observations in which a cause acts positively, and the mean of all in which it acts negatively. Half the difference of these means will give the effect of the cause in question, provided that no other effect happens to vary in the same period or nearly so. Since the moon causes a movement of the ocean, it is evident that its attraction must have some effect upon the atmosphere. The laws of atmospheric tides were investigated by Laplace, but as it would be impracticable by theory to calculate their amounts we can only determine them by observation, as Laplace predicted that they would one day be determined.1 But the oscillations of the barometer thus caused are far smaller than the oscillations due to several other causes. Storms, hurricanes, or changes of weather produce movements of the barometer sometimes as much as a thousand times as great as the tides in question. There are also regular daily, yearly, or other fluctuations, all greater than the desired quantity. To detect and measure the atmospheric tide it was desirable that observations should be made in a place as free as possible from irregular disturbances. On this account several long series of observations were made at St. Helena, where the barometer is far more regular in its movements than in a continental climate. The effect of the moon's attraction was then detected by taking the mean of all the readings when the moon was on the meridian and the similar mean when she was on the horizon. The difference of these means was found to be only 00365, yet it was possible to discover even the variation of this tide according as the moon was nearer to or further from the earth, though this difference was only 00056 inch. It is quite evident that such minute effects could never be discovered in a purely empirical manner. Having no information but the series of observations before us, we could have no clue as to the mode of grouping them which would give so small a difference. In applying this method of means in an extensive manner we must generally then have à priori knowledge as to the periods at which a cause will act in one direction or the other. 1 Essai Philosophique sur les Probabilités, pp. 49, 50. We are sometimes able to eliminate fluctuations and take a mean result by purely mechanical arrangements. The daily variations of temperature, for instance, become imperceptible one or two feet below the surface of the earth, so that a thermometer placed with its bulb at that depth gives very nearly the true daily mean temperature. At a depth of twenty feet even the yearly fluctuations are nearly effaced, and the thermometer stands a little above the true mean temperature of the locality. In registering the rise and fall of the tide by a tide-gauge, it is desirable to avoid the oscillations arising from surface waves, which is very readily accomplished by placing the float in a cistern communicating by a small hole with the sea. Only a general rise or fall of the level is then perceptible, just as in the marine barometer the narrow tube prevents any casual fluctuations and allows only a continued change of pressure to manifest itself. Determination of the Zero point. In many important observations the chief difficulty consists in defining exactly the zero point from which we are to measure. We can point a telescope with great precision to a star and can measure to a second of arc the angle through which the telescope is raised or lowered; but all this precision will be useless unless we know exactly the centre point of the heavens from which we measure, or, what comes to the same thing, the horizontal line 90° distant from it. Since the true horizon has reference to the figure of the earth at the place of observation, we can only determine it by the direction of gravity, as marked either by the plumb-line or the surface of a liquid. The question resolves itself then into the most accurate mode of observing the direction of gravity, and as the plumb-line has long been found hopelessly inaccurate, astronomers generally employ the surface of mercury in repose as the criterion of horizontality. They ingeniously observe the direction of the surface by making a star the index. From the laws of reflection it follows that the angle between the direct ray from a star and that reflected from a surface of mercury will be exactly double the angle between the surface and the direct ray from the star. Hence the horizontal or zero point is the mean between the apparent place of any star or other very distant object and its reflection in mercury. A plumb-line is perpendicular, or a liquid surface is horizontal only in an approximate sense; for any irregu larity of the surface of the earth, a mountain, or even a house must cause some deviation by its attracting power. To detect such deviation might seem very difficult, because every other plumb-line or liquid surface would be equally affected by gravity. Nevertheless it can be detected; for if we place one plumb-line to the north of a mountain, and another to the south, they will be about equally deflected in opposite directions, and if by observations of the same. star we can measure the angle between the plumb-lines, half the inclination will be the deviation of either, after allowance has been made for the inclination due to the difference of latitude of the two places of observation. By this mode of observation applied to the mountain Schiehallion the deviation of the plumb-line was accurately measured by Maskelyne, and thus a comparison instituted between the attractive forces of the mountain and the whole globe, which led to