| MEIRlCAli 
 ETERMINATIONDF HEIGHTS 
 
THE BAROMETRICAL 
 DETERMINATION OF HEIGHTS 
 
 A PRACTICAL METHOD 
 
 OF 
 
 BAROMETRICAL LEVELLING 
 AND HYPSOMETRY 
 
 FOR 
 
 SURVEYORS AND MOUNTAIN CLIMBERS 
 
 BY 
 
 F. J. B. CORDEIRO 
 t< 
 
 Second Edition, Revised and Enlarged 
 
 NEW YORK 
 SPOX & CHAMBERLAIN, 120 LIBERTY ST. 
 
 LONDON 
 
 E. & F. N. SPOX, LIMITED, 57 HAYMARKET 
 1917 
 

 Copyright, 1897 
 
 Copyright, 1917, by 
 
 SPON & CHAMBERLAIN 
 
 BARR & HAYFIELD, Inc., Printers, 159 William St., New York 
 
PREFACE TO SECOND EDITION. 
 
 The methods and formulas given in this book 
 have been recognized by competent authority as 
 the only ones that can give correct and consist- 
 ent results. The older formulas are incorrect, 
 and the results obtained by them very inconsist- 
 ent. Although this book has been in print some 
 years, attempts are still made to determine moun- 
 tain heights by antiquated formulas, and by tak- 
 ing pressures and temperatures at the summit 
 and some other remote point. As an example, 
 a few years ago the determination of the height 
 of Alt. \Yhitney in California was attempted by 
 taking pressures and temperatures simultaneous- 
 ly at the summit and at San Francisco, and then 
 applying Laplace's formula. 
 
 Some distinguished mountain climbers have, 
 within the last few years, made the conquest of a 
 number of very high summits, but unfortunately 
 their height measurements may vary consider- 
 ably from the truth, from the fact that they have 
 used incorrect methods and formulas. It is to. 
 be hoped that the present methods and formulas 
 will become more generally known and adopted, 
 as all others are a waste of time, if accuracy is 
 desired. 
 
 412365 
 
PREFACE TO THE SECOND EDITION 
 
 Common logarithms are used in the computa- 
 tion, p must be reduced to freezing, and also, 
 when possible, to Paris. Practically, the local 
 value of g is usually unknown, but the error 
 from neglecting this correction is slight, and it 
 may in general be ignored. The correction for 
 capillarity is applied to all readings. The deci- 
 mals may be shortened without much error : thus, 
 for quick work, .00003815 gives nearly the same 
 result as the longer decimal. 
 
 Barometers should, if possible, be compared 
 with a standard barometer, both before and after 
 an ascent, f and p must be both taken in mms. 
 of mercury, or inches of mercury. 
 
PREFACE. 
 
 THE discrepancies arising in the calculation of 
 mountain heights by the barometrical formulae 
 which have hitherto been in use have brought 
 this valuable and in many cases only applicable 
 method into disrepute. The fault has lain in the 
 formulae, not in the method, which is one sus- 
 ceptible of great accuracy. These formulae have 
 either been based upon unwarrantable assump- 
 tions or have failed to take account of all the con- 
 ditions obtaining in the problem. 
 
 The present essay \vas originally entered in 
 the Hodgkins Prize Competition, held under the 
 auspices of the Smithsonian Institution, and was 
 awarded honorable mention. In it the important 
 problem of Barometrical Hypsometry, which has 
 not been touched upon since 1851, when it was 
 discjssed by Guyot, has been gone over anew 
 and brought up to date. Important errors in the 
 older formulae have been detected and a new 
 method has been furnished which is rigidly accu- 
 rate in theory and which in practice will give re- 
 liable results under all conditions. 
 
 F. J. B. C. 
 
OTHER WORKS BY F. J. B. CORDEIRO 
 
 THE ATMOSPHERE, Its Characteristics and Dynamics, 
 viii+129 pages, 35 illustrations, 10^4 x l l /2 in., paper 
 binding, $1.50. Cloth, $2.50. 
 
 THE GYROSCOPE, Its Theory and Applications, vii-f-105 
 pages, 19 illustrations, 8*4 x6^ in., cloth, $1.50. 
 
 , THE MECHANICS OF ELECTRICITY, vi+78 pages, 7 
 illustrations, 7^x5J4 "! cloth, $1.25. 
 
THE 
 
 BAROMETRICAL DETERMINATION 
 OF HEIGHTS. 
 
 ONE of the most important applications of the 
 known properties of air has been a deduction 
 from them of a means of finding the vertical 
 height between any two points, and the problem 
 of measuring the vertical distances between any 
 two levels is one that has engaged the attention 
 of a number of mathematicians and physicists for 
 many years. 
 
 Laplace, in the " Mecanique Celeste," gave what 
 at the time was considered a complete solution of 
 the problem ; but as it was based upon several 
 unwarrantable assumptions, and took no account 
 of the aqueous vapor in the atmosphere, it was 
 at best an approximation. 
 
2 Barometrical Determination of Heights, 
 The complete formula as given by him is 
 
 Z= log 18336. 
 
 (l -p; .0028371 cos. 2 L) 
 
 (log rj+ .868589) 
 
 (I + 
 
 h 
 
 where Z = the difference of level in metres ; 
 
 a Earth's mean radius = 6,366,200 metres; 
 L = mean latitude of the two stations. 
 
