% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
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% The Project Gutenberg EBook of On the Study and Difficulties of Mathematics, by
% Augustus De Morgan %
% %
% This eBook is for the use of anyone anywhere at no cost and with %
% almost no restrictions whatsoever. You may copy it, give it away or %
% re-use it under the terms of the Project Gutenberg License included %
% with this eBook or online at www.gutenberg.org %
% %
% %
% Title: On the Study and Difficulties of Mathematics %
% %
% Author: Augustus De Morgan %
% %
% Release Date: February 17, 2013 [EBook #39088] %
% Most recently updated: June 11, 2021 %
% %
% Language: English %
% %
% Character set encoding: UTF-8 %
% %
% *** START OF THIS PROJECT GUTENBERG EBOOK STUDY AND DIFFICULTIES OF MATHEMATICS ***
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The Project Gutenberg EBook of On the Study and Difficulties of Mathematics, by
Augustus De Morgan
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org
Title: On the Study and Difficulties of Mathematics
Author: Augustus De Morgan
Release Date: February 17, 2013 [EBook #39088]
Most recently updated: June 11, 2021
Language: English
Character set encoding: UTF-8
*** START OF THIS PROJECT GUTENBERG EBOOK STUDY AND DIFFICULTIES OF MATHEMATICS ***
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Produced by Andrew D. Hwang. (This ebook was produced using
OCR text generously provided by the University of
California, Berkeley, through the Internet Archive.)
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ON THE
\vfill
{\Large STUDY AND DIFFICULTIES OF \\[4pt]
MATHEMATICS}
\vfill\vfill
{\footnotesize BY}
AUGUSTUS DE~MORGAN
\vfill\vfill\vfill
{\footnotesize THIRD REPRINT EDITION}
\vfill\vfill\vfill
CHICAGO \\
{\small THE OPEN COURT PUBLISHING COMPANY} \\
{\footnotesize LONDON \\
\textsc{Kegan Paul. Trench, Trübner \&~Co., Ltd.} \\
1910}
\end{center}
\PageSep{ii}
%[Blank page]
\PageSep{iii}
\Section[EdNote]{Editor's Note.}
\First{No} apology is needed for the publication of the present new
edition of \Title{The Study and Difficulties of Mathematics},---a
characteristic production of one of the most eminent and luminous
of English mathematical writers of the present century. De~Morgan,
though taking higher rank as an original inquirer than
either Huxley or Tyndall, was the peer and lineal precursor of
these great expositors of science, and he applied to his lifelong task
an historical equipment and a psychological insight which have
not yet borne their full educational fruit. And nowhere have these
distinguished qualities been displayed to greater advantage than in
the present work, which was conceived and written with the full
natural freedom, and with all the fire, of youthful genius. For the
contents and purpose of the book the reader may be referred to
the Author's Preface. The work still contains points (notable
among them is its insistence on the study of logic), which are insufficiently
emphasised, or slurred, by elementary treatises; while
the freshness and naturalness of its point of view contrasts strongly
with the mechanical character of the common text-books. Elementary
instructors and students cannot fail to profit by the general
loftiness of its tone and the sound tenor of its instructions.
The original treatise, which was published by the Society for
the Diffusion of Useful Knowledge and bears the date of~1831, is
now practically inaccessible, and is marred by numerous errata
and typographical solecisms, from which, it is hoped, the present
edition is free. References to the remaining mathematical text-books
of the Society for the Diffusion of Useful Knowledge now
\PageSep{iv}
out of print have either been omitted or supplemented by the mention
of more modern works. The few notes which have been
added are mainly bibliographical in character, and refer, for instance,
to modern treatises on logic, algebra, the philosophy of
mathematics, and pangeometry. For the portrait and autograph
signature of De~Morgan, which graces the page opposite the title,
The Open Court Publishing Company is indebted to the courtesy
of Principal David Eugene Smith, of the State Normal School at
\index{Smith, D. E.}%
Brockport, N.~Y\@.
\Signature{Thomas J. McCormack}
{\textsc{La~Salle}, Ill., Nov.~1, 1898.}
\PageSep{v}
\cleardoublepage
\Section[Preface]{Author's Preface.}
\First{In} compiling the following pages, my object has been to notice
particularly several points in the principles of algebra and
geometry, which have not obtained their due importance in our
elementary works on these sciences. There are two classes of men
who might be benefited by a work of this kind, viz., teachers of
the elements, who have hitherto confined their pupils to the working
of rules, without demonstration, and students, who, having
acquired some knowledge under this system, find their further
progress checked by the insufficiency of their previous methods
and attainments. To such it must be an irksome task to recommence
their studies entirely; I have therefore placed before them,
by itself, the part which has been omitted in their mathematical
education, presuming throughout in my reader such a knowledge
of the rules of algebra, and the theorems of Euclid, as is usually
obtained in schools.
It is needless to say that those who have the advantage of
University education will not find more in this treatise than a little
thought would enable them to collect from the best works now in
use~[1831], both at Cambridge and Oxford. Nor do I pretend to
settle the many disputed points on which I have necessarily been
obliged to treat. The perusal of the opinions of an individual,
offered simply as such, may excite many to become inquirers, who
would otherwise have been workers of rules and followers of dogmas.
They may not ultimately coincide in the views promulgated
by the work which first drew their attention, but the benefit which
they will derive from it is not the less on that account. I am not,
\PageSep{vi}
however, responsible for the contents of this treatise, further than
for the manner in which they are presented, as most of the opinions
here maintained have been found in the writings of eminent
mathematicians.
It has been my endeavor to avoid entering into the purely
metaphysical part of the difficulties of algebra. The student is, in
my opinion, little the better for such discussions, though he may
derive such conviction of the truth of results by deduction from
particular cases, as no \textit{à~priori} reasoning can give to a beginner.
In treating, therefore, on the negative sign, on impossible quantities,
and on fractions of the form~$\frac{0}{0}$, etc., I have followed the
method adopted by several of the most esteemed continental writers,
of referring the explanation to some particular problem, and
showing how to gain the same from any other. Those who admit
such expressions as $-a$, $\sqrt{-a}$, $\frac{0}{0}$, etc., have never produced any
clearer method; while those who call them absurdities, and would
reject them altogether, must, I think, be forced to admit the fact
that in algebra the different species of contradictions in problems
are attended with distinct absurdities, resulting from them as
necessarily as different numerical results from different numerical
data. This being granted, the whole of the ninth chapter of this
work may be considered as an inquiry into the nature of the different
misconceptions, which give rise to the various expressions
above alluded to. To this view of the question I have leaned,
finding no other so satisfactory to my own mind.
The number of mathematical students, increased as it has
been of late years, would be much augmented if those who hold
the highest rank in science would condescend to give more effective
assistance in clearing the elements of the difficulties which they
present. If any one claiming that title should think my attempt
obscure or erroneous, he must share the blame with me, since it is
through his neglect that I have been enabled to avail myself of an
opportunity to perform a task which I would gladly have seen confided
to more skilful hands.
\Signature{Augustus De~Morgan.}{}
\PageSep{vii}
\TableofContents
\iffalse
CONTENTS.
CHAPTER PAGE
Editor's Note iii
Author's Preface v
I. Introductory Remarks on the Nature and Objects of
Mathematics 1
II. On Arithmetical Notation 11
III. Elementary Rules of Arithmetic 20
IV. Arithmetical Fractions 30
V. Decimal Fractions 42
VI. Algebraical Notation and Principles 55
VII. Elementary Rules of Algebra 67
VIII. Equations of the First Degree 90
IX. On the Negative Sign, etc 103
X. Equations of the Second Degree 129
XI. On Roots in General, and Logarithms 158
XII. On the Study of Algebra 175
XIII. On the Definitions of Geometry 191
XIV. On Geometrical Reasoning 203
XV. On Axioms 231
XVI. On Proportion 240
XVII. Application of Algebra to the Measurement of Lines,
Angles, Proportion of Figures, and Surfaces 266
\fi
\PageSep{viii}
%[Blank page]
\MainMatter
\PageSep{1}
\index{Exponents.|See Indices.}%
\index{Symbols@{Symbols|See Signs}.}%
\Chapter[Nature and Objects of Mathematics.]
{I.}{Introductory Remarks on the Nature and
Objects of Mathematics.}
\index{Mathematics!nature, object, and utility of the study of|EtSeq}%
\First{The Object} of this Treatise is---(1)~To point
out to the student of Mathematics, who has not
the advantage of a tutor, the course of study which it
is most advisable that he should follow, the extent to
which he should pursue one part of the science before
he commences another, and to direct him as to the
sort of applications which he should make. (2)~To
treat fully of the various points which involve difficulties
and which are apt to be misunderstood by beginners,
and to describe at length the nature without
going into the routine of the operations.
No person commences the study of mathematics
without soon discovering that it is of a very different
nature from those to which he has been accustomed.
The pursuits to which the mind is usually directed before
entering on the sciences of algebra and geometry,
are such as languages and history, etc. Of these,
neither appears to have any affinity with mathematics;
\PageSep{2}
yet, in order to see the difference which exists between
these studies,---for instance, history and geometry,---it
will be useful to ask how we come by knowledge
in each. Suppose, for example, we feel certain
of a fact related in history, such as the murder of
Cæsar, whence did we derive the certainty? how came
\index{Caesar@{Cæsar}}%
we to feel sure of the general truth of the circumstances
of the narrative? The ready answer to this
question will be, that we have not absolute certainty
upon this point; but that we have the relation of historians,
men of credit, who lived and published their
accounts in the very time of which they write; that
succeeding ages have received those accounts as true,
and that succeeding historians have backed them with
a mass of circumstantial evidence which makes it the
most improbable thing in the world that the account,
or any material part of it, should be false. This is
perfectly correct, nor can there be the slightest objection
to believing the whole narration upon such
grounds; nay, our minds are so constituted, that,
upon our knowledge of these arguments, we cannot
help believing, in spite of ourselves. But this brings
us to the point to which we wish to come; we believe
that Cæsar was assassinated by Brutus and his friends,
not because there is any absurdity in supposing the
contrary, since every one must allow that there is just
a possibility that the event never happened: not because
we can show that it must necessarily have been
that, at a particular day, at a particular place, a successful
\PageSep{3}
adventurer must have been murdered in the
manner described, but because our evidence of the
fact is such, that, if we apply the notions of evidence
which every-day experience justifies us in entertaining,
we feel that the improbability of the contrary
compels us to take refuge in the belief of the fact;
and, if we allow that there is still a possibility of its
falsehood, it is because this supposition does not involve
absolute absurdity, but only extreme improbability.
In mathematics the case is wholly different. It is
true that the facts asserted in these sciences are of a
nature totally distinct from those of history; so much
so, that a comparison of the evidence of the two may
almost excite a smile. But if it be remembered that
acute reasoners, in every branch of learning, have
acknowledged the use, we might almost say the necessity,
of a mathematical education, it must be admitted
that the points of connexion between these pursuits
and others are worth attending to. They are the more
so, because there is a mistake into which several have
fallen, and have deceived others, and perhaps themselves,
by clothing some false reasoning in what they
called a mathematical dress, imagining that, by the
application of mathematical symbols to their subject,
they secured mathematical argument. This could not
have happened if they had possessed a knowledge of
the bounds within which the empire of mathematics
is contained. That empire is sufficiently wide, and
\PageSep{4}
might have been better known, had the time which
has been wasted in aggressions upon the domains of
others, been spent in exploring the immense tracts
which are yet untrodden.
We have said that the nature of mathematical demonstration
\index{Demonstration!mathematical@{mathematical|EtSeq}}%
is totally different from all other, and the
difference consists in this---that, instead of showing
the contrary of the proposition asserted to be only improbable,
it proves it at once to be absurd and impossible.
This is done by showing that the contrary of
the proposition which is asserted is in direct contradiction
to some extremely evident fact, of the truth of
which our eyes and hands convince us. In geometry,
\index{Geometry!study of|EtSeq}%
of the principles alluded to, those which are most
commonly used are---
I. If a magnitude be divided into parts, the whole
is greater than either of those parts.
II. Two straight lines cannot inclose a space.
III. Through one point only one straight line can
be drawn, which never meets another straight line, or
which is \emph{parallel} to it.
It is on such principles as these that the whole of
geometry is founded, and the demonstration of every
proposition consists in proving the contrary of it to be
inconsistent with one of these. Thus, in Euclid, Book~I.,
\index{Euclid}%
Prop.~4, it is shown that two triangles which have
two sides and the included angle respectively equal
are equal in all respects, by proving that, if they are
not equal, two straight lines will inclose a space, which
\PageSep{5}
is impossible. In other treatises on geometry, the
same thing is proved in the same way, only the self-evident
truth asserted sometimes differs in form from
that of Euclid, but may be deduced from it, thus---
Two straight lines which pass through the same
two points must either inclose a space, or coincide
and be one and the same line, but they cannot inclose
a space, therefore they must coincide. Either of these
propositions being granted, the other follows immediately;
it is, therefore, immaterial which of them we
use. We shall return to this subject in treating
specially of the first principles of geometry.
Such being the nature of mathematical demonstration,
what we have before asserted is evident, that
our assurance of a geometrical truth is of a nature
wholly distinct from that which we can by any means
obtain of a fact in history or an asserted truth of metaphysics.
In reality, our senses are our first mathematical
instructors; they furnish us with notions
which we cannot trace any further or represent in any
other way than by using single words, which every
one understands. Of this nature are the ideas to
which we attach the terms number, one, two, three,
etc., point, straight line, surface; all of which, let
them be ever so much explained, can never be made
any clearer than they are already to a child of ten
years old.
But, besides this, our senses also furnish us with
the means of reasoning on the things which we call
\PageSep{6}
by these names, in the shape of incontrovertible propositions,
such as have been already cited, on which,
if any remark is made by the beginner in mathematics,
it will probably be, that from such absurd truisms
as ``the whole is greater than its part,'' no useful result
can possibly be derived, and that we might as
well expect to make use of ``two and two make four.''
This observation, which is common enough in the
mouths of those who are commencing geometry, is
the result of a little pride, which does not quite like
the humble operation of beginning at the beginning,
and is rather shocked at being supposed to want such
elementary information. But it is wanted, nevertheless;
the lowest steps of a ladder are as useful as the
highest. Now, the most common reflection on the
nature of the propositions referred to will convince us
of their truth. But they must be presented to the understanding,
and reflected on by it, since, simple as
they are, it must be a mind of a very superior cast
which could by itself embody these axioms, and proceed
from them only one step in the road pointed out
in any treatise on geometry.
But, although there is no study which presents so
simple a beginning as that of geometry, there is none
in which difficulties grow more rapidly as we proceed,
and what may appear at first rather paradoxical, the
more acute the student the more serious will the impediments
in the way of his progress appear. This
necessarily follows in a science which consists of reasoning
\PageSep{7}
from the very commencement, for it is evident
that every student will feel a claim to have his objections
answered, not by authority, but by argument,
and that the intelligent student will perceive more
readily than another the force of an objection and the
obscurity arising from an unexplained difficulty, as
the greater is the ordinary light the more will occasional
darkness be felt. To remove some of these
difficulties is the principal object of this Treatise.
We shall now make a few remarks on the advantages
to be derived from the study of mathematics,
considered both as a discipline for the mind and a key
to the attainment of other sciences. It is admitted by
all that a finished or even a competent reasoner is not
the work of nature alone; the experience of every day
makes it evident that education develops faculties
which would otherwise never have manifested their
existence. It is, therefore, as necessary to \emph{learn to
reason} before we can expect to be able to reason, as it
is to learn to swim or fence, in order to attain either
of those arts. Now, something must be reasoned
upon, it matters not much what it is, provided that it
can be reasoned upon with certainty. The properties
of mind or matter, or the study of languages, mathematics,
or natural history, may be chosen for this purpose.
Now, of all these, it is desirable to choose the
one which admits of the reasoning being verified, that
is, in which we can find out by other means, such as
measurement and ocular demonstration of all sorts,
\PageSep{8}
whether the results are true or not. When the guiding
property of the loadstone was first ascertained,
and it was necessary to learn how to use this new discovery,
and to find out how far it might be relied on,
it would have been thought advisable to make many
passages between ports that were well known before
attempting a voyage of discovery. So it is with our
reasoning faculties: it is desirable that their powers
should be exerted upon objects of such a nature, that
we can tell by other means whether the results which
we obtain are true or false, and this before it is safe
to trust entirely to reason. Now the mathematics are
peculiarly well adapted for this purpose, on the following
grounds:
1. Every term is distinctly explained, and has but
one meaning, and it is rarely that two words are employed
to mean the same thing.
2. The first principles are self-evident, and, though
derived from observation, do not require more of it
than has been made by children in general.
3. The demonstration is strictly logical, taking
nothing for granted except the self-evident first principles,
resting nothing upon probability, and entirely
independent of authority and opinion.
4. When the conclusion is attained by reasoning,
its truth or falsehood can be ascertained, in geometry
by actual measurement, in algebra by common arithmetical
calculation. This gives confidence, and is
\PageSep{9}
absolutely necessary, if, as was said before, reason is
not to be the instructor, but the pupil.
5. There are no words whose meanings are so
much alike that the ideas which they stand for may
be confounded. Between the meanings of terms there
is no distinction, except a total distinction, and all
adjectives and adverbs expressing difference of degrees
are avoided. Thus it may be necessary to say,
``$A$~is greater than~$B$;'' but it is entirely unimportant
whether $A$~is very little or very much greater than~$B$.
Any proposition which includes the foregoing assertion
will prove its conclusion generally, that is, for all
cases in which $A$~is greater than~$B$, whether the difference
be great or little. Locke mentions the distinctness
\index{Locke}%
of mathematical terms, and says in illustration:
``The idea of two is as distinct from the idea of
three as the magnitude of the whole earth is from
that of a mite. This is not so in other simple modes,
in which it is not so easy, nor perhaps possible for us
to distinguish between two approaching ideas, which
yet are really different; for who will undertake to
find a difference between the white of this paper,
and that of the next degree to it?''
These are the principal grounds on which, in our
opinion, the utility of mathematical studies may be
shown to rest, as a discipline for the reasoning powers.
But the habits of mind which these studies have
a tendency to form are valuable in the highest degree.
The most important of all is the power of concentrating
\PageSep{10}
the ideas which a successful study of them increases
where it did exist, and creates where it did
not. A difficult position, or a new method of passing
from one proposition to another, arrests all the attention
and forces the united faculties to use their utmost
exertions. The habit of mind thus formed soon extends
itself to other pursuits, and is beneficially felt
in all the business of life.
As a key to the attainment of other sciences, the
use of the mathematics is too well known to make it
necessary that we should dwell on this topic. In fact,
there is not in this country any disposition to under-value
them as regards the utility of their applications.
But though they are now generally considered as a
part, and a necessary one, of a liberal education, the
views which are still taken of them as a part of education
by a large proportion of the community are
still very confined.
The elements of mathematics usually taught are
contained in the sciences of arithmetic, algebra, geometry,
and trigonometry. We have used these four divisions
because they are generally adopted, though,
in fact, algebra and geometry are the only two of them
which are really distinct. Of these we shall commence
with arithmetic, and take the others in succession in
the order in which we have arranged them.
\PageSep{11}
\Chapter{II.}{On Arithmetical Notation.}
\index{Arithmetical!notation|EtSeq}%
\index{Notation!arithmetical, decimal|EtSeq}%
\First{THE} first ideas of arithmetic, as well as those of
other sciences, are derived from early observation.
How they come into the mind it is unnecessary
to inquire; nor is it possible to define what we mean
by number and quantity. They are terms so simple,
that is, the ideas which they stand for are so completely
the first ideas of our mind, that it is impossible
to find others more simple, by which we may explain
them. This is what is meant by defining a term; and
here we may say a few words on definitions in general,
which will apply equally to all sciences.
Definition is the explaining a term by means of
\index{Definition}%
others, which are more easily understood, and thereby
fixing its meaning, so that it may be distinctly seen
what it does imply, as well as what it does not. Great
care must be taken that the definition itself is not a
tacit assumption of some fact or other which ought to
be proved. Thus, when it is said that a square is ``a
four-sided figure, all whose sides are equal, and all
\PageSep{12}
whose angles are right angles,'' though no more is
said than is true of a square, yet more is said than is
necessary to define it, because it can be proved that
if a four-sided figure have all its sides equal, and one
only of its angles a right angle, all the other angles
must be right angles also. Therefore, in making the
above definition, we do, in fact, affirm that which
ought to be proved. Again, the above definition,
though redundant in one point, is, strictly speaking,
defective in another, for it omits to state whether the
sides of the figure are straight lines or curves. It
should be, ``a square is a four-sided rectilinear figure,
all of whose sides are equal, and one of whose angles
is a right angle.''
As the mathematical sciences owe much, if not all,
of the superiority of their demonstrations to the precision
with which the terms are defined, it is most essential
that the beginner should see clearly in what a
good definition consists. We have seen that there
are terms which cannot be defined, such as number
and quantity. An attempt at a definition would only
throw a difficulty in the student's way, which is already
done in geometry by the attempts at an explanation
of the terms point, straight line, and others, which
\index{Straight line}%
are to be found in treatises on that subject. A point is
defined to be that ``which has no parts, and which
has no magnitude\Chg{'';}{;''} a straight line is that which
``lies evenly between its extreme points.'' Now, let
any one ask himself whether he could have guessed
\PageSep{13}
what was meant, if, before he began geometry, any
one had talked to him of ``that which has no parts
and which has no magnitude,'' and ``the line which
lies evenly between its extreme points,'' unless he had
at the same time mentioned the words ``point'' and
``straight line,'' which would have removed the difficulty?
In this case the explanation is a great deal
harder than the term to be explained, which must
always happen whenever we are guilty of the absurdity
of attempting to make the simplest ideas yet more
simple.
A knowledge of our method of reckoning, and of
\index{Counting|EtSeq}%
\index{Reckoning|EtSeq}%
writing down numbers, is taught so early, that the
method by which we began is hardly recollected.
Few, therefore, reflect upon the very commencement
of arithmetic, or upon the simplicity and elegance
with which calculations are conducted. We find the
method of reckoning by ten in our hands, we hardly
know how, and we conclude, so natural and obvious
does it seem, that it came with our language, and is
\index{Language}%
a part of it; and that we are not much indebted to
instruction for so simple a gift. It has been well observed,
that if the whole earth spoke the same language,
we should think that the name of any object
was not a mere sign \emph{chosen} to represent it, but was a
sound which had some real connexion with the thing;
and that we should laugh at, and perhaps persecute,
any one who asserted that any other sound would do
as well if we chose to think so. We cannot fall into
\PageSep{14}
this error, because, as it is, we happen to know that
what we call by the sound ``horse,'' the Romans distinguished
as well by that of ``\textit{equus},'' but we commit
a similar mistake with regard to our system of numeration,
because at present it happens to be received
by all civilised nations, and we do not reflect on what
was done formerly by almost all the world, and is done
still by savages. The following considerations will,
perhaps, put this matter on a right footing, and show
that in our ideas of arithmetic we have not altogether
rid ourselves of the tendency to attach ideas of mysticism
\index{Mysticism in numbers}%
to numbers which has prevailed so extensively
in all times.
We know that we have nine signs to stand for the
\index{Decimal!system of numeration|EtSeq}%
\index{Numeration, systems of|EtSeq}%
first nine numbers, and one for nothing, or zero. Also,
that to represent ten we do not use a new sign, but
combine two of the others, and denote it by~$10$, eleven
by~$11$, and so on. But why was the number \emph{ten} chosen
as the limit of our separate symbols---why not nine,
eight, or eleven? If we recollect how apt we are to
count on the fingers, we shall be at no loss to see the
reason. We can imagine our system of numeration
formed thus:---A man proceeds to count a number,
and to help the memory he puts a finger on the table
for each one which he counts. He can thus go as far
as ten, after which he must begin again, and by reckoning
the fingers a second time he will have counted
twenty, and so on. But this is not enough; he must
also reckon the number of times which he has done
\PageSep{15}
this, and as by counting on the fingers he has divided
the things which he is counting into lots of ten each,
he may consider each lot as a unit of its kind, just as
we say a number of sheep is \emph{one} flock, twenty shillings
are \emph{one} pound. Call each lot a \emph{ten}. In this way he
can count a ten of tens, which he may call a hundred,
a ten of hundreds, or a thousand, and so on. The
process of reckoning would then be as follows:---Suppose,
to choose an example, a number of faggots is to
be counted. They are first tied up in bundles of ten
each, until there are not so many as ten left. Suppose
there are seven over. We then count the bundles of
ten as we counted the single faggots, and tie them up
also by tens, forming new bundles of one hundred
each with some bundles of ten remaining. Let these
last be six in number. We then tie up the bundles
of hundreds by tens, making bundles of thousands,
and find that there are five bundles of hundreds remaining.
Suppose that on attempting to tie up the
thousands by tens, we find there are not so many as
ten, but only four. The number of faggots is then $4$~thousands,
$5$~hundreds, $6$~tens, and~$7$.
The next question is, how shall we represent this
\index{Notation!general principle of|EtSeq}%
\index{Numbers, representation of|EtSeq}%
number in a short and convenient manner? It is plain
that the way to do this is a \emph{matter of choice}. Suppose
then that we distinguish the tens by marking their
number with one accent, the hundreds with two accents,
and the thousands with three. We may then
represent this number in any of the following ways:---
\PageSep{16}
$76' 5'' 4'''$, $6' 75'' 4'''$, $6' 4''' 5'' 7$, $4''' 5'' 6' 7$, the whole number
of ways being~$24$. But this is more than we want;
one certain method of representing a number is sufficient.
The most natural way is to place them in order
of magnitude, either putting the largest collection first
or the smallest; thus $4''' 5'' 6' 7$, or $76' 5'' 4'''$. Of these
we choose the first.
In writing down numbers in this way it will soon
\index{Cipher}%[** TN: [sic] variant spelling]
\index{Zero!as a figure}%
be apparent that the accents are unnecessary. Since
the singly accented figure will always be the second
from the right, and so on, the \emph{place} of each number
will point out what accents to write over it, and we
may therefore consider each figure as deriving a value
from the place in which it stands. But here this difficulty
occurs. How are we to represent the numbers
$3''' 3'$, and $4''' 2' 7$ without accents? If we write them
thus, $33$~and~$427$, they will be mistaken for $3' 3$~and~$4'' 2' 7$.
This difficulty will be obviated by placing cyphers
so as to bring each number into the place allotted
to the sort of collection which it represents;
thus, since the trebly accented letters, or thousands,
are in the fourth place from the right, and the singly
accented letters in the second, the first number may
be written~$3030$, and the second~$4027$. The cypher,
which plays so important a part in arithmetic that it
was anciently called the \emph{art of cypher}, or \emph{cyphering},
does not stand for any number in itself, but is merely
employed, like blank types in printing, to keep other
signs in those places which they must occupy in order
\PageSep{17}
to be read rightly. We may now ask what would
have been the case if, instead of ten fingers, men had
had more or less. For example, by what signs would
$4567$ have been represented, if man had nine fingers
instead of ten? We may presume that the method
would have been the same, with the number nine represented
by~$10$ instead of ten, and the omission of the
symbol~$9$. Suppose this number of faggots is to be
counted by nines. Tie them up in bundles of nine,
and we shall find $4$~faggots remaining. Tie these
bundles again in bundles of nine, each of which will,
therefore, contain eighty-one, and there will be $3$~bundles
remaining. These tied up in the same way into
bundles of nine, each of which contains seven hundred
and twenty-nine, will leave $2$~odd bundles, and, as
there will be only six of them, the process cannot be
carried any further. If, then, we represent, by~$1'$, a
bundle of nine, or a \emph{nine}, by~$1''$ a nine of nines, and
so on, the number which we write~$4567$, must be written
$6''' 2'' 3' 4$. In order to avoid confusion, we will
suffer the accents to remain over all numbers which
are not reckoned in tens, while those which are so
reckoned shall be written in the common way. The
following is a comparison of the way in which numbers
in the common system are written, and in the
one which we have just explained:
\begin{align*}
&\begin{array}{l*{13}{c}}
\PadTxt[l]{Nines}{\scriptsize COUNTING BY} \\
\text{Tens}
&1 &2 &3 &4 &5 &6 &7 &8 &9 &10 &11 &12 &13 \\
\text{Nines}
&1 &2 &3 &4 &5 &6 &7 &8 &1' 0 &1' 1 &1' 2 &1' 3 &1' 4
\end{array} \displaybreak[0]\\
\PageSep{18}
&\begin{array}{l*{10}{c}}
\text{Tens}
&14 &15 &16 &17 &18 &19 &20 &30 &40 &50 \\
\text{Nines}
&1' 5 &1' 6 &1' 7 &1' 8 &2' 0 &2' 1 &2' 2 &3' 3 &4' 4 &5' 5
\end{array} \displaybreak[0]\\
%
&\begin{array}{l*{5}{c}}
\text{Tens}
&60 &70 &80 &90 &100 \\
\text{Nines}
&6' 6 &7' 7 &8' 8 &1'' 1' 0 &1'' 2' 1
\end{array}
\end{align*}
We will now write, in the common way, in the
tens' system, the process which we went through in
order to find how to represent the number~$4567$ in
that of the nines, thus:
\[
\begin{array}{l@{\,}r@{\,}rr}
9) & 4567 \\
9) & 507 &\text{---rem.~$4$}. \\
9) & 56 &\text{---rem.~$3$}. \\
9) & 6 &\text{---rem.~$2$}. \\
& 0 &\text{---rem.~$6$}.
& \text{Representation required, $6''' 2'' 3' 4$.}
\end{array}
\]
\setlength{\arraycolsep}{0pt}%
The processes of arithmetic are the same in principle
whatever system of numeration is used. To
show this, we subjoin a question in each of the first
four rules, worked both in the common system, and
in that of the nines. There is the difference, that, in
the first, the tens must be carried, and in the second
the nines.
%[** TN: Each calculation set on its own line in the original]
\begin{gather*}
\begin{array}[t]{rcr}
& \TEntry[SUBTRACTION.]{ADDITION.}\quad \\
636 && 7'' 7' 6 \\
987 &&1''' 3'' 1' 6 \\
403 && 4'' 8' 7 \\
\cline{1-1}
\cline{3-3}
\Strut
2026 &&2''' 7'' 0' 1
\end{array}\qquad %\\
%
\begin{array}[t]{rcr}
& \TEntry{SUBTRACTION.}\quad \\
1384 &&1''' 8'' 0' 7 \\
797 &&1''' 0'' 7' 5 \\
\cline{1-1}
\cline{3-3}
\Strut
587 && 7'' 2' 2
\end{array} \displaybreak[0]\\[8pt]
\PageSep{19}
\begin{array}{rc*{5}{l}}
& \TEntry{MULTIPLICATION.}\quad \\
297 && &&3'' &6' &0 \\
136 && &&1'' &6' &1 \\
\cline{1-1}
\cline{5-7}
\Strut
1782 && &&3 &6 &0 \\
891\Z &&2 &4 &0 &0 \\
297\Z\Z &&3 &6 &0 \\
\cline{1-1}
\cline{3-7}
\Strut
40392 &&6'''' &\DPad{1'''} &\DPad{3''} &\DPad{6'} &0
\end{array} \displaybreak[0]\\[8pt]
%
\begin{array}{rrlr*{7}{l@{\ }}}
\multicolumn{11}{c}{\TEntry{DIVISION.}} \\
633)\ \null & 79125 &\ (125\qquad\null &7'' 7' 3)\ \null
&1\Ord{v} &3\Ord{iv} &0\Ord{iii} &4\Ord{ii} &7\Ord{i}\:\null &6\rlap{\footnotemark}
&\ (1'' 4' 8 \\
& 633\Z\Z & &&&7 &7 &3 \\
\cline{2-2}
\cline{6-9}
\Strut
& 1582\Z & &&&4 &2 &1 &7 \\
& 1266\Z & &&&3 &4 &2 &3 \\
\cline{2-2}
\cline{6-10}
\Strut
& 3165 & &&& &6 &8 &4 &6 \\
& 3165 & &&& &6 &8 &4 &6 \\
\cline{2-2}
\cline{7-10}
& 0 & &&& &&&&0
\end{array}
\end{gather*}
\footnotetext{To avoid too great a number of accents, Roman numerals are put instead
of them; also, to avoid confusion, the accents are omitted after the
first line.}
The student should accustom himself to work
questions in different systems of numeration, which
will give him a clearer insight into the nature of arithmetical
processes than he could obtain by any other
method. When he uses a system in which numbers
are counted by a number greater than ten, he will
want some new symbols for figures. For example, in
the duodecimal system, where twelve is the number
\index{Duodecimal system}%
of figures supposed, twelve will be represented by~$1'0$;
there must, therefore, be a distinct sign for ten and
eleven: a nine and six reversed, thus \reflectbox{$9$}~and~\reflectbox{$6$}, might
be used for these.
\PageSep{20}
\Chapter{III.}{Elementary Rules of Arithmetic.}
\index{Arithmetic!elementary rules of|EtSeq}%
\First{As Soon} as the beginner has mastered the notion
of arithmetic, he may be made acquainted with
the meaning of the algebraical signs $+$,~$-$,~$×$,~$=$, and
\index{Signs!arithmetical and algebraical@{arithmetical and algebraical|EtSeq}}%
also with that for division, or the common way of representing
a fraction. There is no difficulty in these
signs or in their use. Five minutes' consideration will
make the symbol~$5 + 3$ present as clear an idea as the
words ``$5$~added to~$3$.'' The reason why they usually
cause so much embarrassment is, that they are generally
deferred until the student commences algebra,
when he is often introduced at the same time to the
representation of numbers by letters, the distinction
of known and unknown quantities, the signs of which
we have been speaking, and the use of figures as
the exponents of letters. Either of these four things
is quite sufficient at a time, and there is no time more
favorable for beginning to make use of the signs of
operation than when the habit of performing the operations
commences. The beginner should exercise
\PageSep{21}
\index{Brackets}%
\index{Instruction!principles of natural|EtSeq}%
himself in putting the simplest truths of arithmetic in
this new shape, and should write such sentences as
the following frequently:
\begin{gather*}
2 + 7 = 9, \\
6 - 4 = 2, \\
1 + 8 + 4 - 6 = 4 + 2 + 1, \\
2 × 2 + 12 × 12 = 14 × 10 + 2 × 2 × 2.
\end{gather*}
These will accustom him to the meaning of the signs,
just as he was accustomed to the formation of letters
by writing copies. As he proceeds through the rules
of arithmetic, he should take care never to omit connecting
each operation with its sign, and should avoid
confounding operations together and considering them
as the same, because they produce the same result.
Thus $4 × 7$ does not denote the same operation as
$7 × 4$, though the result of both is~$28$. The first is
four multiplied by seven, four taken seven times; the
second is seven multiplied by four, seven taken four
times; and that $4 × 7 = 7 × 4$ is a proposition to be
proved, not to be taken for granted. Again, $\frac{1}{7} × 4$
and $\frac{4}{7}$ are marks of distinct operations, though their
result is the same, as we shall show in treating of
fractions.
The examples which a beginner should choose for
practice should be simple and should not contain very
large numbers. The powers of the mind cannot be
directed to two things at once: if the complexity of
the numbers used requires all the student's attention,
he cannot observe the principle of the rule which he
\PageSep{22}
is following. Now, at the commencement of his career,
a principle is not received and understood by
the student as quickly as it is explained by the instructor.
He does not, and cannot, generalise at all;
he must be taught to do so; and he cannot learn that
a particular fact holds good for \emph{all numbers} unless by
having it shown that it holds good for \emph{some numbers},
and that for those \emph{some numbers} he may substitute
\emph{others}, and use the same demonstration. Until he
can do this himself he does not understand the principle,
and he can never do this except by seeing the
rule explained and trying it himself on small numbers.
He may, indeed, and will, believe it on the word of
his instructor, but this disposition is to be checked.
He must be told, that whatever is not gained by his
own thought is not gained to any purpose; that the
mathematics are put in his way purposely because
they are the only sciences in which he must not trust
the authority of any one. The superintendence of
these efforts is the real business of an instructor in
arithmetic. The merely showing the student a rule
by which he is to work, and comparing his answer
with a key to the book, printed for the preceptor's
private use, to save the trouble which he ought to
bestow upon his pupil, is not teaching arithmetic any
more than presenting him with a grammar and dictionary
is teaching him Latin. When the principle
of each rule has been well established by showing its
application to some simple examples (and the number
\PageSep{23}
of these requisite will vary with the intellect of the
student), he may then proceed to more complicated
cases, in order to acquire facility in computation. The
four first rules may be studied in this way, and these
will throw the greatest light on those which succeed.
The student must observe that all operations in
arithmetic may be resolved into addition and subtraction;
\index{Addition}%
\index{Subtraction}%
that these additions and subtractions might be
made with counters; so that the whole of the rules
consist of processes intended to shorten and simplify
that which would otherwise be long and complex. For
example, multiplication is continued addition of the
\index{Division@{Division|EtSeq}}%
\index{Multiplication@{Multiplication|EtSeq}}%
same number to itself---twelve times seven is twelve
sevens added together. Division is a continued subtraction
of one number from another; the division of
$129$ by~$3$ is a continued subtraction of~$3$ from~$129$, in
order to see how many threes it contains. All other
operations are composed of these four, and are, therefore,
the result of additions and subtractions only.
The following principles, which occur so continually
in mathematical operations that we are, at length,
hardly sensible of their presence, are the foundation
of the arithmetical rules:
I. We do not alter the sum of two numbers by
taking away any part of the first, if we annex that
part to the second. This may be expressed by signs,
in a particular instance, thus:
\[
(20 - 6) + (32 + 6) = 20 + 32.
\]
\PageSep{24}
II. We do not alter the difference of two numbers
by increasing or diminishing one of them, provided
we increase or diminish the other as much. This may
be expressed thus, in one instance:
\begin{align*}
(45 + 7) - (22 + 7) &= 45 - 22\Chg{.}{,} \\
(45 - 8) - (22 - 8) &= 45 - 22.
\end{align*}
III. If we wish to multiply one number by another,
for example $156$ by~$29$, we may break up~$156$ into any
number of parts, multiply each of these parts by~$29$,
and add the results. For example, $156$~is made up of
$100$,~$50$, and~$6$. Then
\[
156 × 29 = 100 × 29 + 50 × 29 + 6 × 29.
\]
IV. The same thing may be done with the multiplier
instead of the multiplicand. Thus, $29$~is made
up of $18$,~$6$, and~$5$. Then
\[
156 × 29 = 156 × 18 + 156 × 6 + 156 × 5.
\]
V. If any two or more numbers be multiplied together,
it is indifferent in what order they are multiplied,
the result is the same. Thus,
\[
10 × 6 × 4 × 3 = 3 × 10 × 4 × 6 = 6 × 10 × 4 × 3, \quad\text{etc.}
\]
VI. In dividing one number by another, for example
$156$ by~$12$, we may break up the dividend, and
divide each of its parts by the divisor, and then add
the results. We may part $156$ into $72$,~$60$, and~$24$;
this is expressed thus:
\[
\frac{156}{12} = \frac{72}{12} + \frac{60}{12} + \frac{24}{12}.
\]
\PageSep{25}
The same thing cannot be done with the divisor. It
is not true that
\[
\frac{156}{12} = \frac{156}{3} + \frac{156}{4} + \frac{156}{5}.
\]
The student should discover the reason for himself.
A prime number is one which is not divisible by
\index{Prime numbers and factors}%
any other number except~$1$. When the process of division
can be performed, it can be ascertained whether
a given number is divisible by any other number, that
is, whether it is prime or not. This can be done by
dividing it by all the numbers which are less than its
half, since it is evident that it cannot be divided into
a number of parts, each of which is greater than its
half. This process would be laborious when the given
number is large; still it may be done, and by this
means the number itself may be \emph{reduced to its prime
factors},\footnote
{The \emph{factors} of a number are those numbers by the multiplication of
which it is made.}
as it is called, that is, it may either be shown
to be a prime number itself or made up by multiplying
several prime numbers together. Thus, $306$ is
$34 × 9$, or $2 × 17 × 9$, or $2 × 17 × 3 × 3$, and has for
its prime factors $2$,~$17$, and~$3$, the latter of which is
repeated twice in its formation. When this has been
done with two numbers, we can then see whether
they have any factors in common, and, if that be the
\index{Greatest common measure@{Greatest common measure|EtSeq}}%
case, we can then find what is called their \emph{greatest
common measure} or \emph{divisor}; that is, the number made
\PageSep{26}
by multiplying all their common factors. It is an evident
truth that, if a number can be divided by the
product of two others, it can be divided by each of
them. If a number can be parted into an exact number
of twelves, it can be parted also into a number of
sixes, twos, or fours. It is also true that, if a number
can be divided by any other number, and the quotient
can then be divided by a third number, the original
number can be divided by the product of the other
two. Thus, $144$~is divisible by~$2$; the quotient,~$72$, is
divisible by~$6$; and the original number is divisible
by $6 × 2$ or~$12$. It is also true that, if two numbers
are prime, their product is divisible by no numbers
except themselves. Thus, $17 × 11$ is divisible by no
numbers except $17$~and~$11$. Though this is a simple
proposition, its proof is not so, and cannot be given
to the beginner. From these things it follows that
the greatest common measure of two numbers (measure
being an old word for divisor) is the product of all
the prime factors which the two possess in common.
For example, the numbers $90$ and $100$, which, when
reduced to their prime factors, are $2 × 5 × 3 × 3$ and
$2 × 2 × 5 × 5$, have the common factors $2$~and~$5$, and
are divisible by $2 × 5$, or~$10$. The quotients are $3 × 3$
and $2 × 5$, or $9$~and~$10$, which have no common factor
remaining, and $2 × 5$, or~$10$, is the greatest common
measure of $90$ and~$100$. The same may be shown in
the case of any other numbers. But the method we
\PageSep{27}
have mentioned of resolving numbers into their prime
factors, being troublesome to apply when the numbers
are large, is usually abandoned for another. It
happens frequently that a method simple in principle
is laborious in practice, and the contrary.
When one number is divided by another, and its
quotient and remainder obtained, the dividend may
be recovered again by multiplying the quotient and
divisor together, and adding the remainder to the product.
Thus $171$ divided by~$27$ gives a quotient~$6$ and
a remainder~$9$, and $171$~is made by multiplying $27$ by~$6$,
and adding~$9$ to the product. That is, $171 = 27 × 6 + 9$.
Now, from this equation it is easy to
show that every number which divides $171$ and $27$
also divides~$9$, that is, every common measure of $171$
and $27$ is also a common measure of $27$ and~$9$. We
can also show that $27$ and $9$ have no common measures
which are not common to $171$ and~$27$. Therefore,
the common measures of $171$ and $27$ are those, and no
others, which are common to $27$ and~$9$; the greatest
common measure of each pair must, therefore, be the
same, that is, the greatest common measure of a divisor
and dividend is also the greatest common measure
of the remainder and divisor. Now take the common
process for finding the greatest common measure
of two numbers; for example, $360$ and $420$, which is
as follows, and abbreviate the words \emph{greatest common
measure} into their initials~$\gcm$:
\PageSep{28}
\[
\begin{array}{rrcrl}
360) & 420 &(& 1\Z\Z \\
& 360 \\
\cline{2-2}
\Strut
& 60 &)& 360 & (6 \\
& & & 360 \\
\cline{4-4}
\Strut
& & & 0
\end{array}
\]
From the theorem above enunciated it appears
that
\begin{gather*}
\text{$\gcm$ of $420$ and $360$ is $\gcm$ of $60$ and $360$;} \\
\text{$\gcm$ of $60$ and $360$ is $60$;}
\end{gather*}
because $60$~divides both $60$ and~$360$, and no number
can have a greater measure than itself. Thus may be
seen the reason of the common rule for finding the
greatest common measure of two numbers.
Every number which can be divided by another
without remainder is called a multiple of it. Thus,
$12$,~$18$, and~$42$ are multiples of~$6$, and the last is a
\emph{common multiple} of $6$~and~$7$, because it is divisible both
by $6$ and~$7$. The only things which it is necessary to
observe on this subject are, (1),~that the product of
two numbers is a common multiple of both; (2),~that
when the two numbers have a common measure greater
than~$1$, there is a common multiple less than their
product; (3),~that when they have no common measure
except~$1$, the least common multiple is their product.
\index{Least common multiple}%
The first of these is evident; the second will
appear from an example. Take $10$ and~$8$, which have
the common measure~$2$, since the first is $2 × 5$ and
the second $2 × 4$. The product is $2 × 2 × 4 × 5$, but
\PageSep{29}
$2 × 4 × 5$ is also a common multiple, since it is divisible
by $2 × 4$, or~$8$, and by $2 × 5$, or~$10$. To find this
common multiple we must, therefore, divide the product
by the greatest common measure. The third
principle cannot be proved in an elementary way, but
the student may convince himself of it by any number
of examples. He will not, for instance, be able to
find a common multiple of $8$~and~$7$ less than $8 × 7$
or~$56$.
\PageSep{30}
\Chapter{IV.}{Arithmetical Fractions.}
\index{Fractions!arithmetical@{arithmetical|EtSeq}}%
\First{When} the student has perfected himself in the
four rules, together with that for finding the
greatest common measure, he should proceed at once
to the subject of fractions. This part of arithmetic is
usually supposed to present extraordinary difficulties;
whereas, the fact is that there is nothing in fractions
so difficult, either in principle or practice, as the rule
for finding the greatest common measure. We would
recommend the student not to attend to the distinctions
of proper and improper, pure or mixed fractions,
etc., as there is no distinction whatever in the rules,
which are common to all these fractions.
When one number, as~$56$, is to be divided by another,
as~$8$, the process is written thus:~$\frac{56}{8}$. By this
we mean that $56$~is to be divided into $8$~equal parts,
and one of these parts is called the quotient. In this
case the quotient is~$7$. But it is equally possible
to divide~$57$ into $8$~equal parts; for example, we can
divide $57$~feet into $8$~equal parts, but the eighth part
\PageSep{31}
of $57$~feet will not be an exact number of feet, since
$57$~does not contain an exact number of eights; a part
of a foot will be contained in the quotient~$\frac{57}{8}$, and this
quotient is therefore called a fraction, or broken number.
If we divide~$57$ into $56$~and~$1$, and take the
eighth part of each of these, whose sum will give the
eighth part of the whole, the eighth of $56$~feet is $7$~feet;
the eighth of $1$~foot is a fraction, which we write~$\frac{1}{8}$,
and $\frac{57}{8}$~is $7 + \frac{1}{8}$, which is usually written~$7\frac{1}{8}$. Both
of these quantities~$\frac{57}{8}$, and~$7\frac{1}{8}$, are called fractions; the
only difference is that, in the second, that part of the
quotient which is a whole number is separated from
the part which is less than any whole number.
There are two ways in which a fraction may be
considered. Let us take, for example,~$\frac{5}{8}$. This means
that $5$~is to be divided into $8$~parts, and $\frac{5}{8}$~stands for
one of these parts. The same length will be obtained
if we divide~$1$ into $8$~parts, and take $5$~of them, or find
$\frac{1}{8} × 5$. To prove this let each of the lines drawn below
represent $\frac{1}{8}$~of an inch; repeat $\frac{1}{8}$~five times, and
repeat the same line eight times.
\[
\ArrayCompress[0.8]
\begin{array}{c*{4}{>{\qquad}c}}
\BAR & \BAR & \BAR & \BAR & \BAR \\
\BAR & \BAR & \BAR & \BAR & \BAR \\
\BAR & \BAR & \BAR & \BAR & \BAR \\
\BAR & \BAR & \BAR & \BAR & \BAR \\
\BAR & \BAR & \BAR & \BAR & \BAR \\
\BAR & \BAR & \BAR & \BAR & \BAR \\
\BAR & \BAR & \BAR & \BAR & \BAR \\
\BAR & \BAR & \BAR & \BAR & \BAR
\end{array}
\]
In each column is $\frac{1}{8}$th~of an inch repeated $8$~times;
that is one inch. There are, then, $5$~inches in all,
\PageSep{32}
since there are five columns. But since there are $8$~lines,
each line is the eighth of $5$~inches, or~$\frac{5}{8}$, but
each line is also $\frac{1}{8}$th~of an inch repeated $5$~times, or
$\frac{1}{8} × 5$. Therefore, $\frac{5}{8} = \frac{1}{8} × 5$; that is, in order to find
$\frac{5}{8}$~inches, we may either divide \emph{five inches} into $8$~parts,
and take \emph{one} of them, or divide \emph{one inch} into $8$~parts,
and take \emph{five} of them. The symbol~$\frac{5}{8}$ is made to stand
for both these operations, since they lead to the same
result.
{\Loosen The most important property of a fraction is, that
if both its numerator and denominator are multiplied
by the same number, the value of the fraction is not
altered; that is, $\frac{3}{5}$~is the same as~$\frac{12}{20}$, or each part is
the same when we divide $12$~inches into $20$~parts, as
when we divide $3$~inches into $5$~parts. Again, we get
the same length by dividing $1$~inch into $20$~parts, and
taking $12$~of them, which we get by dividing $1$~inch
into $5$~parts and taking $3$~of them. This hardly needs
demonstration. Taking $12$~out of~$20$ is taking $3$~out
of~$5$, since for every~$3$ which $12$~contains, there is a~$5$
contained in~$20$. Every fraction, therefore, admits of
innumerable alterations in its form, without any alteration
in its value. Thus, $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} = \frac{5}{10}$, etc.;
$\frac{2}{7} = \frac{4}{14} = \frac{6}{21} = \frac{8}{28}$,~etc.}
On the same principle it is shown that the terms
of a fraction may be divided by any number without
any alteration of its value. There will now be no difficulty
in reducing fractions to a common denominator,
in reducing a fraction to its lowest terms; neither
\PageSep{33}
in adding nor subtracting fractions, for all of which
the rules are given in every book of arithmetic.
We now come to a rule which presents more peculiar
\index{Extension@{Extension of rules and meanings of terms|EtSeq}}%
\index{Notation!extension of}%
\index{Rules!extension of meaning of}%
difficulties in point of principle than any at
which we have yet arrived. If we could at once take
the most general view of numbers, and give the beginner
the extended notions which he may afterwards
attain, the mathematics would present comparatively
few impediments. But the constitution of our minds
will not permit this. It is by collecting facts and
principles, one by one, and thus only, that we arrive
at what are called general notions; and we afterwards
make comparisons of the facts which we have acquired
and discover analogies and resemblances which, while
they bind together the fabric of our knowledge, point
out methods of increasing its extent and beauty. In
the limited view which we first take of the operations
which we are performing, the names which we give
are necessarily confined and partial; but when, after
additional study and reflection, we recur to our former
notions, we soon discover processes so resembling one
another, and different rules so linked together, that
we feel it would destroy the symmetry of our language
if we were to call them by different names. We are
then induced to extend the meaning of our terms, so
as to make two rules into one. Also, suppose that
when we have discovered and applied a rule and given
the process which it teaches a particular name, we
find that this process is only a part of one more general,
\PageSep{34}
which applies to all cases contained under the
first, and to others besides. We have only the alternative
of inventing a new name, or of extending the
meaning of the former one so as to merge the particular
process in the more general one of which it is a
part. Of this we can give an instance. We began
with reasoning upon simple numbers, such as $1$,~$2$,~$3$,
$20$,~etc. We afterwards divided these into parts, of
which we took some number, and which we called
fractions, such as $\frac{2}{3}$,~$\frac{7}{2}$, $\frac{1}{5}$,~etc. Now there is no number
which may not be considered as a fraction in as
many different ways as we please. Thus $7$~is $\frac{14}{2}$ or~$\frac{21}{3}$,
etc.; $12$~is $\frac{144}{12}$, $\frac{72}{6}$,~etc. Our new notion of fraction
is, then, one which includes all our former ideas
of number, and others besides. It is then customary
to represent by the word number, not only our first
notion of it, but also the extended one, of which the
first is only a part. Those to which our first notions
applied we call whole numbers, the others fractional
numbers, but still the name number is applied to both
$2$~and~$\frac{1}{2}$, $3$~and~$\frac{3}{5}$. The rule of which we have spoken
\index{Multiplication@{Multiplication|EtSeq}}%
is another instance. It is called the multiplication of
fractional numbers. Now, if we return to our meaning
of the word multiplication, we shall find that the
multiplication of one fraction by another appears an
absurdity. We multiply a number by taking it several
times and adding these together. What, then, is
meant by multiplying by a fraction? Still, a rule has
been found which, in applying mathematics, it is necessary
\PageSep{35}
to use for fractions, in all cases where multiplication
\Figure[nolabel]{035a}
would have been used had they been whole
numbers. Of this we shall now give a simple example.
Take an oblong figure (which is called a rectangle
in geometry), such as~$ABCD$, and find the magnitudes
of the sides $AB$~and~$BC$ in inches. Draw the
line~$EF$ equal in length to one inch, and the square~$G$,
each of whose sides is one inch. If the lines $AB$
and $BC$ contain an exact number of inches, the rectangle~$ABCD$
contains an exact number of squares,
\Figure[nolabel]{035b}
each equal to~$G$, and the number of squares contained
is found by multiplying the number of inches in~$AB$
by the number of inches in~$BC$. In the present case
the number of squares is $3 × 4$, or~$12$. Now, suppose
another rectangle~$A'B'C'D'$, of which neither of the
sides is an exact number of inches; suppose, for example,
that $A'B'$~is $\frac{2}{3}$~of an inch, and that $B'C'$~is $\frac{5}{7}$~of an
\Figure[nolabel]{035c}
\PageSep{36}
inch. We may show, by reasoning, that we can find
how much $A'B'C'D'$~is of~$G$ by forming a fraction
which has the product of the numerators of $\frac{2}{3}$ and $\frac{5}{7}$
for its numerator, and the product of their denominators
for its denominator; that is, that $A'B'C'D'$ contains
$\frac{10}{21}$~of~$G$. Here then appears a connexion between
the multiplication of whole numbers, and the
formation of a fraction, whose numerator is the product
of two numerators, and its denominator the product
of the corresponding denominators. These operations
will always come together, that is whenever a
question occurs in which, when whole numbers are
given, those numbers are to be multiplied together;
when fractional numbers are given, it will be necessary,
in the same case, to multiply the numerator by
the numerator, and the denominator by the denominator,
and form the result into a fraction, as above.
This would lead us to suspect some connexion between
these two operations, and we shall accordingly
find that when whole numbers are formed into fractions,
they may be multiplied together by this very
rule. Take, for example, the numbers $3$~and~$4$, whose
product is~$12$. The first may be written as~$\frac{15}{5}$, and
the second as~$\frac{8}{2}$. Form a fraction from the product
of the numerators and denominators of these, which
will be~$\frac{120}{10}$, which is~$12$, the product of $3$~and~$4$.
From these considerations it is customary to call
the fraction which is produced from two others in the
manner above stated, the \emph{product} of those two fractions,
\PageSep{37}
and the process of finding the third fraction,
\emph{multiplication}. We shall always find the first meaning
of the word multiplication included in the second, in
all cases in which the quantities represented as fractions
are really whole numbers. The mathematics are
\index{Mathematics!language of|EtSeq}%
not the only branches of knowledge in which it is customary
to extend the meaning of established terms.
