Extract.1

The rule for computing altitudes from Barometrical observation is as mathematically demonstrable as those for trigonometrical calculation; the accuracy of the results deduced from either method of calculation must therefore depend wholly upon the accuracy of the data obtained. In order to calculate the altitude of a mountain by trigonometry (where the angle of ascent is not a right one) we must have two stations, which two stations and the part to be calculated must be in the same perpendicular plane; that is, the first station must be exactly between the mountain & the second Station. the distance between the two Stations must be known. also the angle of elevation of the top of the mountain from each of the Stations and, (in case the two stations are not upon the Same level) the angle of ascent or descent of the base of the mountain above or below the first station. these things being obtained, by the solution of one oblique & one right angled triangle, in each of which a side & its opposite angle are two of the given parts, we obtain the altitude required. the only difficulty here (& this I am inclined to think, will be rather a serious one) is to ascertain correctly the length of the base, (the distance between the two stations.) and the dimensions of the several angles, for either of these being incorrect it is evident the results will be erroneous; The difficulty of accurately measuring a line of any considerable length even on good ground, is I presume, evident to every one who has made the experiment; but when instead of plain open ground the operation is to be performed on ground somewhat uneven, & perhaps covered with bushes, (such as we generally find near the foot of mountains) the difficulty will be greatly increased, so much so as to render an accurate measurement almost impossible; the measuring of the angles is also a very nice operation, and one, in which the most accurate observers owing to a variety of circumstances too numerous to mention, but which I presume are generally known, are very liable to err.

I shall now proceed to make some observations upon the Barometrical method of computation. Torricelli the Disciple of the famous Galileo first proved by experiment that the Atmosphere had weight, this ingenious Philosopher having filled a glass tube closed at one end with Mercury, and then inverting it with the open end downwards in a bason of Mercury, he found that a column of Mercury of a certain height remained suspended in the tube, he therefore very rationally concluded that the weight of the column of Mercury in the tube must be balanced by the pressure of the air upon the surface of the Mercury in the bason, this experiment gave rise to the Barometer, an Instrument for measuring the weight of the atmosphere; by repeated experiments, it has been found that at the mean temperature of 55° a column of air of any given base and of the height of the atmosphere is equal in weight to a column of Mercury of an equal base, & 29½ inches high. This property of the atmosphere, (its gravity) being fully established as well as its elasticity, it follows that the air nearest the surface of the earth being pressed by the weight of the superincumbent atmosphere, must be more dense than the air above it, having a greater weight to sustain; & consequently the air must continually decrease in density & of course in weight in proportion as we ascend above the common surface of the earth.

From what has been said it would seem natural to conclude that, if at the common surface of the earth the air by its pressure will support a column of Mercury in the Barometer 29½ inches high; that at any altitude above the common surface, where the air is less dense, it must support a column less than 29½ inches high, and consequently that in proportion as we ascend the mercury in the Barometer must descend;

The truth of the above conclusion was first experimentally proved by the celebrated Pascal, who thereby substantiated the correctness of the conclusion deduced from the experiment of Torricelli.

Philosophers have demonstrated that the density of the air (at different heights above the earth) decreases in such a manner that when these heights are taken in Arithmetical Progression the corresponding densities decrease in a geometrical progression; and2 since the terms of an arithmetical progression are proportional to the log.s of the terms of a geometrical one, it follows that different altitudes above the earths surface are as the logarithms of the densities, or weights of the air at those altitudes,3 and consequently the difference between the altitudes of any two places is as the difference of the logarithms of the densities of the air at those places. so that if d. denote the density at the altitude A. and (∂) the density at the altitude (a). then by what has been said A is as the log. of d, and (a) is as the log. of ∂. Also the difference of altitude A − a, is as the difference of log. d − log. ∂ or log. d/∂. and if A = 0 or d = the density at the surface of the earth, then it is evident that any altitude as (a) above the surface of the earth is as the log d/∂;4

from what has just been said is derived the method of calculating the heights of mountains or other eminences by means of the Barometer,5 for if by means of this instrument the pressure or density of the air be taken at the foot of any mountain, and again at the top of it, the difference of the logarithms of these two pressures or densities, or the log. of their quotient (which is the same thing) will be as the6 difference of altitudes or as the height of the mountain, supposing the temperature of the air to be the same at both places.7 but since this formula expresses only the relation between different altitudes with respect to their densities it will be necessary to obtain the real altitude which corresponds to any given density, or the density which corresponds to a given altitude,8 and since it has been proved that (a) is as the log. d/∂, assume (h) so that a = h × log. d/∂ where (h) will be of one constant value for all altitudes. Now to determine the value of (h) let a case be taken in which we know the altitude (a) corresponding to a known density ∂; as for instance let (a) be taken equal to one foot or some such small altitude, then because the density (d) may be measured by the pressure of the atmosphere or the uniform column of 27600 feet (which is the altitude of a column of air of the same density throughout as the air at the surface of the earth at the temperature of 55°, that would balance a column of Mercury of equal base with it, and 29½ inches high)9 when the temperature is 55°.

