Charles Davies.

# Key to Davies' Bourdon, with many additional examples, illustrating the algebraic analysis online

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LIBRARY

OF THE

University of California.

GIFT OF

s r

.SJ^Q'.,A^\.AJU<ys^-.

Class

^^yj^

KEY

TO

DAVIES' BOUEDON.

WITH

MANY ADDITIONAL EXAMPLES, ILLUSTRATING
THE ALGEBRAIC ANALYSIS.

By CHARLES DAVIES, LL.D..

AUTHOR OF A FULL COURSE OF MATHEMATICS,

Vfr^,.

- I_

A. S. BARNES & COMPANY,

NEW TOEK, CHICAGO AND NEW OELEANS.

NATIONAL SERIES OF MATHEMATICS,

BY
CHAS. DAVIES, LL.D.

I>AVIES' PRIMARY ARITHMETIC \$ .15

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*** I'he forepfoing are the prices at the Publishers' office. Any volume will be
f jrwarJed by mail, postpaid, on receipt of its price, with one-fourth additional for
postage and mailing fee.

♦ A Key will be furnished at the same price of book.

Copyright, 185'', by Chas. Davies, LL.D.
Copyright, 1879, by A. S. Barnes & Co.

PREFACE.

A wiDS difference of opinion is known to exist among teachers
in regard to the value of a Key to any mathematical work, and
it is perhaps yet undecided whether a Key is a help or a
hindrance.

If a Key is designed to supersede the necessity of investigfw
tion and labor on the part of the teacher; to present to his
mind every combination of thought which ought to be suggested
by a problem, and to permit him to float sluggishly along the
current of ideas developed by the author, it would certainly do
great harm, and should be excluded from every school.

If, on the contrary, a Key is so constructed as to suggest
ideas, both in regard to particular questions and general science,
which the Text-book might not impart; if it develops methods
of solution too particular or to-, elaborate to find a place in the
test ; if it is mainly designed to lessen the mechanical labor of
teaching, rather than the labor of study and investigation ; i*.
may, in the hands of a good teacher, prove a valuable auxiliary.

The Key to Bourdon is intended to answer, precisely, thia

-I Q O /? O o

IT PREFACE.

end. The principles developed in the text are explained and
Ulustrated by means of numerous examples, and these are all
wrought in the Key by methods which accord with and make
evident the principles themselves. The Key, therefore, not only
explains the various questions, but is a commentary on the text
itself.

Nothing is more gratifying to an ambitious teacher than to
push forward the investigations of his pupils beyond the limits
of the text book. To aid him hi an undertaking so useful to
himself and to them, an Appendix has been added, containing a
copious collection of Practical Examples. Mauy of the solutions
are quite curious and in.-tructive ; and taken in connection with
those embraced in the Text, form a full and complete system of
Algebraic Analysis.

The many letters which I have received from Teachers and
Pupils, in regard to the best solutions of new questions, have
suggested the desirableness of furnishing, in the present work,
those which have been most approved. They are a collection
of problems that have special values, and their solutions may be
studied with great profit by every one seeking mathematical
knowledge.

FisHKiLL Landing, )
July, 1S73. \

INTRODUCTION.

ALGEBRA.

1. On an analysis of the subject of Algebra, we Aig«b«.
think it will appear that the subject itself presents no DifficuUi«.
serious difficulties, and that most of the embarrassment
which is experienced by the pupil in gaining a knowl- "°^J7"
edge of its principles, as well as in their applications,
arises from not attending sufficiently to the lanyuage Langua«<^
or signs of the thoughts which are combined in the
reasonings. At the hazard, therefore, of being a little
diffuse, I shall begin with the very elements of the
algebraic language, and explain, with much minute-
ness, the exact signification of tlie characters that stand

Characters
which repre-

for the quantities which are the subjects of the analy- sent quantity

sis ; and also of those signs which indicate the several
operations to be performed on the quantities.

Signs.

2. The quantities which are the subjects of the Huantitie*
algebraic analysis may be divided into two classes : ^°^ divid«a
those which are known or given, and those which are
unknown or sought. The known are uniformily repre-
sented by the first letters of the alphabet, «, b, c, rf,
&c. ; and the unknown by the final letters, r, y, 2,
t', w. &c.

How repre-
sented.

6

mTEODUCTION.

May be in-
creased or
diminisheJ.

