CHAPTER XV.
ANALYSIS OF QUANTITATIVE PHENOMENA.
In the two preceding chapters we have been engaged in considering how a phenomenon may be accurately measured and expressed. So delicate and complex an operation is a measurement which pretends to any considerable degree of exactness, that no small part of the skill and patience of physicists is usually spent upon this work. Much of this difficulty arises from the fact that it is scarcely ever possible to measure a single effect at a time. The ultimate object must be to discover the mathematical equation or law connecting a quantitative cause with its quantitative effect; this purpose usually involves, as we shall see, the varying of one condition at a time, the other conditions being maintained constant. The labours of the experimentalist would be comparatively light if he could carry out this rule of varying one circumstance at a time. He would then obtain a series of corresponding values of the variable quantities concerned, from which he might by proper hypothetical treatment obtain the required law of connection. But in reality it is seldom possible to carry out this direction except in an approximate manner. Before then we proceed to the consideration of the actual process of quantitative induction, it is necessary to review the several devices by which a complicated series of effects can be disentangled. Every phenomenon measured will usually be the sum, difference, or it may be the product or quotient, of two or more different effects, and these must be in some way analysed and separately measured before we possess the materials for inductive treatment.
*Illustrations of the Complication of Effects.*
It is easy to bring forward a multitude of instances to show that a phenomenon is seldom to be observed simple and alone. A more or less elaborate process of analysis is almost always necessary. Thus if an experimentalist wishes to observe and measure the expansion of a liquid by heat, he places it in a thermometer tube and registers the rise of the column of liquid in the narrow tube. But he cannot heat the liquid without also heating the glass, so that the change observed is really the difference between the expansions of the liquid and the glass. More minute investigation will show the necessity perhaps of allowing for further minute effects, namely the compression of the liquid and the expansion of the bulb due to the increased pressure of the column as it becomes lengthened.
In a great many cases an observed effect will be apparently at least the simple sum of two separate and independent effects. The heat evolved in the combustion of oil is partly due to the carbon and partly to the hydrogen. A measurement of the heat yielded by the two jointly, cannot inform us how much proceeds from the one and how much from the other. If by some separate determination we can ascertain how much the hydrogen yields, then by mere subtraction we learn what is due to the carbon; and *vice versâ*. The heat conveyed by a liquid, may be partly conveyed by true conduction, partly by convection. The light dispersed in the interior of a liquid consists both of what is reflected by floating particles and what is due to true fluorescence;[233] and we must find some mode of determining one portion before we can learn the other. The apparent motion of the spots on the sun, is the algebraic sum of the sun’s axial rotation, and of the proper motion of the spots upon the sun’s surface; hence the difficulty of ascertaining by direct observations the period of the sun’s rotation.
[233] Stokes, *Philosophical Transactions* (1852), vol. cxlii. p. 529.
We cannot obtain the weight of a portion of liquid in a chemical balance without weighing it with the containing vessel. Hence to have the real weight of the liquid operated upon in an experiment, we must make a separate weighing of the vessel, with or without the adhering film of liquid according to circumstances. This is likewise the mode in which a cart and its load are weighed together, the *tare* of the cart previously ascertained being deducted. The variation in the height of the barometer is a joint effect, partly due to the real variation of the atmospheric pressure, partly to the expansion of the mercurial column by heat. The effects may be discriminated, if, instead of one barometer tube we have two tubes containing mercury placed closely side by side, so as to have the same temperature. If one of them be closed at the bottom so as to be unaffected by the atmospheric pressure, it will show the changes due to temperature only, and, by subtracting these changes from those shown in the other tube, employed as a barometer, we get the real oscillations of atmospheric pressure. But this correction, as it is called, of the barometric reading, is better effected by calculation from the readings of an ordinary thermometer.
In other cases a quantitative effect will be the difference of two causes acting in opposite directions. Sir John Herschel invented an instrument like a large thermometer, which he called the Actinometer,[234] and Pouillet constructed a somewhat similar instrument called the Pyrheliometer, for ascertaining the heating power of the sun’s rays. In both instruments the heat of the sun was absorbed by a reservoir containing water, and the rise of temperature of the water was exactly observed, either by its own expansion, or by the readings of a delicate thermometer immersed in it. But in exposing the actinometer to the sun, we do not obtain the full effect of the heat absorbed, because the receiving surface is at the same time radiating heat into empty space. The observed increment of temperature is in short the difference between what is received from the sun and lost by radiation. The latter quantity is capable of ready determination; we have only to shade the instrument from the direct rays of the sun, leaving it exposed to the sky, and we can observe how much it cools in a certain time. The total effect of the sun’s rays will obviously be the apparent effect *plus* the cooling effect in an equal time. By alternate exposure in sun and shade during equal intervals the desired result may be obtained with considerable accuracy.[235]
[234] *Admiralty Manual of Scientific Enquiry*, 2nd ed. p. 299.