Therefore 27600 feet will denote the density d at the lower place and 27599 the less density ∂ at one foot above it; consequently 1 = h × log. 27600⁄2759910 = h × ,0000158 = h × ,0000158 × 27600⁄27600 = h × .43429448⁄27600 nearly, = h × .43429448⁄27600 × 2,3025850929940⁄2,3025850929940 = h × 1⁄63551ft.11 (see the accompanying demonstration) therefore for any altitude we have this general theorem vizt a = 63551 × log. d⁄∂ or = 63551 × log. Μ⁄m or = to 10592 × log. Μ⁄m fathoms where Μ is the column of mercury at the bottom, and consequently equal to the weight of the12 atmosphere at that place, and (m) that at the top of the altitude (a).13 This formula is adapted to the mean temperature of 55° but since it has been found by experiment that air expands about the 435th part of its whole bulk for every degree of heat; it follows, that14 for every degree that the mean temperature between the temperatures at the top & bottom of the altitude a, exceeds 55° that altitude must be increased by its 435th part & diminished in the same ratio for every degree the mean is below 55°—

It is also found that Mercury expands about the 1⁄9600th part of its whole bulk for every degree of heat, and since it often happens that there is a considerable difference between the temperature at the top & bottom of a mountain, it would follow that the Mercury in the Barometer, would be more expanded by heat in the one case than the other, and consequently if not rectified would introduce an error into the calculation. in cases of this kind the mercury may be reduced to the same temperature by increasing the column in the coldest temperature or diminishing that in the warmest by its 1⁄9600 part for every degree of difference between the temperatures at the top & bottom of the altitude a,15 but the formula may be rendered much more convenient for practice, by reducing the factor 10592 to 10,000 which we may do provided we change the temperature proportionally from 55°. thus since the difference 592 is the 18th part of the whole factor 10592 and also since 18 is the 24th part [of]16 435, therefore the corresponding change of temperature is 24° which reduces the 55° to 31° & therefore the formula becomes a = 10,000 × log Μ/m where the temperature is 31°, & therefore for every degree above that, the result is to be increased by so many times its 435th part.17

From the foregoing observations I think it is evident that the Barometric mode of calculation is true in theory; & I can discover but one objection that can be urged against it in practice. Every Person accustomed to make observations with a Barometer knows that it is much affected by different currents of air, the Mercury uniformly (with very few exceptions) rising with westerly or northerly winds, and falling when southerly or easterly ones prevail. If then, in making observations different winds should prevail at the upper & lower stations (which may often happen when those stations are at a considerable distance from each other) it would follow that the Mercury in the Barometer being from this cause elevated at one station and depressed at the other more than it ought, the results deduced would be erroneous. In order to remedy this my method has been (in cases where the stations were a considerable distance apart) to have a regular journal kept at the lower one, in which were inserted daily the altitude of the Mercury in the Barometer, the temperature of the air, and the prevailing winds, together with the state of the weather generally, & then by comparing the observations made upon the top of the height to be calculated, I could select therefrom one that agreed with it as it respects the prevailing winds, and state of the weather and from those make out the calculations required. From the preceding observations the following easy rule for the calculation of heights generally (by the Barometer) is deduced.—18

1st Observe the height of the Barometer at the Bottom of any height or depth to be measured; with the temperature of the Mercury by means of a Thermometer attached to the Barometer and also the temperature of the air in the Shade by means of a detached Thermometer.

2d Let the same thing be done at the top of the said height or depth, and at the same time, or as near the same time as may be. And let those Altitudes of the Barometer be reduced to the same temperature, if it be thought necessary, by correcting either the one or the other, that is, augment the height of the Mercury in the colder temperature, or diminish that in the warmer by its 1⁄9600 part for every degree of difference of the two.

3d Take the difference of the common Logarithms of the two heights of the Barometer, corrected as above if necessary, cutting off three figures next the right hand for decimals,19 where the Log. tables go to seven places of figures, or cut off only two figures when the tables go to six places, and so on; or in general remove the decimal point four places, more towards the right hand those on the left hand being fathoms in whole numbers.

4th Correct the number last found for the difference of the temperature of the air, as follows, take half the sum of the two20 temperatures, for the mean one; & for every degree which this differs from the temperature 31.° take so many times the 1⁄435 part of the fathoms above found, & add them if the mean temperature be above 31.° but subtract them if the mean temperature be below 31.° and the sum or difference will be the true altitude in fathoms. or being multiplied by six, it will be the altitude in feet.21

Norwich Vt Augst 20th 1811.22

Demonstration

that h × log. 27600⁄27599 is nearly = to h × ,43429448⁄27600. The number ,43429448 is the modulus of the common system of Logarithms, and is equal to the quotient which arises from dividing 1 by 2,3025850929940. but for the purpose of more clearly investigating the matter, I shall have recourse to the logarithmic curve, the principal property of which is, that the abscisses being taken in arithmetical progression the corresponding Ordinates will be in Geometrical Progression—that is, the Abscisses are the log.s of their corresponding ordinates. Let x represent any abscissa, y its ordinates; & let (a) represent the subtangent23 of the curve, Moreover let the fluxion of x be denoted thus, (ẋ) and the fluxion of y thus, (ẏ). then by similar triangles (the figure being constructed) it will be as, ẏ : ẋ :: y : a, and if instead of ẋ & ẏ be substituted their increments marked, thus (x́)(ý) which may be done (see simpsons Fluxions, page 109, Scholium) the above proportion will stand thus, as, ý : x́ :: y : a; now it is evident from the property of the Curve abovementioned, that if y represent any number x will represent its log. it is also evident that the small quantity, ý, may express the difference between any two numbers very nearly equal, and also that the other small quantity, x́, may denote the difference between the log.s of those numbers. the last proportion therefore, ý : x́ :: y : a, being put into words, will be, as the difference between any two numbers, nearly equal, is to the dif. of their log.s. so is either of those numbers, to the modulus of the system of logarithms. to apply this to the case under consideration, here are two numbers 27600 & 27599 nearly equal to each other, their difference being 1, and the log. of their ratio, .0000158; therefore from what has been said, as 1 : 27600 :: .0000158 : ,43429448 nearly, therefore h × log. 27600⁄27599 is = h × .0000158 = h × .0000158 × 27600⁄27600= h × ,43429448⁄27600 nearly, = h × ,43429448⁄27600 × 2,3025850929940⁄2,3025850929940 = h × 1⁄63551, nearly which is = h⁄63551.—