Five opera-
tions

Quantity is susceptible of "being increased, di-
minished, and measured ; and there are six operations
which can be performed upon a quantity that will
give results differing from the quantity itself, viz.:

1st. To add it to itself or to some other quantity;

2d. To subtract some other quantity from it;

3d. To multiply it by a number;

4th. To divide it;

5th. To raise it to any power;

6th. To extract a root of it.

The cases in which the multiplier or divisor is 1,
are of course excepted; as also the case where a
root is to be extracted of 1.

4. The six signs which denote these operations
Elements ^^.^ ^^^ ^,gjj j^jiq^^jj {q ^g repeated here. These, with

of the ^ "

Algebr^iic the signs of equality and inequality, tlie letters of the

language.

alphabet and the figures which are employed, make up
ts words and ji,g elements of the algebraic Janfruace. The words

phrases :

and phrases of the algebraic, like those of every
How inter- other lanfjuaofe, are to be taken in connection with

preted.

each other, and are not to be interpreted as separate
and isolated symbols

First

Becand
Third.

Fourth
Fifth

Exception.
Signs.

Symbols of
quantity.

5. The symbols of quantity are designed to repre-
sent quantity in general, whether abstract or concrete,
whether known or unknown ; and the signs which in-
dicate the operations to be performed on the quanti-
ties are to be interpreted in a sense equally general.
When the sign plus is written, it indicate* that the
eigns pins and quantity before which it is placed is to be added to
some other quantity : and the sign minus implies the

General.

Esample?.

INTROPrCTION. <

existence of a minuend, from which the subtrahend is

to be taken. One thing should be observed in regard Signs have n*

effect on tho

to the signs which indicate the operations that are to nature of

t 7 I 1 I! ^ quantity.

be performed on quantities, viz. : thev do not at all
affict or change the nature of the quantity before or
lifter 'which they are tvritten, hnt merely indicate u'hat
is to be done ivith the quantity. In Algebra, for ex- i^xamplet:

In Algebra.

ample, the minus sign merely indicates that the quan-
tity before which it is written is to be subtracted from
some other quantity; and in Anal)tical Geometry, that in Analytical

Geometry.

the line before which it falls is estimated in a contrary
direction from that in which it would have been reck-
oned, had it had the sign plus ; but in neither case is
the nature of the quantity itself different from what
it would have been had the sign been plus.

The interpretation of the language of Algebra is Tnterpretat.on

of the

the first thing to which the attention of a pupil should u.iguag*

be directed ; and he should be drilled on the meaning

and import of the symbols, until their significations

and uses are as familiar as the sounds and combina- '*' "«<=««^"'

tions of the letters of the alphabet.

6 Beo-inning with the elements of the language, lUt^menu

ex;<lai.-iP.J

let any number or quantity be designated by the letter

«, and let it be required to add this letter to it.-elf

and find the result or sum. The additiDu will be

expressed by

a -\- a ^z the sum.

But how is the sum to be expressed? By simply Significatie.
reo'ardina a as one a, or la, and then observing that
one a and one a, make tico aV or 2a : hence.

S INTRODUCTION.

a -f- a = 2a; .

and thus we place a figure before a letter to indicate
how many times it is taken. Such figure is called a
Co-effieient Co-effi,cienL

Pwduct: 7. The product of several numbers is indicated

by the sign of multiplication, or by simply writing the
letters which represent the numbers by the side of
each other. Thus,

Iaw indicated

Factor.

ay^bxcxdxf,

or

abcdf.

indicates the continued product of cr, h, c, d, and _/,
and each letter is called a factor of the product :
hence, a factor of a product is one of the multipliers
which produce it. Any figure, as 5, written before a

product, as

oabcdf,

Co-efficient of is the co-eflScient of the product, and shows that the
• P' ^'^ • product is taken 5 times.

E;iiai factors : g^ jf the numbers represented by a, b, c, d, and
»hatthe /, were equal to each other, they would each be
wmel represented by a single letter a, and the product
would then become

How
•xpr«utc

axaxaxaxa = a~;

that is, we indicate the product of several equal fao
tors by simply writing the letter once and placing a
figure above and a Tnle at the right )f it, to indicate

INTRODUCTION.

9

how many times it is taken as a factor. The figure Exponent :
60 written is called an exponent. Hence, on exponent where writun.
\g a simple form of langituge to point mtt how many
equal factors are employed.

