 1st._
_With_ 4°, _on_ .178 Tenths of an Inch above 24 Inches; begin thus:
_With_ 4°, _on_ 24 Inches: then,
_With_ 4°, _on_ 25: then,
_With_ 4°, _on_ 1 Inch above 24, i.e. _on_ the 25th Inch: then,
_With_ 4°, _on_ .1 Tenth above 24: then,
_With_ 4°, _on_ .178 Tenths above 24.
1st Part of _Case the 2d._ To find the Expansion,
_With_ .6 above 4° _on_ 24; begin thus:
_With_ 4° _on_ 24 Inches: then,
_With_ 5° _on_ 24: then,
_With_ 1° above 4°, _on_ 24, i.e. the 5th°: then,
_With_ .1 Tenth above 4°, _on_ 24: then
_With_ .6 Tenths above 4°, _on_ 24.
2d Part of _Case the 2d._ To find the Expansion,
_With_ .6 Tenths above 4° of Heat _on_ .178 Tenths above 24 Inches: to
be done thus:
_The_ +expansion+ _with 4°, on .178 Tenths above 24 Inches, being once
found; divide_ +it+ _by 4: and the Quotient is the Expansion with 1°
above 4°, on .178 Tenths of an Inch above 24 Inches_.
_Then for the Expansion with .1 Tenth above 4°, on .178 Tenths above
24 Inches; add a Cypher and decimal Point to the left of the same
Quotient._
_Then for the Expansion with .6; multiply that Sum into .6, and add a
Cypher and decimal Point._
_The Answer is the_ +part+ _of an Inch, to which .6 Tenths of a Degree
above 4° of Heat, on .178 Tenths of an Inch above 24 Inches, raises the
Barometer_.
_It is true, the_ +part+ _is so minute as to be rejected: yet the Mode
of Proceeding, in order to investigate the Expansion with Precision, is
proper to be retained_.
392. +practice+ of the first Part of _Case the 1st._
For the Expansion _with_ 4°, _on_ 24 Inches; look, in the first Table,
(Sect. 363) and in the left vertical Column, _with_ 4 Degrees of the
Thermometer; and along the upper horizontal Line, _on_ 24 Inches of
Quicksilver in the Tube of the Barometer: the Point of Meeting gives
the Expansion .0097;[128] which, preparatory to Addition,
is to be placed under the 24, .178 thus,
.0097
+practice+ of the 2d Part of _Case the first_.
393. In order to obtain the Expansion, _with_ 4°, of Heat _on_
.178 Tenths of an Inch above 24 Inches of the Barometer; let it be
considered where it ought to be found in the Table: for, Tenths of 1
Inch above 24 Inches, are at some intermediate Point between 24 and 25;
that is, above 24, yet not so high as 25: or more than 24, yet less
than 25.
Look therefore in the Table, _with_ 4 Degrees of Heat, _on_ 24 Inches;
then _with_ 4° _on_ 25 Inches: and the respective Numbers are .0097 and
.0101.
And by taking the Expansion _with_ 4° _on_ 24 Inches, from 4° on 25;
the Remainder will be the Expansion with 4° on 1 Inch above 24 Inches,
viz. on the 25th Inch, thus:
} 25 = .0101 from;
_With_ 4° _on_ }
} 24 = .0097 subtract:
—————
.0004:
This therefore is the Expansion _with_ 4°, _on_ 1 Inch above 24 Inches.
Then _with_ 4°, _on_ .1 Tenth of an Inch above 24 Inches.
The Answer is the same as the former, viz. .0004, with the Addition
of a Cypher and decimal Point to the left, thus; .0004 becomes .00004,
viz. the Expansion _with_ 4°, _on_ .1 Tenth of an Inch above 24 Inches.
Then for the Expansion _with_ 4°, _on_ .178 Tenths, say,
If the Expansion _with_ 4°, _on_ .1 Tenth above 24 Inches gives .00004
Part of an Inch, what will the Expansion _with_ 4°, _on_ .178 give?
Thus; .1 : .00004 :: .178?
