Popular Science Monthly/Volume 46/January 1895/The Barometric Measurements of Heights

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THERE are several methods of measuring the heights of mountains and other elevated portions of the earth's surface above the sea level. Of these maybe mentioned the following: (1) by actual leveling with an engineer's spirit level and graduated staff; (2) by trigonometrical calculation based on the measurement of the angles of elevation observed at the extremities of a carefully measured base line; (3) by observing the temperature of the boiling point of water; and (4) by reading a barometer at the sea level, and again at the top of the mountain or elevation the height of which is to be determined.

The first of these methods is certainly the most accurate, but it involves a considerable amount of labor, and for very high mountains is sometimes impracticable. The second method is sufficiently accurate if carefully carried out and a nearly level plain is available for the measurement of a base line. The third method is not accurate enough to give reliable results. The fourth is the simplest and most expeditious of all. It is especially useful for finding the difference of level between two points at considerable distances apart, and would be sufficiently accurate if certain difficulties could be successfully surmounted. A consideration of this method and the difficulties to be overcome before its accuracy can be relied upon may prove of interest to the general reader.

The principle of the barometric method is as follows: The barometer measures the weight of the atmosphere. The column of mercury in an ordinary mercurial barometer is equal in weight to a column of air of the same diameter and of a height equal to that of the earth's atmosphere. The densest portion of the atmosphere is that close to the earth's surface, and its density diminishes as we ascend. At the top of a mountain, therefore, the pressure of the atmosphere will balance a shorter column of mercury, and hence the mercury descends in the tube. From the difference in height of the mercury at the level of the sea and on the top of the mountain it is possible to calculate the height we have ascended, as will be shown further on.

There are two forms of barometers—namely, the mercurial barometer and the aneroid. Of mercurial barometers there are two forms, the "cistern" and the "siphon." The cistern form is the one most generally used for scientific observations, and is the best for measuring heights. One of the most approved forms of cistern barometers known as "Fortin's barometer"—consists of a glass tube closed at one end and filled with mercury, the lower portion of which dips into another tube of larger diameter which contains a reservoir of mercury forming the "cistern." The bottom of the cistern is formed of leather and fitted with an adjusting screw below, for the purpose of adjusting the level of the mercury in the cistern to an ivory index point above, which marks the zero of the graduated scale. By means of this adjusting screw the mercury may also be raised so as to completely fill the cistern and tube, and thus adapt the instrument for traveling.

We need not discuss here the manufacture of barometers and the filling of the tube with mercury, an operation which must be done carefully so as to exclude air from the tube. Suffice it to say that the best method is to fill the tube gradually, and boil the mercury as we proceed by means of a spirit lamp, in order to drive out all bubbles of air which may be contained in the mercury. The tube may be filled without boiling, but the resulting instrument will not be so accurate as one in which the mercury has been boiled.

To determine the difference of elevation between two places with a mercurial barometer, several points must be attended to. In the first place, the temperature of the barometer and the temperature of the air must be noted at each station. As the mercury in a barometer is affected by heat—in the same way that a thermometer is—the temperature at which the barometer is read must be observed. For this purpose a thermometer is usually attached to the barometer. The temperature should be read as accurately as possible, for an error of one degree Fahrenheit would make a difference of about three feet in the resulting altitude. The reading of the attached thermometer should be first noted, and then the height of the barometer. To do this, first bring the surface of the mercury in the cistern accurately to the index point by means of the adjusting screw. Then tap the tube gently near the top of the column in order to get rid of the adhesion between the mercury and the glass. The height of the mercury may then be read by means of the attached scale and vernier. Sometimes the amount of aqueous vapor in the atmosphere is ascertained by another instrument. The above data being known for two stations, we substitute the values found in one of the barometric formulae, and thus obtain the height, or difference of height, required. Before the barometer readings can be used, this must be reduced to the same temperature—usually 32º F.

Various formulæ have been computed by eminent mathematicians and physicists for calculating the difference of height between two points. These formulæ depend on certain assumptions which, however, can not be considered as rigidly true. The most important of these assumptions is that the atmosphere may be supposed to be in a state of statical equilibrium. But owing to the changes constantly taking place, due to differences of temperature, humidity, winds, etc., this assumption can not be considered correct. The result will therefore be only an approximation to the truth. Assuming, however, a statical equilibrium of the atmosphere, a formula can easily be deduced from known principles. For this purpose we must first ascertain the weight of a cubic inch of air and a cubic inch of mercury at a certain temperature and pressure, and in a given latitude, say 45º. Then, by Boyle and Mariotte's law, connecting the weight of a gas and the pressure, the formula can be obtained for determining the height required. There are several elaborate formulæ used for this purpose. These include terms for altitude, latitude, temperature, and humidity. A correction for altitude is theoretically necessary owing to the diminution in the force of gravity—and, therefore, a decrease in the weight of bodies—with increased distance from the center of the earth, but this correction is comparatively very small, and may, for all practical purposes, be neglected. For the same reason a correction for latitude is mathematically required, owing to the spheroidal figure of the earth; but this, too, is very small, and may be safely neglected. The correction for temperature of the air is, however, very important. This term is easily computed. It is obtained—for the Fahrenheit scale—by deducting 64 from the sum of the observed temperatures at the upper and lower stations, dividing the difference by 900, and adding unity to the result. A correction for humidity of the air is also necessary; but it is doubtful whether it is desirable to complicate the formula by a correction for atmospheric moisture, the laws of which are so imperfectly understood.