[235] Pouillet, *Taylor’s Scientific Memoirs*, vol. iv. p. 45.
Two quantitative effects were beautifully distinguished in an experiment of John Canton, devised in 1761 for the purpose of demonstrating the compressibility of water. He constructed a thermometer with a large bulb full of water and a short capillary tube, the part of which above the water was freed from air. Under these circumstances the water was relieved from the pressure of the atmosphere, but the glass bulb in bearing that pressure was somewhat contracted. He next placed the instrument under the receiver of an air-pump, and on exhausting the air, the water sank in the tube. Having thus obtained a measure of the effect of atmospheric pressure on the bulb, he opened the top of the thermometer tube and admitted the air. The level of the water now sank still more, partly from the pressure on the bulb being now compensated, and partly from the compression of the water by the atmospheric pressure. It is obvious that the amount of the latter effect was approximately the difference of the two observed depressions.
Not uncommonly the actual phenomenon which we wish to measure is considerably less than various disturbing effects which enter into the question. Thus the compressibility of mercury is considerably less than the expansion of the vessels in which it is measured under pressure, so that the attention of the experimentalist has chiefly to be concentrated on the change of magnitude of the vessels. Many astronomical phenomena, such as the parallax or the proper motions of the fixed stars, are far less than the errors caused by instrumental imperfections, or motions arising from precession, nutation, and aberration. We need not be surprised that astronomers have from time to time mistaken one phenomenon for another, as when Flamsteed imagined that he had discovered the parallax of the Pole star.[236]
[236] Baily’s *Account of the Rev. John Flamsteed*, p. 58.
*Methods of Eliminating Error.*
In any particular experiment it is the object of the experimentalist to measure a single effect only, and he endeavours to obtain that effect free from interfering effects. If this cannot be, as it seldom or never can really be, he makes the effect as considerable as possible compared with the other effects, which he reduces to a minimum, and treats as noxious errors. Those quantities, which are called *errors* in one case, may really be most important and interesting phenomena in another investigation. When we speak of eliminating error we really mean disentangling the complicated phenomena of nature. The physicist rightly wishes to treat one thing at a time, but as this object can seldom be rigorously carried into practice, he has to seek some mode of counteracting the irrelevant and interfering causes.
The general principle is that a single observation can render known only a single quantity. Hence, if several different quantitative effects are known to enter into any investigation, we must have at least as many distinct observations as there are quantities to be determined. Every complete experiment will therefore consist in general of several operations. Guided if possible by previous knowledge of the causes in action, we must arrange the determinations, so that by a simple mathematical process we may distinguish the separate quantities. There appear to be five principal methods by which we may accomplish this object; these methods are specified below and illustrated in the succeeding sections.
(1) *The Method of Avoidance.* The physicist may seek for some special mode of experiment or opportunity of observation, in which the error is non-existent or inappreciable.
(2) *The Differential Method.* He may find opportunities of observation when all interfering phenomena remain constant, and only the subject of observation is at one time present and another time absent; the difference between two observations then gives its amount.
(3) *The Method of Correction.* He may endeavour to estimate the amount of the interfering effect by the best available mode, and then make a corresponding correction in the results of observation.
(4) *The Method of Compensation.* He may invent some mode of neutralising the interfering cause by balancing against it an exactly equal and opposite cause of unknown amount.
(5) *The Method of Reversal.* He may so conduct the experiment that the interfering cause may act in opposite directions, in alternate observations, the mean result being free from interference.
I. *Method of Avoidance of Error.*
Astronomers seek opportunities of observation when errors will be as small as possible. In spite of elaborate observations and long-continued theoretical investigation, it is not practicable to assign any satisfactory law to the refractive power of the atmosphere. Although the apparent change of place of a heavenly body produced by refraction may be more or less accurately calculated yet the error depends upon the temperature and pressure of the atmosphere, and, when a ray is highly inclined to the perpendicular, the uncertainty in the refraction becomes very considerable. Hence astronomers always make their observations, if possible, when the object is at the highest point of its daily course, *i.e.* on the meridian. In some kinds of investigation, as, for instance, in the determination of the latitude of an observatory, the astronomer is at liberty to select one or more stars out of the countless number visible. There is an evident advantage in such a case, in selecting a star which passes close to the zenith, so that it may be observed almost entirely free from atmospheric refraction, as was done by Hooke.
