The method of postulates (that is, assumptions or axioms) and definitions is fully adequate to mathematics; as for other sciences, the more complex their subject-matter, the less amenable are they to this method, since, under it, every descriptive or historical fact becomes the subject of a new postulate.

A student of the American Indian languages is naturally led to investigate the wide-spread use of the quinary-vigesimal system of counting which he meets in the whole territory from Alaska along the Pacific Coast to the Orinoco and the Amazon. One is tempted to begin with an inquiry into the concept of number in general, and a study of the binary, ternary, and quaternary systems of counting, but for these subjects I refer to the works of Conant, McGee, Villiers, Eells, and Gibbs. It is interesting to follow the evolution of the concept of number from the stage where there is practically no idea of it, to the counting in pairs, frequent among the Australian and South American tribes, and then to higher units. For the purpose of this paper, however, I shall assume that the stage of counting by fives, and the discovery that the five fingers of the hand are the most practical means of so counting, has been reached. This discovery naturally leads, almost immediately, to the extension of the quinary base to the decimal, i.e., the two hands, and then to the vigesimal, including also the toes of the feet.

Grimm's Law having been given to the world one hundred years ago, the present occasion would seem to lend itself to a critical review of some of its salient features.

Grimm was by no means the first scholar to observe that in the Germanic languages consonants are often at variance with those of the cognate languages. Such discrepancies were especially noted and commented upon by students of Gothic from Franciscus Junius to the Swedish Professor Johannes ab Ihre and James Jamieson, the author of a well-known Etymological Dictionary of the Scottish language. Yet for a long time observations to this effect remained isolated, and amounted to little more than a mixture of truth and errors.

Heretofore, in etymological investigations, in the Indo-European field at least, several processes have been overworked, being used to explain quite too many instances of irregularity. These include the following: (1) ablaut variation within the permissible grades, but without any indication of the reason for the variation; (2) cognation with words found in remote branches of the Indo-European family, but without any near congeners; (3) special phonetic laws, set up to explain some two or three examples, and so narrowly defined that they can apply to very few examples.

In the Conquête de Constantinople of Robert de Clary there occurs twice the expression de ne, used to introduce a negative question. The editor, M. Philippe Lauer, writes it as two words, probably considering it the preposition de plus the negative particle.