9. The division of one quantity by another is indi-
cated by simply writing the divisor below the dividend,
after the manner of a fraction ; by placing it on the
right of the dividend with a horizontal line and two
dots between them ; or by placing it on the right with
a vertical line between them : thus either form of
expression :

DiTiiioB :

how
expresEed

a

a,

or

Three fonna

mdicates the division of b by a.

10. The extraction of a root is indicated by the Roots

sign -y/. This sign, when used by itself indicates the how indicate
lowest root, viz., the square root. If any other root
is to be extracted, as the third, fourth, fifth, &c., the index;

figure marking the degree of the root is written above where written
and at the left of the sign ; as,

^/ cube root, \/ ^fourth root, &c.

The figure so written, is called the Index of the root.
"We have thus given the very simple and general
language by which we indicate each of the five
operations that may be performed on an algebraic
quantity, and every process iit Algebra involves one or
otlu'r of these operations.

Language fsi

the five
opera tian*

10 INTRODUCTION. .

MINUSSIGN.

Algebraic ^j rpj^^ alsrc^braic symbols are divided into two

language ■ ' ^ J

classes entirely distinct from each otlier — viz., the

kow diTiJe: letters that are used to designate the quantities which

are the subjects of the science, and the signs v.hich

are employed to indicate certain operations to be per-

Algebraic formed on those quantities. We have seen that all

processes : ...

the algebraic processes are comprised under addition,
ihei: number, subtraction, multiplication, division, and the extraction
Do not change of roots ; and it is plain, that the nature of a quan-
;he nature of ^. j^ ^^^ ^^ ^^^ changed by prefixing to it the sign

I'le quantitiee. •' o .- a ^^

which indicates either of these operations. The quan-
tity denoted by the letter n, for example, is the same,
in everi/ respect, whatever sign may be prefixed to it ;
that is, whether it be added to another quantity, sub-
tracted from it, whether multiplied or divided by any
number, or whether we extract the square or cube or
Ai ebraic ^ny Other root of it. The algebraic signs, therefore,
"pns: j^jjjj^ j^y regarded merely as indicating opert./lo)ts to

how regarded. _

be performed on quantity, and not as affecting the

nature of the quantities to which they may be prefixed.

Plus and ^,^[e say, indeed, that quantities are plus and minus,

' '°"'' but this is an abbreviated language to express that

they are to be added or subtracted.

I'rinciples of |2_ j^ Algebra, as in Arithmetic and Geometry

tll6 8ClGnC€ *

FromwhM all tlie principles of the science are deduced from tnf
deduced ^j^fi,,iti(,„s and axioms; and the rules for performing

the operations are but directions framed in conformifj
Fxample to such principles. Having, for example, fixed b;

definition, the power of the minus sign, viz., that an;

INTRODUCTION. 11

(jiiamily before which it is written, shall be regarded

a^ to be subtracted from another quantity, we wish to whatwewak

to discover.

discover the process of performing that subtraction, so
a.- ro deduce therefrom a general formula, from which
V. c can fram(; a rule applicable to all similar cases.

SUBTRACTION.

13. Let it be required, for example, to subtract Subtractiom,

from h the difference between a and c. \ .

I b

Now, having written the letters, with j Process.

' ^ \ a — c

their proper signs, the language of Al- ' -

gebra expresses that it is the difference only between

(/ and c, which is to be taken from h ; and if this dif- Diirerenc«.

ference were known, we could make the subtraction at

once. But the nature and generality of the algebraic

symbols, enable us to indicute operations, merely, and Operations

indicated.

we cannot in general make reductions until we come
to the final result. In what genei-al way, therefore,
can we indicate the true difference ?

If we indicate the subtraction of a
from h, we have b — a.; but then we
have taken away too much from b by
the number of units in c;for it was not a, but the dif-
ference between a and c that was to be subtracted
from b. Having taken away too much, the i-emainder
1; too entail by c : hence, if c be added, the true re-
mainder will be expressed by b — a -\- c.

b — a

Final formula
b — a -{• C

Now, by analyzing this result, we see that the sign Analysis l

the result.

of every term of the subtrahend has been changed ;
and what has been si own with respect to these quan-

12

INTRODUCTION.

Oeneraliza-
tion.