 And further 
 
 ( ( h = height of barometer ; 
 
 I Lower \ T = temperature of barometer ; 
 ( t temperature of air ; 
 At station { 
 
 II h' = heiglit of barometer ; 
 Upper < T' = temperature of barometer ; 
 [_ ( t temperature of air ; 
 
 T 
 
 and I-I = h + h'" 
 
 The first parenthesis in the terminal factor is 
 the correction for the difference of temperature 
 of the two levels. It assumes that the problem 
 would be the same if the air between the two 
 levels were of a uniform temperature the mean 
 of what is observed at the two levels. As a 
 matter of fact, if the two stations are remote, a 
 large range of temperatures may be found at in- 
 tervening points. 
 
Barometrical Determination of Heights. 3 
 
 The second parenthesis is the correction for the 
 change of gravity with the latitude. It assumes 
 that gravity increases regularly according, to a 
 law as we go from the equator to the poles a 
 supposition which we now know to be true only 
 in a general way. The third parenthesis is the 
 correction for the decrease of gravity in a vertical 
 direction. It is based upon the Newtonian law 
 that externally to the earth's surface gravity de- 
 creases inversely as the square of the distance 
 from the centre of mass. From careful pendu- 
 lum experiments we know that such a law does 
 not hold near the earth's surface, large masses of 
 matter in different localities causing variations 
 that are not to be accounted for by any simple' 
 law. 
 
 Baily, in his " Astronomical Tables and For- 
 mulae," gives the following formula: 
 
 X - 60345.51 -| i + .ooiiin (t + t'-64) j- 
 
 - X ' +- 0026 95 cos. 2* 
 
 where <$> = latitude ; 
 
 /8 = height of barometer ; } 
 
 T = temperature of mercury; ' at lower station. 
 
 t temperature of air; ) 
 
 #' = height of barometer ; ) 
 T' temperature of mercury; I at 
 t' = temperature of air ; ) 
 
 upper station. 
 
 Feet, inches, and the Fahrenheit scale are here 
 used. Here the same assumption is made in re- 
 
4 Barometrical Determination of Heights. 
 
 gard.to the increase of gravity with the latitude 
 as in Laplace's formula, and no account is taken 
 of the moisture in the air. 
 
 Bessel first introduced in his formula, Astro- 
 nomische Nachrichten, No. 356, a separate cor- 
 rection for the effects of moisture. Laplace's 
 barometrical coefficient is retained, but the cor- 
 rection for change of gravity is considerably 
 modified. 
 
 Elie Ritter in his formula, "Memoires de la So- 
 ciete de Physique de Geneve," tome XIII., p. 343, 
 gave a correction for moisture. The values of the 
 barometrical and thermometrical coefficients are 
 derived from Regnault's determinations, and a 
 new method is proposed for applying the cor- 
 rection due to the. expansion of the air, which is 
 made proportional to the square of the differences 
 between the observed temperatures at each sta- 
 tion. 
 
 Baeyer's formula, Poggendorf's Anna/en der 
 Physik und Chemie, tome XCVIII., p. 371, does 
 not belong to either of the two classes just men- 
 tioned ; for while it keeps Laplace's barometrical 
 and thermometrical coefficients, it corrects the 
 effects of temperature by a method analogous to 
 that of Ritter, and it entirely neglects the effect 
 of aqueous vapor. 
 
 Plantamour in his tables substitutes for La- 
 place's barometrical coefficient that derived from 
 the probably more accurate determination of the 
 relative weight of air and mercury by Regnault, 
 
Barometrical Determination of Heights. 5 
 
 viz., 18404.8 metres. Laplace used the results of 
 Biot and Arago, and the coefficient deduced from 
 it was 1 83 1 7 metres. This coefficient was, however, 
 empirically increased to 18336 metres in order to 
 adjust the results of the formula to those fur- 
 nished by the careful trigonometrical measure- 
 ments made by Ramond for the purpose of test- 
 ing its correctness. 
 
 An error in all these formulae was the assump- 
 tion of an absolutely invariable barometric coef- 
 ficient. The barometric coefficient in Baily's 
 formula, 60345.51, varies under different condi- 
 tions of pressure, temperature, and relative hu- 
 midity, from 55,000 to 65,000, and in fact the chief 
 thing to do before applying any formula is to 
 compute exactly the barometric coefficient. The 
 barometric coefficient, 18336 metres, which was 
 substituted in Laplace's formula in order to make 
 the data of some observations conform to Ra- 
 mond's trigonometrical determinations, would 
 have had to be changed if other observations had 
 been taken under different conditions of the at- 
 mosphere. 
 
 We shall demonstrate this in what follows, 
 elucidating the problem at each step, so that it 
 may be easily understood by any reader having 
 a slight knowledge of mathematics. 
 