Whenever we pass from that which is simple to that
which is complex, we shall see the necessity of carrying
our terms with us and enlarging their meaning,
as we enlarge our own ideas. This is the only method
of forming a language which shall approach in any
\index{Language}%
degree towards perfection; and more depends upon
a well-constructed language in mathematics than in
anything else. It is not that an imperfect language
would deprive us of the means of demonstration, or
cramp the powers of reasoning. The propositions of
Euclid upon numbers are as rationally established as
\index{Euclid}%
any others, although his terms are deficient in analogy,
and his notation infinitely inferior to that which we
\index{Discovery, progress of, dependent on language}%
use. It is the progress of discovery which is checked
by terms constructed so as to conceal resemblances
which exist, and to prevent one result from pointing
out another. The higher branches of mathematics
date the progress which they have made in the last
century and a half, from the time when the genius of
Newton, Leibnitz, Descartes, and Hariot turned the
\index{Descartes}%
\index{Leibnitz}%
\index{Newton}%
attention of the scientific world to the imperfect mechanism
of the science. A slight and almost casual improvement,
\PageSep{38}
made by Hariot in algebraical language,
\index{Hariot}%
has been the foundation of most important branches
of the science.\footnote
{The mathematician will be aware that I allude to writing an equation
in the form
%[** TN: Aligned in the original]
$x^{2} + ax - b = 0$; instead of
$x^{2} + ax = b$.}
The subject of the last articles is of
very great importance, and will often recur to us in
explaining the difficulties of algebraical notation.
The multiplication of $\frac{5}{6}$ by~$\frac{3}{2}$ is equivalent to dividing
$\frac{5}{6}$ into $2$~parts, and taking three such parts. Because
$\frac{5}{6}$ being the same as~$\frac{10}{12}$, or $1$~divided into $12$~parts
and $10$~of them taken, the half of~$\frac{10}{12}$ is $5$~of those
parts, or~$\frac{5}{12}$. Three times this quantity will be $15$~of
those parts, or~$\frac{15}{12}$, which is by our rule the same as
what we have called, $\frac{5}{6}$~multiplied by~$\frac{3}{2}$. But the same
result arises from multiplying $\frac{3}{2}$ by~$\frac{5}{6}$, or dividing~$\frac{3}{2}$
into $6$~parts and taking $5$~of them. Therefore, we find
that $\frac{3}{2}$~multiplied by~$\frac{5}{6}$ is the same as $\frac{5}{6}$~multiplied by~$\frac{3}{2}$,
or $\frac{3}{2} × \frac{5}{6} = \frac{5}{6} × \frac{3}{2}$. This proposition is usually considered
as requiring no proof, because it is received
very early on the authority of a rule in the elements
of arithmetic. But it is not self-evident, for the truth
of which we appeal to the beginner himself, and ask
him whether he would have seen at once that $\frac{5}{6}$~of an
apple divided into $2$~parts and $3$~of them taken, is the
same as $\frac{3}{2}$~of an apple, or one apple and a-half divided
into six parts and $5$~of them taken.
An extension of the same sort is made of the term
division. In dividing one whole number by another,
\index{Division}%
\PageSep{39}
for example, $12$ by~$2$, we endeavor to find how many
\emph{twos} must be added together to make~$12$. In passing
from a problem which contains these whole numbers
to one which contains fractional quantities, for example
$\frac{3}{4}$~and~$\frac{2}{5}$, it will be observed that in place of finding
how many twos make~$12$, we shall have to find
into how many parts $\frac{2}{5}$ must be divided, and how many
of them must be taken, so as to give~$\frac{3}{4}$. If we reduce
these fractions to a common denominator, in which
case they will be $\frac{15}{20}$ and~$\frac{8}{20}$; and if we divide the second
into $8$~equal parts, each of which will be~$\frac{1}{20}$, and
take $15$~of these parts, we shall get~$\frac{15}{20}$, or~$\frac{3}{4}$. The
fraction whose numerator is~$15$, and whose denominator
is~$8$, or~$\frac{15}{8}$, will in these problems take the place
of the quotient of the two whole numbers. In the
same manner as before, it may be shown that this process
is equivalent to the division of one whole number
by another, whenever the fractions are really whole
numbers; for example, $3$~is~$\frac{12}{4}$, and $15$~is~$\frac{30}{2}$. If this
process be applied to $\frac{30}{2}$~and~$\frac{12}{4}$, the result is~$\frac{120}{24}$,
which is~$5$, or the same as $15$~divided by~$3$. This process
is then, by extension, called division: $\frac{15}{8}$~is called
the quotient of~$\frac{3}{4}$ divided by~$\frac{2}{5}$, and is found by multiplying
the numerator of the first by the denominator
of the second for the numerator of the result, and the
denominator of the first by the numerator of the second
for the denominator of the result. That this process
does give the same result as ordinary division in
all cases where ordinary division is applicable, we can
\PageSep{40}
easily show from any two whole numbers, for example,
$12$~and~$2$, whose quotient is~$6$. Now $12$~is~$\frac{36}{3}$, and
$2$~is~$\frac{10}{5}$, and the rule for what we have called division
of fractions will give as the quotient~$\frac{180}{30}$, which is~$6$.
In all fractional investigations, when the beginner
meets with a difficulty, he should accustom himself to
leave the notation of fractions, and betake himself to
their original definition. He should recollect that $\frac{5}{6}$
is $1$~divided into $6$~parts and five of them taken, or the
sixth part of~$5$, and he should reason upon these suppositions,
neglecting all rules until he has established
them in his own mind by reflexion on particular instances.
These instances should not contain large
numbers, and it will perhaps assist him if he reasons
on some given unit, for example a foot. Let $AB$~be
one foot, and divide it into any number of equal parts
($7$~for example) by the points $C$,~$D$, $E$, $F$,~$G$, and~$H$.
\Figure[nolabel]{040}
He must then recollect that each of these parts is $\frac{1}{7}$~of
a foot; that any two of them together are $\frac{2}{7}$~of a
foot; any~$3$,~$\frac{3}{7}$, and so on. He should then accustom
himself, without a rule, to solve such questions as the
following, by observation of the figure, dividing each
part into several equal parts, if necessary; and he
may be well assured that he does not understand the
nature of fractions until such questions are easy to
him.
\PageSep{41}
What is $\frac{1}{4}$~of $\frac{2}{7}$~of a foot? What is $\frac{2}{5}$~of $\frac{1}{3}$~of $\frac{3}{4}$~of a
foot? Into how many parts must $\frac{3}{7}$~of a foot be divided,
and how many of them must be taken to produce
$\frac{14}{15}$~of a foot? What is $\frac{1}{3} + \frac{1}{7}$~of a foot? and so on.
\PageSep{42}
\Chapter{V.}{Decimal Fractions.}
\index{Decimal!fractions|(}%
\index{Fractions!decimal}%
\First{It} is a disadvantage attending rules received without
\index{Rules}%
a knowledge of principles, that a mere difference
of language is enough to create a notion in the mind
of a student that he is upon a totally different subject.
Very few beginners see that in following the rule
usually called practice, they are working the same
questions as were proposed in compound multiplication;---that
the rule of three is only an application of
the doctrine of fractions; that the rules known by the
name of commission, brokerage, interest, etc., are the
same, and so on. No instance, however, is more conspicuous
than that of decimal fractions, which are
made to form a branch of arithmetic as distinct from
ordinary or vulgar fractions as any two parts of the
subject whatever. Nevertheless, there is no single
rule in the one which is not substantially the same as
the rule corresponding in the other, the difference
consisting altogether in a different way of writing the
fractions. The beginner will observe that throughout
\PageSep{43}
the subject it is continually necessary to reduce fractions
to a common denominator: he will see, therefore,
the advantage of always using either the same
denominator, or a set of denominators, so closely connected
as to be very easily reducible to one another.
Now of all numbers which can be chosen the most
easily manageable are $10$,~$100$, $1000$,~etc., which are
called decimal numbers on account of their connexion
\index{Decimal!point}%
with the number ten. All fractions, such as $\frac{75}{100}$,
$\frac{333}{1000}$, $\frac{178699}{10}$, which have a decimal number for the
denominator, are called decimal fractions. Now a
denominator of this sort is known whenever the number
of cyphers in it are known; thus a decimal number
with $4$~cyphers can only be~$10,000$, or ten thousand.
We need not, therefore, write the denominator,
provided, in its stead, we put some mark upon the
numerator, by which we may know the number of
cyphers in the denominator. This mark is for our own
selection. The method which is followed is to point
off from the numerator as many \emph{figures} as there are
\emph{cyphers} in the denominator. Thus $\frac{17334}{1000}$ is represented
by $17.334$; $\frac{229}{1000}$ thus,~$.229$. We might, had we so
pleased, have represented them thus, $17334^{3}$, $229^{3}$;
or thus, $17334_{3}$, $229_{3}$, or in any way by which we
might choose to agree to recollect that the denominator
is $1$~followed by $3$~cyphers. In the common method
this difficulty occurs immediately. What shall be done
when there are not as many figures in the numerator
as there are cyphers in the denominator? How shall
\PageSep{44}
we represent $\frac{88}{10000}$? We must here extend our language
a little, and imagine some method by which,
without essentially altering the numerator, it may be
made to show the number of cyphers in the denominator.
Something of the sort has already been done
in representing a number of tens, hundreds, or thousands,
etc.; for $5$~thousands were represented by~$5000$,
in which, by the assistance of cyphers, the $5$~is made
to stand in the place allotted to thousands. If, in the
present instance, we place cyphers at the beginning of
the numerator, until the number of figures and cyphers
together is equal to the number of cyphers in the denominator,
and place a point before the first cypher,
the fraction $\frac{88}{10000}$ will be represented thus,~$.0088$; by
which we understand a fraction whose numerator is~$88$,
and whose denominator is a decimal number containing
four cyphers.
There is a close connexion between the manner of
representing decimal fractions, and the decimal notation
for numbers. Take, for example, the fraction
$217.3426$ or~$\frac{2173426}{10000}$. You will recollect that $2173426$
is made up of $2000000 + 100000 + 70000 + 3000 +
400 + 20 + 6$. If each of these parts be divided by~$10000$,
and the quotient obtained or the fraction reduced
to its lowest terms, the result is as follows:
\[
\frac{2173426}{10000}
= 200 + 10 + 7
+ \frac{3}{10} + \frac{4}{100} + \frac{2}{1000} + \frac{6}{10000}.
\]
We see, then, that in the fraction $217.3426$ the first
figure~$2$ counts two hundred; the second figure,~$1$,
\PageSep{45}
ten, and the third $7$~units. It appears, then, that all
figures on the left of the decimal point are reckoned
as ordinary numbers. But on the right of that point
we find the figure~$3$, which counts for~$\frac{3}{10}$; $4$,~which
counts for~$\frac{4}{100}$; $2$,~for $\frac{2}{1000}$; and $6$,~for $\frac{6}{10000}$. It appears
therefore, that numbers on the right of the decimal
point decrease as they move towards the right,
each number being one-tenth of what it would have
been had it come one place nearer to the decimal
point. The first figure on the right hand of that point
is so many tenths of a unit, the second figure so many
hundredths of a unit, and so on.
The learner should go through the same investigation
with other fractions, and should demonstrate by
means of the principles of fractions, generally, such
exercises as the following, until he is thoroughly accustomed
to this new method of writing fractions:
\begin{gather*}
\begin{aligned}
.68342 &= .6 + .08 + .003 + .0004 + .00002\Add{,} \\
\text{or}\quad
\frac{68342}{100000}
&= \frac{6}{10} + \frac{8}{100} + \frac{3}{1000}
+ \frac{4}{10000} + \frac{2}{100000}\Add{,}
\end{aligned} \\
.00012 = .0001 + .00002 = \frac{1}{10000} + \frac{2}{100000}\Add{,} \\
\begin{aligned}
%[** TN: Slightly reformatted from the original]
163.499 &= \frac{163429}{1000}
= 163\frac{429}{1000} \\
&= \frac{1634}{10} + \frac{29}{1000}
= \frac{16342}{100} + \frac{9}{1000}, \quad \etc.
\end{aligned}
\end{gather*}
The rules of addition, subtraction, and multiplication
may now be understood. In addition and subtraction,
the keeping the decimal points under one
\PageSep{46}
another is equivalent to reducing the fractions to a
common denominator, as we may show thus: Take
two fractions, $1.5$ and $2.125$, or $\frac{15}{10}$ and $\frac{2125}{1000}$, which,
reducing the first to the denominator of the second,
may be written $\frac{1500}{1000}$ and $\frac{2125}{1000}$. If we add the numerators
together, we find the sum of the fractions $\frac{3625}{1000}$,
or~$3.625$
\[
\begin{array}{c}
2125 \\
1500 \\
\hline
\Strut
3625
\end{array}\qquad\qquad
\begin{array}{l}
2.125 \\
1.5 \\
\hline
\Strut
3.625
\end{array}
\]
The learner can now see the connexion of the rule
given for the addition of decimal fractions with that
for the addition of vulgar fractions. There is the
same connexion between the rules of subtraction. The
principle of the rule of multiplication is as follows:
If two decimal numbers be multiplied together, the
product has as many cyphers as are in both together.
Thus $100 × 1000 = 100000$, $10 × 100 = 1000$,
etc. Therefore the denominator of the product, which
is the product of the denominators, has as many cyphers
as are in the denominators of both fractions,
and since the numerator of the product is the product
of the numerators, the point must be placed in that
product so as to cut off as many decimal places as are
both in the multiplier and the multiplicand. Thus:
\begin{gather*}
\frac{13}{100} × \frac{12}{10} = \frac{156}{1000},
\quad\text{or}\quad .13 × 1.2 = .156; \\
%
\begin{aligned}
\frac{4}{1000} × \frac{6}{100} &= \frac{24}{100000}, \\
\text{or}\quad .004 × .06 &= .00024,\quad \etc.
\end{aligned}
\end{gather*}
\PageSep{47}
It is a general rule, that wherever the number of figures
falls short of what we know ought to be the number
of decimals, the deficiency is made up by cyphers.
It may now be asked, whether all fractions can be
reduced to decimal fractions? It may be answered
that they cannot. It is a principle which is demonstrated
in the science of algebra,---that if a number
be not divisible by a prime number, no multiplication
of that number, by itself, will make it so. Thus $10$~not
being divisible by~$7$, neither $10 × 10$, nor $10 × 10
× 10$, etc., is divisible by~$7$. A consequence of this
is, that since $5$~and~$2$ are the only prime numbers
which will divide~$10$, no fraction can be converted into
a decimal unless its denominator is made up of products,
either of $5$~or~$2$, or of both combined, such as
$5 × 2$, $5 × 5 × 2$, $5 × 5 × 5$, $2 × 2$, etc. To show that
this is the case, take any fraction with such a denominator;
for example, $\dfrac{13}{5 × 5 × 5}$. Multiply the numerator
and denominator by~$2$, once for every~$5$, which is
contained in the denominator, and the fraction will
then become
\[
\frac{13 × 2 × 2 × 2}{5 × 5 × 5 × 2 × 2 × 2},\quad\text{or}\quad
\frac{2 × 2 × 2 × 13}{10 × 10 × 10},
\]
which is~$\frac{140}{1000}$, or~$.104$. In a similar way, any fraction
whose denominator has no other factors than $2$~or~$5$,
can be reduced to a decimal fraction. We first search
for such a number as will, when multiplied by the denominator,
produce a decimal number, and then multiply
\PageSep{48}
both the numerator and denominator by that
number.
No fraction which has any other factor in its denominator
\index{Approximations@{Approximations|EtSeq}}%
\index{Errors!in mathematical computations|EtSeq}%
can be reduced to a decimal fraction exactly.
But here it must be observed that in most
parts of mathematical computation a very small error
is not material. In different species of calculations,
more or less exactness may be required; but even in
the most delicate operations, there is always a limit
beyond which accuracy is useless, because it cannot
be appreciated. For example, in measuring land for
sale, an error of an inch in five hundred yards is not
worth avoiding, since even if such an error were committed,
it would not make a difference which would
be considered as of any consequence, as in all probability
the expense of a more accurate measurement
would be more than the small quantity of land thereby
saved would be worth. But in the measurement of a
line for the commencement of a trigonometrical survey,
an error of an inch in five hundred yards would
be fatal, because the subsequent processes involve
calculations of such a nature that this error would be
multiplied, and cause a considerable error in the final
result. Still, even in this case, it would be useless
to endeavor to avoid an error of one-thousandth part
of an inch in five hundred yards; first, because no instruments
hitherto made would show such an error:
and secondly, because if they could, no material difference
\PageSep{49}
would be made in the result by a correction of
it. Again, we know that almost all bodies are lengthened
in all directions by heat. For example: A brass
ruler which is a foot in length to-day, while it is cold,
will be more than a foot to-morrow if it is warm. The
difference, nevertheless, is scarcely, if at all, perceptible
to the naked eye, and it would be absurd for a
carpenter, in measuring a few feet of mahogany for a
table, to attempt to take notice of it; but in the measurement
of the base of a survey, which is several miles
in length and takes many days to perform, it is necessary
to take this variation into account, as a want of
attention to it may produce perceptible errors in the
result: nevertheless, any error which has not this effect,
it would be useless to avoid even were it possible.
We see, therefore, that we may, instead of a
fraction, which cannot be reduced to a decimal, substitute
a decimal fraction, if we can find one so near
to the former, that the error committed by the substitution
will not materially affect the result. We will
now proceed to show how to find a series of decimal
fractions, which approach nearer and nearer to a given
fraction, and also that, in this approximation, we may
approach as near as we please to the given fraction
without ever being exactly able to reach it.
Take, for example,, the fraction~$\frac{7}{11}$. If we divide
the series of numbers $70$,~$700$, $7000$,~etc., by~$11$, we
shall obtain the following results:
\PageSep{50}
\[
\ArrayCompress[1.2]
\begin{array}{lcrccr}
\frac{70}{11} & \PadTxt{gives quotient}{ gives the quotient} & 6
& \text{, and the remainder } &4 & \text{, and is $6\frac{4}{11}$}\Add{,} \\
\frac{700}{11} & \Ditto & 63 & \Ditto &7 & 63\frac{7}{11}\Add{,} \\
\frac{7000}{11} & \Ditto & 636 & \Ditto &4 & 636\frac{4}{11}\Add{,} \\
\frac{70000}{11} & \Ditto & 6363 & \Ditto &7 & 6363\frac{7}{11}\Add{,} \\
\etc.\Add{,} & & \etc.\Add{,} & & \PadTxt{}{\etc.}
\end{array}
\]
Now observe that if two numbers do not differ by
so much as~$1$, their tenth parts do not differ by so
much as~$\frac{1}{10}$, their hundredth parts by so much as~$\frac{1}{100}$,
their thousandth parts by so much as~$\frac{1}{1000}$, and so on;
and also remember that $\frac{7}{11}$~is the tenth part of~$\frac{70}{11}$, the
hundredth part of~$\frac{700}{11}$, and so on. The two following
tables will now be apparent:
\[
\ArrayCompress[1.2]
\begin{array}{ccrcc}
\frac{70}{11} & \PadTxt[l]{does not differ fro}{ does not differ from} & 6
& \text{ by so much as } &1\Add{,} \\
\frac{700}{11} & \Ditto & 63 & \Ditto &1\Add{,} \\
\frac{7000}{11} & \Ditto & 636 & \Ditto &1\Add{,} \\
\frac{70000}{11} & \Ditto & 6363 & \Ditto &1\Add{,} \\
\etc.\Add{,} & & \etc.\Add{,} & & \PadTxt{}{\etc.}
\end{array}
\]
Therefore
\[
\ArrayCompress[1.2]
\begin{array}{ccrclcrcl}
\frac{7}{11} & \PadTxt[l]{does not differfrom}{ does not differ from}
& \frac{6}{10} & \text{ or } & .6, &\PadTxt{so much as}{by so much as }
& \frac{1}{10} & \text{ or } & .1\Add{,} \\
\frac{7}{11} & \Ditto
& \frac{63}{100} & \Ditto & .63 & \Ditto
& \frac{1}{100} & \Ditto & .01\Add{,} \\
\frac{7}{11} & \Ditto
& \frac{636}{1000} & \Ditto & .636 & \Ditto
& \frac{1}{1000} & \Ditto & .001\Add{,} \\
\frac{7}{11} & \Ditto
& \frac{6363}{10000} & \Ditto & .6363 & \Ditto
& \frac{1}{10000} & \Ditto & .0001\Add{,} \\
\PadTxt[l]{}{\etc.\Add{,}} & & \etc.\Add{,} & & & & \etc.
\end{array}
\]
We have then a series of decimal fractions, viz., $.6$,
$.63$, $.636$, $.6363$, $.63636$, etc., which continually approach
more and more near to~$\frac{7}{11}$, and therefore in
any calculation in which the fraction~$\frac{7}{11}$ appears, any
one of these may be substituted for it, which is sufficiently
near to suit the purpose for which the calculation
is intended. For some purposes $.636$~would be a
\PageSep{51}
\index{Circulating decimals}%
sufficient approximation; for others $.63636363$ would
be necessary. Nothing but practice can show how
far the approximation should be carried in each case.
The division of one decimal fraction by another is
performed as follows: Suppose it required to divide
$6.42$ by~$1.213$. The first of these is~$\frac{642}{100}$, and the second
$\frac{1233}{1000}$. The quotient of these by the ordinary rule
is~$\frac{642000}{121300}$, or~$\frac{6420}{1213}$. This fraction must now be reduced
to a decimal on the principles of the last article, by
the rule usually given, either exactly, or by approximation,
according to the nature of the factors in the
denominator.
When the decimal fraction corresponding to a common
fraction cannot be exactly found, it always happens
that the series of decimals which approximates
to it, contains the same number repeated again and
again. Thus, in the example which we chose, $\frac{7}{11}$~is
more and more nearly represented by the fractions $.6$,
$.63$, $.636$, $.6363$, etc., and if we carried the process on
without end, we should find a decimal fraction consisting
entirely of repetitions of the figures~$63$ after the
decimal point. Thus, in finding~$\frac{1}{7}$, the figures which
are repeated in the numerator are~$142857$. This is
what is commonly called a circulating decimal, and
rules are given in books of arithmetic for reducing
them to common fractions. We would recommend
to the beginner to omit all notice of these fractions,
as they are of no practical use, and cannot be thoroughly
understood without some knowledge of algebra.
\PageSep{52}
It is sufficient for the student to know that he
can always either reduce a common fraction to a decimal,
or find a decimal near enough to it for his purpose,
though the calculation in which he is engaged
requires a degree of accuracy which the finest microscope
will not appreciate. But in using approximate
decimals there is one remark of importance, the necessity
for which occurs continually.
Suppose that the fraction $2.143876$ has been obtained,
and that it is more than sufficiently accurate
for the calculation in which it is to be employed. Suppose
that for the object proposed it is enough that
each quantity employed should be a decimal fraction
of three places only, the quantity $2.143876$ is made up
of~$2.143$, as far as three places of decimals are concerned,
which at first sight might appear to be what
we ought to use, instead of~$2.143876$. But this is not
the number which will in this case give the utmost
accuracy which three places of decimals will admit
of; the common usages of life will guide us in this
case. Suppose a regiment consists of $876$~men, we
should express this in what we call round numbers,
which in this case would be done by saying how many
hundred men there are, leaving out of consideration
the number~$76$, which is not so great as~$100$; but in
doing this we shall be nearer the truth if we say that
the regiment consists of $900$~men instead of~$800$, because
$900$~is nearer to~$876$ than~$800$. In the same
manner, it will be nearer the truth to write $2.144$ instead
\PageSep{53}
of~$2.143$, if we wish to express $2.143876$ as nearly
as possible by three places of decimals, since it will
be found by subtraction that the first of these is nearer
to the third than the second. Had the fraction been
$2.14326$, it would have been best expressed in three
places by~$2.143$; had it been~$2.1435$, it would have
been equally well expressed either by $2.143$ or~$2.144$,
both being equally near the truth; but $2.14351$ is a
little more nearly expressed by~$2.144$ than by~$2.143$.
We have now gone through the leading principles
\index{Commercial arithmetic}%
of arithmetical calculation, considered as a part of
general Mathematics. With respect to the commercial
rules, usually considered as the grand object of
an arithmetical education, it is not within the scope
of this treatise to enter upon their consideration. The
mathematical student, if he is sufficiently well versed
in their routine for the purposes of common life, may
postpone their consideration until he shall have become
familiar with algebraical operations, when he
will find no difficulty in understanding the principles
or practice of any of them. He should, before commencing
the study of algebra, carefully review what
\index{Algebra!advice@{advice on the study of}}%
he has learnt in arithmetic, particularly the reasonings
which he has met with, and the use of the signs which
have been introduced. Algebra is at first only arithmetic
under another name, and with more general
symbols, nor will any reasoning be presented to the
student which he has not already met with in establishing
the rules of arithmetic. His progress in the
\PageSep{54}
former science depends most materially, if not altogether,
upon the manner in which he has attended to
the latter; on which account the detail into which we
have entered on some things which to an intelligent
person are almost self-evident, must not be deemed
superfluous.
When the student is well acquainted with the principles
and practice of arithmetic, and not before, he
should commence the study of algebra. It is usual
\index{Algebra!advice@{advice on the study of}}%
to begin algebra and geometry together, and if the
student has sufficient time, it is the best plan which
he can adopt. Indeed, we see no reason why the elements
of geometry should not precede those of algebra,
and be studied together with arithmetic. In this
case the student should read some treatise which relates
to geometry, first. It is hardly necessary to say
that though we have adopted one particular order,
yet the student may reverse or alter that order so as
to suit the arrangement of his own studies.
We now proceed to the first principles of algebra,
and the elucidation of the difficulties which are found
from experience to be most perplexing to the beginner.
We suppose him to be well acquainted with
what has been previously laid down in this treatise,
particularly with the meaning of the signs $+$,~$-$,~$×$,
and the sign of division.
\index{Decimal!fractions|)}%
\PageSep{55}
\Chapter{VI.}{Algebraical Notation and Principles.}
\index{Algebra!notation of|EtSeq}%
\index{Shorthand symbols}%
\index{Notation!algebraical@{algebraical|EtSeq}}%
\First{Whenever} any idea is constantly recurring,
the best thing which can be done for the perfection
of language, and consequent advancement of
knowledge, is to shorten as much as possible the sign
which is used to stand for that idea. All that we have
accomplished hitherto has been owing to the short
and expressive language which we have used to represent
numbers, and the operations which are performed
upon them. The first step was to write simple
signs for the first numbers, instead of words at full
\index{Signs!arithmetical and algebraical}%
length, such as $8$~and~$7$, instead of eight and seven.
The next was to give these signs an additional meaning,
according to the manner in which they were connected
with one another; thus $187$~was made to represent
one hundred added to eight tens added to seven.
The next was to give by new signs directions when to
perform the operations of addition, subtraction, multiplication,
and division; thus $5 + 8$ was made to represent
$8$~added to~$5$, and so on. With these signs
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reasonings were made, and truths discovered which
are common to all numbers; not at once for every
number, but by taking some example, by reasoning
upon it, and by producing a result; this result led to
a rule which was declared to be a rule which held
equally good for all numbers, because the reasoning
which produced it might have been applied to any
other example as well as to the one which was chosen.
In this way we produced some results, and might have
produced many more; the following is an instance:
half the sum of two numbers added to half their difference,
gives the greater of the two numbers. For example,
take $16$~and~$10$, half their sum is~$13$, half their
difference is~$3$; if we add $13$~and~$3$ we get~$16$, the
greater of the two numbers. We might satisfy ourselves
of the truth of this same proposition for any
other numbers, such as $27$~and~$8$, $15$~and~$19$, and so
on. If we then make use of signs, we find the following
truths:
\begin{alignat*}{2}
\frac{16 + 10}{2} &+ \frac{16 - 10}{2} &&= 16\Chg{.}{,} \\
\frac{27 + 8}{2} &+ \frac{27 - 8}{2} &&= 27\Chg{.}{,} \\
\frac{15 + 9}{2} &+ \frac{15 - 9}{2} &&= 15,
\end{alignat*}
and so on. If, then, we choose any two numbers,
and call them the first and second numbers, and call
that the first number which is the greater of the two,
we have the following:
\PageSep{57}
\index{Literal notation|EtSeq}%
\[
\small
\frac{\text{First No.} + \text{Second No.}}{2}
+ \frac{\text{First No.} - \text{Second No.}}{2}
= \text{First No.}
\]
In this way we might express anything which is true
of all numbers, by writing First~No., Second~No., etc.,
for the different numbers which enter into our proposition,
and we might afterwards suppose the First~No.,
the Second~No., etc., to be any which we please.
In this way we might write down the following assertion,
which we should find to be always true:
\begin{multline*}
(\text{First No.} + \text{Second No.}) ×
(\text{First No.} - \text{Second No.}) \\
= \text{First No.} × \text{First No.} - \text{Second No.} × \text{Second No.}
\end{multline*}
When any sentence expresses that two numbers or
collections of numbers are equal to one another, it is
called an \emph{equation}\footnote
{As now usually defined an \emph{equation} always contains an unknown quantity.
See also \PageRef[p.]{91}.---\Ed.}
thus $7 + 5 = 12$ is an equation, and
the sentences written just above are equations.
Now the next question is, could we not avoid the
trouble of writing First~No., Second~No., etc., so frequently?
This is done by putting letters of the alphabet
to stand for these numbers. Suppose, \eg, we
let $x$~stand for the first number, and $y$~for the second,
the two assertions already made will then be written:
\begin{gather*}
\frac{x + y}{2} + \frac{x - y}{2} = x\Chg{.}{,} \\
(x + y) × (x - y) = x × x - y × y.
\end{gather*}
By the use of letters we are thus enabled to write
sentences which say something of all numbers, with a
\PageSep{58}
very small part only of the time and trouble necessary
for writing the same thing at full length. We now
proceed to enumerate the various symbols which are
used.
1. The letters of the alphabet are used to stand
for numbers, and whenever a letter is used it means
either that any number may be used instead of that
letter, or that the number which the letter stands for
is not known, and that the letter supplies its place in
all the reasonings until it is known.
2. The sign~$+$ is used for addition, as in arithmetic.
Thus $x + z$ is the sum of the numbers represented
by $x$~and~$z$. The following equations are sufficiently
evident:
\begin{gather*}
x + y + z = x + z + y = y + z + x. \\
\text{If $a = b$, then } a + c = b + c,\quad
a + c + d = b + c + d,\quad \etc.
\end{gather*}
3. The sign~$-$ is used for subtraction, as in arithmetic.
The following equations will show its use:
\begin{gather*}
\begin{aligned}
x + a - b - c + e
&= x + a + e - b - c \\
&= a - c + e - b + x.
\end{aligned} \\
\text{If $a = b$,\Add{ then} }
a - c = b - c,\quad
a - c + d = b - c + d,\quad \etc.
\end{gather*}
4. The sign~$×$ is used for multiplication as in
arithmetic, but when two numbers represented by letters
are multiplied together it is useless, since $a × b$
can be represented by putting $a$~and~$b$ together thus,~$ab$.
Also $a × b × c$ is represented by~$abc$; $a × a × a$,
for the present we represent thus,~$aaa$. When two
numbers are multiplied together, it is necessary to
\PageSep{59}
\index{Expressions, algebraical}%
keep the sign~$×$; otherwise $7 × 5$ or~$35$ would be mistaken
for~$75$. It is, however, usual to place a point
between two numbers which are to be multiplied together;
thus $7 × 5 × 3$ is written $7·5·3$. But this
point may sometimes be mistaken for the decimal
point: this will, however, be avoided by always writing
the decimal point at the head of the figure, viz.,
by writing $\frac{23461}{100}$ thus, $234\Dpt 61$.
5. Division is written as in arithmetic: thus, $\dfrac{a}{b}$~signifies
that the number represented by~$a$ is to be divided
by the number represented by~$b$.
6. All collections of numbers are called expressions;
thus, $a + b$, $a + b - c$, $aa + bb - d$, are algebraical
expressions.
7. When two expressions are to be multiplied together,
it is indicated by placing them side by side,
and inclosing each of them in brackets. Thus, if
$a + b + c$ is to be multiplied by $d + e + f$, the product
is written in any of the following ways:
\begin{gather*}
(a + b + c)(d + e + f), \\
[a + b + c][d + e + f], \\
\{a + b + c\}\{d + e + f\}, \\
\overline{a + b + c} · \overline{d + e + f}.
%[** TN: Omitted resh-like grouping mark, which De Morgan never uses]
\end{gather*}
8. That $a$~is greater than~$b$ is written thus, $a > b$.
9. That $a$~is less than~$b$ is written thus, $a < b$.
10. When there is a product in which all the factors
are the same, such as~$xxxxx$, which means that
\PageSep{60}
\index{Indices, theory of}%
five equal numbers, each of which is represented by~$x$,
are multiplied together, the letter is only written
once, and above it is written the number of times
which it occurs, thus $xxxxx$~is written~$x^{5}$. The following
table should be carefully studied by the student:
\begin{align*}
&\text{$x × x$ or $xx$ is written $x^{2}$,} \\
&\qquad\text{and is called the square, or second power of~$x$\Chg{.}{,}} \\
&\text{$x × x × x$ or $xxx$ is written $x^{3}$,} \\
&\qquad\text{and is called the cube, or third power of~$x$\Chg{.}{,}} \\
&\text{$x × x × x × x$ or $xxxx$ is written $x^{4}$,} \\
&\qquad\text{and is called the fourth power of~$x$\Chg{.}{,}} \\
&\text{$x × x × x × x × x$ or $xxxxx$ is written $x^{5}$,} \\
&\qquad\text{and is called the fifth power of~$x$,} \\
&\qquad\qquad\etc., \qquad\qquad\etc., \qquad\qquad\etc.
\end{align*}
There is no point which is so likely to create confusion
in the ideas of a beginner as the likeness between
such expressions as $4x$~and~$x^{4}$. On this account
it would be better for him to omit using the latter expression,
and to put~$xxxx$ in its place until he has
acquired some little facility in the operations of algebra.
If he does not pursue this course, he must recollect
that the~$4$, in these two expressions, has different
names and meanings. In $4x$~it is called a \emph{coefficient},
\index{Coefficient}%
in $x^{4}$~an \emph{exponent} or \emph{index}.
The difference of meaning will be apparent from
the following tables:
\PageSep{61}
\begin{gather*}
\begin{alignedat}{2}
2x &= x + x & x^{2} &= x × x = xx\Add{,} \\
3x &= x + x + x & x^{3} &= x × x × x \quad\text{or}\quad xxx\Add{,} \\
4x &= x + x + x +x\qquad & x^{4} &= x × x × x × x \quad\text{or}\quad xxxx, \\
& \qquad \etc., & & \qquad \etc.
\end{alignedat}
\displaybreak[0] \\
\begin{alignedat}{2}
\text{If } x = 3\Add{,}\quad
2x &= \Z6\Add{,}\quad & x^{2} &= \Z9, \\
3x &= \Z9\Add{,}\quad & x^{3} &= 27, \\
4x &= 12\Add{,}\quad & x^{4} &= 81\Typo{,}{.}
\end{alignedat}
\end{gather*}
The beginner should frequently \emph{write} for himself
such expressions as the following:
\begin{gather*}
4a^{3} b^{2} = aaabb + aaabb + aaabb + aaabb\Chg{.}{,} \\
5a^{4} x = aaaax + aaaax + aaaax + aaaax + aaaax\Chg{.}{,} \\
9a^{2} b^{3} + 4ab^{4} = 9aabbb + 4abbbb\Chg{.}{,} \\
\begin{aligned}
\frac{a^{2} + b^{2}}{a^{2} - b^{2}}
= \frac{aa + bb}{aa - bb}
&= \frac{aa}{aa - bb} + \frac{bb}{aa - bb} \\
&= \frac{aa - cc}{aa - bb} + \frac{bb + cc}{aa - bb}\Chg{.}{,}
\end{aligned} \\
\frac{a^{3} - b^{3}}{a^{2} - b^{2}}
= \frac{aaa - bbb}{aa - bb}
= \frac{aa + ab + bb}{a + b}.
\end{gather*}
With many such expressions every book on algebra
will furnish him, and he should then satisfy himself
of their truth by putting some numbers at pleasure
instead of the letters, and making the results agree
with one another. Thus, to try the expression
\[
\frac{a^{3} - b^{3}}{a - b} = a^{2} + ab + b^{2},
\]
or, which is the same,
\[
\frac{aaa - bbb}{a - b} = a^{2} + ab + b^{2}.
\]
Let $a$~stand for~$6$ and $b$~stand for~$4$, then, if this expression
be true,
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\[
\frac{6·6·6 - 4·4·4}{6 - 4} = 6·6 + 6·4 + 4·4,
\]
which is correct, since each of these expressions is
found, by calculation, to be~$76$.
The student should then exercise himself in the
solution of such questions as the following: What is
\[
a^{2} + b^{2} - \frac{ab}{a + b} + \frac{a}{a + b} - a,
\]
I.~when $a$~stands for~$6$, and $b$~for~$5$, II.~when $a$~stands
for~$13$, and $b$~for~$2$, and so on. He should stop here
until he has, by these means, made the signs familiar
to his eye and their meaning to his mind; nor should
he proceed to any further algebraical operations until
he can readily find the value of any algebraical expression
when he knows the numbers which the letters
stand for. He cannot, at this period of his course,
write too many algebraical expressions, and he must
particularly avoid slurring over the sense of what he
has before him, and must write and rewrite each expression
until the meaning of the several parts forces
itself upon his memory at first sight, without even
the necessity of putting it in words. It is the neglecting
to do this which renders the operations of algebra
\index{Algebra!advice@{advice on the study of}}%
so tedious to the beginner. He usually proceeds to
the addition, subtraction, etc., of symbols, of the
meaning of which he has but an imperfect idea, and
which have been newly introduced to him in such
numbers that perpetual confusion is the consequence.
We cannot, therefore, use too many arguments to induce
\PageSep{63}
him not to mind the drudgery of reducing algebraical
expressions into figures. This is the connecting
link between the new science and arithmetic, and,
unless that link be well fastened, the knowledge which
he has previously acquired in arithmetic will help him
but little in acquiring algebra.
The order of the terms of any algebraical expression
may be changed without changing the value of
that expression. This needs no proof, and the following
are examples of the change:
\begin{multline*}
%[** TN: Moved equality signs to start of each line]
a + b + ab + c + ac - d - e - de - f \\
\begin{aligned}
&= a - d + b - e + ab - de + c - f + ac \\
&= a + b - d - e - de - f + ac + c + ab \\
&= ab + ac - de + a + b + c - e - f - d.
\end{aligned}
\end{multline*}
When the first term changes its place, as in the fourth
of these expressions, the sign~$+$ is put before it, and
should, properly speaking, be written wherever there
is no sign, to indicate that the term in question increases
the result of the rest, but it is usually omitted.
The negative sign is often written before the first
term, as in the expression $-a + b$: but it must be recollected
that this is written on the supposition that
$a$~is subtracted from what comes after it.
When an expression is written in brackets, with
some sign before it, such as $a - (b - c)$, it is understood
that the expression in brackets is to be considered
as one quantity, and that its result or total is to
be connected with the rest by the sign which precedes
the brackets. In this example it is the \emph{difference} of $b$
\PageSep{64}
and~$c$ which is to be subtracted from~$a$. If $a = 12$,
$b = 6$, and $c = 4$, this is~$10$. In the expression $a - b$
made by subtracting $b$ from~$a$, too much has been subtracted
by the quantity~$c$, since it is not~$b$, but $b - c$,
which must be subtracted from~$a$. In order, therefore,
to make $a - (b - c)$, $c$~must be added to~$a - b$, which
gives $a - b + c$. Therefore, $a - (b - c) = a - b + c$.
Similarly
\begin{gather*}
a + b - (c + d - e - f) = a + b - c - d + e + f, \\
(ax^{2} - bx + c) - (dx^{2} - ex + f) \\
= ax^{2} - bx + c - dx^{2} + ex - f.
\end{gather*}
When the positive sign is written before an expression
in brackets, the brackets may be omitted
altogether, unless they serve to show that the expression
in question is multiplied by some other. Thus,
instead of $(a + b + c) + (d + e + f)$, we may write
$a + b + c + d + e + f$, which is, in fact, only saying
that two wholes may be added together by adding together
all the parts of which they are composed. But
the expression $a + (b + c)(d + e)$ must not be written
thus: $a + b + c(d + e)$, since the first expresses that
$(b + c)$ must be multiplied by~$(d + e)$ and the product
added to~$a$, and the second that $c$~must be multiplied
by~$(d + e)$ and the product added to~$a + b$. If $a$,~$b$, $c$,
$d$, and~$e$, stand for $1$,~$2$, $3$, $4$, and~$5$, the first is~$46$ and
the second~$30$.
When two or more quantities have exactly the
same letters repeated the same number of times, such
as $4a^{2} b^{3}$, and~$6a^{2} b^{3}$, they may be reduced into one by
\PageSep{65}
merely adding the coefficients and retaining the same
letters. Thus, $2a + 3a$ is~$5a$, $6bc - 4bc$ is~$2bc$,
$3(x + y) + 2(x + y)$ is~$5(x + y)$. These things are
evident, but beginners are very liable to carry this
farther than they ought, and to attempt to reduce expressions
which do not admit of reduction. For example,
they will say that $3b + ^{2}$ is~$4b$ or~$4b^{2}$, neither
of which is true, except when $b$~stands for~$1$. The expression
$3b + b^{2}$, or $3b + bb$, cannot be made more
simple until we know what $b$~stands for. The following
table will, perhaps, be of service:
\begin{alignat*}{2}
&6a^{2} b^{3} + 3a^{3} b^{2}\ &&\text{is not } 9a^{5} b^{5}\Add{,} \\
&6a^{3} - 4a^{2} &&\text{is not } 2a\Add{,} \\
&2ba + 3b &&\text{is not } 5ab.
\end{alignat*}
Such are the mistakes which beginners almost universally
make, mostly for want of a moment's consideration.
They attempt to reduce quantities which
cannot be reduced, which they do by adding the exponents
of letters as well as their coefficients, or by
collecting several terms into one, and leaving out the
signs of addition and subtraction. The beginner cannot
too often repeat to himself that two terms can
never be made into one, unless both have the same
letters, each letter being repeated the same number
of times in both, that is, having the same index in
both. When this is the case, the expressions may be
reduced by adding or subtracting the coefficients according
to the sign, and affixing the common letters
with their indices. When there is no coefficient, as
\PageSep{66}
in the expression~$a^{2} b$, the quantity represented by~$a^{2} b$
being only taken once, $1$~is called the coefficient.
Thus,
\begin{gather*}
3ab + 4ab + 6ab - ab -7ab = 5ab\Add{,} \\
6xy^{2} + 3xy^{2} - 5xy^{2} + xy^{2} = 5xy^{2}.
\end{gather*}
The student must also recollect that he is not at liberty
to change an index from one letter to another, as
by so doing he changes the quantity represented.
Thus $a^{4} b$ and $ab^{4}$ are quantities totally distinct, the
first representing~$aaaab$ and the second~$abbbb$. The
difference in all the cases which we have mentioned
will be made more clear, by placing numbers at pleasure
instead of letters in the expressions, and calculating
their values; but, in conclusion, the following remark
must be attended to. If it were asserted that the
expression $\dfrac{a^{2} + b^{2}}{a + b}$ is the same as $a + b - \dfrac{2ab}{2a - b}$, and
we wish to proceed to see whether this is always the
case or no, if we commence accidentally by supposing
$b$~to stand for~$2$ and $a$~for~$4$, we shall find that the first
is the same as the second, each being~$3\frac{1}{2}$. But we
must not conclude from this that they are always the
same, at least until we have tried whether they are so,
when other numbers are substituted for $a$~and~$b$. If
we place $6$~and~$8$ instead of $a$~and~$b$, we shall find that
the two expressions are not equal, and therefore we
must conclude that they are not always the same.
Thus in the expressions $3x - 4$ and $2x + 8$, if $x$~stand
for~$12$, these are the same, but if it stands for any
other number they are not the same.
\PageSep{67}
\Chapter{VII.}{Elementary Rules of Algebra.}
\index{Algebra!elementary rules of|EtSeq}%
\First{The} student should be very well acquainted with
the principles and notation hitherto laid down
before he proceeds to the algebraical rules for addition
\index{Addition}%
and subtraction. He should then take some simple
examples of each, and proceed to find the sum
and difference by reasoning as follows. Suppose it is
required to add $c - d$ to~$a - b$. The direction to do
this may either be written in the common way thus:
\[
\begin{array}{rc}
& a - b \\
& c - d \\
\cline{2-2}
\Strut\text{Add }
\end{array}
\]
or more properly thus: Find $(a - b) + (c - d)$.
If we add $c$ to~$a$, or find $a + c$, we have too much;
first, because it is not~$a$ which is to be increased by
$c - d$ but $a - b$; this quantity must therefore be decreased
by~$b$ on this account, or must become $a + c - b$;
but this is still too great, because it is not~$c$ which was
to be added but $c - d$; it must therefore be decreased
by~$d$ on this account, or must become $a + c - b - d$ or
\PageSep{68}
$a - b + c - d$. From a few reasonings of this sort the
rule may be deduced; and not till then should an example
be chosen so complicated as to make the student
lose sight for one moment of his demonstration.
The process of subtraction we have already performed\Typo{.}{,}
and from a few examples of this method the rule may
be deduced.
The multiplication of~$a$ by~$c - d$ is performed thus:
\index{Multiplication@{Multiplication|EtSeq}}%
$a$~is to be taken $c - d$~times. Take it first $c$~times or
find~$ac$. This is too great, because $a$~has been taken
too many times by~$d$. From~$ac$ we must therefore
subtract $d$~times~$a$, or~$ad$, and the result is that
\[
a(c - d) = ac - ad.
\]
This may be verified from arithmetic, in which the
same process is shown to be correct; and this whether
the numbers $a$,~$c$, and~$d$ are whole or fractional. For
example, it will be found that $6(14 - 9)$ or~$6 × 5$ is
the same as $6 × 14 - 6 × 9$, or as $84 - 54$. Also that
$\frac{2}{3}(\frac{1}{7} - \frac{2}{15})$, or $\frac{2}{3} × \frac{1}{105}$ is the same as $\frac{2}{3} × \frac{1}{7} - \frac{2}{3} × \frac{2}{15}$,
or as $\frac{2}{21} - \frac{4}{45}$. Upon similar reasoning the following
equations may be proved:
\begin{gather*}
a(b + c - d) = ab + ac - ad\Chg{.}{,} \\
(p + pq - ar)xz = pxz + pqxz - arxz\Chg{.}{,} \\
(a^{2} + 2b^{2})b^{2},\quad\text{or}\quad
(aa + 2bb)bb = aabb + 2bbbb = a^{2} b^{2} + 2b^{4}.
\end{gather*}
Also when a multiplication has been performed, the
process may be reversed and the factors of it may be
given. Thus, on observing the expression
\PageSep{69}
\begin{align*}
&ab - ac + a^{2}, \\
\text{or}\quad & ab - ac + aa,
\end{align*}
we see that in its formation every term has been multiplied
by~$a$; that is, it has been made by multiplying
\begin{gather*}
b - c + a \text{ by } a, \\
\text{or $a$ by $b - c + a$}.
\end{gather*}
There will now be no difficulty in perceiving that
\begin{gather*}
\begin{aligned}
ac + ad + bc + bd
&= a(c + d) + b(c + d) \\
&= (a + b)(c + d),
\end{aligned} \\
a^{2} - ab^{2} + 2abc - dc + 3a
= a(a - b^{2} + 3) + c(2ba - d).
\end{gather*}
It is proved in arithmetic that if numbers, whether
whole or fractional, are multiplied together, the product
remains the same when the order in which they
are multiplied is changed. Thus $6 × 4 × 3 = 3 × 6 × 4 = 4 × 6 × 3$,
etc., and $\frac{2}{3} × \frac{4}{5} = \frac{4}{5} × \frac{2}{3}$, etc. Also, that a
part of the multiplication may be made, and the partial
product substituted instead of the factors which
produced it, thus, $3 × 4 × 5 × 6$ is $12 × 5 × 6$, or $15 × 4 × 6$,
or $90 × 4$. From these rules two complicated single
terms may be multiplied together, and the product
represented in the most simple manner which the case
admits of. Thus if it be required to multiply
\begin{gather*}
6 a^{3} b^{4} c,\text{ which is } 6\, aaa\, bbbb\, c \\
\text{by $12a^{2} b^{3} c^{3} d$, which is $12\, aa\, bbb\, ccc\, d$},
\end{gather*}
the product is written thus:
\[
6\, aaa\, bbbb\, c\, 12\, aa\, bbb\, ccc\, d,
\]
\PageSep{70}
which multiplication may be performed in the following
order
\[
6 × 12\, aaaaa\, bbbbbbb\, cccc\, d,
\]
which is represented by $72a^{5} b^{7} c^{4} d$. A few examples
of this sort will establish the rule for the multiplication
of such quantities which is usually given in the
treatises on Algebra.
It is to be recollected that in every algebraical
formula which is true of all numbers, any algebraical
expression may be substituted for one of the letters,
provided care is taken to make the substitution wherever
that letter occurs. Thus from the formula:
\[
a^{2} - b^{2} = (a + b)(a - b)
\]
we may deduce the following by making substitutions
for~$a$. If this formula be always true, it is true when
$a$~is equal to~$p + q$, that is, it is true if $p + q$ be put
instead of~$a$ wherever that letter occurs in the formula.
Therefore,
\[
(p + q)^{2} - b^{2} = (p + q + b)(p + q - b).
\]
Similarly,
\begin{gather*}
(b + m)^{2} - b^{2} = (2b + m)m, \\
(x + y)^{2} - (x - y)^{2}
= (x + y + x - y)(x + y - \overline{x - y})
= 4xy,
\end{gather*}
and so on.
We have already established the formula,
\[
(p - q)a = ap - aq.
\]
Instead of~$a$ let us put~$r - s$, and this formula becomes
\[
(p - q)(r - s) = (r - s)p - (r - s)q.
\]
\PageSep{71}
But
\[
(r - s)p = pr - ps, \quad\text{and}\quad
(r - s)q = qr - qs.
\]
Therefore
\begin{align*}
(p - q)(r - s)
&= pr - ps - (qr - qs) \\
&= pr - ps - qr + qs.
\end{align*}
By reasoning in the same way we may prove that
\[
(p - q)(r + s) = pr + ps - qr - qs\Add{.}
\]
A few examples of this sort will establish what is
called the rule of signs in multiplication; viz., that a
term of the multiplicand multiplied by a term of the
multiplier has the sign~$+$ before it if the terms have
the same sign, and $-$~if they have different signs.
But here the student must avoid using an incorrect
mode of expression, which is very common, viz., the
saying that $+$~multiplied by~$+$ gives~$+$; $-$~multiplied
by~$+$ gives~$-$; and so on. He must recollect that
the signs $+$~and~$-$ are not quantities, but \emph{directions}
to add and subtract, and that, as has been well said
by one of the most luminous writers on algebra in our
language, we might as well say, that take away multiplied
by take away gives add, as that $-$~multiplied by~$-$
gives~$+$.\footnote
{Frend, \Title{Principles of Algebra}. The author of this treatise is far from
\index{Frend|FN}%
agreeing with the work which he has quoted in the rejection of the isolated
negative sign which prevails throughout it, but fully concurs in what is there
said of the methods then in use for explaining the difficulties of the negative
sign.}
The only way in which the student should accustom
himself to state this rule is the following: ``In
\PageSep{72}
multiplying two algebraical expressions, multiply each
term of the one by each term of the other, and wherever
two terms are preceded by the same sign put $+$
before the product of the two; when the signs are
different put the sign~$-$ before their product.''
If the student should meet with an equation in
which positive and negative signs stand by themselves,
\index{Minus quantities}%
\index{Negative!quantities}%
such as
\[
+ab × -c = -abc,
\]
let him, for the present, reject the example in which
it occurs, and defer the consideration of such equations
until he has read the explanation of them to
which we shall soon come. Above all, he must reject
the definition still sometimes given of the quantity~$-a$,
that it is less than nothing. It is astonishing that
the human intellect should ever have tolerated such
an absurdity as the idea of a quantity less than nothing;\footnote
{For a full critical and historical discussion of this point, see Duhamel.
\index{Duhamel}%
\Title{Des méthodes dans les sciences de raisonnement}, 2\Ord{me}~partie, chap.~\textsc{xix}. (third
edition, Paris, Gauthier-Villars, 1896).---\Editor.}
above all, that the notion should have outlived
the belief in judicial astrology and the existence of
witches, either of which is ten thousand times more
possible.
These remarks do not apply to such an expression
as $-b + a$, which we sometimes write instead of~$a - b$,
as long as it is recollected that the one is merely used
to stand for the other, and for the present $a$~must be
considered as greater than~$b$.
\PageSep{73}
In writing algebraical expressions, we have seen
\index{Arrangment of algebraical expressions}%
that various arrangements may be adopted. Thus
$ax^{2} - bx + c$ may be written as $c + ax^{2} - bx$, or $-bx + c + ax^{2}$.
Of these three the first is generally chosen,
because the highest power of~$x$ is written first; the
highest but one comes next; and last of all the term
which contains no power of~$x$. When written in this
way the expression is said to be arranged in descending
powers of~$x$; had it been written thus, $c - bx + ax^{2}$,
it would have been arranged in ascending powers of~$x$;
in either case it is said to be arranged in powers
of~$x$, which is called the principal letter. It is usual
to arrange all expressions which occur in the same
question in powers of the same letter, and practice
must dictate the most convenient arrangement. Time
and trouble is saved by this operation, as will be evident
from multiplying two unarranged expressions together,
and afterwards doing the same with the same
expressions properly arranged.
In multiplying two arranged expressions together,
while collecting such terms into one as will admit of
it, it will always be evident that the first and last of
all the products contain powers of the principal letter
which are found in no other part, and stand in the
product unaltered by combination with any other
terms, while in the intermediate products there are
often two or more which contain the same power of
the principal letter, and can be reduced into one.
This will be evident in the following examples:
\PageSep{74}
%[** TN: Set landscape in the original; re-broken, terms aligned by degree]
\begin{align*}
&\begin{array}{*{7}{l}}
\text{Multiply} & x^{6} &{}- 3x^{5} &{}+\Z x^{4} \\
\text{\quad By} & x^{4} &{}- 2x^{2} &{}+\Z x \\
\cline{2-4}
\Strut
\text{The product is\quad}
& x^{10} &{}- 3x^{9} &{}+\Z x^{8} \\
& & &{}-2x^{8} &{}+ 6x^{7} &{}- 2x^{6} \\
& & & &{}+\Z x^{7} &{}- 3x^{6} &{}+ x^{5} \\
\cline{2-7}
\Strut
\text{Or}
& x^{10} &{}- 3x^{9} &{}-\Z x^{8} &{}+ 7x^{7} &{}- 5x^{6} &{}+ x^{5}\rlap{\Add{.}}
\end{array} \displaybreak[0] \\[8pt]
%
&\begin{array}{*{7}{l}}
\text{Multiply} & ax^{3} &{}+ bx^{2} &{}+ cx \\
\text{\quad By} & dx^{2} &{}+ ex &{}+\Z f \\
\cline{2-4}
\Strut
\text{The product is\quad}
& ad x^{5} &{}+ bd x^{4} &{}+ cd x^{3} \\
& &{}+ ae x^{4} &{}+ be x^{3} &{}+ ce x^{2} \\
& & &{}+ af x^{3} &{}+ bf x^{2} &{}+ cf x \\
\cline{2-6}
\Strut
\text{Or\quad}
& ad x^{5} &{}+ \PadTo[l]{bd x^{4}}{(bd + ae)x^{4}} \\
& & &{}+ \PadTo[l]{cd x^{3}}{(cd + be + af)x^{3}} \\
& & & &{}+ \PadTo[l]{ce x^{2}}{(ce + bf)x^{2}} \\
& & & & &{} + cf x\rlap{.}
\end{array}
\end{align*}
It is plain from the rule of multiplication, that the
highest power of~$x$ in a product must be formed by
multiplying the highest power in one factor by the
highest power in the other, or when the two factors
have been arranged in descending powers, the \emph{first}
power in one by the first power in the other. Also,
that the lowest power of~$x$, or should it so happen,
\PageSep{75}
the term in which there is no power of~$x$, is made by
multiplying the last terms in each factor. These being
the highest and lowest, there can be no other such
power, consequently neither of these terms can coalesce
with any other, as is the case in the intermediate
products. This remark will be of most convenient
application in division, to which we now come.
