The first two views are in the text. The third is deduced thus: If we divide 9 into 9 equal parts, each is 1, and 7 of the parts are 7; consequently the fraction which 7 is of 9 is ⁷/₉. The fourth view follows immediately: For _a time_ is only a word used to express one of the repetitions which take place in multiplication, and we allow ourselves, by an easy extension of language, to speak of a portion of a number as being that number taken a _part of a time_. The fifth view is nothing more than a change of words: A number reduced to ⁷/₉ of its amount has every 9 converted into a 7, and any fraction of a 9 which may remain over into the corresponding fraction of 7. This is completely proved when we prove the equation ⁷/₉ of _a_ = 7 times _a_/9. The sixth, seventh, and eighth views are illustrated in the chapter on proportion. When the student comes to algebra, he will find that, in all the applications of that science, fractions such as _a_/_b_ most frequently require that _a_ and _b_ should be themselves supposed to be fractions. It is, therefore, of importance that he should learn to accommodate his views of a fraction to this more complicated case. 2½ Suppose we take -----. 4³/₅ We shall find that we have, in this case, a better idea of the views from and after the third inclusive, than of the first and second, which are certainly the most simple ways of conceiving ⁷/₉. We have no notion of the (4³/₅)th part of 2½, 1 ( 3 ) nor of 2 ---(4---)ths 2 ( 5 ) of a unit; indeed, we coin a new species of adjective when we talk of the (4³/₅)th part of anything. But we can readily imagine that 2½ is some fraction of 4³/₅; that the first is _some_ part of a time the second; that there must be _some_ multiplier which turns every 4³/₅ in a number into 2½; and so on. Let us now see whether we can invent a distinct mode of applying the first and second views to such a compound fraction as the above. We can easily imagine a fourth part of a length, and a fifth part, meaning the lines of which 4 and 5 make up the length in question; and there is also in existence a length of which four lengths and two-fifths of a length make up the original length in question. For instance, we might say that 6, 6, 2 is a division of 14 into 2⅓ equal parts--2 equal parts, 6, 6, and a third of a part, 2. So we might agree to say, that the (2⅓)th, or (2⅓)rd, or (2⅓)st (the reader may coin the adjective as he pleases) part of 14 is 6. If we divide the line A B into eleven equal parts in C, D, E, &c., we must then say that A C is the 11th part, [Illustration]