 If the earth were a mere level plane, the verti- 
 cal distance above it of any point would be a 
 simple matter to determine ; but the earth is not 
 a sphere, nor a spheroid. Strictly it is only ap- 
 
6 Barometrical Determination of Heights. 
 
 proximately a regular figure. If the sea were at 
 rest, a figure. very nearly corresponding to its sur- 
 face would be an ellipsoid of revolution, having an 
 equatorial semi-diameter of 20,926,200 feet, and a 
 polar semi-diameter of 20,854,900 feet, giving an 
 
 eliipticity of 293,465.* 
 
 From a comparison of the different measured 
 arcs of meridians, Colonel Clarke found that the 
 surface most nearly agreeing with the sea-level is 
 an ellipsoid (not of revolution) having for its equa- 
 torial section an ellipse with a major semi-axis at 
 8 \V. Ion. of 20,926,629 feet, and a minor semi- 
 axis of 20,854,477 feet. 
 
 If the air also were at perfect rest it would dis- 
 tribute itself about the earth under the influence 
 of gravity, so that we could trace out in it sur- 
 faces of equal density, or in fact equipotential 
 surfaces, and these surfaces would be more or 
 less similar and similarly situated to the surface 
 of the earth as just considered. In the strictest 
 sense, then, the height of a point above the sur- 
 face of the earth would be the perpendicular dis- 
 tance between the standard equipotential surface 
 and the point. Practically,' however, in measur- 
 ing heights, we determine the perpendicular dis- 
 tance between the equipotential surface passing 
 through the upper point and that passing through 
 some lower point of reference. 
 
 * Colonel Clarke, Geodesy, p. 319. 
 
Barometrical Determination of Heights. 7 
 
 In measuring heights we have three methods at 
 our disposal : 
 
 I. That by levelling. 
 IT. That by vertical angles. 
 
 III. That by weighing a vertical column of air 
 between the levels, /.<?., the barometric method. 
 
 Without entering into a discussion of the com- 
 parative merits of the different systems, it may 
 be stated that in determining mountain heights 
 with any degree of accuracy, the barometric 
 method is usually the only practicable one. In 
 determining the height of a mountain like Chim- 
 borazo, as Mr. Whymper did, it is wholly out of 
 the question to level it, and vertical angles taken 
 at great distances can only lead to an approxima- 
 tion. 
 
 The determination of a height by a barometer 
 is equivalent to the weighing of a column of air 
 under varying conditions. Reduced to its sim- 
 plest terms the atmosphere is supposed to be ab- 
 solutely at rest, of a uniform temperature tlnough- 
 out, absolutely dry, and the value of g the force 
 of gravity at the place to be known. Then it 
 will be easy to obtain the weight of a vertical col- 
 umn of air between any two points, and thereby 
 the height of this column. 
 
 By Mariotte's law we know that the density of 
 air is proportional to the pressure to which it is 
 subjected. If we know accurately the weight of 
 each cubic foot of air in the column, the pressure 
 of the whole column per square foot will be ar- 
 
8 Barometrical Determination of Heights. 
 
 rived at by simply adding the individual weights. 
 Such a summation, when the elements vary ac- 
 cording to a law, can be easily effected by the in- 
 tegral calculus. 
 
 Let the weight of a cubic foot of air which is 
 the increment or increase of pressure for every 
 foot that we descend be represented by Sp. Now 
 this weight or increment is itself proportional to 
 the pressure by Mariotte's law and also propor- 
 tional to the height of the element, -which is here 
 supposed to be the unit of measure one foot. 
 Represent this element of height by Sh, then Sp 
 = c.p. Sh, where c is some constant. 
 
 Or Sh = k - Summing our elements, or in 
 other words, integrating, we have 
 
 .-. h = k log F where p is the pressure given 
 
 by the barometer at the lower station, p x the 
 pressure at the upper station and h is the height. 
 
 If we let h = i, then k= 
 
 where w represents the weight of a cubic foot of 
 air, expressed, of course, in the same terms as 
 
Barometrical Determination of Heights. 9 
 
 the pressure. If then we determine once for all 
 the weight of a cubic foot of air at the pressure 
 p, for the temperature t and value g of gravity, 
 we have a value of k which can be used in all 
 calculations where the conditions remain the 
 
 same. But the above formula, h = k log - 
 
 holds only provided the conditions remain the 
 same throughout the whole column of air a 
 state of affairs that never occurs in nature except 
 for narrow limits. In ascending from one level to 
 another we pass through many varying condi- 
 tions. One layer of the atmosphere may be much 
 warmer than another immediately above or be- 
 low it. 
 
 The relative humidity varies from point to 
 point and from hour to hour. The value of g does 
 not vary greatly for different levels at the same 
 place, but varies sufficiently at places widely 
 removed from each other on the earth's surface 
 to be taken into account. Violent commotions 
 of the atmosphere are another disturbing factor 
 on the pressure, the swirls and eddies of a storm 
 often causing the barometer to jump up and 
 down a hundredth of an inch or more. Under 
 these conditions the attempt to weigh a long 
 column of air to determine a great height by 
 taking conditions (pressure, temperature, and 
 relative humidity) at the base and summit of a 
 mountain simultaneously, and then taking the 
 mean of the latter two is futile, although baromet- 
 
io Barometrical Determination of Heights. 
 
 rical formulae have usually been applied in this 
 manner. 
 
 There remains, however, a means by which a 
 very close estimate can be arrived at in spite of 
 all these obstacles. That is to take the condi- 
 tions (t, p, and r.h.) at frequent intervals, so that 
 we can be reasonably sure that from one point to 
 another the conditions of our original formula 
 hold to a close approximation. 
 
