Chapter 2

# Measurements and Metrology

by Lucien Poincaré

## 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.

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.

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]

§ 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.

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