# Theory of Heat (Maxwell, 4th edition)/Chapter 4

CHAPTER IV.
ELEMENTARY DYNAMICAL PRINCIPLES.
In the first part of this treatise we have confined ourselves to the explanation of the method of ascertaining the temperature of bodies, which we call thermometry, and the method of measuring the quantity of heat which enters or leaves a body, and this we call calorimetry. Both of these are required in order to study the effects of heat upon bodies; but we cannot complete this study without making measurements of a mechanical kind, because heat and mechanical force may act on the same body, and the actual result depends on both actions. I propose, therefore, to recall to the student's memory some of those dynamical principles which he ought to bring with him to the study of heat, and which are necessary when he passes from purely thermal phenomena, such as we have considered, to phenomena involving pressure, expansion, &c., and which will enable him afterwards to proceed to the study of thermodynamics proper, in which the relations of thermal phenomena among themselves are deduced from purely dynamical principles.
The most important step in the progress of every ​science is the measurement of quantities. Those whose curiosity is satisfied with observing what happens have occasionally done service by directing the attention of others to the phenomena they have seen; but it is to those who endeavour to find out how much there is of anything that we owe all the great advances in our knowledge.
Thus every science has some instrument of precision, which may be taken as a material type of that science which it has advanced, by enabling observers to express their results as measured quantities. In astronomy we have the divided circle, in chemistry the balance, in heat the thermometer, while the whole system of civilised life may be fitly symbolised by a foot rule, a set of weights, and a clock. I shall, therefore, make a few remarks on the measurement of quantities.
Every quantity is expressed by a phrase consisting of two components, one of these being the name of a number, and the other the name of a thing of the same kind as the quantity to be expressed, but of a certain magnitude agreed on among men as a standard or unit.
Thus we speak of two days, of forty-eight hours.
Each of these expressions has a numerical part and a denominational part, the numerical part being a number, whole or fractional, and the denominational part being the name of the thing, which is to be taken as many times as is indicated by the number.
If the numerical part is the number one, then the quantity is the standard quantity itself, as when we say one pound, or one inch, or one day. A quantity of which the numerical part is unity is called a unit. When the numerical part is some other number, the quantity is still said to be referred to, or to be expressed in terms of that quantity which would be denoted if the number were one, and which is called the unit.
In all cases the unit is a quantity of the same kind as the quantity which is expressed by means of it.
In many cases several units of the same kind are in use, ​as miles, yards, feet, and inches, as measures of length; cubic yards, gallons, and fluid ounces, as measures of capacity; besides the endless variety of units which have been adopted by different nations, and by different districts and different trades in the same nation.
When a quantity given in terms of one unit has to be expressed in terms of another, we find the number of times the second unit is contained in the first, and multiply this by the given number.
Hence the numerical part of the expression of the same quantity varies inversely as the unit in which it is to be expressed, as in the example, two days and forty-eight hours, which mean the same thing.
There are many quantities which can be defined in terms of standard quantities of a different kind. In this case we make use of derived units. For instance, as soon as we have fixed on a measure of length, we may define by means of it not only all lengths, but also the area of any surface, and the content of any space. For this purpose, if the foot is the unit of length, we construct, by Euclid I. 46, a square whose side is a foot, and express all areas in terms of this square foot, and by constructing a cube whose edge is a foot we have defined a cubic foot as a unit of capacity.
We also express velocities in miles an hour, or feet in a second, &c.
In fact, all quantities with which we have to do in dynamics may be expressed in terms of units derived by definition from the three fundamental units—of Length, Mass, and Time.
STANDARD OF LENGTH.
It is so important to mankind that these units should be well defined that in all civilised nations they are defined by the State with reference to material standards, which are preserved with the utmost care. For instance, in this country it was enacted by Parliament 'that the straight line or ​distance between the centres of the transverse lines in the two gold plugs in the bronze bar deposited in the office of the Exchequer shall be the genuine standard yard at 62° F., and if lost it shall be replaced by means of its copies.'