 The occasions are very rare, especially when 
 the weather is at all settled, where any very sud- 
 den changes take place for an ascent of 500 feet 
 or more. But even then it is easy to multiply 
 stations and take the readings of the barometer, 
 thermometer, and psychrometer. Our total 
 height will then be the algebraic sum of our ele- 
 ments. As a correction we can take readings 
 again on descending and average the heights by 
 ascent and descent. 
 
 All this necessitates the separate determination 
 of the constant k for each interval of ascent, and 
 this requires that besides knowing the extreme 
 pressures we also compute the weight of a cubic 
 foot of air as found either at the top or bottom of 
 the interval. 
 
 The mercurial barometer is the only instrument 
 at he present time that we can rely upon for 
 taking pressures. The aneroid barometer is a 
 misnomer ; it should be called the aneroid baro- 
 scope. The best of these instruments are entire- 
 ly unreliable and irregular in their action. Mr. 
 
Barometrical Determination of Heights. n 
 
 Whymper (" How to Use the Aneroid Barom- 
 eter ") has done an important service in showing 
 conclusively that they are worse than useless, as 
 they always underread the mercurial barometer, 
 and in a most irregular way at that. 
 
 The mercurial barometer should be suspended 
 from a tripod and exposed to the atmosphere for 
 about ten minutes, when several readings should 
 be taken, and the average entered as the reading 
 for the station at that time. The readings of the 
 thermometer and psychrometer are taken simul- 
 taneously. The height of the mercurial column 
 is carefully reduced afterward to freezing, and 
 the correction for capillarity is applied. Where 
 great accuracy is aimed at, or where there is rea- 
 son to believe that the value of g at the place 
 differs considerably from tlie standard value 
 that of Paris, for our purposes, since most of our 
 data were established at that place by Regnault 
 we must apply this correction also. 
 
 Besides a barometer two other instruments only 
 are required a thermometer and a hygrometer. 
 The thermometer attached to the barometer is 
 sufficient to determine the temperature of the at- 
 mosphere. 
 
 Regnault found that the weight of a litre of dry 
 air at Paris under a pressure of 760 mm, of mer- 
 cury was 1.293233 grammes. This figure is the 
 base upon which all our calculations rest. The 
 force of gravity at Paris being g p 32.1747 foot- 
 seconds, or 980.94 dynes the standard of height 
 
12 Barometrical Determination of Heights. 
 
 of our barometer, 760 mm., will vary with the 
 force of gravity. Consequently, the weight of a 
 litre of dry air at any other place x will be 
 
 cr 
 
 I - 2 93 2 33 - grammes. 
 
 The value of g has been determined with a con- 
 siderable degree of accuracy at various points of 
 the earth's surface. Where the value has been 
 determined near some place where the altitude 
 is to be determined, it is best to rely upon that 
 value. Laplace's and the other formulae assume 
 that this value varies according to a regular law 
 with the latitude, and also with the altitude above 
 the sea-level. Actually such a law exists only to 
 a very limited extent. 
 
 While it is probable that the decrease of gravity 
 as we ascend directly into the air (/.<?., by bal- 
 loon), would conform more or less nearly to the 
 law of the inverse squares, such a law is by no 
 means the case on a mountain-slope. Menden- 
 hall found that the value of g on the summit of 
 Fuji-yama was greater than at the base. On 
 many mountains it may be slightly less at the 
 top, but wanting definite measurements it would 
 be safer to assume that there is no appreciable 
 change. 
 
 The following table will give some idea of the 
 variability of g. 
 
Barometrical Determination of Heights. 13 
 
 TABLE I. 
 
 VARIABILITY OF GRAVITY AT: 
 
 Greenwich, g = 32.1912 ) , . A 
 
 Paris, 1 = 32.1747 J- f eet -seconds. 
 
 Washington, g = 980. 100 ~] 
 
 Mt. Hamilton, Cal., g = 979.651 I 
 
 Honolulu, H. I., g = 978.936 I 
 
 Waikiki, H. I. , g = 978.922 } dynes. 
 
 Kawaihfe, H. I., g = 978.803 I 
 
 Kalaieha, H. I., g = 978490 | 6660 ft. 
 
 Mauna Kea, H. I., g = 978.060 J summit. 
 
 The last three determinations are for the slopes 
 of Mauna Kea, the first, Kawaihae, at the base, the 
 second, Kalaieha, half way up, and the third at 
 the summit. In general, the value of g is less on 
 islands than on continents. 
 
