MEASUREMENTS
§ 1. METROLOGY
Not so very long ago, the scholar was often content with qualitative
observations. Many phenomena were studied without much trouble being
taken to obtain actual measurements. But it is now becoming more and
more understood that to establish the relations which exist between
physical magnitudes, and to represent the variations of these
magnitudes by functions which allow us to use the power of
mathematical analysis, it is most necessary to express each magnitude
by a definite number.
Under these conditions alone can a magnitude be considered as
effectively known. "I often say," Lord Kelvin has said, "that if you
can measure that of which you are speaking and express it by a number
you know something of your subject; but if you cannot measure it nor
express it by a number, your knowledge is of a sorry kind and hardly
satisfactory. It may be the beginning of the acquaintance, but you are
hardly, in your thoughts, advanced towards science, whatever the
subject may be."
It has now become possible to measure exactly the elements which enter
into nearly all physical phenomena, and these measurements are taken
with ever increasing precision. Every time a chapter in science
progresses, science shows itself more exacting; it perfects its means
of investigation, it demands more and more exactitude, and one of the
most striking features of modern physics is this constant care for
strictness and clearness in experimentation.
A veritable science of measurement has thus been constituted which
extends over all parts of the domain of physics. This science has its
rules and its methods; it points out the best processes of
calculation, and teaches the method of correctly estimating errors and
taking account of them. It has perfected the processes of experiment,
co-ordinated a large number of results, and made possible the
unification of standards. It is thanks to it that the system of
measurements unanimously adopted by physicists has been formed.
At the present day we designate more peculiarly by the name of
metrology that part of the science of measurements which devotes
itself specially to the determining of the prototypes representing the
fundamental units of dimension and mass, and of the standards of the
first order which are derived from them. If all measurable quantities,
as was long thought possible, could be reduced to the magnitudes of
mechanics, metrology would thus be occupied with the essential
elements entering into all phenomena, and might legitimately claim the
highest rank in science. But even when we suppose that some magnitudes
can never be connected with mass, length, and time, it still holds a
preponderating place, and its progress finds an echo throughout the
whole domain of the natural sciences. It is therefore well, in order
to give an account of the general progress of physics, to examine at
the outset the improvements which have been effected in these
fundamental measurements, and to see what precision these improvements
have allowed us to attain.
§ 2. THE MEASURE OF LENGTH
To measure a length is to compare it with another length taken as
unity. Measurement is therefore a relative operation, and can only
enable us to know ratios. Did both the length to be measured and the
unit chosen happen to vary simultaneously and in the same degree, we
should perceive no change. Moreover, the unit being, by definition,
the term of comparison, and not being itself comparable with anything,
we have theoretically no means of ascertaining whether its length
varies.
If, however, we were to note that, suddenly and in the same
proportions, the distance between two points on this earth had
increased, that all the planets had moved further from each other,
that all objects around us had become larger, that we ourselves had
become taller, and that the distance travelled by light in the
duration of a vibration had become greater, we should not hesitate to
think ourselves the victims of an illusion, that in reality all these
distances had remained fixed, and that all these appearances were due
to a shortening of the rule which we had used as the standard for
measuring the lengths.
From the mathematical point of view, it may be considered that the two
hypotheses are equivalent; all has lengthened around us, or else our
standard has become less. But it is no simple question of convenience
and simplicity which leads us to reject the one supposition and to
accept the other; it is right in this case to listen to the voice of
common sense, and those physicists who have an instinctive trust in
the notion of an absolute length are perhaps not wrong. It is only by
choosing our unit from those which at all times have seemed to all men
the most invariable, that we are able in our experiments to note that
the same causes acting under identical conditions always produce the
same effects. The idea of absolute length is derived from the
principle of causality; and our choice is forced upon us by the
necessity of obeying this principle, which we cannot reject without
declaring by that very act all science to be impossible.
Similar remarks might be made with regard to the notions of absolute
time and absolute movement. They have been put in evidence and set
forth very forcibly by a learned and profound mathematician, M.
Painlevé.
On the particularly clear example of the measure of length, it is
interesting to follow the evolution of the methods employed, and to
run through the history of the progress in precision from the time
that we have possessed authentic documents relating to this question.
