Archimedes also was so perfectly acquainted with the buoyancy of bodies immersed in water, that he could not fail to perceive the existence of a parallel effect in air. Yet throughout the early middle ages the light of true science could not contend with the glare of the Peripatetic doctrine. The genius of Galileo and Newton was required to convince people of the simple truth that all matter is heavy, but that the gravity of one substance may be overborne by that of another, as one scale of a balance is carried up by the preponderating weight in the opposite scale. It is curious to find Newton gravely explaining the difference of absolute and relative gravity, as if it were a new discovery proceeding from his theory. More than a century elapsed before other apparent exceptions to the Newtonian philosophy were explained away.

Newton himself allowed that the motion of the apsides of the moon's orbit appeared to be irreconcilable with the law of gravity, and it remained for Clairaut to remove the difficulty by more complete mathematical analysis. There must always remain, in the motions of the heavenly bodies, discrepancies of some amount between theory and observation; but such discrepancies have so often yielded in past times to prolonged investigation that physicists now regard them as merely apparent exceptions, which will afterwards be found to agree with the law of gravity.

The most beautiful instance of an apparent exception, is found in the total reflection of light, which occurs when a beam of light within a medium falls very obliquely upon the boundary separating it from a rarer medium. The general law is that when a ray strikes the limit between two media of different refractive indices, part of the light is reflected and part is refracted; but when the obliquity of the ray within the denser medium passes beyond a certain point, there is a sudden apparent breach of continuity, and the whole of the light is reflected. A clear reason can be given for this exceptional conduct of the light. According to the law of refraction, the sine of the angle of incidence bears a fixed ratio to the sine of the angle of refraction, so that the greater of the two angles, which is always that in the less dense medium, may increase up to a right angle;

1 Principia, bk. ii. Prop. 20. Corollaries, 5 and 6.

but when the media differ in refractive power, the less angle cannot become a right angle, as this would require the sine of an angle to be greater than the radius. It might seem that this is an exception of the kind described below as a limiting exception, by which a law is shown to be inapplicable beyond certain limits; but in the explanation of the exception according to the undulatory theory, we find that there is really no breach of the general law. When an undulation strikes a point in a bounding surface, spherical waves are produced and spread from the point. The refracted ray is the resultant of an infinite number of such spherical waves, and the bending of the ray at the common surface of two media depends upon the comparative velocities of propagation of the undulations in those media. But if a ray falls very obliquely upon the surface of a rarer medium, the waves proceeding from successive points of the surface spread so rapidly as never to intersect, and no resultant wave will then be produced. We thus perceive that from similar mathematical conditions arise distinct apparent effects.

There occur from time to time failures in our best grounded predictions. A comet, of which the orbit has been well determined, may fail, like Lexell's Comet, to appear at the appointed time and place in the heavens. In the present day we should not allow such an exception to our successful predictions to weigh against our belief in the theory of gravitation, but should assume that some unknown body had through the action of gravitation deflected the comet. As Clairaut remarked, in publishing his calculations concerning the expected reappearance of Halley's Comet, a body which passes into regions so remote, and which is hidden from our view during such long periods, might be exposed to the influence of forces totally unknown to us, such as the attraction of other comets, or of planets too far removed from the sun to be ever perceived by us. In the case of Lexell's Comet it was afterwards shown, curiously enough, that its appearance was not one of a regular series of periodical returns within the sphere of our vision, but a single exceptional visit never to be repeated, and probably due to the perturbing powers of Jupiter. This solitary visit became a strong confirmation of the law of gravity with which it seemed to be in conflict.

Singular Exceptions.

Among the most interesting of apparent exceptions are those which I call singular exceptions, because they are more or less analogous to the singular cases or solutions which occur in mathematical science. A general mathematical law embraces an infinite multitude of cases which perfectly agree with each other in a certain respect. It may nevertheless happen that a single case, while really obeying the general law, stands out as apparently different from all the rest. The rotation of the earth upon its axis gives to all the stars an apparent motion of rotation from east to west; but while countless thousands obey the rule, the Pole Star alone seems to break it. Exact observations indeed show that it also revolves in a small circle, but a star might happen for a short time to exist so close to the pole that no appreciable change of place would be caused by the earth's rotation. It would then constitute a perfect singular exception; while really obeying the law, it would break the terms in which it is usually stated. In the same way the poles of every revolving body are singular points.