 Owing to the ellipticity of the earth and the 
 effect of centrifugal force becoming less as we ap- 
 proach the poles, gravity tends to increase from 
 the equator toward the poles. Let the elliptic 
 section through the poles be represented by 
 
 r' 2 cos. 2 a r 2 sin. 2 a a 2 b 2 
 
 a 2 b 2 ~ r ' ' r = b 2 + c 2 sin. 2 a 
 
 By the law of inverse squares, 
 k k 
 
 g =- 2 . '. g = ^j (b 2 + c 2 sin. 2 a) = k' + k" sin. 2 a 
 
 Therefore, the increment of g from the equator 
 to the poles is proportional to the square of the 
 sine of the latitude. Considering the effect of 
 centrifugal force we have C. F K cos. a. Re- 
 solving this in the direction of gravity its result- 
 
14 Barometrical Determination of Heights. 
 
 ant becomes K cos. 2 a .. g (the resultant force of 
 gravity) = f (actual force) K (i sin. 2 a) .*. g 
 = K' + K" sin.'V 
 
 Thus the increase of g from diminution of the 
 centrifugal force varies according to the same law 
 and we can combine the formulae thus: g = g e 
 (i +.005133 sin. 2 L), where g e is the force of 
 gravity at the equator. An approximate value 
 of g e is 32.088, but, as we have seen, it is not con- 
 stant. The formula given above, with the coeffi- 
 cient .005133, was deduced originally by Clairaut. 
 Measurements show that gravity conforms to a 
 certain degree with this law, and it is advisable 
 where definite determinations are wanting to ap- 
 ply such a correction. It must be borne in mind, 
 however, that gravity may be less on an island 
 situated far to the north than at a continental 
 point near the equator. 
 
 The weight of a litre of pure mercury Regnault 
 found to be 13596 grammes. Let us suppose that 
 at Paris, the air being perfectly -dry, the tempera- 
 ture is o Cent, and the pressure p expressed in 
 inches of mercury. By our barometrical formula 
 
 i 
 k = p + w, where w is the weight of a cubic 
 
 foot of air expressed in inches of mercury. There- 
 fore, 
 
Barometrical Determination of Heights, 15 
 
 But suppose the temperature and relative hu- 
 midity are not zero. As the temperature increases 
 the weight of a cubic foot of air becomes less for 
 ihe same pressure. The coefficient of expansion 
 of air and most gases is .00367 per degree Cent. 
 
 vv t = 
 
 I + a t ' I + .00367 
 
 The relative humidity also affects the weight of a 
 given volume of air. Moist air is always lighter 
 than dry air. The relative humidity is defined 
 to be the ratio of the pressure of the vapor in a 
 given volume of air to the pressure of the vapor 
 in the same volume at saturation. 
 
 Let V denote a volume of air, H its pressure, 
 f the pressure of the vapor in it and t the tem- 
 perature. Tiie entire gaseous mass may be di- 
 vided into two parts, a volume V of dry air at 
 temperature t and pressure H f, the weight of 
 which is 
 
 i H-f 
 
 V x 1.20^2-?^ x - - x and a vol- 
 
 i -f a t 760 
 
 ume V of aqueous vapor at the temperature t and 
 pressure f. Since the density of aqueous vapor is 
 f that of dry* air, the weight of this latter mass is 
 
 J V x 1.293233 x -X 
 
 i Hh a t 760 
 The sum of these two weights is the weight re- 
 quired, viz. : V x 1.293233 X 
 
 i -t- a t 
 
16 Barometrical Determination of Heights. 
 
 The pressures in mm. of mercury at satura- 
 tion for different temperatures of aqueous vapor 
 have been carefully determined by Regnault and 
 are given in the following table : 
 
 TABLE II. 
 
 PRESSURES IN MM. OF MERCURY AT SATURATION FOR DIF- 
 FERENT TEMPERATURES OF AQUEOUS VAPOR. 
 
 Temperatures, 
 Cent. 
 
 Force of 
 vapor, mm. 
 
 Temperatures, 
 Cent. 
 
 Force of 
 vapor, mm. 
 
 32 
 
 0.32 
 
 10 
 
 9.17 
 
 2O 
 
 93 
 
 '5 
 
 12.70 
 
 10 
 
 2.08 
 
 20 
 
 17-39 
 
 5 
 
 3*" 
 
 . 25 
 
 23-55 
 
 
 
 4.60 
 
 30 
 
 31-55 
 
 5 
 
 6.53 
 
 40 
 
 5Ll6 
 
 A more complete table is given at the end. 
 
 The force of the vapor in the atmosphere is 
 obtained by observing the dew-point, which 
 gives us the temperature at which saturation oc- 
 curs for the given pressure. The actual force of 
 the vapor will be the tension of saturated vapor at 
 the dew-point. An efficient instrument for deter- 
 mining the dew-point is Dines's Hygrometer. Or 
 the relative humidity may be obtained from ta- 
 bles by readings of the wet and dry bulb hygrom- 
 eter. The results of this latter instrument, 
 sometimes known 0s August's Psychrometer, are 
 far from accurate. 
 
 We can thus, from the foregoing formula, de- 
 termine the weight of a cubic foot of moist air 
 
Barometrical Determination of Heights. 17 
 
 under any given conditions of temperature, pres- 
 sure, and humidity. 
 
