In order that we may gain a true understanding of the kind, degree, and value of the knowledge which we acquire by experimental investigation, it is requisite that we should be fully conscious of its approximate character. We must learn to distinguish between what we can know and cannot know—between the questions which admit of solution, and those which only seem to be solved. Many persons may be misled by the expression exact science, and may think that the knowledge acquired by scientific methods admits of our reaching absolutely true laws, exact to the last degree. There is even a prevailing impression that when once mathematical formulæ have been successfully applied to a branch of science, this portion of knowledge assumes a new nature, and admits of reasoning of a higher character than those sciences which are still unmathematical.
The very satisfactory degree of accuracy attained in the science of astronomy gives a certain plausibility to erroneous notions of this kind. Some persons no doubt consider it to be proved that planets move in ellipses, in such a manner that all Kepler’s laws hold exactly true; but there is a double error in any such notions. In the first place, Kepler’s laws are not proved, if by proof we mean certain demonstration of their exact truth. In the next place, even assuming Kepler’s laws to be exactly true in a 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 prove that it moved in a perfect 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 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 cannot be verified by observation.
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 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 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 Newton.1 Even at the present day discrepancies exist between the observed dimensions of the planetary orbits and their theoretical magnitudes, after making allowance for all disturbing causes.2 Nothing is more certain in scientific method than that approximate coincidence alone can be expected. In the measurement of continuous quantity perfect correspondence must be accidental, and should give rise to suspicion rather than to satisfaction.
One remarkable result of the approximate character of our observations is that we could never 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 that any planet, even if undisturbed by other bodies, would have a circle for its orbit; but if the orbit were a circle we could never prove the entire