to describe that portion of her orbit lying above the plane of the ecliptic. The power of attraction of the sun will manifestly exert itself in such a manner, as to cause the moon to deviate from its old orbit, and to describe a new orbit, which will lie in all its points a little nearer to the plane of the ecliptic. The moon will not, therefore, rise in this superior part of her orbit as high above the plane of the ecliptic as she did before her motion was disturbed by the sun; and in descending to pass through her node, she will clearly reach the plane of the ecliptic quicker than she did when undisturbed, and pass through her node at a point nearer to herself than that occupied by the former node; in other language, the old node comes up to meet the advancing moon, and thus takes up a retrograde motion.

Let us now examine the motion of the moon in that portion of her orbit lying beneath the plane of the ecliptic, and most remote from the sun. Here the sun's disturbing influence will be diminished somewhat, in consequence of the increased distance at which it operates; but its effect will manifestly be to cause the moon to descend more rapidly, and to reach a lower point beneath the ecliptic than when undisturbed, increasing the inclination of the plane of the orbit, and causing the moon to reach her ascending node at a point earlier than when undisturbed, and thus producing a retrocession or retrograde motion of the line of nodes. Thus it appears that, in the long run, the sun's disturbing influence will tend to change within certain limits the angle of inclination of the moon's orbit; and, indeed, if the earth were fixed in position, would finally destroy this inclination. entirely, reducing the plane of the moon's orbit to absolute coincidence with that of the earth; but, as the moon is carried by the earth around the sun, and as the moon's orbit in the course of an entire revolution of the earth is thus presented to the sun at opposite points of the orbit under reverse circumstances, there is a compensation accomplished, so far as the angle of inclination is concerned, and also a artial compensation in the retrogression of the line of

nodes of the moon's orbit, but not such as to prevent, in the end, a complete revolution of the moon's nodes in a period which we have already seen amounts to eighteen years and two hundred and nineteen days.

We have thus attempted to present a general account of the effect of a disturbing force. These same principles may be extended yet further, and will give a general idea of the effects produced by the planets and their satellites upon each other.

If we return for a moment to the hypothesis that the carth is the only planet revolving about the sun, the magnitude of its orbit, as well as the length and position of the line of apsides, will remain for ever fixed. If, however, we introduce into our system a new planet revolving in an orbit interior to that of the earth, whatever force is exerted upon the earth by the attractive power of this new planet, will go to reinforce the power exerted by the sun; and hence the disturbing influence of the planet will tend to diminish the magnitude of the earth's orbit, and to decrease its periodic time. If the disturbing planet revolve in the same direction with the earth, by applying the reasoning hitherto used, we shall find that its effect will be to cause the perihelion point of the earth's orbit to advance and retreat during the revolution of the disturbing body, always leaving, however, a slight preponderance of the advancing movement over the retrograde.

In case the disturbing body revolve in an orbit exterior to that of the earth, then its effect will be to expand the earth's orbit and to increase the periodic time, while the influence exerted upon the position of the line of apsides will, in the long run, produce an advance.

The reasoning hitherto employed with reference to the inclination of the moon's orbit to the ecliptic is directly applicable to the effect produced by any planet upon the inclination of the orbit of any other planet, as referred to a fixed plane. Take the earth for example, and let us consider the effect of any planet, either interior or exterior,

M

upon the inclination of this plane to any fixed plane. So long as the disturbing body is revolving in that part of its orbit lying below the plane of the ecliptic, the tendency of the disturbing force will be to draw the earth from its undisturbed path below the plane of a fixed ecliptic; while this effect will be reversed, whenever the disturbing planet shall pass through the plane of the ecliptic, and commence the description of that part of its orbit which lies above this plane.