 At the pressure p, reduced to Paris, tempera- 
 ture t and vapor tension f, the weight of a cubic 
 foot of moist air, expressed in inches of mercury, 
 is 
 
 12 I P If 
 
 ^^.^^..-.vx vx \X"o 
 
 i.zy^zjj A - 
 P-ff 
 
 13596 ' i +.003671 
 x .0000381468. 
 
 i 
 
 29.9218 
 
 i + ,oo367t 
 k 
 
 log( P 
 
 p f f 
 X oooo 
 
 381468^ 
 
 i 4- .003671 
 
 where t is Centigrade and p expressed in inches 
 of mercury reduced to Paris and freezing, and 
 corrected for capillarity. The corrections for ca- 
 pillarity and freezing are taken from carefully 
 prepared tables which are easily obtained. The 
 key, then, to all our barometric determinations 
 is the computation of the coefficient k. Hav- 
 ing obtained this for a range where the varia- 
 tions of this factor are inappreciable, it remains 
 only to get the height by applying the formula 
 
 h = k log -2s- 
 
 The computations are made after the ascent 
 from the data collected. These can be taken with- 
 out much trouble during the ascent and descent. 
 The readings should be carefully recorded in a 
 
1 8 Barometrical Determination of Heights. 
 
 systematic manner, so that no doubt can apper- 
 tain to them. Asa rule, one man should carry the 
 barometer or barometers, and nothing else. He 
 should be intrusted with the readings of the mer- 
 curial column and the thermometer attached. 
 Another man should carry and record, simulta- 
 neously, the hygrometer. At the point of depart- 
 ure the station from which the heights are to be 
 estimated the tripod should be set up, the ba- 
 rometer suspended, and after exposing it to the 
 air for ten minutes or more, to make sure that the 
 thermometer attached and all the parts of the ba- 
 rometer are at the same temperature as the air, 
 the readings should be taken. At the same time 
 the hygrometer should be read. The body of the 
 observer should not be too close to the instru- 
 ments. It is well to take several readings at each 
 station and average them. Whenever in the as- 
 cent (or descent) it becomes evident that the 
 conditions are changing, a stop should be made, 
 station noted, and readings taken. Whether the 
 conditions are changing or not can be determined 
 from time to time without a stop by simply noting 
 the temperature or glancing at the wet and 
 dry bulbs, if a psychrometer is taken. When- 
 ever a prolonged stop is made, as at dinner or on 
 camping for the night, the readings should be 
 taken directly after the halt and again on starting 
 out, taking this station as a new point of depart- 
 ure. It is possible that a long ascent may be 
 made without a perceptible change of conditions, 
 
Barometrical Determination of Heights. 19 
 
 &nd in this way tedious computations may be 
 avoided. 
 
 Such is the problem of measuring heights by the 
 barometer a process perfectly intelligible at every 
 step, and consisting of weighing every layer of the 
 atmosphere through which the ascent is made. 
 
 Since in a mixture of one or more gases in a 
 given space, each gas behaves dynamically the 
 same as if it occupied the space alone, it is evi- 
 dent that where the conditions remain the same, 
 we can compute heights by three different sets of 
 data. We can consider the air as a mixture of 
 dry air and vapor, as has been done in the fore- 
 going discussion, using the formula h = k log 
 
 Px 
 
 where k is derived from the weight of a cubic foot 
 of moist air under the obtaining conditions, and 
 p and p x are the actual barometric pressures ob- 
 served. Or we may weigh a column of dry air be- 
 tween the two stations, using the formula h = k' 
 
 log J ___2 where p and p x are the barometric 
 
 Px *x 
 
 readings, and f f x are the vapor tensions at the 
 extreme stations, while k' is derived from the 
 weight of a cubic foot of dry air at pressure 
 p f and temperature t. 
 
 Again, making use of the vapor alone, we can 
 
 employ the formula h = k" log -- , where k" is 
 
 *x 
 
 derived from the weight of a cubic foot of vapor at 
 the pressure f and temperature t. The last two 
 
2o Barometrical Determination of Heights. 
 
 methods will not be so reliable as the chief 
 method, since the tension f, which is derived from 
 the dew-point, cannot be measured so closely as 
 the pressure p by a good barometer. Still, all 
 these methods may be employed as checks on the 
 others. 
 
 A practical example is appended illustrating 
 the method in practice and showing within what 
 limits of accuracy heights can be thus determined. 
 A seven-place table of logarithms will be advis- 
 able for the computations. 
 
 Barometer at lower station read 30.2784, and at 
 upper station 30.224, corrected for gravity, but 
 not for capillarity and temperature. Tempera- 
 ture at both stations 15 C., and f at lower station 
 was .2434 inch. The error of the barometer was 
 very slight, if any. 
 
 30.2784 
 
 .085 correction for freezing point. 
 
 30.1934 
 + .037 correction for capillarity. 
 
 30.2304 corrected reading for lower station. 
 
 30.224 
 
 .085 correction for freezing point. 
 
 correction for capillarity. 
 
 30.176 corrected reading for upper station. 
 
 f f = .091 inch. 
 
 .'. p | f = 30 176 .091 30.085 
 
 I -I- .00367 t= 1.055 
 
 log 30.085 = 1.4783500 
 
 log 1.055 = - 02 3 2 5 2 5 
 
 log .0000381468 = 5.5814527 
 
 .'.log TV "03671 x -000038 1 468 = ^0365502 
 
Barometrical Determination of Heights. 21 
 
 This logarithm corresponds to .001087 inch. 
 This shows that under the conditions a cubic foot 
 of air weighed .001087 inch of mercury. 
 
 P + .001087 = 30.1771 
 log 30. 1771= i.479 6 775 
 
 log 30. 176 1.4796617 
 .0000158 
 
 = 63291.1 = k 
 
 log p x = 1.4804438 
 log p 2 = 1.4796617 
 
 .0007821 x k = 
 
 It will be noticed that the last four logarithms, 
 though actually natural logarithms in the for- 
 mulae, are here computed as common logarithms, 
 since the modulus cancels out. 
 
