of Goethe’s anti-mathematical hypothesis; but the field of conjecture remains open.  On the other hand, Newton’s proof that the solar system is controlled by a central force, was supported by the demonstration that a force having any other direction could not have results agreeing with Kepler’s second law of the planetary motions, namely, that, as a planet moves in its orbit, the areas described by a line drawn from the sun to the planet are proportional to the times occupied in the planet’s motion.  When a planet is nearest to the sun, the area described by such a line is least for any given distance traversed by the planet; and then the planet moves fastest:  when the planet is furthest from the sun, the area described by such a line is greatest for an equal distance traversed; and then the planet moves slowest.  This law may be deduced from the hypothesis of a central force, but not from any other; the proof, therefore, as Mill says, satisfies the method of Difference.

Apparently, to such completeness of demonstration certain conditions are necessary:  the possibilities must lie between alternatives, such as A or not-A, or amongst some definite list of cases that may be exhausted, such as equal, greater or less.  He whose hypothesis cannot be brought to such a definite issue, must try to refute whatever other hypotheses are offered, and naturally he will attack first the strongest rivals.  With this object in view he looks about for a “crucial instance,” that is, an observation or experiment that stands like a cross (sign-post) at the parting of the ways to guide us into the right way, or, in plain words, an instance that can be explained by one hypothesis but not by another.  Thus the phases of Venus, similar to those of the Moon, but concurring with great changes of apparent size, presented, when discovered by Galileo, a crucial instance in favour of the Copernican hypothesis, as against the Ptolemaic, so far at least as to prove that Venus revolved around the Sun inside the orbit of the Earth.  Foucault’s experiment determining the velocity of Light (cited in the last chapter) was at first intended as an experimentum crucis to decide between the corpuscular and undulatory theories; and answered this purpose, by showing that the velocity of a beam passed through water was less than it should be by the former, but in agreement with the latter doctrine (Deschanel:  Sec. 813).

Perhaps experiments of this decisive character are commonest in Chemistry:  chemical tests, says Herschel, “are almost universally crucial experiments.”  The following is abridged from Playfair (Encycl.  Met., Diss. III.):  The Chemists of the eighteenth century observed that metals were rendered heavier by calcination; and there were two ways of accounting for this:  either something had been added in the process, though what, they could not imagine; or, something had been driven off that was in its nature light, namely, phlogiston.  To decide between