In the isolating type the formula of a word is simply R, and that of a sentence R+R+R, &c., where R stands for “root.” If we represent by r those roots whose sense has become obscured so as to pass into the state of prefixes and suffixes significant only of relationship between other words, we shall have a formula of agglutination, Rr, Rrr, rR, rRr, &c. Lastly, the essence of an inflecting language consists in the power of a root to express, by modification of its own form, its various relations to other roots. Not that the roots of all words are necessarily modified; for they often remain as they do in agglutinating tongues. But they _may_ be modified, and “languages in which relations may be thus expressed, not only by suffixes and prefixes, but also by a modification of the form of the roots, are inflectional languages.” Therefore, if we represent this power of inflectional change on the part of the root itself by the symbol ^x, the agglutinating formula Rr may become R^{xr}. Moreover, the modifying elements may also be inflected, words thus yielding such formulæ as Rr^x, Rrr^x, &c. Such, then, are the three main groups or orders of language. But in addition to them we must notice three others, which have been shown to be clearly separable. These three additional groups are the Polysynthetic, the Incorporating, and the Analytic. The _Polysynthetic_ (= _Incapsulating_) order is found among certain savages, especially on the continent of America, where, according to Duponceau, more or less distinctive adherence to this type is to be met with from Greenland to Chili. The peculiarity of such languages consists in the indefinite composition of words by syncope and ellipsis. That is to say, sentences are formed by the running together of compound words of inordinate length, and in the process of fusion the constituent words are so much abbreviated as often to be represented by no more than a single intercalated letter. For example, the Greenland _aulisariartorasuarpok_, “he-hastened-to-go-afishing,” is made up of _aulisar_, “to fish,” _peartor_, “to be engaged in anything,” _pinnesuarpok_, “he hastens:” and the Chippeway _totoccabo_, “wine,” is formed of _toto_, “milk,” with _chominabo_, “a bunch of grapes.” Thus, polysynthesis consists of fusion with contraction, some of the component words losing their first, and others their last syllables. Moreover, composition of this kind further differs from that which occurs in many other types of language (_e.g._ our adjectival _never-to-be-forgotten_), in that the constituent parts may never have attained the rank of independent words, which can be set apart and employed by themselves.