This history has been written in a masterly way by one of the
physicists who have in our days done the most by their personal
labours to add to it glorious pages. M. Benoit, the learned Director
of the International Bureau of Weights and Measures, has furnished in
various reports very complete details on the subject, from which I
here borrow the most interesting.
We know that in France the fundamental standard for measures of length
was for a long time the _Toise du Châtelet_, a kind of callipers
formed of a bar of iron which in 1668 was embedded in the outside wall
of the Châtelet, at the foot of the staircase. This bar had at its
extremities two projections with square faces, and all the _toises_ of
commerce had to fit exactly between them. Such a standard, roughly
constructed, and exposed to all the injuries of weather and time,
offered very slight guarantees either as to the permanence or the
correctness of its copies. Nothing, perhaps, can better convey an idea
of the importance of the modifications made in the methods of
experimental physics than the easy comparison between so rudimentary a
process and the actual measurements effected at the present time.
The _Toise du Châtelet_, notwithstanding its evident faults, was
employed for nearly a hundred years; in 1766 it was replaced by the
_Toise du Pérou_, so called because it had served for the measurements
of the terrestrial arc effected in Peru from 1735 to 1739 by Bouguer,
La Condamine, and Godin. At that time, according to the comparisons
made between this new _toise_ and the _Toise du Nord_, which had also
been used for the measurement of an arc of the meridian, an error of
the tenth part of a millimetre in measuring lengths of the order of a
metre was considered quite unimportant. At the end of the eighteenth
century, Delambre, in his work _Sur la Base du Système métrique
décimal_, clearly gives us to understand that magnitudes of the order
of the hundredth of a millimetre appear to him incapable of
observation, even in scientific researches of the highest precision.
At the present date the International Bureau of Weights and Measures
guarantees, in the determination of a standard of length compared with
the metre, an approximation of two or three ten-thousandths of a
millimetre, and even a little more under certain circumstances.
This very remarkable progress is due to the improvements in the method
of comparison on the one hand, and in the manufacture of the standard
on the other. M. Benoit rightly points out that a kind of competition
has been set up between the standard destined to represent the unit
with its subdivisions and multiples and the instrument charged with
observing it, comparable, up to a certain point, with that which in
another order of ideas goes on between the gun and the armour-plate.
The measuring instrument of to-day is an instrument of comparison
constructed with meticulous care, which enables us to do away with
causes of error formerly ignored, to eliminate the action of external
phenomena, and to withdraw the experiment from the influence of even
the personality of the observer. This standard is no longer, as
formerly, a flat rule, weak and fragile, but a rigid bar, incapable of
deformation, in which the material is utilised in the best conditions
of resistance. For a standard with ends has been substituted a
standard with marks, which permits much more precise definition and
can be employed in optical processes of observation alone; that is, in
processes which can produce in it no deformation and no alteration.
Moreover, the marks are traced on the plane of the neutral fibres[2]
exposed, and the invariability of their distance apart is thus
assured, even when a change is made in the way the rule is supported.
[Footnote 2: The author seems to refer to the fact that in the
standard metre, the measurement is taken from the central one of three
marks at each end of the bar. The transverse section of the bar is an
X, and the reading is made by a microscope.--ED.]
Thanks to studies thus systematically pursued, we have succeeded in
the course of a hundred years in increasing the precision of measures
in the proportion of a thousand to one, and we may ask ourselves
whether such an increase will continue in the future. No doubt
progress will not be stayed; but if we keep to the definition of
length by a material standard, it would seem that its precision cannot
be considerably increased. We have nearly reached the limit imposed by
the necessity of making strokes of such a thickness as to be
observable under the microscope.
It may happen, however, that we shall be brought one of these days to
a new conception of the measure of length, and that very different
processes of determination will be thought of. If we took as unit, for
instance, the distance covered by a given radiation during a
vibration, the optical processes would at once admit of much greater
precision.
Thus Fizeau, the first to have this idea, says: "A ray of light, with
its series of undulations of extreme tenuity but perfect regularity,
may be considered as a micrometer of the greatest perfection, and
particularly suitable for determining length." But in the present
state of things, since the legal and customary definition of the unit
remains a material standard, it is not enough to measure length in
terms of wave-lengths, and we must also know the value of these
wave-lengths in terms of the standard prototype of the metre.