Multiply the two last Terms, thus:
.00004
.178
——————
00032
00028
00004
———————
0000712:
and, as in Multiplication of Decimals, the Product must have as many
decimal Places, as are in the Factors; a Cypher must be added to the
left Hand, thus: .00000712: but having divided that Product by the
first Term .1, viz. a Decimal, the Answer is a Cypher less; viz.
.0000712.
This Answer is the Expansion _with_ 4°, _on_ .178 Tenths of an Inch
above 24 Inches: prepare it for _Addition_, as the former,
24.178
.0097
.0000712
+practice+ of the first Part of _Case the 2d._
394. For the Expansion of .6 Tenths of a Degree of Heat, (more than
the 4 Degrees) on 24 Inches of the _coldest_ Barometer; it shoud be
considered where such Tenths can lie in the Table.
Now .6 Tenths of 1 Degree, (more than the 4°) are at some intermediate
Point of the Thermometer between 1 and 2 Degrees: above 1; yet not so
high as 2: or more than 1; yet less than 2.
Therefore .6 Tenths of 1 Degree above 4 Degrees, are somewhere between
the 4th and 5th Degree: above 4; yet not so high as 5: or more than 4;
yet less than 5.
Look in the Table (Section 363); first _with_ 4 Degrees of Heat, _on_
24 Inches, and then _with_ 5 Degrees of Heat _on_ 24 Inches; and the
respective Numbers are .0097 and .0121: and by taking the Expansion
_with_ 4 Degrees _on_ 24 Inches, from the Expansion _with_ 5 Degrees
_on_ the same 24 Inches; the Remainder will be the Expansion _with_ 1
Degree above 4° _on_ 24 Inches: viz.
_with_ {5° = .0121} _on_ 24 Inches, as in whole
{4° = .0097} Numbers.
—————
Remainder, .0024
This therefore is the Expansion _with_ 1 Degree of Heat, above 4, viz.
_with_ the 5th Degree, _on_ 24 Inches of the Barometer.
Then say, if 1 Degree of the Thermometer (above 4, viz. the 5th Degree)
gives by Expansion, a certain additional Height, or Part of an Inch,
viz. .0024, _on_ 24 Inches of the Barometer; what Height will 6 Degrees
give? Answer 6 Times _more_.
Multiply the 2d and 3d Terms, and divide by the first, thus;
1 : .0024 :: 6?
6
—————
.0144
is the Expansion, or Height, in Parts of an Inch, for 6 Degrees.
And farther, to proportion for the Decimal; say as .1 Tenth of a Degree
gives a certain Tenth of the former .0024, in additional Height, viz.
.00024; what Height will .6 Tenths give? Answer, .00144.
Prepare this _Height_ for Addition to the Numbers already found.
+practice+ of the 2d Part of _Case the 2d._
395. To find the Expansion of .6 above 4° on .178 above 24 Inches.
The Expansion _with_ 4° _on_ .178 is already found to be .0000712:
divide it by 4, and the Answer is .0000178, viz. the Expansion _with_
1° _on_ .178 above 24 Inches:
And, for the Expansion with .1 Tenth; the Answer, with the Addition of
a Cypher and decimal Point to the left, becomes .00000178.
Lastly, for the Expansion with .6, say,
If .1 : .00000178 :: .6?
Multiply the 2d and 3d Terms, and divide by first:
.00000178
.6
——————————
.000001068.
The Answer is a Decimal less, viz. .00001068; i.e. the Decimal of an
Inch, to which .6 Tenths of a Degree above 4 Degrees of Heat, on .178
Tenths of an Inch above 24 Inches, raises the Barometer: which, after
all, is so inconsiderable, that it may be fairly rejected.
Yet the Rules by which these Deductions are made, may be useful in
other Cases.
Prepare for Addition, as before.
The Decimals, in the Answers, may be omitted, when they exceed four
Places.
[Sidenote: 5th Step.]
396. 5th Step. To proceed with the second Example.
Place the different Expansions now found, above each other, Units,
Tens, &c. under Units, Tens, &c. preparatory to Addition, thus;
For the Expansion _with_ 4°, .6 _on_ 24, .178:
1st. _with_ 4°, _on_ 24, .0097
2d. _with_ .6 _on_ 24, .00144
3d. _with_ 4°, _on_ .178 .0000712
4th. _with_ .6 _on_ .178 .00001068
—————————
The Expansions with 4°,.6 added = .01122188
To the Sum add the Height of
the _colder_ Barometer 24.178
———————————
24.1892|
The Answer is Height of the _colder_ Barometer, now equal in
Temperature to the _warmer_: (rejecting all but the four first
Decimals.)