Astronomers endeavour to render their clocks as accurate as possible, by removing the source of variation. The pendulum is perfectly isochronous so long as its length remains invariable, and the vibrations are exactly of equal length. They render it nearly invariable in length, that is in the distance between the centres of suspension and oscillation, by a compensatory arrangement for the change of temperature. But as this compensation may not be perfectly accomplished, some astronomers place their chief controlling clock in a cellar, or other apartment, where the changes of temperature may be as slight as possible. At the Paris Observatory a clock has been placed in the caves beneath the building, where there is no appreciable difference between the summer and winter temperature.
To avoid the effect of unequal oscillations Huyghens made his beautiful investigations, which resulted in the discovery that a pendulum, of which the centre of oscillation moved upon a cycloidal path, would be perfectly isochronous, whatever the variation in the length of oscillations. But though a pendulum may be easily rendered in some degree cycloidal by the use of a steel suspension spring, it is found that the mechanical arrangements requisite to produce a truly cycloidal motion introduce more error than they remove. Hence astronomers seek to reduce the error to the smallest amount by maintaining their clock pendulums in uniform movement; in fact, while a clock is in good order and has the same weights, there need be little change in the length of oscillation. When a pendulum cannot be made to swing uniformly, as in experiments upon the force of gravity, it becomes requisite to resort to the third method, and a correction is introduced, calculated on theoretical grounds from the amount of the observed change in the length of vibration.
It has been mentioned that the apparent expansion of a liquid by heat, when contained in a thermometer tube or other vessel, is the difference between the real expansion of the liquid and that of the containing vessel. The effects can be accurately distinguished provided that we can learn the real expansion by heat of any one convenient liquid; for by observing the apparent expansion of the same liquid in any required vessel we can by difference learn the amount of expansion of the vessel due to any given change of temperature. When we once know the change of dimensions of the vessel, we can of course determine the absolute expansion of any other liquid tested in it. Thus it became an all-important object in scientific research to measure with accuracy the absolute dilatation by heat of some one liquid, and mercury owing to several circumstances was by far the most suitable. Dulong and Petit devised a beautiful mode of effecting this by simply avoiding altogether the effect of the change of size of the vessel. Two upright tubes full of mercury were connected by a fine tube at the bottom, and were maintained at two different temperatures. As mercury was free to flow from one tube to the other by the connecting tube, the two columns necessarily exerted equal pressures by the principles of hydrostatics. Hence it was only necessary to measure very accurately by a cathetometer the difference of level of the surfaces of the two columns of mercury, to learn the difference of length of columns of equal hydrostatic pressure, which at once gives the difference of density of the mercury, and the dilatation by heat. The changes of dimension in the containing tubes became a matter of entire indifference, and the length of a column of mercury at different temperatures was measured as easily as if it had formed a solid bar. The experiment was carried out by Regnault with many improvements of detail, and the absolute dilatation of mercury, at temperatures between 0° Cent. and 350°, was determined almost as accurately as was needful.[237]
[237] Jamin, *Cours de Physique*, vol. ii. pp. 15–28.
The presence of a large and uncertain amount of error may render a method of experiment valueless. Foucault devised a beautiful experiment with the pendulum for demonstrating popularly the rotation of the earth, but it could be of no use for measuring the rotation exactly. It is impossible to make the pendulum swing in a perfect plane, and the slightest lateral motion gives it an elliptic path with a progressive motion of the axis of the ellipse, which disguises and often entirely overpowers that due to the rotation of the earth.[238]
[238] *Philosophical Magazine*, 1851, 4th Series, vol. ii. *passim*.
Faraday’s laborious experiments on the relation of gravity and electricity were much obstructed by the fact that it is impossible to move a large weight of metal without generating currents of electricity, either by friction or induction. To distinguish the electricity, if any, directly due to the action of gravity from the greater quantities indirectly produced was a problem of excessive difficulty. Baily in his experiments on the density of the earth was aware of the existence of inexplicable disturbances which have since been referred with much probability to the action of electricity.[239] The skill and ingenuity of the experimentalist are often exhausted in trying to devise a form of apparatus in which such causes of error shall be reduced to a minimum.
[239] Hearn, *Philosophical Transactions*, 1847, vol. cxxxvii. pp. 217–221.
In some rudimentary experiments we wish merely to establish the existence of a quantitative effect without precisely measuring its amount; if there exist causes of error of which we can neither render the amount known or inappreciable, the best way is to make them all negative so that the quantitative effects will be less than the truth rather than greater. Grove, for instance, in proving that the magnetisation or demagnetisation of a piece of iron raises its temperature, took care to maintain the electro-magnet by which the iron was magnetised at a lower temperature than the iron, so that it would cool rather than warm the iron by radiation or conduction.[240]
[240] *The Correlation of Physical Forces*, 3rd ed. p. 159.