Whenever the laws of nature are reduced to a mathematical form we may expect to meet with singular cases, and, as all the physical sciences will meet in the mathematical principles of mechanics, there is no part of nature where we may not encounter them. In mechanical science the motion of rotation may be considered an exception to the motion of translation. It is a general law that any number of parallel forces, whether acting in the same or opposite directions, will have a resultant which may be substituted for them with like effect. This resultant will be equal to the algebraic sum of the forces, or the difference of those acting in one direction and the other; it will pass through a point which is determined by a simple formula, and which may be described as the mean point of all the points of application of the parallel forces (p. 364). Thus we readily determine the resultant of parallel forces except in one peculiar case, namely, when two forces are equal and opposite but not in the same straight line. Being equal and opposite the amount of the resultant is nothing, yet, as the forces are not in the same

straight line, they do not balance each other. Examining the formula for the point of application of the resultant, we find that it gives an infinitely great magnitude, so that the resultant is nothing at all, and acts at an infinite distance, which is practically the same as to say that there is no resultant. Two such forces constitute what is known in nechanical science as a couple, which occasions rotatory instead of rectilinear motion, and can only be neutralised by an equal and opposite couple of forces.

The best instances of singular exceptions are furnished by the science of optics. It is a general law that in passing through transparent media the plane of vibration of polarised light remains unchanged. But in certain liquids, some peculiar crystals of quartz, and transparent solid media subjected to a magnetic strain, as in Faraday's experiment (pp. 588, 630), the plane of polarisation is rotated. in a screw-like manner. This effect is so entirely sui generis, so unlike any other phenomena in nature, as to appear truly exceptional; yet mathematical analysis shows it to be only a single case of much more general laws. As stated by Thomson and Tait,1 it arises from the composition of two uniform circular motions. If while a point is moving round a circle, the centre of that circle move upon another circle, a great variety of curious curves will be produced according as we vary the dimensions of the circles, the rapidity or the direction of the motions. When the two circles are exactly equal, the rapidities nearly so, and the directions opposite, the point will be found to move gradually round the centre of the stationary circle, and describe a curious star-like figure connected with the molecular motions out of which the rotational power of the media rises. Among other singular exceptions in optics may be placed the conical refraction of light, already noticed (p. 540), arising from the peculiar form assumed by a wave of light when passing through certain doublerefracting crystals. The laws obeyed by the wave are exactly the same as in other cases, yet the results are entirely sui generis. So far are such cases from contradicting the law of ordinary cases, that they afford the best opportunities for verification.

1 Treatise on Natural Philosophy, vol. i. p. 50.

In astronomy singular exceptions might occur, and in an approximate manner they do occur. We may point to the rings of Saturn as objects which, though undoubtedly obeying the law of gravity, are yet unique, as far as our obser vation of the universe has gone. They agree, indeed, with the other bodies of the planetary system in the stability of their movements, which never diverge far from the mean position. There seems to be little doubt that these rings are composed of swarms of small meteoric stones; formerly they were thought to be solid continuous rings, and mathematicians proved that if so constituted an entirely exceptional event might have happened under certain circumstances. Had the rings been exactly uniform all round, and with a centre of gravity coinciding for a moment with that of Saturn, a singular case of unstable equilibrium would have arisen, necessarily resulting in the sudden collapse of the rings, and the fall of their debris upon the surface of the planet. Thus in one single case the theory of gravity would give a result wholly unlike anything else known in the

mechanism of the heavens.

It is possible that we might meet with singular exceptions in crystallography. If a crystal of the second or dimetric system, in which the third axis is usually unequal to either of the other two, happened to have the three axes equal, it might be mistaken for a crystal of the cubic system, but would exhibit different faces and dissimilar properties. There is, again, a possible class of diclinic crystals in which two axes are at right angles and the third axis inclined to the other two. This class is chiefly remarkable for its non-existence, since no crystals have yet been proved to have such axes. It seems likely that the class would constitute only a singular case of the more general triclinic system, in which all three axes are inclined to each other at various angles. Now if the diclinic form were merely accidental, and not produced by any general law of molecular constitution, its actual occurrence would be infinitely improbable, just as it is infinitely improbable that any star should indicate the North Pole with perfect exactness.

In the curves denoting the relation between the temperature and pressure of water there is, as shown by Professor J. Thomson, one very remarkable point entirely unique, at which alone water can remain in the three