[Sidenote: 6th Step.]
397. 6th Step. Place the Barometers _now_ of the same Temperature, i.e.
_equal_ to the warmer, in one View, thus:
1st. the _upper_ Barometer, 24.1892
2d. the _lower_ Barometer, 28.1328
END OF THE FIRST STAGE.
_The 7th Step applied in the second Example._
[Sidenote: _7th Step._]
398. Find the Height, in Feet, in the 2d Column of the 2d Table,
corresponding to Inches and Tenths of the _upper_ barometric Tube, in
the 1st. Column of the same Table, thus: (Sect. 371.)
The Barometer standing at 24.1892; it must be considered where, in the
2d Column of the 2d Table, a Height corresponding to _such_ Inches
and Tenths can lie: and the Answer is, somewhere _above_ 24 Inches .1
Tenth, but not so high as 24 Inches .2 Tenths: 24 Inches .1892 Tenths,
being _more_ than 24 Inches .1 Tenth, but _less_ than 24 Inches .2
Tenths.
First then, look in the 1st Column for Inches 24, .1 Tenth; and the
corresponding Height in Feet is 7388.0: but the Height for 24, .2, in
the 2d Column, beneath the former Number, is _only_ 7280.1.
[Sidenote: 8th Step.]
399. 8th Step. Subtract the latter from the former and the Remainder
is 107.9, the same as in the 3d Column: viz. the Height, in Feet and
Tenths, corresponding to one Tenth only, namely, the ist Tenth above
Inches 24, .1 Tenth: with the Temperature of 31.24 of Farenheit, for
which sole Purpose the 2d Table is calculated.
A new Question _then_ arises, viz. what are the Heights in Feet and
Tenths, corresponding to the remaining Tenths or Decimals of an Inch
above Inches 24, .1 Tenth,
viz. .08
.009
.0002? which is to be resolved, by Application of the 3d Table,
or Table for _Tenths_, which see, (Section 373.)
[Sidenote: _9th Step._]
400. _9th Step applied in the 2d. Example._
First for the _upper_ Barometer.
Look in the Table for Tenths, in the left vertical Column with 107,
(rejecting the .9, as too minute;) and along the horizontal Line at the
top, with 8: and find the Answer _gradually_, thus:
1st. With 107, and 8, (as a whole Number,) answering to .08: which, in
the Place of Meeting, gives 86 Feet.
2d. With 107, and 9, (as a whole Number,) answering to .009: which, in
the Place of Meeting, gives 97.
3d. With 107, and 2, (as a whole Number,) answering to 0002: which, in
the Place of Meeting, gives 21.
Place them in View, and add, and bring them back again into Decimals,
thus:
With 107 and 8, answering to .08 giving 86. Feet
and 9, to .009 9.7
and 2, to .0002 .21
—————
95.9|1
(Next: with the 9, _if required_; which was before rejected:) but there
being no .9 Tenths in the left Vertical, call it 90, and allow for it
in each Answer by moving the decimal Point two Places to the left,
thus: with
90, and 8, answering to .08 giving 72 = .72
and 9, to .009 81 = .081
and 2, to .0002 18 = .0018
————
To .8|00|28
Add the former Sum 95.9|
—————
Total = 96.7)
Which 95.9 is the _Height_ in Feet and Tenths corresponding to .0892
Decimals of an Inch above Inches 24 .1 Tenth: and 24 .1 gave Feet
7388.0 in _Height_; therefore an additional _Height_, of so many Tenths
of an Inch of Quicksilver in the Tube of the Barometer, must give in
Feet, a _less_ Height of the Barometer elevated above the _imaginary_
Level indicated at 32 Inches.
