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 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, on occasion, be identical with the word, it does not follow that it may always, or