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,
a d the (5½)th, a e the (3⅔)th, a f the (2¾)th, a g the (2⅕)th, a h the (1⅚)th, a i the (1⁴/₇)th, a k the (1⅜)th, a l the (1²/₉)th, a m the (1⅒)th, and a b itself the 1st part of a b. The reader may refuse the language if he likes (though it is not so much in defiance of etymology as talking of multiplying by ½); but when a b is called 1, he must either call a f 1/(2¾), or make one definition of one class of fractions and another of another. Whatever abbreviations they may choose, all persons will agree that a/b is a direction to find such a fraction as, repeated b times, will give 1, and then to take that fraction a times.
So, to get 2½/4⅗, the simplest way is to divide the whole unit into 46 parts; 10 of these parts, repeated 4⅗ times, give the whole. The
4⅗th is then ¹⁰/₄₆, and 2½ such parts is ²⁵/₄₆, or a c. The student should try several examples of this mode of interpreting complex fractions.
But what are we to say when the denominator itself is less than unity, as in 3¼/⅖? Are we to have a (⅖)th part of a unit? and what is it? Had there been a 5 in the denominator, we should have taken the part of which 5 will make a unit. As there is ⅖ in the denominator, we must take the part of which ⅖ will be a unit. That part is larger than a unit; it is 2½ units; 2½ is that of which ⅖ is 1. The above fraction then directs us to repeat 2½ units 3¼ times. By