differed in regard to the water of crystallisation.2 Vanadium again appeared to be an exception to the laws of isomorphism, until it was proved by Professor Roscoe, that what Berzelius supposed to be metallic vanadium was really an oxide of vanadium.3
Apparent but Congruent Exceptions.
Not unfrequently a law of nature will present results in certain circumstances which appear to be entirely in conflict with the law itself. Not only may the action of the law be much complicated and disguised, but it may in various ways be reversed or inverted, so that careless observers are misled. Ancient philosophers generally believed that while some bodies were heavy by nature, others, such as flame, smoke, bubbles, clouds, &c., were essentially light, or possessed a tendency to move upwards. So acute an inquirer as Aristotle failed to perceive the true nature of buoyancy, and the doctrine of intrinsic lightness, expounded in his works, became the accepted view for many centuries. It is true that Lucretius was aware why flame tends to rise, holding that—
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.4 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; 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,