The authorised copies here referred to are those which are preserved at the Royal Mint, the Royal Society of London, the Royal Observatory at Greenwich, and the New Palace at Westminster. Other copies have been made with great care, and with these all measures of length must be compared.
The length of the Parliamentary standard was chosen so as to be as nearly as possible equal to that of the best standard yards formerly used in England. The State, therefore, endeavoured to maintain the standard of its ancient magnitude, and by its authority it has defined the actual magnitude of this standard with all the precision of which modern science is capable.
The mètre derives its authority as a standard from a law of the French Republic in 1795.
It is defined to be the distance between the ends of a rod of platinum made by Borda, the rod being at the temperature of melting ice. This distance was chosen without reference to any former measures used in France. It was intended to be a universal and not a national measure, and was derived from Delambre and Mechain's measurement of the size of the earth. The distance measured along the earth's surface from the pole to the equator is nearly ten million of mètres. If, however, in the progress of geodesy, a different result should be obtained from that of Delambre, the mètre will not be altered, but the new result will be expressed in the old mètres. The authorised standard of length is therefore not the terrestrial globe, but Borda's ​platinum rod, which is much more likely to be accurately measured.
The value of the French system of measures does not depend so much on the absolute values of the units adopted as on the fact that all the units of the same kind are connected together by a decimal system of multiplication and division, so that the whole system, under the name of the metrical system, is rapidly gaining ground even in countries where the old national system has been carefully defined.
The mètre is 39.37043 British inches.
STANDARD OF MASS.
By the Act above cited a weight of platinum marked 'P. S. 1844, 1 lb.,' deposited in the office of the Exchequer, 'shall be the legal and genuine standard measure of weight, and shall be and be denominated the Imperial Standard Pound Avoirdupois, and shall be deemed to be the only standard measure of weight from which all other weights and other measures having reference to weight shall be derived, computed, and ascertained, and one equal seven-thousandth part of such pound avoirdupois shall be a grain, and five thousand seven hundred and sixty such grains shall be and be deemed to be a pound troy. If at any time hereafter the said Imperial Standard Pound Avoirdupois be lost or in any manner destroyed, defaced, or otherwise injured, the Commissioners of Her Majesty's Treasury may cause the same to be restored by reference to or adoption of any of the copies aforesaid, or such of them as may remain available for that purpose.'
The construction of this standard was entrusted to Professor W. H. Miller, who has given an account of the methods employed in a paper, which may be here referred to as a model of scientific accuracy.
​The French standard of mass is the Kilogramme des Archives, made of platinum by Borda, and is intended to represent the mass of a cubic décimètre of distilled water at the temperature 4° C.
The actual determination of the density of water is an operation which requires great care, and the differences between the results obtained by the most skilful observers, though small, are a thousand times greater than the differences of the results of a comparison of standards by weighing them. The differences of the values of the density of water as found by careful observers are as much as a thousandth part of the whole, whereas the method of weighing admits of an accuracy of within one part in five millions.
Hence the French standards, though originally formed to represent certain natural quantities, must be now considered as arbitrary standards, of which copies are to be taken by direct comparison. The French or metric system has the advantage of a uniform application of the decimal method, and it is also in many cases convenient to remember that a cubic mètre of water is a tonne, a cubic décimètre a kilogramme, a cubic centimètre a gramme, and a cubic millimètre a milligramme, the water being at its maximum density or at about 4° C.
In 1826 the British standard of mass was defined by saying that a cubic inch of water at 62° F. contains 252.458 grains, and though this is no longer a legal definition, we may take it as a rough statement of a fact, that a cubic inch of water weighs about 252.5 grains, a cubic foot about 1,000 ounces avoirdupois, and a cubic yard about three-quarters of a ton. Of these estimates the second is the furthest from the truth.
Professor Miller has compared the British and French standards, and finds the Kilogramme des Archives equal to 15432.34874 grains.
