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 tendency to error.

The general principle of the subject 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 results of observation 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 these determinations, so that by a simple mathematical process we may distinguish the separate quantities. There appear to be five principal methods in 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 exact observations then gives its amount.

(3) The Method of Correction. He may endeavour to estimate the amount of the interfering force 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 neutralizing the interfering force by balancing against it an exactly equal and opposite force of unknown

amount.

(5) The Method of Reversal. He may so conduct the experiment that the interfering force may act in opposite directions, in alternate observations, the mean result being free from interference.

1. Method of Avoidance of Error.

Astronomers always seek opportunities of observation when errors will have the smallest effect. In spite of elaborate observations and long continued theoretical investigation, it is not found possible to assign any satisfactory law to the refractive power of the atmosphere. Although the apparent change of place of a heavenly body thus produced, 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. It was ingeniously suggested by Wallis that the parallax of the fixed stars might perhaps be detected by observations of the greatest azimuth east and west of some

circumpolar star, since the refractive power of the atmosphere which affects only the altitude would thus be entirely avoided f.

Astronomers also 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 clocks 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 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 avoid. Hence astronomers seek to reduce the error to the smallest amount by maintaining their clock pendulums in uniform movements; and in fact while a clock is in good order and has the same weights, there need be little change in the length of oscillation.

f Grant, History of Physical Astronomy,' p. 548.

Montucla, Histoire des Mathématiques,' vol. ii. p. 420.

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 now 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 h.

The presence of a large and uncertain amount of error may often render a method of experiment valueless. Foucault's beautiful mode of demonstrating the rotation of the earth by the motion of a pendulum was thus frustrated. The slightest lateral disturbance of the pendulum gave it an elliptical path with a progressive motion of the axis of the ellipse, and this motion of an unknown amount disguised and overpowered that due to the rotation of the earthi. Faraday's laborious experiments on the relation of gravity and electricity were much obstructed, too, by the fact that it is almost impossible to move a large weight of iron or even lead without generating currents of electricity, either by friction or induction. To distinguish the electricity directly due to the action of gravity from the greater quantities indirectly produced would have been 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 to the action of electricity with much probability k. The skill and ingenuity of the experimentalist are often exhausted in devising a form of apparatus in which such causes of error shall be reduced to a minimum.

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

i 'Philosophical Magazine,' 1851, 4th Serics, vol. ii. passim.

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* Hearn, Philosophical Transactions,' 1847, vol. cxxxvii. pp. 217-221.