This was determined in 1894 by M. Michelson and M. Benoit in an
experiment which will remain classic. The two physicists measured a
standard length of about ten centimetres, first in terms of the
wave-lengths of the red, green, and blue radiations of cadmium, and
then in terms of the standard metre. The great difficulty of the
experiment proceeds from the vast difference which exists between the
lengths to be compared, the wave-lengths barely amounting to half a
micron;[3] the process employed consisted in noting, instead of this
length, a length easily made about a thousand times greater, namely,
the distance between the fringes of interference.
[Footnote 3: I.e. 1/2000 of a millimetre.--ED.]
In all measurement, that is to say in every determination of the
relation of a magnitude to the unit, there has to be determined on the
one hand the whole, and on the other the fractional part of this
ratio, and naturally the most delicate determination is generally that
of this fractional part. In optical processes the difficulty is
reversed. The fractional part is easily known, while it is the high
figure of the number representing the whole which becomes a very
serious obstacle. It is this obstacle which MM. Michelson and Benoit
overcame with admirable ingenuity. By making use of a somewhat similar
idea, M. Macé de Lépinay and MM. Perot and Fabry, have lately effected
by optical methods, measurements of the greatest precision, and no
doubt further progress may still be made. A day may perhaps come when
a material standard will be given up, and it may perhaps even be
recognised that such a standard in time changes its length by
molecular strain, and by wear and tear: and it will be further noted
that, in accordance with certain theories which will be noticed later
on, it is not invariable when its orientation is changed.
For the moment, however, the need of any change in the definition of
the unit is in no way felt; we must, on the contrary, hope that the
use of the unit adopted by the physicists of the whole world will
spread more and more. It is right to remark that a few errors still
occur with regard to this unit, and that these errors have been
facilitated by incoherent legislation. France herself, though she was
the admirable initiator of the metrical system, has for too long
allowed a very regrettable confusion to exist; and it cannot be noted
without a certain sadness that it was not until the _11th July 1903_
that a law was promulgated re-establishing the agreement between the
legal and the scientific definition of the metre.
Perhaps it may not be useless to briefly indicate here the reasons of
the disagreement which had taken place. Two definitions of the metre
can be, and in fact were given. One had for its basis the dimensions
of the earth, the other the length of the material standard. In the
minds of the founders of the metrical system, the first of these was
the true definition of the unit of length, the second merely a simple
representation. It was admitted, however, that this representation had
been constructed in a manner perfect enough for it to be nearly
impossible to perceive any difference between the unit and its
representation, and for the practical identity of the two definitions
to be thus assured. The creators of the metrical system were persuaded
that the measurements of the meridian effected in their day could
never be surpassed in precision; and on the other hand, by borrowing
from nature a definite basis, they thought to take from the definition
of the unit some of its arbitrary character, and to ensure the means
of again finding the same unit if by any accident the standard became
altered. Their confidence in the value of the processes they had seen
employed was exaggerated, and their mistrust of the future
unjustified. This example shows how imprudent it is to endeavour to
fix limits to progress. It is an error to think the march of science
can be stayed; and in reality it is now known that the ten-millionth
part of the quarter of the terrestrial meridian is longer than the
metre by 0.187 millimetres. But contemporary physicists do not fall
into the same error as their forerunners, and they regard the present
result as merely provisional. They guess, in fact, that new
improvements will be effected in the art of measurement; they know
that geodesical processes, though much improved in our days, have
still much to do to attain the precision displayed in the construction
and determination of standards of the first order; and consequently
they do not propose to keep the ancient definition, which would lead
to having for unit a magnitude possessing the grave defect from a
practical point of view of being constantly variable.
We may even consider that, looked at theoretically, its permanence
would not be assured. Nothing, in fact, proves that sensible
variations may not in time be produced in the value of an arc of the
meridian, and serious difficulties may arise regarding the probable
inequality of the various meridians.
For all these reasons, the idea of finding a natural unit has been
gradually abandoned, and we have become resigned to accepting as a
fundamental unit an arbitrary and conventional length having a
material representation recognised by universal consent; and it was
this unit which was consecrated by the following law of the 11th July
1903:--
"The standard prototype of the metrical system is the international
metre, which has been sanctioned by the General Conference on Weights
and Measures."