Division is in all respects the reverse of multiplication.
\index{Division}%
In dividing $a$ by~$b$ we find the answer to this
question: If $a$~be divided into $b$~equal parts, what is
the magnitude of each of those parts? The quotient
is, from the definition of a fraction, the same as the
fraction~$\dfrac{a}{b}$, and all that remains is to see whether that
fraction can be represented by a simple algebraical
expression without fractions or not; just as in arithmetic
\index{Fractions!arithmetical}%
\index{Fractions!algebraical}%
the division of $200$ by~$26$ is the reduction of the
fraction $\frac{200}{26}$ to a whole number, if possible. But we
must here observe that a distinction must be drawn
between algebraical and arithmetical fractions. For
example, $\dfrac{a + b}{a - b}$~is an algebraical fraction, that is, there
is no expression without fractions which is always
equal to~ $\dfrac{a + b}{a - b}$. But it does not follow from this that
the number which $\dfrac{a + b}{a - b}$ represents is always an arithmetical
fraction; the contrary may be shown. Let $a$~stand
for~$12$, and $b$~for~$6$, then $\dfrac{a + b}{a - b}$ is~$3$. Again,
$a^{2} + ab$ is a quantity which does not contain algebraical
fractions, but it by no means follows that it may
not represent an arithmetical fraction. To show that
\PageSep{76}
it may, let $a = \frac{1}{2}$ and $b = 2$, then $a^{2} + ab = 1\frac{1}{4}$ or~$\frac{5}{4}$.
Other examples will clear up this point if any doubt
yet exist in the mind of the student. Nevertheless,
the following propositions of arithmetic and algebra,
\index{Arithmetic!compared with algebra}%
which only differ in this, that ``\emph{whole number}'' in the
\index{Simple expression}%
\index{Whole number}%
arithmetical proposition is replaced by ``\emph{simple expression}''\footnote
{By a simple expression is meant one which does not contain the principal
letter in the denominator of any fraction.}
in the algebraical one, connect the two
subjects and render those demonstrations which are
in arithmetic confined to whole numbers, equally true
in algebra as far as regards simple expressions:
\Comparison{%
The sum, difference, or product
of two whole numbers, is a
whole number.%
}{%
The sum, difference, or product
of two simple expressions
is a simple expression.}
%
\Comparison{%
One number is said to be a
measure of another when the
quotient of the two is a whole
number.%
}{%
One expression is said to be a
measure of another when the
quotient of the two is a simple
expression.}
%
\Comparison{%
The greatest common measure
of two whole numbers is the
greatest whole number which
measures both, and is the product
of all the prime numbers
which will measure both.%
}{%
The greatest common measure
of two expressions is the
common measure which has the
highest exponents and coefficients,
and is the product of all
prime simple expressions which
measure both.}
%
\Comparison{%
When one number measures
two others, it measures their
sum, difference, and product.%
}{%
When one expression measures
two others, it measures
their sum, difference, and product.}
%
\Comparison{%
In the division of one number
by another, the remainder is
measured by any number which
measures the dividend and divisor.%
}{%
In the division of one expression
by another, the remainder
is measured by any expression
which measures the dividend
and divisor.}
\PageSep{77}
\Comparison{%
A fraction is not altered by
multiplying or dividing both its
numerator and denominator by
the same quantity.%
}{%
A fractional expression is not
altered by multiplying or dividing
both its numerator and denominator
by the same expression.}
In the term \emph{simple expression} are included those
\index{Simple expression}%
quantities which contain arithmetical fractions, provided
there is no algebraical quantity, or quantity represented
by letters in the denominator; thus $\frac{1}{4} ab + \frac{1}{2}$
is called a simple expression. We now proceed to
the division of one simple expression by another, and
we will take first the case where neither quantity contains
more than one term. For example, what is
$42a^{4} b^{3} c$ divided by $6a^{2} bc$? that is, what quantity must
be multiplied by~$6a^{2} bc$, in order to produce $42 a^{4} b^{3} c$.
This last expression written at length, is $42\, aaaa\, bbb\, c$,
and $42$~is~$6 × 7$. We can then separate this expression
into the product of two others, one of which shall be
$6a^{2} bc$, or~$6aabc$; it will then be $6aabc × 7aabb$,
and it is $7aabb$ which must be multiplied by~$6aabc$
in order to produce $42a^{4} b^{3} c$. A few examples worked
in this way, will lead the student to the rule usually
given in all cases but one, to which we now come.
We have represented $cc$, $ccc$, $cccc$, etc., by $c^{2}$, $c^{3}$, $c^{4}$,
etc., and have called them the second, third, fourth,
etc., powers of~$c$. The extension of this rule would
lead us to represent~$c$ by~$c^{1}$, and call it the first power
of~$c$. Again, we have represented $c + c$, $c + c + c$,
$c + c + c + c$, etc.\ by $2c$, $3c$, $4c$, and have called $2$, $3$,
$4$, etc., the coefficients of~$c$. The extension of this
\PageSep{78}
rule would lead us to write $c$~thus, $1c$, or, rather, if we
attend to the last remark,~$1c^{1}$. This instance leads us
to observe the gradual progress of our language. We
begin with the quantity~$c$ by itself; we proceed in our
course, shortening by new signs the more complicated
combinations of~$c$, and the original quantity~$c$ forces
itself anew upon our attention as a part of the series,
\[
\text{$c$, $2c$, $3c$, $4c$, etc., and $c$, $c^{2}$, $c^{3}$, $c^{4}$, etc.,}
\]
in each of which, except the first, there is a distinct
figure, which is called a coefficient or exponent, according
to its situation. We then deduce rules in
which the terms coefficient or exponent occur, but
which, of course, cannot apply to the first term in
each series, because, as yet, it has neither coefficient
nor exponent. Among such rules are the following:
%[** TN: Next three items in-line in the original]
I.~To add two terms of the first series, add the coefficients,
and affix to the sum the letter~$c$. Thus
$4c + 3c = 7c$.
{\Loosen II.~To multiply two terms of the second
series, add the exponents, and make this sum the
exponent of~$c$. Thus $c^{4} × c^{3} = c^{7}$.}
III.~To divide a term
of the second series by one which comes before it, subtract
the exponent of the divisor from the exponent
of the dividend, and make this difference the exponent
of~c. Thus,
\[
\frac{c^{7}}{c^{4}} = c^{3}.
\]
These rules are intelligible for all terms of the
series except the first, to which, nevertheless, they
will apply if we agree that $1c^{1}$~shall represent~$c$, as
will be evident by applying either of them to find
\PageSep{79}
$4c + c$, $c^{4} × c$, or~$\dfrac{c^{4}}{c}$. We therefore \emph{agree} that $1c^{1}$~shall
stand for~$c$, and although $c$~is not written thus, it must
be remembered that $c$~is to be considered as having
the coefficient~$1$ and the exponent~$1$, which is an
amendment and enlargement of our algebraical language,
\index{Analogy, in language of algebra}%
\index{Language}%
\index{Notation!algebraical}%
derived from experience. It may be said that
this is all superfluous, because, if $c^{2}$~stand for~$cc$, and
$c^{3}$~for~$ccc$, what can $c^{1}$~stand for but~$c$? But it must
be recollected that, since the symbol~$c^{1}$ has not yet received
a meaning, we are at liberty to make it stand
for anything which we please, for example, for~$\dfrac{1 + c}{c}$,
or $c - c^{2}$, or any other. If we did this, there would,
indeed, be a great violation of analogy, that is, what
$c^{1}$~stands for would not be as like that which $c^{2}$~has
been made to stand for, as the meaning of~$c^{3}$ is to
that of~$c^{4}$; but, nevertheless, we should not be led to
any incorrect results as long as we remembered to
make $c^{1}$ always stand for the same thing. These remarks
are here introduced in order to show the manner
in which analogy is followed in extending the language
of algebra, and to prove that, after a certain
period, we may rather be said to discover new symbols
than to make them. The immense importance of this
branch of the subject makes it necessary that it should
be fully and early understood by all who intend to
pursue their mathematical studies to any depth. To
illustrate it still further, we subjoin another instance,
which has not been noticed in its proper place.
\PageSep{80}
The signs $+$~and~$-$ were first used to connect one
quantity with others, and to show what arithmetical
operations were performed on other quantities by
means of the first. But the first quantity on which
we begin the operation is not preceded by any sign,
not being considered as added to or subtracted from
any previous one. Rules were afterwards deduced for
\index{Extension@{Extension of rules and meanings of terms}|(}%
\index{Notation!extension of}%
\index{Rules!extension of meaning of}%
\index{Symbols@{Symbols, invention of}|(}%
the addition and subtraction of the total result of several
expressions in which these signs occur, as follows:
To add two expressions, form a third, which has
all the quantities in the first two, with the same signs.
To subtract one expression from another, change
the sign of each term of the subtrahend, and proceed
as in the last rule.
The only terms in which these rules do not apply
are those which have no sign, viz., the first of each.
But they will apply to those terms, and will produce
correct results, if we place the sign~$+$ before each of
them. We are thus led to see that an algebraical
term which has no sign is equivalent in all operations
to one which is preceded by the sign~$+$. We, therefore,
consider this sign as prefixed, though it is not
always written, and thus we are furnished with a
method of containing under one rule that which would
otherwise require two.
From these considerations the following appears
to be the best and most natural course of proceeding
in the invention of additional symbols. When a rule
has been discovered which is not quite general, and
\PageSep{81}
which only fails in its application to a few instances,
annex such additional symbols to those already in use,
or change and modify these so as to make the rule
applicable in all cases, provided always this can be
done without making the same symbol stand for two
different things, and without any violation of analogy.
If the rule itself, by its application to any case, should
produce a new symbol hitherto unexplained, it is a
sign that the rule has been applied to a case which
was never intended to fall under it when it was made.
For the solution of this case we must have recourse
to first principles, but when, by these means, the result
has been found, it will be best to agree that the
new symbol furnished by the rule shall stand for the
result furnished by the principle, by which means the
generality of the rule will be attained and the analogy
of language will not be injured. Of this the following
is a remarkable instance:
To divide $c^{8}$ by~$c^{5}$ the rule tells us to subtract $5$
\index{Zero!exponents}%
from~$8$, and make the result the exponent of~$c$, which
gives the quotient~$c^{3}$. If we \emph{apply the same rule} to divide
$c^{6}$ by~$c^{6}$, since $6$ subtracted from~$6$ leaves~$0$, the
result is~$c^{0}$, a new symbol, to which we have attached
no meaning. The fact is that the rule was formed
from observation of different powers of~$c$, and was
never intended to apply to the division of a power of~$c$
by the same power. If we apply the common principles
to the division of $c^{6}$ by~$c^{6}$, the result is~$1$. We,
therefore, agree that $c^{0}$~shall stand for~$1$, and the least
\index{Symbols@{Symbols, invention of}|)}%
\PageSep{82}
inspection will show that this agreement does not affect
the truth of any result derived from the rule. If,
in the solution of any problem, the symbol~$c^{0}$ should
appear, we must consider it is a sign that we have, in
the course of the investigation, divided a power of~$c$
by itself by the common rule, without remarking that
the quotient is~$1$. We must, therefore, replace $c^{0}$ by~$1$,
but it is entirely indifferent at what stage of the
process this is done.
Several extensions might be noticed, which are
made almost intuitively, to which these observations
will apply. Such, for example, is the multiplication
and division of any number by~$1$, which is not contemplated
in the definition of these operations. Such
is also the continual use of~$0$ as a quantity, the addition
and subtraction of it from other quantities, and
the multiplication of it by others, neither of which
were contemplated when these operations were first
thought of.
\index{Extension@{Extension of rules and meanings of terms}|)}%
We now proceed to the principles on which more
complicated divisions are performed. The question
proposed in division, and the manner of answering it,
may be explained in the following manner. Let $A$~be
an expression which is to be divided by~$H$, and let $Q$~be
the quotient of the two. By the meaning of division,
if there be no remainder $A = QH$, since the quotient
is the expression which must multiply the divisor,
in order to produce the dividend. Now let the
\PageSep{83}
quotient be made up of different terms, $a$,~$b$,~$c$, etc.,
let it be $a + b - c + d$. That is, let
\begin{align*}
A &= QH\Add{,}
\Tag{(1)} \\
Q &= a + b - c + d.
\Tag{(2)}
\end{align*}
By putting, instead of~$Q$ in~\Eq{(1)}, that which is equal
to it in~\Eq{(2)}, we find
\[
A = (a + b - c + d)H = aH + bH - cH + dH\Add{.}
\Tag{(3)}
\]
Now suppose that we can by any method find the
term~$a$ of the quotient, that is, that we can by trial or
otherwise find one term of the quotient. In~\Eq{(3)}, when
the term~$a$ is found, since $H$~is known, the term~$aH$
is found. Now if two quantities are equal, and from
them we subtract the same quantity, the remainders
will be equal. Subtract $aH$ from the equal quantities
$A$ and $aH + bH - cH + dH$, and we shall find
\[
A - aH = bH - cH + dH = (b - c + d)H.
\Tag{(4)}
\]
If, then, we multiply the term of the quotient found
by the divisor, and subtract the product from the dividend,
and call the remainder~$B$; then
\[
B = (b - c + d)H.
\Tag{(5)}
\]
That is, if $B$~be made a dividend, and $H$~still continue
the divisor, the quotient is $b - c + d$, or all the first
quotient, except the part of it which we have found.
We then proceed in the same manner with this new
dividend, that is, we find $b$ and also~$bH$, and subtract
it from~$B$, and let $B - bH$ be represented by~$C$, which
gives by the process which has just been explained
\[
C = (-c + d)H = -cH + dH.
\Tag{(6)}
\]
We now come to a negative term of the quotient.
\PageSep{84}
Let us suppose that we have found~$c$, and that its sign
in the quotient is~$-$. If two quantities are equal, and
we add the same quantity to both, the sums are equal.
Let us therefore add~$cH$ to both the equal quantities
in~\Eq{(6)}, and the equation will become
\[
C + cH = dH;
\Tag{(7)}
\]
or if we denote $C + cH$ by~$D$, this is
\[
D = dH.
\]
There is only one term of the quotient remaining, and
if that can be found the process is finished. But as
we cannot know when we have come to the last term,
we must continue the same process, that is, subtract
$dH$ from~$D$, in doing which we shall find that $dH$~is
equal to~$D$, or that the remainder is nothing. This
indicates that the quotient is now exhausted and that
the process is finished.
We will now apply this to an example in which
the quotient is of the same form as that in the last
process, namely, consisting of four terms, the third of
which has the negative sign. This is the division of
\[
x^{4} - y^{4} - 3x^{2} y^{2} + x^{3} y + 2xy^{3} \quad\text{by}\quad x - y.
\]
Arrange the first quantity in descending powers of~$x$
which will make it stand thus:
\[
x^{4} + x^{3} y - 3x^{2} y^{2} + 2xy^{3} - y^{4}\Add{.}
\Tag{(A)}
\]
One term of the quotient can be found immediately,
for since it has been shown that the term containing
the highest power of~$x$ in a product is made up of
nothing but the product of the terms containing the
highest powers of~$x$ which occur in the multiplier and
\PageSep{85}
multiplicand, and considering that the expression~\Eq{(A)}
is the product of~$x - y$ and the quotient, we shall recover
the highest power of~$x$ in the quotient by dividing~$x^{4}$,
the highest power of~$x$ in~\Eq{(A)}, by~$x$, its highest
power in~$x - y$. This division gives~$x^{3}$ as the first
term of the quotient. The following is the common
process, and with each line is put the corresponding
step of the process above explained, of which this is
an example:
\[
\small
\begin{array}{lr<{\ }*{5}{l}>{\ }l}
& (H) & & &\qquad(A) & & & (a) \\
&x - y) & x^{4} &{}+\Z x^{3} y &{}- 3x^{2} y^{2} &{}+ 2xy^{3} &{}- y^{4} & (x^{3} \\
(aH)
& \text{\qquad Subtract}
& x^{4} &{}-\Z x^{3} y \\
\cline{3-7}
& & &&&&& \quad(b) \\
(B) & \text{Second dividend}
& &\PadTo[r]{{}-2}{2}x^{3} y &{}- 3x^{2} y^{2} &{}+ 2xy^{3} &{}- y^{4} & (+2x^{2} y \\
(bH) & \text{\qquad Subtract}
& &\PadTo[r]{{}-2}{2}x^{3} y &{}- 2x^{2} y^{2} \\
\cline{4-7}
& & &&&&& \quad(c) \\
(C) & \text{Third dividend}
& &&{}-\Z x^{2} y^{2} &{}+ 2x y^{3} &{}- y^{4} & (-xy^{2} \\
(cH) & \text{\qquad Subtract}
& &&{}-\Z x^{2} y^{2} &{}+\Z x y^{3} \\
\cline{5-7}
& & &&&&& \quad(d) \\
(D) & \text{Fourth dividend}
& &&&\phantom{{}+2} x y^{3} &{}- y^{4} & (+y^{3} \\
(dH) & \text{\qquad Subtract}
& &&&\phantom{{}+2} x y^{3} &{}- y^{4} \\
\cline{6-7}
\Strut
&&&&&& 0
\end{array}
\]
The whole quotient is therefore $x^{3} + 2x^{2} y - xy^{2} + y^{3}$.
\PageSep{86}
The second and following terms of the quotient
are determined in exactly the same manner as the
first. In fact, this process is not the finding of a quotient
directly from the divisor and dividend, but one
term is first found, and by means of that term another
dividend is obtained, which only differs from the first
in having one term less in the quotient, viz., that
which was first found. From this second dividend
one term of its quotient is found, and so on until we
obtain a dividend whose quotient has only one term,
the finding of which finishes the process. It is usual
also to neglect all the terms of the first dividend,
except those which are immediately wanted, taking
down the others one by one as they become necessary.
This is a very good method in practice but should be
avoided in explaining the principle, since the first
subtraction is made from the whole dividend, though
the operation may only affect the form of some part
of it.
If the student will now read attentively what has
been said on the greatest common measure of two
\index{Greatest common measure}%
numbers, and then examine the connexion of whole
numbers in arithmetic and simple expressions in algebra
with which we commenced the subject of division,
he will see that the greatest algebraical common measure
of two expressions may be found in exactly the
same manner as the same operation is performed in
arithmetic. He must also recollect that the greatest
common measure of two expressions $A$~and~$B$ is not
\PageSep{87}
altered by multiplying or dividing either of them, $A$,~for
example, by any quantity, provided that quantity
has no measure in common with~$B$. For example,
the greatest common measure of $a^{2} - x^{2}$ and $ba^{3} - bx^{3}$
is the same with that of $2a^{2} - 2x^{2}$ and $a^{3} - x^{3}$, since
though a new measure is now introduced into the first
and taken away from the second, nothing is introduced
or taken away which is common to both. The same
observation applies to arithmetic also. For example,
take the numbers $162$ and~$180$. We may, without
altering their greatest common measure, multiply the
first by~$7$ and the second by~$11$, etc. The rule for
finding the greatest common measure should be practised
with great attention by all who intend to proceed
beyond the usual stage in algebra. To others it is not
of the same importance, as the necessity for it never
occurs in the lower branches of the science.
In proceeding to the subject of fractions, it must
\index{Fractions!algebraical|(}%
be observed that, in the same manner as in arithmetic,
when there is a remainder which cannot be further
divided by the divisor, that is, where the dividend is
so reduced that no simple term multiplied by the first
term of the divisor will give the first term of the remainder,
as in the case where the divisor is $a^{2} x + bx^{2}$
and the remainder $ax + b$; in this case a fraction
must be added to the quotient, whose numerator is
this remainder, and whose denominator is the divisor.
Thus, in dividing $a^{4} + b^{4}$ by $a + b$, the quotient is
$a^{3} - a^{2} b + ab^{2} - b^{3}$, and the remainder~$2b^{4}$, whence
\PageSep{88}
\[
\frac{a^{4} + b^{4}}{a + b}
= a^{3} - a^{2} b + ab^{2} - b^{3} + \frac{2b^{4}}{a + b}.
\]
The arithmetical rules for the addition, etc., of fractions
hold equally good when the numerators and denominators
are themselves fractions. Thus $\dfrac{\,\frac{3}{4}\,}{\frac{2}{7}}$ and $\dfrac{\,\frac{1}{5}\,}{\frac{3}{2}}$
are added, etc., exactly in the same way as $\frac{2}{5}$ and~$\frac{3}{7}$,
the sum of the second being
\[
\frac{7 × 2 + 5 × 3}{5 × 7}
\]
and that of the first
\[
\frac{\frac{3}{2} × \frac{3}{4} + \frac{2}{7} × \frac{1}{5}}{\frac{2}{7} × \frac{3}{2}}.
\]
The rules for the addition, etc., of algebraic fractions
are exactly the same as in arithmetic; for both the
numerator and denominator of every algebraic fraction
stands either for a whole number or a fraction,
and therefore the fraction itself is either of the same
form as $\frac{6}{7}$ or~$\dfrac{\,\frac{2}{3}\,}{\frac{4}{5}}$. Nevertheless the student should attend
to some examples of each operation upon algebraic
fractions, by way of practice in the previous
operations. As the subject is not one which presents
\index{Formulæ, important|(}%
any peculiar difficulties, we shall now pass on to the
subject of equations, concluding this article with a
list of formulas which it is highly desirable that the
student should commit to memory before proceeding
to any other part of the subject.
\begin{gather*}
(a + b) + (a - b) = 2a\Add{,}
\Tag{(1)} \\
(a + b) - (a - b) = 2b\Add{,}
\Tag{(2)} \\
a - (a - b) = b\Add{,}
\Tag{(3)} \displaybreak[0]\\
\PageSep{89}
%
(a + b)^{2} = a^{2} + 2ab + b^{2}\Add{,}
\Tag{(4)} \\
(a - b)^{2} = a^{2} - 2ab + b^{2}\Add{,}
\Tag{(5)} \\
(2ax + b)^{2} = 4a^{2} x^{2} + 4abx + b^{2}\Add{,}
\Tag{(6)} \\
(a + b)(a - b) = a^{2} - b^{2}\Add{,}
\Tag{(7)} \displaybreak[0]\\
%
\left.
\begin{gathered}
(x + a)(x + b) = x^{2} + (a + b)x + ab\Add{,} \\
(x - a)(x - b) = x^{2} - (a + b)x + ab\Add{,}
\end{gathered}
\right\}
\Tag{(8)} \displaybreak[0]\\
%
\frac{a}{b} = \frac{ma}{mb}\Add{,}
\Tag{(9)} \displaybreak[0]\\
a + \frac{c}{d} = \frac{ad + c}{d},\qquad
a - \frac{c}{d} = \frac{ad - c}{d}\Add{,}
\Tag{(10)} \\[8pt]
\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd},\qquad
\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd},\qquad
\Tag{(11)} \\[8pt]
\frac{a}{b} × c = \frac{ac}{b} = \frac{a}{\,\dfrac{b}{c}\,},\qquad
\frac{a}{b} × \frac{c}{d} = \frac{ac}{bd}\Add{,}
\Tag{(12)} \displaybreak[0]\\[8pt]
\frac{\,\dfrac{a}{b}\,}{c} = \frac{a}{bc} = \frac{\,\dfrac{a}{c}\,}{b}\Add{,}
\Tag{(13)} \\[8pt]
\frac{\,\dfrac{a}{b}\,}{\dfrac{c}{d}} = \frac{ad}{bc}
= \frac{\,\dfrac{a}{c}\,}{\dfrac{b}{d}}\Add{,}
\Tag{(14)} \\[8pt]
\frac{1}{\,\dfrac{a}{b}\,} = \frac{b}{a}\Add{.}
\Tag{(15)}
\end{gather*}
\index{Formulæ, important|)}%
\index{Fractions!algebraical|)}%
\PageSep{90}
\Chapter{VIII.}{Equations of the First Degree.}
\index{Equations!first@{of the first degree}|(}%
\index{Equations!identical}%
\First{We} have already defined an equation, and have
come to many equations of different sorts. But
all of them had this character, that they did not depend
upon the particular number which any letter
stood for, but were equally true, whatever numbers
might be put in place of the letters. For example, in
the equation
\[
\frac{a^{2} - 1}{a + 1} = a - 1
\]
the truth of the assertion made in this algebraical sentence
is the same, whether $a$~be considered as representing
$1$,~$2$,~$2\frac{1}{2}$, etc., or any other number or fraction
whatever. The second side of this equation is, in
fact, the result of the operation pointed out on the
first side. On the first side you are directed to divide
$a^{2} - 1$ by~$a + 1$; the second side shows you the result
of that division. An equation of this description is
called an \emph{identical} equation, because, in fact, its two
\index{Identical equations}%
sides are but different ways of writing down the same
\PageSep{91}
number. This will be more clearly seen in the identical
equations
\[
a + a = 2a,\quad
7a - 3a + b = 4a - 3b + 4b,\quad\text{and}\quad
\frac{a}{b} × b = a.
\]
The whole of the formulæ at the end of the last
article are examples of identical equations. There is
not one of them which is not true for all values which
can be given to the letters which enter into them, provided
only that whatever a letter stands for in one
part of an equation, it stands for the same in all the
other parts.
If we take, now, such an equation as $a + 1 = 8$, we
\Pagelabel{91}%
\index{Equations!condition@{of condition}}%
have an equation which is no longer true for every
value which can be given to its algebraic quantities.
It is evident that the only number which $a$~can represent
consistently with this equation is~$7$, as any other
supposition involves absurdity. This is a new species
of equation, which can only exist in some particular
case, which particular case can be found from
the equation itself. The solution of every problem
leads to such an equation, as will be shown hereafter,
and, in the elements of algebra, this latter species of
equation is of most importance. In order to distinguish
them from identical equations, they are called
\emph{equations of condition}, because they cannot be true when
the letters contained in them stand for any number
whatever, and their very existence makes a condition
which the letters contained must fulfil. The solution
of an equation of condition is the process of finding
\PageSep{92}
what number the letter must stand for in order that
the equation may be true. Every such solution is a
process of reasoning, which, setting out with supposing
the truth of the equation, proceeds by self-evident
steps, making use of the common rules of arithmetic
and algebra. We shall return to the subject of the
solution of equations of condition, after showing, in a
few instances, how we come to them in the solution
of problems. In equations of condition, the quantity
whose value is determined by the equation is usually
represented by one of the last letters of the alphabet,
and all others by some of the first. This distinction
is necessary only for the beginner; in time he must
learn to drop it, and consider any letter as standing
for a quantity known or unknown, according to the
conditions of the problem.
In reducing problems to algebraical equations no
\index{Equations!reducing problems to|EtSeq}%
\index{Problems!reducing of, to equations|EtSeq}%
general rule can be given. The problem is some property
of a number expressed in words by which that
number is to be found, and this property must be
written down as an equation in the most convenient
way. As examples of this, the reduction of the following
problems into equations is given:
I. What number is that to which, if $56$ be added,
the result will be $200$ diminished by twice that number?
Let $x$~stand for the number which is to be found.
Then $x + 56 = 200 - 2x$.
If, instead of $56$, $200$, and~$2$, any other given numbers,
\PageSep{93}
$a$,~$b$, and~$c$, are made use of in the same manner,
the equation which determines~$x$ is
\[
x + a = b - cx.
\]
II. Two couriers set out from the same place, the
second of whom goes three miles an hour, and the
first two. The first has been gone four hours, when
the second is sent after him. How long will it be before
he overtakes him?
Let $x$~be the number of hours which the second
must travel to overtake the first. At the time when
this event takes place, the first has been gone $x + 4$~hours,
and will have travelled $(x + 4)2$, or $2x + 8$~miles.
The second has been gone $x$~hours, and will
have travelled $3x$~miles. And, when the second overtakes
the first, they have travelled exactly the same
distance, and, therefore,
\[
3x = 2x + 8.
\]
If, instead of these numbers, the first goes $a$~miles
an hour, the second~$b$, and $c$~hours elapse before the
second is sent after the first,
\[
bx = ax + ac.
\]
Four men, $A$, $B$, $C$, and~$D$, built a ship which
cost~£$2607$, of which $B$~paid twice as much as~$A$, $C$~paid
as much as $A$~and~$B$, and $D$~as much as $C$~and~$B$.
What did each pay?
\begin{align*}
\text{Suppose that } &A \text{ paid $x$~pounds,} \\
\text{then } &B \text{ paid $2x$\dots\Add{,}} \\
&C \text{ paid $x + 2x$ or $3x$\dots\Add{,}} \\
&D \text{ paid $2x + 3x$ or $5x$\dots\Add{.}}
\end{align*}
\PageSep{94}
All together paid $x + 2x + 3x + 5x$, or~$11x$, therefore
\[
11x = 2607.
\]
There are two cocks, from the first of which a cistern
is filled in $12$~hours, and the second in~$15$. How
long would they be in filling it if both were opened
together?
Let $x$~be the number of hours which would elapse
before it was filled. Then, since the first cock fills
the cistern in $12$~hours, in one hour it fills $\frac{1}{12}$~of it, in
two hours~$\frac{2}{12}$, etc., and in $x$~hours~$\frac{x}{12}$. Similarly, in
$x$~hours, the second cock fills $\frac{x}{15}$of the cistern. When
the two have exactly filled the cistern, the sum of
these fractions must represent a whole or~$1$, and,
therefore,
\[
\frac{x}{12} + \frac{x}{15} = 1.
\]
If the times in which the two can fill the cistern are $a$~and
$b$~hours, the equation becomes
\[
\frac{x}{a} + \frac{y}{b} = 1.
\]
A person bought $8$~yards of cloth for £$3$~$2$\textit{s.}, giving
$9$\textit{s.}~a yard for some of it and $7$\textit{s.}~a yard for the rest;
how much of each sort did he buy?
Let $x$~be the number of yards at~$7$\textit{s.} Then $7x$~is
the number of shillings they cost. Also $8 - x$ is the
number of yards at~$9$\textit{s.}, and $(8 - x)9$, or $72 - 9x$, is
the number of shillings they cost. And the sum of
\PageSep{95}
these, or $7x + 72 - 9x$, is the whole price, which is
£$3$~$2$\textit{s.}, or $62$~shillings, and, therefore,
\[
7x + 72 - 9x = 62.
\]
These examples will be sufficient to show the
method of reducing a problem to an equation. Assuming
a letter to stand for the unknown quantity, by
means of this letter the same quantity must be found
in two different forms, and these must be connected
by the sign of equality. However, the reduction into
equations of such problems as are usually given in the
\index{Problems!general disciplinary utility of}%
treatises on algebra rarely occurs in the applications
of mathematics. The process is a useful exercise of
ingenuity, but no student need give a great deal of
time to it. Above all, let no one suppose, because he
finds himself unable to reduce to equations the conundrums
with which such books are usually filled, that,
therefore, he is not made for the study of mathematics,
and should give it up. His future progress depends
in no degree upon the facility with which he
discovers the equations of problems; we mean as far
as power of comprehending the subsequent sciences
is concerned. He may never, perhaps, make any considerable
step for himself, but, without doing this, he
may derive all the benefits which the study of mathematics
can afford, and even apply them extensively.
There is nothing which discourages beginners more
than the difficulty of reducing problems to equations,
and yet, as respects its utility, if there be anything
in the elements which may be dispensed with, it is
\PageSep{96}
\index{Signs!rule of}%
this. We do not wish to depreciate its utility as an
exercise for the mind, or to hinder all from attempting
to conquer the difficulties which present themselves;
but to remind every one that, if he can read
and understand all that is set before him, the essential
benefit derived from mathematical studies will be
gained, even though he should never make one step
for himself in the solution of any problem.
We return now to the solution of equations of condition.
\index{Equations!condition@{of condition|EtSeq}}%
Of these there are various classes. Equations
of the first degree, commonly called simple equations,
are those which contain only the first power of the unknown
quantity. Of this class are all the equations
to which we have hitherto come in the solution of
problems. The principle by which they are solved is,
that two equal quantities may be increased or diminished,
multiplied, or divided by any quantity, and the
results will be the same. In algebraical language,
if $a = b$, $a + c = b + c$, $a - c = b - c$, $ac = bc$, and
$\dfrac{a}{c} = \dfrac{b}{c}$. In every elementary book it is stated that
any quantity may be removed from one side of the
equation to the other, provided its sign be changed.
This is nothing but an application of the principle
just stated, as may be shown thus: Let $a + b - c = d$,
add~$c$ to both quantities, then
\[
a + b - c + c = d + c \quad\text{or}\quad a + b = d + c.
\]
Again subtract $b$ from both quantities, then $a + b - c - b = d - b$,
or $a - c = d - b$. Without always repeating
\PageSep{97}
the principle, it is derived from observation,
that its effect is to remove quantities from one side of
an equation to another, changing their sign at the
same time. But the beginner should not use this rule
until he is perfectly familiar with the manner of using
the principle. He should, until he has mastered a
good many examples, continue the operation at full
length, instead of using the rule, which is an abridgment
of it. In fact it would be better, and not more
prolix, to abandon the received phraseology, and in
the example just cited, instead of saying ``bring the
term~$b$ to the other side of the equation,'' to say ``subtract~$b$
from both sides,'' and instead of saying ``bring~$c$
to the other side of the equation,'' to say ``add~$c$ to
both sides.''
Suppose we have the fractions $\frac{3}{4}$, $\frac{1}{7}$, and~$\frac{5}{14}$. If we
\index{Fractions!algebraical|(}%
multiply them all by the product of the denominators
$4 × 7 × 14$, or~$392$, all the products will be whole numbers.
They will be $\dfrac{3 × 392}{4}$, $\dfrac{1 × 392}{7}$, and $\dfrac{5 × 392}{14}$,
and since $392$~is measured by~$4$, $3 × 392$~is also measured
by~$4$, and $\dfrac{3 × 392}{4}$~is a whole number, and so on.
But any common multiple of $4$,~$7$, and~$14$ will serve
as well. The least common multiple will therefore be
the most convenient to use for this purpose. The
least common multiple of $4$,~$7$, and~$14$ is~$28$, and if the
three fractions be multiplied by~$28$, the results will be
whole numbers. The same also applies to algebraic
fractions. Thus $\dfrac{a}{b}$, $\dfrac{c}{de}$, and~$\dfrac{e}{bdf}$, will become simple
\PageSep{98}
expressions, if they are multiplied by $b × de × bdf$, or
$b^{2} d^{2} ef$. But the most simple common multiple of $b$,~$de$,
and~$dbf$, is~$bdef$, which should be used in preference
to~$b^{2} d^{2} ef$.
This being premised, we can now reduce any equation
which contains fractions to one which does not.
For example, take the equation
\[
\frac{x}{3} + \frac{2x}{5} = \frac{7}{10} - \frac{3 - 2x}{6}.
\]
If we multiply both these equal quantities by any
other, the results will be equal. We choose, then,
the least quantity, which will convert all the fractions
into simple quantities, that is, the least common multiple
of the denominators $3$,~$5$,~$10$, and~$6$, which is~$30$.
If we multiply both equal quantities by~$30$, the equation
becomes
\[
\frac{30x}{3} + \frac{60x}{5} = \frac{210}{10} - \frac{30(3 - 2x)}{6}.
\Tag{(1)}
\]
But $\dfrac{30x}{3}$ is $\dfrac{30}{3} × x$ or~$10x$, $\dfrac{60x}{5}$~is $\dfrac{60}{5} × x$, or~$12x$, etc.;
so that we have
\begin{align*}
10x + 12x &= 21 - 5(3 - 2x),
\Tag{(2)} \\
\text{or\quad}
10x + 12x &= 21 - (15 - 10x),
\Tag{(3)} \\
\text{or\quad}
10x + 12x &= 21 - 15 + 10x.
\Tag{(4)}
\end{align*}
Beginners very commonly mistake this process, and
forget that the sign of subtraction, when it is written
before a fraction, implies that the whole result of
the fraction is to be subtracted from the rest. As
long as the denominator remains, there is no need to
\PageSep{99}
signify this by putting the numerator between brackets,
but when the denominator is taken away, unless
this be done, the sign of subtraction belongs to the
first term of the numerator only, and not to the whole
expression. The way to avoid this mistake would be
to place in brackets the numerators of all fractions
which have the negative sign before them, and not to
remove those brackets until the operation of subtraction
has been performed, as is done in equation~\Eq{(4)}.
The following operations will afford exercise to the
\index{Formulæ, important}%
student, sufficient, perhaps, to enable him to avoid
this error:
\begin{gather*}
a + \frac{b - c + d - e}{f}
= \frac{af + b - c + d - e}{f}, \\
a - \frac{b - c + d - e}{f}
= \frac{af - b + c - d + e}{f}, \\
a + b + \frac{(a - b)^{2}}{a + b} = \frac{2a^{2} + 2b^{2}}{a + b}, \\
a + b - \frac{(a - b)^{2}}{a + b} = \frac{4ab}{a + b}.
\end{gather*}
\index{Fractions!algebraical|)}%
We can now proceed with the solution of the equation.
Taking up the equation~\Eq{(4)} which we have deduced
from it, subtract~$10x$ from both sides, which
gives $10x + 12x - 10x = 21 - 15$, or $12x = 6$: divide
these equal quantities by~$12$, which gives $\dfrac{12x}{12} = \dfrac{6}{12}$, or
$x = \frac{1}{2}$. This is the only value which $x$~can have so as
to make the given equation true, or, as it is called, to
\emph{satisfy} the equation. If instead of~$x$ we substitute~$\frac{1}{2}$,
we shall find that
\PageSep{100}
\[
\frac{\,\frac{1}{2}\,}{3} + \frac{2 × \frac{1}{2}}{5}
= \frac{7}{10} - \frac{3 - 2 × \frac{1}{2}}{6},\quad\text{or}\quad
\frac{1}{6} + \frac{1}{5} = \frac{7}{10} - \frac{2}{6};
\]
this we find to be true, since
\[
\frac{1}{6} + \frac{1}{5} \text{ is } \frac{11}{30}, \quad\text{and}\quad
\frac{7}{10} - \frac{2}{6} = \frac{22}{60},\quad\text{and}\quad
\frac{11}{30} = \frac{22}{60}.
\]
In these equations of the first degree there is one unknown
quantity and all the others are known. These
known quantities may be represented by letters, and,
as we have said, the first letters of the alphabet are
commonly used for that purpose. We will now take
an equation of exactly the same form as the last, putting
letters in place of numbers:
\[
\frac{x}{a} + \frac{bx}{c} = \frac{d}{c} - \frac{f - gx}{h}.
\]
The solution of this equation is as follows: multiply
both quantities by~$aceh$, the most simple multiple
of the denominators, it then becomes:
\begin{gather*}
\frac{acehx}{a} + \frac{abcehx}{c}
= \frac{acdeh}{e} - \frac{aceh(f - gx)}{h}, \\
\text{or},\quad
cehx + abcehx = acdh - ace(f - gx), \\
\text{or},\quad
cehx + abcehx = acdh - acef + acegx.
\end{gather*}
Subtract $acegx$ from both sides, and it becomes
\begin{align*}
cehx + abehx - acegx &= acdh - acef, \\
\text{or},\quad
(ceh + abeh - aceg)x &= acdh - acef.
\end{align*}
Divide both sides by $ceh + abeh - aceg$, which gives
\[
x = \frac{acdh - acef}{ceh + abeh - aceg}.
\]
The steps of the process in the second case are exactly
the same as in the first; the same reasoning establishes
\PageSep{101}
them both, and the same errors, are to be
avoided in each. If from this we wish to find the solution
of the equation first given, we must substitute
$3$~for~$a$, $2$~for~$b$, $5$~for~$c$, $7$~for~$d$, $10$~for~$e$, $3$~for~$f$, $2$~for~$g$,
and $6$~for~$h$, which gives for the value of~$x$,
\begin{gather*}
\frac{3 × 5 × 7 × 6 - 3 × 5 × 10 × 3}
{5 × 10 × 6 + 3 × 2 × 10 × 6 - 3 × 5 × 10 × 2}, \\
\text{or},\quad
\frac{3 × 5 × 12}{3 × 2 × 10 × 6},\quad
\text{or},\quad
\frac{180}{360},
\end{gather*}
which is~$\frac{1}{2}$, the same as before.
If in one equation there are two unknown quantities,
\index{Indeterminate problems}%
the condition is not sufficient to fix the values of
the two quantities; it connects them, nevertheless, so
that if one can be found the other can be found also.
For example, the equation $x + y = 8$ admits of an infinite
number of solutions, for take $x$ to represent any
whole number or fraction less than~$8$, and let $y$~represent
what $x$~wants of~$8$, and this equation is satisfied.
If we have another equation of condition existing between
the same quantities, for example, $3x - 2y = 4$;
this second equation by itself has an infinite number
of solutions: to find them, $y$~may be taken at pleasure,
and $x = \dfrac{4 + 2y}{3}$. Of all the solutions of the second
equation, one only is a solution of the first; thus there
is only one value of $x$~and~$y$ which satisfies both the
equations, and the finding of these values is the solution
of the equations. But there are some particular
cases in which every value of $x$~and~$y$ which satisfies
one of the equations satisfies the other also; this happens
\PageSep{102}
whenever one of the equations can be deduced
from the other. For example, when $x + y = 8$, and
$4x - 29 = 3 - 4y$, the second of these is the same as
$4x + 4y = 3 + 29$, or $4x + 4y = 32$, which necessarily
follows from the first equation.
If the solution of a problem should lead to two
equations of this sort, it is a sign that the problem
admits of an infinite number of solutions, or is what
is called an indeterminate problem. The solution of
equations of the first degree does not contain any peculiar
difficulty; we shall therefore proceed to the
consideration of the isolated negative sign.
\index{Equations!first@{of the first degree}|)}%
\PageSep{103}
\Chapter{IX.}{On the Negative Sign, etc.}
\index{Negative!sign@{sign, isolated|EtSeq}}%
\index{Subtractions, impossible|(}%
\First{If} we wish to say that $8$~is greater than~$5$ by the
number~$3$, we write this equation $8 - 5 = 3$. Also
to say that $a$~exceeds~$b$ by~$c$, we use the equation $a - b = c$.
As long as some numbers whose value we know
are subtracted from others equally known, there is no
fear of our attempting to subtract the greater from
the less; of our writing $3 - 8$, for example, instead of
$8 - 3$. But in prosecuting investigations in which letters
occur, we are liable, sometimes from inattention,
sometimes from ignorance as to which is the greater
of two quantities, or from misconception of some of
the conditions of a problem, to reverse the quantities
in a subtraction, for example to write $a - b$ where $b$~is
the greater of two quantities, instead of~$b - a$. Had
we done this with the sum of two quantities, it would
have made no difference, because $a + b$ and $b + a$ are
the same, but this is not the case with $a - b$ and $b - a$.
For example, $8 - 3$ is easily understood; $3$~can be
taken from~$8$ and the remainder is~$5$; but $3 - 8$ is an
\PageSep{104}
impossibility, it requires you to take from~$3$ more than
there is in~$3$, which is absurd. If such an expression
as $3 - 8$ should be the answer to a problem, it would
denote either that there was some absurdity inherent
in the problem itself, or in the manner of putting it
into an equation. Nevertheless, as such answers will
occur, the student must be aware what sort of mistakes
give rise to them, and in what manner they affect
the process of investigation.
\index{Subtractions, impossible|)}%
We would recommend to the beginner to make
\index{Experience, mathematical}%
\index{Induction, mathematical}%
experience his only guide in forming his notions of
these quantities, that is, to draw his rules from the
observation of many results, not from any theory.
The difficulties which encompass the theory of the
negative sign are explained at best in a manner which
would embarrass him: probably he would not see the
difficulties themselves; too easy belief has always
been the fault of young students in mathematics, and
it is a great point gained to get them to start an objection.
We shall observe the effect of this error in
denoting a subtraction on every species of investigation
to which we have hitherto come, and shall deduce
rules which the student will recollect are the results
of experience, not of abstract reasoning. The
extensions to which he will be led have rendered Algebra
much more general than it was before, have
made it competent to the solution of many, very many
questions which it could not have touched had they
not been attended to. They do, in fact, constitute
\PageSep{105}
part of the groundwork of modern Algebra and should
be considered by the student who is desirous of making
his way into the depths of the science with the
highest degree of attention. If he is well practised in
the ordinary rules which have hitherto been explained,
few difficulties can afterwards embarrass him, except
those which arise from some confusion in the notions
which he has formed upon this part of the subject.
For brevity's sake we hereafter use this phrase.
\index{Change of algebraical form|EtSeq}%
\index{Form, change of in algebraical expressions|EtSeq}%
Where the signs of every term in an expression are
changed, it is said to have changed its form. Thus
$+a - b$ and $+b - a$ are in different forms, and if $a$~be
greater than~$b$, the first is the correct form and the
second incorrect. An extension of a rule is made by
which such a quantity as $3 - 8$ is written in a different
way. Suppose that $+3 - 8$ is connected with any
other number thus, $56 + 3 - 8$. This may be written
$56 + 3 - (3 + 5)$, or $56 + 3 - 3 - 5$, or $56 - 5$. It appears,
then, that $+3 - 8$, connected with any number
is the same as $-5$ connected with that number; from
this we say that $+3 - 8$, or $3 - 8$ is the same thing
as~$-5$, or $3 - 8 = -5$. This is another way of writing
the equation $8 - 3 = 5$, and indicates equally that
$8$~is greater than~$5$ by~$3$. In the same way, $a - b = -c$
indicates that $b$~is greater than~$a$ by the quantity~$c$.
If $a$~be nothing, this equation becomes $-b = -c$,
which indicates that $b = c$, since if the equation $a - b = -c$
be written in its true form $b - a = c$, and if
\PageSep{106}
$a = 0$, then $b = c$. We can now understand the following
equations:
\begin{gather*}
a - b + c - d = -e,\quad\text{or}\quad b + d - a - c = e, \\
2ab - a^{2} - b^{2} = -d - e,\quad\text{or}\quad a^{2} + b^{2} - 2ab = d + e.
\end{gather*}
We must not commence any operations upon such
\index{Errors!in algebraical suppositions, corrected by a change of signs|EtSeq}%
\index{Mistaken suppositions|EtSeq}%
an equation as $a - b = -c$, until we have satisfied ourselves
of the manner in which they should be performed,
by reference to the correct form of the equation.
This correct form is $b - a = c$. This gives
$d + b - a = d + c$, or $d - (a - b) = d + c$. Write instead
of $a - b$ its symbol~$-c$, and then $d - (-c) = d + c$.
Here we have performed an operation with
$a - b$, which is no quantity, since $a$~is less than~$b$, but
this is done because our present object is, in applying
the common rules to such expressions, to watch the
results and exhibit them in their real forms. The first
side $d - (-c)$ is in a form in which we can attach no
meaning to it, and the second side gives its real form
$d + c$. The meaning of this expression is, that if with
$a - b$, which we think to be a quantity, but which is
not, since $a$~is less than~$b$, we follow the algebraical
rule in subtracting $a - b$ from~$d$, we shall thereby get
the same result as if we had added the real quantity
$b - a$ to~$d$. If we make use of the form $d - (-c)$, it
is because we can use it in such a manner as never to
lose sight of its connexion with its real form $d + c$,
and because we can establish rules which will lead us
to the end of a process without any error, except those
\PageSep{107}
which we can correct as certainly at the end as at the
beginning.
The rule by which we proceed, and which we shall
establish by numerous examples, is, that wherever
two like signs come together, the corresponding part
of the real form has a positive sign, and wherever two
unlike signs come together, the real form has a negative
sign. Thus the real form of $d - (-c)$ is $d + c$.
Again, take the real form $b - a = c$ of the equation
$a - b = -c$, and it follows that $d - (b - a) = d - c$,
or $d - b + a = d - c$, or $d + a - b = d - c$, or $d + (a - b) = d - c$.
This is $d + (-c) = d - c$, another
case in which the rule is verified. Again, multiply
together $a - b$ and~$m - n$, the product is $am - an - bm + bn$.
This is the same product as arises from
multiplying $b - a$ by $n - m$, written in a different order.
If, then, $b - a = c$, and $n - m = p$, or $a - b = -c$,
and $m - n = -p$, we find that $(-c) × (-p) = cp$.
By which result we mean that a mistake, in the
form of both $a - b$ and $m - n$, will not produce a mistake
in the form of their product, which remains what
it would have been had the mistake not been made.
Again
\begin{gather*}
(n - m)(b - a) = bn - bm - an + am\Add{,} \\
(n - m)(a - b) = an - am - bn + bm.
\end{gather*}
If the first product be real and equal to~$P$, the second
is represented by~$-P$. The first is~$cp$, the second is
$(-c) × p$, which gives
\PageSep{108}
\[
(-c) × p = -cp.
\]
That is, a mistake in the form of one factor only alters
the form of the product. To distinguish the right
form from the wrong one, we may prefix~$+$ to the
first, and $-$~to the second, and we may then recapitulate
the results, and add others, which the student
will now be able to verify.
The sign~$+$ placed before single quantities shows
that the form of the quantity is correct; the sign~$-$
shows that it has been mistaken or changed.
\begin{gather*}
\begin{alignedat}{2}
a + (+b) &= a + b\Add{,} & a + (-b) &= a - b\Add{,} \\
a - (+b) &= a - b\Add{,} & a - (-b) &= a + b\Add{,} \\
(+a) × (+b) &= +ab\Add{,}\qquad & (+a) × (-b) &= -ab\Add{,}
\end{alignedat} \\
(-a) × (-b) = +ab = (+a) × (+b)\Add{,} \displaybreak[0] \\
\begin{aligned}
\frac{+a}{+b} &= +\frac{a}{b}\Add{,} \\
\frac{+a}{-b} &= -\frac{a}{b} = \frac{-a}{+b}\Add{,} \\
\frac{-a}{-b} &= +\frac{a}{b}\Add{,}
\end{aligned} \displaybreak[0] \\
\begin{alignedat}{3}
& -a × -a &&= && +a^{2}\Add{,} \\
& -a × -a × -a &&= +a^{2} × -a = &&-a^{3}\Add{,} \\
& -a × -a × -a × -a &&= -a^{3} × -a = &&+a^{4}\Add{,} \\
& \qquad\qquad\etc.\Add{,} &&&&\etc.
\end{alignedat}
\end{gather*}
We see, then, that a change in the form of any
quantity changes the form of those powers whose exponent
is an odd number, but not of those whose exponent
is an even number. By these rules we shall
\PageSep{109}
be able to tell what changes would be made in an expression
by altering the forms of any of its letters. It
may be fairly asked whether we are not changing the
meaning of the signs $+$~and~$-$, in making $+a$~stand
for an expression in which we do not alter the signs,
and $-a$~for one in which the signs are altered. The
change is only in name, for since the rule of addition
is, ``annex the expressions which are to be added
without altering the signs of either,'' or ``annex the
expressions without altering the form of either;'' the
quantity $a + b$, which is the sum of the two expressions
$a$~and~$b$, stands for the same as $+a + b$, in
which the new notion of the sign~$+$ is used, viz., the
expressions $a$~and~$b$ are annexed with unaltered forms,
which is denoted by writing together $+a$~and~$+b$.
Again, the rule for subtraction is, ``change the sign
of the subtrahend or expression which is to be subtracted,
and annex the result to the other expression,''
or ``change the form of the subtrahend and annex it
to the other,\Add{''} which, the expressions being $a$~and~$b$, is
written $a - b$, which answers equally well to the second
notion of the sign~$-$, since $+a - b$ indicates that
$a$~and~$b$ are to be annexed, the first without, the second
with a change of form. These ideas of the signs
$+$~and~$-$ give, therefore, in practice, the same results
as the former ones, and, in\Note{[sic], no "the"} future, the two meanings
may be used indiscriminately. But when a single
term is used, such as $+a$~or~$-a$, the last acquired
notions of $+$~and~$-$ are always understood.
\PageSep{110}
This much being premised, we can see, by numberless
instances, that, if the form of a quantity is to
be changed, it matters nothing whether it is changed
at the beginning of the process, or whether we wait
till the end, and then follow the rules above mentioned.
This is evident to the more advanced student,
from the nature of the rules themselves, but the
beginner should satisfy himself of this fact from experience.
We now give a proof of this, as far as one
expression can prove it, in the solution of the equations,
\begin{align*}
\frac{a^{2}}{b} + ax &= \frac{a^{2} x}{b} + a - b\Add{,} \\
\text{and}\quad
\frac{a^{2}}{b} - ax &= \frac{a^{2} x}{b} - a - b\Add{,}
\end{align*}
which two equations only differ in the form in which
$a$~appears. For, if the form of~$a$ in the first equation
be altered, that of $\dfrac{a^{2}}{b}$ and $\dfrac{a^{2} x}{b}$ is unaltered, $+ax$~becomes~$-ax$,
and $+a$~becomes~$-a$. We now solve
the two equations in opposite columns.
\[
\begin{gathered}
\frac{a^{2}}{b} + ax = \frac{a^{2} x}{b} + a - b\Add{,} \\
a^{2} + abx = a^{2} x + ab - b^{2}\Add{,} \\
\begin{aligned}
a^{2} - ab + b^{2}
&= a^{2} x - abx \\
&= (a^{2} - ab)x\Add{,}
\end{aligned} \\
x = \frac{a^{2} - ab + b^{2}}{a^{2} - ab}\Add{;}
\end{gathered}\qquad
\begin{gathered}
\frac{a^{2}}{b} - ax = \frac{a^{2} x}{b} - a - b\Add{,} \\
a^{2} - abx = a^{2} x - ab - b^{2}\Add{,} \\
\begin{aligned}
a^{2} + ab + b^{2}
&= a^{2} x + abx \\
&= (a^{2} + ab)x\Add{,}
\end{aligned} \\
x = \frac{a^{2} + ab + b^{2}}{a^{2} + ab}.
\end{gathered}
\]
The only difference between these expressions
\PageSep{111}
\index{Solutions, general algebraical}%
arises from the different form of~$a$ in the two. If, in
either of them, $-a$~be put instead of~$+a$, and the
rules laid down be followed, the other will be produced.
