The practice and theory of arithmetic

THE

PRACTICE AND THEORY

OF

ARITHMETIC,

CONTAINING

DEFINITIONS OF TERMS, AND RULES OF OPERATIONS,

With Examples of Methods,

AND THE PRINCIPLES EXPLAINED IN PROPOSITIONS,

THE WHOLE FORMING

A COMPLETE TREATISE

ON

PRACTICAL AND THEORETICAL ARITHMETIC.

BY THE

REV. W. F. GREENFIELD, M. A.

LATE SCHOLAR OF PEMBROKE COLLEGE, CAMBRIDGE,

SECOND AND MATHEMATICAL MASTER IN QUEEN ELIZABETH'S SCHOOL, IPSWICH.

LONDON:

LONGMAN AND CO. PATERNOSTER ROW:

IPSWICH:

PAWSEY, OLD BUTTER MARKET.

18. c. 24.

PREFACE.

The writer of the present work believes that all persons will agree in allowing that the ends, which ought to be kept in view in teaching any branch of Science, are these:-1st, That Principles be well understood; 2nd, That Definitions of terms be known; 3rd, That Rules for the application of the Principles to Practice be learnt; 4th, That neat methods of working be adopted; 5th, That a facility in expressing by writing the Principles, and the reasoning by which the Rules are established, be acquired. The first is necessary, because no Science can be said to be taught, unless a knowledge and comprehension of Principles be secured : the second is necessary, because, if definitions are not known, all reasoning on the subject will be as it were expressed in an unknown language: the third is required, because the main object of all Science is Practical utility, and it cannot be expected that on all occasions printed Rules can be before the eye: the fourth is desirable, if not necessary, in the same degree as a neat method of writing is so: the fifth will appear an object of great importance, if it be considered that the best method of testing a person's knowledge of a subject is by requiring him to write his ideas upon it; if he understand it, he will have no difficulty in expressing himself correctly; but if he do not, he will certainly shew the fact by his defective reasoning. Now Definitions and Rules can only be learnt accurately from books, because they require to be expressed in precise terms in order to fix themselves in the memory; neat methods of working too must be shewn to the eye; but Principles cannot be learnt thoroughly from a book, without oral explanations, those contained in a book being necessarily concise, and such as themselves require elucidation and illustration: on the other hand, it is found that Principles may very well be taught, by a competent teacher, without a book, by repeated oral explanations, by illustrations, and by continual interrogations put to the learner in every variety of form. And that this is by far the better method of teaching will appear on these grounds, because it may be thereby ensured that no advance is made from one step to another without every previous step being understood, and because a fuller explanation is made compulsory on the part of the teacher, than he would be inclined to give, if any part could be supplied by a book, and because young pupils are very apt to overlook all the explanatory

portion of a book, being anxious for the most part to come as quickly, as may be, to the actual work of the subject, which is always more attractive, and generally pays best in a school. For these reasons it appears that oral teaching is preferable; nor will it occupy any longer time of the teacher, for teaching by book must be accompanied by oral teaching to be of any use, and this must be full and complete to be effective; if it be not full, the teacher will find that he will have to do his work over and over again, and that his pupils will make but slow and imperfect progress. And further, in a large school where many are taught in a class, it frequently happens that it is assumed that all understand a Principle, which one can explain according to a book, which all have given them to study, whereas the assumption ought certainly to be, that no one understands anything at all about it.

From what has preceded, it will have been seen that, in the opinion of the writer, the best mode of teaching any science in a school is by orally explaining, illustrating, and questioning upon, the principles; by supplying definitions and rules to be learnt by heart; by placing before the pupils neat methods of setting down their work; and by requiring them to explain the reasons of the processes vivâ voce and by writing.

The first portion of the present work contains the Rules and Definitions of Arithmetic, with examples of processes worked out in full, and in the precise form in which they ought to be shewn to a Master. The second portion of the work contains the Principles, exhibited in Propositions, as specimens of the form in which the pupil should give written explanations of processes. The former of these portions is all that is required in elementary classes, the second will furnish a useful exercise for more advanced pupils, and will serve as a good introduction to the "bookwork" of Algebra, and containing, as it does, a gradual and complete explanation of principles, will be found to be an important assistant both to the student and to the teacher.

The writer has found, that in most works on Arithmetic the Rules and Definitions are spread over many pages of explanation, and in many instances not distinctly or connectedly given, so that the pupil has to spend time in seeking for them, and in connecting them. He has therefore exhibited every Rule and Definition distinctly and connectedly, in such a form as may be easily committed to memory; and it is hoped that the two portions of the work will be found to contain all that is necessary for imparting an accurate knowledge of both the Practice and Principles of Arithmetic in the hands of a competent instructor.