§ 3. THE MEASURE OF MASS
On the subject of measures of mass, similar remarks to those on
measures of length might be made. The confusion here was perhaps still
greater, because, to the uncertainty relating to the fixing of the
unit, was added some indecision on the very nature of the magnitude
defined. In law, as in ordinary practice, the notions of weight and of
mass were not, in fact, separated with sufficient clearness.
They represent, however, two essentially different things. Mass is the
characteristic of a quantity of matter; it depends neither on the
geographical position one occupies nor on the altitude to which one
may rise; it remains invariable so long as nothing material is added
or taken away. Weight is the action which gravity has upon the body
under consideration; this action does not depend solely on the body,
but on the earth as well; and when it is changed from one spot to
another, the weight changes, because gravity varies with latitude and
altitude.
These elementary notions, to-day understood even by young beginners,
appear to have been for a long time indistinctly grasped. The
distinction remained confused in many minds, because, for the most
part, masses were comparatively estimated by the intermediary of
weights. The estimations of weight made with the balance utilize the
action of the weight on the beam, but in such conditions that the
influence of the variations of gravity becomes eliminated. The two
weights which are being compared may both of them change if the
weighing is effected in different places, but they are attracted in
the same proportion. If once equal, they remain equal even when in
reality they may both have varied.
The current law defines the kilogramme as the standard of mass, and
the law is certainly in conformity with the rather obscurely expressed
intentions of the founders of the metrical system. Their terminology
was vague, but they certainly had in view the supply of a standard for
commercial transactions, and it is quite evident that in barter what
is important to the buyer as well as to the seller is not the
attraction the earth may exercise on the goods, but the quantity that
may be supplied for a given price. Besides, the fact that the founders
abstained from indicating any specified spot in the definition of the
kilogramme, when they were perfectly acquainted with the considerable
variations in the intensity of gravity, leaves no doubt as to their
real desire.
The same objections have been made to the definition of the
kilogramme, at first considered as the mass of a cubic decimetre of
water at 4° C., as to the first definition of the metre. We must
admire the incredible precision attained at the outset by the
physicists who made the initial determinations, but we know at the
present day that the kilogramme they constructed is slightly too heavy
(by about 1/25,000). Very remarkable researches have been carried out
with regard to this determination by the International Bureau, and by
MM. Macé de Lépinay and Buisson. The law of the 11th July 1903 has
definitely regularized the custom which physicists had adopted some
years before; and the standard of mass, the legal prototype of the
metrical system, is now the international kilogramme sanctioned by the
Conference of Weights and Measures.
The comparison of a mass with the standard is effected with a
precision to which no other measurement can attain. Metrology vouches
for the hundredth of a milligramme in a kilogramme; that is to say,
that it estimates the hundred-millionth part of the magnitude studied.
We may--as in the case of the lengths--ask ourselves whether this
already admirable precision can be surpassed; and progress would seem
likely to be slow, for difficulties singularly increase when we get to
such small quantities. But it is permitted to hope that the physicists
of the future will do still better than those of to-day; and perhaps
we may catch a glimpse of the time when we shall begin to observe that
the standard, which is constructed from a heavy metal, namely,
iridium-platinum, itself obeys an apparently general law, and little
by little loses some particles of its mass by emanation.
§ 4. THE MEASURE OF TIME
The third fundamental magnitude of mechanics is time. There is, so to
speak, no physical phenomenon in which the notion of time linked to
the sequence of our states of consciousness does not play a
considerable part.
Ancestral habits and a very early tradition have led us to preserve,
as the unit of time, a unit connected with the earth's movement; and
the unit to-day adopted is, as we know, the sexagesimal second of mean
time. This magnitude, thus defined by the conditions of a natural
motion which may itself be modified, does not seem to offer all the
guarantees desirable from the point of view of invariability. It is
certain that all the friction exercised on the earth--by the tides,
for instance--must slowly lengthen the duration of the day, and must
influence the movement of the earth round the sun. Such influence is
certainly very slight, but it nevertheless gives an unfortunately
arbitrary character to the unit adopted.
We might have taken as the standard of time the duration of another
natural phenomenon, which appears to be always reproduced under
identical conditions; the duration, for instance, of a given luminous
vibration. But the experimental difficulties of evaluation with such a
unit of the times which ordinarily have to be considered, would be so
great that such a reform in practice cannot be hoped for. It should,
moreover, be remarked that the duration of a vibration may itself be
influenced by external circumstances, among which are the variations
of the magnetic field in which its source is placed. It could not,
therefore, be strictly considered as independent of the earth; and the
theoretical advantage which might be expected from this alteration
would be somewhat illusory.