Rumford’s celebrated experiment to prove that heat was generated out of mechanical force in the boring of a cannon was subject to the difficulty that heat might be brought to the cannon by conduction from neighbouring bodies. It was an ingenious device of Davy to produce friction by a piece of clock-work resting upon a block of ice in an exhausted receiver; as the machine rose in temperature above 32°, it was certain that no heat was received by conduction from the support.[241] In many other experiments ice may be employed to prevent the access of heat by conduction, and this device, first put in practice by Murray,[242] is beautifully employed in Bunsen’s calorimeter.
[241] *Collected Works of Sir H. Davy*, vol. ii. pp. 12–14. *Elements of Chemical Philosophy*, p. 94.
[242] *Nicholson’s Journal*, vol. i. p. 241; quoted in *Treatise on Heat*, Useful Knowledge Society, p. 24.
To observe the true temperature of the air, though apparently so easy, is really a very difficult matter, because the thermometer is sure to be affected either by the sun’s rays, the radiation from neighbouring objects, or the escape of heat into space. These sources of error are too fluctuating to allow of correction, so that the only accurate mode of procedure is that devised by Dr. Joule, of surrounding the thermometer with a copper cylinder ingeniously adjusted to the temperature of the air, as described by him, so that the effect of radiation shall be nullified.[243]
[243] Clerk Maxwell, *Theory of Heat*, p. 228. *Proceedings of the Manchester Philosophical Society*, Nov. 26, 1867, vol. vii. p. 35.
When the avoidance of error is not practicable, it will yet be desirable to reduce the absolute amount of the interfering error as much as possible before employing the succeeding methods to correct the result. As a general rule we can determine a quantity with less inaccuracy as it is smaller, so that if the error itself be small the error in determining that error will be of a still lower order of magnitude. But in some cases the absolute amount of an error is of no consequence, as in the index error of a divided circle, or the difference between a chronometer and astronomical time. Even the rate at which a clock gains or loses is a matter of little importance provided it remain constant, so that a sure calculation of its amount can be made.
2. *Differential Method.*
When we cannot avoid the existence of error, we can often resort with success to the second mode by measuring phenomena under such circumstances that the error shall remain very nearly the same in all the observations, and neutralise itself as regards the purposes in view. This mode is available whenever we want a difference between quantities and not the absolute quantity of either. The determination of the parallax of the fixed stars is exceedingly difficult, because the amount of parallax is far less than most of the corrections for atmospheric refraction, nutation, aberration, precession, instrumental irregularities, &c., and can with difficulty be detected among these phenomena of various magnitude. But, as Galileo long ago suggested, all such difficulties would be avoided by the differential observation of stars, which, though apparently close together, are really far separated on the line of sight. Two such stars in close apparent proximity will be subject to almost exactly equal errors, so that all we need do is to observe the apparent change of place of the nearer star as referred to the more distant one. A good telescope furnished with an accurate micrometer is alone needed for the application of the method. Huyghens appears to have been the first observer who actually tried to employ the method practically, but it was not until 1835 that the improvement of telescopes and micrometers enabled Struve to detect in this way the parallax of the star α Lyræ. It is one of the many advantages of the observation of transits of Venus for the determination of the solar parallax that the refraction of the atmosphere affects in an exactly equal degree the planet and the portion of the sun’s face over which it is passing. Thus the observations are strictly of a differential nature.
By the process of substitutive weighing it is possible to ascertain the equality or inequality of two weights with almost perfect freedom from error. If two weights A and B be placed in the scales of the best balance we cannot be sure that the equilibrium of the beam indicates exact equality, because the arms of the beam may be unequal or unbalanced. But if we take B out and put another weight C in, and equilibrium still exists, it is apparent that the same causes of erroneous weighing exist in both cases, supposing that the balance has not been disarranged; B then must be exactly equal to C, since it has exactly the same effect under the same circumstances. In like manner it is a general rule that, if by any uniform mechanical process we get a copy of an object, it is unlikely that this copy will be precisely the same as the original in magnitude and form, but two copies will equally diverge from the original, and will therefore almost exactly resemble each other.
Leslie’s Differential Thermometer[244] was well adapted to the experiments for which it was invented. Having two equal bulbs any alteration in the temperature of the air will act equally by conduction on each and produce no change in the indications of the instrument. Only that radiant heat which is purposely thrown upon one of the bulbs will produce any effect. This thermometer in short carries out the principle of the differential method in a mechanical manner.
[244] Leslie, *Inquiry into the Nature of Heat*, p. 10.
