272 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[July, 



stations; d = the difference of the degrees shown by Fahrenheit's 

 thermometer attached to the barometer; /= the mean of the two 

 temperatures shown by the detached thermometers exposed for a few 

 minutes to the open air in the shade, at the two stations. The sign 

 minus takes place when the attached thermometer is highest at the 

 lower station, and the sign plus when it is lowest at that station. 

 100O0 (iog. m — iog. M) was the expression formerly given to find the 

 altitude in fathoms, m M being the heights of the mercury at the base 

 and summit of any eminence. This formula is very easily applied, 

 and not far from the truth when an allowance is made fur the increase 

 of temperature above 31 c , for this is the degree of temperature to 

 which the above formula is calculated, or rather adopted. As air 

 expands very nearly ^ part of its bulk with every degree of heat, 

 and suffers the same contraction with every degree of cold, the follow- 

 ing rule was usually given. Rule. — Observe the height of the mer- 

 cury at the bottom of the object to be measured, and again at the top, 

 as also the degree of the thermometer at both these situations, and 

 half the sum of these two last may be accounted the mean tempera- 

 ture. Then multiply the difference of the logs, of the two heights of 

 the barometer by 10000, and correct the result by adding or subtract- 

 ing so many times its 435th part as the degrees of the mean tempe- 

 rature are more or less than 81°'; the last number will be the altitude 

 in fathoms. 



We are too apt to say, when two or more phenomena happen to- 

 gether, that one' is caused by the other ; where all may be governed 

 by some unknown phenomena. The writer of this article agrees with 

 Mr. Pasley, that the philosophy is false which teaches that expansion 

 is caused by heat, for without fire or heat water is expanded as it 

 becomes ice, and air in the air pump vacuum ; solids require fire as a 

 means, but the expanding cause itself is perfectly distinct from fire. 

 However, when experiments show that certain phenomena increase 

 and decrease together, one may be taken as an index, if not as a func- 

 tion of the other ; but great care ought to hi- taken not to draw gene- 

 ral inferences from limited experiments. At some future time we 

 shall explain why the attached and detached thermometers differ, and 

 also show how they may he made to agree, and at present proceed to 

 the more immediate object of the communication. To illustrate the 

 rule just given, we shall add another example. 



III. If the heights of the barometer at the bottom and top of a hill 

 are 29 - 37 and 20-59 inches respectively, and the mean temperature 

 26°, what is the height'! 



Log. 29-37 = 1-4679 



Log. 26-59= 1-1247 1x3 



0-0481866 x 10000 = 431*856. Then 31 -26° = 

 5 degrees, therefore by the rule, 431-850 - 431-856 X tjt = 431-856 

 — 4-S64 = 426-892 = the height in fathoms. 



We shall now investigate the last 

 formula, and give an outline of the 

 theory upon which this proposition is 

 founded. Let EAR represent part of 

 the surface of the earth, and A T a co- 

 lumn of the atmosphere. Conceive this 

 column to be divided into a number of 

 equal and infinitely small parts, as AB, 

 BC, CD, &c, in each of which we may 

 suppose the density to be uniform, be- 

 cause they are infinitely small. Now since the density of the air is 

 always directly as the compressing force, therefore we have the den- 

 sity of the air in any of the portions AB, BC, &c, as the weight of 

 the column of the atmosphere above that place ; that is, if P repre- 

 sents generally the pressure, D the density of any place, P' the 

 pressure at any other place, and D' its corresponding density, we 

 shall have P : P' : : D : D' ; that is, the pressure is to the density 

 in a constant ratio, and may be represented by n to 1 ; therefore, 



P : D : : P' ; D' : ; n ; 1, consequently abstractedly speaking, 



D = - P', 



D" = - P ', &c. 



That is, the density at any place is equal to, or rather may be mea- 

 sured by, the nth of the pressure of the column of the atmosphere 

 above that place, or by the n th of the compressing force. Hence if 

 we make P stand for the pressure at the surface A, and let each of 



the parts AB, BC, CD, &c, be equal to 1, then will - P represent the 



weight or pressure of the part AB, and therefore, 



P P =z P =: the pressure at B, and — ; — P = the density 



n n n- 



or weight of BC. In the same way, 



: — P = the pressure at C, j — P = the pressure at D, &c. 



So that the pressure, and consequently the density, will decrease in a 

 geometrical progression, as the altitudes increase in an arithmetical 



progression. Calling the density at the surface d' , and the several 

 altitudes 1, 2, 3, 4, &c. we shall have the following corresponding 

 series : 

 Altitudes .. 0, 1, 2, 3, &c. 



m—1 jji-2 j»!-3 



Corresponding densities d d" 



Dividing the latter series by d n we have, 

 Altitudes ..0, 1, i 



Corresg. densities 1 d d~ 



,&c. 



3, 



j— 3 



4, &c. 

 ,-4 



',&c. 



This is strictly analogous to the property of logarithms. In fact 

 the several altitudes form a peculiar system of logarithms of which 

 the reciprocals of the corresponding densities are the natural num- 

 bers : from this circumstance they have been denominated atmospheric 

 logarithms. From a similar circumstance the Napierian are termed 

 hyperbolic logarithms, because they express the areas contained 

 between the asyptotle and curve of au hyperbola. We shall write 

 these atmospheric logarithms with large letters, thus, "Log," to 

 distinguish them from the Briggean or common logarithms, which are 

 written "log. ., " or simply "log." and also from the hyperbolic or 

 Naperian which are denoted by "log. " Let A, a, represent any 



two altitudes, and D, d, their corresponding densities, then will A = 

 — Log. D, and a =. — Log. d; 



d 



A — a = Log. d — Log. D = Log. 



D 



Now it is a well known property in logarithms, that by assuming 

 different values for the base, there will be as many different systems 

 of logarithms ; and it is equally well known, that in all the various 

 system;, of logarithms, the logarithms of the same numbers can be 

 converted from one system to another, by a constant multiplier or 

 modulus. The object of our present inquiry is to determine a con- 

 stant multiplier that shall convert the common logarithm of a number 

 into the atmospheric logarithm of the same number. To accomplish 

 this, let 



d , d 



— = x log. p, 



d 



U 



A — a = x log. 



D' 



Then making a =z o, or which is the same, if we suppose d to repre- 

 sent the density of the atmosphere at the surface of the earth, we 

 d 



D 



shall have A = x log. 



In order to find x let us take the height of a homogeneous atmos- 

 phere, when the temperature shown by the thermometer is 31°, and 

 the height of the barometer 29J inches at 26057 feet; then the den- 

 sity at the surface, and one foot above it, will be 