Perhaps in the future recourse may be had to very different phenomena.
Thus Curie pointed out that if the air inside a glass tube has been
rendered radioactive by a solution of radium, the tube may be sealed
up, and it will then be noted that the radiation of its walls
diminishes with time, in accordance with an exponential law. The
constant of time derived by this phenomenon remains the same whatever
the nature and dimensions of the walls of the tube or the temperature
may be, and time might thus be denned independently of all the other
units.
We might also, as M. Lippmann has suggested in an extremely ingenious
way, decide to obtain measures of time which can be considered as
absolute because they are determined by parameters of another nature
than that of the magnitude to be measured. Such experiments are made
possible by the phenomena of gravitation. We could employ, for
instance, the pendulum by adopting, as the unit of force, the force
which renders the constant of gravitation equal to unity. The unit of
time thus defined would be independent of the unit of length, and
would depend only on the substance which would give us the unit of
mass under the unit of volume.
It would be equally possible to utilize electrical phenomena, and one
might devise experiments perfectly easy of execution. Thus, by
charging a condenser by means of a battery, and discharging it a given
number of times in a given interval of time, so that the effect of the
current of discharge should be the same as the effect of the output of
the battery through a given resistance, we could estimate, by the
measurement of the electrical magnitudes, the duration of the interval
noted. A system of this kind must not be looked upon as a simple _jeu
d'esprit_, since this very practicable experiment would easily permit
us to check, with a precision which could be carried very far, the
constancy of an interval of time.
From the practical point of view, chronometry has made in these last
few years very sensible progress. The errors in the movements of
chronometers are corrected in a much more systematic way than
formerly, and certain inventions have enabled important improvements
to be effected in the construction of these instruments. Thus the
curious properties which steel combined with nickel--so admirably
studied by M.Ch.Ed. Guillaume--exhibits in the matter of dilatation
are now utilized so as to almost completely annihilate the influence
of variations of temperature.
§ 5. THE MEASURE OF TEMPERATURE
From the three mechanical units we derive secondary units; as, for
instance, the unit of work or mechanical energy. The kinetic theory
takes temperature, as well as heat itself, to be a quantity of energy,
and thus seems to connect this notion with the magnitudes of
mechanics. But the legitimacy of this theory cannot be admitted, and
the calorific movement should also be a phenomenon so strictly
confined in space that our most delicate means of investigation would
not enable us to perceive it. It is better, then, to continue to
regard the unit of difference of temperature as a distinct unit, to be
added to the fundamental units.
To define the measure of a certain temperature, we take, in practice,
some arbitrary property of a body. The only necessary condition of
this property is, that it should constantly vary in the same direction
when the temperature rises, and that it should possess, at any
temperature, a well-marked value. We measure this value by melting ice
and by the vapour of boiling water under normal pressure, and the
successive hundredths of its variation, beginning with the melting
ice, defines the percentage. Thermodynamics, however, has made it
plain that we can set up a thermometric scale without relying upon any
determined property of a real body. Such a scale has an absolute value
independently of the properties of matter. Now it happens that if we
make use for the estimation of temperatures, of the phenomena of
dilatation under a constant pressure, or of the increase of pressure
in a constant volume of a gaseous body, we obtain a scale very near
the absolute, which almost coincides with it when the gas possesses
certain qualities which make it nearly what is called a perfect gas.
This most lucky coincidence has decided the choice of the convention
adopted by physicists. They define normal temperature by means of the
variations of pressure in a mass of hydrogen beginning with the
initial pressure of a metre of mercury at 0° C.
M.P. Chappuis, in some very precise experiments conducted with much
method, has proved that at ordinary temperatures the indications of
such a thermometer are so close to the degrees of the theoretical
scale that it is almost impossible to ascertain the value of the
divergences, or even the direction that they take. The divergence
becomes, however, manifest when we work with extreme temperatures. It
results from the useful researches of M. Daniel Berthelot that we must
subtract +0.18° from the indications of the hydrogen thermometer
towards the temperature -240° C, and add +0.05° to 1000° to equate
them with the thermodynamic scale. Of course, the difference would
also become still more noticeable on getting nearer to the absolute
zero; for as hydrogen gets more and more cooled, it gradually exhibits
in a lesser degree the characteristics of a perfect gas.