 Therefore the height between the two stations 
 is by the computations 49.5 feet. The height as 
 previously determined by levelling was 50 feet. 
 A number of determinations of the same height 
 were made under widely differing conditions of 
 the atmosphere and the results all came out within 
 half a foot of the correct height, 50 feet some 
 of them being remarkably close. 
 
 Such short distances are very severe tests for a 
 barometer. Where the heights are greater the 
 proportional error will be less. By no refinement 
 is it possible to measure with a barometer much 
 less than a foot, since, as we have seen, the weight 
 of a cubic foot of air is equivalent to only .001 
 (4- or ) inch of mercury, and our barometers 
 do not read finer than thousandths. By careful 
 
22 Barometrical Determination of Heights. 
 
 work it is probable that our error need never ex- 
 ceed 5 feet in 1000. By our method also of mul- 
 tiple stations, the errors, which are as liable to 
 be above as below, will be apt to eliminate 
 themselves in the summation. 
 
 The altitude of Mont Blanc was computed by 
 Delcros, from measurements taken by MM. Bra- 
 vais and Martins, to be 4810 metres, which came 
 strikingly close to the result obtained by a geo- 
 detic survey, viz. : 4809.6. But such a coinci- 
 dence must not deceive us. It is a mere coinci- 
 dence, as Delcros's formula is not susceptible of 
 such accuracy particularly as it wholly neglects 
 the moisture of the air. 
 
TABLE III. 
 
 ELASTIC FORCE OF SATURATED AQUEOUS VAPOR. EXPRESSED 
 IN INCHES OF MERCURY AND FAHRENHEIT TEMPERATURES. 
 
 Temp. 
 
 Force. ' Temp. 
 
 Force. Temp. Force. 
 
 
 i 
 
 
 '- 51 
 
 .0087 13 
 
 0783 1 57 -4655 
 
 ~ 30 
 
 .OOQ2 
 
 14 
 
 .0818 58 .4825 
 
 - 29 
 
 .0098 
 
 15 
 
 .0857 59 .5000 
 
 -- 28 
 
 .OIO4 
 
 16 
 
 .0898 
 
 60 .5179 
 
 27 
 
 .OIIO 
 
 17 
 
 .0940 
 
 61 .5365 
 
 -- 26 
 
 .0117 
 
 18 
 
 .0984 
 
 62 -.5558 
 
 - 25 - 
 
 .0124 ' 19 
 
 .1030 
 
 63 .5756 
 
 - 24 
 
 .0131 20 
 
 .1078 64 -5962 
 
 * 23 
 
 .0138 
 
 21 
 
 .1128 65 .6173 
 
 22 
 
 .0146 
 
 22 
 
 .1179 66 .6392 
 
 21 
 
 .0154 
 
 23 
 
 .1233 67 .6616 
 
 20 
 
 .0163 
 
 24 
 
 .1289 68 .6847 
 
 19 
 
 .OI7I 
 
 25 
 
 .1347 69 .7084 
 
 - 18 
 
 .Ol8l 
 
 26 
 
 1407 70 .7329 
 
 - -7 
 
 .OigO 
 
 27 
 
 .1469 71 .7583 
 
 16 
 
 .O2OO 
 
 28 
 
 .1534 72 .7844 
 
 15 
 
 .0210 
 
 29 ' 
 
