accelerate the motion of bodies passing through it, somewhat in the mode which Aristotelians attributed falsely to the air. Thus we can frame the notion of negative density, and Newton could reason exactly concerning it, although no such thing exists 9.

In every direction of thought we may meet ultimately with similar failures of analogy. A moving point generates a line, a moving line generates a surface, a moving surface generates a solid, but what does a moving solid generate? When we compare a polyhedron, or manysided solid, with a polygon, or plane figure of many sides, the volume of the first is analogous to the area o the second; the face of the solid answers to the side of the polygon; the edge of the solid to the point of the figure ; but the corner, or junction of edges in the polyhedron, is left wholly unrepresented in the plane of the polygon. Even if we attempted to draw the analogies in some other manner, we should still find a geometrical notion embodied in the solid which has no representative in the plain figurer.

Faraday was able to frame some notion of matter in a fourth condition, which should be to gas what gas is to liquids. Such substance, he thought, would not fall far short of radiant matter, by which apparently he meant the supposed caloric or matter assumed to constitute heat, according to the Corpuscular Theory. Even if we could frame the notion, matter in such a state cannot be known to exist, and recent discoveries concerning the continuity of the solid, liquid, and gaseous states remove the basis of the speculation.

From these and many other instances which might be

9 ‘Principia,' bk. II. Section II. Prop. X.

r De Morgan, 'Cambridge Philosophical Transactions,' vol. xi. Part ii. p. 246.

Life of Faraday,' vol. i. p. 216.

adduced, we learn that analogical reasoning leads us to the conception of many things which, so far as we can ascertain, do not exist. In this way great perplexities have arisen in the use of language and mathematical symbols. All language depends upon analogy; for we join and arrange words so that they may represent the corresponding junctions or arrangements of things and their qualities. But in the use of language we are obviously capable of forming many combinations of words to which no corresponding meaning apparently exists. The same difficulty arises in the use of mathematical signs, and mathematicians have needlessly puzzled themselves about the square root of a negative quantity, which is, in many applications of algebraic calculation, simply a sign without any analogous meaning, there being a failure of analogy.

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In other sciences, as well as in that of the heavens, exceptional events are commonly the points from which we start to explore new regions of knowledge. It has been beautifully said that Wonder is the daughter of Ignorance, but the mother of Invention; and though the most familiar and slight events, if fully examined, will afford endless food for wonder and for wisdom, yet it is the few peculiar and unlooked-for events which most often lead a scientific mind into a course of discovery. It is true, indeed, that it requires much philosophy to observe things which are too near to us.

The high scientific importance attaching, then, to exceptions, renders it desirable that we should carefully consider the various modes in which an exception may be disposed of; while some new facts will be found to confirm the very laws to which at first sight they seem clearly opposed, others will cause us to limit the generality of our previous statements. In some cases the exception may be proved to be no exception; occasionally it will prove fatal to our previous most confident speculations ; and there are some new phenomena which, without really destroying any of our former theories, open to us wholly new fields of scientific investigation. The study of this subject is especially interesting and important, because, as I have before said (vol. ii. p. 233), no important theory can be built up complete and perfect all at once. When unexplained phenomena present themselves as objections to the theory, it will often demand the utmost judgment and sagacity to assign to them their proper place and force. The acceptation or rejection of a theory will entirely depend upon discriminating the one insuperable contra

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amerate at least eight different classes or kinds neptical phenomena, to one or other of which any supposed exception to the known laws of nature will ultimately be referred; they may be briefly described as below, and will be sufficiently illustrated in the succeeding sections.

(1) Imaginary, or false exceptions, that is, facts, objects, or events which are not really what they are supposed to be.

(2) Apparent, but congruent exceptions, which, though apparently in conflict with a law of nature, are really in agreement with it.

(3) Singular exceptions, which really agree with a law of nature, but exhibit remarkable and unique results of it. (4) Divergent exceptions, which really proceed from the ordinary action of known processes of nature, but which are excessive in amount or monstrous in character.

(5) Accidental exceptions, arising from the interference of some entirely distinct but known law of nature.

(0) Novel and unexplained exceptions, which lead to the discovery of a new series of laws and phenomena, modifying or disguising the effects of previously known tows, without being inconsistent with them.

Limiting exceptions, showing the falsity of a supposed law in cases to which it had been extended, but not cheng its truth in other cases.

3 Contradictory or real exceptions which lead us to e occlusion that a supposed hypothesis or theory is in og on to the phenomena of nature, and must therefore A abandoned.