To study the lower regions which border on that kind of pole of cold
towards which are straining the efforts of the many physicists who
have of late years succeeded in getting a few degrees further forward,
we may turn to a gas still more difficult to liquefy than hydrogen.
Thus, thermometers have been made of helium; and from the temperature
of -260° C. downward the divergence of such a thermometer from one of
hydrogen is very marked.
The measurement of very high temperatures is not open to the same
theoretical objections as that of very low temperatures; but, from a
practical point of view, it is as difficult to effect with an ordinary
gas thermometer. It becomes impossible to guarantee the reservoir
remaining sufficiently impermeable, and all security disappears,
notwithstanding the use of recipients very superior to those of former
times, such as those lately devised by the physicists of the
_Reichansalt_. This difficulty is obviated by using other methods,
such as the employment of thermo-electric couples, such as the very
convenient couple of M. le Chatelier; but the graduation of these
instruments can only be effected at the cost of a rather bold
extrapolation.
M.D. Berthelot has pointed out and experimented with a very
interesting process, founded on the measurement by the phenomena of
interference of the refractive index of a column of air subjected to
the temperature it is desired to measure. It appears admissible that
even at the highest temperatures the variation of the power of
refraction is strictly proportional to that of the density, for this
proportion is exactly verified so long as it is possible to check it
precisely. We can thus, by a method which offers the great advantage
of being independent of the power and dimension of the envelopes
employed--since the length of the column of air considered alone
enters into the calculation--obtain results equivalent to those given
by the ordinary air thermometer.
Another method, very old in principle, has also lately acquired great
importance. For a long time we sought to estimate the temperature of a
body by studying its radiation, but we did not know any positive
relation between this radiation and the temperature, and we had no
good experimental method of estimation, but had recourse to purely
empirical formulas and the use of apparatus of little precision. Now,
however, many physicists, continuing the classic researches of
Kirchhoff, Boltzmann, Professors Wien and Planck, and taking their
starting-point from the laws of thermodynamics, have given formulas
which establish the radiating power of a dark body as a function of
the temperature and the wave-length, or, better still, of the total
power as a function of the temperature and wave-length corresponding
to the maximum value of the power of radiation. We see, therefore, the
possibility of appealing for the measurement of temperature to a
phenomenon which is no longer the variation of the elastic force of a
gas, and yet is also connected with the principles of thermodynamics.
This is what Professors Lummer and Pringsheim have shown in a series
of studies which may certainly be reckoned among the greatest
experimental researches of the last few years. They have constructed a
radiator closely resembling the theoretically integral radiator which
a closed isothermal vessel would be, and with only a very small
opening, which allows us to collect from outside the radiations which
are in equilibrium with the interior. This vessel is formed of a
hollow carbon cylinder, heated by a current of high intensity; the
radiations are studied by means of a bolometer, the disposition of
which varies with the nature of the experiments.
It is hardly possible to enter into the details of the method, but the
result sufficiently indicates its importance. It is now possible,
thanks to their researches, to estimate a temperature of 2000° C. to
within about 5°. Ten years ago a similar approximation could hardly
have been arrived at for a temperature of 1000° C.
§ 6. DERIVED UNITS AND THE MEASURE OF A QUANTITY OF ENERGY
It must be understood that it is only by arbitrary convention that a
dependency is established between a derived unit and the fundamental
units. The laws of numbers in physics are often only laws of
proportion. We transform them into laws of equation, because we
introduce numerical coefficients and choose the units on which they
depend so as to simplify as much as possible the formulas most in use.
A particular speed, for instance, is in reality nothing else but a
speed, and it is only by the peculiar choice of unit that we can say
that it is the space covered during the unit of time. In the same way,
a quantity of electricity is a quantity of electricity; and there is
nothing to prove that, in its essence, it is really reducible to a
function of mass, of length, and of time.
Persons are still to be met with who seem to have some illusions on
this point, and who see in the doctrine of the dimensions of the units
a doctrine of general physics, while it is, to say truth, only a
doctrine of metrology. The knowledge of dimensions is valuable, since
it allows us, for instance, to easily verify the homogeneity of a
formula, but it can in no way give us any information on the actual
nature of the quantity measured.