3. *Method of Correction.*
Whenever the result of an experiment is affected by an interfering cause to a calculable amount, it is sufficient to add or subtract this amount. We are said to correct observations when we thus eliminate what is due to extraneous causes, although of course we are only separating the correct effects of several agents. The variation in the height of the barometer is partly due to the change of temperature, but since the coefficient of absolute dilatation of mercury has been exactly determined, as already described (p. 341), we have only to make calculations of a simple character, or, what is better still, tabulate a series of such calculations for general use, and the correction for temperature can be made with all desired accuracy. The height of the mercury in the barometer is also affected by capillary attraction, which depresses it by a constant amount depending mainly on the diameter of the tube. The requisite corrections can be estimated with accuracy sufficient for most purposes, more especially as we can check the correctness of the reading of a barometer by comparison with a standard barometer, and introduce if need be an index error including both the error in the affixing of the scale and the effect due to capillarity. But in constructing the standard barometer itself we must take greater precautions; the capillary depression depends somewhat upon the quality of the glass, the absence of air, and the perfect cleanliness of the mercury, so that we cannot assign the exact amount of the effect. Hence a standard barometer is constructed with a wide tube, sometimes even an inch in diameter, so that the capillary effect may be rendered almost zero.[245] Gay-Lussac made barometers in the form of a uniform siphon tube, so that the capillary forces acting at the upper and lower surfaces should balance and destroy each other; but the method fails in practice because the lower surface, being open to the air, becomes sullied and subject to a different force of capillarity.
[245] Jevons, Watts’ *Dictionary of Chemistry*, vol. i. pp. 513–515.
In mechanical experiments friction is an interfering condition, and drains away a portion of the energy intended to be operated upon in a definite manner. We should of course reduce the friction in the first place to the lowest possible amount, but as it cannot be altogether prevented, and is not calculable with certainty from any general laws, we must determine it separately for each apparatus by suitable experiments. Thus Smeaton, in his admirable but almost forgotten researches concerning water-wheels, eliminated friction in the most simple manner by determining by trial what weight, acting by a cord and roller upon his model water-wheel, would make it turn without water as rapidly as the water made it turn. In short, he ascertained what weight concurring with the water would exactly compensate for the friction.[246] In Dr. Joule’s experiments to determine the mechanical equivalent of heat by the condensation of air, a considerable amount of heat was produced by friction of the condensing pump, and a small portion by stirring the water employed to absorb the heat. This heat of friction was measured by simply repeating the experiment in an exactly similar manner except that no condensation was effected, and observing the change of temperature then produced.[247]
[246] *Philosophical Transactions*, vol. li. p. 100.
[247] *Philosophical Magazine*, 3rd Series, vol. xxvi. p. 372.
We may describe as *test experiments* any in which we perform operations not intended to give the quantity of the principal phenomenon, but some quantity which would otherwise remain as an error in the result. Thus in astronomical observations almost every instrumental error may be avoided by increasing the number of observations and distributing them in such a manner as to produce in the final mean as much error in one way as in the other. But there is one source of error, first discovered by Maskelyne, which cannot be thus avoided, because it affects all observations in the same direction and to the same average amount, namely the Personal Error of the observer or the inclination to record the passage of a star across the wires of the telescope a little too soon or a little too late. This personal error was first carefully described in the *Edinburgh Journal of Science*, vol. i. p. 178. The difference between the judgment of observers at the Greenwich Observatory usually varies from 1/100 to 1/3 of a second, and remains pretty constant for the same observers.[248] One practised observer in Sir George Airy’s pendulum experiments recorded all his time observations half a second too early on the average as compared with the chief observer.[249] In some observers it has amounted to seven or eight-tenths of a second.[250] De Morgan appears to have entertained the opinion that this source of error was essentially incapable of elimination or correction.[251] But it seems clear, as I suggested without knowing what had been done,[252] that this personal error might be determined absolutely with any desirable degree of accuracy by test experiments, consisting in making an artificial star move at a considerable distance and recording by electricity the exact moment of its passage over the wire. This method has in fact been successfully employed in Leyden, Paris, and Neuchatel.[253] More recently, observers were trained for the Transit of Venus Expeditions by means of a mechanical model representing the motion of Venus over the sun, this model being placed at a little distance and viewed through a telescope, so that differences in the judgments of different observers would become apparent. It seems likely that tests of this nature might be employed with advantage in other cases.
[248] *Greenwich Observations for* 1866, p. xlix.
[249] *Philosophical Transactions*, 1856, p. 309.
[250] Penny *Cyclopædia*, art. *Transit*, vol. xxv. pp. 129, 130.
