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THE PRINCIPLES OF ARITHMETIC.

OPINIONS OF THE PRESS.

“Teachers and others who wish to grasp the fundamental principles on which the various rules of Arithmetic rest, may be greatly assisted by this text-book, which explains the reasons of every operation, from the simplest to the most complicated, and treats of every branch of the subject."-Athenæum. Sui generis

as an exposition of the rationale of Arithmetic, it stands alone."-Irish Teachers' Journal.

An admirable work original in method and treatment.”School Board Chronicle.

“Eminently explanatory."-Daily Review.
“Scholarly ... thoroughly exhaustive.”-Schoolmaster.
“Greatly needed."-National Schoolmaster.

“We have never met any work so well calculated to impart a correct knowledge of the principles of the science.”—Morning and Evening Mail.

“Stands at the head of all works upon Arithmetic that we have seen."— Irish Times.

“We must congratulate Dr. O'Sullivan on having produced a very able and useful work.”Daily Express.

“The principles laid down, and the method adopted by the author, have the combined merit of novelty and of scientific precision, as well as of clearness and systematic arrangement."-Londonderry Standard. “A luminous work

most comprehensive and philosophical.”-Cork Examiner.

“The work of a man who is thoroughly master of his subject, and who knows how to treat it .. destined to become our standard treatise.”Freeman's Journal.

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BY THE SAME AUTHOR.

THE PRACTICE OF ARITHMETIC:

A COMPANION VOLUME

(IN TWO PARTS)

TO

THE PRINCIPLES OF ARITHMETIC.

OPINIONS OF THE PRESS.

“We can warmly congratulate Professor O'Sullivan on the completion of a work which, taken in conjunction with his PRINCIPLES, forms unquestionably the most valuable treatise on Arithmetic published in the English tongue." -Irish Teachers' Journal.

“Will furnish abundant practice in every branch of Arithmetic."--Athencum.

“Graduated with great skill, so as to bring forward the pupil steadily from the simplest exercises to the most difficult that could be set by an Examining Board.”—Cork Examiner.

“The work is marked by conspicuous ability in its original conception, as well as by rare ingenuity and fertility of resources in its illustrations and general details.”Daily Express.

“Professor O'Sullivan's immense educational experience has enabled him to exactly adapt his work to the needs and capacities of all classes of students. His language is always accurate and (what is of no less importance) clear. Nobody who makes a diligent use of these books can fail to become an intelligent and expert Arithmetician.”Morning and Evening Mail.

“The most exhaustive treatise on Arithmetic ever published.”— Educational Journal.

Sanctioned by the Commissioners of National Education, Ireland.

THE

PRINCIPLES

OF

A Ꭱ Ꭵ Ꭲ Ꮋ Ꮇ Ꭼ T I C:

Comprebensive Text-Book

FOR THE USE OF

TEACHERS AND ADVANCED PUPILS.

BY
D. O'SULLIVAN, Ph.D., M.R.I.A.;
ONE OF THE PROFESSORS IN THE TRAINING DEPARTMENT OF THE

COMMISSIONERS OF NATIONAL EDUCATION, IRELAND;
EXAMINER IN ARITHMETIC TO THE COMMISSIONERS OF INTERMEDIATE

EDUCATION, IRELAND;

ETC.

EIGHTH EDITION-REVISED AND ENLARGED.

DUBLIN:
M. H. GILL & SON, SACKVILLE-STREET;
SULLIVAN, BROTHERS, MARLBOROUGH-STREET.

LONDON: SIMPKIN, MARSHALL, & CO.
EDINBURGH: JOHN MENZIES & CO.

1883.

[All Rights Reserved.]

1802.6.lt

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“Arithmetic is the branch of pure science which specially belongs to the people. It is the compass of their household economy: their resources and their expenditure, their gains and their losses, are balanced and measured by it : luxury and enjoyment are tempered, abstemiousness and self-denial promoted, by the homely deductions it enables them to make: wages and food, taxes and rent, are felt and understood by it: time is noted-the span of the past and of the future estimated : in short, it is to the poor the first coalescence of the impalpable with the material, the first and most general subjection of their intellect to a process of abstract reasoning."-SIR PATRICK J. KEENAN. K.C.M.G., C.B., &c.

PREFACE TO THE FIRST EDITION.

This book, which is detailed as well as comprehensive, and which is the result of a long and varied experience, will, I hope, enable persons of ordinary capacity and intelligence to acquire for themselves, by diligent and patient study, a sound knowledge of the rationale of Arithmetic. The want of such a treatise has in numerous instances been complained of by the Irish National Teachers, with whomas Inspector of National Schools, as Superintendent of the Central Model Schools, and as Professor-I have for many years been officially connected.

In the following pages Arithmetic is reduced to a series of distinct “Principles," which—for facility of reference—are printed in large type, and numbered. The student is asked to take nothing for granted. Every Principle is logically established, and is moreover illustrated by a sufficient number of carefullyprepared examples.

In the arrangement and treatment of the Principles I have exercised my own judgment. Many of the demonstrations are original; and some of them, in which the employment of figures is supplemented by that of general symbols, may be acceptable to the Algebraist as well as to the Arithmetician.

D. O'S. DUBLIN,

August, 1872.

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