Note 124 — Chapter LXXIII

The Foundation of the fourth Table.

(Ph. Tr. for 1777, Part 2d, Pages 564, and 566,)—From the Mean of a Series of Experiments with a Manòmeter, or Instrument to measure the Rarity and Density of the Atmosphere, depending on the Action of Heat and Cold, it was found, that when the Portion of a Tube containing Air (at the Temperature of freezing by Farenheit, and Pressure of 30​1⁄2 Inches[125] by a common Barometer) was divided into 1000 Parts; the Volume of Air within it, encreased nearly in a certain Proportion, as each Degree of Temperature encreased; viz. at a Mean, 2.43, or simply (by rejecting the 2d Decimal as too minute) 2.4: that is, a 1000 Parts of Air became by Expansion with one Degree of the Thermometer, equal to 1002.43: i. e. the Portion of Air occupying 1000 Parts, did, with the Addition of one Degree of Heat, occupy 1002.43 Parts: that is (by rejecting the 2d Decimal 3 as too minute) occupied two Parts and 4 Tenths more than the thousand.

Construction of the fourth Table.

Supposing therefore that the Portion of the Tube containing Air, was one Foot in Length of Height, divided also into a thousand Parts; one Degree of Heat would encrease or expand it two Parts and four Tenths more than the thousand Parts into which the Foot was divided.

CAUTION.

The fourth Table properly consists of only nine horizontal Columns of thousands, in Breadth; which Columns are extended in Length to one hundred Lines, corresponding to 100 Degrees of Heat.

The Table is here divided, in order that it may conform to the Size of the Pages: by which Means the Formation of each vertical Number by the following Rule, (which renders the Table self-evident) might without this Caution, have been attended with some Difficulty.

The vertical Columns below the Figures expressing each thousand, shew the Expansion of Air on each respective thousand, with the corresponding Degrees of Temperature indicated by the Thermometer in the vertical Column to the left Hand.

Example the first: to find the Expansion of Air on one thousand Feet, with one Degree of Temperature; the Answer in the Table is 2.4, or 2.43: i. e. 2 Feet and 4 Tenths of a Foot, rejecting the 2d Decimal as too minute.

Example the second: to find the Expansion on 8 thousand Feet, with 99 Degrees of Heat: the Answer is 1924.56: and so of the Rest.

Thus any of the vertical Numbers shewing the Expansion, may be readily formed, by doubling, first, the Number immediately under each thousand in the horizontal Line, for the nine first thousands, (of which the Breadth of the Table properly consists, exclusive of the thermometric Column) for the Expansion below it: and, afterwards, for each Expansion immediately below the former, by adding, to the Expansion last found, the Number immediately under its respective thousand.

First Example: to find the vertical Number for the Expansion under the first thousand, viz. 1000, with 2 Degrees of Heat: the Number under 1000 is 2.43: double this: and the Answer is 4.86.

Second Example: suppose the Expansion last found be that on one thousand Feet with 24 Degrees of Heat; viz. 58.32: and the Expansion on the same thousand, with one Degree of Heat more, viz. on 25 Degrees, be required; add the Expansion

on one thousand Feet, with 24 Degrees, viz. 58.32

to the Expansion on the same 1000, with 1 Degree, viz. 2.43

———

and the Answer is, by Addition, 60.75

Third Example: supposing the Expansion last found to be the Expansion on 9000 Feet with 99 Degrees of Heat, which in the Table is 2165.1.

It is required to find the Expansion on the same 9000 Feet, with 100 Degrees of Heat; add to the Expansion last found,

viz. 2165.13, the Expansion on the same 9000 Feet,

viz. 21.87 with one Degree of Heat, and

———

2187.00 is the Answer by Addition.

Any vertical Number shewing the Expansion may likewise be found, first, by multiplying the first Figure, or Number, of the given thousand Feet (in the horizontal Line,) into the Answer or Expansion on the first thousand Feet, with one Degree of Heat: for Example;

To find the Expansion on 9000 Feet with one Degree of Heat.

The Expansion on 1000 Feet, with 1 Degree of Heat (from whence, all the other Expansions are derived) being 2.43; multiply that Number by 9, the first Figure of the given thousand Feet, and the Answer or Expansion with 1 Degree of Heat, is 21.87: hence all the Answers or Expansions, immediately under the horizontal Line of thousands, are formed.

Then 2dly, any other vertical Number or Expansion may be formed by multiplying the Expansion immediately under the given thousand Feet in the horizontal Line, into the given Number of Degrees: for Example;

To find the Expansion on 9000 Feet, with 50 Degrees.

The Expansion with one Degree on 9000, is 21.87: therefore the Expansion with 50°, is 50 Times more, viz. 1093.50, and so of the Rest.

These different Methods serve to prove the Answers, and to elucidate the Table.