[251] Ibid. art. *Observation*, p. 390.
[252] *Nature*, vol. i. p. 85.
[253] *Nature*, vol. i. p 337. See references to the Memoirs describing the method.
Newton employed the pendulum for making experiments on the impact of balls. Two balls were hung in contact, and one of them, being drawn aside through a measured arc, was then allowed to strike the other, the arcs of vibration giving sufficient data for calculating the distribution of energy at the moment of impact. The resistance of the air was an interfering cause which he estimated very simply by causing one of the balls to make several complete vibrations without impact and then marking the reduction in the lengths of the arcs, a proper fraction of which reduction was added to each of the other arcs of vibration when impact took place.[254]
[254] *Principia*, Book I. Law III. Corollary VI. Scholium. Motte’s translation, vol. i. p. 33.
The exact definition of the standard of length is one of the most important, as it is one of the most difficult questions in physical science, and the different practice of different nations introduces needless confusion. Were all standards constructed so as to give the true length at a fixed uniform temperature, for instance the freezing-point, then any two standards could be compared without the interference of temperature by bringing them both to exactly the same fixed temperature. Unfortunately the French metre was defined by a bar of platinum at 0°C, while our yard was defined by a bronze bar at 62°F. It is quite impossible, then, to make a comparison of the yard and metre without the introduction of a correction, either for the expansion of platinum or bronze, or both. Bars of metal differ too so much in their rates of expansion according to their molecular condition that it is dangerous to infer from one bar to another.
When we come to use instruments with great accuracy there are many minute sources of error which must be guarded against. If a thermometer has been graduated when perpendicular, it will read somewhat differently when laid flat, as the pressure of a column of mercury is removed from the bulb. The reading may also be somewhat altered if it has recently been raised to a higher temperature than usual, if it be placed under a vacuous receiver, or if the tube be unequally heated as compared with the bulb. For these minute causes of error we may have to introduce troublesome corrections, unless we adopt the simple precaution of using the thermometer in circumstances of position, &c., exactly similar to those in which it was graduated. There is no end to the number of minute corrections which may ultimately be required. A large number of experiments on gases, standard weights and measures, &c., depend upon the height of the barometer; but when experiments in different parts of the world are compared together we ought as a further refinement to take into account the varying force of gravity, which even between London and Paris makes a difference of ·008 inch of mercury.
The measurement of quantities of heat is a matter of great difficulty, because there is no known substance impervious to heat, and the problem is therefore as difficult as to measure liquids in porous vessels. To determine the latent heat of steam we must condense a certain amount of the steam in a known weight of water, and then observe the rise of temperature of the water. But while we are carrying out the experiment, part of the heat will escape by radiation and conduction from the condensing vessel or calorimeter. We may indeed reduce the loss of heat by using vessels with double sides and bright surfaces, surrounded with swans-down wool or other non-conducting materials; and we may also avoid raising the temperature of the water much above that of the surrounding air. Yet we cannot by any such means render the loss of heat inconsiderable. Rumford ingeniously proposed to reduce the loss to zero by commencing the experiment when the temperature of the calorimeter is as much below that of the air as it is at the end of the experiment above it. Thus the vessel will first gain and then lose by radiation and conduction, and these opposite errors will approximately balance each other. But Regnault has shown that the loss and gain do not proceed by exactly the same laws, so that in very accurate investigations Rumford’s method is not sufficient. There remains the method of correction which was beautifully carried out by Regnault in his determination of the latent heat of steam. He employed two calorimeters, made in exactly the same way and alternately used to condense a certain amount of steam, so that while one was measuring the latent heat, the other calorimeter was engaged in determining the corrections to be applied, whether on account of radiation and conduction from the vessel or on account of heat reaching the vessel by means of the connecting pipes.[255]
[255] Graham’s *Chemical Reports and Memoirs*, Cavendish Society, pp. 247, 268, &c.
4. *Method of Compensation.*
There are many cases in which a cause of error cannot conveniently be rendered null, and is yet beyond the reach of the third method, that of calculating the requisite correction from independent observations. The magnitude of an error may be subject to continual variations, on account of change of weather, or other fickle circumstances beyond our control. It may either be impracticable to observe the variation of those circumstances in sufficient detail, or, if observed, the calculation of the amount of error may be subject to doubt. In these cases, and only in these cases, it will be desirable to invent some artificial mode of counterpoising the variable error against an equal error subject to exactly the same variation.
