(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.

1. 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 o° Cent. and 350°, was determined almost as accurately as was needful.1

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.2

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. The skill and ingenuity of the experimentalist

3

1 Jamin, Cours de Physique, vol. ii. pp. 15-28.

2 Philosophical Magazine, 1851, 4th Series, vol. ii. passim.

3 Hearn, Philosophical Transactions, 1847, vol. cxxxvii. pp. 217

-221.

are often exhausted in trying to devise a form of apparatus in which such causes of error shall be reduced to a minimum.

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.1

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.2 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, is beautifully employed in Bunsen's calorimeter.

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

The Correlation of Physical Forces, 3rd ed. p. 159.

2 Collected Works of Sir H. Davy, vol. ii. pp. 12—14. Elements of Chemical Philosophy, p. 94.

3 Nicholson's Journal, vol. i. p. 241; quoted in Treatise on Heat Useful Knowledge Society. p. 24.

adjusted to the temperature of the air, as described by him, so that the effect of radiation shall be nullified.1

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 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.

Clerk Maxwell, Theory of Heat, p. 228. Proceedings of the Manchester Philosophical Society, Nov. 26. 1867, vol. vii. p. 35.