A MANUAL OF IMPROVED METHODS OF WORKING SOME OF INCLUDING NUMEROUS EXAMPLES IN PROPORTION, VULGAR PREFACE. A LONG course of practical experience, both as a teacher and as an examiner, has made the author of this little manual familiar with some general sources of imperfection and of failure in the higher arithmetical work of schools and public examinations, sources continuing to occasion the imperfection and the failure, notwithstanding the provision of so many well-planned volumes of Arithmetic, well stored with good examples. The present publication is an attempt not only to remove such imperfection, and to avert, as far as possible, such failure, but also to follow up with yet further improvement the important contributions of improvement which arithmetical method has from time to time received since the middle of the present century. A persuasion has for a considerable time existed and been acted upon, that questions such as are proposed under the Rule of Simple Proportion should always be solved by what is called the Unitary Method, rather than according to the long established method of arranging into 1st, 2nd, and 3rd terms, whence the title Rule of Three is derived. We dislike the title Rule of Three; for, as a descriptive phrase, it seems to indicate an artificial dogmatism in which the reasoning employed is merely mechanical. The Unitary Method is undoubtedly preferable as an intellectual and demonstrative system. While, however, we do not allow that the method of stating can without much inconvenience be quite discarded, we think the usual procedure of the Unitary Method occupies an amount of space which is often inconvenient; and we have endeavoured in the present work to make much less space contain equally rational and intelligible forms of solution for questions in Proportion and its various branches of application. In accordance with the title of this work we believe that the peculiar considerations and illustrations which we have introduced regarding Vulgar Fractions will be found very usefully supplementary to what is usually taught concerning that department of Arithmetic. The old rules of Alligation, with their artificial linking processes, have been very properly omitted from modern treatises; but since the author of this manual, in his 'Modern Arithmetic' (p. 154), called attention to an intellectual method of treating certain useful Alligation problems, questions of the kind have been frequently proposed in Examination Papers in Arithmetic. The subject is here much simplified in the chapter entitled Averaging of Rates. Problems relating to Scales of Notation have also of late years been frequently proposed in arithmetical Examination Papers, although the subject is seldom met with in treatises of Arithmetic,* being *An excellent chapter on Scales of Notation is given in Elsee's Arithmetic.' · usually committed to the care of Algebra. But it can be very easily taught by ordinary Arithmetic; and we have given it a place in this work not only as provision for test Examinations, but on account of its utility in promoting a proper understanding and appreciation of our ordinary decimal system of notation. For many of those students who are preparing for Civil Service or other public examination, the specimen we have given at p. 51 of a good method of presenting the work of a particular Paper is likely to be of considerable service. But we may here interpose two cautions, which our frequent inspection of the written work of candidates has shown to be in many instances very needful : : One of the most common faults in their work is a wrong use of the sign of equality: as when, in subtracting from the sum of and %, the work is done thus: 3+8=23-72=&c., which wrongly asserts the sum of and to be equal to the difference of and; the proper form of work is +2, hence 23-7-38; or better as follows: + = 36 = 36 4 5 7 16+30-21 25 9 6 I 2 Again: although when mixed fractions are given for multiplication or division, it is generally most convenient to reduce them to improper fractions, many candidates improperly think the same preparation to be expedient in addition and subtraction; but in this case such previous reduction makes very clumsy work: it is a waste of time, and often very irksome to an examiner. Let the student cultivate familiarity with the following better methods: |