covered by investigation. The exceptional cases, in which
deduction is really useful, occur where certain limits to the number and
combination of qualities happen to be known, as they may be in human
institutions, or where there are mathematical conditions. Thus, we might
be able to classify orders of Architecture, or the classical metres and
stanzas of English poetry; though, in fact, these things are too free,
subtle and complex for deductive treatment: for do not the Arts grow
like trees? The only sure cases are mathematical; as we may show that
there are possible only three kinds of plane triangles, four conic
sections, five regular solids.

Sec. 5. The rules for _testing_ a Division are as follows:

1. Each Sub-class, or Species, should comprise less than the Class, or
Genus, to be divided. This provides that the division shall be a real
one, and not based upon an attribute common to the whole class; that,
therefore, the first rule for making a division shall have been adhered
to. But, as in Sec. 4, we are here met by a logical difficulty. Suppose
that the class to be divided is A, and that we attempt to divide upon
the attribute B, into AB and Ab; is this a true division, if we do not
know any A that is not B? As far as our knowledge extends, we have not
divided A at all. On the other hand, our knowledge of concrete things is
never exhaustive; so that, although we know of no A that is not B, it
may yet exist, and we have seen that it is a logical caution not to
assume what we do not know. In a deductive classification, at least, it
seems better to regard every attribute as a possible ground of division.
Hence, in the above division of 'All Things,'--'Not-phenomenal,'
'Extended-Not-resistant,' 'Resistant-Not-gravitating,' appear as
negative classes (that is, classes based on the negation of an
attribute), although their real existence may be doubtful. But, if this
be justifiable, we must either rewrite the first test of a division
thus: 'Each sub-class should _possibly_ comprise less than the class to
be divided'; or else we must confine the test to (a) thoroughly
empirical divisions, as in dividing Colour into Red and Not-red, where
we know that both sub-classes are real; and (b) divisions under
demonstrable conditions--as in dividing the three kinds of triangles by
the quality equilateral, we know that it is only applicable to
acute-angled triangles, and do not attempt to divide the right-angled or
obtuse-angled