We cannot weigh an object with great accuracy unless we make a correction for the weight of the air displaced by the object, and add this to the apparent weight. In very accurate investigations relating to standard weights, it is usual to note the barometer and thermometer at the time of making a weighing, and, from the measured bulks of the objects compared, to calculate the weight of air displaced; the third method in fact is adopted. To make these calculations in the frequent weighings requisite in chemical analysis would be exceedingly laborious, hence the correction is usually neglected. But when the chemist wishes to weigh gas contained in a large glass globe for the purpose of determining its specific gravity, the correction becomes of much importance. Hence chemists avoid at once the error, and the labour of correcting it, by attaching to the opposite scale of the balance a dummy sealed glass globe of equal capacity to that containing the gas to be weighed, noting only the difference of weight when the operating globe is full and empty. The correction, being the same for both globes, may be entirely neglected.[256]
[256] Regnault’s *Cours Elémentaire de Chimie*, 1851, vol i. p. 141.
A device of nearly the same kind is employed in the construction of galvanometers which measure the force of an electric current by the deflection of a suspended magnetic needle. The resistance of the needle is partly due to the directive influence of the earth’s magnetism, and partly to the torsion of the thread. But the former force may often be inconveniently great as well as troublesome to determine for different inclinations. Hence it is customary to connect together two equally magnetised needles, with their poles pointing in opposite directions, one needle being within and another without the coil of wire. As regards the earth’s magnetism, the needles are now *astatic* or indifferent, the tendency of one needle towards the pole being balanced by that of the other.
An elegant instance of the elimination of a disturbing force by compensation is found in Faraday’s researches upon the magnetism of gases. To observe the magnetic attraction or repulsion of a gas seems impossible unless we enclose the gas in an envelope, probably best made of glass. But any such envelope is sure to be more or less affected by the magnet, so that it becomes difficult to distinguish between three forces which enter into the problem, namely, the magnetism of the gas in question, that of the envelope, and that of the surrounding atmospheric air. Faraday avoided all difficulties by employing two equal and similar glass tubes connected together, and so suspended from the arm of a torsion balance that the tubes were in similar parts of the magnetic field. One tube being filled with nitrogen and the other with oxygen, it was found that the oxygen seemed to be attracted and the nitrogen repelled. The suspending thread of the balance was then turned until the force of torsion restored the tubes to their original places, where the magnetism of the tubes as well as that of the surrounding air, being the same and in the opposite directions upon the two tubes, could not produce any interference. The force required to restore the tubes was measured by the amount of torsion of the thread, and it indicated correctly the difference between the attractive powers of oxygen and nitrogen. The oxygen was then withdrawn from one of the tubes, and a second experiment made, so as to compare a vacuum with nitrogen. No force was now required to maintain the tubes in their places, so that nitrogen was found to be, approximately speaking, indifferent to the magnet, that is, neither magnetic nor diamagnetic, while oxygen was proved to be positively magnetic.[257] It required the highest experimental skill on the part of Faraday and Tyndall, to distinguish between what is apparent and real in magnetic attraction and repulsion.
[257] Tyndall’s *Faraday*, pp. 114, 115.
Experience alone can finally decide when a compensating arrangement is conducive to accuracy. As a general rule mechanical compensation is the last resource, and in the more accurate observations it is likely to introduce more uncertainty than it removes. A multitude of instruments involving mechanical compensation have been devised, but they are usually of an unscientific character,[258] because the errors compensated can be more accurately determined and allowed for. But there are exceptions to this rule, and it seems to be proved that in the delicate and tiresome operation of measuring a base line, invariable bars, compensated for expansion by heat, give the most accurate results. This arises from the fact that it is very difficult to determine accurately the temperature of the measuring bars under varying conditions of weather and manipulation.[259] Again, the last refinement in the measurement of time at Greenwich Observatory depends upon mechanical compensation. Sir George Airy, observing that the standard clock increased its losing rate 0·30 second for an increase of one inch in atmospheric pressure, placed a magnet moved by a barometer in such a position below the pendulum, as almost entirely to neutralise this cause of irregularity. The thorough remedy, however, would be to remove the cause of error altogether by placing the clock in a vacuous case.
[258] See, for instance, the Compensated Sympiesometer, *Philosophical Magazine*, 4th Series, vol. xxxix. p. 371.
[259] Grant, *History of Physical Astronomy*, pp. 146, 147.
We thus see that the choice of one or other mode of eliminating an error depends entirely upon circumstances and the object in view; but we may safely lay down the following conclusions. First of all, seek to avoid the source of error altogether if it can be conveniently done; if not, make the experiment so that the error may be as small, but more especially as constant, as possible. If the means are at hand for determining its amount by calculation from other experiments and principles of science, allow the error to exist and make a correction in the result. If this cannot be accurately done or involves too much labour for the purposes in view, then throw in a counteracting error which shall as nearly as possible be of equal amount in all circumstances with that to be eliminated. There yet remains, however, one important method, that of Reversal, which will form an appropriate transition to the succeeding chapters on the Method of Mean Results and the Law of Error.