From these legal definitions it will be seen that what is generally called a standard of weight is a certain piece of ​platinum—that is, a particular body the quantity of matter in which is taken and defined by the State to be a pound or a kilogramme.
The weight strictly so called—that is, the tendency of this body to move downwards—is not invariable, for it depends on the part of the world where it is placed, its weight being greater at the poles than at the equator, and greater at the level of the sea than at the top of a mountain.
What is really invariable is the quantity of matter in the body, or what is called in scientific language the mass of the body, and even in commercial transactions what is generally aimed at in weighing goods is to estimate the quantity of matter, and not to determine the force with which they tend downwards.
In fact, the only occasions in common life in which it is required to estimate weight considered as a force is when we have to determine the strength required to lift or carry things, or when we have to make a structure strong enough to support their weight. In all other cases the word weight must be understood to mean 'the quantity of the thing as determined by the process of weighing against 'standard weights.'
As a great deal of confusion prevails on this subject in ordinary language, and still greater confusion has been introduced into books on mechanics by the notion that a pound is a certain force, instead of being, as we have seen, a certain piece of platinum, or a piece of any other kind of matter equal in mass to the piece of platinum, I have thought it worth while to spend some time in defining accurately what is meant by a pound and a kilogramme.
ON THE UNIT OF TIME.
All nations derive their measures of time from the apparent motions of the heavenly bodies. The motion of rotation of the earth about its axis is very nearly indeed uniform, and the measure of time in which one day is equal ​to the time of revolution of the earth about its axis, or more exactly to the interval between successive transits of the first point of Aries, is used by astronomers under the name of sidereal time.
Solar time is that which is given by a sun-dial, and is not uniform. A uniform measure of time, agreeing with solar time in the long run, is called mean solar time, and is that which is given by a correct clock. A solar day is longer than a sidereal day. In all physical researches mean solar time is employed, and one second is generally taken as the unit of time.
The evidence upon which we form the conclusion that two different portions of time are or are not equal can only be appreciated by those who have mastered the principles of dynamical reasoning. I can only here assert that the comparison, for example, of the length of a day at present with the length of a day 3,000 years ago is by no means an unfruitful enquiry, and that the relative length of these (lays may be determined to within a small fraction of a second. This shows that time, though we conceive it merely as the succession of our states of consciousness, is capable of measurement, independently, not only of our mental states, but of any particular phenomenon whatever.
ON MEASUREMENTS FOUNDED ON THE THREE FUNDAMENTAL UNITS.
In the measurement of quantities differing in kind from the three units, we may either adopt a new unit independently for each new quantity, or we may endeavour to define a unit of the proper kind from the fundamental units. In the latter case we are said to use a system of units. For instance, if we have adopted the foot as a unit of length, the systematic unit of capacity is the cubic foot.
The gallon, which is a legal measure in this country, is unsystematic considered as a measure of capacity, as it ​contains the awkward number of 277.274 cubic inches. The gallon, however, is never tested by a direct measurement of its cubic contents, but by the condition that it must contain ten pounds of water at 62° F.
Definition of Density.—The density of a body is measured by the number of units of mass in unit of volume of the substance.
For instance, if the foot and the pound be taken as fundamental units, then the density of anything is the number of pounds in a cubic foot. The density of water is about 62.5 pounds to the cubic foot. In the metric system, the density of water is one tonne to the stère, one kilogramme to the litre, one gramme to the cubic centimètre, and one milligramme to the cubic millimètre.
We shall sometimes have to use the word rarity, to signify the inverse of density, that is, the volume of unit of mass of a substance.
Definition of Specific Gravity.—The specific gravity of a body is the ratio of its density to that of some standard substance, generally water.
Since the specific gravity of a body is the ratio of two things of the same kind, it is a numerical quantity, and has the same value, whatever national units are employed by those who determine it. Thus, if we say that the specific gravity of mercury is about 13.5, we state that mercury is about thirteen and a half times heavier than an equal bulk of water, and this fact is independent of the way in which we measure either the mass or the volume of the liquids.