Magnitudes to which we attribute like dimensions may be qualitatively
irreducible one to the other. Thus the different forms of energy are
measured by the same unit, and yet it seems that some of them, such as
kinetic energy, really depend on time; while for others, such as
potential energy, the dependency established by the system of
measurement seems somewhat fictitious.
The numerical value of a quantity of energy of any nature should, in
the system C.G.S., be expressed in terms of the unit called the erg;
but, as a matter of fact, when we wish to compare and measure
different quantities of energy of varying forms, such as electrical,
chemical, and other quantities, etc., we nearly always employ a method
by which all these energies are finally transformed and used to heat
the water of a calorimeter. It is therefore very important to study
well the calorific phenomenon chosen as the unit of heat, and to
determine with precision its mechanical equivalent, that is to say,
the number of ergs necessary to produce this unit. This is a number
which, on the principle of equivalence, depends neither on the method
employed, nor the time, nor any other external circumstance.
As the result of the brilliant researches of Rowland and of Mr
Griffiths on the variations of the specific heat of water, physicists
have decided to take as calorific standard the quantity of heat
necessary to raise a gramme of water from 15° to 16° C., the
temperature being measured by the scale of the hydrogen thermometer of
the International Bureau.
On the other hand, new determinations of the mechanical equivalent,
among which it is right to mention that of Mr. Ames, and a full
discussion as to the best results, have led to the adoption of the
number 4.187 to represent the number of ergs capable of producing the
unit of heat.
In practice, the measurement of a quantity of heat is very often
effected by means of the ice calorimeter, the use of which is
particularly simple and convenient. There is, therefore, a very
special interest in knowing exactly the melting-point of ice. M.
Leduc, who for several years has measured a great number of physical
constants with minute precautions and a remarkable sense of precision,
concludes, after a close discussion of the various results obtained,
that this heat is equal to 79.1 calories. An error of almost a calorie
had been committed by several renowned experimenters, and it will be
seen that in certain points the art of measurement may still be
largely perfected.
To the unit of energy might be immediately attached other units. For
instance, radiation being nothing but a flux of energy, we could, in
order to establish photometric units, divide the normal spectrum into
bands of a given width, and measure the power of each for the unit of
radiating surface.
But, notwithstanding some recent researches on this question, we
cannot yet consider the distribution of energy in the spectrum as
perfectly known. If we adopt the excellent habit which exists in some
researches of expressing radiating energy in ergs, it is still
customary to bring the radiations to a standard giving, by its
constitution alone, the unit of one particular radiation. In
particular, the definitions are still adhered to which were adopted as
the result of the researches of M. Violle on the radiation of fused
platinum at the temperature of solidification; and most physicists
utilize in the ordinary methods of photometry the clearly defined
notions of M. Blondel as to the luminous intensity of flux,
illumination (_éclairement_), light (_éclat_), and lighting
(_éclairage_), with the corresponding units, decimal candle, _lumen_,
_lux_, carcel lamp, candle per square centimetre, and _lumen_-hour.[4]
[Footnote 4: These are the magnitudes and units adopted at the
International Congress of Electricians in 1904. For their definition
and explanation, see Demanet, _Notes de Physique Expérimentale_
(Louvain, 1905), t. iv. p. 8.--ED.]
§ 7. MEASURE OF CERTAIN PHYSICAL CONSTANTS
The progress of metrology has led, as a consequence, to corresponding
progress in nearly all physical measurements, and particularly in the
measure of natural constants. Among these, the constant of gravitation
occupies a position quite apart from the importance and simplicity of
the physical law which defines it, as well as by its generality. Two
material particles are mutually attracted to each other by a force
directly proportional to the product of their mass, and inversely
proportional to the square of the distance between them. The
coefficient of proportion is determined when once the units are
chosen, and as soon as we know the numerical values of this force, of
the two masses, and of their distance. But when we wish to make
laboratory experiments serious difficulties appear, owing to the
weakness of the attraction between masses of ordinary dimensions.