In all the barometric formulæ which have been proposed the first term is constant, and common to all. It is known as the "barometric coefficient," and is 5·744m/a, where m is the "weight of a cubic inch of mercury at the sea level in latitude 45º at 30º F. when the barometer reads 29·92 inches," and a the weight of a cubic inch of dry air under the same conditions of latitude, temperature, and pressure. Various values of this constant have been found, depending on the values assumed for m and a. Arago and Biot found m/a 10,467. This makes the "barometric coefficient" 60,122·4 feet. Raymond's value, namely 60,158·6 feet, was found by comparing the values given by the formulae with the results of actual leveling with a spirit level. His observations were, however, few in number, and, although his coefficient is frequently used, it is probably the least accurate of all the determinations. In Laplace's formula, Raymond's constant is used. Babinet used the constant 60,334, and in Baily's formula the constant is 60,346. In Williamson's formulæ the constant is 60,384, which is the value found by Regnault, and is probably the most accurate of all. Sometimes the coefficient in the formula is given as 10,000 fathoms, which is roughly correct.

We will now consider the errors underlying the barometric measurement of heights, which render the method inapplicable in cases where great accuracy is required. The most important of these sources of error is probably that due to what is called the "barometric gradient," a term frequently used in meteorological reports. Taking three points at which the barometric pressure is, the same, if the atmosphere was in a state of statical equilibrium these points would lie on the same level plane. But usually this plane is not level, but inclined, and the inclination of the plane is termed the "barometric gradient." For a number of points the surface on which they lie would not be a plane at all, but an undulating surface. These surfaces for different heights are never parallel, and frequently slope in opposite directions. Allowance can not be fully made for this disturbing cause, but the error can, to some extent, be eliminated by making a number of simultaneous observations at the two stations, and taking the mean of the results.

Another cause of error is due to variations in the temperature of the air. It is generally assumed that the mean temperature of the column of air between two stations, one vertically over the other, is the mean of the temperatures at the upper and lower stations, but this is not always the case. The error may be partially eliminated by making observations at intermediate stations, but can not be entirely overcome. High winds also cause a variation in the height of the barometer.

In addition to the errors mentioned, there are, of course, errors of observation, and instrumental errors. The former may be caused by imperfect adjustment of the zero point, and erroneous reading of the mercury on the scale. These errors are, however, usually small, and may with care be neglected. The instrumental errors are due chiefly to imperfect graduation of the scales of the barometer and attached thermometer, the impurity of the mercury, and to air in the tube. These errors may be corrected by comparison with a standard instrument.

The form of barometer known as the aneroid is also frequently used for the determination of heights, a graduated scale being added for this purpose. This scale is graduated by means of one of the barometric formulæ already referred to. The aneroid barometer usually consists of a metallic box from which the air has been exhausted, and differences of atmospheric pressure are recorded by a system of levers which act on an index hand which marks the reading on a graduated scale. In some forms of aneroid the box is not completely exhausted of air, and these are called "compensated aneroids," but the name is misleading, some of these instruments being more sensitive to changes of temperature than those not compensated. The aneroid is a very handy instrument and easily used, but for the purpose of measuring heights it is much inferior to the mercurial barometer. In some instruments the altitude scale is fixed at a certain reading, say thirty or thirty-one inches, and in others it is movable, and can be adjusted to any reading required. The latter seems the most convenient plan. In either case it is clear that absolute elevations above the sea level can not be determined with this instrument with any approach to accuracy, as there is no way of making the necessary corrections for variations in pressure, temperature, etc. The aneroid barometer should, therefore, be used only for finding differences of elevation, and for this purpose it will give fairly good approximate results in cases where extreme accuracy is not required.

To show the degree of accuracy attainable by the barometric method, two examples may be cited. From readings of a mercurial barometer at the summit of Mont Blanc and at the Geneva Observatory made by MM. Bravais and Martins in the year 1844, the height of the mountain above the level of the sea was computed to be 4,815·9 metres, or 15,800'44 feet. Corabeuf found by trigonometrical measurement a height of 15,783 feet, or 17'44 feet less than that indicated by the barometer.

The height of Mount Washington, in the United States, was found by a spirit level to be 6,293 feet above sea level, while the barometric method gave 6,291·7 feet, a close approximation. In some other cases, however, much larger differences have been found, and the good agreements quoted above may perhaps be considered as accidental.—Gentleman's Magazine.


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