5. *Method of Reversal.*
The fifth method of eliminating error is most potent and satisfactory when it can be applied, but it requires that we shall be able to reverse the apparatus and mode of procedure, so as to make the interfering cause act alternately in opposite directions. If we can get two experimental results, one of which is as much too great as the other is too small, the error is equal to half the difference, and the true result is the mean of the two apparent results. It is an unavoidable defect of the chemical balance, for instance, that the points of suspension of the pans cannot be fixed at exactly equal distances from the centre of suspension of the beam. Hence two weights which seem to balance each other will never be quite equal in reality. The difference is detected by reversing the weights, and it may be estimated by adding small weights to the deficient side to restore equilibrium, and then taking as the true weight the geometric mean of the two apparent weights of the same object. If the difference is small, the arithmetic mean, that is half the sum, may be substituted for the geometric mean, from which it will not appreciably differ.
This method of reversal is most extensively employed in practical astronomy. The apparent elevation of a heavenly body is observed by a telescope moving upon a divided circle, upon which the inclination of the telescope is read off. Now this reading will be erroneous if the circle and the telescope have not accurately the same centre. But if we read off at the same time both ends of the telescope, the one reading will be about as much too small as the other is too great, and the mean will be nearly free from error. In practice the observation is differently conducted, but the principle is the same; the telescope is fixed to the circle, which moves with it, and the angle through which it moves is read off at three, six, or more points, disposed at equal intervals round the circle. The older astronomers, down even to the time of Flamsteed, were accustomed to use portions only of a divided circle, generally quadrants, and Römer made a vast improvement when he introduced the complete circle.
The transit circle, employed to determine the meridian passage of heavenly bodies, is so constructed that the telescope and the axis bearing it, in fact the whole moving part of the instrument, can be taken out of the bearing sockets and turned over, so that what was formerly the western pivot becomes the eastern one, and *vice versâ*. It is impossible that the instrument could have been so perfectly constructed, mounted, and adjusted that the telescope should point exactly to the meridian, but the effect of the reversal is that it will point as much to the west in one position as it does to the east in the other, and the mean result of observations in the two positions must be free from such cause of error.
The accuracy with which the inclination of the compass needle can be determined depends almost entirely on the method of reversal. The dip needle consists of a bar of magnetised steel, suspended somewhat like the beam of a delicate balance on a slender axis passing through the centre of gravity of the bar, so that it is at liberty to rest in that exact degree of inclination in the magnetic meridian which the magnetism of the earth induces. The inclination is read off upon a vertical divided circle, but to avoid error arising from the centring of the needle and circle, both ends are read, and the mean of the results is taken. The whole instrument is now turned carefully round through 180°, which causes the needle to assume a new position relatively to the circle and gives two new readings, in which any error due to the wrong position of the zero of the division will be reversed. As the axis of the needle may not be exactly horizontal, it is now reversed in the same manner as the transit instrument, the end of the axis which formerly pointed east being made to point west, and a new set of four readings is taken.
Finally, error may arise from the axis not passing accurately through the centre of gravity of the bar, and this error can only be detected and eliminated on changing the magnetic poles of the bar by the application of a strong magnet. The error is thus made to act in opposite directions. To ensure all possible accuracy each reversal ought to be combined with each other reversal, so that the needle will be observed in eight different positions by sixteen readings, the mean of the whole of which will give the required inclination free from all eliminable errors.[260]
[260] Quetelet, *Sur la Physique du Globe*, p. 174. Jamin, *Cours de Physique*, vol. i. p. 504.
There are certain cases in which a disturbing cause can with ease be made to act in opposite directions, in alternate observations, so that the mean of the results will be free from disturbance. Thus in direct experiments upon the velocity of sound in passing through the air between stations two or three miles apart, the wind is a cause of error. It will be well, in the first place, to choose a time for the experiment when the air is very nearly at rest, and the disturbance slight, but if at the same moment signal sounds be made at each station and observed at the other, two sounds will be passing in opposite directions through the same body of air and the wind will accelerate one sound almost exactly as it retards the other. Again, in trigonometrical surveys the apparent height of a point will be affected by atmospheric refraction and the curvature of the earth. But if in the case of two points the apparent elevation of each as seen from the other be observed, the corrections will be the same in amount, but reversed in direction, and the mean between the two apparent differences of altitude will give the true difference of level.
In the next two chapters we really pursue the Method of Reversal into more complicated applications.