We see, then, that a simple alteration of the
form of~$a$ in the original equation produces no other
change in the result, or in any one of the steps which
lead to that result, except a simple alteration in the
form of~$a$. From this it follows that, having the solution
of an equation, we have also the solution of all
the equations which can be formed from it, by altering
the form of the different known quantities which are
contained in it. And, as all problems can be reduced
to equations, the solution of one problem will lead us
to the solution of others, which differ from the first in
producing equations in which some of the known
quantities are in different forms. Also, in every identical
equation, the form of one or more of its quantities
may be altered throughout, and the equation will
still remain identically true. For example,
\[
\frac{a^{3} - b^{3}}{a - b} = a^{2} + ab + b^{2}\Add{.}
\]
Change $+b$ into~$-b$, and this equation will become
\[
\frac{a^{3} + b^{3}}{a + b} = a^{2} - ab + b^{2},
\]
which last, common division will show to be true.
Again, suppose than when $a$,~$b$, and~$c$ are in a
given form, which we denote by $+a$,~$+b$, and~$+c$,
the solution of a problem is,
\PageSep{112}
\[
x = \frac{b^{2} - 4ac}{a + c - b}.
\]
The following table will show the alterations which
take place in~$x$ when the forms of $a$,~$b$, and~$c$ are
changed in different manners, and the verification of
it will be an exercise for the student.
\[
\begin{array}{c<{\qquad}r}
\TEntry{FORMS OF $a$, $b$, AND~$c$.} &
\TEntry{VALUES OF~$x$.} \\
+a,\ +b,\ +c & \dfrac{b^{2} - 4ac}{a + c - b}\Add{,} \\
+a,\ +b,\ -c & \dfrac{b^{2} + 4ac}{a - c - b}\Add{,} \\
+a,\ -b,\ -c & \dfrac{b^{2} + 4ac}{a - c + b}\Add{,} \\
-a,\ +b,\ -c & -\dfrac{b^{2} - 4ac}{b + a + c}\Add{,} \\
-a,\ -b,\ -c & \dfrac{b^{2} - 4ac}{b - a - c}.
\end{array}
\]
Also, the expression for~$x$ may be written in the
following different ways, the forms of $a$,~$b$, and~$c$ remaining
the same:
\[
\frac{b^{2} - 4ac}{a + c - b},\quad
-\frac{b^{2} - 4ac}{b - a - c},\quad
-\frac{4ac - b^{2}}{a + c - b},\quad
\frac{4ac - b^{2}}{b - a - c}.
\]
We now proceed to apply these principles to the
solution of the following problems:
\Pagelabel{112}%
\index{Courier, problem of the two|EtSeq}%
\index{Problems!of the two couriers|EtSeq}%
\Figure[nolabel]{112}
Two couriers, $A$~and~$B$, in the course of a journey
between the towns $C$~and~$D$, are at the same moment
\PageSep{113}
of time at $A$~and~$B$. $A$~goes $m$~miles, and $B$, $n$~miles
an hour. At what point between $C$~and~$D$ are they
together? It is evident that the answer depends upon
whether they are going in the same or opposite directions,
whether $A$~goes faster or slower than~$B$, and so
on. But all these, as we shall see, are included in
the same general problem, the difference between them
corresponding to the different forms of the letters
which we shall have occasion to use. After solving
the different cases which present themselves, each
upon its own principle, we shall compare the results
in order to establish their connexion. Let the distance~$AB$
be called~$a$.
\textit{Case first.}---Suppose that they are going in the
same direction from $C$~to~$D$, and that $m$~is greater than~$n$.
They will then meet at some point between $B$~and~$D$.
Let that point be~$H$, and let $AH$~be called~$x$.
Then $A$~travels through~$AH$, or~$x$, in the time during
which $B$~travels through~$BH$, or $x - a$. But, since $A$~goes
$m$~miles an hour, he travels the distance~$x$ in
$\dfrac{x}{m}$~hours. Again, $B$~travels the distance $x - a$ in $\dfrac{x - a}{n}$~hours.
These times are the same, and, therefore,
\begin{align*}
\frac{x}{m} = \frac{x - a}{n}\Add{,} \quad\text{or}\quad
x &= \frac{ma}{m - n} = AH \\
\text{and}\quad
x - a &= \frac{na}{m - n} = BH.
\end{align*}
The time which elapses before they meet is
\[
\frac{x}{m} \quad\text{or}\quad \frac{a}{m - n}.
\]
\PageSep{114}
\textit{Case second.}---Suppose them now moving in the
same direction as before, but let $B$~move faster than~$A$.
They never will meet after they come to $A$ and~$B$,
since $B$~is continually gaining upon~$A$, but they
must have met at some point before reaching $A$ and~$B$.
Let that point be~$H$, and, as before, let $AH = x$.
\Figure[nolabel]{114}
Then since $A$~travels through~$HA$ or~$x$ in the time
during which $B$~travels through~$HB$, or $x + a$, in the
same manner as in the last case, we show that
\begin{align*}
\frac{x}{m} = \frac{x + a}{n}\Add{,} \quad\text{or}\quad
x &= \frac{ma}{n - m} = AH \\
\text{and}\quad
x + a &= \frac{na}{n - m} = BH.
\end{align*}
The time elapsed is~$\dfrac{a}{n - m}$.
\textit{Case third.}---If they are moving from $D$ to~$C$, and
if $B$~moves faster than~$A$, the point~$H$ is the same as
in the last case, since, if having in the last case arrived
at $A$ and~$B$, they move back again at the same
rate, they will both arrive at the point~$H$ together.
The answers in this case are therefore the same as in
the last.
\textit{Case fourth.}---Similarly, if they are moving from $D$
to~$C$, and $A$~moves faster than~$B$, the answers are the
same as in the first case, since this is a reverse of the
first case, as the third is of the second. We reserve
\PageSep{115}
for the present the case in which they move equally
fast, as another species of difficulty is involved which
has no connexion with the present subject. We shall
return to it hereafter.
\textit{Case fifth.}---Suppose them now moving in contrary
directions, viz.: $A$~towards~$D$ and $B$~towards~$C$.
Whether $A$~moves faster or slower than~$B$, they must
now meet somewhere between $A$ and~$B$; as before let
them meet in~$H$, and let $AH = x$.
\Figure[nolabel]{115}
Then $A$~moves through~$AH$, or~$x$, in the same time as
$B$~moves through~$BH$, or $a - x$. Therefore
\begin{align*}
\frac{x}{m} &= \frac{a - x}{n}, \quad\text{or} \\
x &= \frac{ma}{m + n}\Add{,} \\
a - x &= \frac{na}{m + n}\Add{.}
\end{align*}
The time elapsed is~$\dfrac{a}{m + n}$.
\textit{Case sixth.}---Let them be moving in contrary directions,
but let $A$~be moving towards~$C$, and $B$~towards~$D$.
They will then have met somewhere between $A$
and~$B$, and as this is only the reverse of the last case,
just as the fourth is of the first, or the third of the
second, the answers are the same. We now exhibit
the results of these different cases in a table, stating
\PageSep{116}
the circumstances of each case, and also whether the
time of meeting is before or after the instant which
finds them at $A$~and~$B$.
%[** TN: Not floated in the original]
\begin{table}[hbt!]
\scriptsize
\Pagelabel{116}
\[
\begin{array}{l|*{3}{c|}>{\ }l}
\hline\hline
\TEntry{Circumstances of the case.} &
\TEntry[the point~$H$.]{Direction of \\ the point~$H$.} &
\TEntry[of~$AH$.]{Value \\ of~$AH$.} &
\TEntry[of~$AH$.]{Value \\ of~$BH$.} &
\TEntry[meeting]{Time of \\ meeting} \\
\hline
\Strut[16pt]
1.~\Bigl\{\!\!\TEntry[Both move from $C$ to~$D$,]
{Both move from $C$ to~$D$, \\ $A$~moves faster than~$B$.} &
\TEntry[Between]{Between \\ $B$~and~$D$.} &
\dfrac{ma}{m - n} & \dfrac{na}{m - n} & \dfrac{a}{m - n} \text{ after.} \\
%
2.~\Bigl\{\!\!\TEntry[Both move from $C$ to~$D$,]
{Both move from $C$ to~$D$, \\ $A$~moves slower than~$B$.} &
\TEntry[Between]{Between \\ $A$~and~$C$.} &
\dfrac{ma}{n - m} & \dfrac{na}{n - m} & \dfrac{a}{n - m} \text{ before.} \\
%
3.~\Bigl\{\!\!\TEntry[Both move from $D$ to~$C$,]
{Both move from $D$ to~$C$, \\ $A$~moves slower than~$B$.} &
\TEntry[Between]{Between \\ $A$~and~$C$.} &
\dfrac{ma}{n - m} & \dfrac{na}{n - m} & \dfrac{a}{n - m} \text{ after.} \\
%
4.~\Bigl\{\!\!\TEntry[Both move from $D$ to~$C$,]
{Both move from $D$ to~$C$, \\ $A$~moves faster than~$B$.} &
\TEntry[Between]{Between \\ $B$~and~$D$.} &
\dfrac{ma}{m - n} & \dfrac{na}{m - n} & \dfrac{a}{m - n} \text{ before.} \\
%
5.~\Bigl\{\!\!\TEntry[$A$ moves towards $D$ and]
{$A$~moves towards $D$ and \\ $B$~towards~$C$.\qquad\qquad} &
\TEntry[Between]{Between \\ $A$~and~$B$.} &
\dfrac{ma}{m + n} & \dfrac{na}{m + n} & \dfrac{a}{m + n} \text{ after.} \\
%
6.~\Bigl\{\!\!\TEntry[$A$ moves towards $C$ and]
{$A$~moves towards $C$ and \\ $B$~towards~$D$.\qquad\qquad} &
\TEntry[Between]{Between \\ $A$~and~$B$.} &
\dfrac{ma}{m + n} & \dfrac{na}{m + n} & \dfrac{a}{m + n} \text{ before.} \\
\hline\hline
\end{array}
\]
\end{table}
Now $\dfrac{a}{m - n}$ and $\dfrac{a}{n - m}$ are the same quantity written in
different forms, for $n - m$ is $-(m - n)$; and according
to the rules
\[
\frac{a}{n - m} = -\frac{a}{m - n}.
\]
Similarly
\[
\frac{ma}{n - m} = -\frac{ma}{m - n},
\]
and so on.
We see also, that in the first and second cases, which
differ in this, that $AH$~falls to the right in the first,
and to the left in the second, the forms of~$AH$ are
different, there being $\dfrac{ma}{m - n}$ in the first, and $-\dfrac{ma}{m - n}$
\PageSep{117}
\index{Form, change of in algebraical expressions|EtSeq}%
in the second. Again, in the same cases, in the first
of which the time of meeting is \emph{after}, and in the second
\emph{before} the moment of being at $A$~and~$B$, we see a
difference of form in the value of that time; in the
first it is~$\dfrac{a}{m - n}$, and in the second~$-\dfrac{a}{m - n}$, or~$\dfrac{a}{n - m}$.
The same remarks apply to the third and fourth examples.
Again, in the first and fifth cases, which only
differ in this, that $B$~is moving towards~$D$ in the first,
and in the contrary direction towards~$C$ in the fifth,
the values of~$AH$, and of the time, may be deduced
from the first by changing the form of~$n$, and writing~$+n$,
instead of~$-n$. The expression for~$BH$ in the
first, if the form of~$n$ be likewise changed, becomes
$-\dfrac{na}{m + n}$, which is the value of~$BH$ in the fifth, but in
a different form. But we observe that $BH$~falls to the
left of~$B$ in the fifth, whereas it fell to the right in the
first. Again, in the first and sixth examples, which
differ in this that $A$~moves towards~$D$ in the first and
towards~$C$ in the sixth, the value of~$AH$ in the sixth
may be deduced from that of~$AH$ in the first by
changing the form of~$m$, which change makes $AH$~become
$\dfrac{-ma}{-m - n}$, or~$\dfrac{-ma}{-(m + n)}$, or~$\dfrac{ma}{m + n}$. If we alter the
value of the time in the first, in the same manner, it
becomes $\dfrac{a}{-m - n}$, or~$-\dfrac{a}{m + n}$, which is of a different
form from that in the sixth; but it must also be observed
that the first is after and the other before the
moment when they are at $A$~and~$B$. In the fifth and
sixth examples which differ in this, that the direction
\PageSep{118}
in which both are going is changed, since in the fifth
they move towards one another, and in the sixth away
from one another, the values of $AH$ and $BH$ in the
one may be deduced from those in the other by a
change of form, both in $m$~and~$n$, which gives the
same values as before. But if $m$~and~$n$ change their
forms in the expression for the time, the value in the
sixth case is $\dfrac{a}{-m - n}$, or~$-\dfrac{a}{m + n}$. Also the time in
%** (Table of Page 116 Repeated.)
\begin{table}[hbt!]
\scriptsize
\[
\begin{array}{l|*{3}{c|}>{\ }l}
\hline\hline
\TEntry{Circumstances of the case.} &
\TEntry[the point~$H$.]{Direction of \\ the point~$H$.} &
\TEntry[of~$AH$.]{Value \\ of~$AH$.} &
\TEntry[of~$AH$.]{Value \\ of~$BH$.} &
\TEntry[meeting]{Time of \\ meeting} \\
\hline
\Strut[16pt]
1.~\Bigl\{\!\!\TEntry[Both move from $C$ to~$D$,]
{Both move from $C$ to~$D$, \\ $A$~moves faster than~$B$.} &
\TEntry[Between]{Between \\ $B$~and~$D$.} &
\dfrac{ma}{m - n} & \dfrac{na}{m - n} & \dfrac{a}{m - n} \text{ after.} \\
%
2.~\Bigl\{\!\!\TEntry[Both move from $C$ to~$D$,]
{Both move from $C$ to~$D$, \\ $A$~moves slower than~$B$.} &
\TEntry[Between]{Between \\ $A$~and~$C$.} &
\dfrac{ma}{n - m} & \dfrac{na}{n - m} & \dfrac{a}{n - m} \text{ before.} \\
%
3.~\Bigl\{\!\!\TEntry[Both move from $D$ to~$C$,]
{Both move from $D$ to~$C$, \\ $A$~moves slower than~$B$.} &
\TEntry[Between]{Between \\ $A$~and~$C$.} &
\dfrac{ma}{n - m} & \dfrac{na}{n - m} & \dfrac{a}{n - m} \text{ after.} \\
%
4.~\Bigl\{\!\!\TEntry[Both move from $D$ to~$C$,]
{Both move from $D$ to~$C$, \\ $A$~moves faster than~$B$.} &
\TEntry[Between]{Between \\ $B$~and~$D$.} &
\dfrac{ma}{m - n} & \dfrac{na}{m - n} & \dfrac{a}{m - n} \text{ before.} \\
%
5.~\Bigl\{\!\!\TEntry[$A$ moves towards $D$ and]
{$A$~moves towards $D$ and \\ $B$~towards~$C$.\qquad\qquad} &
\TEntry[Between]{Between \\ $A$~and~$B$.} &
\dfrac{ma}{m + n} & \dfrac{na}{m + n} & \dfrac{a}{m + n} \text{ after.} \\
%
6.~\Bigl\{\!\!\TEntry[$A$ moves towards $C$ and]
{$A$~moves towards $C$ and \\ $B$~towards~$D$.\qquad\qquad} &
\TEntry[Between]{Between \\ $A$~and~$B$.} &
\dfrac{ma}{m + n} & \dfrac{na}{m + n} & \dfrac{a}{m + n} \text{ before.} \\
\hline\hline
\end{array}
\]
\caption{(Table of \PageRef[Page]{116} Repeated.)}
\end{table}
the fifth case is after the moment at which they are
at $A$~and~$B$, and in the sixth case it is before. From
these comparisons we deduce the following general
conclusions:
1. If we take the first case as a standard, we may,
from the values which it gives, deduce those which
hold good in all the other cases. If a second case be
taken, and it is required to deduce answers to the
\PageSep{119}
second case from those of the first, this is done by
changing the sign of all those quantities whose directions
are opposite in the second case to what they are
in the first, and if any answer should appear in a negative
form, such as~$\dfrac{ma}{m - n}$, when $m$~is less than~$n$,
which may be written thus $-\dfrac{ma}{n - m}$, it is a sign that
the quantity which it represents is different in direction
in the first and second cases. If it be a right
line measured from a given point in all the cases,
such as~$AH$, it is a sign that $AH$~falls on the left in
the second case, if it fell on the right in the first case,
and the converse. If it be the time elapsed between
the moment in which the couriers are at $A$~and~$B$ and
their meeting, it is a sign that the moment of meeting
is before the other, in the second case, if it were after
it in the first, and the converse. We see, then, that
these six cases can be all contained in one if we apply
this rule, and it is indifferent which of the cases is
taken as the standard, provided the corresponding
alterations are made to determine answers to the rest.
This detail has been entered into in order that the
student may establish from his own experience the
general principle which will conclude this part of the
subject. Further illustration is contained in the following
problem:
A workman receives $a$~shillings a day for his labor
\index{Problems!of loss and gain as illustrating changes of sign}%
or a proportion of $a$~shillings for any part of a day
which he works. His expenses are $b$~shillings every
\PageSep{120}
day, whether he works or no, and after $m$~days he
finds that he has gained $c$~shillings. How many days
did he work? Let $x$ be that number of days, $x$~being
either whole or fractional; then for his work he receives
$ax$~shillings, and during the $m$~days his expenditure
is $bm$~shillings, and since his gain is the difference
between his receipts and expenditure:
\begin{gather*}
ax - bm = c\Add{,} \\
\text{or}\quad
x = \frac{bm + c}{a}\Add{.}
\end{gather*}
Now suppose that he had worked so little as to lose $c$~shillings
instead of gaining anything. The equation
from which $x$~is derived is now
\[
bm - ax = c,
\]
which, when its form is changed, becomes
\[
ax - bm = -c,
\]
an equation which only differs from the former in having
$-c$~written instead of~$c$. The solution of the equation
is
\[
x = \frac{bm - c}{a},
\]
which only differs from the former in having $-c$ instead
of~$+c$. It appears then that we may alter the
solution of a problem which proceeds upon the supposition
of a gain into the solution of one which supposes
an equal loss, by changing the form of the expression
which represents that gain; and also that if
the answer to a problem which we have solved upon
the supposition of a gain should happen to be negative,
\PageSep{121}
suppose it~$-c$, we should have proceeded upon
the supposition that there is a loss and should in that
case have found a loss,~$c$. When such principles as
these have been established, we have no occasion to
correct an erroneous solution by recommencing the
whole process, but we may, by means of the form of
the answer, set the matter right at the end. The
principle is, that a negative solution indicates that
the nature of the answer is the very reverse of that
which it was supposed to be in the solution; for example,
if the solution supposes a line measured in
feet in one direction, a negative answer, such as~$-c$,
indicates that $c$~feet must be measured in the opposite
direction; if the answer was thought to be a number
of days \emph{after} a certain epoch, the solution shows that
it is $c$~days before that epoch; if we supposed that $A$~was
to receive a certain number of pounds, it denotes
that he is to pay $c$~pounds, and so on, In deducing
this principle we have not made any supposition as
to what $-c$~is; we have not asserted that it indicates
the subtraction of~$c$ from~$0$; we have derived the result
from observation only, which taught us first to
deduce rules for making that alteration in the result
which arises from altering~$+c$ into~$-c$ at the commencement;
and secondly, how to make the solution
of one case of a problem serve to determine those of
all the others. By observation then the student must
acquire his conviction of the truth of these rules, reserving
all metaphysical discussion upon such \Typo{quantyties}{quantities}
\PageSep{122}
\index{Zero!its varying significance as an algebraical result|EtSeq}%
as $+c$ and~$-c$ to a later stage, when he will be
better prepared to understand the difficulties of the
subject. We now proceed to another class of difficulties,
which are generally, if possible, as much misconceived
by the beginner as the use of the negative sign.
Take any fraction~$\dfrac{a}{b}$. Suppose its numerator to
remain the same, but its denominator to decrease, by
which means the fraction itself is increased. For example,
$\dfrac{5}{12}$~is greater than~$\dfrac{5}{20}$~or the twelfth part of~$5$
is greater than its twentieth part. Similarly, $\dfrac{2\frac{1}{2}}{4\frac{1}{6}}$~is
greater than~$\dfrac{2\Typo{\frac{1}{3}}{\frac{1}{2}}}{\frac{27}{2}}$, etc. If, then, $b$~be diminished more
and more, the fraction~$\dfrac{a}{b}$ becomes greater and greater,
and there is no limit to its possible increase. To show
this, suppose that $b$~is a part of~$a$, or that $b = \dfrac{a}{m}$. Then
$\dfrac{a}{b}$ or~$\dfrac{a}{\,\frac{a}{m}\,}$ is~$m$. Now since $b$~may diminish so as to be
equal to any part of~$a$, however small, that is, so as
to make $m$~any number, however great, $\dfrac{a}{b}$~which is
$= m$ may be any number however great. This diminution
of~$b$, and the consequent increase of~$\dfrac{a}{b}$, may be
carried on to any extent, which we may state in these
words: As the quantity~$b$ becomes nearer and nearer
to~$0$, the fraction~$\dfrac{a}{b}$ increases, and in the interval in
which $b$~passes from its first magnitude to~$0$, the fraction~$\dfrac{a}{b}$
passes from its first value through every possible
greater number. Now, suppose that the solution
of a problem in its most general form is~$\dfrac{a}{b}$, but that
\PageSep{123}
in one particular case of that problem $b$~is~$= 0$. We
have then instead of a solution~$\dfrac{a}{0}$, a symbol to which
we have not hitherto given a meaning.
To take an instance: return to the problem of the
\index{Fractions!singular values of|EtSeq}%
\index{Infinite quantity, meaning of|EtSeq}%
\index{Singular values|EtSeq}%[** TN: Moved frompage 122]
two couriers, and suppose that they move in the same
direction from $C$~to~$D$ (\textit{Case first}) at the same rate, or
that $m = n$. We find that $AH = \dfrac{ma}{m - n}$ or $\dfrac{ma}{n - m}$ or
$\dfrac{ma}{0}$. On looking at the equation which produced this
result we find that it becomes $\dfrac{x}{m} = \dfrac{x - a}{m}$, or $x = x - a$,
which is impossible. On looking at the manner in
which this equation was formed, we find that it was
made on the supposition that $A$~and~$B$ are together at
some point, which in this case is also impossible, since
if they move at the same rate, the same distance which
separated them at one moment will separate them at
any other, and they will never be together, nor will
they ever have been together on the other side of~$A$.
The conclusion to be drawn is, that such an equation
as $x = \dfrac{a}{0}$ indicates that the supposition from which $x$~was
deduced can never hold good. Nevertheless in
the common language of algebra it is said that they
meet at an infinite distance, and that $\dfrac{a}{0}$~is infinite.
This phrase is one which in its literal meaning is an
absurdity, since there is no such thing as an infinite
number, that is a number which is greater than any
other, because the mind can set no bounds to the
magnitude of the numbers which it can conceive, and
\PageSep{124}
whatever number it can imagine, however great, it
can imagine the next to it. But as the use of the
phrase is very general, the only method is to attach a
meaning which shall not involve absurdity or confusion
of ideas. The phrase used is this: When
$c = b$, $\dfrac{a}{c - b} = \dfrac{a}{0}$ and is infinitely great. The student
should always recollect that this is an abbreviation of
the following sentence. ``The fraction $\dfrac{a}{c - b}$ becomes
greater and greater as $c$~approaches more and more
near to~$b$; and if $c$, setting out from a certain value,
should change gradually until it becomes equal to~$b$,
the fraction $\dfrac{a}{c - b}$ setting out also from a certain value,
will attain any magnitude however great, before $c$~becomes
equal to~$b$.'' That is, before a fraction can assume
the form~$\dfrac{a}{0}$, it must increase without limit. The
symbol~$\infty$ is used to denote such a fraction, or in general
any quantity which increases without limit. The
following equation will tend to elucidate the use of
this symbol. In the problem of the two couriers, the
equation which gave the result $\dfrac{ma}{0}$ was $\dfrac{x}{m} = \dfrac{x - a}{m}$, or
$x = x - a$, which is evidently impossible. Nevertheless,
the larger $x$~is taken the more near is this equation
to the truth, as may be proved by dividing both
sides by~$x$, when it becomes $1 = 1 - \dfrac{a}{x}$, which is never
exactly true. But the fraction~$\dfrac{a}{x}$ decreases as $x$~increases,
and by taking $x$~sufficiently great may be reduced
to any degree of smallness. For example, if it
\PageSep{125}
is required that $\dfrac{a}{x}$ should be as small as~$\dfrac{1}{10000000}$ of a
unit, take $x$~as great as~$10000000a$, and the fraction
becomes $\dfrac{a}{10000000a}$, or~$\dfrac{1}{10000000}$. But as $\dfrac{a}{x}$~becomes
smaller and smaller, the equation $1 = 1 - \dfrac{a}{x}$ becomes
nearer and nearer the truth, which is expressed
by saying that when $1 = 1 - \dfrac{a}{x}$, or $x = x - a$, the solution
is $x = \infty$. In the solution of the problem of
the two couriers this does not appear to hold good,
since when $m = n$ and $x = \dfrac{ma}{0}$ the same distance~$a$
always separates them, and no travelling will bring
them nearer together. To show what is meant by
saying that the greater $x$ is, the nearer will it be a solution
of the problem, suppose them to have travelled
at the same rate to a great distance from~$C$. They
\Figure[nolabel]{125}
can never come together unless $CA$~becomes equal to~$CB$,
or $A$~coincides with~$B$, which never happens,
since the distance~$AB$ is always the same. But if we
suppose that they have met, though an error always
will arise from this false supposition, it will become
less and less as they travel farther and farther from~$C$.
For example, let $CA = 10000000 AB$, then the
supposing that they have met, or that $B$~and~$A$ coincide,
or that $BA = 0$, is an error which involves no
more than $\dfrac{1}{10000000}$ of~$AC$; and though $AB$~is always
of the same numerical magnitude, it grows smaller
\PageSep{126}
and smaller in comparison with~$AC$, as the latter
grows greater and greater.
Let us suppose now that in the problem of the two
\index{Fractions!evanescent|(}%
couriers they move in the same direction at the same
rate, as in the case we have just considered, but that
moreover they set out from the same point, that is,
let $a = 0$. It is now evident that they will always be
together, that is, that any value of~$x$ whatever is an
answer to the question. On looking at the value of~$AH$,
or~$\dfrac{ma}{m - n}$, we find the numerator and denominator
both equal to~$0$, and the value of~$AH$ appears in
the form~$\dfrac{0}{0}$. But from the problem we have found
that one value cannot be assigned to~$AH$, since every
point of their course is a point where they are together.
The solution of the following equation will
further elucidate this. Let
\begin{align*}
ax + by &= c\Add{,} \\
dx + ey &=f,
\end{align*}
from which, by the common method of solution, we
find
\[
x = \frac{ce - bf}{ae - bd},\qquad
y = \frac{af - cd}{ae - bd}.
\]
Now, let us suppose that $ce = bf$ and $ae = bd$. Dividing
the first of these by the second, we find
\[
\frac{ce}{ae} = \frac{bf}{bd},\quad\text{or}\quad
\frac{c}{a} = \frac{f}{d},\quad\text{or}\quad
cd = af.
\]
The values both of $x$~and~$y$ in this case assume the
form~$\dfrac{0}{0}$; to find the cause of this we must return to
\PageSep{127}
the equations. If we divide the first of these by~$c$,
and the second by~$f$, we find that
\begin{alignat*}{2}
\frac{a}{c} x &+ \frac{b}{c} y &&= 1\Chg{.}{,} \\
\frac{d}{f} x &+ \frac{e}{f} y &&= 1.
\end{alignat*}
But the equations $ce = bf$ and $cd = af$ give us $\dfrac{b}{c} = \dfrac{e}{f}$
and $\dfrac{a}{c} = \dfrac{d}{f}$, that is, these two are, in fact, one and
the same equation repeated, from which, as has been
explained before, an infinite number of values of $x$
and~$y$ can be found; in fact, any value may be given
to~$x$ provided $y$~be then found from the equation. We
see that in these instances, when the value of any
quantity appears in the form~$\dfrac{0}{0}$, that quantity admits
of an infinite number of values, and this indicates that
the conditions given to determine that quantity are
not sufficient. But this is not the only cause of the
appearance of a fraction in the form~$\dfrac{0}{0}$. Take the
identical equation
\[
\frac{a^{2} - b^{2}}{c(a - b)} = \frac{a + b}{c}.
\]
When $a$~approaches towards~$b$, $a + b$~approaches towards~$2b$,
and $a^{2} - b^{2}$ and $a - b$ approach more and
more nearly towards~$0$. If $a = b$ the equation assumes
this form:
\[
\frac{0}{0} = \frac{2b}{c}.
\]
\PageSep{128}
This may be explained thus: if we multiply the numerator
and denominator of the fraction~$\dfrac{A}{B}$ by $a - b$
(which does not alter its value) it becomes $\dfrac{Aa - Ab}{Ba - Bb}$.
If in the course of an investigation this has been
done when the two quantities $a$~and~$b$ are equal to
one another, the fraction~$\dfrac{A}{B}$ or $\dfrac{Aa - Ab}{Ba - Bb}$ will appear
in the form~$\dfrac{0}{0}$. But since the result would have been
$\dfrac{A}{B}$ had that multiplication not been performed, this
last fraction must be used instead of the unmeaning
form~$\dfrac{0}{0}$. Thus the fraction $\dfrac{a^{2} - b^{2}}{c(a - b)}$ or $\dfrac{(a + b)(a - b)}{c(a - b)}$
is the fraction~$\dfrac{a + b}{c}$ after its numerator and denominator
have been multiplied by~$a - b$, and may be used
in all cases except that in which $a = b$. When the
form~$\dfrac{0}{0}$ occurs, the problem must be carefully examined
in order to ascertain the reason.
\index{Fractions!evanescent|)}%
\PageSep{129}
\Chapter{X.}{Equations of the Second Degree.}
\index{Equations!second@{of the second degree|EtSeq}}%
\index{Quadratic!equations|EtSeq}%
\index{Roots|EtSeq}%
\First{Every} operation of algebra is connected with another
which is exactly opposite to it in its effects.
Thus addition and subtraction, multiplication and division,
are reverse operations, that is, what is done
by the one is undone by the other. Thus $a + b - b$ is~$a$,
and $\dfrac{ab}{b}$~is~$a$. Now in connexion with the raising of
powers is a contrary operation called the extraction
of roots. The term \emph{root} is thus explained: We have
seen that $aa$, or~$a^{2}$, is called the square of~$a$; from
which $a$~is called the square root of~$a^{2}$. As $169$~is
called the square of~$13$, $13$~is called the square root of~$169$.
The following table will show how this phraseology
is carried on.
\begin{align*}
&\text{$a$ is called the square root of $a^{2}$, denoted by $\sqrt{a^{2}}$}\Add{,} \\
&\text{$a$ \Ditto[is] \Ditto[called] \Ditto[the]
\PadTxt[r]{square}{cube} root of $a^{3}$,
\Ditto[denoted] \Ditto[by] $\sqrt[3]{a^{3}}$}\Add{,} \\
&\text{$a$ \Ditto[is] \Ditto[called] \Ditto[the]
\PadTxt[r]{square}{fourth} root of $a^{4}$,
\Ditto[denoted] \Ditto[by] $\sqrt[4]{a^{4}}$}\Add{,} \\
&\text{$a$ \Ditto[is] \Ditto[called] \Ditto[the]
\PadTxt[r]{square}{fifth} root of $a^{5}$,
\Ditto[denoted] \Ditto[by] $\sqrt[5]{a^{5}}$}\Add{,} \\
&\etc.\Add{,}
\PadTxt{is called the square root of $a^{2}$, denoted by}{etc.\Add{,}} \etc.
\end{align*}
\PageSep{130}
If $b$~stand for~$a^{5}$, $\sqrt[5]{b}$~stands for~$a$, and the foregoing
table may be represented thus:
\begin{alignat*}{2}
\text{If } a^{2} &= b,\quad a &&= \sqrt{b}\Chg{;}{,} \\
\text{if } a^{3} &= b,\quad a &&= \sqrt[3]{b},\quad\etc.
\end{alignat*}
The usual method of proceeding is to teach the
\index{Approximations}%
student to extract the square root of any algebraical
quantity immediately after the solution of equations
of the first degree. We would rather recommend him
to omit this rule until he is acquainted with the solution
of equations of the second degree, except in the
cases to which we now proceed. In arithmetic, it
must be observed that there are comparatively very
few numbers of which the square root can be extracted.
For example, $7$~is not made by the multiplication
either of any whole number or fraction by
itself. The first is evident; the second cannot be
readily proved to the beginner, but he may, by taking
a number of instances, satisfy himself of this, that no
fraction which is really such, that is whose numerator
is not measured by its denominator, will give a whole
number when multiplied by itself, thus $\frac{4}{2} × \frac{4}{3}$ or~$\frac{16}{9}$ is
not a whole number, and so on. The number~$7$,
therefore, is neither the square of a whole number, nor
of a fraction, and, properly speaking, has no square
root. Nevertheless, fractions can be found extremely
near to~$7$, which have square roots, and this degree
of nearness may be carried to any extent we please.
Thus, if required, between $7$~and $7\frac{1}{1000000000}$ could
be found a fraction which has a square root, and the
\PageSep{131}
fraction in the last might be decreased to any extent
whatever, so that though we cannot find a fraction
whose square is~$7$, we may nevertheless find one whose
square is as near to~$7$ as we please. To take another
example, if we multiply $1.4142$ by itself the product
is $1.99996164$, which only differs from~$2$ by the very
small fraction~$.00003836$, so that the square of~$1.4142$
is very nearly~$2$, and fractions might be found whose
squares are still nearer to~$2$. Let us now suppose the
following problem. A man buys a certain number of
yards of stuff for two shillings, and the number of
yards which he gets is exactly the number of shillings
which he gives for a yard. How many yards does he
buy? Let $x$~be this number, then $\dfrac{2}{x}$~is the price of
one yard, and $x = \dfrac{2}{x}$ or $x^{2} = 2$. This, from what we
have said, is impossible, that is, there is no exact
number of yards, or parts of yards, which will satisfy
the conditions; nevertheless, $1.4142$~yards will nearly
do it, $1.4142136$ still more nearly, and if the problem
were ever proposed in practice, there would be no
difficulty in solving it with sufficient nearness for any
purpose. A problem, therefore, whose solution contains
a square root which cannot be extracted, maybe
rendered useful by approximation to the square root.
Equations of the second degree, commonly called
quadratic equations, are those in which there is the
second power, or square of an unknown quantity:
such as $x^{2} - 3 = 4x^{2} - 15$, $x^{2} + 3x = 2x^{2} - x - 1$, etc.
\PageSep{132}
\index{Factoring@{Factoring|EtSeq}}%
\index{Theory of equations@{Theory of equations|EtSeq}}%
By transposition of their terms, they may always be
reduced to one of the following forms:
\begin{align*}
&ax^{2} + b = 0\Add{,} \\
&ax^{2} - b = 0\Add{,} \displaybreak[0]\\
&ax^{2} + bx + c = 0\Add{,} \\
&ax^{2} - bx + c = 0\Add{,} \displaybreak[0]\\
&ax^{2} + bx - c = 0\Add{,} \\
&ax^{2} - bx - c = 0.
\end{align*}
For example, the two equations given above, are
equivalent to $3x^{2} - 12 = 0$, and $x^{2} - 4x - 1 = 0$, which
agree in form with the second and last. In order to
proceed to each of these equations, first take the equation
$x^{2} = a^{2}$. This equation is the same as $x^{2} - a^{2} = 0$,
or $(x + a)(x - a) = 0$. Now, in order that the product
of two or more quantities may be equal to nothing,
it is sufficient that \emph{one} of those quantities be nothing,
and therefore a value of~$x$ may be derived from
either of the following equations:
\begin{align*}
x - a &= 0\Add{,} \\
\text{or}\quad
x + a &= 0\Add{,}
\end{align*}
the first of which gives $x = a$, and the second $x = -a$.
To elucidate this, find $x$ from the following equation:
\[
(3x + a)(a^{3} + x^{3}) = (x^{2} + ax)(a^{2} + ax + 2x^{2})\Add{;}
\]
develop this equation, and transpose all its terms on
one side, when it becomes
\begin{gather*}
x^{4} - 2a^{2} x^{2} + a^{4} + 2a^{3} x - 2ax^{3} = 0\Add{,} \\
\text{or}\quad
(x^{2} - a^{2})^{2} - 2ax(x^{2} - a^{2}) = 0\Add{,} \\
\text{or}\quad
(x^{2} - a^{2})(x^{2} - 2ax - a^{2}) = 0.
\end{gather*}
This last equation is true when $x^{2} - a^{2} = 0$, or when
\PageSep{133}
$x^{2} = a^{2}$, which is true either when $x= +a$, or $x = -a$.
If in the original equation $+a$~is substituted instead
of~$x$, the result is $4a × 2a^{3} = 2a^{2} × 4a^{2}$; if $-a$~be
substituted instead of~$x$, the result is $0 = 0$, which
show\Note{[sic], not "shows"} that $+a$~and~$-a$ are both correct values of~$x$.
We have here noticed, for the first time, an equation
of condition which is capable of being solved by more
than one value of~$x$. We have found two, and shall
find more when we can solve the equation $x^{2} - 2ax - a^{2} = 0$,
or $x^{2} - 2ax = a^{2}$. Every equation of the second
degree, if it has one value of~$x$, has a second, of
which $x^{2} = a^{2}$ is an instance, where $x = ±a$, in which
by the double sign~$±$ is meant, that either of them
may be used at pleasure. We now proceed to the solution
of $ax^{2} - bx + c = 0$. In order to understand
the nature of this equation, let us suppose that we
take for~$x$ such a value, that $ax^{2} - bx + c$, instead of
being equal to~$0$, is equal to~$y$, that is
\[
y = ax^{2} - bx + c\Add{,}\footnotemark
\Tag{(1)}
\]
\footnotetext{In the investigations which follow, $a$,~$b$, and~$c$ are considered as having
the sign which is marked before them, and no change of form is supposed to
take place.}%
in which the value of~$y$ depends upon the value given
to~$x$, and changes when $x$~changes. Let $m$~be one of
those quantities which, when substituted instead of~$x$,
makes $ax^{2} - bx + c$ equal to nothing, in which case $m$~is
called a root of the equation,
\[
ax^{2} - bx + c = 0\Add{,}
\Tag{(2)}
\]
and it follows that
\[
am^{2} - bm + c = 0\Add{.}
\Tag{(3)}
\]
\PageSep{134}
Subtract \Eq{(3)} from~\Eq{(1)}, the result of which is
\[
y = a(x^{2} - m^{2}) - b(x - m) = (x - m)(a\, \overline{x + m} - b).
\]
Here $y$~is evidently equal to~$0$, when $x = m$, as we
might expect from the supposition which we made;
but it is also nothing when $a(x + m) - b = 0$; there
is, therefore, another value of~$x$, for which $y = 0$; if
we call this~$n$ we find it from the equation $a(n + m) - b = 0$,
\[
\text{or}\quad
n + m = \frac{b}{a}\Add{.}
\Tag{(4)}
\]
In \Eq{(3)} substitute for~$b$ its value derived from~\Eq{(4)}, from
which $b = a(n + m)$; it then becomes
\[
am^{2} - am(n + m) + c = 0,\quad\text{or}\quad
c - amn = 0,
\]
which gives
\[
mn = \frac{c}{a}.
\Tag{(5)}
\]
Substitute in \Eq{(1)} the values of $b$ and~$c$ derived from
\Eq{(4)}~and~\Eq{(5)}, which gives
\begin{align*}
y &= ax^{2} - a(m + n)x + amn \\
&= a(x^{2} - \overline{m + n}\, x + mn).
\end{align*}
Now the second factor of this expression arises from
multiplying together $\overline{x - m}$ and~$\overline{x - n}$; therefore,
\[
y = a(x - m)(x - n)\Add{.}
\Tag{(6)}
\]
To take an example of this, let $y = 4x^{2} - 5x + 1$.
Here when $x = 1$, $y = 4 - 5 + 1 = 0$, and therefore
$m = 1$. If we divide $4x^{2} - 5x + 1$ by~$x - 1$, the quotient
(which is without remainder) is $4x - 1$, and
therefore
\[
y = (x - 1)(4x - 1).
\]
This is also nothing when $4x - 1 = 0$, or when $x$~is~$\frac{1}{4}$.
Therefore $n = \frac{1}{4}$, and $y = 4(x - 1)(x - \frac{1}{4})$, a result
\PageSep{135}
coinciding with that of~\Eq{(6)}. If, therefore, we can find
one of the values of~$x$ which satisfy the equation
$ax^{2} - bx + c = 0$, we can find the other and can divide
$ax^{2} - bx + c$ into the factors $a$,~$x - m$ and $x - n$, or
\[
ax^{2} - bx + c = a(x - m)(x - n).
\]
If we multiply $x + m$ by $x + n$, the only difference between
$(x + m)(x + n)$ and $(x - m)(x - n)$ is in the
sign of the term which contains the first power of~$x$.
If, therefore,
\[
ax^{2} - bx + c = a(x - m)(x - n),
\]
it follows that
\[
ax^{2} + bx + c = a(x + m)(x + n).
\]
We now take the expression $ax^{2} - bx - c$. If there
is one value of~$x$ which will make this quantity equal
to~$0$, let this be~$m$, and let
%[** TN: Reformatted]
\begin{align*}
y &= ax^{2} - bx - c\Add{.} \\
\intertext{Then}
0 &= am^{2} - bm - c, \\
\intertext{from which}\quad
y &= a(x^{2} - m^{2}) - b(x - m) \\
&= (x - m)(a\, \overline{x + m} - b) \\
&= (x - m)(ax + am - b).
\end{align*}
Let $\dfrac{am - b}{a}$ be called~$n$, or let $am - b = an$; then
\begin{align*}
y &= (x - m)(ax + an) \\
&= a(x - m)(x + n).
\end{align*}
As an example, it may be shown that
\[
3x^{2} - x - 2 = 3(x - 1)(x + \tfrac{2}{3}).
\]
Again, with regard to $ax^{2} + bx - c$, since $(x + m)(x - n)$
only differs from $(x - m)(x + n))$ in the sign
\PageSep{136}
of the term which contains the first power of~$x$, it is
evident that
\begin{align*}
\text{if}\quad
ax^{2} - bx - c &= a(x - m)(x + n)\Add{,} \\
ax^{2} + bx - c &= a(x + m)(x - n).
\end{align*}
Results similar to those of the first case may be obtained
for all the others, and these results may be arranged
in the following way. In the first and third,
$m$~is a quantity, which, when substituted for~$x$, makes
$y = 0$, and in the second and fourth $m$~and~$n$ are the
same as in the first and third.
\begin{gather*}
\text{1st}\qquad
y = ax^{2} - bx + c = a(x - m)(x - n)\Add{,} \\
m + n = \frac{b}{a}\Add{,} \qquad mn = \frac{c}{a}. \displaybreak[0] \\
%
\text{2d}\qquad
y = ax^{2} + bx + c = a(x + m)(x + n)\Add{,} \\
m + n = \frac{b}{a}\Add{,} \qquad mn = \frac{c}{a}. \displaybreak[0] \\
%
\text{3d}\qquad
y = ax^{2} - bx - c = a(x - m)(x + n)\Add{,} \\
m - n = \frac{b}{a}\Add{,} \qquad mn = \frac{c}{a}. \displaybreak[0] \\
%
\text{4th}\qquad
y = ax^{2} + bx - c = a(x + m)(x - n)\Add{,} \\
m - n = \frac{b}{a}\Add{,} \qquad mn = \frac{c}{a}.
\end{gather*}
We must now inquire in what cases a value can be
found for~$x$, which will make $y = 0$ in these different
expressions, and in this consists the solution of equations
of the second degree.
Let
\[
y = ax^{2} - bx + c\Add{,}
\Tag{(1)}
\]
\PageSep{137}
\index{Quadratic!roots, discussion of the character of|EtSeq}%
\index{Roots|EtSeq}%
and observe that $(2ax - b)^{2} = 4a^{2} x^{2} - 4abx + b^{2}$.
Multiply both sides of~\Eq{(1)} by~$4a$, which gives
\[
4ay = 4a^{2} x^{2} - 4abx + 4ac\Add{.}
\Tag{(2)}
\]
Add $b^{2}$ to the first two terms of the second side of~\Eq{(2)},
and subtract it from the third, which will not alter the
whole, and this gives
\[
4ay = 4a^{2} x^{2} - 4abx + b^{2} + 4ac - b^{2}
= (2ax - b)^{2} + ac - b^{2}\Add{.}
\Tag{(3)}
\]
Now it must be recollected that the square of any
quantity is positive whether that quantity is positive
or negative. This has been already sufficiently explained
in saying that a change of the form of any expression
does not change the form of its square. Common
multiplication shows that $(c - d)^{2}$ and $(d - c)^{2}$
are the same thing; and, since one of these must be
positive, the other must be also positive. Whenever,
therefore, we wish to say that a quantity is positive, it
can be done by supposing it equal to the square of an
algebraical quantity. In equation~\Eq{(3)} there are three
distinct cases to be considered.
I. When $b^{2}$~is greater than~$4ac$, that is, when
$b^{2} - 4ac$~is positive, let $b^{2} - 4ac = k^{2}$, which expresses
the condition.
Then
\[
4ay = (2ax - b)^{2} - k^{2}\Add{,}
\Tag{(4)}
\]
and we determine those values of~$x$, which make $y = 0$,
from the equation,
\[
(2ax - b)^{2} - k^{2} = 0.
\]
We have already solved such an equation, and we
find that
\PageSep{138}
\[
2ax - b = ±k,
\]
{\Loosen where either sign may be taken. This shows that $y$
or $ax^{2} - bx + c$ is equal to nothing either when}
\begin{align*}
\text{instead of~$x$ is put}\quad
\frac{b + k}{2a} &= \frac{b + \sqrt{b^{2} - 4ac}}{2a} = m, \\
\text{or}\quad
\frac{b - k}{2a} &= \frac{b - \sqrt{b^{2} - 4ac}}{2a} = n,
\end{align*}
the second values being formed by putting, instead of~$k$
its value $\sqrt{b^{2} - 4ac}$. They are both positive quantities,
because $k^{2}$~being equal to $b^{2} - 4ac$ is less than~$b^{2}$,
and therefore $k$~is less than~$b$, and therefore $\dfrac{b + k}{2a}$
and $\dfrac{b - k}{2a}$ are both positive. These are the quantities
which we have called $m$~and~$n$ in the former investigations,
and, therefore,
\begin{multline*}
ax^{2} - bx + c
= a(x - m)(x - n) \\
= a\biggl(x - \frac{b + \sqrt{b^{2} - 4ac}}{2a}\biggr)
\biggl(x - \frac{b - \sqrt{b^{2} - 4ac}}{2a}\biggr).
\end{multline*}
Actual multiplication of the factors will show that this
is an identical equation.
II. When $b^{2}$, instead of being greater than~$4ac$, is
\index{Perfect square}%
equal to it, or when $b^{2} - 4ac = 0$ and $k = 0$. In this
case the values of $m$~and~$n$ are equal, each being $\dfrac{b}{2a}$
and
\[
y = ax^{2} - bx + c = a(x - m)(x - n)
= a\biggl(x - \frac{b}{2a}\biggr)^{2}.
\]
In this case $y$~is said, in algebra, to be a perfect
square, since its square root can be extracted, and is
$\sqrt{a}\biggl(x - \dfrac{b}{2a}\biggr)$. Arithmetically speaking, this would
\PageSep{139}
not be a perfect square unless $a$~was a number whose
square root could be extracted, but in algebra it is
usual to call any quantity a perfect square with respect
to any letter, which, when reduced, does not
contain that letter under the sign~$\sqrt{\Z}$. This result is
one which often occurs, and it must be recollected
that when $b^{2} - 4ac = 0$, $ax^{2} - bx + c$ is a perfect
square.
III. When $b^{2}$ is less than~$4ac$, or when $b^{2} - 4ac$
is negative and $4ac - b^{2}$ positive, let $4ac - b^{2} = k^{2}$,
and equation~\Eq{(3)} becomes
\[
4ay = (2ax - b)^{2} + k^{2}.
\]
In this case no value of~$x$ can ever make $y = 0$, for
the equation $v^{2} + w^{2} = 0$ indicates that $v^{2}$~is equal to
$w^{2}$ with a contrary sign, which cannot be, since all
squares have the same sign. The values of~$x$ are said,
in this case, to be impossible, and it indicates that
there is something absurd or contradictory in the conditions
of a problem which leads to such a result.
Having found that whenever
\[
ax^{2} - bx + c = a(x - m)(x - n),
\]
it follows that $ax^{2} + bx + c = a(x + m)(x + n)$, we
know that
(1) when $b^{2}$~is greater than~$4ac$,
\[
ax^{2} + bx + c =
a\biggl(x + \frac{b + \sqrt{b^{2} - 4ac}}{2a}\biggr)
\biggl(x + \frac{b - \sqrt{b^{2} - 4ac}}{2a}\biggr);
\]
(2) when $b^{2} = 4ac$,
\PageSep{140}
\[
ax^{2} + bx + c = a\biggl(x + \frac{b}{2a}\biggr)^{2},
\]
and $y$~is a perfect square;
(3) when $b^{2}$~is less than~$4ac$, $ax^{2} + bx + c$ cannot
be divided into factors.
Now, let
\[
y = ax^{2} - bx - c\Add{.}
\Tag{(1)}
\]
As before,
\begin{align*}
4ay &= 4a^{2} x^{2} - 4abx + b^{2} - 4ac - b^{2} \\
&= (2ax - b)^{2} - (b^{2} + 4ac)\Add{.}
\Tag{(2)}
\end{align*}
Let $b^{2} + 4ac = k^{2}$. Then
\[
4ay = (2ax - b)^{2} - k^{2}.
\Tag{(3)}
\]
{\Loosen Therefore $y$~is~$0$ when $(2ax - b)^{2} = k^{2}$, or when
$2ax - b = ±k$.}
That is,
\begin{align*}
m &= \frac{b + k}{2a} = \frac{b + \sqrt{b^{2} + 4ac}}{2a}\Add{,} \\
n &= \frac{b - k}{2a} = \frac{b - \sqrt{b^{2} + 4ac}}{2a}.
\end{align*}
Now, because $b^{2}$~is less than $b^{2} + 4ac$, $b$~is less
than $\sqrt{b^{2} + 4ac}$, therefore $n$~is a negative quantity.
Leaving, for the present, the consideration of the
negative quantity, we may decompose~\Eq{(3)} into factors
by means of the general formula
\[
p^{2} - q^{2} = (p - q)(p + q),
\]
which gives
\begin{align*}
4ay &= (2ax - b - k)(2ax - b + k) \\
&= 4a^{2}\biggl(x - \frac{k + b}{2a}\biggr)
\biggl(x + \frac{k - b}{2a}\biggr)\Add{,}
\end{align*}
from which $y$~or
\PageSep{141}
\[
ax^{2} - bx - c
= a\biggl(x - \frac{\sqrt{b^{2} + 4ac} + b}{2a}\biggr)
\biggl(x + \frac{\sqrt{b^{2} + 4ac} - b}{2a}\biggr)\Add{.}
\]
Therefore, from what has been proved before,
\[
ax^{2} + bx - c
= a\biggl(x + \frac{\sqrt{b^{2} + 4ac} + b}{2a}\biggr)
\biggl(x - \frac{\sqrt{b^{2} + 4ac} - b}{2a}\biggr)\Add{.}
\]
The following are some examples, of the truth of
which the student should satisfy himself, both by reference
to the ones just established, and by actual
multiplication:
\begin{gather*}
\begin{aligned}
2x^{2} - 7x + 3
&= 2\biggl(x - \frac{7 + \sqrt{49 - 24}}{4}\biggr)
\biggl(x - \frac{7 - \sqrt{49 - 24}}{4}\biggr) \\
&= 2(x - 3)(x - \tfrac{1}{2})\Add{,}
\end{aligned} \displaybreak[0]\\
%
3x^{2} - 6x + 1
= 3\biggl(x - \frac{3 + \sqrt{6}}{3}\biggr)
\biggl(x - \frac{3 - \sqrt{6}}{3}\biggr)\Add{,}\footnotemark \\
5\tfrac{1}{2} x^{2} - 22x + 22 = 5\tfrac{1}{2}(x - 2)^{2}\Add{,} \\
5x^{2} + 9x - 7
= 5\biggl(x + \frac{\sqrt{221} + 9}{10}\biggr)
\biggl(x - \frac{\sqrt{221} - 9}{10}\biggr).
\end{gather*}
\footnotetext{Recollect that $\sqrt{24} = \sqrt{6 × 4} = \sqrt{6} × \sqrt{4} = 2\sqrt{6}$.}
If we collect together the different results at which
\index{Formulæ, important|(}%
we have arrived, to which species of tabulation the
student should take care to accustom himself, we have
the following:
\begin{multline*}
ax^{2} + bx + c \\
= a\biggl(x + \frac{b + \sqrt{b^{2} - 4ac}}{2a}\biggr)
\biggl(x + \frac{b - \sqrt{b^{2} - 4ac}}{2a}\biggr)\Add{,}
\Tag{(A)}
\end{multline*}
\PageSep{142}
\begin{multline*}
ax^{2} - bx + c \\
= a\biggl(x - \frac{b + \sqrt{b^{2} - 4ac}}{2a}\biggr)
\biggl(x - \frac{b - \sqrt{b^{2} - 4ac}}{2a}\biggr)\Add{,}
\Tag{(B)}
\end{multline*}
\begin{multline*}
ax^{2} + bx - c \\
= a\biggl(x + \frac{\sqrt{b^{2} + 4ac} + b}{2a}\biggr)
\biggl(x - \frac{\sqrt{b^{2} + 4ac} - b}{2a}\biggr)\Add{,}
\Tag{(C)}
\end{multline*}
\begin{multline*}
ax^{2} - bx - c \\
= a\biggl(x - \frac{\sqrt{b^{2} + 4ac} + b}{2a}\biggr)
\biggl(x + \frac{\sqrt{b^{2} + 4ac} - b}{2a}\biggr)\Add{.}
\Tag{(D)}
\end{multline*}
\index{Formulæ, important|)}%
These four cases may be contained in one, if we
apply those rules for the change of signs which we
have already established. For example, the first side
of~\Eq{(C)} is made from that of~\Eq{(A)} by changing the sign
of~$c$; the second side of~\Eq{(C)} is made from that of~\Eq{(A)}
in the same way. We have also seen the necessity
of taking into account the negative quantities which
satisfy an equation, as well as the positive ones; if
we take these into account, each of the four forms of
$ax^{2} + bx + c$ can be made equal to nothing by two
values of~$x$. For example, in~\Eq{(1)}, when
\begin{gather*}
ax^{2} + bx + c = 0\Add{,} \\
\begin{aligned}
\text{either}\quad
x + \frac{b - \sqrt{b^{2} - 4ac}}{2a} &= 0 \\
\text{or}\quad
x + \frac{b + \sqrt{b^{2} - 4ac}}{2a} &= 0\Add{.}
\end{aligned}
\end{gather*}
If we call the values of~$x$ derived from the equations
$m$~and~$n$, we find that
\begin{alignat*}{2}
m &= \frac{-b + \sqrt{b^{2} - 4ac}}{2a}\Add{,}\qquad &
n &= \frac{-b - \sqrt{b^{2} - 4ac}}{2a}\Add{.}
\Tag{(A')} \displaybreak[0]\\
\PageSep{143}
\intertext{\indent In the cases marked \Eq{(B)}, \Eq{(C)}, and~\Eq{(D)}, the results
are}
m &= \frac{b + \sqrt{b^{2} - 4ac}}{2a}\Add{,}\qquad &
n &= \frac{b - \sqrt{b^{2} - 4ac}}{2a}\Add{,}
\Tag{(B')} \displaybreak[0]\\
m &= \frac{-b + \sqrt{b^{2} + 4ac}}{2a}\Add{,}\qquad &
n &= \frac{-b - \sqrt{b^{2} + 4ac}}{2a}\Add{,}
\Tag{(C')} \displaybreak[0]\\
m &= \frac{b + \sqrt{b^{2} + 4ac}}{2a}\Add{,}\qquad &
n &= \frac{b - \sqrt{b^{2} + 4ac}}{2a}\Add{,}
\Tag{(D')}
\end{alignat*}
and in all the four cases the form of $ax^{2} + bx + c$
which is used, is the same as the corresponding form
of
\[
a(x - m)(x - n)
\]
and the following results may be easily obtained. In~\Eq{(A')}
both $m$~and~$n$, if they exist at all, are negative.
