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ON A PLAN PECULIARLY CALCULATED TO
FACILITATE THE IMPROVEMENT OF THE PUPIL, AND REDUCE THE
LABOUR OF THE TEACHER;
TO WHICH IS ADDED, AN IMPROVED RULE FOR THE
EXTRACTION OF THE CUBE ROOT,
BY WHICH THE OPERATION IS PERFORMED
As easily, accurately, and expeditiously, as that of the Square Root ;
EXERCISES IN MENTAL CALCULATION.
BY W. 'WATSON,
AND MAY BE HAD OF ALL BOOKSELLERS.
(Price only 25. strongly bound in Sheep, or 2s. 6d. in Roan.)
RECENTLY PUBLISHED, BY THE SAME AUTHOR, Second Edition, much improved and enlarged, price Three Shillings, neatly bound,
AN EASY AND COMPREHENSIVE
INTRODUCTION TO ALGEBRA,
DESIGNED FOR THE USE OF SCHOOLS;
With plain and familiar Examples, and numerous Notes and Observations,
intended as Aids to Private Students.
ADVERTISEMENT TO THE SECOND EDITION. “ The slightly altered arrangement, together with the great extension of ele. mentary Exercises, in the present edition, have both been suggested by increased experience in Algebraical teaching.
“ The various cases of Equations have also been re-modelled and much enlarged, besides the addition of an entire new case of Pure Equations.
“ The Diophantine department too has likewise been very much extended, and several new forms added; so that now it is not only the most complete, but the most extensive popular treatise extant.
“ Thronghout the whole work it has been the Author's aim to make each department copious, without being redundant; easy, without being simple; and smart, without being recondite. How far he has succeeded, is left for the judicious and candid public to determine.
“ W. WATSON." Beverley, June, 1844.
[ENTERED AT STATIONERS' HALL.]
PREFACE TO THE FIRST EDITION.
In no department of science, for the last half century, has the press teemed with a greater number of publications, than on the subject of Arithmetic :-almost every succeeding writer has discovered, in the imperfections of former works, a sufficient motive for presenting a new treatise to the public.
Without disparaging the merits of others, to the Author of the following pages there still appears room for improvement. Comparatively few are even tolerably calculated for the purposes of tuition, especially in Seminaries for young ladies and Classical Academies, where the time devoted to Arithmetic is necessarily limited. In the works alluded to, the prevailing fault appears to be the want of simplicity; even the Questions comprising the leading Exercises are often involved and intricate, and so obscurely expressed as to battle the attempts of a youthful inquirer fully to comprehend them; this obseurity tends to deter from inquiry, and check the mental efforts of young beginners. In the following treatise, the Author has en. deavoured to avoid these imperfections: the Questions are nearly all new, and their arrangement and gradation such, as to enable the Pupil to prosecute his studies with but very little help from his teacher; Examples are given to illustrate the operation of each Rule, and to every two preceding elementary Rules, Exercises are annexed, so arranged as to encourage thought, and habituate the mind to comparison.
For the convenience of Teachers, all the Examples in Simple and Compound Addition are so contrived, that the Sum of each is equal to three times the last line; and those of the Exercises are placed beneath each Question. Experience has convinced the Author of the utility of the latter provision to the learner, who is encouraged to new exertions, by seeing that he has solved a question correctly. The doubt and uncertainty inseparable from the plan of withholding the results is avoided, together with the inevitable consequences of mental weariness and aversion.
In order to render the present Treatise an acceptable and useful addition to the library of the Working Mechanic, it is hoped the Rules of Practical Mensuration, the great number of Exercises, and Explanatory Notes, with a familiar description of the Sliding Rule, the construction of its lines, its application to the solution of Arithmetical Questions, Timber Measure, &c. will not be deemed out of place.
To promote the advancement of the Pupil, the Author would recommend, as of paramount importance, frequent catechetical examinations and cyphering in class ; the former, although it affects inquiry only, tends to impress the rules on the memory, and affords the Tutor a sort of intellectual burometer; the latter, by exciting emulation, elicits the powers of the mind, and thus greatly accelerates the process of calculation.
ADVERTISEMENT TO THE FOURTH EDITION. The present edition has been carefully revised and corrected, and such additions made as the Author trusts will be deemed an improvement. With this object in view, he has complied with the expressed wishes of several eminent Teachers, by adding a number of Geometrical Problems, which renders the Mensuration unique and complete.
W. WATSON. Bererley, January, 1845,
CHARACTERS USED IN ARITHMETIC.
+ Plus, or more. The sign of Addition.
Minus, or less. The sign of Subtraction. X Into, or Multiplied by. The sign of Multiplication. • Divided by.
The sign of Division. = Equal to. T'he sign of Equality; as, 4+2=6. : is to :: so is :
The sign of Proportion. V
The sign of the Square Root. Thus v5 or 57 are expressions for the square root of 5, and x 5 5$ are expressions for the cube root of 5.
Also, 5 is the cube root of the square of 5. And 52, 53, and 54, is the square, cube, and biquadrate of 5. .. Ergo, or therefore. > Angle.
DEFINITIONS. 1. A prime number, is one that has no divisor between itself and unity.
2. A composite number, is one that has two or more divisors, commonly called factors.
3. A perfect number, is equal to the sum of all its quotients.
4. The figures or digits by which all Arithmetical operations are performed are 1,2,3,4,5,6,7,8,9,0; the last is called a cypher or naught, and is of no value when it stands alone, but when put to the right of any integral number increases its value ten-fold.
5. The unit, or first digit, can neither multiply nor divide.
6. A common measure between two or more numbers, is a number that will divide them all without a remainder.
7. Multiple, a number is said to be a multiple of a less number, when it contains it a certain number of times.
8. Numbers are said to be prime to each other when they admit not of a common divisor: and if one be divided by any number, the quotient is the reciprocal of that number