[Sidenote: _10th. Step._]
401. 10th. Step. Subtract the _Height_ in Feet, corresponding to the
_Expansion_ on .0892 Tenths of an Inch, (_less_ than Inches 24.2
Tenths, of the _upper_ barometric Tube,) from the _Height_, in Feet,
corresponding to _the Expansion on_ Inches 24.1 Tenth of the same
barometric Tube, continuing at the Standard Heat,[129]
viz. 7388.0
95.9
——————
The Remainder 7292.1 gives the real, viz. the _less_ Height of the
_upper_ Barometer, at 24.1892 with the Standard Temperature.
Repeat the same Process, viz. the 9th. and 10th. Steps, for the _lower_
Barometer, thus:
For the lower Barometer in the 2d. Example.
First, Find the Height, in _Feet_, of the lower Barometer, standing
at Inches 28.1318 Tenths, in the 2d. Column of the 2d. Table,
corresponding to Inches and Tenths of the Quicksilver in the barometric
Tube, in the first Column of the same Table, thus:
The lower Barometer standing at 28.1318; it must be considered, where
in the 2d. Column of the 2d. Table, a Height corresponding to such
Inches and Tenths can lye: and the Answer is, somewhere above 28
Inches, .1 Tenth, but not so high as 28 Inches .2 Tenths: 28.1318
Tenths being more than 28 Inches .1 Tenth, yet less than 28 Inches .2
Tenths.
First, then, look, in the first Column for 28.1, and the corresponding
Height, in Feet, is 3386.6:
but the Height for 28.2, is only 3294.0:
——————
subtracting the less from the greater;
the Remainder is 92.6,
the same as in the 3d. Column, viz. the Height, in Feet and Tenths,
corresponding to _one Tenth only_ above 28.1.
Having therefore found that Feet 92.6 Tenths, are the Height,
corresponding to one Tenth only above Inches 28.1 Tenth, of the lower
Barometer, with the Temperature of freezing; for which _sole_ Purpose,
the 2d Table is calculated;—a new Question arises, viz. what are the
Heights, in Feet and Tenths, corresponding to the remaining Decimals
above 28.1, viz.
.03
.001
.0008; to be resolved by Application of the third Table, or Table for
Tenths, which see, (in Section 373.)
Look in the 3d. Table, with 92, (omitting the .6 as too minute) and with
3 answering to .03, which gives 28 = Feet 28.
1 to .001, 9 = .9
8 to .0008, 74 = .74
——————
29.6|4
Which 29.6 is the _Height_ in Feet and Tenths corresponding to .0318
Tenths above Inches 28.1 Tenth: and Inches 28.1 Tenth gave Feet 3386.6
Tenths in Height: therefore an additional Height of so many Tenths or
Decimals of an Inch of Quicksilver in the Tube of the Barometer, must
give in Feet, a _less_ Height of the _lower_ Barometer, elevated above
the _imaginary_ Level indicated by the Quicksilver resting in the Tube
at 32 Inches.[130]
402. Therefore subtract the _Height_, in Feet, corresponding to the
_Expansion on_ .0318 Tenths of an Inch (_less_ than Inches 28.2
Tenths of the _lower_ barometric Tube,) from the Height, in Feet,
corresponding to the _Expansion on_ 28.1 Tenth of the same Barometer,
viz.
3386.6
29.6
——————
and the Remainder - 3357.0, gives the _real_ Height in Feet of the
lower Barometer, at 28.1318 when above the _imaginary_ Level, and with
the Temperature of _freezing_ by the second Table.
403. Then, by taking the Number of Feet and Tenths _above_ the
imaginary Level, (indicated by the Quicksilver, in both Tubes, resting
at 32 Inches) answering to the _Expansion on_ Inches and Tenths of
the _lower_ Tube, from the Number of Feet, &c. by the former Process,
answering to that of the _upper_ Tube; viz.
_upper_ 7292.1
_lower_ 3357.0
——————
the remaining Feet 3935.1 Tenth is the _Height_, by which the
_Station_ of the _upper_ Barometer exceeds the _Station_ of the
_lower_; both being at the Temperature of 31°.24 on Farenheit’s Scale.
See Section 371.
END OF THE SECOND STAGE
* * * * *
[Sidenote: 11th Step.]
Section 404. 11th Step.
(See the Practice in the 1st Example, Sect. 376.)
_Air_-Thermom. +above+ was 56°.