Slil«

titles IS equally true of all others standing in the ?ame
relation : hence, we have the following general rule
for the subtraction of algebraic quantities ;

Change the sign of every term tf the stibtrahendy or
conceive it to be changed, and then vnite the quantities
as in addition.

MULTIPLICATION.

Mnltiphca- |^^ j^gj ^g j^^^ consider the case of raultiplicatioTi,

tion.

and let it be required to multiply a — b by c. Th*»
algebraic language expresses that the
Signification difference between a and b is to be

of ths

taken as many times as there are

language.

Process.

a -b
c
ac — be

lt> nature.

Priaeipls for
tk* signs.

units in c. If we knew this differ-
ence, we could at once perform the multiplication.
But by what general process is it to be performed
without finding that difference ? If we fake a, c times,
the product will be ac ; but as it was only the differ-
ence between a and b, that was to be multiplied by i\
this product ac will be too great by b taken c times ;
that is, the true product will be expressed by ac — be-.
hence, we see, that,

If a quantity having a plus Sign be multiplied by
another quantity having also a plus sign^ the sign of
the product will be plus ; and if a quantity having a
minus sign be multiplied by a quantity having a plus
sign, the sign of the product will be minus.

6*aeta,\caM»: 15. Let us^ now take the most general ca.se, viz.,
that in which it is required to multipy a — b by c - d.

INTRODUCTION.

13

a

~h

c

-d

ac

-be

— ad -\- hd

ac

— be — ad -\- bd

\\M form.

Lei US again observe that the algebraic language
'lenotes that a — h is to be ta-
ker as many times as there
are units in c — d ; and if these
two differences were known,
their product would at once
form the product required.

First : let us take a — b as First step.

many times as there are units in c ; this product,
from what has already been shown, is equal to ac — be.
But since the multiplier is not c, but c — d, it follows
that this product is too large by a — b taken d times ;
that is, by ad — bd \ hence, the first product dimin- Second step:
ished by this last, will give the true product. But, by
the rule for subtraction, this difference is found by How tuken.
changing the signs of the subtrahend, and then uniting
all the terms as in addition : hence, the true product
is expressed by ac — be — ud -{- bd.

By analyzing this result, and employing an abbre- Analysis oi

. xho result.

viated language, we have the following general prin-
ciple to which the signs conform in multiplication, viz. :

Plus mulliplied by plus gives plus in the product ; General

VrincipU.

plus multiplied by minus gives minus; minus mul-
tiplied by plus gives mimis ; and minus multiplied by
minus gives plus in the product.

16. The remark is often made by pupils that the R«maA.
above reasoning appears very satisfactory so long as
the quantities are presented under the above form ;

. Particnlar

but why will — !> mulliplied by — - give plus 'id * ^a^e

14 INTKUDUCTION.

How can the product of two negative quantities stand-

mg alone be plus ?
MiauBsign In the first place, the minus sign being prefixed to

/ and d, shows that in an algvhraic sense they do not

ta interpre- ^'tand by themselves, but are connected with other quan-

'°° tities ; and if they are not so connected, the minus

sign makes no difference ; for, it in no case affects the

quaatity, but merely points out a connection with other

quantities. Besides, the product determined a^ove.

being independent of any particular value attributed

Foim jf the ^^ ^he letters a, b, c, and d, must be of such a form as

product : ^^ ^g ^^.^^^ j-^^. j^jj y^lues ; and hence for the case in

must be irue

forquantiiiei which « and c are each equal to zero. Making this

of any value

supposition, the product reduces to the form of 4- bJ.
Signs in The rulcs for the signs in division are readily deduced

division. , -, „ . . „,... ,. ... ,

trom the definition ot division, and the principles al-
ready laid down.

ZERO AND INFINITY.

Zero and 17. The tcrms zero and infinity have given rise to

"''y- much discussion, and been regarded as presenting diffi-
culties not easily removed. It may not be easy to
frame a form of language that shall convey to a mind,

Ideas not but little versed in mathematical science, the preci.>-e
ideas which these terms are designed to express ; bu(
we are unwilling to suppose that the ideas themselvei
are beyond the grasp of an ordinary intellect. The
terms are used to designate the two limils of Space
and Number.

