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]
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.
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.
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]
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.