of giving precedence to the larger.  In German and other Teutonic languages the inverse method is continued still further.  Here 25 is fuenf und zwanzig, 5 and 20; 92 is zwei und neunzig, 2 and 90, and so on to 99.  Above 100 the order is made direct, as in English.  Of course, this mode of formation between 20 and 100 is permissible in English, where “five and twenty” is just as correct a form as twenty-five.  But it is archaic, and would soon pass out of the language altogether, were it not for the influence of some of the older writings which have had a strong influence in preserving for us many of older and more essentially Saxon forms of expression.

Both the methods described above are found in all parts of the world, but what I have called the direct is far more common than the other.  In general, where the smaller number precedes the larger it signifies multiplication instead of addition.  Thus, when we say “thirty,” i.e. three-ten, we mean 3 x 10; just as “three hundred” means 3 x 100.  When the larger precedes the smaller, we must usually understand addition.  But to both these rules there are very many exceptions.  Among higher numbers the inverse order is very rarely used; though even here an occasional exception is found.  The Taensa Indians, for example, place the smaller numbers before the larger, no matter how far their scale may extend.  To say 1881 they make a complete inversion of our own order, beginning with 1 and ending with 1000.  Their full numeral for this is yeha av wabki mar-u-wab mar-u-haki, which means, literally, 1 + 80 + 100 x 8 + 100 x 10.[54] Such exceptions are, however, quite rare.

One other method of combination, that of subtraction, remains to be considered.  Every student of Latin will recall at once the duodeviginti, 2 from 20, and undeviginti, 1 from 20, which in that language are the regular forms of expression for 18 and 19.  At first they seem decidedly odd; but familiarity soon accustoms one to them, and they cease entirely to attract any special attention.  This principle of subtraction, which, in the formation of numeral words, is quite foreign to the genius of English, is still of such common occurrence in other languages that the Latin examples just given cease to be solitary instances.

The origin of numerals of this class is to be found in the idea of reference, not necessarily to the last, but to the nearest, halting-point in the scale.  Many tribes seem to regard 9 as “almost 10,” and to give it a name which conveys this thought.  In the Mississaga, one of the numerous Algonquin languages, we have, for example, the word cangaswi, “incomplete 10,” for 9.[55] In the Kwakiutl of British Columbia, 8 as well as 9 is formed in this way; these two numbers being matlguanatl, 10 — 2, and nanema, 10 — 1, respectively.[56] In many of the languages of British Columbia we find a similar formation for 8 and 9, or for 9 alone.  The same formation occurs