_Air_-Thermom. +below+ was 63.9
—————
Whole Heat 119.9(0 adding a Cypher)
Half Heat 59.95
Standard-Heat 31.24
which deduct; and there ——————
remains each Moiety, 28.71
above the Standard-Heat.
[Sidenote: _12th Step._]
405. 12th Step. (See the Practice in the first Example, Section 377.)
By the fourth Table, find the Expansion of Air, _with_ 28.71, (more
than the Standard-Temperature) _on_ Feet 3935, .1 Tenth, gradually,
thus:
406.
_First_ _with_ 28° _on_ Feet 3000 = 204.1[131]
900 as 9000 = 612.3
30 3000 = 204.1
5 5000 = 340.1
.1 1000 = 68.0
Note: 1st. The decimal Point in the Answer corresponding to the Place
of _Thousands_, in the Question, is to remain, as taken from the Table
calculated for thousand Feet, thus: 204.1.
2d. For _Hundreds_ in the Question, remove the decimal Point _one
Place_ in the Answer, thus: 612.3 becomes 61.23:
3d. For _Tens_, _two_ Places, thus: 204.1 becomes 2.041:
4th. For _Units_, _three_ Places, thus: 340.1 becomes .3401:
5th. And for each _Decimal_, a Place more, by adding Cyphers to the
left, if wanted, thus: 68.0 becomes .00680.
407. Place the plain and decimated Answers, in one View, and add the
latter together, thus:
204.1 = the same 204.1
612.3 = becomes 61.23
204.1 = 2.041
340.1 = .3401
68.0 = .00680
—————————
viz. Expansion of Air _with_} 267.7|179
28° _on_ 3935.1 }
408. _Second_, _with_ .71° _on_ Feet 3000 = 517.5
900 as 9000 = 1552.7
30 3000 = 517.5
5 5000 = 862.6
.1 1000 = 172.5
In order to decimate these Answers, it must be observed that the
Expansion was not _with_ 71 Degrees, but with .71 _Tenths_ of a Degree
of Heat; therefore the decimal Point corresponding to 3000 Feet in the
Question, must in the Answer be removed _two_ Places to the left, thus:
517.5 becomes 5.175: for the 100, three Places: for
1.5527 the 10, _four_ Places: and so
.05175 on.
.008626
.0001725
——————————
6.7|882485
The Expansion with .71 being found, viz.
Feet 6.7 Tenths; add it to the Expansion on
28 Feet already found, viz.
267.7
—————
274.4 Answer.
Which _Height_ in Feet and Tenths, corresponding to the _Expansion_
of Air with 28°.71 Tenths of a Degree of Heat more than the Standard
31°.24, being added to the _Height_ in Feet and Tenths, corresponding
to the _Expansion on Inches_ of the Quicksilver in the _upper_
Barometer, with the Standard-Heat, already found,
viz. 3935.1
gives the _real Height_ of the _Mountain_, 274.4
——————
or _upper Station_, sought. 4209.5
END OF THE THIRD STAGE.
* * * * *
_The second Example_ briefly _stated: referring to the Sections._
[Sidenote: Section, 391.]
409. Below: Barometer 28.1318.
Attached Thermometer 61°.8; Air ditto 63.9.
Above: Barom. 24.178.
Attached Thermometer 57°.2; Air ditto 56°. Degrees of Heat, viz. 4°.6
to be added to the _colder_ Barometer at Inches 24.178 Tenths, by the
first Table, viz. .0112
Parts of an Inch of the Quicksilver
in the Barometer,
raised by 4°.6 of Heat. ———————
The Sum 24.1892
is the +point+, in Inches and Tenths of an Inch, at which the upper
Barometer _now_ rests, being of _equal_ Heat with the lower.
_End of the first Stage._
[Sidenote: Section, 399.]
By the 2d. Table, find the _Height_, in Feet and Tenths, corresponding
to the _said_ +point+ when at the Standard-Heat; gradually, thus: the
_Height_ corresponding to Feet 24.1 is 7388.0: then with the Difference
107.9, (rejecting the .9).
[Sidenote: Section, 400.]
Find the Height by the 3d. Table corresponding to
.08 86.0 }
.009 9.7 } = Feet 95.9 Tenths.
