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THE

ART AND PRACTICAL APPLICATION

OF

ARITHMETIC

BY JOHN & THOMAS FLINT

NEW EDITION

GLASGOW

DAVID ROBERTSON, ST VINCENT STREET.
EDINBURGH OLIVER AND BOYD. LONDON: HOULSTON AND WRIGHT.

1802

MDCCCLXII.

566.6.195.

"The surest way for a learner, in this as in all other cases, is not to advance by jumps and large strides; let that which he sets himself to learn next be indeed the next-that is, as nearly conjoined with what he knows already as is possible; let it be distinct, but not remote from it; let it be new, and what he did not know before, that the understanding may advance; but let it be as little at once as may be, that its advances may be clear and sure. All the ground that it gets this way it will hold. This distinct gradual growth in knowledge is firm and sure; it carries its own light with it, in every step of its progression, in an easy and orderly train, than which there is nothing of more use to the understanding. And though this, perhaps, may seem a very slow and lingering way to knowledge, yet I dare confidently affirm, that whoever will try it in himself, or any one he will teach, shall find the advances greater in this method, than they would in the same space of time have been in any other he could have taken. The greatest part of true knowledge lies in a distinct perception of things in themselves distinct. And some men give more clear light and knowledge by the bare distinct stating of a question, than others by talking of it in gross whole hours together. In this, they who so state a question, do no more but separate and disentangle the parts of it one from another, and lay them, when so disentangled, in their due order. This often, without any more ado, resolves the doubt, and shows the mind where the truth lies."

"Nobody is made any thing by hearing of rules, or laying them up in his memory; practice must settle the habit of doing without reflecting on the rule."-Locke on the Conduct of the Understanding.

BODLEIAN LIBRARY

23.JUN. 1

OXFORD UNIVERSITY!

PREFACE.

CONSIDERING the primary object of all teaching to consist in preparing the pupil for the efficient discharge of those duties which are likely to devolve upon him in after life, and being convinced that, in the study of Arithmetic, the Art, and its Practical Applications, in this point of view, are entitled to claim the highest position, and to demand the first consideration of the Arithmetician, either as such merely, or in the higher capacity of Teacher; and seeing that they have not received the share of attention which was due to them, owing chiefly to the exaggerated im portance attached to the study of the Science, we have devoted ourselves entirely to the improvement of this branch of the subject, and the mode of teaching it. Accordingly, we have endeavoured to improve, in the first place, the Analysis of the Art, and to point out distinctly what are the real elements, to show the natural connection between them, and especially to indicate (principally, however, by example,) the perfect synthetical reconstruction, or the way by which the pupil may most easily be led from one step to another, until he is put in possession of the whole. We have also endeavoured to give method, and its proper place in the art of teaching to revision. The general principles which have guided us in the execution of this design are well expressed in the preceding quotations from Locke.

Nothing is so likely to produce in the Pupil's mind a permanent aversion to the study, as frequent failures; and there is nothing more likely to cause them, than a neglect of the fundamental principle of method, which consists in so graduating the exercises that the pupil is carried, without unnecessary trouble, by steps almost imperceptible, from what is undoubtedly the simplest, to that which is sufficiently difficult.

It is an essential principle, in the successful cultivation of every art, that the Pupil should ave acquired considerable facility in one exercise, before passing to another which is more difficult, and in none is its observance more necessary, or its neglect more prejudical, than in that of Arithmetic; for in it facility, in an advanced step, generally demands a corresponding facility in that which precedes it.

To secure the requisite facility, the exercises on each step of the art have been made very numerous.

Should any circumstance prevent a Teacher from affording his class the mental training involved in the Preparatory Exercises, he must remember that it is even more necessary in his case to adopt a system of

slate exercises, which, like the following, should be numerous, and in which each set should form a distinct but easy step in the art; because the importance of such exercises is obviously enhanced by there being no previous mental preparation, all being made to depend upon them.

Our mode of presenting Reduction, Compound Addition, Compound Multiplication, and various other parts, on the principle of analysis, will be found to be an improvement in the art of teaching; but we think the 44th, 45th, 46th, 47th, 48th, and 49th pages should not be studied until after the 65th.

The process to which we have given the name of Compound Reduction is, we believe, very much used by business men, though in a somewhat less orderly form.

The last section of the 59th page, the first of the 60th, the first two of the 63d, and the whole of the 95th, will be found to be useful and metho dical preparations for the calculation of Interest.

The 3d of the 63d page, the whole of the 72d, and the 1st nalt of the 73d, are very necessary preparations for Proportion. Proportion is the great difficulty of the practical art, and, at the same time the most important part of it; hence the analysis made of it at page 95. The answers to the Proportion are given down to the lowest denomination, as an exercise in Reduction and Division.

In conclusion, we beg leave to repeat, that we think the following Exercises will be found to be superior to those commonly given in elementary books, both in the analysis made of the subject, and in their nearer approach to the perfect synthetical progression, and, we may add, in the observance of method in the revisions.

In issuing a new edition of this work, it may be necessary for me to state that it was originally compiled by my late brother and myself for the use of our own pupils; and we have found that, without such a book, we could not have taught so efficiently in classes the large number of pupils which we have had under our care. The work has been found so essential, in the case of large classes especially, that I deemed it advisable to introduce the higher branches, so that the pupil might be furnished with a complete Arithmetical Text-Book. And if, after a trial, the additions made be found to be as useful as the first editions have proved, I have only to state, that to teachers having large numbers the Book simply requires to be known to secure its introduction to general use. Most of the calculations have been thoroughly verified in the actual work of instruction; but any hints as to inaccuracies or defects will meet with careful consideration.

JOHN FLINT.

GLASGOW, November, 1861.

ARITHMETICAL EXERCISES.

COUNTING TABLE,

In which the digits are added, by construction, successively to every number between 1 and 100, in the form of continued additions of the same digit.

FIRST SERIES BY One.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22, &c. to 100

SECOND SERIES BY Two.

1357 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

THIRD SERIES BY Three.

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54

FOURTH SERIES BY Four.

}

&c. to 100

&c. to 100

15 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90

3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92

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