Definition of Uniform Velocity.—The velocity of a body moving uniformly is measured by the number of units of length travelled over in unit of time.
Thus we speak of a velocity of so many feet or mètres per second.
Definition of Momentum.—The momentum of a body is measured by the product of the velocity of the body into the number of units of mass in the body
​Definition of Force.—Force is whatever changes or tends to change the motion of a body by altering either its direction or its magnitude; and a force acting on a body is measured by the momentum it produces in its own direction in unit of time.
The unit of force is that force which if it acted on unit of mass for unit of time would produce in it unit of velocity.
For the British unit of force the name of Poundal has been proposed by Prof. James Thomson. It is that force which, if it acted for a second on a pound, would produce in it a velocity of one foot per second.
In the centimetre-gramme-second system, adopted by the Committee on Units of the British Association, the unit of force is the Dyne. A dyne acting for one second on a gramme would give it a velocity of one centimetre per second.
The weight of any body at London, acting on that body for a second, would produce in it a velocity of 32.1889 feet per second. Hence the weight of a pound at London is 32.1889 poundals.
At Paris the velocity of a body after falling freely for one second is 980.868 centimes per second. Hence the weight of a gramme at Paris is 980.868 dynes.
It is so convenient, especially when all our experiments are conducted in the same place, to express forces in terms of the weight of a pound or a gramme, that in all countries the first measurements of forces were made in this way, and a force was described as a force of so many pounds weight or grammes weight. It was only after the measurements of forces made by persons in different parts of the world had to be compared that it was found that the weight of a pound or a gramme is different in different places, and depends on the intensity of gravitation, or the attraction of the earth; so that for purposes of accurate comparison all forces must be reduced to absolute or dynamical measure as explained above. We shall distinguish the measure by ​comparison with weight as the gravitation measure of force. To reduce forces expressed in gravitation measure to absolute measure, we must multiply the number denoting the force in gravitation measure by the value of the intensity of gravity expressed in the same metrical system. The value of the intensity of gravity is a very important number in all scientific calculations, and it is generally denoted by the letter g. The number g may be defined in any of the following ways, which are all equivalent:
g is a number expressing the velocity produced in a falling body in unit of time.
g is a number expressing twice the distance through which a body falls in unit of time.
g is a number expressing the weight of unit of mass in absolute measure.
The value of g is generally determined at any place by experiments with the pendulum. These experiments require great care, and the description of them does not belong to our present subject. The value of g may be found with sufficient accuracy for the present state of science by means of the formula,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikisource.org/v1/":): {\displaystyle g = G (1-0.0025659 \cos 2 \lambda) { 1 − ( 2 − 3) ( ); }}
In this formula, |G is the intensity of gravity a the mean level of the sea in latitude 45°:
G=32.1703 poundals to the pound, or 9.80533 dynes to the gramme.
$\lambda$ is the latitude of the place. The formula shows that the force of gravity at the level of the sea increases from the equator to the poles. The last factor of the formula expresses, according to the calculations of Poisson, the effect of the height of the place of observation above the level of the sea in diminishing the force of gravity. The symbol $\rho$ represents the mean density of the whole earth, which is probably about 51/2 times that of water. $\rho '$ ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/101 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/102 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/103 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/104 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/105 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/106 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/107 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/108 ​Page:Theory of Heat, James Clerk Maxwell, Fourth Edition.djvu/109 ​body to be in the form of kinetic energy—that is, that the particles of the hot body are in actual though invisible motion—will be discussed afterwards.
* ↑ 18 & 19 Vict. c. 72, July 30, 1855.
* ↑ 'Mètre conforme à la loi du 18 Germinal, an III. Présenté le 4 Messidor, an VII.
* ↑ In the same places as the Standards of Length.
* ↑ Phil. Trans. 1856, p. 753.
* ↑ Traité de Mécanique, t. ii. p. 629.