18. Assuming any two points in space, and joining

INTRODUCTION. 15

them by a straight line, the distance between the points
will be truly indicated by the length of this line, and
this length may be expressed numerically by the num-
ber of times which the line contains a known unit. If
nDw, the points are made to approach each other, the ii'.usttation,

showing lh«

length of the line will diminish as the points come meaning of
nearer and nearer together, until at length, when the ^ *™
two points become one, the length of the line will
disappear, having attained its limit, which is called
^ero. If, on the contrary, the points recede from each
other, the length of the line joining them will con-
tinually increase ; but so long as the length of the lUustraiicn,

. showing the

line can be expressed in terms of a known unit ot me.nmgof

. f • , -r» i -c ii the term

measure, it is not infinite. But, if we suppose the ,„fi„j,j.
points removed, so that any known unit of measure
would occupy no appreciable portion of the line, then
the length of the line is said to be Infinite.

19. Assuming one as the unit of number, and ad-
mitting the self-evident truth that it may be increased
or diminished, we shall have no difficulty in under-
standing the import of the terms zero and infinity. The term*

Zero and In-

as applied to number. For, if we suppose the unit fmity applied

,,,...,,, ,. . . ,1 to numberg.

one to be continually diminished, by division or other-
wise, the fractional units thus arising will be less and niustratioa.
less, and in proportion as we continue the divisions,
they will continue to diminish. Now, the limit or
boundary to which these very small fractions approach,
is called Zero, or notliing. So long as the fractional Zwo :
number forms an appreciable part of one, it is not
zero, but a finite fiaction : and the term zero is only

^.^

16

tUnstr&tion.

Infinity ;

The terms,

how
employed.

Xtu limits.

Why limits?

INTIIODUCTION.

Definition of
a limit.

Of Space and
Number.

Subject of

equatious :

how divided.

First part:

applicable to that which forms no appreciable part of
the standard.

If, on the other liand, we suppose a number to be
continually increased, the relation of this number to the
unit will be constantly changing. So long as the num-
ber can be expressed in terms of the unit one, it ia
finite, and not infinite ; but when the unit one forms
no appreciable part of the number, the term infinite
is used to express that state of value, or rather, that
limit of value.

20. The terras zero and infinity are therefore em-
ployed to designate the limits to which decreasing and
increasing quantities may be made to approach nearer
than any assignable quantity; but these limits cannot
be compared, in respect to magnitude, with any known
standard, so as to give a finite ratio.

21. It may, perhaps, appear somewhat paradoxical,
that zero and infinity should be defined as '• the limits
of number and space" when they are in themselves
not measurable. But a limit is that " which sets bounds
to, or circumscribes ;" and as all finite space and finite
number (and such only are implied by the terms Space
and Number), are contained between zero and infinity,
we employ these terms to designate the limits of Num-
ber and Space.

OP THE EQUATION.

22. The subject of equations is divided into two
parts. The first, consists in finding the equation ; that
is, in the process of expressing the relations existing'

INTRODUCTION.

17

Solution

Discussion ol
an equation

Statement :
what it ia.

between the quantities considered, by means of the
algebraic symbols and formula. This is called the
Statement of the proposition. The second is purely s-atomsor
deductive, and consists, in Algebra, in what is called " "" ^^
the solution of the equation, or finding the value of
the unknown quantity ; and in the other branches of
analysis, it consists in the discussion of the equation ;
that is, in the drawing out from the equation every
proposition which it is capable of expressing.

23. Making the statement, or finding the equation,
is merely analyzing the problem, and expressing its
elements and their relations in the language of analy-
sis. It is, in truth, collating the facts, noting their
bearing and connection, and inferring some general
law or principle which leads to the formation of an
equation.

The condition of equality between two quantities ^^quaiity of

111. „ tw quanti-

IS expressed by the sign of equality, which is placed ties:

between them. The quantity on the left of the si<^n ^°^ ^"^

'^ pressed.

of equality is called the first member, and that on i>t .nembe,
the right, the second member of the equation. Hence, -'' "lember
an equation is merely a proposition expressed aWe- I'ropositioo
braically, in which equahty is predicated of one quan-
tity as compared with another It is the ffreat formula
!jf Alsrebra.

24. Every quantity is either abstract or concrete :
hence, an equation which is a general formula for
expressing equality, must be either abstract or con-
crete.

2

Abstract.
Zoncteie.

18

INTRODUCTION.

Abstract
•qnation.

Online LibraryCharles DaviesKey to Davies' Bourdon, with many additional examples, illustrating the algebraic analysis → online text (page 1 of 11)