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TKe Library of

CollegeOntario ofTKe

Education

TorontoTKe University of

by

James Amossn,v %1

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LOVELL'S SERIES OF SCHOOL BOOKS.

ELEMENTS

ALGEBRA;

DESIGNED FOR THE USE OF

CAHADIAH GRAMMAR AND COMMON SCHOOLS,

HERBERT SANGSTER,BY JOHN M.A.,

AND LECTURER INMATHEMATICAL MASTER CHEMISTRY AND NATDRAI,

IN THE NORMAL SCHOOL FOR UPl'ERPmLOSOPHY CANADA.

IHoutvcat

PRINTED AND PUBLISHED BY JOHN LOVELL,

AND SOLD BY K. MILLER.

ADAM MILLER, 62 KING STREET EAST.

1864.according to tlie Act of the Provincial Pjirlianient, iaBntered,

tlie year one thousand eight hundred and sixty-four, by

Lov?:Lr,, in the OfScc of the RegistrarJohn of the Province

of Canada.

____

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ti"68 ninth256 four'hPREFACE.

respectfullyfollowing Treatise is submitted by theThe

teachers of Canuda, in the confident beliefauthor to the

lighten the labor of the instructor,that it will materially

thesame time, facilitate pupil's progress andand, at the

comprehension of the principles of the sciencehis thorough

algebra. It is the earnest hope of the author that itof

same flattering reception,may meet with the and very

general introduction into the schools of the country, that

kindly accordedhis fellow-teachers have so to his previous

productions.

of the differentThe order of succession chapters depends

mainly on their importance and difficulty, andof course

one that appearsthat here adopted is the preferable to the

but, as every chapter is nearly independent of theauthor ;

easily modify theothers, the teacher can arrangement to

suit himself.

toThe aim of the work is embrace all that can be pro-

in the time usually -allottedfitably discussed to a comnic-n

grammar school course; and, indeed, this volume willand

at least much the as isbe found to contain as of subject,

for the ordinary degree of B. A. inrequired to be read

British and Canadian Universities. Chapters on con-the

series, probabilities, andtinued fractions, logarithmicIV PREFACE.

the general theory of eciuutions were prepared, but, in

accordance with the advice of some oi' the leadin*^ educators

of the province, they were omitted as unsuited to tlie de.siti;n

of tothe work, and the requii emonts of cduimon or gram-

mar school.*.

The author has approaclied the subject with the cou-

viction, founded on many years' experience as a teacher of

mathematics, that the science of algebra tries, beyond all

others, the powers and patience of the learner. The ])upil

is commonly introduced to it while his mind is yet in an

undeveloped state its

; language is new to him, and he is

unprepared by previous training to comprehend its

abstractions. The difficulties which thus beset his path

are, of course, for the most part, only to be overcome by

his perseverance,own aided by the knowledge and ingen-

uity of his instructor, yet it appears to the author that

alsovery much depends upon the style and thoroughness

and adaptation of the text-book employed. Accordingly

in the preparatiou of this volume no pains have been spared

rendering the statementof principles,in and the demonstra-

tion of theorems as clear and concise as possible, or iu

illustrating each rule numerousfully by examples carefully

worked out and explained, or in selecting and arranging the

examples of an exercise so as to begin with the simple,

and gradually pass on to the more difficult.

The author hopes that while he has insisted upon

great thoroughness by numerous and appropriate problems,

at the same time, rendered pupil'she has, the advancement

easy and certain by the many explanations and illustra-

tions introduced.

The great majority of the problems and exercises are

new,—being now published for the first time, but there arePKEFACE. V

familiar to the teacher. In select-number alreadyalso a

in every case rigidlyauthor has, he believes,those theino;

by Todhunter, Colenso, andthe rule, adoptedadhered to

alreadya problem unless it hadof not insertingothers,

which case itleast two British authors—inappeared in at

property.regarded as commonis to be

of the pupils of ourthe fact that very manyKecognizing

schools study with the view of com-and grammarcommon

of our excellent Cana-education at some onepleting their

author has, at the end of the book,universities, thedian

embracingcollection of problems and theorems,introduced a

the pass and honor workothers all or nearly all ofamong

examination papershas been given on thein algebra which

the last eight or tenuniversity of Toronto duringof the

shew the pupil the style ofyears. These will serve to

universities,expected to answer at ourquestions he is

measure prepare him forwill, at the same time, in aand

his examinations.

I

introducing his pupils toteacher would think ofAs no

some extent at least, first drillingarithmetic without, to

intelligent teachernotation and numeration, so notliem in

drill his pupils in algebraic notation andwill neglect to

before inti'oducing them to the ordinary rules.numeration

referred to exercises ii, iii,The teacher is re.«pectfully

is recommended to extend and continue theseand iv, and

acquainteduntil his pupil is thoroughly and practically

definitions.with the

Well knowing the great inconvenience to both teacher

mistakes in a work onand pupils of inaccuracies and

algebra, the author has subjected this treatise to a searching

markedrevision and he believes that the few corrections

;

on the back of the title page are the only errors in thePREFACE.VI

exercises and anHWors of the workletter-press of the

is respectfully recommended to cause hisThe teaeher

eitjht trifling alterations therepupils to make the six or

in the body of the work with pen and ink.indicated

more difficultA key, containing full solutions to all the

press and will be issued almost immediatelyproblems, is in

1864.Toronto, January,