Microscopic forces, so to speak, have to be observed, and therefore
all the causes of errors have to be avoided which would be unimportant
in most other physical researches. It is known that Cavendish was the
first who succeeded by means of the torsion balance in effecting
fairly precise measurements. This method has been again taken in hand
by different experimenters, and the most recent results are due to Mr
Vernon Boys. This learned physicist is also the author of a most
useful practical invention, and has succeeded in making quartz threads
as fine as can be desired and extremely uniform. He finds that these
threads possess valuable properties, such as perfect elasticity and
great tenacity. He has been able, with threads not more than 1/500 of
a millimetre in diameter, to measure with precision couples of an
order formerly considered outside the range of experiment, and to
reduce the dimensions of the apparatus of Cavendish in the proportion
of 150 to 1. The great advantage found in the use of these small
instruments is the better avoidance of the perturbations arising from
draughts of air, and of the very serious influence of the slightest
inequality in temperature.
Other methods have been employed in late years by other experimenters,
such as the method of Baron Eötvös, founded on the use of a torsion
lever, the method of the ordinary balance, used especially by
Professors Richarz and Krigar-Menzel and also by Professor Poynting,
and the method of M. Wilsing, who uses a balance with a vertical beam.
The results fairly agree, and lead to attributing to the earth a
density equal to 5.527.
The most familiar manifestation of gravitation is gravity. The action
of the earth on the unit of mass placed in one point, and the
intensity of gravity, is measured, as we know, by the aid of a
pendulum. The methods of measurement, whether by absolute or by
relative determinations, so greatly improved by Borda and Bessel, have
been still further improved by various geodesians, among whom should
be mentioned M. von Sterneek and General Defforges. Numerous
observations have been made in all parts of the world by various
explorers, and have led to a fairly complete knowledge of the
distribution of gravity over the surface of the globe. Thus we have
succeeded in making evident anomalies which would not easily find
their place in the formula of Clairaut.
Another constant, the determination of which is of the greatest
utility in astronomy of position, and the value of which enters into
electromagnetic theory, has to-day assumed, with the new ideas on the
constitution of matter, a still more considerable importance. I refer
to the speed of light, which appears to us, as we shall see further
on, the maximum value of speed which can be given to a material body.
After the historical experiments of Fizeau and Foucault, taken up
afresh, as we know, partly by Cornu, and partly by Michelson and
Newcomb, it remained still possible to increase the precision of the
measurements. Professor Michelson has undertaken some new researches
by a method which is a combination of the principle of the toothed
wheel of Fizeau with the revolving mirror of Foucault. The toothed
wheel is here replaced, however, by a grating, in which the lines and
the spaces between them take the place of the teeth and the gaps, the
reflected light only being returned when it strikes on the space
between two lines. The illustrious American physicist estimates that
he can thus evaluate to nearly five kilometres the path traversed by
light in one second. This approximation corresponds to a relative
value of a few hundred-thousandths, and it far exceeds those hitherto
attained by the best experimenters. When all the experiments are
completed, they will perhaps solve certain questions still in
suspense; for instance, the question whether the speed of propagation
depends on intensity. If this turns out to be the case, we should be
brought to the important conclusion that the amplitude of the
oscillations, which is certainly very small in relation to the already
tiny wave-lengths, cannot be considered as unimportant in regard to
these lengths. Such would seem to have been the result of the curious
experiments of M. Muller and of M. Ebert, but these results have been
recently disputed by M. Doubt.
In the case of sound vibrations, on the other hand, it should be noted
that experiment, consistently with the theory, proves that the speed
increases with the amplitude, or, if you will, with the intensity. M.
Violle has published an important series of experiments on the speed
of propagation of very condensed waves, on the deformations of these
waves, and on the relations of the speed and the pressure, which
verify in a remarkable manner the results foreshadowed by the already
old calculations of Riemann, repeated later by Hugoniot. If, on the
contrary, the amplitude is sufficiently small, there exists a speed
limit which is the same in a large pipe and in free air. By some
beautiful experiments, MM. Violle and Vautier have clearly shown that
any disturbance in the air melts somewhat quickly into a single wave
of given form, which is propagated to a distance, while gradually
becoming weaker and showing a constant speed which differs little in
dry air at 0° C. from 331.36 metres per second. In a narrow pipe the
influence of the walls makes itself felt and produces various effects,
in particular a kind of dispersion in space of the harmonics of the
sound. This phenomenon, according to M. Brillouin, is perfectly
explicable by a theory similar to the theory of gratings.