theoretical point of view, the planets never move according to those laws. Even if we could observe the motions of a planet, of a perfect globular form, free from all perturbing or retarding forces, we could never perfectly prove that it moved in an ellipse. To prove the elliptical form we should have to measure infinitely small angles, and infinitely small fractions of a second; we should have to perform impossibilities. All we can do is to show that the motion of an unperturbed planet approaches very nearly to the form of an ellipse, and the more nearly the more accurately our observations are made. But if we go on to assert that the path is an ellipse we pass beyond our data, and make an assumption which may be more or less probable, but cannot be proved, in the strict sense of that term.

But, secondly, as a matter of fact no planet does move in a perfect ellipse, or manifest the truth of Kepler's laws exactly. The very law of gravity prevents its own results from being clearly exhibited, because the mutual perturbations of the planets distort the elliptical paths. Those laws again hold exactly true only of infinitely small planetary bodies, and when two great globes, like the sun and Jupiter, attract each other, the law must be modified. The periodic time is then shortened in the ratio of the square root of the number expressing the sun's mass, to that of the sum of the numbers expressing the masses of the sun and planet, as was shown by Newtona. Even at the present day discrepancies exist between the observed dimensions of the planet's orbits and their theoretical magnitudes, after making allowance for all disturbing causes b. Nothing, in fact, is more certain in scientific method than that approximate coincidence can alone be expected. In the measurement of continuous quantity a Principia,' bk. III. Prop. 15.

b See Lockyer's 'Lessons in Elementary Astronomy,' p. 301.

perfect correspondence must be purely accidental, and should give rise to suspicion rather than to satisfaction.

One remarkable result of the approximate character of our observations is that we never could prove the existence of perfectly circular or parabolic movement, even if it existed. The circle is a singular case of the ellipse, for which the eccentricity is zero; it is infinitely improbable than any planet, even if undisturbed by other bodies, should have a circle for its orbit; but if the orbit were a circle we could never prove the entire absence of eccentricity. All that we could do would be to declare the divergence from the circular form to be inappreciable. Delambre was unable to detect the slightest ellipticity in the orbit of Jupiter's first satellite, but he could only infer that the orbit was nearly circular. The parabola is the singular limit between the ellipse and the hyperbola. As there are elliptic and hyperbolic comets, so we might conceive the existence of a parabolic comet. Indeed if an undisturbed comet fell towards the sun from an infinite distance it would move in a parabola; but we could never prove that it so moved.

Substitution of Simple IIypotheses.

In truth men never can solve problems fulfilling the complex circumstances of nature. All laws and explanations are in a certain sense hypothetical, and apply exactly to nothing which we can know to exist. In place of the actual objects which we see and feel, the mathematician invariably substitutes imaginary objects, only partially resembling those represented, but so devised that the discrepancies may not be of an amount to alter seriously the character of the solution. When we probe the matter to the bottom physical astronomy is as hypothetical as Euclid's elements. There may exist in nature perfect

straight lines, triangles, circles, and other regular geometrical figures; to our science it is a matter of indifference whether they do or do not exist, because in any case they must be beyond our powers of appreciation. If we submitted a perfect circle to the most rigorous scrutiny and measurement, it is impossible that we should discover whether it were perfect or not. Nevertheless in geometry we argue concerning perfect rectilineal figures and curves, and the conclusions apply to existing objects so far as we can assure ourselves that they agree with the hypothetical conditions of our reasoning. Now this is in reality all that we can do in the most perfect of the sciences of nature.

Doubtless in astronomy we meet with the nearest approximation to actual conditions. The law of gravity is not a complex one in itself, and we believe it with much. probability to be exactly true; but we cannot calculate out in any one case its accurate results. The law asserts that every particle of matter in the universe attracts every other particle, with a force depending on the masses of the particles and their distance. We cannot then know the force acting on any one particle unless we know the masses and distances and positions of all the other particles in the universe. The physical astronomer has from the first made a sweeping assumption, namely, that all the other millions of existing systems exert no perturbing effects in our planetary system, that is to say, no effects in the least appreciable. Thus the problem becomes at once hypothetical, because there is little doubt that gravitation between our sun and planets and other systems must exist in some degree. But even when they consider the relations of our planetary bodies inter se, all their processes are grossly approximative. In the first place they assume that each of the planets is a perfect ellipsoid, with a smooth surface and a homogeneous interior. That this assumption is untrue every mountain and valley, every

sea, every mine affords conclusive evidence. If the astronomer is to make his calculations perfect, he must not only take account of the Himalayas and the Andes, the Atlantic and Pacific, but the attraction of every hill, nay, every ant-hill, must be separately calculated, nor must the attractive power of any grain of sand be neglected. So far are they from having yet considered any local inequality of the surface, that they have not yet decided upon the general form of the earth; it is yet a matter of speculation whether or not the earth is an ellipsoid with three unequal axes. If, as is probable, the globe is proved to be irregularly compressed in some directions, the calculations of astronomers will have to be repeated and refined, in order that they may approximate to the attractive power of such a body. If we cannot accurately learn the form of our own earth, how can we expect to ascertain that of the moon, the sun, and other planets, in some of which are probably irregularities of greater proportional

amount.

The science of physical astronomy is yet in a further way merely approximative and hypothetical. Given perfectly homogeneous ellipsoids acting upon each other according to the law of gravity, the best mathematicians have never and perhaps never will determine exactly the resulting movements. Even when three bodies simultaneously attract each other the complication of effects is so great that only approximate calculations can be made. Astronomers have not even attempted the general problem of the simultaneous attractions of four, five, six, or more bodies, resolving the general problem into so many different problems of three bodies. The principle upon which the calculations of physical astronomy proceed, is to neglect every effect which could not lead to any quantity appreciable in observation, and the quantities rejected e Thomson and Tait, Treatise on Natural Philosophy,' vol. i. p. 646.

are indefinitely more numerous and complex than the few larger terms which are retained. All then is merely approximate.

Concerning other branches of physical science the same general statements are even more evidently true. We speak and calculate about inflexible bars, inextensible lines, heavy points, homogeneous substances, uniform spheres, perfect fluids and gases, and we deduce an infinite number of beautiful theorems; but all is hypothetical. There is no such thing as an inflexible bar, an inextensible line, nor any one of the other perfect objects of mechanical science; they are to be classed with those other almost mythical existences, the straight line, triangle, circle, rectangle, &c., about which Euclid so freely discoursed. Take the simplest operation considered in statics—the use of a crowbar in raising a heavy stone, and we shall find, as Thomson and Tait have pointed out, that we neglect far more than we observed. If we suppose the bar to be quite rigid, the fulcrum and stone perfectly hard, and the points of contact real points, we might give the true relation of the forces. But in reality the bar must bend, and the extension and compression of different parts involve us in difficulties. Even if the bar be homogeneous in all its parts, there is no mathematical theory capable of determining with accuracy all that goes on; if, as is infinitely more probable, the bar is not homogeneous, the complete solution will be indefinitely more complicated, but hardly more hopeless. No sooner had we determined the change of form according to simple mechanical principles, than we should discover the interference of thermodynamic principles. Compression produces heat and extension cold, and thus the conditions of the problem are modified throughout. In attempting a fourth approximation we should have to allow for the conduction d Treatise on Natural Philosophy,' vol. i. pp. 337, &c.