a probable estimate of the earth's density. In some cases it is actually better to determine the zero point by the average of equally diverging quantities than by direct observation. In delicate weighings by a chemical balance it is requisite to ascertain exactly the point at which the beam comes to rest, and when standard weights are being compared the position of the beam is ascertained by a carefully divided scale viewed through a microscope. But when the beam is just coming to rest, friction, small impediments or other accidental causes may readily obstruct it, because it is near the point at which the force of stability becomes infinitely small. Hence it is found better to let the beam vibrate and observe the terminal points of the vibrations. The mean between two extreme points will nearly indicate the position of rest. Friction and the resistance of air tend to reduce the vibrations, so that this mean will be erroneous by half the amount of this effect during a half vibration. But by taking several observations we may determine this retardation and allow for it. Thus if a, b, c be the readings of the terminal B B points of three excursions of the beam from the zero of the scale, then (a + b) will be about as much erroneous in one direction as (b+c) in the other, so that the mean of these two means, or (a + 2 b + c), will be exceedingly near to the point of rest. A still closer approximation may be made by taking four readings and reducing them by the formula (a + 2 b+ 2 c + d). The accuracy of Baily's experiments, directed to determine the density of the earth, entirely depended upon this mode of observing oscillations. The balls whose gravitation was measured were so delicately suspended by a torsion balance that they never came to rest. The extreme points of the oscillations were observed both when the heavy leaden attracting ball was on one side and on the other. The difference of the mean points when the leaden ball was on the right hand and that when it was on the left hand gave double the amount of the deflection. A beautiful instance of avoiding the use of a zero point is found in Mr. E. J. Stone's observations on the radiant heat of the fixed stars. The difficulty of these observations arose from the comparatively great amounts of heat which were sent into the telescope from the atmosphere, and which were sufficient to disguise almost entirely the feeble heat rays of a star. But Mr. Stone fixed at the focus of his telescope a double thermo-electric pile of which the two parts were reversed in order. Now any disturbance of temperature which acted uniformly upon both piles produced no effect upon the galvanometer needle, and when the rays of the star were made to fall alternately upon one pile and the other, the total amount of the deflection represented double the heating power of the star. Thus Mr. Stone was able to detect with much certainty a heating effect of the star Arcturus, which even when concentrated by the telescope amounted only to o°02 Fahr., and which represents a heating effect of the direct ray of only about 0°00000137 Fahr., equivalent to the heat which would be received from a three-inch cubic vessel full of boiling water at the distance of 400 yards. It is probable that 1 Gauss, Taylor's Scientific Memoirs, vol. ii. p. 43, &c. 2 Proceedings of the Royal Society, vol. xviii. p. 159 (Jan. 13, 1870). Philosophical Magazine (4th Series), vol. xxxix. p. 376. Mr. Stone's arrangement of the pile might be usefully employed in other delicate thermometric experiments subject to considerable disturbing influences. Determination of Maximum Points. We employ the method of means in a certain number of observations directed to determine the moment at which a phenomenon reaches its highest point in quantity. In noting the place of a fixed star at a given time there is no difficulty in ascertaining the point to be observed, for a star in a good telescope presents an exceedingly small disc. In observing a nebulous body which from a bright centre fades gradually away on all sides, it will not be possible to select with certainty the middle point. In many such cases the best method is not to select arbitrarily the supposed middle point, but points of equal brightness on either side, and then take the mean of the observations of these two points for the centre. As a general rule, a variable quantity in reaching its maximum increases at a less and less rate, and after passing the highest point begins to decrease by insensible degrees. The maximum. may indeed be defined as that point at which the increase or decrease is null. Hence it will usually be the most. indefinite point, and if we can accurately measure the phenomenon we shall best determine the place of the maximum by determining points on either side at which the ordinates are equal. There is moreover this advantage in the method that several points may be determined with the corresponding ones on the other side, and the mean of the whole taken as the true place of the maximum. But this method entirely depends upon the existence of symmetry in the curve, so that of two equal ordinates one shall be as far on one side of the maximum as the other is on the other side. The method fails when other laws of variation prevail. In tidal observations great difficulty is encountered in fixing the moment of high water, because the rate at which the water is then rising or falling, is almost imperceptible. Whewell proposed, therefore, to note the time at which the water passes a fixed point somewhat below the maximum both in rising and falling, and take the mean |