I say, if they exist at all, because it has been shown
that if $b^{2} - 4ac$ is negative, the quantity $ax^{2} + bx + c$
cannot be divided into factors at all, since $\sqrt{b^{2} - 4ac}$
is then no algebraical quantity, either positive or negative.
In \Eq{(B')} both, if they exist at all, are positive.
In \Eq{(C')} there are always real values for $m$~and~$n$,
since $b^{2} + 4ac$ is always positive; one of these values
is positive, and the other negative, and the negative
one is numerically the greatest.
In \Eq{(D')} there are also real values of $m$~and~$n$, one
\index{Extension@{Extension of rules and meanings of terms}|(}%
\index{Notation!extension of}%
\index{Rules!extension of meaning of}%
positive and the other negative, of which the positive
one is numerically the greatest. Before proceeding
any further, we must notice an extension of a phrase
\PageSep{144}
which is usually adopted. The words \emph{greater} and \emph{less},
\index{Algebraically@\emph{Algebraically greater}|(}%
\index{Greater@{\emph{Greater} and \emph{less}}, the meaning of}%
\index{Numerically@{\emph{Numerically greater}}}%
as applied to numbers, offer no difficulty, and from
them we deduce, that if $a$~be greater than~$b$, $a - c$ is
greater than $b - c$, as long as these subtractions are
possible, that is, as long as $c$~can be taken both from
$a$~and~$b$. This is the only case which was considered
when the rule was made, but in extending the meaning
of the word \emph{subtraction}, and using the symbol~$-3$
to stand for $5 - 8$, the principle that if $a$~be greater
than~$b$, $a - c$ is greater than $b - c$, leads to the following
results. Since $6$~is greater than~$4$, $6 - 12$~is greater
than $4 - 12$, or $-6$~is greater than~$-8$; again $6 - 6$~is
greater than $4 - 6$, or $0$~is greater than~$-2$. These
results, particularly the last, are absurd, as has been
noticed, if we continue to mean by the terms \emph{greater}
and \emph{less}, nothing more than is usually meant by them
in arithmetic; but in extending the meaning of one
term, we must extend the meaning of all which are
connected with it, and we are obliged to apply the
terms \emph{greater} and \emph{less} in the following way. Of two
algebraical quantities with the same or different signs,
that one is the greater which, when both are connected
with a number numerically greater than either of them,
gives the greater result. Thus $-6$~is said to be greater
than~$-8$, because $20 - 6$~is greater than $20 - 8$, $0$~is
greater than~$-4$, because $6 + 0$ is greater than $6 - 4$;
$+12$~is greater than~$-30$, because $40 + 12$ is greater
than $40 - 30$. Nevertheless $-30$~is said to be \emph{numerically}
greater than~$+12$, because the number contained
\PageSep{145}
in the first is greater than that in the second. For this
reason it was said, that in~\Eq{(C)}, the negative quantity
was \emph{numerically} greater, than the positive, because any
positive quantity is in algebra called greater than any
negative one, even though the number contained in
the first should be less than that in the second. In
the same way $-14$~is said to lie between $+3$~and~$-20$,
being less than the first and greater than the
second. The advantage of these extensions is the
same as that of others; the disadvantage attached to
them, which it is not fair to disguise, is that, if used
without proper caution, they lead the student into
erroneous notions, which some elementary works, far
from destroying, confirm, and even render necessary,
by adopting these very notions as definitions; as for
example, when they say that a negative quantity is
one which is less than nothing; as if there could be
such a thing, the usual meaning of the word less being
considered, and as if the student had an idea of a
quantity less than nothing already in his mind, to
which it was only necessary to give a name.
The product $(x - m)(x - n)$ is positive when
$(x - m)$ and $(x - n)$ have the same, and negative when
they have different signs. This last can never happen
except when $x$~lies between $m$~and~$n$, that is, when $x$~is
algebraically greater than the one, and less than
the other. The \hyperref[table:146]{following table} will exhibit this, where
different products are taken with various signs of $m$~and~$n$,
and three values are given to~$x$ one after the
\index{Algebraically@\emph{Algebraically greater}|)}%
\index{Extension@{Extension of rules and meanings of terms}|)}%
\PageSep{146}
other, the first of which is less than both $m$~and~$n$,
the second between both, and the third greater than
both.
\begin{table}[hbt!]
\[
\begin{array}{lcc}
\multicolumn{1}{c}{\TEntry{PRODUCT.}} &
\TEntry{VALUE OF~$x$.} &
\TEntry[PRODUCT WITH]{VALUE OF THE \\ PRODUCT WITH \\ ITS SIGN.} \\
(x - 4)(x - 7) & +\Z1 & + 18 \\
m = +4 & +\Z5 & -\Z2 \\
n = +7 & + 10 & + 18 \\
%
(x + 10)(x - 3) & -12 & + 30 \\
m = -10 & -\Z7 & - 30 \\
n = +3 & +\Z4 & + 14 \\
%
(x + 2)(x + 12) & - 13 & + 11 \\
m = -2 & -\Z6 & - 24 \\
n = -12 & -\Z1 & + 11
\end{array}
\]
\phantomsection\label{table:146}
\end{table}
The student will see the reason of this, and perform
a useful exercise in making two or three tables
of this description for himself. The result is that
$(x - m)(x - n)$ is negative when $x$~lies between $m$~and~$n$;
is nothing when $x$~is either equal to~$m$ or to~$n$, and
positive when $x$~is greater than both, or less than
both. Consequently, $a(x - m)(x - n)$ has the same
sign as a when $x$~is greater than both $m$~and~$n$, or less
than both, and a different sign from~$a$ when $x$~lies between
both. But whatever may be the signs of $a$,~$b$,
and~$c$, if there are two quantities $m$~and~$n$, which make
\[
ax^{2} + bx + c = a(x - m)(x - n),
\]
that is, if the equation $ax^{2} + bx + c = 0$ has real roots,
the expression $ax^{2} + bx + c$ always has the same sign
\PageSep{147}
as~$a$ for all values of~$x$, except when $x$~lies between
these roots.
{\Loosen It only remains to consider those cases in which
$ax^{2} + bx + c$ cannot be decomposed into different factors,
which happens whenever $b^{2} - 4ac$ is~$0$, or negative.
In the first case when $b^{2} - 4ac = 0$, we have}
\begin{align*}
ax^{2} + bx + c &= a\biggl(x + \frac{b}{2a}\biggr)^{2}\Add{,} \\
ax^{2} - bx + c &= a\biggl(x - \frac{b}{2a}\biggr)^{2}\Add{,}
\end{align*}
{\Loosen and as these expressions are composed of factors, one
of which is a square, and therefore positive, they have
always the same sign as the other factor, which is~$a$.
When $b^{2} - 4ac$ is negative, we have proved that if
$y = ax^{2} ± bx + c$, $4ay = (2ax ± b)^{2} + k^{2}$, where $k^{2} = 4ac - b^{2}$,
and therefore $4ay$~being the sum of two
squares is always positive, that is, $ax^{2} ± bx + c$ has
the same sign as~$a$, whatever may be the value of~$x$.
When $c = 0$, the expression becomes $ax^{2} + bx$, or
$x(ax + b)$, which is nothing either when $x = 0$, or
when $ax + b = 0$ and $x = -\dfrac{b}{a}$; the general expressions
for $m$~and~$n$ become in this case $\dfrac{-b + \sqrt{b^{2}}}{2a}$ and
$\dfrac{-b - \sqrt{b^{2}}}{2a}$, which give the same results.}
When $b = 0$, the expression is reduced to $ax^{2} + c = 0$,
which is nothing when $x =±\sqrt{-\dfrac{c}{a}}$, which is
not possible, except when $c$~and~$a$ have different signs.
In this case, that is, when the expression assumes the
form $ax^{2} - c$, it is the same as
\PageSep{148}
\[
a\biggl(x - \sqrt{\frac{c}{a}}\biggr)\biggl(x + \sqrt{\frac{c}{a}}\biggr).
\]
The same result might be deduced by making $b = 0$
in the general expressions for $m$~and~$n$.
{\Loosen When $a = 0$, the expression is reduced to $bx + c$,
which is made equal to nothing by one value of~$x$ only,
that is~$-\dfrac{c}{b}$. If we take the general expressions
for $m$~and~$n$, and make $a = 0$ in them, that is, in
$\dfrac{-b + \sqrt{b^{2} - 4ac}}{2a}$, and $\dfrac{-b - \sqrt{b^{2} - 4ac}}{2a}$, we find as
the results $\dfrac{0}{0}$ and~$\Chg{\dfrac{-2b}{0}}{-\dfrac{2b}{0}}$. These have been already explained.
The first may either indicate that any value
of~$x$ will solve the problem which produced the equation
$ax^{2} + bx + c = 0$, or that we have applied a rule
to a case which was not contemplated in its formation,
and have thereby created a factor in the numerator
and denominator of~$x$, which, in attempting to
apply the rule, becomes equal to nothing. The student
is referred to the problem of the two couriers,
solved in the preceding part of this treatise. The
latter is evidently the case here, because in returning
to the original equation, we find it reduced to $bx + c = 0$,
which gives a rational value for~$x$, namely,~$-\dfrac{c}{b}$.
The second value, or~$-\dfrac{2b}{0}$, which in algebraical language
is called infinite, may indicate, that though
there is no other value of~$x$, except~$-\dfrac{c}{b}$, which
solves the equation, still that the greater the number
which is taken for~$x$, the more nearly is a second solution
\PageSep{149}
obtained. The use of these expressions is to
point out the cases in which there is anything remarkable
in the general problem; to the problem itself we
must resort for further explanation.}
The importance of the investigations connected
with the expression $ax^{2} + bx + c$, can hardly be overrated,
at least to those students who pursue mathematics
to any extent. In the higher branches, great
familiarity with these results is indispensable. The
student is therefore recommended not to proceed until
he has completely mastered the details here given,
which have been hitherto too much neglected in English
works on algebra.
In solving equations of the second degree, we have
\index{Impossible quantities|EtSeq}%
obtained a new species of result, which indicates that
the problem cannot be solved at all. We refer to
those results which contain the square root of a negative
\index{Negative!squares}%
quantity. We find that by multiplication the
squares of $c - d$ and of $d - c$ are the same, both being
$c^{2} -2cd + d^{2}$. Now either $c - d$ or $d - c$ is positive,
and since they both have the same square, it appears
that the squares of all quantities, whether positive or
negative, are positive. It is therefore absurd to suppose
that there is any quantity which $x$~can represent,
and which satisfies the equation $x^{2} = -a^{2}$, since that
would be supposing that~$x^{2}$, a positive quantity, is
equal to the negative quantity~$-a^{2}$. The solution is
then said to be impossible, and it will be easy to show
an instance in which such a result is obtained, and
\PageSep{150}
also to show that it arises from the absurdity of the
problem.
Let a number~$a$ be divided into any two parts, one
of which is greater than the half, and the other less.
Call the first of these $\dfrac{a}{2} + x$, then the second must be
$\dfrac{a}{2} - x$, since the sum of both parts must be~$a$. Multiply
these parts together, which gives
\[
\biggl(\frac{a}{2} + x\biggr)\biggl(\frac{a}{2} - x\biggr),
\quad\text{or}\quad
\biggl(\frac{a}{2}\biggr)^{2} - x^{2}.
\]
As $x$~diminishes, this product increases, and is greatest
of all when $x = 0$, that is, when the two parts, into
which $a$~is divided, are $\dfrac{a}{2}$ and~$\dfrac{a}{2}$, or when the number~$a$
is halved. In this case the product of the parts is
$\dfrac{a}{2} × \dfrac{a}{2}$, or~$\dfrac{a^{2}}{4}$, and a number~$a$ can never be divided
into two parts whose product is greater than~$\dfrac{a^{2}}{4}$. This
being premised, suppose that we attempt to divide
the number~$a$ into two parts, whose product is~$b$. Let
$x$~be one of these parts, then $a - x$~is the other, and
their product is $ax - x^{2}$.
We have, therefore,
\begin{gather*}
ax - x^{2} = b\Add{,} \\
\text{or}\quad
x^{2} - ax + b = 0.
\end{gather*}
If we solve this equation, the two roots are the two
parts required, since from what we have proved of
the expression $x^{2} - ax + b$ the sum of the roots is~$a$
and their product~$b$. These roots are
\PageSep{151}
\index{Imaginary quantities|EtSeq}%
\[
\frac{a}{2} + \sqrt{\frac{a^{2}}{4} - b} \quad\text{and}\quad
%[** TN: Moved footnote marker after comma]
\frac{a}{2} - \sqrt{\frac{a^{2}}{4} - b},\footnotemark
\]
\footnotetext{The general expressions for $m$~and~$n$ give $\dfrac{a ± \sqrt{a^{2} - 4b}}{2}$ as the roots of
$x^{2} - ax + b = 0$.}%
which are impossible when $\dfrac{a^{2}}{4} - b$ is negative, or when
\index{Negative!squares}%
$b$~is greater than~$\dfrac{a^{2}}{4}$, which agrees with what has just
been proved, that no number is capable of being divided
into two parts whose product is greater than~$\dfrac{a^{2}}{4}$.
We have shown the symbol~$\sqrt{-a}$ to be void of
meaning, or rather self-contradictory and absurd.
Nevertheless, by means of such symbols, a part of
algebra is established which is of great utility. It
depends upon the fact, which must be verified by experience,
that the common rules of algebra may be
applied to these expressions without leading to any
false results. An appeal to experience of this nature
appears to be contrary to the first principles laid down
at the beginning of this work. We cannot deny that
it is so in reality, but it must be recollected that this
is but a small and isolated part of an immense subject,
to all other branches of which these principles
apply in their fullest extent. There have not been
wanting some to assert that these symbols may be
used as rationally as any others, and that the results
derived from them are as conclusive as any reasoning
could make them. I leave the student to discuss this
question as soon as he has acquired sufficient knowledge
to understand the various arguments: at present
\PageSep{152}
let him proceed with the subject as a part of the
mechanism of algebra, on the assurance that by careful
attention to the rules laid down he can never be
led to any incorrect result. The simple rule is, apply
all those rules to such expressions as~$\sqrt{-a}$, $a + \sqrt{-b}$,
etc., which have been proved to hold good for such
quantities as~$\sqrt{a}$, $a + \sqrt{b}$, etc. Such expressions as
the first of these are called \emph{imaginary}, to distinguish
them from the second, which are called \emph{real}; and it
must always be recollected that there is no quantity,
either positive or negative, which an imaginary expression
can represent.
{\Loosen It is usual to write such symbols as~$\sqrt{-b}$ in a different
form. To the equation $-b = b × (-1)$ apply
the rule derived from the equation $\sqrt{xy} = \sqrt{x} × \sqrt{y}$,
which gives $\sqrt{-b} = \sqrt{b} × \sqrt{-1}$, of which the first
factor is real and the second imaginary. Let $\sqrt{b} = c$,
then $\sqrt{-b} = c\sqrt{-1}$. In this way all expressions
may be so arranged that $\sqrt{-1}$~shall be the only imaginary
quantity which appears in them. Of this reduction
the following are examples:}
\begin{align*}
\sqrt{-24} &= \sqrt{24}\sqrt{-1} = 2\sqrt{6}\sqrt{-1}\Add{,} \\
\sqrt{-a^{2}} &= a\sqrt{-1}\Add{,} \displaybreak[0]\\
\sqrt{-a} × \sqrt{-a} &= -a\Add{,} \displaybreak[0]\\
\sqrt{2ab - a^{2} - b^{2}} &= (a - b)\sqrt{-1}\Add{,} \\
\sqrt{-a^{2}} × \sqrt{-b^{2}} &= a\sqrt{-1} × b\sqrt{-1} = -ab.
\end{align*}
The following tables exhibit other applications of
the rules:
\PageSep{153}
\begin{align*}
\PadTo[l]{c^{2}}{c} &= a\sqrt{-1} & c^{7} &= -a^{7} \sqrt{-1}\Add{,} \\
c^{2} &= -a^{2} & c^{8} &= a^{8},\quad\etc.\Add{,} \\
c^{3} &= -a^{3}\sqrt{-1}\qquad & c^{4n-3} &= a^{4n-3} \sqrt{-1}\Add{,} \\
c^{4} &= a^{4} & c^{4n-2} &= -a^{4n-2}\Add{,} \\
c^{5} &= a^{5}\sqrt{-1}\qquad & c^{4n-1} &= -a^{4n-1} \sqrt{-1}\Add{,} \\
c^{6} &= -a^{6} & c^{4n} &= a^{4n}.
\end{align*}
The powers of such an expression as~$a\sqrt{-1}$ are
therefore alternately real and imaginary, and are positive
and negative in pairs.
\begin{gather*}
(a + b\sqrt{-1})^{2} = a^{2} - b^{2} + 2ab\sqrt{-1}\Add{,} \\
(a - b\sqrt{-1})^{2} = a^{2} - b^{2} - 2ab\sqrt{-1}\Add{,} \\
(a + b\sqrt{-1})(a - b\sqrt{-1}) = a^{2} + b^{2}\Add{,} \\
\frac{a + b\sqrt{-1}}{a - b\sqrt{-1}}
= \frac{a^{2} - b^{2}}{a^{2} + b^{2}} + \frac{2ab\sqrt{-1}}{a^{2} + b^{2}}\Add{,} \\
(a + b\sqrt{-1})(c + d\sqrt{-1})
= ac - bd + (ad + bc)\sqrt{-1}.
\end{gather*}
Let the roots of the equation $ax^{2} + bx + c$ be impossible,
that is, let $b^{2} - 4ac$ be negative and equal
to~$-k^{2}$. Its roots, as derived from the rules established
when $b^{2} - 4ac$ was positive, are
\begin{gather*}
\frac{-b + \sqrt{-k^{2}}}{2a} \quad\text{and}\quad
\frac{-b - \sqrt{-k^{2}}}{2a}, \quad\text{or} \\
-\frac{b}{2a} + \frac{k}{2a}\sqrt{-1} \quad\text{and}\quad
-\frac{b}{2a} - \frac{k}{2a}\sqrt{-1}.
\end{gather*}
Take either of these instead of~$x$; for example, let
\[
x = -\frac{b}{2a} + \frac{k}{2a}\sqrt{-1}.
\]
\PageSep{154}
\begin{gather*}
\text{Then}\quad
ax^{2} = \frac{b^{2}}{4a} - \frac{bk}{2a}\sqrt{-1} - \frac{k^{2}}{4a}\Add{,} \\
bx = -\frac{b^{2}}{2a} + \frac{bk}{2a}\sqrt{-1}\Add{,} \\
c = c\Add{.}
\end{gather*}
Therefore, $ax^{2} + bx + c = \dfrac{b^{2}}{4a} - \dfrac{k^{2}}{4a} - \dfrac{b^{2}}{4a} + c$, in
which, if $4ac - b^{2}$ be substituted instead of~$k^{2}$, the result
is~$0$. It appears, then, that the imaginary expressions
which take the place of the roots when $b^{2} - 4ac$
is negative, will, if the ordinary rules be applied, produce
the same results as the roots. They are thence
called imaginary roots, and we say that every equation
of the second degree has two roots, either both
real or both imaginary. It is generally true, that
wherever an imaginary expression occurs the same results
will follow from the application of these expressions
in any process as would have followed had the
proposed problem been possible and its solution real.
When an equation arises in which imaginary and
real expressions occur together, such as $a + b\sqrt{-1} = c + d\sqrt{-1}$,
when all the terms are transferred on one
side, the part which is real and that which is imaginary
must each of them be equal to nothing. The
equation just given when its left side is transposed
becomes $a - c + (b - d)\sqrt{-1} = 0$. Now, if $b$~is not
equal to~$d$, let $b - d = e$; then $a - c + e\sqrt{-1} = 0$, and
$\sqrt{-1} = \dfrac{c - a}{e}$; that is, an imaginary expression is
equal to a real one, which is absurd. Therefore, $b = d$
\PageSep{155}
and the original equation is thereby reduced to $a = c$.
This goes on the supposition that $a$,~$b$,~$c$, and~$d$ are
real. If they are not so there is no necessary absurdity
in $\sqrt{-1} = \dfrac{c - a}{e}$. If, then, we wish to express
that two possible quantities $a$~and~$b$ are respectively
equal to two others $c$~and~$d$, it may be done at once by
the equation
\[
a + b\sqrt{-1} = c + d\sqrt{-1}\Add{.}
\]
The imaginary expression~$\sqrt{-a}$ and the negative expression~$-b$
have this resemblance, that either of
them occurring as the solution of a problem indicates
some inconsistency or absurdity. As far as real meaning
is concerned, both are equally imaginary, since
$0 - a$~is as inconceivable as~$\sqrt{-a}$. What, then, is the
difference of signification? The following problems
will elucidate this. A father is fifty-six, and his son
twenty-nine years old: when will the father be twice
as old as the son? Let this happen $x$~years from the
present time; then the age of the father will be $56 + x$,
and that of the son $29 + x$; and therefore, $56 + x
= 2(29 + x) = 58 + 2x$, or $x = -2$. The result is absurd;
nevertheless, if in the equation we change the
sign of~$x$ throughout it becomes $56 - x = 58 - 2x$, or
$x = 2$. This equation is the one belonging to the
problem: a father is~$56$ and his son $29$~years old;
when \emph{was} the father twice as old as the son? the answer
to which is, two years ago. In this case the
negative sign arises from too great a limitation in the
\PageSep{156}
terms of the problem, which should have demanded
how many years have elapsed or will elapse before the
father is twice as old as his son?
Again, suppose the problem had been given in this
last-mentioned way. In order to form an equation, it
will be necessary either to suppose the event past or
future. If of the two suppositions we choose the
wrong one, this error will be pointed out by the negative
form of the result. In this case the negative result
will arise from a mistake in reducing the problem
to an equation. In either case, however, the result
may be interpreted, and a rational answer to the question
may be given. This, however, is not the case in
a problem, the result of which is imaginary. Take
the instance above solved, in which it is required to
divide~$a$ into two parts, whose product is~$b$. The resulting
equation is
\begin{gather*}
x^{2} - ax + b = 0\Add{,}
\quad\text{or}\quad
x = \frac{a}{2} ± \sqrt{\frac{a^{2}}{4} - b},
\end{gather*}
the roots of which are imaginary when $b$~is greater
than~$\dfrac{a^{2}}{4}$. If we change the sign of~$x$ in the equation
it becomes
\begin{gather*}
x^{2} + ax + b = 0\Add{,}
\quad\text{or}\quad
x = -\frac{a}{2} ± \sqrt{\frac{a^{2}}{4} - b},
\end{gather*}
and the roots of the second are imaginary, if those of
the first are so. There is, then, this distinct difference
\PageSep{157}
between the negative and the imaginary result. When
the answer to a problem is negative, by changing the
sign of~$x$ in the equation which produced that result,
we may either discover an error in the method of
forming that equation or show that the question of the
problem is too limited, and may be extended so as to
admit of a satisfactory answer. When the answer to
a problem is imaginary this is not the case.
\PageSep{158}
\Chapter[Roots in General, and Logarithms.]
{XI.}{On Roots in General, and Logarithms.}
\index{Indices, theory of@{Indices, theory of|EtSeq}}%
\index{Powers, theory of|EtSeq}%
\index{Roots|EtSeq}%
\First{The} meaning of the terms \emph{square root}, \emph{cube root},
\emph{fourth root}, etc., has already been defined. We
now proceed to the difficulties attending the connexion
of the roots of~$a$ with the powers of~$a$. The following
table will refresh the memory of the student
with respect to the meaning of the terms:
\[
\small
\begin{array}{*{2}{l}>{\quad}lr}
\multicolumn{2}{c}{\TEntry{NAME OF~$x$.}} &
\multicolumn{2}{c}{\TEntry{NAME OF~$x$.}} \\
\text{Square of $a$} & x = aa & \text{Square Root of $a$} & xx = a \\
\text{Cube of $a$} & x = aaa & \text{Cube Root of $a$} & xxx = a \\
\text{Fourth Power of $a$\quad} & x = aaaa &\text{Fourth Root of $a$} & xxxx = a \\
\text{Fifth Power of $a$} & x = aaaaa & \text{Fifth Root of $a$} & xxxxx = a
\end{array}
\]
The different powers and roots of~$a$ have hitherto
been expressed in the following way:
\begin{gather*}
\text{Powers}\quad
a^{2}\quad a^{3}\quad a^{4}\quad a^{5}\ \dots\quad
a^{m}\ \dots\quad a^{m+n}, \quad\etc. \\
\text{Roots}\quad
\sqrt[2]{a}\footnotemark\quad
\sqrt[3]{a}\quad
\sqrt[4]{a}\quad
\sqrt[5]{a}\quad
\sqrt[m]{a}\quad
\sqrt[m+n]{a},\quad\etc.
\end{gather*}
which series are connected together by the following
equation, $(\sqrt[n]{a})^n = a$.
\footnotetext{The $2$ is usually omitted, and the square root is written thus~$\sqrt{a}$.}
\PageSep{159}
There has hitherto been no connexion between the
manner of expressing powers and roots, and we have
found no properties which are common both to powers
and roots. Nevertheless, by the extension of rules,
we shall be led to a method of denoting the raising of
powers, the extraction of roots, and combinations of
the two, to which algebra has been most peculiarly
indebted, and the importance of which will justify the
length at which it will be treated here.
Suppose it required to find the cube of $2a^{2} b^{3}$; that
\index{Notation!algebraical}%
is, to find $2a^{2} b^{3} × 2a^{2} b^{3} × 2a^{2} b^{3}$. The common rules
of multiplication give, as the result, $8a^{6} b^{9}$, which is
expressed in the following equation,
\[
(2a^{2} b^{3})^{3} = 8a^{6} b^{9}.
\]
Similarly
\begin{gather*}
(3a^{4} b^{3})^{4} = 81a^{16} b^{12}, \\
\biggl(\frac{1}{2}\, \frac{b^{4}}{a}\biggr)^{6}
= \frac{1}{64}\, \frac{b^{24}}{a^{6}};
\end{gather*}
and the general rule by which any single term may be
raised to the power whose index is~$n$, is: Raise the coefficient
to the power~$n$, and multiply the index of
every letter by~$n$, that is,
\[
(a^{p} b^{q} c^{r})^{n} = a^{np} b^{nq} c^{nr}.
\]
In extracting the root of any simple term, we are
guided by the manner in which the corresponding
power is found. The rule is: Extract the required root
of the coefficient, and divide the index of each letter
by the index of the root. Where these divisions do
not give whole numbers as the quotients, the expression
\PageSep{160}
whose root is to be extracted does not admit of
the extraction without the introduction of some new
symbol. For example, extract the fourth root of
$16a^{12} b^{8} c^{4}$, or find $\sqrt[4]{16a^{12} b^{8} c^{4}}$. The expression here
given is the same as the following:
\[
2a^{3} b^{2} c × 2a^{3} b^{2} c × 2a^{3} b^{2} c × 2a^{3} b^{2} c,
\]
or $(2a^{3} b^{2} c)^{4}$, the fourth root of which is~$2a^{3} b^{2} c$, conformably
to the rule.
Any root of a product, such as~$AB$, may be extracted
\index{Factoring}%
by extracting the root of each of its factors.
Thus, $\sqrt[3]{AB} = \sqrt[3]{A} \sqrt[3]{B}$. For, raise $\sqrt[3]{A} \sqrt[3]{B}$ to the
third power, the result of which is,
\begin{gather*}
\sqrt[3]{A} \sqrt[3]{B} × \sqrt[3]{A} \sqrt[3]{B} × \sqrt[3]{A} \sqrt[3]{B}, \\
\text{or}\quad
\sqrt[3]{A} \sqrt[3]{A} \sqrt[3]{A} × \sqrt[3]{B} \sqrt[3]{B} \sqrt[3]{B}, \\
\text{or}\quad AB.
\end{gather*}
{\Loosen In the same way it may be proved generally, that
$\sqrt[n]{ABC} = \sqrt[n]{A} \sqrt[n]{B} \sqrt[n]{C}$.} The most simple way of representing
any root of any expression is the dividing it
into two factors, one of which is the highest which it
admits of whose root can be extracted by the rule just
given. For example, in finding $\sqrt[3]{16a^{4} b^{7} c}$ we must
observe that $16$~is $8 × 2$, $a^{4}$~is $a^{3} × a$, $b^{7}$~is $b^{6} × b$, and
the expression is $8a^{3} b^{6} × 2abc$, the cube root of which,
found by extracting the cube root of each factor, is
$2ab^{2} \sqrt[3]{2abc}$. The second factor has no cube root
which can be expressed by means of the symbols
hitherto used, but when the numbers which $a$,~$b$, and~$c$
stand for are known, $\sqrt[3]{2abc}$ maybe found either
\PageSep{161}
exactly, or, when that is not possible, by approximation.
We find that a power of a power is found by affixing,
as an index, the product of the indices of the two
powers. Thus $(a^{2})^{4}$ or $a^{2} × a^{2} × a^{2} × a^{2}$ is~$a^{8}$, or~$a^{4×2}$.
This is the same as $(a^{4})^{2}$, which is $a^{4} × a^{4}$, or~$a^{8}$.
Therefore, generally $(a^{m})^{n} = (a^{n})^{m} = a^{mn}$. In the same
manner, a root of a root is the root whose index is the
product of the indices of the two roots. Thus
\[
\sqrt[3]{\sqrt[2]{a}} = \sqrt[6]{a}.
\]
For since $a = \sqrt[6]{a} \sqrt[6]{a} \sqrt[6]{a} × \sqrt[6]{a} \sqrt[6]{a} \sqrt[6]{a}$, the square
root of~$a$ is~$\sqrt[6]{a} \sqrt[6]{a} \sqrt[6]{a}$, the cube root of which is~$\sqrt[3]{a}$.
This is the same as~$\sqrt[2]{\sqrt[3]{a}}$, and generally
\[
\sqrt[m]{\sqrt[n]{a}} = \sqrt[n]{\sqrt[m]{a}} = \sqrt[mn]{a}.
\]
{\Loosen Again, when a power is raised and a root extracted,
it is indifferent which is done first. Thus $\sqrt[3]{a^{2}}$~is the
same thing as~$(\sqrt[3]{a})^{2}$. For since $a^{2} = a × a$, the cube
root may be found by taking the cube root of each of
these factors, that is $\sqrt[3]{a^{2}} = \sqrt[3]{a} × \sqrt[3]{a} = (\sqrt[3]{a})^{2}$, and
generally}
\[
\sqrt[n]{a^{m}} = (\sqrt[n]{a})^{m}.
\]
In the expression $\sqrt[n]{a^{m}}$, $n$~and~$m$ may both be multiplied
by any number, without altering the expression,
that is
\[
\sqrt[np]{a^{mp}} = (\sqrt[n]{a})^{m}.
\]
To prove this, recollect that
\[
\sqrt[np]{a^{mp}} = \sqrt[n]{\sqrt[p]{a^{mp}}}.
\]
\PageSep{162}
But $a^{mp}$~is $(a^{m})^{p}$, and by definition $\sqrt[p]{(a^{m})^{p}} = a^{m}$. Therefore
$\sqrt[np]{a^{mp}} = \sqrt[n]{a^{m}}$. This multiplication is equivalent
to raising a power of~$\sqrt[n]{a^{m}}$, and afterwards reducing
the result to its former value, by extracting the correspending
root, in the same way as $\dfrac{mp}{np}$ signifies that $\dfrac{m}{n}$
has been multiplied by~$p$, and the result has been restored
to its former value by dividing it by~$p$.
The following equations should be established by
the student to familiarise him with the notation and
principles hitherto laid down.
\begin{gather*}
\sqrt[n]{(a - b)^{n-2}} × \sqrt[3n]{(a - b)^{6}} = a - b\Add{,} \\
\Squeeze{\sqrt[n+m]{(a +b)^{n-m}} × \sqrt[n-m]{(a - b)^{n+m}}
= (a^{2} - b^{2}) \biggl(\frac{\sqrt[n-m]{a - b}}{\sqrt[n+m]{a + b}}\biggr)^{2m}\Add{,}} \\
\sqrt[n]{\frac{ab}{cd}}
= \frac{\sqrt[n]{ab}}{\sqrt[n]{cd}}
= \frac{\sqrt[n]{a}\sqrt[n]{b}}{\sqrt[n]{c}\sqrt[n]{d}}
= \sqrt[n]{\frac{a}{c}} × \sqrt[n]{\frac{b}{d}}\Add{,} \\
\sqrt[n]{\frac{a}{b}}
= \frac{\sqrt[n]{ab^{n-1}}}{b}
= \frac{a}{\sqrt[n]{a^{n-1}b}}.
\end{gather*}
The quantity $\sqrt[n]{a^{m}}$ is a simple expression when $m$~can
be divided by~$n$, without remainder, for example
$\sqrt[2]{a^{12}} = a^{6}$, $\sqrt[5]{a^{20}} = a^{4}$, and in general, whenever $m$~can
be divided by~$n$ without remainder, $\sqrt[n]{a^{m}} = a^{\efrac{m}{n}}$. This
symbol, viz., a letter which has an exponent appearing
in a fractional form, has not hitherto been used.
We may give it any meaning which we please, provided
it be such that when $\dfrac{m}{n}$~is fractional in form only,
and not in reality, that is, when $m$~is divisible by~$n$,
\PageSep{163}
and the quotient is~$p$, $a^{\efrac{m}{n}}$~shall stand for~$a^{p}$, or
$aaa\dots (p)$.\footnote
{This is a notation in common use, and means that $aaa\dots$ is to be
\index{Notation!extension of}%
continued until it has been repeated $p$~times. Thus
\begin{gather*}
a + a + a + \dots (p) = pa, \\
a × a × a × \dots (p) = a^{p}.
\end{gather*}}
It will be convenient to let $a^{\efrac{m}{n}}$~always
\index{Extension@{Extension of rules and meanings of terms}}%
\index{Rules!extension of meaning of}%
stand for~$\sqrt[n]{a^{m}}$, in which case the condition alluded to
is fulfilled, since when $\dfrac{m}{n} = p$, $a^{\efrac{m}{n}}$~or $\sqrt[n]{a^{m}} = a^{p}$. This
extension of a rule, the advantages of which will soon
\index{Formulæ, important|(}%
\index{Fractional exponents@{Fractional exponents|EtSeq}}%
be apparent, is exemplified in the following table,
which will familiarise the student with the different
cases of this new notation:
\begin{gather*}
\text{$a^{\efrac{1}{2}}$ stands for $\sqrt[2]{a^{1}}$ or $\sqrt{a}$}\Add{,} \\
\text{$a^{\efrac{1}{3}}$ stands for $\sqrt[3]{a}$}\Add{,} \\
\text{$a^{\efrac{1}{4}}$ stands for $\sqrt[4]{a}$}\Add{,} \displaybreak[0]\\
%
\text{$a^{\efrac{2}{3}}$ stands for $\sqrt[3]{a^{2}}$ or $(\sqrt[3]{a})^{2}$}\Add{,} \\
\text{$a^{\efrac{7}{5}}$ stands for $\sqrt[5]{a^{7}}$ or $(\sqrt[5]{a})^{7}$}\Add{,} \displaybreak[0]\\
%
\text{$a^{\efrac{m+n}{m-n}}$ stands for $\sqrt[m-n]{a^{m+n}}$}\Add{,} \displaybreak[0]\\
\text{$(p + q)^{\efrac{m-n}{2}}$ stands for $\sqrt{(p + q)^{m-n}}$}\Add{,} \\
\text{$(c^{\efrac{m}{n}})^{\efrac{p}{q}}$ stands for $\sqrt[q]{(\sqrt[n]{c^{m}})^{p}}$}\Add{,} \\
\text{$(a^{\efrac{1}{n}})^{\efrac{1}{q}}$ stands for $\sqrt[q]{\sqrt[n]{a}}$}\Add{.}
\end{gather*}
The results at which we have arrived in this chapter,
translated into this new language, are as follows:
\begin{gather*}
(x^{\efrac{1}{n}})^{n} = (x^{n})^{\efrac{1}{n}} = x\Add{,}
\Tag{(1)} \\
(ABC)^{\efrac{1}{n}} = A^{\efrac{1}{n}} B^{\efrac{1}{n}} C^{\efrac{1}{n}}\Add{,}
\Tag{(2)} \\
\PageSep{164}
(a^{\efrac{1}{n}})^{\efrac{1}{q}} = a^{\efrac{1}{nq}}\Add{,}
\Tag{(3)} \\
(a^{m})^{\efrac{1}{n}} = (a^{\efrac{1}{n}})^{m}
= a^{\efrac{m}{n}}\Add{,}
\Tag{(4)} \\
a^{\efrac{m}{n}} = a^{\efrac{mp}{nq}}\Add{.}
\Tag{(5)}
\end{gather*}
The advantages resulting from the adoption of this
notation, are, (1)~that time is saved in writing algebraical
expressions; (2)~all rules which have been
shown to hold good for performing operations upon
such quantities as~$a^{m}$, hold good also for performing
the same operations upon such quantities as~$a^{\efrac{m}{n}}$, in
which the exponents are fractional. The truth of this
last assertion we proceed to establish.
Suppose it required to multiply together $a^{\efrac{m}{n}}$ and~$a^{\efrac{l}{n}}$,
\index{Multiplication}%
or $\sqrt[n]{a^{m}}$ and~$\sqrt[n]{a^{l}}$. From~\Eq{(2)} this is $\sqrt[n]{a^{m} × a^{l}}$, or
$\sqrt[n]{a^{m+l}}$, or~$a^{\efrac{m+l}{n}}$. Suppose it now required to multiply
$a^{\efrac{m}{n}}$~and~$a^{\efrac{p}{q}}$. From~\Eq{(5)} the first of these is the same as~$a^{\efrac{mq}{nq}}$,
and the second is the same as~$a^{\efrac{np}{nq}}$. The product
of these by the last case is~$a^{\efrac{mq+np}{nq}}$, or~$\sqrt[nq]{a^{mq+np}}$. But
$\dfrac{mq + np}{nq}$ is $\dfrac{m}{n} + \dfrac{p}{q}$, and therefore
\[
a^{\efrac{m}{n}} × a^{\efrac{p}{q}}
= a^{\efrac{m}{n}+\efrac{p}{q}}\Add{.}
\Tag{(6)}
\]
This is the same result as was obtained when the
indices were whole numbers. The rule is: To multiply
together two powers of the same quantity, add
the indices, and make the sum the index of the product.
It follows in the same way that
\PageSep{165}
\index{Division|(}%
\[
\frac{a^{\efrac{m}{n}}}{a^{\efrac{p}{q}}}
= a^{\efrac{m}{n}-\efrac{p}{q}}
= a^{\efrac{mq-\Chg{pn}{np}}{nq}}
= \sqrt[nq]{a^{mq-\Chg{pn}{np}}}\Add{,}
\]
or, to divide one power of a quantity by another, subtract
the index of the divisor from that of the dividend,
and make the difference the index of the result.
{\Loosen Suppose it required to find~$(a^{\efrac{m}{n}})^{p}$. It is evident
that $a^{\efrac{m}{n}} × a^{\efrac{m}{n}} = a^{\efrac{m}{n}+\efrac{m}{n}} = a^{\efrac{2m}{n}}$, or $(a^{\efrac{m}{n}})^{2} = a^{\efrac{2m}{n}}$. Similarly
$(a^{\efrac{m}{n}})^{3} = a^{\efrac{3m}{n}}$, and so on. Therefore $(a^{\efrac{m}{n}})^p = a^{\efrac{mp}{n}}$.}
Again to find $(a^{\efrac{m}{n}})^{\efrac{1}{q}}$, or~$\sqrt[q]{a^{\efrac{m}{n}}}$. Let this be~$a^{\efrac{x}{y}}$.
Then $a^{\efrac{x}{y}} = \sqrt[q]{a^{\efrac{m}{n}}}$, or $(a^{\efrac{x}{y}})^{q} = a^{\efrac{m}{n}}$, or $a^{\efrac{xq}{y}} = a^{\efrac{m}{n}}$. Therefore
$\dfrac{xq}{y} = \dfrac{m}{n}$, or $\dfrac{x}{y} = \dfrac{m}{nq}$, and $(a^{\efrac{m}{n}})^{\efrac{1}{q}} = a^{\efrac{m}{nq}}$.
Again to find $(a^{\efrac{m}{n}})^{\efrac{p}{q}}$, or~$\sqrt[q]{(a^{\efrac{m}{n}})^{p}}$. Apply the last
two rules, and it appears that $(a^{\efrac{m}{n}})^{p} = a^{\efrac{mp}{n}}$, and
$\sqrt[q]{a^{\efrac{mp}{n}}} = a^{\efrac{mp}{nq}}$. Therefore $(a^{\efrac{m}{n}})^{\efrac{p}{q}} = a^{\efrac{mp}{nq}} = a^{\efrac{m}{n}×\efrac{p}{q}}$.
The rule is: To raise one power of a quantity to
another power, multiply the indices of the two powers
together, and make the product the index of the result.
All these rules are exactly those which have
been shown to hold good when the indices are whole
numbers. But there still remains one remarkable extension,
which will complete this subject.
We have proved that whether $m$~and~$n$ be whole
or fractional numbers, $\dfrac{a^{n}}{a^{m}} = a^{m-n}$. The only cases
which have been considered in forming this rule are
\PageSep{166}
\index{Indices, theory of}%
\index{Negative!indices}%
\index{Zero!exponents}%
those in which $m$~is greater than~$n$, being the only
ones in which the subtraction indicated is possible. If
we apply the rule to any other case, a new symbol is
produced, which we proceed to consider. For example,
suppose it required to find~$\dfrac{a^{3}}{a^{7}}$. If we apply the
rule, we find the result~$a^{3-7}$, or~$a^{-4}$, for which we have
hitherto no meaning. As in former cases, we must
apply other methods to the solution of this case, and
when we have obtained a rational result, $a^{-4}$~may be
used in future to stand for this result. Now the fraction~$\dfrac{a^{3}}{a^{7}}$
is the same as~$\dfrac{1}{a^{4}}$, which is obtained by dividing
both its numerator and denominator by~$a^{3}$. Therefore
$\dfrac{1}{a^{4}}$~is the rational result, for which we have obtained~$a^{-4}$
by applying a rule in too extensive a manner.
Nevertheless, if $a^{-4}$~be made to stand for~$\dfrac{1}{a^{4}}$, and
$a^{-m}$~for~$\dfrac{1}{a^{m}}$, the rule will always give correct results,
and the general rules for multiplication, division, and
raising of powers remain the same as before. For
example, $a^{-m} × a^{-n}$ is $\dfrac{1}{a^{m}} × \dfrac{1}{a^{n}}$, or~$\dfrac{1}{a^{m} a^{n}}$, which is
$\dfrac{1}{a^{m+n}}$, or~$a^{-(m+n)}$, or~$a^{-m-n}$. Similarly
\[
\text{$\dfrac{a^{-m}}{a^{-n}}$, or
$\dfrac{\,\dfrac{1}{a^{m}}\,}{\dfrac{1}{a^{n}}}$, is
$\dfrac{a^{n}}{a^{m}}$, or
$a^{n-m}$, or
$a^{-m-(-n)}$.}
\]
Again
\[
\text{$(a^{m})^{-n}$ is
$\dfrac{1}{(a^{m})^{n}}$, or
$\dfrac{1}{a^{mn}}$, or
$a^{-mn}$,}
\]
and so on.
\PageSep{167}
It has before been shown that $a^{0}$~stands for~$1$ whenever
\index{Logarithms|EtSeq}%
it occurs in the solution of a problem. We can
now, therefore, assign a meaning to the expression~$a^{m}$,
whether $m$~be whole or fractional, positive, negative,
or nothing, and in all these cases the following rules
hold good:
\begin{gather*}
a^{m} × a^{n} = a^{m+n}\Add{,} \\
\frac{a^{m}}{a^{n}} = a^{m-n} = a^{m} a^{-n}\Add{,} \\
(a^{m})^{n} = (a^{n})^{m} = a^{mn}.
\end{gather*}
\index{Formulæ, important|)}%
The student can now understand the meaning of such
an expression as~$10^{.301}$, where the index or exponent
is a decimal fraction. Since $.301$~is~$\frac{301}{1000}$, this stands
for $\sqrt[1000]{(10)^{301}}$, an expression of which it would be impossible
to calculate the value by any method which
the student has hitherto been taught, but which may
be shown by other processes to be very nearly equal
to~$2$.
Before proceeding to the practice of logarithmic
calculations, the student should thoroughly understand
the meaning of fractional and negative indices,
and be familiar with the operations performed by
means of them. He should work many examples of
multiplication and division in which they occur, for
\index{Division|)}%
which he can have recourse to any elementary work.
The rules are the same as those to which he has been
accustomed, substituting the addition, subtraction,
and so forth, of fractional indices, instead of these
which are whole numbers.
\PageSep{168}
In order to make use of logarithms, he must provide
\index{Tables, mathematical, recommended}%
himself with a table. Either of the following
works may be recommended to him:
[1. Bruhns, \Title{A New Manual of Logarithms to Seven
\Pagelabel{168}%
\index{Bruhns}%
Places of Decimals} (English preface, Leipsic).
2. Schrön, \Title{Seven-Figure Logarithms} (English edition,
\index{Schrön}%
London).
3. Bremiker's various editions of \Title{Vega's Logarithmic
\index{Bremiker}%
\index{Vega}%
Tables} (Weidmann, Berlin). With English preface.\Chg{]}{}
4. Callet, \Title{Tables portatives de Logarithmes}. (Last
\index{Callet}%
impression, Paris, 1890).
5. V. Caillet, \Title{Tables des Logarithmes et Co-Logarithmes
\index{Caillet}%
des nombres} (Paris).
6. Hutton's \Title{Mathematical Tables} (London).
\index{Hutton}%
7. Chambers's \Title{Mathematical Tables} (Edinburgh).
8. The American six-figure Tables of Jones, of
\index{Jones}%
Wells, and of Haskell.
\index{Haskell}%
\index{Wells}%
For fuller bibliographical information on the subject
of tables of logarithms, see the \Title{Encyclopædia Britannica},
Article ``Tables,'' Vol.~XXIII.---\Ed.]\footnote
{The original text of De~Morgan, for which the above paragraph has
been substituted, reads as follows: ``Either of the following works may be
recommended to him: (1)~Taylor's \Title{Logarithms}. (2)~Hutton's \Title{Logarithms}.
\index{Taylor}%
(3)~Babbage, \Title{Logarithms of Numbers}; Callet, \Title{Logarithms of Sines, Cosines},
\index{Babbage}%
etc. (4)~Bagay, \Title{Tables Astronomiques et Hydrographiques}. The first and last
\index{Bagay}%
of these are large works, calculated for the most accurate operations of
spherical trigonometry and astronomy. The second and third are better
suited to the ordinary student. For those who require a pocket volume there
are Lalande's and Hassler's Tables, the first published in France, the second
\index{Hassler}%
\index{Lallande}%
in the United States.''---\Ed.}
The limits of this treatise will not allow us to enter
\PageSep{169}
\index{Interpolation|(}%
into the subject of the definition, theory, and use of
logarithms, which will be found fully treated in the
standard text-books of Arithmetic, Algebra, and Trigonometry.
There is, however, one consideration connected
with the tables, which, as it involves a principle
of frequent application, it will be well to explain
here. On looking into any table of logarithms it will
be seen, that for a series of numbers the logarithms
increase in arithmetical progression, as far as the first
seven places of decimals are concerned; that is, the
difference between the successive logarithms continues
the same. For example, the following is found from
any tables:
%[** TN: Log changed to log throughout; see preamble for an alternative macro]
\begin{align*}
\Log 41713 &= 4.6202714\Add{,} \\
\Log 41714 &= 4.6202818\Add{,} \\
\Log 41715 &= 4.6202922\Add{.}
\end{align*}
The difference of these successive logarithms and of
almost all others in the same page is~$.0000104$. Therefore
in this the addition of~$1$ to the number gives an
addition of~$.0000104$ to the logarithm. It is a general
rule that when one quantity depends for its value upon
another, as a logarithm does upon its number, or an
algebraical expression, such as $x^{2} + x$ upon the letter
or letters which it contains, if a very small addition be
made to the value of one of these letters, in consequence
of which the expression itself is increased or
diminished; generally speaking, the increment\footnote
{When any quantity is increased, the quantity by which it is increased is
\index{Increment}%
called its \emph{increment}.}
of the
\PageSep{170}
\index{Greatness and smallness}%
\index{Proportions}%
expression will be very nearly proportional to the increment
of the letter whose value is increased, and the
more nearly so the smaller is the increment of the letter.
We proceed to illustrate this. The product of
two fractions, each of which is less than unity, is itself
less than either of its factors. Therefore the square,
cube, etc., of a fraction less than unity decrease, and
the smaller the fraction is the more rapid is that decrease,
as the following examples will show:
\begin{align*}
\text{Let}\quad
\PadTo[l]{x^{2}}{x} &= .01\Add{,} &
\text{Let}\quad
\PadTo[l]{x^{2}}{x} &= .00001\Add{,} \\
x^{2} &= .0001\Add{,} &
x^{2} &= .0000000001\Add{,} \\
x^{3} &= .000001\Add{,} &
x^{3} &= .000000000000001\Add{,} \\
& \etc. &
&\qquad \etc.
\end{align*}
Now quantities are compared, not by the actual
difference which exists between them, but by the number
of times which one contains the other, and, of two
quantities which are both very small, one may be very
great as compared with the other. In the second example
$x^{2}$~and~$x^{3}$ are both small fractions \Typo{whem}{when} compared
with unity; nevertheless, $x^{2}$~is very great when
compared with~$x^{3}$, being $100,000$~times its magnitude.
This use of the words small and great sometimes embarrasses
the beginner; nevertheless, on consideration,
it will appear to be very similar to the sense in
which they are used in common life. We do not form
our ideas of smallness or greatness from the actual
numbers which are contained in a collection, but from
the proportion which the numbers bear to those which
\PageSep{171}
\index{Approximations@{Approximations|EtSeq}}%
are usually found in similar collections. Thus of $1000$~men
we should say, if they lived in one village, that
it was extremely large; if they formed a regiment,
that it was rather large; if an army, that it was utterly
insignificant in point of numbers. Hence, in
such an expression as $Ah + Bh^{2} + Ch^{3}$, we may, if $h$~is
very small, reject $Bh^{2} + Ch^{3}$, as being very small compared
with~$Ah$. An error will thus be committed, but
a very small one only, and which becomes smaller as
$h$~becomes smaller.
{\Loosen Let us take any algebraical expression, such as
$x^{2} + x$, and suppose that $x$~is increased by a very small
quantity~$h$. The expression then becomes $(x + h)^{2} + (x + h)$,
or $x^{2} + x + (2x + 1)h + h^{2}$. But it was $x^{2} + x$;
therefore, in consequence of $x$~receiving the increment~$h$,
$x^{2} + x$~has received the increment $(2x + 1)h + h^{2}$,
for which $(2x + 1)h$~may be written, since $h$~is very
small. This is proportional to~$h$, since, if $h$~were
doubled, $(2x + 1)h$~would be doubled; also, if the
first were halved the second would be halved, etc. In
general, if $y$~is a quantity which contains~$x$, and if $x$~be
changed into~$x + h$, $y$~is changed into a quantity of
the form $y + Ah + Bh^{2} + Ch^{3} + \etc.$; that is, $y$~receives
an increment of the form $Ah + Bh^{2} + Ch^{3} + \etc$.
If $h$~be very small, this may, without sensible error,
be reduced to its first term, viz.,~$Ah$, which is proportional
to~$h$. The general proof of this proposition belongs
to a higher department of mathematics; nevertheless,
the student may observe that it holds good in
\PageSep{172}
all the instances which occur in elementary treatises
on arithmetic and algebra.}
For example:
\[
(x + h)^{m}
= x^{m} + mx^{m-1}h + m\, \frac{m - 1}{2}\, x^{m-2}h^{2} + \etc.
\]
Here $A = mx^{m-1}$, $B = m\, \dfrac{m - 1}{2}\, x^{m-2}$, etc.; and if $h$~be
very small, $(x + h)^{m} = x^{m} + mx^{m-1} h$, nearly.
Again, $e^{h} = 1 + h + \dfrac{h^{2}}{2} + \dfrac{h^{3}}{2·3} + \etc$. Therefore,
$e^{x} × e^{h}$ or $e^{x+h} = e^x + e^{x} h + \dfrac{e^{x}}{2}\, h^{2} + \etc$. And if $h$~be
very small, $e^{x+h} = e^{x} + e^{x}h$, nearly.
Again, $\Log(1 + n') = M(n' - \frac{1}{2} n'^{2} + \frac{1}{3}n'^{3} - \etc.)$.
To each side add~$\Log x$, recollecting that
\[
\Log x + \Log(1 + n') = \Log x(1 + n') = \Log(x + xn'),
\]
and let
\[
xn' = h\quad\text{or}\quad n' = \frac{x}{h}.
\]
Making these substitutions, the equation becomes
\[
\Log(x + h) = \Log x + \frac{M}{x}\, h - \frac{M}{2x^{2}}\, h^{2} + \etc.
\]
If $h$~is very small, $\Log(x + h) = \Log x + \dfrac{M}{x}\, h$.
We can now apply this to the logarithmic example
with which we commenced this subject. It appears
that
\begin{alignat*}{2}
&\Log 41713 &&= 4.6202714\Add{,} \\
&\Log (41713 + 1) &&= 4.6202714 + .0000104\Add{,} \\
&\Log (41713 + 2) &&= 4.6202714 + .0000104 × 2.
\end{alignat*}
From which, and the considerations above-mentioned,
\PageSep{173}
\index{Proportional parts}%
\[
\Log (41713 + h) = \Log 41713 + .0000104 × h,
\]
which is extremely near the truth, even when $h$~is a
much larger number, as the tables will show. Suppose,
then, that the logarithm of~$41713.27$ is required.
Here $h = .27$. It therefore only remains to calculate
$.0000104 × .27$, and add the result, or as much of it
as is contained in the first seven places of decimals,
to the logarithm of~$41713$. This trouble is saved in
the tables in the following manner. The difference of
the successive logarithms is written down, with the
exception of the cyphers at the beginning, in the
column marked~$D$ or~\emph{Diff.}, under which are registered
the tenths of that difference, or as much of them as is
contained in the first seven decimal places, increasing
the seventh figure by~$1$ when the eighth is equal to or
greater than~$5$, and omitting the cyphers to save room.