 .1600 73 .8113 
 
 - 14 
 
 .O22I 
 
 30 
 
 .1668 74 -8391 
 
 - 13 
 
 .O232 
 
 31 
 
 1739 75 -8676 
 
 12 
 
 .0244 
 
 32 
 
 .1811 76 .8970 
 
 II 
 
 .0257 
 
 33 
 
 .1883 77 .9272 
 
 IO 
 
 .0270 
 
 34 
 
 .1959 78 .9582 
 
 - 9 
 
 .0283 
 
 35 
 
 .2037 79 .9902 
 
 g 
 
 .0297 36 
 
 .2119 So 
 
 .0232 
 
 7 
 
 .0312 i 37 
 
 .2204 Si 
 
 .0572 
 
 - 6 
 
 .0327 38 
 
 .2291 82 
 
 .0922 
 
 - 5 
 
 '0343 | 39 
 
 .2382 83 
 
 .1281 
 
 - 4 
 
 0359 40 
 
 .2476 
 
 84 
 
 .1651 
 
 - 3 
 
 .0376 41 
 
 2572 85 
 
 .2031 
 
 -* 2 
 
 395 
 
 42 
 
 .2672 86 
 
 .2421 
 
 I 
 
 0454 43 
 
 .2775 87 
 
 .2821 
 
 O 
 
 .0434 44 
 
 .2882 : 88 
 
 3234 
 
 I 
 
 0454 45 
 
 .2993 89 
 
 3659 
 
 2 
 
 .0476 
 
 46 
 
 .3108 
 
 90 
 
 .4097 
 
 3 
 
 .0498 
 
 47 
 
 -3228 
 
 91 
 
 4546 
 
 4 
 
 .0521 
 
 48 
 
 3.35* 
 
 9 2 
 
 .5008 
 
 5 
 
 0545 
 
 49 
 
 3477 
 
 93 
 
 .5482 
 
 6 
 
 .0570 
 
 50 
 
 .3608 
 
 94 
 
 .5969 
 
 7 
 
 0597 
 
 51 
 
 3743 
 
 95 
 
 .6468- 
 
 S 
 
 .0625 
 
 52 
 
 .3882 
 
 96 
 
 .6980 
 
 9 
 
 0654 53 
 
 .4027 
 
 97 
 
 .7508 
 
 IO 
 
 .0684 54 
 
 .4176 
 
 98 
 
 .8050 
 
 ii 
 
 0716 55 
 
 4331 
 
 99 
 
 .8607 
 
 12 
 
 .0749 56 
 
 .4490 
 
 100 
 
 .9179 
 
24 Barometrical Determination of Heights. 
 
 TABLE IV. 
 
 Pressures of Aqueous Vapor in Mms. of Mercury 
 at Saturation, for Different Temperatures Cent. 
 
 Temp. 
 
 Press. 
 
 Temp. 
 
 Press. 
 
 Temp. 
 
 Press. 
 
 Temp. 
 
 Press. 
 
 10 
 
 2.0/8 
 
 12 
 
 10.457 
 
 29 
 
 29.782 90 
 
 52545 
 
 8 
 
 2.456 
 
 13 
 
 11.062 
 
 30 
 
 31.548 91 
 
 54578 
 
 6 
 
 2.890 
 
 14 
 
 11.906 
 
 31 
 
 33405 
 
 92 
 
 566.76 
 
 4 
 
 3.387 
 
 15 
 
 12.699 
 
 32 
 
 35-359 
 
 93 
 
 588.41 
 
 2 
 
 3-955 
 
 16 
 
 13.635 
 
 33 
 
 37-410 
 
 94 
 
 610.74 
 
 O 
 
 4.600 
 
 17 
 
 14.421 
 
 34 
 
 39.565 
 
 95 
 
 63378 
 
 4- i 
 
 4.940 
 
 18 
 
 15-357 
 
 35 
 
 41.827 
 
 96 
 
 657.54 
 
 2 
 
 5-302 
 
 19 
 
 16.346 
 
 40 
 
 51.160 
 
 97 
 
 682.03 
 
 3 
 
 5-687 
 
 20 
 
 I7.39I 
 
 45 
 
 7L39I 
 
 98 
 
 707.26 
 
 4 
 
 6.097 
 
 21 
 
 18.495 
 
 50 
 
 91.982 
 
 98.5 
 
 720.15 
 
 5 
 
 6.534 
 
 22 
 
 19.659 
 
 55 
 
 II7479 
 
 99 
 
 733-91 
 
 6 
 
 6.998 
 
 23 
 
 20.888 
 
 60 
 
 149.321 
 
 99-5 
 
 746.50 
 
 7 
 
 7-492 
 
 24 
 
 22.184 
 
 65 
 
 186.945 
 
 IOO 
 
 760.00 
 
 8 
 
 8.017 
 
 25 
 
 23-550 
 
 70 
 
 233-093:100.5 
 
 773-71 
 
 9 
 
 8-574 
 
 26 
 
 24.998 
 
 75 
 
 288.517 
 
 101 
 
 787.63 
 
 10 
 
 9.165 
 
 27 
 
 26.505 
 
 80 
 
 355-40 
 
 102 
 
 8l6.I7 
 
 IT 
 
 9.792 28 
 
 28.101 85 
 
 433-41 
 
 104 
 
 875-69 
 
Barometrical Determination of Heights. 25 
 
 STANDARD BAROMETRICAL FORMULAS 
 
 h = height 
 
 I _ k i og . 
 
 PQ k = barometrical coefficient 
 
 p^ p = pressure at lower station 
 
 p 4 = pressure at upper station 
 
 k = 
 
 3* 
 i 
 
 log (i + .000125156. 
 
 i -f- .003671 
 t is Cent. Gives height in metres. 
 
 k = 
 
 3f 
 i 
 
 8 P 
 log (i + .0000381468. 
 
 ) 
 _J 
 
 .00204 (t 32) 
 t is Fahr. Gives height in feet. 
 
 k = 
 
 i 
 
 8p 
 log (i -f .0000381468 . 
 
 i + .003671 
 t is Cent. Gives height in feet. 
 
26 Barometrical Determination of Heights. 
 
 NOTE ON BOILING THERMOMETERS. 
 
 By determining the temperature at which 
 pure water boils, we can, from the table giving 
 the tension of saturated vapor at various tem- 
 peratures find the pressures of the atmosphere. 
 If we are provided with thermometers which can 
 be depended upon to give the boiling point within 
 i/ioo of a degree Cent, (which is doubtful), 
 we shall be able by this method to determine dif- 
 ferences of level of about 10 feet, since at lower 
 levels, an ascent of about 1080 feet makes a 
 difference. of i Cent, in the boiling point. But 
 by an ordinary mercurial barometer, we can, 
 with less trouble, determine differences of level 
 of one foot. Hence, the mercurial barometer 
 remains the most reliable and accurate instru- 
 ment we possess at present for the measurement 
 of pressures. 
 
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