_singing_, _singer_ each conveys a perfectly definite and intelligible
idea, though the idea is disconnected and is therefore functionally of
no practical value. We recognize immediately that these words are of two
sorts. The first word, _sing_, is an indivisible phonetic entity
conveying the notion of a certain specific activity. The other words all
involve the same fundamental notion but, owing to the addition of other
phonetic elements, this notion is given a particular twist that modifies
or more closely defines it. They represent, in a sense, compounded
concepts that have flowered from the fundamental one. We may, therefore,
analyze the words _sings_, _singing_, and _singer_ as binary expressions
involving a fundamental concept, a concept of subject matter (_sing_),
and a further concept of more abstract order--one of person, number,
time, condition, function, or of several of these combined.

If we symbolize such a term as _sing_ by the algebraic formula A, we
shall have to symbolize such terms as _sings_ and _singer_ by the
formula A + b.[1] The element A may be either a complete and independent
word (_sing_) or the fundamental substance, the so-called root or
stem[2] or "radical element" (_sing-_) of a word. The element b (_-s_,
_-ing_, _-er_) is the indicator of a subsidiary and, as a rule, a more
abstract concept; in the widest sense of the word "form," it puts upon
the fundamental concept a formal limitation. We may term it a
"grammatical element" or affix. As we shall see later on, the
grammatical element or the grammatical increment, as we had better put
it, need not be suffixed to the radical element. It may be a prefixed
element (like the _un-_ of _unsingable_), it may be inserted into the
very body of the stem (like the _n_ of the Latin _vinco_ "I conquer" as
contrasted with its absence in _vici_ "I have conquered"), it may be the
complete or partial repetition of the stem, or it may consist of some
modification of the inner form of the stem (change of vowel, as in