From the above reasoning it is clearly manifest that, as not a solitary planet or satellite is moving undisturbed under the attractive power of its primary body, not one of the heavenly bodies describes rigorously an elliptic orbit; nor does the line joining the sun with any planet sweep over precisely equal areas in equal times; neither are the squares of the periodic times of the planets exactly proportioned to the cubes of their mean distances from the sun. In short, every law of Kepler, whereby perfect harmony seemed to be introduced among the heavenly bodies, is now seen to fail, in consequence of the law of universal gravitation, and we find ourselves surrounded by a problem of wonderful grandeur, but of almost infinite complexity. Before this problem can be fully solved, we must measure the distance which separates every planet from the sun, and which divides every satellite from its primary; we must weigh the sun and all his planets and every satellite; we must determine the exact periods of revolution of each of these revolving worlds; and, when all this is accomplished, to trace out the reciprocal influences of each upon the other demands powers of reasoning far transcending the abilities of the most powerful genius, and hence the mind must either forego the resolution of this problem, or prepare for itself some mental machinery which shall give to thought and reason the same mechanical advantages which are obtained for the physical powers of the body by the invention and construction of the mighty engines of modern mechanics.

This has actually been accomplished in the discovery and gradual perfection of a branch of mathematics called the infinitesimal analysis. Up to the time of Newton, the mind employed alone the reasoning of geometry in the examination and discussion of the problems presented in the heavens. Even Newton himself was content to publish to the world the results of the application of the law of gravitation to the movement of the planets and their satellites under a geometrical form, exhibiting, in the use of these old methods, a sort of gigantic power which has ever remained as a monument of his wonderful ability.

He was, however, fully conscious of the fact, that the mind demanded for its use, in a full investigation of the physical universe-in the pursuit of these flying worlds, journeying through space amid such a crowd of disturbing influences-a far more subtle, pliable, and powerful mental machinery than that furnished in the cumbrous forms of geometrical reasoning. Conscious of this want, the genius. of Newton supplied the deficiency, and gave to the world the infinitesimal analysis, which, as improved and extended by the successors of the great English philosopher, has enabled man to accomplish results which seem to place him almost among the gods.

The plan of our work does not permit any attempt to explain the nature and powers of this new method of reasoning. We can only illustrate imperfectly the difference between the use of geometry and analysis. The demonstration of a problem by geometry demands that the mind shall comprehend and hold the first step in the train of reasoning; then, while the first is held, the second must be comprehended; and while intently holding these two steps, the third must be mastered and held, while the mind advances to the fourth step: thus progressing with a constantlyaccumulating weight oppressing the attention, and tending to crush and destroy further effort to advance; till, finally, the steps become so numerous and complex, that only those possessed of a genius of surpassing vigour are able to reach

in safety the last step, and thus grasp the full demonstration of the problem. Such is the reasoning of geometry. That of analysis is entirely different. Here the great effort is put forth to master fully and perfectly the conditions of the problem, and then to fasten upon the problem thus mastered the analytical machinery demanded in its resolution. This once accomplished, the mind puts forth its energy and accomplishes the first step, and may there stop and rest, in the full confidence that what has been gained can never be lost. Days, even months may pass, before the problem be resumed; but in this lapse of time there is no loss, and the investigation may be taken up precisely where it was left off; and so one step after another may be taken, each dependent on the other, but each in some sense stereotyped as the mind advances, and remaining fixed without the putting forth of any mental effort to retain it. In short, geometry demands a vigour of mind sufficient to grasp, and hold at the same instant, every link in the longest and most complex chain of reasoning, while analysis only requires a power of genius sufficient to deal with individual links in succession; thus, in the end, reaching the conclusion by short and comparatively easy mental marches.

CHAPTER XI.

INSTRUMENTAL ASTRONOMY.

Method for obtaining the Mass of the Sun.-For getting the Mass of a Planet with a Satellite.-For Weighing a Planet having no Satellite.For Weighing the Satellites.-Planetary Distances to be measured.Intervals between Primaries and their Satellites to be obtained.Intensity and Direction of the Impulsive Forces to be determined.These Problems all demand Instrumental Measures.-Differential Places. -Absolute Places.-The Transit-Instrument. -Adjustments. - Instrumental Errors.-Corrections due to various Causes.-American Method of Transits.-Meridian-Circle.-The Declinometer.

THE general reasoning presented in the preceding chapter can only be reduced to exact application after having