From this table of tenths the table of hundredth parts
may be made by striking off the last figure, making
the usual change in the last but one, when the last is
equal to or greater than~$5$, and placing an additional
cypher. The logarithm of~$41713.27$ is, therefore, obtained
in the following manner:
\[
\begin{array}{r@{\,}c@{\,}r}
\Log 41713\phantom{.99} &=& 4.6202714 \\
.0000104 × .2\Z &=& .0000021 \\
.0000104 × .07 &=& .0000007 \\
\cline{3-3}
\Strut
\Log 41713.27 &=& 4.620274\rlap{\Add{.}}
\end{array}
\]
This, when the useless cyphers and parts of the operation
are omitted, is the process given in all the books
of logarithms. If the logarithm of a number containing
\PageSep{174}
more than seven significant figures be sought, for
example $219034.717$, recourse must be had to a table,
in which the logarithms are carried to more than seven
places of decimals. The fact is, that in the first seven
places of decimals there is no difference between
$\Log 219034.7$ and $\Log 219034.717$. For an excellent
treatise on the practice of logarithms the reader may
\index{Babbage}%
consult the preface to Babbage's \Title{Table of Logarithms}.\footnote
{Copies of Babbage's \Title{Table of Logarithms} are now scarce, and the reader
may accordingly be referred to the prefaces of the treatises mentioned \Typo{no}{on}
\PageRef{168}. The article on ``Logarithms, Use of'' in the \Title{English Cyclopedia},
\index{English@{\Title{English Cyclopedia}}}%
may also be consulted with profit.---\Ed.}
\index{Interpolation|)}%
\PageSep{175}
\Chapter{XII.}{On the Study of Algebra.}
\index{Algebra!advice@{advice on the study of|EtSeq}}%
\index{Mathematics!advice on study of}%
\First{In} this chapter we shall give the student some advice
as to the manner in which he should prosecute
his studies in algebra. The remaining parts of
this subject present a field infinite in its extent and in
the variety of the applications which present themselves.
By whatever name the remaining parts of
the subject may be called, even though the ideas on
which they are based may be geometrical, still the
mechanical processes are algebraical, and present continual
applications of the preceding rules and developments
of the subjects already treated. This is the
case in Trigonometry, the application of Algebra to
Geometry, the Differential Calculus, or Fluxions, etc.
I. The first thing to be attended to in reading any
algebraical treatise, is the gaining a perfect understanding
of the different processes there exhibited,
and of their connexion with one another. This cannot
be attained by a mere reading of the book, however
great the attention which may be given. It is
\PageSep{176}
impossible, in a mathematical work, to fill up every
process in the manner in which it must be filled up in
the mind of the student before he can be said to have
completely mastered it. Many results must be given,
of which the details are suppressed, such are the additions,
multiplications, extractions of the square root,
etc., with which the investigations abound. These
must not be taken on trust by the student, but must
be worked by his own pen, which must never be out
of his hand while engaged in any algebraical process.
The method which we recommend is, to write the
whole of the symbolical part of each investigation,
filling up the parts to which we have alluded, adding
only so much verbal elucidation as is absolutely necessary
to explain the connexion of the different steps,
which will generally be much less than what is given
in the book. This may appear an alarming labor to
one who has not tried it, nevertheless we are convinced
that it is by far the shortest method of proceeding,
since the deliberate consideration which the
act of writing forces us to give, will prevent the confusion
and difficulties which cannot fail to embarrass
the beginner if he attempt, by mere perusal only, to
understand new reasoning expressed in new language.
If, while proceeding in this manner, any difficulty
should occur, it should be written at full length, and
it will often happen that the misconception which occasioned
the embarrassment will not stand the trial to
which it is thus brought. Should there be still any
\PageSep{177}
matter of doubt which is not removed by attentive reconsideration,
the student should proceed, first making
a note of the point which he is unable to perceive.
To this he should recur in his subsequent progress,
whenever he arrives at anything which appears to
have any affinity, however remote, to the difficulty
which stopped him, and thus he will frequently find
himself in a condition to decypher what formerly
appeared incomprehensible. In reasoning purely geometrical,
there is less necessity for committing to writing
the whole detail of the arguments, since the symbolical
language is more quickly understood, and the
subject is in a great measure independent of the mechanism
of operations; but, in the processes of algebra,
there is no point on which so much depends, or on
which it becomes an instructor more strongly to insist.
II. On arriving at any new rule or process, the
\index{Binomial theorem, exercises in|EtSeq}%
student should work a number of examples sufficient
to prove to himself that he understands and can apply
the rule or process in question. Here a difficulty will
occur, since there are many of these in the books, to
which no examples are formally given. Nevertheless,
he may choose an example for himself, and his previous
knowledge will suggest some method of proving
whether his result is true or not. For example, the
development of~$(a + x)^{\efrac{7}{3}}$ will exercise him in the use
of the binomial theorem; when he has obtained the
series which is equivalent to~$(a + x)^{\efrac{7}{3}}$, let him, in the
\PageSep{178}
same way, develop~$(a + x)^{\efrac{2}{3}}$; the product of these,
since $\frac{7}{3} + \frac{2}{3} = 3$, ought to be the same as the development
of $(a + x)^{3}$, or as $a^{3} + 3a^{2} x + 3ax^{2} + x^{3}$. He
may also try whether the development of~$(a + x)^{\efrac{1}{2}}$ by
the binomial theorem, gives the same result as is obtained
by the extraction of the square root of~$a + x$.
Again, when any development is obtained, it should
be seen whether the development possesses all the
properties of the expression from which it has been
derived. For example, $\dfrac{1}{1 - x}$~is proved to be equivalent
to the series
\[
1 + x + x^{2} + x^{3} +\Chg{,}{}\ \etc.,\quad \textit{ad infinitum}.
\]
This, when multiplied by~$1 - x$, should give~$1$; when
multiplied by~$1 - x^{2}$, should give~$1 + x$, because
\[
\frac{1}{1 - x} × (1 - x) = 1,\qquad
\frac{1}{1 - x} × (1 - x^{2}) = 1+x,\quad \etc.
\]
Again,
\begin{align*}
a^{x} &= 1 + x\Log a + \frac{x^{2}(\Log a)^{2}}{2} + \frac{x^{3}(\Log a)^{3}}{2·3}
+ \dots \textit{ad inf.}\Add{,} \\
a^{y} &= 1 + y\Log a + \frac{y^{2}(\Log a)^{2}}{2} +\Chg{,}{} \etc.\Add{,} \\
a^{x+y} &= 1 + (x + y)\Log a + \frac{(x + y)^{2}(\Log a)^{2}}{2} +\Chg{,}{} \etc.
\end{align*}
Now, since $a^{x} × a^{y} = a^{x+y}$, the product of the two
first series should give the third. Many other instances
of the same sort will suggest themselves, and
a careful attention to them will confirm the demonstration
of the several theorems, which, to a beginner,
\PageSep{179}
is often doubtful, on account of the generality of the
reasoning.
III. Whenever a demonstration appears perplexed,
\index{Demonstration!inductive}%
\index{Induction, mathematical}%
on account of the number and generality of the symbols,
let some particular case be chosen, and let the
same demonstration be applied. For example, if the
binomial theorem should not appear sufficiently plain,
the same reasoning may be applied to the expansion
of~$(1 + x)^{\efrac{2}{3}}$, or any other case, which is there applied
to~$(1 + x)^{\efrac{m}{n}}$. Again, the general form of the product
$(x + a)$, $(x + b)$, $(x + c)$, etc.,~\dots\ containing $n$~factors,
will be made apparent by taking first two, then three,
and four factors, before attempting to apply the reasoning
which establishes the form of the general product.
The same applies particularly to the theory of
permutations and combinations, and to the doctrine
of probabilities, which is so materially connected with
it. In the theory of equations it will be advisable at
\index{Theory of equations}%
first, instead of taking the general equation of the
form
\[
x^{n} + Ax^{n-1} + Bx^{n-2} + \dots + Lx + M = 0,
\]
to choose that of the third, or at most of the fourth
degree, or both, on which to demonstrate all the
properties of expressions of this description. But in
all these cases, when the particular instances have
been treated, the general case should not be neglected,
since the power of reasoning upon expressions such
as the one just given, in which all the terms cannot
\PageSep{180}
be written down, on account of their indeterminate
number, must be exercised, before the student can
proceed with any prospect of success to the higher
branches of mathematics.
IV. When any previous theorem is referred to, the
reference should be made, and the student should
satisfy himself that he has not forgotten its demonstration.
If he finds that he has done so, he should
not grudge the time necessary for its recovery. By
so doing, he will avoid the necessity of reading over
the subject again, and will obtain the additional advantage
of being able to give to each part of the subject
a time nearly proportional to its importance,
whereas, by reading a book over and over again until
he is a master of it, he will not collect the more prominent
parts, and will waste time upon unimportant
details, from which even the best books are not free.
The necessity for this continual reference is particularly
felt in the Elements of Geometry, where allusion
is constantly made to preceding propositions, and
where many theorems are of no importance, considered
as results, and are merely established in order to
serve as the basis of future propositions.
V. The student should not lose any opportunity
of exercising himself in numerical calculation, and
particularly in the use of the logarithmic tables. His
power of applying mathematics to questions of practical
utility is in direct proportion to the facility which
he possesses in computation. Though it is in plane
\index{Computation}%
\PageSep{181}
and spherical trigonometry that the most direct numerical
applications present themselves, nevertheless
the elementary parts of algebra abound with useful
practical questions. Such will be found resulting from
the binomial theorem, the theory of logarithms, and
that of continued fractions. The first requisite in this
branch of the subject, is a perfect acquaintance with
the arithmetic of decimal fractions; such a degree of
acquaintance as can only be gained by a knowledge
of the principles as well as of the rules which are deduced
from them. From the imperfect manner in
which arithmetic is usually taught, the student ought
in most cases to recommence this study before proceeding
to the practice of logarithms.
VI. The greatest difficulty, in fact almost the only
one of any importance which algebra offers to the reason,
is the use of the isolated negative sign in such
\index{Negative!sign@{sign, isolated}}%
expressions as $-a$,~$a^{-x}$, and the symbols which we
have called imaginary. It is a remarkable fact, that
the first elements of the mathematics, sciences which
demonstrate their results with more certainty than any
others, contain difficulties which have been the subjects
of discussion for centuries. In geometry, for
example, the theory of parallel lines has never yet
\index{Parallels, theory of}%
been freed from the difficulty which presented itself to
Euclid, and obliged him to assume, instead of proving,
\index{Euclid}%
the 12th~axiom of his first book. Innumerable as have
been the attempts to elude or surmount this obstacle,
no one has been more successful than another. The
\PageSep{182}
\index{Instruction!faulty}%
elements of fluxions or the differential calculus, of
mechanics, of optics, and of all the other sciences, in
the same manner contain difficulties peculiar to themselves.
These are not such as would suggest themselves
to the beginner, who is usually embarrassed by
the actual performance of the operations, and no ways
perplexed by any doubts as to the foundations of the
rules by which he is to work. It is the characteristic
of a young student in the mathematical sciences, that
he sees, or fancies that he sees, the truth of every result
which can be stated in a few words, or arrived at
by few and simple operations, while that which is long
is always considered by him as abstruse. Thus while
he feels no embarrassment as to the meaning of the
equation $+a × -a = -a^{2}$, he considers the multiplication
of $a^{m} + a^{n}$ by $b^{m} + b^{n}$ as one of the difficulties
of algebra. This arises, in our opinion, from the manner
in which his previous studies are usually conducted.
From his earliest infancy, he learns no fact
from his own observation, he deduces no truth by the
exercise of his own reason. Even the tables of arithmetic,
which, with a little thought and calculation, he
might construct for himself, are presented to him
ready made, and it is considered sufficient to commit
them to memory. Thus a habit of examination is not
formed, and the student comes to the science of algebra
fully prepared to believe in the truth of any rule
which is set before him, without other authority than
the fact of finding it in the book to which he is recommended.
\PageSep{183}
It is no wonder, then, that he considers the
difficulty of a process as proportional to that of remembering
and applying the rule which is given,
without taking into consideration the nature of the
reasoning on which the rule was founded. We are
not advocates for stopping the progress of the student
by entering fully into all the arguments for and against
such questions, as the use of negative quantities, etc.,
which he could not understand, and which are inconclusive
on both sides; but he might be made aware
that a difficulty does exist, the nature of which might
be pointed out to him, and he might then, by the consideration
of a sufficient number of examples, treated
separately, acquire confidence in the results to which
the rules lead. Whatever may be thought of this
method, it must be better than an unsupported rule,
such as is given in many works on algebra.
It may perhaps be objected that this is induction,
\index{Induction, mathematical}%
a species of reasoning which is foreign to the usually
received notions of mathematics. To this it may be
answered, that inductive reasoning is of as frequent
occurrence in the sciences as any other. It is certain
that most great discoveries have been made by means
of it; and the mathematician knows that one of his
most powerful engines of demonstration is that peculiar
species of induction which proves many general
truths by demonstrating that, if the theorem be true
in one case, it is true for the succeeding one. But the
beginner is obliged to content himself with a less rigorous
\PageSep{184}
species of proof, though equally conclusive, as
far as moral certainty is concerned. Unable to grasp
the generalisations with which the more advanced
student is familiar, he must satisfy himself of the
truth of general theorems by observing a number of
particular simple instances which he is able to comprehend.
For example, we would ask any one who
has gone over this ground, whether he derived more
certainty as to the truth of the binomial theorem from
the general demonstration (if indeed he was suffered
\index{Demonstration!mathematical}%
to see it so early in his career), or from observation
of its truth in the particular cases of the development
of $(a + b)^{2}$, $(a + b)^{3}$, etc., substantiated by ordinary
multiplication. We believe firmly, that to the mass
of young students, general demonstrations afford no
conviction whatever; and that the same may be said
of almost every species of mathematical reasoning,
when it is entirely new. We have before observed,
that it is necessary to learn to reason; and in no case
is the assertion more completely verified than in the
study of algebra. It was probably the experience of
the inutility of general demonstrations to the very
young student that caused the abandonment of reasoning
which prevailed so much in English works on
elementary mathematics. Rules which the student
\index{Rules!mechanical}%
could follow in practice supplied the place of arguments
which he could not, and no pains appear to
have been taken to adopt a middle course, by suiting
the nature of the proof to the student's capacity. The
\PageSep{185}
objection to this appears to have been the necessity
which arose for departing from the appearance of rigorous
demonstration. This was the cry of those who,
not having seized the spirit of the processes which
they followed, placed the force of the reasoning in the
forms. To such the authority of great names is a
strong argument; we will therefore cite the words of
Laplace on this subject.
\index{Laplace}%
``Newton extended to fractional and negative
\index{Fractional exponents}%
\index{Indices, theory of}%
\index{Negative!indices}%
\index{Newton}%
powers the analytical expression which he had found
for whole and positive ones. You see in this extension
one of the great advantages of algebraic language
which expresses truths much more general than those
which were at first contemplated, so that by making
the extension of which it admits, there arises a multitude
of new truths out of formulæ which were founded
upon very limited suppositions. At first, people were
afraid to admit the general consequences with which
analytical formulæ furnished them; \emph{but a great number
of examples having verified them}, we now, without fear,
yield ourselves to the guidance of analysis through all
the consequences to which it leads us, and the most
happy discoveries have sprung from the boldness.
We must observe, however, that precautions should
be taken to avoid giving to formulæ a greater extension
than they really admit, and that it is always well
to demonstrate rigorously the results which are obtained.''
We have observed that beginners are not disposed
\PageSep{186}
\index{Signs!rule of}%
to quarrel with a rule which is easy in practice, and
verified by examples, on account of difficulties which
occur in its establishment. The early history of the
sciences presents occasion for the same remark. In
the work of Diophantus, the first Greek writer on algebra,
\index{Diophantus}%
we find a principle equivalent to the equations
$+a × -b = -ab$, and $-a × -b = +ab$, admitted
as an axiom, without proof or difficulty. In the Hindoo
\index{Hindu algebra}%[** TN: [sic] variant spelling]
works on algebra, and the Persian commentators
upon them, the same thing takes place. It appears,
that struck with the practical utility of the rule, and
certain by induction of its truth, they did not scruple
to avail themselves of it. A more cultivated age, possessed
of many formulæ whose developments presented
striking examples of an universality in algebraic
language not contemplated by its framers, set
itself to inquire more closely into the first principles
of the science. Long and still unfinished discussions
have been the result, but the progress of nations has
exhibited throughout a strong resemblance to that of
individuals.
VII. The student should make for himself a syllabus
\index{Syllabi, mathematical}%
of results only, unaccompanied by any demonstration.
It is essential to acquire a correct memory for
algebraical formulæ, which will save much time and
labor in the higher departments of the science. Such
a syllabus will be a great assistance in this respect,
and care should be taken that it contain only the most
useful and most prominent formulæ. Whenever that
\PageSep{187}
can be done, the student should have recourse to the
system of tabulation, of which he will have seen several
examples in this treatise. In this way he should
write the various forms which the roots of the equation
$ax^{2} + bx + c = 0$ assume, according to the signs
of $a$,~$b$, and~$c$, etc. Both the preceptor and the pupil,
but especially the former, will derive great advantage
\index{Lacroix}%
\index{Instruction!books on mathematical}%
from the perusal of Lacroix, \Title{Essais sur l'Enseignement
en général et sur celui des Mathématiques en particulier},\footnote
{The books mentioned in the present passage, while still very valuable,
are now not easily procurable and, besides, do not give a complete idea of
the subject in its modern extent. A recent work on the \Title{Philosophy and Teaching
of Mathematics} is that of C.~A. Laisant (\Title{La Mathématique. Philosophie-Enseignement},
\index{Laisant}%
\index{Mathematics!philosophy of}%
Paris, 1898, Georges Carré et C. Naud, publishers.) Perhaps
the most accessible and useful work in English for the elements is David
Eugene Smith's new book \Title{The Teaching of Elementary Mathematics}. (New
\index{Smith, D. E.}%
York: The Macmillan Company, 1900). Mention might be made also of W.~M.
Gillespie's translation from Comte's \Title{Cours de Philosophie Positive}, under the
\index{Comte}%
title of \Title{The Philosophy of Mathematics} (New York: Harpers, 1851), and of the
\Title{Cours de Méthodologie Mathématique} of Félix Dauge (Deuxième edition, revue
\index{Dauge, F.}%
et augmentee. Gand, Ad.\ Hoste; Paris, Gauthier-Villars, 1896). The recent
work of Freycinet on the \Title{Philosophy of the Sciences} (Paris, 1896, Gauthier-Villars)
\index{Freycinet}%
will be found valuable. One of the best and most comprehensive of the
modern works is that of Duhamel, \Title{Des Méthodes dans les Sciences de Raisonnement},
\index{Duhamel}%
(5~parts, Paris, Gauthier-Villars), a work giving a comprehensive exposition
of the foundations of all the mathematical sciences. The chapters in
\index{Duhring@{Dühring}}%
Dühring's \Title{Kritische Geschichte der Prinzipien der Mechanik} and his \Title{Neue
Grundmittel} on the study of mathematics and mechanics is replete with original,
but hazardous, advice, and may be consulted as a counter-irritant to
the traditional professional views of the subject. The articles in the \Title{English
Cyclopedia}, by De~Morgan himself, contain refreshing hints on this subject.
\index{Demorgan@{De Morgan}}%
\index{English@{\Title{English Cyclopedia}}}%
But the greatest inspiration is to be drawn from the works of the masters
themselves; for example, from such works as Laplace's Introduction to the
\Title{Calculus of Probabilities}, or from the historical and philosophical reflexions
that uniformly accompany the later works of Lagrange. The same remark
\index{Lagrange}%
applies to the later mathematicians of note.---\Ed.}
Condillac, \Title{La Langue des Calculs}, and the various articles
\index{Condillac}%
on the elements of algebra in the French Encyclopedia,
which are for the most part written by
D'Alembert. The reader will here find the first principles
\index{Dalembert@{D'Alembert}}%
\PageSep{188}
of algebra, developed and elucidated in a masterly
manner. A great collection of examples will be
found in most elementary works, but particularly in
Hirsch, \Title{Sammlung von Beispielen}, etc., translated into
\index{Hirsch}%
English under the title of \Title{Self-Examinations in Algebra},
etc., London: Black, Young and Young, 1825.\footnote
{Hirsch's \Title{Collection}, enlarged and modernised, can be obtained in various
recent German editions. The old English translations of the original
are not easily procured.---\Ed.}
The
\index{Algebras@{\emph{Algebras}, bibliographical list of}|(}%
student who desires to carry his algebraical studies
farther than usual, and to make them the stepping-stone
to a knowledge of the higher mathematics,
\index{French language}%
\index{German language}%
should be acquainted with the French language.\footnote
{German is now of as much importance as French. But the French text-books
still retain their high standard.---\Ed.}
A
knowledge of this, sufficient to enable him to read the
simple and easy style in which the writers of that nation
treat the first principles of every subject, may be
acquired in a short time. When that is done, we recommend
\index{Bourdon}%
to the student the algebra of M. Bourdon,\footnote
{Bourdon's \Title{Elements of Algebra} is still used in France, having appeared
in 1895 in its eighteenth edition, with notes by M.~Prouhet (Gauthier-Villars,
Paris.) A more elementary French work of a modern character is that of
J.~Collin (Second edition, 1888, Paris, Gauthier-Villars). A larger and more
\index{Collin, J.}%
complete treatise which begins with the elements and extends to the higher
branches of the subject is the \Title{Traité d'Algèbre} of H.~Laurent, in four small
\index{Laurent, H.}%
volumes (Gauthier-Villars, Paris). This work contains a large collection of
examples. Another elementary work is that of C.~Bourlet, \Title{Lecons d'Algèbre
\index{Bourlet, C.}%
Elémentaire}, Paris, Colin, 1896. A standard and exhaustive work on higher
algebra is the \Title{Cours d'Algèbre Supérieure}, of J.~A. Serret, two large volumes
\index{Serret, J. A.}%
(Fifth edition, 1885, Paris, Gauthier-Villars).
The number of American and English text-books of the intermediate and
higher type is very large. Todhunter's \Title{Algebra and Theory of Equations}
\index{Todhunter}%
(London: Macmillan \&~Co.) were for a long time the standards in England
and this country, but have now (especially the first-mentioned) been virtually
superseded. An excellent recent text-book for beginners, and one that skilfully
introduces modern notions, is the \Title{Elements of Algebra} of W.~W. Beman
\index{Beman, W. W.}%
and D.~E. Smith (Boston, 1900). Fisher and Schwatt's elementary text-books
\index{Fisher and Schwatt}%
of algebra are also recommendable from both a practical and theoretical
point of view. Valuable are C.~Smith's \Title{Treatise on Algebra} (London: Macmillan),
and Oliver, Wait, and Jones's \Title{Treatise on Algebra} (Ithaca, N.~Y.,
\index{Oliver, Waite, and Jones}%
1887), also Fine's \Title{Number System of Algebra} (Boston: Leach). The best English
\index{Fine, H. B.}%
work on the theory of equations is Burnside and Panton's (Longmans).
\index{Burnside, W. S.}%
\index{Panton, A. W.}%
A very exhaustive presentation of the subject from the modern point
of view is the \Title{Algebra} of Professor George Chrystal (Edinburgh: Adam and
\index{Chrystal, Prof.}%
Charles Black, publishers), in two large volumes of nearly six hundred pages
each. Recently Professor Chrystal has published a more elementary work
entitled \Title{Introduction to Algebra} (same publishers).
A few German works may also be mentioned in this connexion, for the
benefit of readers acquainted with that language. Professor Hermann Schubert
\index{Schubert, H.}%
has, in various forms, given systematic expositions of the elementary
principles of arithmetic, (\eg, see his \Title{Arithmetik und Algebra}, Sammlung
Göschen, Leipsic,---an extremely cheap series containing several other elementary
mathematical works of high standard; also, for a statement of
Schubert's views in English consult his \Title{Mathematical Recreations}, Chicago,
1898). Professor Schubert has recently begun the editing of a new and larger
series of mathematical text-books called the \Title{Sammlung Schubert} (Leipsic:
Göschen), which contains three works treating of algebra. In this connexion
may be mentioned also Matthiessen's admirable \Title{Grundzüge der antiken und
\index{Matthiessen}%
modernen Algebra} (Leipsic: Teubner) for literal equations. The following
are all excellent: (1) Otto Biermann's \Title{Elemente der höheren Mathematik}
\index{Biermann, O.}%
(Leipsic, 1895); (2)~Petersen's \Title{Theorie der algebraischen Gleichungen} (Copenhagen:
\index{Petersen}%
Höst; also in French, Paris: Gauthier-Villars); (3)~Richard Baltzer's
\index{Baltzer, R.}%
\Title{Elemente der Mathematik} (2~vols., Leipsic: Hirzel); (4)~Gustav Holzmüller's
\index{Holzmüller, G.}%
\Title{Methodisches Lehrbuch der Elementarmathematik} (3~parts, Leipsic: Teubner);
(5)~Werner Jos.\ Schüller's \Title{Arithmetik und Algebra für höhere Schulen und
\index{Schüller, W. J.}%
Lehrerseminare, besonders zum Selbstunterricht}, etc. (Leipsic, 1891, Teubner);
(6) Oskar Schlömilch's \Title{Handbuch der algebraischen Analysis} (Frommann,
\index{Schlömilch, O.}%
Stuttgart); (7)~Eugen Netto's \Title{Vorlesungen über Algebra} (Leipsic: Teubner, 2~vols.);
\index{Netto, E.}%
(8)~Heinrich Weber's \Title{Lehrbuch der Algebra} (Braunschweig: Vieweg, 2~vols).
\index{Weber, H.}%
This last work is the most advanced treatise that has yet appeared.
A French translation has been announced.---\Ed.---April, 1902.}
\PageSep{189}
a work of eminent merit, though of some difficulty to
the English student, and requiring some previous
habits of algebraical reasoning.
\index{Algebras@{\emph{Algebras}, bibliographical list of}|)}%
VIII. The height to which algebraical studies
should be carried, must depend upon the purpose to
which they are to be applied. For the ordinary purposes
of practical mathematics, algebra is principally
useful as the guide to trigonometry, logarithms, and
the solution of equations. Much and profound study
\PageSep{190}
is not therefore requisite; the student should pay
great attention to all numerical processes and particularly
to the methods of approximation which he will
find in all the books. His principal instrument is the
table of logarithms of which he should secure a knowledge
both theoretical and practical. The course which
should be adopted preparatory to proceeding to the
higher branches of mathematics is different. It is still
of great importance that the student should be well
acquainted with numerical applications; nevertheless,
he may omit with advantage many details relative to
the obtaining of approximative numerical results, particularly
in the theory of equations of higher degrees
\index{Theory of equations}%
than the second. Instead of occupying himself upon
these, he should proceed to the application of algebra
to geometry, and afterwards to the differential calculus.
When a competent knowledge of these has
been obtained, he may then revert to the subjects
which he has neglected, giving them more or less attention
according to his own opinion of the use which
he is likely to have for them. This applies particularly
to the theory of equations, which abounds with
processes of which very few students will afterwards
find the necessity.
We shall proceed in the next number to the difficulties
which arise in the study of Geometry and Trigonometry.
\PageSep{191}
\Chapter{XIII.}{On the Definitions of Geometry.}
\index{Geometry!definitions and study of|EtSeq}%
\First{In} this treatise on the difficulties of Geometry and
Trigonometry, we propose, as in the former part
of the work, to touch on those points only which, from
novelty in their principle, are found to present difficulties
to the student, and which are frequently not
sufficiently dwelt upon in elementary works. Perhaps
it may be asserted, that there are no difficulties in
geometry which are likely to place a serious obstacle
in the way of an intelligent beginner, except the temporary
embarrassment which always attends the commencement
of a new study; that, for example, there
is nothing in the elements of pure geometry comparable,
in point of complexity, to the theory of the negative
sign, of fractional indices, or of the decomposition
of an expression of the second degree into factors.
This may be true; and were it only necessary to study
the elements of this science for themselves, without
reference to their application, by means of algebra, to
higher branches of knowledge, we should not have
\PageSep{192}
thought it necessary to call the attention of our readers
to the points which we shall proceed to place before
them. But while there is a higher study in which
elementary ideas, simple enough in their first form,
are so generalised as to become difficult, it will be an
assistance to the beginner who intends to proceed
through a wider course of pure mathematics than
forms part of common education, if his attention is
early directed, in a manner which he can comprehend,
to future extensions of what is before him.
The reason why geometry is not so difficult as algebra,
\index{Algebra!nature of the reasoning in}%
is to be found in the less general nature of the
symbols employed. In algebra a general proposition
respecting numbers is to be proved. Letters are taken
which may represent any of the numbers in question,
and the course of the demonstration, far from making
any use of a particular case, does not even allow that
any reasoning, however general in its nature, is conclusive,
unless the symbols are as general as the arguments.
We do not say that it would be contrary to
good logic to form general conclusions from reasoning
on one particular case, when it is evident that the
same considerations might be applied to any other,
but only that very great caution, more than a beginner
can see the value of, would be requisite in deducing
the conclusion. There occurs also a mixture of general
and particular propositions, and the latter are
liable to be mistaken for the former. In geometry on
the contrary, at least in the elementary parts, any
\PageSep{193}
proposition may be safely demonstrated by reasonings
on any one particular example. For though in proving
a property of a triangle many truths regarding
that triangle may be asserted as having been proved
before, none are brought forward which are not general,
that is, true for all instances of the same kind.
It also affords some facility that the results of elementary
geometry are in many cases sufficiently evident
\index{Geometry!elementary ideas of|EtSeq}%
of themselves to the eye; for instance, that two sides
of a triangle are greater than the third, whereas in
algebra many rudimentary propositions derive no evidence
from the senses; for example, that $a^{3} - b^{3}$ is
always divisible without remainder by~$a - b$.
The definitions of the simple terms \emph{point}, \emph{line}, and
\index{Line}%
\emph{surface} have given rise to much discussion. But the
difficulties which attend them are not of a nature to
embarrass the beginner, provided he will rest content
with the notions which he has already derived from
observation. No explanation can make these terms
more intelligible. To them may be added the words
\emph{straight line}, which cannot be mistaken for one moment,
\index{Straight line}%
unless it be by means of the attempt to explain
them by saying that a straight line is ``that which lies
evenly between its extreme points.''
The line and surface are distinct species of magnitude,
as much so as the yard and the acre. The first
is no part of the second, that is, no number of lines
can make a surface. When therefore a surface is divided
into two parts by a line, the dividing line is not
\PageSep{194}
to be considered as forming a part of either. That
the idea of the line or boundary necessarily enters
into the notion of the division is very true; but if we
conceive the line abstracted, and thus get rid of the
idea of division, neither surface is increased or diminished,
which is what we mean when we say that the
line is not a part of the surface. The same considerations
apply to a point, considered as the boundary of
the divisions of a line.
\index{Point, geometrical|(}%
The beginner may perhaps imagine that a line is
made up of points, that is, that every line is the sum
of a number of points, a surface the sum of a number
of lines, and so on. This arises from the fact, that
the things which we draw on paper as the representatives
of lines and points, have in reality three dimensions,
two of which, length and breadth, are perfectly
visible. Thus the point, such as we are obliged to
represent it, in order to make its position visible, is
in reality a part of our line, and our points, if sufficiently
multiplied in number and placed side by side,
would compose a line of any length whatever. But
taking the mathematical definition of a point, which
denies it all magnitude, either in length, breadth, or
thickness, and of a line, which is asserted to possess
length only without breadth or thickness, it is easy to
show that a point is no part of a line, by making it
appear that the shortest line can be cut in as many
points as the longest, which may be done in the following
manner. Let $AB$~be any straight line, from
\PageSep{195}
the ends of which, $A$~and~$B$, draw two lines, $AF$~and~$CB$,
parallel to one another. Consider $AF$ as produced
without limit, and in~$CB$ take any point~$C$, from
which draw lines $CE$,~$CF$, etc., to different points in~$AF$.
It is evident that for each point~$E$ in~$AF$ there
is a distinct point in~$AB$, viz., the intersection of~$CE$
with~$AB$;---for, were it possible that two points, $E$~and~$F$
in~$AF$, could be thus connected with the same
point of~$AB$, it is evident that two straight lines would
enclose a space, viz., the lines $CE$~and~$CF$, which
\Figure{195}% ** Fig. 1.
both pass through~$C$, and would, were our supposition
correct, also pass through the same point in~$AB$.
There can then be taken as many points in the finite
or unbounded line~$AB$ as in the indefinitely extended
line~$AF$.
\index{Point, geometrical|)}%
The next definition which we shall consider is that
of a \emph{plane surface}. The word \emph{plane} or \emph{flat} is as hard
\index{Plane surface}%
to define, without reference to any thing but the idea
we have of it, as it is easy to understand. Nevertheless
the practical method of ascertaining whether or
no a surface is plane, will furnish a definition, not
\PageSep{196}
such, indeed, as to render the nature of a plane surface
more evident, but which will serve, in a mathematical
point of view, as a basis on which to rest the
propositions of solid geometry. If the edge of a ruler,
known to be perfectly straight, coincides with a surface
throughout its whole length, in whatever direction
\index{Direction}%
it may be placed upon that surface, we conclude
that the surface is plane. Hence the definition of a
plane surface is that in which, any two points being
taken, the straight line joining these points lies wholly
upon the surface.
Two straight lines have a relation to one another
independent altogether of their length. This we commonly
express (for among the most common ideas are
found the germ of every geometrical theory) by saying
that they are in the same or different \emph{directions}. By
the \emph{direction} of the needle we ascertain the \emph{direction} in
which to proceed at sea, and by the \emph{direction} in which
the hands of a clock are placed we tell the hour. It
remains to reduce this common notion to a more precise
form.
Suppose a straight line~$OA$ to be given in magnitude
\index{Angle!def@{definition of|EtSeq}}%
and position, and to remain fixed while another
line~$OB$, at first coincident with~$OA$, is made to move
round~$OA$, so as continually to vary its direction with
respect to~$OA$. The process of opening a pair of compasses
will furnish an illustration of this, but the two
lines need not be equal to one another. In this case
the opening made by the two will continually increase,
\PageSep{197}
and this opening is a species of magnitude, since one
opening may be compared with another, so as to ascertain
which of the two is the greater. Thus if the
figure~$CPD$ be removed from its place, without any
other change, so that the point~$P$ may fall on~$O$, and
the line~$PC$ may lie upon and become a part of~$OA$,
or $OA$~of~$PC$, according to which is the longer of the
two, then if the opening~$CPD$ is the same as the opening~$AOB$,
$PD$~will lie upon~$OB$ at the same time as
$PC$~lies upon~$OA$. But if $PD$~does not then lie upon~$OB$,
but falls between $OB$ and~$OA$, the opening~$CPD$
\Figure{197}% ** Fig. 2.
is less than the opening~$AOB$, and if $PD$~does not
fall between $OA$ and~$OB$, or on~$OB$, the opening~$CPD$
is greater than the opening~$BOA$. To this species
of magnitude, the opening of two lines, the name
of angle is given, that is $BO$~is said to make an angle
with~$OA$. The difficulty here arises from this magnitude
being one, the measure of which has seldom fallen
under observation of those who begin geometry.
Every one has measured one line by means of another,
and has thus made a number the representative of a
length; but few, at this period of their studies, have
\PageSep{198}
been accustomed to the consideration, that one opening
may be contained a certain number of times in
another, or may be a certain fraction of another.
Nevertheless we may find measures of this new species
of magnitude either by means of time, length, or
number.
One magnitude is said to be a measure of another,
\index{Measures}%
when, if the first be doubled, trebled, halved, etc., the
second is doubled, trebled, or halved, etc.; that is,
when any fraction or multiple of the first corresponds
to the same fraction or multiple of the second in the
same manner as the first does to the second. The two
quantities need not be of the same kind: thus, in the
barometer the height of the mercury (a length) measures
the pressure of the atmosphere (a weight); for if
the barometer which yesterday stood at $28$~inches, to-day
stands at $29$~inches, in which case the height of
yesterday is increased by its $28$th~part, we know that
the atmospheric pressure of yesterday is increased by
its $28$th~part to-day. Again, in a watch, the \emph{number
of hours} elapsed since twelve o clock is measured by
the \emph{angle} which a hand makes with the position it occupied
at twelve o'clock. In the spring balances a
\emph{weight} is measured by an \emph{angle}, and many other similar
instances might be given.
This being premised, suppose a line which moves
round another as just described, to move uniformly,
that is, to describe equal openings or angles in equal
times. Suppose the line~$OA$ to move completely
\PageSep{199}
round, so as to reassume its first position in twenty-four
hours. Then in twelve hours the moving line
will be in the position~$OB$, in six hours it will be in~$OC$,
and in eighteen hours in~$OD$. The line~$OC$ is
that which makes equal angles with $OA$ and~$OB$, and
is said to be at right angles, or perpendicular to $OA$
and~$OB$. Again, $OA$~and~$OB$ which are in the same
\Figure{199}% ** Fig. 3.
right line, but on opposite sides of the point~$O$, evidently
make an opening or angle which is equal to
the sum of the angles $AOC$ and~$COB$, or equal to two
right angles. A line may also be said to make with
itself an opening equal to four right angles, since
after revolving through four right angles, the moving
line reassumes its original position. We may even
carry this notion farther: for if the moving line be in
\PageSep{200}
the position~$OE$ when $P$~hours have elapsed, it will
recover that position after every twenty-four hours,
that is, for every additional four right angles described;
so that the angle~$AOE$ is equally well represented
by any of the following angles:
\begin{align*}
\text{$4$ right angles} &+ AOE\Add{,} \\
\text{$8$ right angles} &+ AOE\Add{,} \\
\text{$12$ right angles} &+ AOE,\quad \etc.
\end{align*}
These formulæ which suppose an opening greater
than any \emph{apparent} opening, and which take in and
represent the fact that the moving line has attained
its position for the second, third, fourth, etc., time,
since the commencement of the motion, are not of
any use in elementary geometry; but as they play an
important part in the application of algebra to the
theory of angles, we have thought it right to mention
them here.
It is plain also that we may conceive the line~$OE$
to make two openings or angles with the original position~$OA$:
(1)~that through which it has moved to recede
from~$OA$; (2)~that through which it must move
to reach $OA$ again. The first (in the position in which
we have placed~$OA$) is what is called in geometry the
angle~$AOE$; the second is more simply described as
composed of the openings or angles $EOC$, $COB$,
$BOD$, $DOA$, and is not used except in the application
of algebra above mentioned.\footnote
{But use is made of it in some modern text-books of elementary geometry.---\Ed.}
Of the two angles just
\PageSep{201}
alluded to, one must be less than two right angles,
and the second greater; the first is the one usually
referred to.
It is plain that the angle or opening made by two
lines does not depend upon their length but upon
their position; if either be shortened or lengthened,
the angle still remains the same; and if while the angle
increases or decreases one of the straight lines
containing it is diminished, the angle so contained
may have a definite and given magnitude at the moment
\Figure{201}% ** Fig. 4.
when the straight line disappears altogether and
becomes nothing. For example, take two points of
any curve~$AB$, and join $A$~and~$B$ by a straight line.
Let the point~$B$ move towards~$A$; it is evident that
the angle made by the moving line with~$AB$ increases
continually, while as much of one of the lines containing
it as is intercepted by the curve, diminishes without
limit. When this intercepted part disappears entirely,
the line in which it would have lain had it had
any length, has reached the line~$AG$, which is called
the tangent of the curve.
\PageSep{202}
In elementary geometry two equal angles lying on
different sides of a line, such as $AOE$, $AOH$ (\Fig{3}),
would be considered as the same. In the application
of algebra, they would be considered as having different
signs, for reasons stated at length in \PageRef[pages]{112} et~seq.,
of the first part of this Treatise. It is also common
in the latter branch of the science to measure
angles in one direction only; for example, in \Fig[Figure]{3}
the angles made by $OE$, $OF$, $OG$, and~$OH$, if measured
upwards from~$OA$, would be the openings through
which a line must move \emph{in the same direction} from~$OA$,
to attain those positions; and the second, third, and
fourth angles would be greater than one, two, and
three right angles respectively.
We proceed to the method of reasoning in geometry,
or rather to the method of reasoning in general,
since there is, or ought to be, no essential difference
between the manner of deducing results from first
principles, in any science.
\PageSep{203}
\Chapter{XIV.}{On Geometrical Reasoning.}
\index{Assertions, logical|EtSeq}%
\index{Geometrical reasoning and proof|EtSeq}%
\index{Logic of mathematics|(}%
\index{Particular affirmative and negative}%
\index{Propositions|EtSeq}%
\index{Reasoning!geometrical|EtSeq}%
\index{Universal affirmative and negative}%
\First{It} is evident that all reasoning, of what form soever,
can be reduced at last to a number of simple propositions
or assertions; each of which, if it be not self-evident,
depends upon those which have preceded it.
Every assertion can be divided into three distinct
parts. Thus the phrase, ``all right angles are equal,''
consists of: (1)~the \emph{subject} spoken of, viz., right angles,
\index{Subject}%
which is here spoken of universally, since every
right angle is a part of the subject; (2)~the \emph{copula}, or
\index{Copula}%
manner in which the two are joined together, which
is generally the verb \emph{is}, or \emph{is equal to}, and can always
be reduced to one or the other: in this case the copula
is affirmative; (3)~the \emph{predicate}, or thing asserted
\index{Predicate}%
of the subject, viz., equal angles. The phrase, thus
divided, stands as written below under~1, and is called
a \emph{universal affirmative}. The second is called a \emph{particular
affirmative} proposition; the third a \emph{universal negative};
the fourth a \emph{particular negative}:
1. All right angles are equal (magnitudes).
\PageSep{204}
2. Some triangles are equilateral (figures).
3. No circle is convex to its diameter.
4. Some triangles are not equilateral (figures).
Many assertions appear in a form which, at first
sight, cannot be reduced to one of the preceding; the
following are instances of the change which it is necessary
to make in them:
1. Parallel lines never meet, or parallel lines are
lines which never meet.
2. The angles at the base of an isosceles triangle
are equal, or an isosceles triangle is a triangle having
the angles at the base equal.
The different species of assertions, and the arguments
which are compounded of them, may be distinctly
conceived by referring them all to one species
of subject and predicate. Since every assertion, generally
speaking, includes a number of individual cases
in its subject, let the points of a circle be the subject
and those of a triangle the predicate. These figures
being drawn, the four species of assertions just alluded
to are as follows:
1. Every point of the circle is a point of the triangle,
or the circle is contained in the triangle.
2. Some points of the circle are points of the triangle,
or part of the circle is contained in the triangle.
3. No point of the circle is a point of the triangle,
or the circle is entirely without the triangle.
\PageSep{205}
4. Some points of the circle are not points of the
triangle, or part of the circle is outside the triangle.
On these we observe that the second follows from
the first, as also the fourth from the third, since that
which is true of all is true of some or any; while the
first and third do not follow from the second and
fourth, \emph{necessarily}, since that which is true of some
only need not be true of all. Again, the second and
fourth are not necessarily inconsistent with each other
for the same reason. Also two of these assertions
must be true and the others untrue. The first and
the third are called \emph{contraries}, while the first and
\index{Contraries}%
fourth, and the second and third are \emph{contradictory}.
\index{Contradictory}%
The \emph{converse} of a proposition is made by changing the
\index{Converse}%
predicate into the subject, and the subject into the
predicate. No mistake is more common than confounding
together a proposition and its converse, the
tendency to which is rather increased in those who
begin geometry, by the number of propositions which
they find, the converses of which are true. Thus all
the definitions are necessarily conversely true, since
the identity of the subject and predicate is not merely
asserted, but the subject is declared to be a name
\emph{given} to \emph{all} those magnitudes which have the properties
laid down in the predicate, and to no others.
Thus a square is a four-sided figure having equal
sides and one right angle, that is, let every four-sided
figure having, etc., \emph{be called} a square, and let no other
figure be called by that name, whence the truth of the
\PageSep{206}
converse is evident. Also many of the facts proved
in geometry are conversely true. Thus all equilateral
triangles are equiangular, from which it is proved that
\emph{all} equiangular triangles are equilateral. Of the first
species of assertion, the universal affirmative, the converse
is not necessarily true. Thus ``every point in
figure~$A$ is a point of~$B$,'' does not imply that ``every
point of~$B$ is a point of~$A$,'' although this may be the
case, and is, if the two figures coincide entirely. The
second species, the particular affirmative, is conversely
true, since if some points of~$A$ are points of~$B$, some
points of~$B$ are also points of~$A$. The first species of
assertion is conversely true, if the converse be made
to take the form of the second species: thus from
``all right angles are equal,'' it may be inferred that
``\emph{some} equal magnitudes are right angles.'' The third
species, the universal negative, is conversely true,
since if ``no point of~$B$ is a point of~$A$,'' it may be inferred
that ``no point of~$A$ is a point of~$B$.'' The
fourth species, the particular negative, is not necessarily
conversely true. From ``some points of~$A$ are
not points of~$B$,'' or $A$~is not entirely contained within~$B$,
we can infer nothing as to whether $B$~is or is not
entirely contained in~$A$. It is plain that the converse
of a proposition is not necessarily true, if it says more
either of the subject or predicate than was said before.
Now ``every equilateral triangle is equiangular,'' does
not speak of all equiangular triangles, but asserts that
\emph{among} all possible equiangular triangles are to be
\PageSep{207}
found \emph{all} the equilateral ones. There may then, for
anything to the contrary to be discovered in our assertion,
be classes of equiangular triangles not included
under this assertion, of which we can therefore
say nothing. But in saying ``no right angles are unequal,''
that which we exclude, we exclude from all
unequal angles, and therefore ``no unequal angles are
right angles'' is not more general than the first.
The various assertions brought forward in a geometrical
demonstration must be derived in one of the
following ways:
I. \textit{From definition.} This is merely substituting,
\index{Definition}%
instead of a description, the name which it has been
agreed to give to whatever bears that description. No
definition ought to be introduced until it is certain
that the thing defined is really possible. Thus though
parallel lines are defined to be ``lines which are in
the same plane, and which being ever so far produced
never meet,'' the mere agreement to call such lines,
should they exist, by the name of parallels, is no sufficient
ground to assume that they do exist. The definition
is therefore inadmissible until it is really shown
that there are such things as lines which being in the
same plane never meet. Again, before applying the
name, care must be taken that all the circumstances
connected with the definition have been attended to.
Thus, though in plane geometry, where all lines are
in one plane, it is sufficient that two lines would never
meet though ever so far produced, to call them parallel,
\PageSep{208}
yet in solid geometry the first circumstance must
be attended to, and it must be shown that lines are in
the same plane before the name can be applied. Some
of the axioms come so near to definitions in their nature,
\index{Axioms}%
that their place may be considered as doubtful.
Such are, ``the whole is greater than its part,'' and
``magnitudes which entirely coincide are equal to one
another.''
II. \textit{From hypothesis.} In the statement of every
\index{Hypothesis}%
proposition, certain connexions are supposed to exist
from which it is asserted that certain consequences
will follow. Thus ``in an isosceles triangle the angles
at the base are equal,'' or, ``if a triangle be isosceles
the angles at the base will be equal.'' Here the hypothesis
or supposition is that the triangle has two
equal sides, the consequence asserted is that the angles
at the base or third side will be equal. The consequence
being only asserted to be true when the angle
is isosceles, such a triangle is supposed to be taken
as the basis of the reasonings, and the condition that
its two sides are equal, when introduced in the proof,
is said to be introduced by hypothesis.
In order to establish the result it may be necessary
to draw other lines, etc., which are not mentioned in
the first hypothesis. These, when introduced, form
what is called the construction.
There is another species of hypothesis much in
use, principally when it is required to deduce the converse
of a theorem from the theorem itself. Instead
\PageSep{209}
of proving the consequence directly, the contradictory
of the consequence is assumed to hold good, and if
from this new hypothesis, supposed to exist together
with the old one, any evidently absurd result can be
derived, such as that the whole is greater than its
part, this shows that the two hypotheses are not consistent,
and that if the first be true, the second cannot
be so. But if the second be not true, its contradictory
is true, which is what was required to be proved.
III. \textit{From the evidence of the assertions themselves.}
\index{Self-evidence}%
The propositions thus introduced without proof are
only such as are in their nature too simple to admit
of it. They are called axioms. But it is necessary
to observe, that the claim of an assertion to be
called an axiom does not depend only on its being
self-evident. Were this the case many propositions
which are always proved might be assumed; for example,
that two sides of a triangle are greater than
the third, or that a straight line is the shortest distance
between two points. In addition to being self-evident,
it must be incapable of proof by any other
means, and it is one of the objects of geometry to reduce
the demonstrations to the least possible number
of axioms. There are only two axioms which are distinctly
geometrical in their nature, viz., ``two straight
lines cannot enclose a space,'' and ``through each
point outside a line, not more than one parallel to
that line can be drawn.'' All the rest of the propositions
commonly given as axioms are either arithmetical
\PageSep{210}
in their nature; such as ``the whole is greater
than its part,'' ``the doubles of equals are equals,''
etc.; or mere definitions, such as ``magnitudes which
entirely coincide are equal\Chg{'';}{;''} or theorems admitting
of proof, such as ``all right angles are equal.'' There
is however one more species of self-evident proposition,
the postulate or self-evident problem, such as
\index{Postulate}%
the possibility of drawing a right line, etc.
IV. \textit{From proof already given.} What has been
proved once may be always taken for granted afterwards.
It is evident that this is merely for the sake
of brevity, since it would be possible to begin from
the axioms and proceed direct to the proof of any one
proposition, however far removed from them; and
this is an exercise which we recommend to the student.
Thus much for the legitimate use of any single
assertion or proposition. We proceed to the manner
of deducing a third proposition from two others.
It is evident that no assertion can be the direct
\index{Syllogisms|EtSeq}%
and necessary consequence of two others, unless those
two contain something in common, or which is spoken
of in both. In many, nay most, cases of ordinary conversation
and writing, we leave out one of the assertions,
which is, usually speaking, very evident, and
make the other assertion followed by the consequence
of both. Thus, ``Geometry is useful, and therefore
ought to be studied,'' contains not only what is expressed,
but also the following, ``That which is useful
ought to be studied;'' for were this not admitted, the
\PageSep{211}
former assertion would not be necessarily true. This
may be written thus:
\begin{quote}
Every thing useful is what ought to be studied.
Geometry is useful, therefore geometry is
what ought to be studied.
\end{quote}
This, in its present state, is called a syllogism, and
may be compared with the following, from which it
only differs in the \emph{things} spoken of, and not in the
\emph{manner} in which they are spoken of.
\begin{quote}
Every point of the circle is a point of the triangle.
The point~$B$ is a point of the circle.
Therefore the point~$B$ is a point of the triangle.
\end{quote}
Here a connexion is established between the point~$B$
and the points of the triangle (viz., that the first is
one of the second) by comparing them with the points
of the circle; that which is asserted of every point of
the circle in the first can be asserted of the point~$B$,
because from the second $B$~is one of these points.
Again, in the former argument, whatever is asserted
of every thing useful is true of geometry, because geometry
is useful.
The common term of the two propositions is called
\index{Premisses|EtSeq}%
the \emph{middle term}, while the \emph{predicate} and \emph{subject} of the
conclusion are called the \emph{major} and \emph{minor} terms, respectively.
The two first assertions are called the
\emph{major} and \emph{minor premisses}, and the last the \emph{conclusion}.
\PageSep{212}
Suppose now the two premisses and conclusion of the
syllogism just quoted to be varied in every possible
way from affirmative to negative, from universal to
particular, and \textit{vice versa}, where the number of changes
will be $4 × 4 × 4$, or~$64$ (called moods); since each
\index{Moods, logical|EtSeq}%
proposition may receive four different forms, and each
form of one may be compounded with any of the other
two. And these may be still further varied, if instead
of the middle term being the subject of the first, and
the predicate of the second, this order be reversed, or
if the middle term be the subject of both, or the predicate
of both, which will give four different figures, as
they are called, to each of the sixty-four moods above
mentioned. But of these very few are correct deductions,
and without entering into every case we will
state some general rules, being the methods which
common reason would take to ascertain the truth or
\index{Logics@{\emph{Logics}, bibliographical list of}}
falsehood of any one of them, collected and generalised.\footnote
{Whately's \Title{Logic}, page~76, third edition. A work which should be read
\index{Whately}%
by all mathematical students. [Whately's \Title{Logic} is procurable in modern editions,
many of which were, until recently, widely read in our academies and
colleges. The following works in which the same material is presented in a
shape more comforming to modern methods may be mentioned: T.~Fowler's
\Title{Elements of Deductive Logic}; Bain's \Title{Logic}; Venn's \Title{Empirical Logic and Symbolical
\index{Bain}%
\index{Fowler, T.}%
\index{Venn}%
Logic}; Keynes's \Title{Formal Logic}; Carveth Read's \Title{Logic, Deductive and
\index{Keynes}%
\index{Read, Carveth}%
Inductive}; Mill's \Title{System of Logic} (a discussion rather than a presentation).
\index{Mill, J. S.}%
Strictly contemporary logic will be found represented in the following works
in English: Jevons's \Title{Principles of Science and Studies in Deductive Logic};
\index{Jevons}%
Bradley's \Title{Principles of Logic}; Sidgwick's \Title{Process of Argument}; Bosanquet's
\index{Bosanquet}%
\index{Bradley}%
\index{Sidgwick}%
\Title{Logic: or, the Morphology of Knowledge}; and the same author's \Title{Essentials of
Logic}; Sigwart's \Title{Logic}, recently translated from the German; and Ueberweg's
\index{Sigwart}%
\index{Ueberweg}%
\Title{System of Logic and History of Logical Doctrines}.---\Ed.]}
I. The middle term must be the same in both
\PageSep{213}
premisses, by what has just been observed; since in
the comparison of two things with one and the same
third thing, in order to ascertain their connexion or
discrepancy, consists the whole of reasoning. Thus,
the deduction without further process of the equation
$a^{2} + b^{2} = c^{2}$ from the proposition, which proves that
the sum of the \emph{squares} described on the sides of a
right-angled triangle is equal to the square on its hypothenuse,
$a$,~$b$, and~$c$ being the number of linear units
in the sides and the hypothenuse, is incorrect, since
syllogistically stated the argument would stand thus:
\begin{gather*}
\left.
\begin{aligned}
&\settowidth{\TmpLen}{The sum of the \emph{squares} of the}
\parbox[b]{\TmpLen}{The sum of the \emph{squares} of the \\
\null\quad lines $a$~and~$b$} \\
&\qquad\text{and} \\
&\text{the \emph{square} of the line~$c$}
\end{aligned}
\right\} \text{ are equal quantities,} \displaybreak[0]\\
%
\left.
\begin{aligned}
&a^{2} + b^{2} \\
&\quad\text{and} \\
&c^{2}
\end{aligned}
\right\} \text{ are }
\left\{
\begin{aligned}
&\text{the sum of the \emph{squares} of $a$~and~$b$,} \\
&\qquad\text{and} \\
&\text{the \emph{square} of~$c$.}
\end{aligned}\right. \displaybreak[0]\\
%
\text{Therefore}\qquad
\left.
\begin{aligned}
&a^{2} + b^{2} \\
&\quad\text{and} \\
&c^{2}
\end{aligned}
\right\}
\text{ are equal quantities.}
\end{gather*}
Here the term \emph{square} in the major premiss has its
geometrical, and in the minor its algebraical sense,
being in the first a geometrical figure, and in the second
an arithmetical operation. The term of comparison
is not therefore the same in both, and the conclusion
does not therefore follow from the premisses.