.0002 .2 }
Which Height subtract from 7388.0
95.9
——————
And there remains, in Feet, 7292.1
The Height corresponding to Inches 24.1892 Tenths of the _upper_
Barometer, with the Standard Temperature of 31.24; for which sole
Purpose the 2d. Table is calculated.
Repeat the last Process with the _lower_ Barometer, resting at 28.1318,
gradually, thus:
[Sidenote: Section, 401.]
By the 2d. Table, find the _Height_ corresponding to 28.1, which is
3386.61; then with the Difference 92.6 (rejecting the .6) find the
corresponding _Height_, by the 3d. Table for the remaining Tenths or
Decimals of an Inch, above 28.1, viz.
.03 28.0 }
.001 .9 } = Feet 29.6 Tenths.
.0008 .7 }
[Sidenote: Section 402.]
Which _Height_ subtract from 3386.6
29.6
——————
And there remains, 3357.0 viz. the _Height_ in Feet
corresponding to Inches 28.1318 Tenths of the lower Barometer, with the
Standard Temperature of 31.24, for which sole Purpose the 2d. Table is
calculated.
[Sidenote: Section 403.]
Subtract the _Height_ in Feet, corresponding to Inches of Quicksilver
in the upper Barometer,
viz. 7292.1 from ditto in _lower_ Barometer,
viz. 3357.0 and there remains the _Height_ in Feet
—————— of the upper Barometer at the Standard-Temperature
viz. 3935.1 of 31.24.
_End of the second Stage._
[Sidenote: Section, 404.]
On which Number of Feet, viz. 3935.1, by the 4th Table, find the
_Height_, with 28°.71 of Heat:
_With_ 28°. _on_ Feet 3935.1 = 267.7 and
_With_ .71 _on_ the same = 6.7
—————
Sum 274.4: which
Height, more than the Standard-Heat,
being _added_ to 3935.1
the Height, with the Standard, ——————
gives the true Height, viz. 4209.5.
_End of the third Stage._
CHAPTER LXXVII.
PRACTICE OF THE THIRD EXAMPLE,
REFERRING TO THE SECTIONS.[132]
Section 410. _Below_: Barom. Inches 30, .0168:
Attached Therm. 60°.6; Air-ditto, 60°.2:
_Above_: Barom. - - Inches 29, .5218:
Attached Therm. 56°.6; Air-ditto, 57°.
Subtract the _colder_ ————— from the _warmer_,
and there remains 4° of Heat to be added to the _colder_ Barometer; to
give it an _equal_ Temperature: which is to be done by _the 1st Table_,
thus:
[Sidenote: Section, 356.]
To find the Expansion _with_ 4° of Heat, _on_ the _colder_ Barometer;
(which, as before, is the _upper_ Barometer) standing at Inches 29,
.5218 Tenths.
_First_, _with_ 4° _on_ 29 Inches = .0117:
_2d_, _with_ 4° _on_ .5218 Tenths _above_ 29 Inches:
In order to obtain which, begin
_with_ 4° _on_ 29 = .0117
then _with_ 4° _on_ 30 = .0121
Subtract for the Expansion _with_ —————
4° _on_ 1 Inch above 29, and there
remains .0004.
[Sidenote: Section 362.]
Then for the Expansion _with_ 4° _on_ .1 Tenth of an Inch above 29
Inches; add a Cypher and
decimal Point, viz. .00004:
[Sidenote: Section 363.]
Then for the Expansion on .5128, multiply
the two last Terms, and divide —————
the Product by the first Term
.1: the Answer is .0002|0872
Add the Expansion _with_ 4° _on_
29 Inches, just found, .0117
to the Inches of the _colder_
Barometer, viz. 29.5218
———————
Answer; Inches 29.5337 Tenths of the _colder_
Barometer, are _now_ expanded equally with the _warmer_: (rejecting the
Decimals as in Section 395.)
Place the Barometers, thus:
_Upper_ Barometer, 29.5337
_Lower_ Barometer, 30.0168
_End of the first Stage._
[Sidenote: Section 371.]
