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 versa. 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.

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.

Quetelet, Sur la Physique du Globe, p. 174. Jamin, Cours de Physique, vol. i. p. 504.

CHAPTER XVI.

THE METHOD OF MEANS.

ALL results of the measurement of continuous quantity can be only approximately true. Were this assertion doubted, it could readily be proved by direct experience. If any person, using an instrument of the greatest precision, makes and registers successive observations in an unbiassed manner, it will almost invariably be found that the results differ from each other. When we operate with sufficient care we cannot perform so simple an experiment as weighing an object in a good balance. without getting discrepant numbers. Only the rough and careless experimenter will think that his observations agree, but in reality he will be found to overlook the differences. The most elaborate researches, such as those undertaken in connection with standard weights and measures, always render it apparent that complete coincidence is out of the question, and that the more accurate our modes of observation are rendered, the more numerous are the sources of minute error which become apparent. We may look upon the existence of error in all measurements as the normal state of things. It is absolutely impossible to eliminate separately the multitude of small disturbing influences, except by balancing them off against each other. Even in drawing a mean it is to be expected that we shall come near the truth rather than exactly to it. In the measurement of continuous quantity, absolute coincidence, if it seems to occur, must be only apparent, and is no indication of precision. It is one of the most embarrassing things we can meet when experimental

results agree too closely. Such coincidences should raise our suspicion that the apparatus in use is in some way restricted in its operation, so as not really to give the true result at all, or that the actual results have not been faithfully recorded by the assistant in charge of the apparatus.

If then we cannot get twice over exactly the same result, the question arises, How can we ever attain the truth or select the result which may be supposed to approach most nearly to it? The quantity of a certain phenomenon is expressed in several numbers which differ from each other; no more than one of them at the most can be true, and it is more probable that they are all false. It may be suggested, perhaps, that the observer should select the one observation which he judged to be the best made, and there will often doubtless be a feeling that one or more results were satisfactory, and the others less trustworthy. This seems to have been the course adopted by the early astronomers. Flamsteed, when he had made several observations of a star, probably chose in an arbitrary manner that which seemed to him nearest to the truth.1

When Horrocks selected for his estimate of the sun's semi-diameter a mean between the results of Kepler and Tycho, he professed not to do it from any regard to the idle adage, "Medio tutissimus ibis," but because he thought it from his own observations to be correct. But this method will not apply at all when the observer has made a number of measurements which are equally good in his opinion, and it is quite apparent that in using an instrument or apparatus of considerable complication the observer will not necessarily be able to judge whether slight causes have affected its operation or not.

In this question, as indeed throughout inductive logic, we deal only with probabilities. There is no infallible mode of arriving at the absolute truth, which lies beyond the reach of human intellect, and can only be the distant object of our long-continued and painful approximations. Nevertheless there is mode pointed out alike by common sense and the highest mathematical reasoning, which is

1 Baily's Account of Flamsteed, p. 376.

The Transit of Venus across the Sun, by Horrocks, London, 1859,

P. 146.

more likely than any other, as a general rule, to bring us near the truth. The apioτov μéтpov, or the aurea mediocritas, was highly esteemed in the ancient philosophy of Greece. and Rome; but it is not probable that any of the ancients should have been able clearly to analyse and express the reasons why they advocated the mean as the safest course. But in the last two centuries this apparently simple question of the mean has been found to afford a field for the exercise of the utmost mathematical skill. Roger Cotes, the editor of the Principia, appears to have had some insight into the value of the mean; but profound mathematicians such as De Moivre, Daniel Bernoulli, Laplace, Lagrange, Gauss, Quetelet, De Morgan, Airy, Leslie Ellis, Boole, Glaisher, and others, have hardly exhausted the subject.

Several uses of the Mean Result.

The elimination of errors of unknown sources, is almost always accomplished by the simple arithmetical process of taking the mean, or, as it is often called, the average of several discrepant numbers. To take an average is to add the several quantities together, and divide by the number of quantities thus added, which gives a quotient lying among, or in the middle of, the several quantities. Before however inquiring fully into the grounds of this procedure, it is essential to observe that this one arithmetical process is really applied in at least three different cases, for different purposes, and upon different principles, and we must take great care not to confuse one application of the process with another. A mean result, then, may have any one of the following significations.

(1) It may give a merely representative number, expressing the general magnitude of a series of quantities, and serving as a convenient mode of comparing them with other series of quantities. Such a number is properly called The fictitious mean or The average result.

(2) It may give a result approximately free from disturbing quantities, which are known to affect some results in one direction, and other results equally in the opposite direction. We may say that in this case we get a Precise mean result.