257. These are the principal rules employed in the application of arithmetic to commerce. There are others, which, as no one who understands the principles here laid down can fail to see, are virtually contained in those which have been given. Such is what is commonly called the Rule of Exchange, for such questions as the following: If 20 shillings be worth 25½ francs, in France, what is £160 worth? This may evidently be done by the Rule of Three. The rules here given are those which are most useful in common life; and the student who understands them need not fear that any ordinary question will be above his reach. But no student must imagine that from this or any other book of arithmetic he will learn precisely the modes of operation which are best adapted to the wants of the particular kind of business in which his future life may be passed. There is no such thing as a set of rules which are at once most convenient for a butcher and a banker’s clerk, a grocer and an actuary, a farmer and a bill-broker; but a person with a good knowledge of the _principles_ laid down in this work, will be able to examine and meet his own future wants, or, at worst, to catch with readiness the manner in which those who have gone before him have done so for themselves. APPENDIX TO THE FIFTH EDITION OF DE MORGAN’S ELEMENTS OF ARITHMETIC. I. ON THE MODE OF COMPUTING. The rules in the preceding work are given in the usual form, and the examples are worked in the usual manner. But if the student really wish to become a ready computer, he should strictly follow the methods laid down in this Appendix; and he may depend upon it that he will thereby save himself trouble in the end, as well as acquire habits of quick and accurate calculation. I. In numeration learn to connect each primary decimal number, 10, 100, 1000, &c. not with the place in which the unit falls, but with the number of ciphers following. Call ten a _one-cipher_ number, a hundred a _two-cipher_ number, a million a _six-cipher_ number, and so on. If _five_ figures be cut off from a number, those that are left are hundred-thousands; for 100,000 is a _five-cipher_ number. Learn to connect tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, &c. with 1, 2, 3, 4, 5, 6, &c. in the mind. What is a _seventeen-cipher_ number? For every 6 in seventeen say _million_, for the remaining 5 say _hundred-thousand_: the answer is a hundred thousand millions of millions. If twelve places be cut off from the right of a number, what does the remaining number stand for?--_Answer_, As many millions of millions as there are units in it when standing by itself.