411. _By the 2d Table_, and in the 2d Column, find the _Height_ of
each Barometer, _with_ the _Standard-Heat_, in Feet and Tenths,
corresponding to the Inches and +nearest+ Tenth _above_ and _below_ the
Point required: and
First of the _upper_, at 29.5337:
The Inches and _nearest Tenth_ is above
Feet.
29.5, corresp. to 2119.7 } Difference
and below 29.6, cor. to 2031.5 } between .5 and .6
—————— } above 29 Inches.
88|.2 }
[Sidenote: Section 373.]
412. _By the 3d Table_, _with_ the _Difference_ 88 Feet, find the
_Expansion_ on the remaining Decimals, above 29.5, viz. on .0337, thus:
on 03 = 26 decimated 26.
003 = 26 2.6
0007 = 62 .62
—————
Feet 29.22
From the _Height_ corresponding to 29.5
viz. Feet 2119.7 Tenths,
subtract the 29.22, i.e. Height cor. to .0338
and there ————————— ———————
remains 2090.4|8, the _Height_ cor. to 29.5338
with Expansion of the Standard-Heat.
413. Repeat the 4 last Steps for the _lower_ Barometer, at 30.0168.
1st. The Inches and _nearest_ Tenth is _above_
30. corresp. to Feet 1681.7 } Difference of
and _below_ 30.1 cor. 1595.0 } .1 above 30
—————— } Inches.
86|.7
2d. Then with 86 Feet, find the _Expansion_ on the remaining Decimals,
above 30,
viz. .0168, thus: on 01 = 9 9.
006 = 52 5.2
0008 = 69 .69
—————
Feet 14.89
414. (3d.) From the _Height_ corresponding to
30 Inches, viz. Feet 1681.7 Tenths,
subtract the Height 14.89 corresp. to .0168,
———————
and there remains 1666.8|1, the Height corresp.
to 30.0168, with _Expansion_ of the Standard-Heat.
4th. From the _upper_ Height, at 2090.48
Subtract the _lower_ Height, at 1666.81
———————
And there remains the Height 423.67 in Feet
and Tenths of the upper Barometer, with the
Standard Temperature.
_End of the second Stage._
[Sidenote: Section 374.]
415. Detached Therm. _above_ 57°
Detached ditto, _below_ 60.2
———————
_Whole_ Heat 117.2
_Half_ Heat 58.6(0 adding a Cypher)
_Standard_ Heat 31.24
——————
which being deducted, leaves 27°.36, viz. Degrees
of Heat more than the _Standard_, for each Barometer.
[Sidenote: Section 380.]
416. By the 4th Table, find the Expansion of Air, with 27°.36, on Feet
423.67 Tenths.
[Sidenote: Section 406.]
_First_, _with_ 27°, _on_ 423.67, thus:
viz. _on_ 400 as 4000 = 262.4 decimated 26.24
20 as 2000 = 131.2 1.312
3 as 3000 = 196.8 .1968
.6 as 6000 = 393.6 .03936
.07 as 7000 = 459.2 .004592
—————————
Expansion = 27.692752
[Sidenote: Section 407.]
_Second_, _with_ .36 _on_ the _same_, thus:
on 400 as 4000 = 349.9 decimated .3499
20 as 2000 = 174.9 .01749
3 as 3000 = 262.4 .002624
.6 as 6000 = 524.8 .0005248
.07 as 7000 = 612.3 .00006123
—————————
Expansion = .37050003
Add the former 27.692752
———————————
Height in Feet 28.06325203
417. Which Height for Expansion of Air, _with more than_ the Standard
Heat, being +added+[133] to the Height, for Expansion of the Barometer,
_with_ the Standard-Heat, gives the true Height of the upper Barometer,
at the given Heat.
For _Expansion_ of _Air_ above Standard Heat,
Height in Feet 28.0
For _Expansion_ of _Barometer_,
with Standard: Height in Feet 423.6
—————
418. True Height of the _upper_ Barometer 451.6
_Lower_ Barometer 1 Foot above the Water 1.0
Height of the Top of the Cross above the
Gallery 50.0
—————
Height of the Top of the Cross above the
Tyber 502.6
Height of the same, measured the same
Day geometrically, was Feet 502.9
_End of the last Stage._
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.