The same error is committed if all that can be contained
under the middle term be not spoken of either
in the major or minor premiss. For if each premiss
\PageSep{214}
mentions only a part of the middle term, these parts
may be different, and the term of comparison really
different in the two, though passing under the same
name in both. Thus,
\begin{quote}
All the triangle is in the circle, \\
All the square is in the circle,
\end{quote}
proves nothing, since the square may, consistently
with these conditions, be either wholly, partly, or not
at all contained in the triangle. In fact, as we have
before shown, each of these assertions speaks of a part
of the circle only. The following is of the same kind:
\begin{quote}
Some of the triangle is in the circle. \\
Some of the circle is not in the square, etc.
\end{quote}
II. If both premisses are negative, no conclusion
can be drawn. For it can evidently be no proof either
of agreement or disagreement that two things both
disagree with a third. Thus the following is inconclusive:
\begin{quote}
None of the circle is in the triangle. \\
None of the square is in the circle.
\end{quote}
III. If both premisses are particular, no conclusion
can be drawn, as will appear from every instance that
can be taken, thus:
\begin{quote}
Some of the circle is in the triangle. \\
Some of the square is not in the circle,
\end{quote}
proves nothing.
IV. In forming a conclusion, where a conclusion
can be formed, nothing must be asserted more generally
\PageSep{215}
in the conclusion than in the premisses. Thus, if
from the following,
\begin{quote}
All the triangle is in the circle, \\
All the circle is in the square,
\end{quote}
we would draw a conclusion in which the square
should be the subject, since the whole square is not
mentioned in the minor premiss, but only part of it,
the conclusion must be,
\begin{quote}
Part of the square is in the triangle.
\end{quote}
V. If either of the premisses be negative, the conclusion
must be negative. For as both premisses cannot
be negative, there is asserted in one premiss an
agreement between the term of the conclusion and
the middle term, and in the other premiss a disagreement
between the other term of the conclusion, and
the same middle term. From these nothing can be
inferred but a disagreement or negative conclusion.
Thus, from
\begin{quote}
None of the circle is in the triangle, \\
All the circle is in the square,
\end{quote}
can only be inferred,
\begin{quote}
Some of the square is \emph{not} in the triangle.
\end{quote}
VI. If either premiss be particular, the conclusion
must be particular. For example, from
\begin{quote}
None of the circle is in the triangle, \\
Some of the circle is in the square,
\end{quote}
we deduce,
\begin{quote}
\emph{Some} of the square is not in the triangle.
\end{quote}
If the student now applies these rules, he will find
\PageSep{216}
that of the sixty-four moods eleven only are admissible
in any case; and in applying these eleven moods
to the different figures he will also find that some of
them are not admissible in every figure, and some not
necessary, on account of the conclusion, though true,
not being as general as from the premisses it might be.
This he may do either by reasoning or by actual inspection
of the figures, drawn and arranged according
to the premisses. The admissible moods are nineteen
in number, and are as follows, where $A$~at the beginning
of a proposition signifies that it is a universal
affirmative, $E$~a universal negative, $I$~a particular
affirmative, $O$~a particular negative.
Figure~I. The middle term is the subject of the
\index{Figures, logical|EtSeq}%
major, and the predicate of the minor premiss.
\begin{gather*}
\begin{Syllogism}
\Syl{1.\footnotemark}{A}{All the}{\Circ}{is in the}{\Tri}{,}
\Syl{}{A}{All the}{\Sqr}{is in the}{\Circ}{,}
\Syl{\Thus}{A}{All the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{2.}{E}{None of the}{\Circ}{is in the}{\Tri}{,}
\Syl{}{A}{All the}{\Sqr}{is in the}{\Circ}{,}
\Syl{\Thus}{E}{None of the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{3.}{A}{All the}{\Circ}{is in the}{\Tri}{,}
\Syl{}{I}{Some of the}{\Sqr}{is in the}{\Circ}{,}
\Syl{\Thus}{I}{Some of the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{4.}{E}{None of the}{\Circ}{is in the}{\Tri}{,}
\Syl{}{I}{Some of the}{\Sqr}{is in the}{\Circ}{,}
\Syl{\Thus}{O}{Some of the}{\Sqr}{is not in}{\Tri}{.}
\end{Syllogism}
\end{gather*}
\footnotetext{This, and~3, are the most simple of all the combinations, and the most
frequently used, especially in geometry.}
\PageSep{217}
Figure~II. The middle term is the predicate of
both premisses.
\begin{gather*}
\begin{Syllogism}
\Syl{1.}{E}{None of the}{\Tri}{is in the}{\Circ}{,}
\Syl{}{A}{All the}{\Sqr}{is in the}{\Circ}{,}
\Syl{\Thus}{E}{None of the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{2.}{A}{All the}{\Tri}{is in the}{\Circ}{,}
\Syl{}{E}{None of the}{\Sqr}{is in the}{\Circ}{,}
\Syl{\Thus}{E}{None of the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{3.}{E}{None of the}{\Tri}{is in the}{\Circ}{,}
\Syl{}{I}{Some of the}{\Sqr}{is in the}{\Circ}{,}
\Syl{\Thus}{O}{Some of the}{\Sqr}{is not in}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{4.}{A}{All the}{\Tri}{is in the}{\Circ}{,}
\Syl{}{O}{Some of the}{\Sqr}{is not in}{\Circ}{,}
\Syl{\Thus}{O}{Some of the}{\Sqr}{is not in}{\Tri}{.}
\end{Syllogism}
\end{gather*}
Figure~III. The middle term is the subject of both
premisses.
\begin{gather*}
\begin{Syllogism}
\Syl{1.}{A}{All the}{\Circ}{is in the}{\Tri}{,}
\Syl{}{A}{All the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{I}{Some of the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{2.}{I}{Some of the}{\Circ}{is in the}{\Tri}{,}
\Syl{}{A}{All the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{I}{Some of the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{3.}{A}{All the}{\Circ}{is in the}{\Tri}{,}
\Syl{}{I}{Some of the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{I}{Some of the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{4.}{E}{None of the}{\Circ}{is in the}{\Tri}{,}
\Syl{}{A}{All the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{O}{Some of the}{\Sqr}{is not in}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{5.}{O}{Some of the}{\Circ}{is not in}{\Tri}{,}
\PageSep{218}
\Syl{}{A}{All the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{O}{Some of the}{\Sqr}{is not in}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{6.}{E}{None of the}{\Circ}{is in the}{\Tri}{,}
\Syl{}{I}{Some of the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{O}{Some of the}{\Sqr}{is not in}{\Tri}{.}
\end{Syllogism}
\end{gather*}
Figure~IV. The middle term is the predicate of
the major, and the subject of the minor premiss.
\begin{gather*}
\begin{Syllogism}
\Syl{1.}{A}{All the}{\Tri}{is in the}{\Circ}{,}
\Syl{}{A}{All the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{I}{Some of the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{2.}{A}{All the}{\Tri}{is in the}{\Circ}{,}
\Syl{}{E}{None of the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{E}{None of the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{3.}{I}{Some of the}{\Tri}{is in the}{\Circ}{,}
\Syl{}{A}{All the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{I}{Some of the}{\Sqr}{is in the}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{4.}{E}{None of the}{\Tri}{is in the}{\Circ}{,}
\Syl{}{A}{All the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{O}{Some of the}{\Sqr}{is not in}{\Tri}{.}
\end{Syllogism} \\
\begin{Syllogism}
\Syl{5.}{E}{None of the}{\Tri}{is in the}{\Circ}{,}
\Syl{}{I}{Some of the}{\Circ}{is in the}{\Sqr}{,}
\Syl{\Thus}{O}{Some of the}{\Sqr}{is not in}{\Tri}{.}
\end{Syllogism}
\end{gather*}
We may observe that it is sometimes possible to
condense two or more syllogisms into one argument,
thus:
\begin{align*}
&\text{Every $A$ is $B$ (1),} \\
&\text{Every $B$ is $C$ (2),} \\
&\text{Every $C$ is $D$ (3),} \\
&\text{Every $D$ is $E$ (4),} \\
\text{Therefore }
&\text{Every $A$ is $E$ (5),}
\end{align*}
\PageSep{219}
is equivalent to three distinct syllogisms of the form
Fig.~1.; these syllogisms at length being (1),~(2),~$a$;
$a$,~(3),~$b$; $b$,~(4),~(5).
The student, when he has well considered each of
these, and satisfied himself, first by the rules, and
afterwards by inspection, that each of them is legitimate;
and also that all other moods, not contained
in the above, are not allowable, or at least do not give
the most general conclusion, should form for himself
examples of each case, for instance of Fig.~III,~3:
\begin{quote}
The axioms constitute part of the basis of
geometry.
Some of the axioms are grounded on the evidence
of the senses.
$\Thus$~Some evidence derived from the senses is
part of the basis of geometry.
\end{quote}
He should also exercise himself in the first principles
of reasoning by reducing arguments as found in
books to the syllogistic form. Any controversial or
argumentative work will furnish him with a sufficient
number of instances.
\emph{Inductive} reasoning is that in which a universal
\index{Inductive reasoning}%
proposition is proved by proving separately every one
of its particular cases. As where, for example, a
figure, $ABCD$, is proved to be a rectangle by proving
each of its angles separately to be a right angle, or
proving all the premisses of the following, from which
the conclusion follows necessarily:
\PageSep{220}
\index{Geometrical reasoning and proof|EtSeq}%
\begin{quote}
The angles at $A$, $B$, $C$, and~$D$ are all the angles
of the figure~$ABCD$.
\begin{align*}
&\text{$A$ is a right angle,} \\
&\text{$B$ is a right angle,} \\
&\text{$C$ is a right angle,} \\
&\text{$D$ is a right angle,}
\end{align*}
Therefore all the angles of the figure $ABCD$
are right angles.
\end{quote}
This may be considered as one syllogism of which
the minor premiss is,
\begin{quote}
$A$, $B$, $C$, and $D$ are right angles,
\end{quote}
where each part is to be separately proved.
Reasoning \textit{\Chg{a}{à}~fortiori}, is that contained in Fig.~I.~1.
in a different form, thus: $A$~is greater than~$B$, $B$~is
greater than~$C$; \textit{\Chg{a}{à}~fortiori} $A$~is greater than~$C$; which
may be also stated as follows:
\begin{quote}
The whole of $B$ is contained in~$A$,
The whole of $C$ is contained in~$B$,
Therefore $C$~is contained in~$A$.
\end{quote}
The premisses of the second do not necessarily imply
as much as those of the first; the complete reduction
we leave to the student.
The elements of geometry present a collection of
such reasonings as we have just described, though in
a more condensed form. It is true that, for the convenience
of the learner, it is broken up into distinct
propositions, as a journey is divided into stages; but
nevertheless, from the very commencement, there is
nothing which is not of the nature just described. We
\PageSep{221}
present the following as a specimen of a geometrical
proposition reduced nearly to a syllogistic form. To
avoid multiplying petty syllogisms, we have omitted
some few which the student can easily supply.
\index{Pythagorean proposition|EtSeq}%
\textit{Hypothesis.}---$ABC$~is a right-angled triangle the
right angle being at~$A$.
\textit{Consequence.}---The squares on $AB$ and~$AC$ are together
equal to the square on~$BC$.
\textit{Construction}: Upon $BC$
and~$BA$ describe squares,
produce~$DB$ to meet~$EF$,
produced, if necessary, in~$G$,
and through~$A$ draw
$HAK$ parallel to~$BD$.
\subsubsection*{\centering\normalfont\itshape Demonstration.}
I. Conterminous sides
of a square are at right
angles to one another.
(Definition.)
\Wrapfigure{r}{1.75in}{221}
$EB$ and $BA$ are conterminous
sides of a square.
(Construction.)
$\Thus$ $EB$ and $BA$ are at right angles.
II. A similar syllogism to prove that $DB$ and $BC$
are at right angles, and another to prove that $GB$ and
$BC$ are at right angles.
III. Two right lines drawn perpendicular to two
other right lines make the same angle as those others
\PageSep{222}
(already proved); $EB$~and $BG$ and $AB$ and~$BC$ are
two right lines, etc., (I.~II\Add{.}).
$\Thus$ The angle~$EBG$ is equal to~$ABC$.
IV. All sides of a square are equal. (Definition.)
$AB$ and $BE$ are sides of a square. (Construction.)
$\Thus$ $AB$ and $BE$ are equal.
V. All right angles are equal. (Already proved.)
$BEG$ and $BAC$ are right angles. (Hypothesis and
construction.)
$\Thus$ $BEG$ and $BAC$ are equal angles.
VI. Two triangles having two angles of one equal
to two angles of the other, and the interjacent sides
equal, are equal in all respects. (Proved.)
$BEG$ and $BAC$ are two triangles having $BEG$ and
$EBG$ respectively equal to $BAC$ and $ABC$ and the
sides $EB$ and $BA$ equal. (III.~IV.~V.)
$\Thus$ The triangles $BEG$, $BAC$ are equal in all respects.
VII. $BG$ is equal to~$BC$. (VI.)
$BC$ is equal to~$BD$. (Proved as~IV.)
$\Thus$ $BG$ is equal to~$BD$.
VIII. A four-sided figure whose opposite sides
are parallel is a parallelogram. (Definition.) $BGHA$
and $BPKD$ are four-sided figures, etc. (Construction.)
$\Thus$ $BGHA$ and $BPKD$ are parallelograms.
IX. Parallelograms upon the same base and between
the same parallels are equal. (Proved.) $EBAF$
and $BGHA$, are parallelograms, etc. (Construction.)
\PageSep{223}
$\Thus$ $EBAF$ and $BGHA$ are equal.
X. Parallelograms on equal bases and between the
same parallels, are equal. (Proved.)
$BGHA$ and $BDKP$ are parallelograms, etc. (Construction.)
$\Thus$ $BGHA$ and $BDKP$ are equal.
XI. $EBAF$ is equal to $BGHA$. (IX.)
$BGHA$ is equal to~$BDKP$. (X.)
$\Thus$ $EBAF$ (that is the square on~$AB$) is equal to~$BDKP$.
XII. A similar argument from the commencement
to prove that the square on~$AC$ is equal to the rectangle~$CPK$.
XIII. The rectangles $BK$ and~$CK$ are together
equal to the square on~$BC$. (Self-evident from the
construction.)
The squares on $BA$ and $AC$ are together equal to
the rectangles $BK$ and~$CK$. (Self-evident from XI\Add{.}
and~XII.)
$\Thus$ The squares on $BA$ and $AC$ are together equal
to the square on~$BC$.
Such is an outline of the process, every step of
which the student must pass through before he has
understood the demonstration. Many of these steps
are not contained in the book, because the most ordinary
intelligence is sufficient to suggest them, but the
least is as necessary to the process as the greatest.
Instead of writing the propositions at this length, the
\PageSep{224}
student is recommended to adopt the plan which we
now \hyperref[table:224]{lay before him}.
\begin{table}[hp!]
\small
\[
\phantomsection\label{table:224}
\begin{array}{l@{\ }r@{\quad}c@{\quad}l}
\text{Hyp.} & 1 & & \Reason{$ABC$ is a triangle, right-angled at~$A$.} \\
\text{Constr.}
& 2 & a & \Reason{On $BA$ describe a square $BAFE$.} \\
& 3 & a & \Reason{On $BC$ describe a square.} \\
& 4 & & \Reason{Produce $BD$ to meet~$EF$, produced if necessary, in~$G$.} \\
& 5 & b & \Reason{Through $A$ draw $HAK$ parallel to~$BD$.} \\
\text{Demonst.}
& 6 & 2, \text{Def.} & \Reason{$EBA$ is a right angle.} \\
& 7 & 3 & \Reason{$GBC$ is a right angle.} \\
& 8 & 6, 7, c & \Reason{$\angle EBG$ is equal to~$\angle ABC$.} \\
& 9 & 2, 1, d & \Reason{$\angle BEG$ is equal to~$\angle BAC$.} \\
&10 & 2 & \Reason{$EB$ is equal to $AB$.} \\
&11 &8, 9, 10, e & \Reason{The triangles $BEG$ and $ABC$ are equal.} \\
&12 &11, 3 & \Reason{$BG$ is equal to~$BD$.} \\
&13 &5, 2, \text{Def.} & \Reason{$AHGB$ is a parallelogram.} \\
&14 &5, 3, \text{Def.} & \Reason{$BPDK$ is a parallelogram.} \\
&15 &13, 2, f & \Reason{$AHGB$ and $ABEF$ are equal.} \\
&16 &13, 14, g & \Reason{$AHGB$ and $BPDK$ are equal.} \\
&17 &15, 16 & \Reason{$BPDK$ and the square on $AB$ are equal.} \\
&18 &\settowidth{\TmpLen}{By similar}
\biggl\{\parbox{\TmpLen}{\centering By similar \\ reasoning}\biggr\}
& \Reason{$CPK$ and the square on $CA$ are equal.} \\
\PageSep{225}
&19 & 17, 18 & \Reason{The square on $BC$ is equal to the squares on $BA$ and~$AC$.}
\end{array}
\]
\ifthenelse{\not\boolean{ForPrinting}}{%
\end{table}
\begin{table}[ht!]
\small
}{}
\[
\begin{array}{c@{\quad}l}
a, b & \Reason[0.85]{Here refer to the necessary problems.} \\
c & \Reason[0.85]{If two lines be drawn at right angles to
two others, the angles made by the
first and second pair are equal.} \\
d & \Reason[0.85]{All right angles are equal.} \\
e & \Reason[0.85]{Two triangles which have two angles of
one equal to two angles of the other,
and the interjacent sides equal, are
equal in all respects.} \\
f, g & \Reason[0.85]{Parallelograms on the same or equal
bases, and between the same parallels,
are equal.}
\end{array}
\]
\end{table}
The explanation of this is as follows: the whole
proposition is divided into distinct assertions, which
are placed in separate consecutive paragraphs, which
paragraphs are numbered in the first column on the
left; in the second column on the left we state the
reasons for each paragraph, either by referring to the
preceding paragraphs from which they follow, or the
preceding propositions in which they have been
proved. In the latter case a letter is placed in the
column, and at the end, the enunciation of the proposition
there used is written opposite to the letter. By
this method, the proposition is much shortened, its
more prominent parts are brought immediately under
notice, and the beginner, if he recollect the preceding
propositions perfectly well, is not troubled by the
\PageSep{226}
repetition of prolix enunciations, while in the contrary
case he has them at hand for reference.
In all that has been said, we have taken instances
only of direct reasoning, that is, where the required
\index{Direct reasoning}%
\index{Reasoning!direct and indirect}%
result is immediately obtained without any reference
to what might have happened if the result to be proved
had not been true. But there are many propositions
in which the only possible result is one of two things
which cannot be true at the same time, and it is more
easy to show that one is \emph{not} the truth, than that the
other \emph{is}. This is called indirect reasoning; not that
\index{Indirect reasoning}%
it is less satisfactory than the first species, but because,
as its name imports, the method does not appear
so direct and natural. There are two propositions
of which it is required to show that whenever
the first is true the second is true; that is, the first
being the hypothesis the second is a necessary conclusion
from it, whence the hypothesis in question, and
anything contradictory to, or inconsistent with, the
conclusion cannot exist together. In indirect reasoning,
we suppose that, the original hypothesis existing
and being true, something inconsistent with or contradictory
to the conclusion is true also. If from combining
the consequences of these two suppositions,
something evidently erroneous or absurd is deduced,
it is plain that there is something wrong in the assumptions.
Now care is taken that the only doubtful
point shall be the one just alluded to, namely, the
supposition that one proposition and the contradictory
\PageSep{227}
of the other are true together. This then is incorrect,
that is, the first proposition cannot exist with anything
contradictory to the second, or the second must exist
wherever the first exists, since if any proposition be
not true its contradictory must be true, and \textit{vice versa}.
This is rather embarrassing to the beginner, who finds
that he is required to admit, for argument's sake, a
proposition which the argument itself goes to destroy.
But the difficulty would be materially lessened, if instead
of assuming the contradictory of the second
proposition positively, it were hypothetically stated,
and the consequences of it asserted with the verb
``would be,'' instead of ``is.'' For example: suppose
it to be known that if $A$ is~$B$, then $C$~must be~$D$, and
it is required to show indirectly that when $C$~is not~$D$,
$A$~is not~$B$. This put into the form in which such a
proposition would appear in most elementary works,
is as follows.
It being granted that if $A$ is~$B$, $C$~is~$D$, it is required
to show that when $C$~is not~$D$, $A$~is not~$B$. If
possible, let $C$~be not~$D$, and let $A$~be~$B$. Then by
what is granted, since $A$~is~$B$, $C$~is~$D$; but by hypothesis
$C$~is not~$D$, therefore both $C$~\emph{is}~$D$ and \emph{is not}~$D$,
which is absurd; that is, it is absurd to suppose
that $C$~\emph{is not}~$D$ and $A$~\emph{is}~$B$, consequently when $C$~is
not~$D$, $A$~is not~$B$. The following, which is exactly
the same thing, is plainer in its language. Let $C$~be
not~$D$. Then if $A$~\emph{were}~$B$, $C$~\emph{would be}~$D$ by the proposition
granted. But by hypothesis $C$~is not~$D$, etc.
\PageSep{228}
This sort of indirect reasoning frequently goes by the
name of \textit{reductio ad~absurdum}.
In all that has gone before we may perceive that
the validity of an argument depends upon two distinct
considerations,---(1)~the truth of the relations assumed,
or represented to have been proved before; (2)~the
manner in which these facts are combined so as to
produce new relations; in which last the \emph{reasoning}
properly consists. If either of these be incorrect in
any single point, the result is certainly false; if both
be incorrect, or if one or both be incorrect in more
points than one, the result, though not at all to be depended
on, is not certainly false, since it may happen
and has happened, that of two false reasonings or
facts, or the two combined, one has reversed the effect
of the other and the whole result has been true; but
this could only have been ascertained after the correction
of the erroneous fact or reasoning. The same
thing holds good in every species of reasoning, and it
must be observed, that however different geometrical
argument may be in form from that which we employ
daily, it is not different in reality. We are accustomed
to talk of mathematical \emph{reasoning} as above all
other, in point of accuracy and soundness. This, if
by the term \emph{reasoning} we mean the comparing together
of different ideas and producing other ideas from the
comparison, is not correct, for in this view mathematical
reasonings and all other reasonings correspond
exactly. For the real difference between mathematics
\PageSep{229}
and other studies in this respect we refer the student
to the first chapter of this treatise.
In what then, may it be asked, does the real advantage
of mathematical study consist? We repeat
again, in the actual certainty which we possess of the
truth of the facts on which the whole is based, and
the possibility of verifying every result by actual measurement,
and not in any superiority which the method
of reasoning possesses, since there is but one method
of reasoning. To pursue the illustration with which
we opened this work (page the first), suppose this
point to be raised, was the slaughter of Cæsar justifiable
\index{Caesar@{Cæsar}}%
or not? The actors in that deed justified themselves
by saying, that a tyrant and usurper, who meditated
the destruction of his country's liberty, made it
the duty of every citizen to put him to death, and that
Cæsar was a tyrant and usurper, etc. Their \emph{reasoning}
was perfectly correct, though proceeding on premisses
then extensively, and now universally, denied. The
first premiss, though correctly used in this reasoning,
is now asserted to be false, on the ground that it is
the duty of every citizen to do nothing which would,
were the practice universal, militate against the general
happiness; that were each individual to act upon
his own judgment, instead of leaving offenders to the
law, the result would be anarchy and complete destruction
of civilisation, etc. Now in these reasonings
and all others, with the exception of those which occur
in mathematics, it must be observed that there
\PageSep{230}
are no premisses so certain, as never to have been
denied, no first principles to which the same degree of
evidence is attached as to the following, that ``no
two straight lines can enclose a space.'' In mathematics,
therefore, we reason on certainties, on notions
to which the name of innate can be applied, if it can
be applied to any whatever. Some, on observing that
we dignify such simple consequences by the name of
reasoning, may be loth to think that this is the process
to which they used to attach such ideas of difficulty.
There may, perhaps, be many who imagine
that reasoning is for the mathematician, the logician,
etc., and who, like the Bourgeois Gentilhomme, may
\index{Bourgeois gentilhomme, the}%
be surprised on being told, that, well or ill, they have
been reasoning all their lives. And yet such is the
fact; the commonest actions of our lives are directed
by processes exactly identical with those which enable
us to pass from one proposition of geometry to another.
A porter, for example, who being directed to
carry a parcel from the city to a street which he has
never heard of, and who on inquiry, finding it is in
the Borough, concludes that he must cross the water
to get at it, has performed an act of reasoning, differing
nothing in kind from those by a series of which,
did he know the previous propositions, he might be
convinced that the square of the hypothenuse of a
right-angled triangle is equal to the sum of the squares
of the sides.
\index{Logic of mathematics|)}%
\PageSep{231}
\Chapter{XV.}{On Axioms.}
\index{Axioms@{Axioms|EtSeq}}%
\index{Parallels, theory of|(}%
\First{Geometry}, then, is the application of strict logic
to those properties of space and figure which
are self-evident, and which therefore cannot be disputed.
But the rigor of this science is carried one
step further; for no property, however evident it may
be, is allowed to pass without demonstration, if that
can be given. The question is therefore to demonstrate
all geometrical truths with the smallest possible
number of assumptions. These assumptions are called
\index{Assumptions}%
axioms, and for an axiom it is requisite: (1)~that it
should be self-evident; (2)~that it should be incapable
of being proved from the other axioms. In fulfilling
these conditions, the number of axioms which are
really geometrical, that is, which have not equal reference
to Arithmetic, is reduced to two, viz., two
straight lines cannot enclose a space, and through a
given point not more than one parallel can be drawn
to a given straight line. The first of these has never
been considered as open to any objection; it has
\PageSep{232}
always passed as perfectly self-evident.\footnote
{But see J.~B. Stallo, \Title{Concepts and Theories of Modern Physics}, New York,
\index{Stallo, J. B.}%
1884, p.~242, p.~208 et~seq., and p.~248 et~seq. For popular philosophical discussions
of the subject of Axioms generally, in the light of modern psychology
and pangeometry, the reader may consult the following works: Helmholtz's
\index{Helmholtz}%
``Origin and Meaning of Geometrical Axioms,'' \Title{Mind}, Vol.~III., p.~215,
and the article in the same author's \Title{Popular Lectures on Scientific Subjects},
Second Series, London, 1881, pp.~27--71; W.~K. Clifford's \Title{Lectures and Essays},
\index{Clifford}%
Vol.~I., p.~297, p.~317; Duhamel, \Title{Des Méthodes dans les Sciences de Raisonnement},
\index{Duhamel}%
Part~2; and the articles ``Axiom'' and ``Measurement'' in the \Title{Encyclopedia
Britannica}, Vol.~XV\@. See also Riemann's Essay on the \Title{Hypotheses
\index{Riemann}%
Which Lie at the Basis of Geometry}, a translation of which is published in
Clifford's \Title{Works}, pp.~55--69. For part of the enormous technical literature of
this subject cf.\ Halsted's \Title{Bibliography of Hyper-Space and Non-Euclidean
\index{Halsted}%
Geometry}, \Title{American Journal of Mathematics}, Vol.~I., pp.~261 et~seq., and Vol.~II.,
pp.~65 et~seq. Much, however, has been written subsequently to the date
of the last-mentioned compilation, and translations of Lobachévski and Bolyai,
\index{Bolyai}%
\index{Lobachévski}%
for instance, may be had in the \Title{Neomonic Series} of Dr.~G.~B. Halsted
(Austin, Texas). A full history of the theory of parallels till recent times
is given in Paul Stäckel's \Title{Theorie der Parallellinien von Euklid bis auf Gauss}
\index{Stäckel, Paul}%
(Leipsic, 1895). Of interest are the essays of Prof.~J. Delb\oe{}uf on \Title{The Old
\index{Delboeuf@{Delb\oe{}uf, J.}}%
and the New Geometries} (\Title{Revue Philosophique}, 1893--1895), and those of Professor
Poincaré and of other controversialists in the recent volumes of the
\index{Poincaré, H.}%
\Title{Revue de Métaphysique et de Morale}, where valuable bibliographical references
will be found to literature not mentioned in this note. See also P.~Tannery
\index{Tannery, P.}%
in the recent volumes of the \Title{Revue générale} and the \Title{Revue philosophique},
Poincaré in \Title{The Monist} for October, 1898, and B.~A.~W. Russell's \Title{Foundations
\index{Russell, B. A. W.}%
of Geometry} (Cambridge, 1897). In Grassmann's \Title{Ausdehnungslehre} (1844), ``assumptions''
\index{Assumptions}%
\index{Grassmann}%
and ``axioms'' are replaced by purely formal (logical) ``predications,''
which presuppose merely the consistency of mental operations. (See
\Title{The Open Court}, Vol.~II. p.~1464, Grassmann, ``A Flaw in the Foundation of
Geometry,'' and Hyde's \Title{Directional Calculus}, Ginn \&~Co., Boston). Dr.~Paul
\index{Carus, Paul}%
\index{Hyde}%
Carus in his \Title{Primer of Philosophy} (Chicago), p.~51 et~seq., has treated the subject
of Axioms at length, from a similar point of view. On the psychological
side, consult Mach's \Title{Analysis of the Sensations} (Chicago, 1897), and the bibliographical
\index{Mach, E.}%
references and related discussions in such works as James's \Title{Psychology}
\index{James, W.}%
\index{Jodl, F.}%
and Jodl's \Title{Psychology} (Stuttgart, 1896).---\Ed.}
It is on this
account made the proposition on which are grounded
all reasonings relative to the straight line, since the
definition of a straight line is too vague to afford any
information. But the second, viz., that through a
given point not more than one parallel can be drawn
to a given straight line, has always been considered
as an assumption not self-evident in itself, and has
\PageSep{233}
therefore been called the defect and disgrace of geometry.
We proceed to place it on what we conceive to
be the proper footing.
By taking for granted the arithmetical axioms only,
with the first of those just alluded to, the following
propositions may be strictly shown.
I. One perpendicular, and only one, can be let fall
from any point~$A$ to a given line~$CD$. Let this be~$AB$.
II. If equal distances $BC$~and~$BD$ be taken on
both sides of~$B$, $AC$~and~$AD$ are equal, as also the
angles $BAC$ and~$BAD$.
\Figure{233}% ** Fig. 6.
III. Whatever may be the length of $BC$ and~$BD$,
the angles $BAC$ and~$BAD$ are each less than a right
angle.
IV. Through $A$ a line may be drawn parallel to~$CD$
(that is, by definition, never meeting~$CD$, though
the two be ever so far produced), by drawing any line~$AD$
and making the angle~$DAE$ equal to the angle~$ADB$,
which it is before shown how to do.
From proposition~IV. we should at first see no
\PageSep{234}
reason against there being as many parallels to~$CD$,
to be drawn through~$A$, as there are different ways of
taking~$AD$, since the direction for drawing a parallel
to~$CD$ is, ``take \emph{any line}~$AD$ cutting~$CD$ and make
the angle~$DAE$ equal to~$ADB$.'' But this our senses
immediately assure us is impossible.
\index{Euclid|(}%
It appears also a proposition to which no degree
of doubt can attach, that if the straight line~$AB$, produced
indefinitely both ways, set out from the position~$AB$
and revolve round the point~$A$, moving first
towards~$AE$; then the point of intersection~$D$ will
first be on one side of~$B$ and afterwards on the other,
and there will be one position where there is no point
of intersection either on one side or the other, and \emph{one
such position only}. This is in reality the assumption of
Euclid; for having proved that $AE$ and $BF$ are parallel
when the angles $BDA$ and $DAE$ are equal, or,
which is the same thing, when $EAD$ and $ADF$ are
together equal to two right angles, he further assumes
that they will be parallel in no other case, that is, that
they will meet when the angles $EAD$ and $ADF$ are
together greater or less than two right angles; which
is really only assuming that the parallel which he has
found is the only one which can be drawn. The remaining
part of his axiom, namely, that the lines $AE$
and~$DF$, if they meet at all, will meet upon that side
of~$DA$ on which the angles are less than two right
angles, is not an assumption but a consequence of his
proposition which shows that any two angles of a
\PageSep{235}
\index{Infinite spaces, compared|EtSeq}%
triangle are together less than two right angles, and
which is established before any mention is made of
parallels. It has been found by the experience of
\Pagelabel{235}%
two thousand years that some assumption of this sort
is indispensable. Every species of effort has been
made to avoid or elude the difficulty, but hitherto
without success, as some assumption has always been
involved, at least equal, and in most cases superior,
in difficulty to the one already made by Euclid. For
example, it has been proposed to define parallel lines
as those which are equidistant from one another at
every point. In this case, before the name \emph{parallel}
can be allowed to belong to any thing, it must be
proved that there are lines such that a perpendicular
to one is always perpendicular to the other, and that
the parts of these perpendiculars intercepted between
the two are always equal. A proof of this has never
been given without the previous assumption of something
equivalent to the axiom of Euclid. Of this last,
indeed, a proof has been given, but involving considerations
not usually admitted into geometry, though
it is more than probable that had the same come
down to us, sanctioned by the name of Euclid, it
\index{Euclid|)}%
would have been received without difficulty. The
Greek geometer confines his notion of equal magnitudes
to those which have boundaries. Suppose this
notion of equality extended to all such spaces as can
be made to coincide entirely in all their extent, what
ever that extent may be; for example, the unbounded
\PageSep{236}
spaces contained between two equal angles whose
sides are produced without end, which by the definition
of equal angles might be made to coincide entirely
by laying the sides of one angle upon those of the
other. In the same sense we may say, that, one
angle being double another, the space contained by
the sides of the first is double that contained by the
sides of the second, and so on. Now suppose two
\Figure{236}% ** Fig. 7.
lines $Oa$ and~$Ob$, making any angle with one another,
and produced \textit{ad infinitum}.\footnote
{Every line in this figure must be produced \textit{ad infinitum}, from that extremity
at which the small letter is placed.}
On~$Oa$ take off the equal
spaces $OP$, $PQ$, $QR$, etc., \textit{ad infinitum}, and draw the
lines $Pp$, $Qq$, $Rr$, etc., so that the angles $OPp$, $OQq$,
etc., shall be equal to one another, each being such
as with~$bOP$ will make two right angles. Then $Ob$,
$Pp$, $Qq$, etc., are parallel to one another, and the infinite
\PageSep{237}
spaces $bOPp$, $pPQq$, $qQRr$, etc., can be made
to coincide, and are equal. Also no finite number
whatever of these spaces will fill up the infinite space~$bOa$,
since $OP$, $PQ$, etc., may be contained \textit{ad infinitum}
upon the line~$Oa$. Let there be any line~$Ot$, such
that the angles $tOP$ and $pPO$ are together less than
two right angles, that is, less than $bOP$ and~$pPO$;
whence $tOP$~is less than~$bOP$ and $tO$~falls between
$bO$ and~$aO$. Take the angles $tOv$, $vOw$, $wOx$, each
equal to~$bOt$, and continue this until the last line~$Oz$
falls beneath~$Oa$, so that the angle~$bOz$ is greater than~$bOa$.
That this is possible needs no proof, since it is
manifest that any angle being continually added to
itself the sum will in time exceed any other given angle;
again, the infinite spaces $bOt$, $tOv$, etc., are all
equal. Now on comparing the spaces $bOt$ and~$bOPp$,
we see that a certain number of the first is more than
equal to the space~$bOa$, while no number whatever of
the second is so great. We conclude, therefore, that
the space~$bOt$ is greater than~$bOPp$, which cannot be
unless the line~$Ot$ cuts~$Pp$ at last; for if $Ot$~did never
cut~$Pp$, the space~$bOt$ would evidently be less than~$bOPp$,
as the first would then fall entirely within the
second. Therefore two lines which make with a third
angles together less than two right angles will meet if
sufficiently produced. [See Note on \PageRef{239}.]
\Pagelabel{237}%
\index{Parallels, theory of|)}%
This demonstration involves the consideration of
a new species of magnitude, namely, the whole space
contained by the sides of an angle produced without
\PageSep{238}
limit. This space is unbounded, and is greater than
any number whatever of finite spaces, of square feet,
for example. No comparison, therefore, as to magnitude
can be instituted between it and any finite space
whatever, but that affords no reason against comparing
this magnitude with others of the same kind.
Any thing may become the subject of mathematical
reasoning, which can be increased or diminished
by other things of the same kind; this is, in fact, the
definition given of the term magnitude; and geometrical
reasoning, in all other cases at least, can be applied
as soon as a criterion of equality is discovered.
Thus the angle, to beginners, is a perfectly new species
\index{Angle!def@{definition of}}%
of magnitude, and one of whose measure they
have no conception whatever; they see, however, that
it is capable of increase or diminution, and also that
two of the kind can be equal, and how to discover
whether this is so or not, and nothing more is necessary
for them. All that can be said of the introduction
of the angle in geometry holds with some, (to us
it appears an equal force,) with regard to these unlimited
spaces; the two are very closely connected, so
much so, that the term angle might even be defined
as ``the unlimited space contained by two right lines,''
without alteration in the truth of any theorem in which
the word \emph{angle} is found. But this is a point which
cannot be made very clear to the beginner.
The real difficulties of geometry begin with the
theory of proportion, to which we now proceed. The
\PageSep{239}
points of discussion which we have hitherto raised,
are not such as to embarrass the elementary student,
however much they may perplex the metaphysical inquirer
into first principles. The theory to which we
are coming abounds in difficulties of both classes.
\bigskip
{\small
[\textsc{Note to \PageRef[Page]{237}.}---The demonstration given on \PageRefs{235}{237}
\Pagelabel{239}%
is now regarded as fallacious by mathematicians; the considerations
that apply to finite aggregates not being transferable to
infinite aggregates,---for example, it is not true for infinite aggregates
that the part is always less than the whole\Add{.} Even Plato is
\index{Plato}%
cited for the assertion that equality is only to be predicated of
finite magnitudes. See the modern works on the Theory of the
Infinite. The demonstration in question is not De~Morgan's, but
M.~Bertrand's.---\Ed.]}
\index{Bertrand}%
\PageSep{240}
\Chapter{XVI.}{On Proportion.}
\index{Proportions!theory of|(}%
\First{In} the first elements of geometry, two lines, or two
surfaces, are mentioned in no other relation to
one another than that of equality or non-equality.
Nothing but the simple fact is announced that one
magnitude is equal to, greater than, or less than another,
except occasionally when the sum of two equal
magnitudes is said to be double one of them. Thus
in proving that two sides of a triangle are together
greater than the third, the fact that they are \emph{greater}
is the essence of the proposition; no measure is given
of the excess, nor does anything follow from the theorem
as to whether it is, or may be, small or great.
We now come to the doctrine of proportion in which
geometrical magnitude is considered in a new light.
The subject has some difficulties, which have been
materially augmented by the almost universal use, in
this country at least,\footnote
{In England.}
of the theory laid down in the
\index{Euclid!theory@{his theory of proportion}}%
fifth book of Euclid.\footnote
{See Todhunter's \Title{Euclid} (Macmillan, London).---\Ed.}
Considered as a complete conquest
\PageSep{241}
over a great and acknowledged difficulty of principle,
this book of Euclid well deserves the immortality
of which its existence, at the present moment, is
the guarantee; nay, had the speculations of the mathematician
been wholly confined to geometrical magnitude,
it might be a question whether any other notions
would be necessary. But when we come to apply
arithmetic to geometry, it is necessary to examine well
the primary connexion between the two; and here
difficulties arise, not in comprehending that connexion
so much as in joining the two sciences by a chain of
demonstration as strong as that by which the propositions
of geometry are bound together, and as little
open to cavil and disputation.
The student is aware that before pronouncing upon
\index{Measuring|EtSeq}%
the connexion of two lines with one another, it is necessary
to \emph{measure} them, that is, to refer them to some
third line, and to observe what number of times the
third is contained in the other two. Whether the two
first are equal or not is readily ascertained by the use
of the compasses, on principles laid down with the
utmost strictness in Euclid and other elementary
works. But this step is not sufficient; to say that two
lines are not equal, determines nothing. There are
an infinite number of ways in which one line may be
greater or less than a given line, though there is only
one in which the other can be equal to the given one.
We proceed to show how, from the common notion
\PageSep{242}
of measuring a line, the more strict geometrical method
\index{Line}%
is derived.
To measure the line~$AB$, apply to it another line
\index{Approximations@{Approximations|EtSeq}}%
(the edge of a ruler), which is divided into equal parts
(as inches), each of which parts is again subdivided
into ten equal parts, as in the figure. This division is
made to take place in practice until the last subdivision
gives a part so small that anything less may be
neglected as inconsiderable. Thus a carpenter's rule
is divided into tenths or eighths of inches
only, while in the tube of a barometer a
process must be employed which will
mark a much less difference. In talking
of accurate measurement, therefore, anywhere
but in geometry, or algebra, we
only mean accurate
%[** TN: Width-dependent line break]
\Wrapfigure{l}{0.75in}{242}%** Fig. 8.
as far as the senses
are concerned, and as far as is necessary
for the object in view. The ruler in the
figure shows that the line~$AB$ contains
more than two and less than three inches; and closer
inspection shows that the excess above two inches is
more than sixth-tenths of an inch, and less than
seven. Here, in practice, the process stops; for, as
the subdivision of the ruler was carried only to tenths
of inches, because a tenth of an inch is a quantity
which may be neglected in ordinary cases, we may
call the line two inches and six-tenths, by doing
which the error committed is less than one-tenth of
an inch. In this way lines may be compared together
\PageSep{243}
with a common degree of correctness; but this is not
enough for the geometer. His notions of accuracy
are not confined to tenths or hundredths, or hundred-millionth
parts of any line, however small it may be
at first. The reason is obvious; for although to suit
the eye of the generality of readers, figures are drawn
in which the least line is usually more than an inch,
yet his theorems are asserted to remain true, even
though the dimensions of the figure are so far diminished
as to make the whole imperceptible in the
strongest microscope. Many theorems are obvious
upon looking at a moderately-sized figure; but the
reasoning must be such as to convince the mind of
their truth when, from excessive increase or diminution
of the scale, the figures themselves have past the
boundary even of imagination. The next step in the
process of measurement is as follows, and will lead us
to the great and peculiar difficulty of the subject.
The inch, the foot, and the other lengths by which
we compare lines with one another, are perfectly arbitrary.
There is no reason for their being what they
are, unless we adopt the commonly received notion
that our inch is derived from our Saxon ancestors,
who observed that a barley corn is always of the same
length, or nearly so, and placed three of them together
as a common standard of measure, which they called
an inch. Any line whatever may be chosen as the
standard of measure, and it is evident that when two
or more lines are under consideration, exact comparisons
\PageSep{244}
of their lengths can only be obtained from a line
which is contained an exact number of times in them
all. For even exact fractional measures are reduced
to the same denominator, in order to compare their
magnitudes. Thus, two lines which contain $\frac{2}{11}$ and $\frac{3}{7}$
of a foot, are better compared by observing that $\frac{2}{11}$
and $\frac{3}{7}$ being $\frac{14}{77}$ and~$\frac{33}{77}$, the given lines contain one
$77$th~part of a foot $14$~and $33$~times respectively. Any
line which is contained an exact number of times in
another is called in geometry a measure of it, and a
common measure of two or more lines is that which
is contained an exact number of times in each.
\index{Arithmetical!notion of proportion|EtSeq}%
Again, a line which is measured by another is called
a multiple of it, as in arithmetic.
The same definition, \textit{mutatis mutandis}, applies to
surfaces, solids, and all other magnitudes; and though
in our succeeding remarks we use lines as an illustration,
it must be recollected that the reasoning applies
equally to every magnitude which can be made the
subject of calculation.
In order that two quantities may admit of comparison
\index{Comparison of quantities|EtSeq}%
as to magnitude, they must be of the same
sort; if one is a line, the other must be a line also.
Suppose two lines $A$~and~$B$ each of which is measured
by the line~$C$; the first containing it five times and
the second six. These lines $A$~and~$B$, which contain
the same line~$C$ five and six times respectively, are
said to have to one another the ratio of five to six, or
to be in the proportion of five to six. If then we denote
\PageSep{245}
the first by~$A$,\footnote
{The student must distinctly understand that the common meaning of
algebraical terms is departed from in this chapter, wherever the letters are
large instead of small. For example, $A$, instead of meaning the number of
\emph{units} of some sort or other contained in the line~$A$, stands for \emph{the line $A$ itself},
and $mA$ (the small letters throughout meaning \emph{whole numbers}) stands for the
line made by taking~$A$, $m$~times. Thus such expressions as $mA + B$, $mA - nB$,
etc., are the only ones admissible. $AB$, $\dfrac{A}{B}$, $A^{2}$, etc., are unmeaning, while $\dfrac{A}{m}$
is the line which is contained $m$~times in~$A$, or the $m$th~part of~$A$. The capital
letters throughout stand for concrete quantities, not for their representations
in abstract numbers.}
and the second by~$B$, and the
common measure by~$C$, we have
\begin{alignat*}{3}
A &= 5C, \quad &&\text{or}\quad &6A &= 30C, \\
B &= 6C, &&\text{or} &5B &= 30C, \\
\text{whence}\quad
6A &= 5B, &&\text{or} &6A &- \Z5B = 0.
\end{alignat*}
Generally, when $mA - nB = 0$, the lines, or whatever
they are, represented by $A$~and~$B$, are said to be
in the proportion of $n$~to~$m$, or to have the ratio of $n$~to~$m$.
Let there be two other magnitudes $P$~and~$Q$, of
the same kind with one another, either differing from
the first in kind or not, (thus $A$~and~$B$ may be lines,
and $P$~and~$Q$ surfaces, etc.,) and let them contain a
common measure~$R$, just as $A$~and~$B$ contain~$C$, viz.:
Let $P$~contain~$R$ five times, and let $Q$~contain~$R$ six
times, we have by the same reasoning
\[
6P - 5Q = 0,
\]
and $P$ and~$Q$, being also in the ratio of five to six, as
well as $A$~and~$B$, are said to be proportional to $A$~and~$B$,
which is denoted thus
\[
A : B :: P : Q,
\]
by which \emph{at present} all we mean is this, that there are
\PageSep{246}
\index{Incommensurables|EtSeq}%
some two whole numbers $m$~and~$n$ such that, at the
same time
\begin{alignat*}{2}
mA &- nB &&= 0, \\
mP &- nQ &&= 0.
\end{alignat*}
\Wrapfigure{l}{1.875in}{246}% ** Fig. 9.
Nothing more than this would be necessary for the
formation of a complete theory of proportion, if the
common measure, which we have supposed to exist
in the definition, did always really exist. We have,
however, no right to assume
that two lines $A$~and~$B$,
whatever may be
their lengths, both contain
some other line an
exact number of times.
We can, moreover, produce
a direct instance in
which two lines have no
common measure what
ever, in the following
manner.
Let $ABC$ be an isosceles right-angled triangle, the
side~$BC$ and the hypothenuse have no common measure
whatever. If possible let $D$~be a common measure
of $BC$ and~$AB$; let $BC$ contain~$D$, $n$~times, and
let $AB$ contain~$D$, $m$~times. Let $E$~be the square described
on~$D$. Then since $AB$ contains~$D$, $m$~times,
the square described on~$AB$ contains, $m × m$ or $m^{2}$~times.
Similarly the square described on~$BC$ contains~$E$
$n × n$ or $n^{2}$~times. But, because $AB$~is an isosceles
\PageSep{247}
right-angled triangle, the square on~$AB$ is double
that on~$BC$, whence $m × m = 2(n × n)$ or $m^{2} = 2n^{2}$. To
prove the impossibility of this equation (when $m$~and~$n$
are whole numbers), observe that $m^{2}$~must be an
even number, since it is twice the number~$n^{2}$. But
$m × m$ cannot be an even number unless $m$~is an even
number, since an odd number multiplied by itself
produces an odd number.\footnote
{Every odd number, when divided by~$2$, gives a remainder~$1$, and is therefore
of the form $2p + 1$ where $p$~is a whole number. Multiply $2p + 1$ by itself,
which gives $4p^{2} + 4p + 1$, or $2(2p^{2} + 2p) + 1$, which is an odd number, since,
when divided by~$2$, it gives the quotient $2p^{2} + 2p$, a whole number, and the
remainder~$1$.}
Let $m$ (which has been
shown to be even) be double~$m'$ or $m = 2m'$. Then
$2m' × 2m' = 2n^{2}$ or $4m'^{2} = 2n^{2}$ or $n^{2} = 2m'^{2}$. By repeating
the same reasoning we show that $n$~is even. Let
it be~$2n'$. Then $2n' × 2n' = 2m'^{2}$ or $m'^{2} = 2n'^{2}$. By the
same reasoning $m'$~and~$n'$ are both even, and so on \textit{ad
infinitum}. This reasoning shows that the whole numbers
which satisfy the equation $n^{2} = 2m^{2}$ (if such there
be) are divisible by~$2$ without remainder, \textit{ad infinitum}.
The absurdity of such a supposition is manifest: there
are then no such whole numbers, and consequently no
common measure to $BA$ and~$BC$.
Before proceeding any further, it will be necessary
to establish the following proposition.
If the greater of two lines $A$~and~$B$ be divided into
$m$~equal parts, and one of these parts be taken away;
if the remainder be then divided into $m$~equal parts,
and one of them be taken away, and so on,---the remainder
\PageSep{248}
of the line~$A$ shall in time become less than
the line~$B$, how small soever the line~$B$ may be.
Take a line which is less than~$B$, and call it~$C$. It
is evident that, by a continual addition of the same
quantity to~$C$, this last will come in time to exceed~$A$;
and still more will it do so if the quantity added to~$C$
be increased at each step. To simplify the proof we
suppose that $20$~is the number of equal parts into
which $A$ and its remainders are successively divided,
so that $19$~out of the $20$~parts remain after subtraction.
Divide $C$ into $19$~equal parts and add to~$C$ a line
equal to one of these parts. Let the length of~$C$, so
increased, be~$C'$. Divide $C'$~into $19$~equal parts and
let~$C'$, increased by its $19$th~part, be~$C''$. Now, since
we add more and more each time to~$C$, in forming~$C'$,
$C''$, etc, we shall in time exceed~$A$. Let this have
been done, and let $D$ be the line so obtained, which
is greater than~$A$. Observe now that $C'$~contains $19$,
and $C''$, $20$~of the same parts, whence $C'$~is made by
dividing~$C''$ into $20$~parts and removing one of them.
The same of all the rest. Therefore we may return
from $D$ to~$C$ by dividing~$D$ into $20$~parts, removing
one of them, and repeating the process continually.
But $C$~is less than~$B$ by hypothesis. If then we can,
by this process, reduce~$D$ below~$B$, still more can we
do so with~$A$, which is less than~$D$, by the same
method.
This depends on the obvious truth, that if, at the
end of any number of subtractions ($D$~being taken),
\PageSep{249}
we have left~$\dfrac{p}{q}\, D$, at the end of the same number of
subtractions ($A$~being taken), we shall have~$\dfrac{p}{q}\, A$, since
the method pursued in both cases is the same. But
since $A$~is less than~$D$, $\dfrac{p}{q}\, A$~is less than~$\dfrac{p}{q}\, D$, which becomes
equal to~$C$, therefore $\dfrac{p}{q}\, A$ becomes less than~$C$.\footnote
{Algebraically, let $a$~be the given line, and let $\dfrac{1}{m}$th~part of the remainder
be removed at every subtraction. The first quantity taken away is~$\dfrac{a}{m}$ and the
remainder $a - \dfrac{a}{m}$ or $a\Bigl(1 - \dfrac{1}{m}\Bigr)$, whence the second quantity removed is
$\dfrac{a}{m}\Bigl(1 - \dfrac{1}{m}\Bigr)$, and the remainder $\Bigl(1 - \dfrac{a}{m}\Bigr)\Bigl(1 - \dfrac{1}{m}\Bigr)$ or $a\Bigl(1 - \dfrac{1}{m}\Bigr)^{2}$.
Similarly, the $n$th~remainder is $a\Bigl(1 - \dfrac{1}{m}\Bigr)^n$. Now, since $1 - \dfrac{1}{m}$ is less
than unity, its powers decrease, and a power of so great an index may be
taken as to be less than any given quantity.}
We now resume the isosceles right-angled triangle.
The lines $BC$ and~$AB$, which were there shown to
have no common measure, are called \emph{incommensurable}
quantities, and to their existence the theory of proportion
owes its difficulties. We can nevertheless
show that $A$ and~$B$ being incommensurable, a line can
be found as near to~$B$ as we please, either greater or
less, which is commensurable with~$A$. Let $D$~be any
line taken at pleasure, and therefore as small as we
please. Divide~$A$ into two equal parts, each of those
parts into two equal parts, and so on. We shall thus
at last find a part of~$A$ which is less than~$D$. Let this
part be~$E$, and let it be contained $m$~times in~$A$. In
the series $E$, $2E$, $3E$, etc., we shall arrive at last at
two consecutive terms, $pE$~and~$(p + 1)E$ of which the
first is less, and the second greater than~$B$. Neither of
these differs from~$B$ by so much as~$E$; still less by so
much as~$D$; and both $pE$ and $(p + 1)E$ are commensurable
\PageSep{250}
with~$A$, that is with~$mE$, since $E$~is a common
measure of both. If therefore $A$~and~$B$ are incommensurable,
a third magnitude can be found, either greater
or less than~$B$, differing from~$B$ by less than a given
quantity, which magnitude shall be commensurable
with~$A$.
We have seen that when $A$~and~$B$ are incommensurable,
there are no whole values of $m$~and~$n$, which
will satisfy the equation $mA - nB = 0$; nevertheless,
we can prove that values of $m$~and~$n$ can be found
which will make $mA - nB$ less than any given magnitude~$C$,
of the same kind, how small soever it may be.
Suppose, that for certain values of $m$~and~$n$,\footnote
{It is necessary here to observe, that in speaking of the expression $mA - nB$
we more frequently refer to its form than to any actual value of it, derived
from supposing $m$~and~$n$ to have certain known values. When we say that
$mA - nB$ can be made smaller than~$C$, we mean that some values can be
given to $m$~and~$n$ such that $mA - nB < C$, or that \emph{some} multiple of~$B$ subtracted
from some multiple of~$A$ is less than~$C$. The following expressions are all of
the same form, viz., that of some multiple of~$B$ subtracted from some multiple
of~$A$:
\begin{align*}
mA &- nB\Add{,} \\
mpA &- (np + 1)B\Add{,} \\
2mA &- 4mB,\ \etc.,\ \etc.
\end{align*}}
we find
$mA - nB = E$, and let the first multiple of~$E$, which
is greater than~$B$, be~$pE$, so that $pE = B + E'$ where
$E'$~is less than~$E$, for were it greater, $(p - 1)E$~or
$pE - E$, which is $B + (E' - E)$, would be greater
than~$B$, which is against the supposition.
The equation $mA - nB = E$ gives
\[
pmA - pnB = pE = B + E',
\]
whence
\[
pmA - (pn + 1)B = E'.
