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 [Illustration] 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 extending our word ‘multiplication’ to the taking of a part of a time, all multiplications are also divisions, and all divisions multiplications, and all the terms connected with either are subject to be applied to the results of the other. If 2⅓ yards cost 3½ shillings, how much does one yard cost? In such a case as this, the student looks at a more simple question. If 5 yards cost 10 shillings, he sees that each yard costs ¹⁰/₅, or 2 shillings, and, concluding that the same process will give the true result when the data are fractional, he forms 3½/2⅓, reduces it by rules to ³/₂ or 1½, and concludes that 1 yard costs 18 pence. The answer happens to be correct; but he is not to suppose that this rule of copying for fractions whatever is seen to be true of integers is one which requires no demonstration. In the above question we want money which, repeated 2⅓ times, shall give 3½ shillings. If we divide the shilling into 14 equal parts, 6 of these parts repeated 2⅓ times give the shilling. To get 3½ times as much by the same repetition, we must take 3½ of these 6 parts at each step, or 21 parts. Hence, ²¹/₁₄, or 1½, is the number of shillings in the price. APPENDIX V. ON CHARACTERISTICS. When the student comes to use logarithms, he will find what follows very useful. In the mean while, I give it merely as furnishing a rapid rule for finding the place of a decimal point in the quotient before the division is commenced.