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If we symbolize such a term as _sing_ by the algebraic formula A, wenshall have to symbolize such terms as _sings_ and _singer_ by thenformula 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

_sung_ and _song_; change of consonant as in _dead_ and _death_; change of accent; actual abbreviation).

Each and every one of these types of grammatical element or modification has this peculiarity, that it may not, in the vast majority of cases, be used independently but needs to be somehow attached to or welded with a radical element in order to convey an intelligible notion. We had better, therefore, modify our formula, A + b, to A + (b), the round brackets symbolizing the

incapacity of an element to stand alone.

The grammatical element, moreover, is not only non-existent except as associated with a radical

one, it does not even, as a rule, obtain its measure of significance unless it is associated with a particular class of radical elements. Thus, the _-s_ of English _he hits_ symbolizes an utterly different notion from the _-s_ of _books_, merely because _hit_ and _book_ are differently classified as to function.

We must hasten to observe, however, that while the radical element may,onoccasion, beidentical with the word, it does not follow that it may always, or even customarily, be used as a word. Thus, the _hort-_ "garden" of such Latin forms as _hortus_, _horti_, and _horto_ is as much of an abstraction, though one yielding a more easily apprehended significance, than the

ing of singing.

Neither exists as an independently intelligible and satisfying element of speech. Both the radical element, as such, and the grammatical element, therefore, are reached only by a process of abstraction. It seemed proper to symbolize _sing-er_ as A + (b);

_hort-us_ must be symbolized as (A) + (b).