\]
\PageSep{251}
Let
\[
pm = m' \quad\text{and}\quad
pn + 1 = n',
\]
whence
\[
m'A - n'B = E'.
\]
We have therefore found a difference of multiples
which is less than~$E$. Let $p'E'$~be the first multiple
of~$E'$ which is greater than~$B$, where $p'$~must be \emph{at
least as great as~$p$}, since $E$~being greater than~$E'$, it
cannot take \emph{more}\footnote
{It may require \emph{as many}. Thus it requires as many of~$7$ as of~$8$ to exceed~$33$,
though $7$~is less than~$8$.}
of~$E$ than of~$E'$ to exceed~$B$. Let
\[
p'E' = B + E'',
\]
then, as before,
\[
m'p'A - (n'p' + 1)B = E'',
\]
or
\[
m''A - n''B = E'';
\]
we have therefore still further diminished the difference
of the multiples; and the process may be repeated
any number of times; it only remains to show
that the diminution may proceed to any extent.
This will appear superfluous to the beginner, who
will probably imagine that a quantity diminished at
every step, must, by continuing the number of steps,
at last become as small as we please. Nevertheless
if any number, as~$10$, be taken and its square root extracted,
and the square root of that square root, and
so on, the result will not be so small as unity, although
ten million of square roots should have been extracted.
Here is a case of continual diminution, in which the
\index{Diminution, not necessarily without limit}%
diminution is not \emph{without limit}. Again, from the point~$D$
\PageSep{252}
in the line~$AB$ draw~$DE$, making an angle with~$AB$
less than half a right angle. Draw $BE$ perpendicular
%[** TN: Moved up from end of paragraph]
\Figure{252}% ** Fig. 10.
to~$AB$, and take $BC = BE$. Draw $CF$ perpendicular
to~$AB$, and take $CC' = CF$, and so on. The
points $C$, $C'$, $C''$, etc., will always be further from~$A$
than $D$~is; and all the lines $AC$, $AC'$, $AC''$, etc.,
though diminished at every step, will always remain
greater than~$AD$. Some such species of diminution,
for anything yet proved to the contrary, may take
place in~$mA - nB$.
To compare the quantities $E$, $E'$, etc., we have
the equations
\begin{align*}
\PadTo{p''E''}{pE} &= B + E'\Add{,} \\
\PadTo{p''E''}{p'E'} &= B + E''\Add{,} \\
p''E'' &= B + E'''\Add{,} \\
\etc.\Add{,} & \qquad \etc.
\end{align*}
The numbers $p$, $p'$, $p''$, etc., do not diminish; the
lines $E$, $E'$, $E''$, etc., diminish at every step. If then
we can show that $p$, $p'$, etc., can only remain the same
for a finite number of steps, and must then increase,
and after the increase can only remain the same for
another finite number of steps, and then must increase
again, and so on, we show that the process can be
continued, until one of them is as great as we please;
\PageSep{253}
let this be~$p^{(z)}$, where $z$~is not an exponent, but marks
the number which our notation will have reached, and
indicates the $(z + 1)$\Ord{th}~step of the process. Let $E^{(z)}$~be
the corresponding remainder from the former step.
Then, since $p^{(z)}E^{(z)}$~is the first multiple of~$E^{(z)}$, which
exceeds the given quantity~$B$, if $p^{(z)}$~can be as great as
we please, $E^{(z)}$~can be as small as we please. To show
that $p^{(z)}$~can be as great as we please, observe, that $p$,
$p'$, $p''$, etc., must remain the same, or increase, since,
as appears from their method of formation, they cannot
diminish. Let them remain the same for some
steps, that is, let $p = p' = p''$, etc. The equations become
\begin{alignat*}{2}
&pE &&= B + E'\Add{,} \\
&pE' &&= B + E''\Add{,} \\
&pE'' &&= B + E'''\Add{,} \\
&\etc.\Add{,} && \qquad \etc.
\end{alignat*}
Then by subtraction,
\begin{alignat*}{4}
&E' &&- E'' &&= p(E - E')\Add{,} \\
&E'' &&- E''' &&= p(E' - E'') &&= pp(E - E')\Add{,} \\
&E''' &&- E'''' &&= p(E'' - E''') &&= ppp(E - E')\Add{,} \\
&&&\quad \etc.\Add{,} &&&&\qquad \etc.
\end{alignat*}
Now,
{\small
\begin{alignat*}{4}
&E &&- E'' &&= E - E' + E' - E''
&&= (E - E')(1 + p)\Add{,} \\
&E &&- E''' &&= E - E' + E' - E'' + E'' - E'''
&&= (E - E')(1 + p + p^{2})\Add{,} \\
& && &&\quad \etc.\Add{,} \qquad\qquad \etc.\Add{,} &&\qquad \etc.
\end{alignat*}}%
Generally,
\begin{align*}
E - E^{(w)} &= E - E' + E' - E'' + \dots + E^{(w-1)} - E^{(w)} \\
&= (E - E')(1 + p + p^{2} + \dots + p^{w-1}),
\end{align*}
\PageSep{254}
which is derived from $w$~steps of the process. Now,
if this can go on \textit{ad infinitum}, it can go on until
$1 + p + p^{2} + \dots + p^{w-1}$ is as great as we please; for,
since $p$~is not less than unity, the continual addition
of its powers will, in time, give a sum exceeding any
\emph{given} number. This is absurd, from the step at which
$1 + p + p^{2} + \dots + p^{w-1}$ becomes greater than the number
of times which $E - E'$ is contained in~$E$; for,
from the above equation, $E - E'$~is contained in
$E - E^{(w)}$, $1 + p + p^{2} + \dots + p^{w-1}$ times; and it is contradictory
to suppose that $E - E'$ should be contained
in $E - E^{(w)}$ more times than it is contained in~$E$.
To take an example: suppose that $B$~is $55$~feet,
and $E$~is $54$~feet; the first equation is
\[
2 × 54^{f} = 55^{f} + 53^{f},
\]
where $E' = 53^{f}$ and $E - E' = 1^{f}$, and is contained in~$E$
$54$~times. If, then, we continue the process, $2$~cannot
maintain its present place through so many steps
of the process as will, if the same number of terms be
taken, give $1 + 2 + 2^{2} + 2^{3} +\Chg{,}{} \etc.$, greater than~$54$;
that is, it cannot be the same for \emph{six} steps. And we
find, on actually performing the operations,
\begin{align*}
2 × 54^{f} &= 55 + 53^{f}\Add{,} \\
2 × 53^{f} &= 55 + 51^{f}\Add{,} \displaybreak[0]\\
2 × 51^{f} &= 55 + 47^{f}\Add{,} \displaybreak[0]\\
2 × 47^{f} &= 55 + 39^{f}\Add{,} \displaybreak[0]\\
2 × 39^{f} &= 55 + 23^{f}\Add{,} \\
3 × 23^{f} &= 55 + 14^{f}\Add{.}
\end{align*}
We do not say that $p$, $p'$, etc., \emph{will} remain the
\PageSep{255}
same until $1 + p + p^{2} + \dots$ would be greater than the
number of times which $E$ contains $E - E'$, but only
that they cannot remain the same longer. By repetition
of the same process, we can show that a further
and further increase must take place, and so on until
we have attained a quantity greater than any given
one. And it has already been shown to be a consequence
of this, that $mA - nB$ can be diminished to
any extent we please. Similarly it may be shown that
when $A$~and~$B$ are incommensurable, $mA - nB$ may
be brought as near as we please to any other quantity~$C$,
of the same kind as $A$~and~$B$, so as not to differ
from~$C$ by so much as a given quantity~$E$. For let $m$~and~$n$
be taken, by the last case, so that $mA - nB$ may
be less than~$E$, and let $mA - nB$, in this case, be
equal to~$E'$. Let $C$~lie between $pE'$ and $(p + 1)E'$,
neither of which can differ from~$C$ by so much as~$E'$,
and therefore not by so much as~$E$. Then since
\[
mA - nB = E';
\]
therefore
\[
pmA - pnB = pE',
\]
and
\[
(p + 1)mA - (p + 1)nB = (p + 1)E'.
\]
Both which last expressions differ from~$C$ by a quantity
less than~$E$, the first being less and the second
greater than~$C$, and both are of the \emph{form} $mA - nB$, $m$~and~$n$
being changed for other numbers.
The common ideas of proportion are grounded
entirely upon the false notion that all quantities of
the same sort are commensurable. That the supposition
is practically correct, if there are any limits to
\PageSep{256}
the senses, may be shown, for let any quantity be rejected
as imperceptible, then since a quantity can be
found as near to~$B$ as we please, which is commensurable
with~$A$, the difference between $B$~and its approximate
commensurable magnitude, may be reduced below
the limits of perceptible quantity. Nevertheless,
inaccuracy to some extent must infest all \emph{general} conclusions
drawn from the supposition that $A$~and~$B$
being two magnitudes, whole numbers, $m$~and~$n$, can
\emph{always} be found such that $mA - nB = 0$. We have
shown that this can be brought as near to the truth as
we please, since $mA - nB$ can be made as small as we
please. This, however, is not a perfect answer, at
least it wants the unanswerable force of all the preceding
reasonings in geometry. A definition of proportion
should therefore be substituted, which, while
it reduces itself, in the case of commensurable quantities
to the one already given, is equally applicable
to the case of incommensurables. We proceed to examine
the definition already given with a view to this
object.
Resume the equations
\begin{alignat*}{2}
mA - nB &= 0,\quad\text{or}\quad & A &= \frac{n}{m}\, B\Add{,} \\
mP - nQ &= 0,\quad\text{or}\quad & P &= \frac{n}{m}\, Q\Add{.}
\end{alignat*}
If we take any other expression of the same sort
$\dfrac{n'}{m'}\, B$ and $\dfrac{n'}{m'}\, Q$, it is plain that, according as the arithmetical
fraction~$\dfrac{n}{m}$ is greater than, equal to, or less
\PageSep{257}
than~$\dfrac{n'}{m'}$, so will $\dfrac{n}{m}\, B$~be greater than, equal to, or less
than $\dfrac{n'}{m'}\, B$, and the same of $\dfrac{n}{m}\, Q$ and~$\dfrac{n'}{m'}\, Q$. Let the
symbol
\[
\GLT{x}{y}{z}{w}
\]
be the abbreviation of the following sentence: ``when
$x$~is greater than~$y$, $z$~is greater than~$w$; when $x$~is
equal to~$y$, $z$~is equal to~$w$; when $x$~is less than~$y$, $z$~is
less than~$w$.'' The following conclusions will be evident:
If
\[
\GLT{a}{b}{c}{d} \quad\text{and}\quad \GLT{a}{b}{e}{f\Add{,}}
\]
\Chg{Then}{then}
\[
\GLT{c}{d}{e}{f\Add{.}}
\Tag{(1)}
\]
And from the first of these alone it follows that
\[
\GLT{ma}{mb}{nc}{nd\Add{.}}
\Tag{(2)}
\]
We have just noticed the following:
\[
\GLT{\frac{n}{m}}{\frac{n'}{m'}}
{\Strut[24pt]\frac{n}{m}\, B}{\Strut[24pt]\frac{n'}{m'}\, B}
\quad\text{and}\quad
\GLT{\frac{n}{m}}{\frac{n'}{m'}}
{\Strut[24pt]\frac{n}{m}\, Q}{\Strut[24pt]\frac{n'}{m'}\, Q\Add{.}}
\]
Therefore~(1)
\[
\GLT{\frac{n}{m}\, B}{\frac{n'}{m'}\, B}
{\Strut[24pt]\frac{n}{m}\, Q}{\Strut[24pt]\frac{n'}{m'}\, Q}
\quad\text{or}\quad
\GLT{A}{\frac{n'}{m'}\, B}{P}{\Strut[24pt]\frac{n'}{m'}\, Q\Add{.}}
\]
Therefore (2)
\[
\GLT{m'A}{n'B}{m'P}{n'Q\Add{.}}
\]
\PageSep{258}
Or, if four magnitudes are proportional, according to
the common notion, it follows that the same multiples
of the first and third being taken, and also of the second
and fourth, the multiple of the first is greater
than, equal to, or less than, that of the second, according
as that of the third is greater than, equal to,
or less than, that of the fourth. This property\footnote
{It would be expressed algebraically by saying that if $mA - nB$ and
$mP - nQ$ are nothing for the same values of $m$~and~$n$, they are either both
positive or both negative, for every other value of $m$~and~$n$.}
necessarily follows from the equations
\begin{alignat*}{2}
mA &- nB &&= 0\Add{,} \\
mP &- nQ &&= 0\Add{,}
\end{alignat*}
but it does not therefore follow that the equations are
necessary consequences of the property, since the latter
may possibly be true of incommensurable quantities,
of which, by definition, the former is not. The
existence of this property is Euclid's definition of proportion:
\index{Euclid!theory@{his theory of proportion|EtSeq}}%
he says, let four magnitudes, two and two,
of the same kind, \emph{be called proportional}, when, if equi-multiples
be taken of the first and third, etc., repeating
the property just enunciated. What is lost and
gained by adopting Euclid's definition may be very
simply stated; the gain is an entire freedom from all
the difficulties of incommensurable quantities, and
even from the necessity of inquiring into the fact of
their existence, and the removal of the inaccuracy attending
the supposition that, of two quantities of the
same kind, each is a determinate arithmetical fraction
of the other; on the other hand, there is no obvious
\PageSep{259}
connexion between Euclid's definition and the ordinary
and well-established ideas of proportion; the
definition itself is made to involve the idea of infinity,
since \emph{all possible multiples} of the four quantities enter
into it; and lastly, the very existence of the four
quantities, called proportional, is matter for subsequent
demonstration, since to a beginner it cannot
but appear very unlikely that there are any magnitudes
which satisfy the definition. The last objection
is not very strong, since the learner could read the
first proposition of the sixth book immediately after
the definition, and would thereby be convinced of the
existence of proportionals; the rest may be removed
by showing another definition, more in consonance
with common ideas, and demonstrating that, if four
magnitudes fall under either of these definitions, they
fall under the other also. The definition which we
propose is as follows: ``Four magnitudes, $A$,~$B$, $P$,
and~$Q$, of which $B$~is of the same kind as~$A$, and $Q$
as~$P$, are said to be proportional, if magnitudes $B + C$
and $Q + R$ can be found \emph{as near as we please} to $B$ and~$Q$,
so that $A$, $B + C$, $P$ and~$Q + R$, are proportional
according to the common notion, that is, if whole
numbers $m$~and~$n$ can satisfy the equations
\begin{alignat*}{2}
mA &- n(B + C) &&= 0\Add{,} \\
mP &- n(Q + R) &&= 0.\text{''}
\end{alignat*}
We have now to show that Euclid's definition follows
from the one just given, and also that the last
follows from Euclid's, that is, if there are four magnitudes
\PageSep{260}
which fall under either definition, they fall under
the other also. Let us first suppose that Euclid's
definition is true of $A$,~$B$, $P$, and~$Q$, so that
\[
\GLT{mA}{nB}{mP}{nQ\Add{.}}
\]
This being true, it will follow that we can take $m$~and~$n$,
so as not only to make $mA - nB$ less than a given
magnitude~$E$, which may be as small as we please,
but also so that $mP - nQ$ shall at the same time be
less than a given magnitude~$F$, however small this
last may be. For if not, while $m$~and~$n$ are so taken
as to make $mA - nB$ less than~$E$ (which it has been
proved can be done, however small $E$~may be) suppose,
if possible, that the same values of $m$~and~$n$ will
never make $mP - nQ$ less than some certain quantity~$F$,
and let $E$~be the first multiple of~$F$ which exceeds~$Q$,
and also let $E$~be taken so small that $pE$~shall be
less than~$B$, still more then shall $p(mA - nB)$, or
$pmA - pnB$ be less than~$B$. But since $pF$~is greater
than~$Q$, and $mP - nQ$ is by hypothesis greater than~$F$,
still more shall $mpP - npQ$ be greater than~$Q$.
We have then, if our last supposition be correct, some
value of $mp$ and~$n$p, for which
\[
\text{$mpA - npB$ is less than $B$,}
\]
while
\[
\text{$mpP - npQ$ is greater than $Q$,}
\]
or
\begin{gather*}
\text{$mpA$ is less than $(np + 1)B$,} \\
\text{$mpP$ is greater than $(np + 1)Q$,}
\end{gather*}
\PageSep{261}
which is contrary to our first hypothesis respecting
$A$,~$B$, $P$, and~$Q$, that hypothesis being Euclid's definition
of proportion, from which if
\begin{gather*}
\text{$mpA$ is less than $(np + 1)B$\Add{,}} \\
\text{$mpP$ is \emph{less} than $(np + 1)Q$.}
\end{gather*}
We must therefore conclude that if the four quantities
$A$,~$B$, $P$, and~$Q$ satisfy Euclid's definition of proportion,
then $m$~and~$n$ may be so taken that $mA - nB$ and
$mP - nQ$ shall be as small as we please.
Let
\begin{alignat*}{4}
mA &- nB &&= E \quad&\text{and}\quad& E &&= nC\Add{,} \\
mP &- nQ &&= F &\text{and}\quad& F &&= nR.
\end{alignat*}
Then
\begin{alignat*}{2}
mA &- n(B + C) &&= 0\Add{,} \\
mP &- n(Q + R) &&= 0,
\end{alignat*}
and since $E$ and $F$ can, by properly assuming $m$~and~$n$,
be made as small as we please, much more can the
same be done with $C$~and~$R$, consequently we can produce
$B + C$ and $Q + R$ as near as we please to $B$~and~$Q$,
and proportional to $A$~and~$P$, according to the
common arithmetical notion. In the same way it may
be proved, that on the same hypothesis $B - C$ and
$Q - R$ can be found as near to $B$~and~$Q$ as we please,
and so that $A$, $B - C$, $P$ and $Q - R$ are proportional
according to the ordinary notion. It only remains to
show that if the last-mentioned property be assumed,
Euclid's definition of proportion will follow from it.
That is, if quantities can be exhibited as near to $P$
and~$Q$ as we please, which are proportional to $A$ and~$B$,
according to the ordinary notion, it follows that
\PageSep{262}
\[
\GLT{mA}{nB}{mP}{nQ.}
\]
For let $B + C$ and $Q + R$ be two quantities, such that
\begin{alignat*}{2}
fA &- g(B + C) &&= 0\Add{,} \\
fP &- g(Q + R) &&= 0,
\end{alignat*}
in which, by the hypothesis, $f$~and~$g$ can be so taken
that $C$ and~$R$ are as small as we please. We have already
shown that in this case ($m$~and~$n$ being any
numbers whatever) $mA$~is never greater or less than
$n(B + C)$, without $mP$~being at the same time the
same with regard to~$n(Q + R)$. That is, if
\[
\text{$mA$ is greater than $nB + nC$,}
\]
then
\[
\text{$mP$ is greater than $nQ + nR$.}
\]
Take some \emph{given}\footnote
{It is very necessary to recollect that the relations just expressed are
true for every value of $m$~and~$n$; and therefore true for any particular case.
In this investigation $f$~and~$g$ may both be very great in order that $C$~and~$R$
may be sufficiently small, and we must suppose them to vary with the values
we give to $C$~and~$R$, or rather the limits which we assign to them; but $m$~and~$n$
are \emph{given}.}
values for $m$~and~$n$, fulfilling the
first condition; then, since $C$~and~$R$ may be as small
as we please, the same is true of $nC$ and~$nR$; if then
\begin{gather*}
\text{$mA$ is greater than $nB$\Add{,}} \\
\text{$mP$ is greater than $nQ$.}
\end{gather*}
For if not, let $mA = nB + x$, while $mP = nQ - y$, $x$~and~$y$
being some definite magnitudes. Then if
\begin{gather*}
nB + x > nB + nC\Add{,} \\
nQ - y > nQ + nR,
\end{gather*}
which last equation is evidently impossible; therefore
if $mA > nB$, $mP > nQ$. In the same way it may be
\PageSep{263}
proved that if $mA < nB$, $mP < nQ$, etc., so that Euclid's
definition is shown to be a necessary consequence
of the one proposed.
\index{Exhaustions, method of|(}%
The definition of proportion which we have here
given, and the methods by which we have established
its identity with the one in use, bear a close analogy
to the process used by the ancients, and denominated
by the moderns the \emph{method of exhaustions}. We have
seen that the common definition of proportion fails in
certain cases where the magnitudes are what we have
called incommensurable, but at the same time we
have shown that though in this case we can never
take $m$~and~$n$, so that $mA = nB$, or $mA - nB = 0$, we
can nevertheless find $m$~and~$n$, so that $mA$~shall differ
from~$nB$ by a quantity less than any which we please
to assign. We therefore extend the definition of the
word proportion, and make it embrace not only those
magnitudes which fulfil a given condition, but also
others, of which it is impossible that they should fulfil
that condition, provided always, that whatever magnitudes
we call by the name of proportionals, they must
be such as to admit of other magnitudes being taken
as near as we please to the first, which are proportional,
according to the common arithmetical notion.
It is on the same principle that in algebra we admit
the existence of such a quantity as~$\sqrt{2}$, and use it in
the same manner as a definite fraction, although there
is no such fraction in reality as, multiplied by itself,
will give~$2$ as the product. But, however small a
\PageSep{264}
quantity we may name, we can assign a fraction which,
multiplied by itself, shall differ less from~$2$ than that
quantity.
Having established the properties of rectilinear
figures, as far as their proportions are concerned, it is
necessary to ascertain the properties of curvilinear
figures in this respect. And here occurs a difficulty
of the same kind as that which met us at the outset,
for no rectilinear figure, how small soever its sides
may be, or how great soever their number, can be
called curvilinear. Nevertheless, it may be shown
that in every curve a rectilinear figure may be inscribed,
whose area and perimeter shall differ from
the area and perimeter of the curve by magnitudes
less than any assigned magnitudes. The circle is the
only curve whose properties are considered in elementary
geometry, and the proposition in question is discussed
in all standard treatises on geometry. Indeed,
for this or any other curve the proposition is almost
self-evident. This being granted, the properties of
curvilinear figures are established by help of the following
theorem.
If $A$, $B$, $C$, and~$D$ are always proportional, and of
these, if $C$~and~$D$ may be made as near as we please to
$P$~and~$Q$, than which they are always both greater or
both less, then $A$,~$B$, $P$, and~$Q$ are proportional.
Let $C = P + P'$, and $D = Q + Q'$, where by hypothesis
$P'$~and~$Q'$ may be made as small as we please,
and $A$,~$B$, $P + P'$, and $Q + Q'$ are proportionals. If
\PageSep{265}
$A$,~$B$, $P$, and~$Q$ are not proportionals, let $P$ and $Q + R$
be proportional to $A$~and~$B$. Then, since $A$~and~$B$
are proportional to $P + P'$ and $Q + Q'$, and also to $P$
and $Q + R$, therefore
\[
P + P' : Q + Q' :: P : Q + R
\]
in which all the magnitudes are of the same kind.
Now let $P'$~and~$Q'$ be so taken that $Q'$~is less than~$R$,
which may be done, since by hypothesis $Q'$~can be as
small as we please. Hence $Q + Q'$ is less than $Q + R$,
and therefore $P + P'$ is less than~$P$, which is absurd.
In the same way it may be proved that $P$~is not to
$Q - R$ in the proportion of $A$~to~$B$, and consequently
$P$~is to~$Q$ in the proportion of $A$~to~$B$. This theorem,
with those which prove that the surfaces, solidities,
areas, and lengths, of curve lines and surfaces, may
be represented as nearly as we please by the surfaces,
etc., of rectilinear figures and solids, form the method
of exhaustions.\footnote
{For a classical example, see Prop.~II. of the twelfth book of Euclid
\index{Euclid}%
(Simson's edition). Consult also Beman and Smith's \Title{Plane and Solid Geometry}
\index{Beman, W. W.}%
\index{Smith, D. E.}%
(Ginn \&~Co., Boston), pp.~144--145, and~190.---\Ed.}
In this method are the first germs of
that theory which, under the name of Fluxions, or the
\index{Fluxions}%
Differential Calculus, contains the principles of all
\index{Differential calculus}%
the methods of investigation now employed, whether
in pure or mixed mathematics.
\index{Exhaustions, method of|)}%
\index{Proportions!theory of|)}%
\PageSep{266}
%[** TN: Ties force line breaks to match the original w/o interfering with ToC]
\Chapter[On the Application of Algebra, etc.]
{XVII.}{Application of Algebra to the Measurement
of~Lines, Angles, Proportion~of
Figures,~and~Surfaces.}
\index{Algebra!applied to the measurement of lines, angles, proportion of figures and surfaces|(}%
\index{Measures@{Measures|EtSeq}}%
\index{Measurement, of lines, angles, proportion of figures, and surfaces|(}%
\First{We} have already defined a measure, and have noticed
several instances of magnitudes of one
kind being measured by those of another. But the
most useful measure, and that with which we are most
familiar, is number. We express one line by the number
of times which another line is repeated in it, or if
the second is not exactly contained in the first, by the
greatest number of the second contained in the first,
together with the fraction of the second, which will
complete the first. Thus, suppose the line~$A$ contains~$B$
$m$~times, with a remainder which can be formed by
dividing~$B$ into $q$~parts, and taking $p$~of them. Then
$B$~is to~$A$ in the proportion of $1$~to~$m + \dfrac{p}{q}$, or as $q$~to
$mq + p$, and if $B$~be a fixed line, which is used for the
comparison of all lines whatsoever, then the line~$A$ is
$m + \dfrac{p}{q}$, or $\dfrac{mq + p}{q}$, if it be understood that for every
unit in~$m$, $B$~is to be taken, and also that for $\dfrac{p}{q}$ the
\PageSep{267}
same fraction of~$B$ is to be taken that $\dfrac{p}{q}$ is of unity.
In this case $B$~is called the \emph{linear unit}.
\index{Linear unit}%
\index{Approximations}%
\index{Continued fractions|(}%
\index{Fractions!continued|(}%
\index{Greatest common measure@{Greatest common measure|EtSeq}}%
But here we suppose that a line~$B$ being taken,
the ratio of any other line~$A$ to~$B$ can be expressed by
that of the whole numbers $mq + p$ to~$q$, which we have
shown in some cases to be impossible. If we take
one of these cases, $mA - nB$, though it can never be
made equal to nothing, can be made as small as we
please, by properly assuming $m$~and~$n$. Let $mA - nB = E$,
then $A = \dfrac{n}{m}\, B + \dfrac{E}{m}$, and since $\dfrac{E}{m}$~can be made as
small as we please, $A$~can be represented as nearly as
we please by a fraction~$\dfrac{n}{m}$, where $B$~is the linear unit.
Hence, in practice an approximation may be found to
the value of~$A$, sufficient for any purpose whatever,
in the following manner, which will be easily understood
by the student who has a tolerable facility in
performing the operations of algebra. Let
\begin{gather*}
\text{$A$ contain $B$, $p$~times with a remainder~$P$,} \\
\text{$B$ contain $P$, $q$~times with a remainder~$Q$,} \\
\text{$P$ contain $Q$, $r$~times with a remainder~$R$,}
\end{gather*}
and so on. If the two magnitudes are commensurable,
this operation will end by one of the remainders
becoming nothing. For, let $A$~and~$B$ have a common
measure~$E$, then $P$~has the same measure, for $P$~is
$A - pB$, of which both $A$ and~$pB$ contain~$E$ an exact
number of times. Again, because $B$~and~$P$ contain
the common measure~$E$, $Q$~has the same measure,
and so on. All the remainders are therefore multiples
\PageSep{268}
of~$E$, and if $E$~be the linear unit, are represented by
whole numbers. Now, if a whole number be continually
diminished by a whole number, it must, if the
operation can be continued without end, eventually
become nothing. If, therefore, the remainder never
disappears, it is a sign that the magnitudes $A$~and~$B$
are incommensurable. Nevertheless, approximate
whole numbers can be found whose ratio is as near as
we please to the ratio of $A$~and~$B$.
From the suppositions above mentioned it appears
that
\begin{alignat*}{2}
A &= pB &&+ P\Add{,}\footnotemark
\Tag{(a)} \\
B &= qP &&+ Q\Add{,}
\Tag{(b)} \\
P &= rQ &&+ R\Add{,}
\Tag{(c)} \\
Q &= sR &&+ S\Add{,}
\Tag{(d)} \\
R &= tS &&+ T\Add{,}
\Tag{(e)} \\
&\etc., &&\etc.
\end{alignat*}
\footnotetext{Throughout these investigations the capital letters represent the lines
themselves, and not the numbers of units, which represent them, while the
small letters are whole numbers, as in the last chapter.}%
Substitute in~\Eq{(b)} the value of~$P$ derived from~\Eq{(a)}, find~$Q$
from the result, and substitute the values of $P$ and~$Q$
in~\Eq{(c)}; find a value of~$R$ from the result, and substitute
the values of $Q$~and~$R$ in~\Eq{(d)}, and so on, which
give the following series of equations:
\begin{alignat*}{2}
A &={}& pB + P\Add{,} \\
qA &={}& (pq + 1)B - Q\Add{,} \\
(qr + 1)A &={}& (pqr + p + r)B + R\Add{,} \\
(qrs + q + s)A &={}& \Squeeze{(pqrs + ps + rs + pq + 1)B - S\Add{,}} \\
(qrst + qt + st + qr + 1)A &={} \\
&& \llap{$(pqrst + pst + rst + pqt + pqr + p + r + t)B +T\Add{.}$}
\end{alignat*}
\PageSep{269}
{\Loosen On inspection it will be found that the coefficients
\index{Convergent fractions|EtSeq}%
of $A$~and~$B$ in these equations may be formed by a
very simple law. In each a letter is introduced which
was not in the preceding one, and every coefficient is
formed from the two preceding, by multiplying the
one immediately preceding by the new letter, and adding
to the product the one which comes before that.
Thus the third coefficient of~$B$ is $pqr + p + r$; the
new letter is~$r$, and the two preceding coefficients are
$pq + 1$ and~$p$, and $pqr + p + r = (pq + 1)r + p$. The
remainders enter also with signs alternately positive
and negative. Let $x$,~$x'$ and~$x''$ be the $n$\Ord{th}, $(n + 1)$\Ord{th},
and $(n + 2)$\Ord{th} numbers of the series $p$,~$q$,~$r$, etc., and
$X$,~$X'$, and~$X''$ the corresponding remainders. Let
the corresponding equations be}
\begin{alignat*}{3}
&a &&A = b &&B + X\Add{,} \\
&a' &&A = b' &&B - X'\Add{,} \\
&a''&&A = b'' &&B + X''\Add{.}
\end{alignat*}
Here $n$~must be supposed odd, since, were it even,
the first equation would be $aA = bB - X$, as will be
seen by reference to the equations deduced. Hence,
from the law of formation of the coefficients, $x''$~being
the new letter in the last equation,
\begin{alignat*}{2}
a'' &= a'x'' &&+ a\Add{,} \\
b'' &= b'x'' &&+ b.
\end{alignat*}
{\Loosen Eliminate $x''$ from these two, the result of which
is $a''b - ab'' = ab' - a'b$, the first side of which is
\PageSep{270}
the numerator of $\dfrac{b'}{a'} - \dfrac{b''}{a''}$, and the second of $\dfrac{b'}{a'} - \dfrac{b}{a}$.
It appears then that $\dfrac{b'}{a'}$~is either greater than both
$\dfrac{b}{a}$ and $\dfrac{b''}{a''}$ or less than both, since $\dfrac{b'}{a'} - \dfrac{b''}{a''}$ and $\dfrac{b'}{a'} - \dfrac{b}{a}$
will both have the same sign, the numerators being
the same and the denominators positive. It may also
be proved that $\dfrac{b''}{a''}$ lies between $\dfrac{b}{a}$ and $\dfrac{b'}{a'}$ by means of
the following lemma.}
The fraction $\dfrac{m + n}{p + q}$ must lie between $\dfrac{m}{p}$ and~$\dfrac{n}{q}$; for
let $\dfrac{m}{p}$~be the greater of the two last, or $\dfrac{m}{p} > \dfrac{n}{q}$, then
$mq > np$, or $\dfrac{mq}{mp} > \dfrac{np}{mp}$, or $\dfrac{q}{p} > \dfrac{n}{m}$, and $1 + \dfrac{q}{p} > 1 + \dfrac{n}{m}$;
therefore $\dfrac{1 + \frac{n}{m}}{1 + \frac{q}{p}}$ is less than unity, and any fraction
multiplied by this is diminished. But
\[
\frac{m + n}{p + q}\text{ is }
\frac{m}{p} × \frac{1 + \frac{n}{m}}{1 + \frac{q}{p}},
\]
and is therefore less than~$\dfrac{m}{p}$, the greater of the two.
In the same way it may be proved to be greater than~$\dfrac{n}{q}$,
the least of the two.
This being premised, since $\dfrac{b''}{a''} = \dfrac{b'x'' + b}{a'x'' + a'}$, it lies
between $\dfrac{b'x''}{a'x''}$ and~$\dfrac{b}{a}$ or between $\dfrac{b'}{a'}$ and~$\dfrac{b}{a}$.
Call the coefficients of $A$~and~$B$ in the series of
equations, $a_{1}$,~$a_{2}$,~etc., $b_{1}$,~$b_{2}$,~etc., and form the series
of fractions $\dfrac{b_{1}}{a_{1}}$, $\dfrac{b_{2}}{a_{2}}$, $\dfrac{b_{3}}{a_{3}}$, etc. The two first of these
will be $\dfrac{p}{1}$ and~$\dfrac{pq + 1}{q}$, of which the second is the
\PageSep{271}
greater, since it is $p + \dfrac{1}{q}$. Hence by what has been
proved $\dfrac{b_{3}}{a_{3}}$~is less than $\dfrac{b_{2}}{a_{2}}$ and greater than~$\dfrac{b_{1}}{a_{1}}$; and
every fraction is greater or less than the one which
comes before it, according as the number of its equation
is even or odd. Again, as the numerator of the
difference of two successive fractions $\dfrac{a''}{b''}$ and $\dfrac{a'}{b'}$, is the
same as that of $\dfrac{a'}{b'}$ and~$\dfrac{a}{b}$, whatever the numerator of
the first difference is, the same must be that of the
second, third, etc., and of all the rest. But the numerator
of the difference of $\dfrac{p}{1}$ and $\dfrac{pq + 1}{q}$ is~$1$; therefore
either $ab' - a'b$, or $a'b - ab'$, is~$1$ according as $\dfrac{b'}{a'}$
or $\dfrac{b}{a}$~is the greater of the two, that is according as $n$~is
odd or even.\footnote
{We might say that $ab' - a'b$ is alternately $+1$~and~$-1$; but we wish to
avoid the use of the isolated negative sign.}
Now since the $n$\Ord{th} and $(n + 1)$\Ord{th} equations,
$n$~being odd, are
\begin{alignat*}{3}
&a &&A = b &&B + X \\
\text{and}\quad
&a'&&A = b'&& B - X';
\end{alignat*}
by eliminating~$A$ we have
\begin{align*}
(ab' - a'b)B &= a'X + aX' \\
\text{or}\quad B &= a'X + aX'
\end{align*}
{\Loosen since $ab' - a'b = 1$; and since the remainders decrease
and the coefficients increase, $a' > a$ and $X > X'$,
whence $2aX < a'X + aX'$, or $2aX' < B$ and $X' < \dfrac{B}{2a}$;
the remainder therefore which comes in the $(n + 1)$\Ord{th}
equation is less than the part of~$B$ arising from dividing
it into twice as many equal parts as there are
\PageSep{272}
units in the $n$\Ord{th}~coefficient of~$A$; and as this number
of units may increase to any amount whatever, by
carrying the process far enough, $\dfrac{B}{2a}$~may be made as
small as we please, and \textit{à~fortiori}, the remainders may
be made as small as we please.}
The same theorem may be proved in a similar
way, if we begin at an even step of the process. Resuming
the equations
\begin{alignat*}{3}
&a &&A = b &&B + X\Add{,} \\
&a' &&A = b' &&B - X'\Add{,} \\
&a''&&A = b'' &&B + X''\Add{,}
\end{alignat*}
we obtain from the second,
\[
A = \frac{b'}{a'}\, B - \frac{X'}{a'};
\]
and since $X' < \dfrac{B}{2a}$, $\dfrac{X'}{a'} < \dfrac{B}{2aa'}$, or if $B$~be taken as
the linear unit, $\dfrac{b'}{a'}$~will express the line~$A$ with an error
less than~$\dfrac{1}{2aa'}$, which last may be made as small as
we please by continuing the process.
It is also evident that $\dfrac{b}{a}$~is too small, while $\dfrac{b'}{a'}$~is
too great; and since $X$ and~$X'$ are less than~$B$,
$aA < bB + B$, or $\dfrac{b + 1}{a}$~is too great, while $a'A > b'B - B$,
or $\dfrac{b' - 1}{a'}$, is too small. Again, $A - \dfrac{b}{a}\, B = \dfrac{X}{A}$ and
$\dfrac{b'}{a'}\, B - A = \dfrac{X'}{a'}$. Now $X' < X$ and $a' > a$; whence
$\dfrac{X'}{a'} < \dfrac{X}{a}$; that is, $\dfrac{b'}{a'}\, B$ exceeds~$A$ by a less quantity
than $\dfrac{b}{a}\, B$ falls short of it, so that $\dfrac{b'}{a'}$~is a nearer representation
of~$A$ than~$\dfrac{b}{a}$, though on a different side of it.
\PageSep{273}
We have thus shown how to find the representation
of a line by means of a linear unit, which is incommensurable
with it, to any degree of nearness
which we please. This, though little used in practice,
is necessary to the theory; and the student will
see that the method here followed is nearly the same
as that of continued fractions in algebra.\footnote
{See Lagrange's \Title{Elementary Mathematics} (Chicago, 1898), p.~2 et~seq.---\Ed.}
\index{Continued fractions|)}%
\index{Fractions!continued|)}%
We now come to the measurement of an angle;
\index{Angle!measure of|EtSeq}%
\index{Sexagesimal system of angular measurement}%
and here it must be observed that there are two distinct
measures employed, one exclusively in theory,
and one in practice. The latter is the well-known division
of the right angle into $90$~equal parts, each of
which is one degree; that of the degree into $60$~equal
parts, each of which is one minute; and of the minute
into $60$~parts, each of which is one second. On these
it is unnecessary to enlarge, as this division is perfectly
arbitrary, and no reason can be assigned, as far as theory
is concerned, for conceiving the right angle to be
so divided. But it is far otherwise with the measure
which we come to consider, to which we shall be naturally
led by the theorems relating to the circle. Assume
any angle, $AOB$, as the angular unit, and any
other angle, $AOC$ (\Fig{11}). Let $r$~be the number\footnote
{It must be recollected that the word number means both \emph{whole} and
\emph{fractional} number.}
of
%[** TN: Moved up from end of paragraph]
\Figure{274}% ** Fig. 11.
linear units contained in the radius~$OA$, and $t$~and~$s$
the lengths, or number of units contained in the arcs
$AB$ and~$AC$. Then since the angles $AOB$ and~$AOC$
\PageSep{274}
are proportional to the arcs $AB$ and~$AC$, or to the
numbers $t$~and~$s$, we have
\[
\text{Angle $AOC$ is $\dfrac{s}{t}$ of the angle $AOB$;}
\]
and the angle~$AOB$ being the angular unit, the number~$\dfrac{s}{t}$
is that which expresses the angle~$AOC$. This
number is the same for the same angle, whatever
circle is chosen; in the circle~$FD$ the proportion of
the arcs $DE$ and~$DF$ is the same as that of $AB$ and~$AC$:
for since similar arcs of different circles are proportional
to their radii,
\begin{align*}
&AB : DE :: OA : OD\Add{.} \\
\intertext{Also }
&AC : DF :: OA : OD\Add{,} \\
\Thus\
&AB : DE :: AC : DF;
\end{align*}
therefore the proportion of $DF$ to~$DE$ is that of $s$~to~$t$,
and $\dfrac{s}{t}$~is the measure of the angle~$DOF$,\Chg{---}{ }$DOE$ being
the unit, as before. It only remains to choose the
angular unit~$AOB$, and here that angle naturally presents
itself, whose arc is equal to the radius in length.
This, from what is proved in Geometry, will be the
\PageSep{275}
\index{Angular units|(}%
same for all circles, since in two circles, arcs which
have the same ratio (in this case that of equality) to
their radii, subtend the same angle. Let $t = r$, then
$\dfrac{s}{r}$~is the number corresponding to the angle whose arc
is~$s$. This is the number which is always employed
in theory as the measure of an angle, and it has the
advantage of being independent of all linear units;
for suppose $s$~and~$r$ to be expressed, for example, in
feet, then $12s$ and $12r$ are the numbers of inches in
the same lines, and by the common theory of fractions
$\dfrac{s}{r} = \dfrac{12s}{12r}$. Generally, the alteration of the unit
does not affect the number which expresses the \emph{ratio}
of two magnitudes. When it is said that the angle
$= \dfrac{\arc}{\radius}$, it is only meant that, \emph{on one particular supposition},
(namely, that the angle~$1$ is that angle whose
arc is equal to the radius,) the number of these \emph{units}
in any other angle is found by dividing the number of
\emph{linear} units in its arc by the number of \emph{linear} units in
the radius. It only remains to give a formula for finding
the number of degrees, minutes, and seconds in
an angle, whose theoretical measure is given. It is
proved in geometry that the ratio of the circumference
of a circle to its diameter, or that of half the circumference
to its radius, though it cannot be expressed
exactly, is between $3.14159265$ and $3.14159266$. Taking
the last of these, which will be more than a sufficient
approximation for our purpose, it follows that
the radius being~$r$, one-half of the circumference is
\PageSep{276}
$r × 3.14159266$; and one-fourth of the circumference,
or the arc of a right angle, is $r × 1.57079633$. Hence
the number of units above described, in a right angle,
is $\dfrac{\arc}{\radius}$, or $1.57079633$. And the number of seconds
in a right angle is $90 × 60 × 60$, or $324000$. Hence if
$\theta$~be an angle expressed in units of the first kind, and
$A$~the number of seconds in the same angle, the proportion
of $A$~to~$324000$ will also be that of $\theta$~to
$1.57079633$. To understand this, recollect that the
proportion of any angle to the right angle is not altered
by changing the units in which both are expressed,
so that the numbers which express the two
for one unit, are proportional to the like numbers for
another.
Hence
\begin{align*}
&A : 324000 :: \theta : 1.57079633\Typo{:}{;} \\
\intertext{or } &A = \frac{324000}{1.57079633} × \theta; \\
\intertext{or } &A = 206265 × \theta, \text{ very nearly}.
\end{align*}
Suppose, for example, the number of seconds in the
theoretical unit itself is required. Here $\theta = 1$ and
$A = 206265$; similarly if $A$~be~$1$, $\theta = \dfrac{1}{206265}$, which
is the expression for the angle of one second referred
to the other unit. In this way, any angle, whose
number of seconds is given, may be expressed in
terms of the angle whose arc is equal to the radius,
which, for distinction, might be called the \emph{theoretical}
\index{Beman, W. W.}%
\index{Radian}%
\index{Smith, D. E.}%
unit.\footnote
{Also called a \emph{radian}. See Beman and Smith's \Title{Geometry}, p.~192.---\Ed.}
This unit is used without exception in analysis;
\index{Angular units|)}%
\PageSep{277}
thus, in the formula, for what is called in trigonometry
the sine of~$x$, viz.:
\[
\sin x = x - \frac{x^{3}}{2·3} + \frac{x^{5}}{2·3·4·5} -\Chg{,}{} \etc.
\]
If $x$~be an angle of one second, it is not~$1$ which must
be substituted for~$x$, but~$\dfrac{1}{206265}$.
The number $3.14159265$, etc., is called~$\pi$, and is
\index{Pi@{$\pi$}}%
the measure, in theoretical units, of two right angles.
Also $\dfrac{\pi}{2}$~is the measure of one right angle; but it must
not be confounded, as is frequently done, with~$90°$.
It is true that they stand for the same angle, but on
different suppositions with respect to the unit; the
unit of the first being very nearly $\dfrac{206265}{60 × 60}$ times that of
the second.
There are methods of ascertaining the value of
one magnitude by means of another, which, though it
varies with the first, is not a measure of it, since the
increments of the two are not proportional; for example,
when, if the first be doubled, the second, though
it changes in a definite manner, is not doubled. Such
is the connexion between a number and its common
logarithm, which latter increases much more slowly
than its number; since, while the logarithm changes
from $0$ to~$1$, and from $1$ to~$2$, the number changes
from $1$ to~$10$, and from $10$ to~$100$, and so on.
Now, of all triangles which have the same angles,
\index{Triangles, measurement of proportions of|EtSeq}%
the proportions of the sides are the same. If, therefore,
any angle~$CAB$ be given, and from any points
\PageSep{278}
\index{Trigonometrical ratios|EtSeq}%
$B$,~$B'$,~$B''$, etc., in one of its sides, and $b$,~$b'$, etc., in
the other, perpendiculars be let fall on the remaining
side, the triangles $BAC$, $B'AC'$, $bAc$, etc., having a
right angle in all, and the angle~$A$ common, are equiangular;
that is, one angle being given, which is not
a right angle, the proportions of every right-angled
triangle in which that angle occurs are given also;
and, \textit{vice versa}, if the proportion, or ratio of any two
sides of a right-angled triangle are given, the angles
of the triangle are given.
\Figure{278}% ** Fig. 12.
To these ratios names are given; and as the ratios
themselves are connected with the angles, so that
one of either set being given, viz., ratios or angles,
all of both are known, their names bear in them the
name of the angle to which they are supposed to be
referred. Thus, $\dfrac{BC}{AB}$, or $\dfrac{\text{side opposite to $A$}}{\text{hypothenuse}}$, is called
the \emph{sine} of~$A$; while $\dfrac{AC}{AB}$, or $\dfrac{\text{side opposite to $B$}}{\text{hypothenuse}}$, or the
sine of~$B$, the complement\footnote
{When two angles are together equal to a right angle, each is called the
complement of the other. Generally, complement is the name given to one
part of a whole relatively to the rest. Thus, $10$~being made of $7$ and~$3$, $7$~is
the \emph{complement} of~$3$ \emph{to}~$10$.}
of~$A$; is called the \emph{cosine}
\PageSep{279}
of~$A$. The following table expresses the names which
are given to the six ratios,
$\dfrac{BC}{AB}$\Add{,}
$\dfrac{AC}{AB}$\Add{,}
$\dfrac{BC}{AC}$\Add{,}
$\dfrac{AC}{BC}$\Add{,}
$\dfrac{AB}{AC}$ and
$\dfrac{AB}{BC}$, relatively to both angles, with the abbreviations
made use of. The terms \emph{opp.}, \emph{adj.}, and \emph{hyp.}, stand
for, \emph{opposite side}, \emph{adjacent side}, and \emph{hypothenuse}, and
refer to the angle last mentioned in the table.
\[
\scriptsize
\setlength{\arraycolsep}{2pt}
\begin{array}{c|l|c|l|c|l|l}
\hline\hline
\TEntry[RATIO]{THE \\ RATIO} &
\multicolumn{1}{c|}{\TEntry{IS THE}} &
\TEntry{BEING} &
\multicolumn{1}{c|}{\TEntry{OR}} &
\TEntry{BEING} &
\multicolumn{2}{c}{\TEntry[THESE ARE]{THESE ARE \\ WRITTEN}} \\
\hline
\Strut[14pt]
\dfrac{BC}{AB} &
\text{sine of $A$} &
\dfrac{\opp.}{\hyp.} &
\text{cosine of $B$} &
\dfrac{\adj.}{\hyp.} &
\sin A & \cos B \\
%
\dfrac{AC}{AB} &
\text{cosine of $A$} &
\dfrac{\adj.}{\hyp.} &
\text{sine of $B$} &
\dfrac{\opp.}{\hyp.} &
\cos A & \sin B \\
%
\dfrac{BC}{AC} &
\text{tangent of $A$} &
\dfrac{\opp.}{\adj.} &
\text{cotangent of $B$} &
\dfrac{\adj.}{\opp.} &
\tan A & \cot B \\
%
\dfrac{AC}{BC} &
\text{cotangent of $A$} &
\dfrac{\adj.}{\opp.} &
\text{tangent of $B$} &
\dfrac{\opp.}{\adj.} &
\cot A & \tan B \\
%
\dfrac{AB}{AC} &
\text{secant of $A$} &
\dfrac{\hyp.}{\adj.} &
\text{cosecant of $B$} &
\dfrac{\hyp.}{\opp.} &
\sec A & \cosec B \\
%
\dfrac{AB}{BC} &
\text{cosecant of $A$} &
\dfrac{\hyp.}{\opp.} &
\text{secant of $B$} &
\dfrac{\hyp.}{\adj.} &
\cosec A & \sec B \\
\hline\hline
\end{array}
\]
If all angles be taken, beginning from one minute,
and proceeding through $2'$,~$3'$, etc., up to $45°$, or~$2700'$,
and tables be formed by a calculation, the nature of
which we cannot explain here, of their sines, cosines,
and tangents, or of the logarithms of these, the proportions
of every right-angled triangle, one of whose
angles is an exact number of minutes, are registered.
\PageSep{280}
We say sines, cosines, and tangents only, because it
is evident, from the table above made, that the cosecant,
secant, and cotangent of any angle, are the
reciprocals of its sine, cosine, and tangent, respectively.
Again, the table need only include~$45°$, instead
of the whole right angle, because, the sine of an
angle above~$45°$ being the cosine of its complement,
which is less than~$45°$, is already registered. Now, as
all rectilinear figures can be divided into triangles,
and every triangle is either right-angled, or the sum
or difference of two right-angled triangles, a table of
this sort is ultimately a register of the proportions of
all figures whatsoever. The rules for applying these
tables form the subject of trigonometry, which is one
of the great branches of the application of algebra to
geometry. In a right-angled triangle, whose angles
do not contain an exact number of minutes, the proportions
may be found from the tables by the method
explained in \hyperref[chapter:XI.]{Chapter~XI}. of this treatise. It must be
observed, that the sine, cosine, etc., are not \emph{measures}
of their angle; for, though the angle is given when
either of them is given, yet, if the angle be increased
in any proportion, the sine is not increased in the
same proportion. Thus, $\sin 2A$ is not double of~$\sin A$.
The measurement of surfaces may be reduced to
\index{Surfaces, measurement of incommensurable|EtSeq}%
the measurement of rectangles; since every figure
may be divided into triangles, and every triangle is
half of a rectangle on the same base and altitude. The
superficial unit or quantity of space, in terms of which
\PageSep{281}
\index{Approximations@{Approximations|EtSeq}}%
\index{Incommensurables|EtSeq}%
it is chosen to express all other spaces, is perfectly
arbitrary; nevertheless, a common theorem points out
the convenience of choosing, as the superficial unit,
the square on that line which is chosen as the linear
unit. If the sides of a rectangle contain $a$~and~$b$ units,
the rectangle itself contains $ab$~of the squares described
on the unit. This proposition is true, even
when $a$~and~$b$ are fractional. Let the number of units
in the sides be $\dfrac{m}{n}$ and~$\dfrac{p}{q}$, and take another unit which
is $\dfrac{1}{nq}$~of the first, or is obtained by dividing the first
unit into $nq$~parts, and taking one of them. Then,
by the proposition just quoted, the square described
on the larger unit contains $nq × nq$ of that described
on the smaller. Again, since $\dfrac{m}{n}$ and $\dfrac{p}{q}$ are the same
fractions as $\dfrac{mp}{nq}$ and~$\dfrac{np}{nq}$, they are formed by dividing
the first unit into $nq$~parts, and taking one of these
parts $mq$~and $np$~times; that is, they contain $mq$~and
$np$~of the smaller unit; and, therefore, the rectangle
contained by them, contains $mq × np$ of the square
described on the smaller unit. But of these there are
$nq × nq$ in the square on the longer unit; and, therefore,
$\dfrac{mq × np}{nq × nq}$, or~$\dfrac{mp × nq}{nq × nq}$, or~$\dfrac{mp}{nq}$, is the number of
the larger squares contained in the rectangle. But
$\dfrac{mp}{nq}$~is the algebraical product of $\dfrac{m}{n}$ and~$\dfrac{p}{q}$. This proposition
is true in the following sense, where the sides
of the rectangle are incommensurable with the unit.
Whatever the unit may be, we have shown that, for
\PageSep{282}
any incommensurable magnitude, we can go on finding
$b$~and~$a$, two whole numbers, so that $\dfrac{b}{a}$~is too little, and
$\dfrac{b + 1}{a}$~too great: until $a$~is as great as we please. Let
$AB$ and $AC$ be the sides of a rectangle~$AK$, and let
them be incommensurable with the unit~$M$. Let the
lines $AF$ and~$AG$, containing $\dfrac{b}{a}$ and $\dfrac{b + 1}{a}$ units, be
respectively less and greater than~$AC$; and let $AD$
and~$AE$, containing $\dfrac{c}{d}$ and $\dfrac{c + 1}{d}$ units, be respectively
\Figure{282}% ** Fig. 13.
less and greater than~$AB$; and complete the figure.
The rectangles $AH$ and $AI$ contain, respectively,
\index{Square@{\emph{Square}, the term}}%
$\dfrac{b}{a} × \dfrac{c}{d}$ and $\dfrac{b + 1}{a} × \dfrac{c + 1}{d}$ square units,\footnote
{``Square unit'' is the abbreviation of ``square described on the unit.''}
and the
first is less than the given rectangle, and the second
greater; consequently the given rectangle does not
differ from either, so much as they differ from one
another. But the difference of $AH$ and $AI$ is
\begin{gather*}
\text{$\dfrac{(b + 1)(c + 1)}{ad} - \frac{bc}{ad}$, or
$\dfrac{b + c + 1}{ad}$}, \\
\text{or }
\frac{b}{ad} + \frac{c}{ad} + \frac{1}{ad}, \\
\PageSep{283}
\text{or }
\frac{1}{d}\, \frac{b}{a} + \frac{1}{a}\, \frac{c}{d} + \frac{1}{ad}.
\end{gather*}
Proceed through two,\footnote
{This is done, because, by proceeding one step at a time, $\dfrac{b}{a}$~is alternately
too little and too great to represent~$AC$; whereas we wish the successive
steps to give results always less than~$AC$.}
four, six, etc., steps of the
approximation. The linear unit being~$M$, the results
will be such, that $\dfrac{b}{a}\, M$ will be always less than~$AC$,
but continually approaching to it. Hence $\dfrac{1}{d}\, \dfrac{b}{a}\, M$ is
always less than~$\dfrac{AC}{d}$; and since $AC$~remains the same,
and $d$~is a number which may increase as much as we
please, by carrying on the approximation, $\dfrac{AC}{d}$~and
\textit{à~fortiori} $\dfrac{1}{d}\, \dfrac{b}{a}\, M$ may be made as small a line as we
please; that is, $\dfrac{1}{d}\, \dfrac{b}{a}$ may be made as small as we
please, and so may $\dfrac{1}{a}\, \dfrac{c}{d}$ in the same manner. Also
$\dfrac{1}{ad}$~may be made as small as we please; and therefore,
also, the sum $\dfrac{1}{d}\, \dfrac{b}{a} + \dfrac{1}{a}\, \dfrac{c}{d} + \dfrac{1}{ad}$. But this number,
when the unit is the square unit, represents the
difference of the rectangles $AH$ and~$AI$, and is greater
than the difference of $AK$ and~$AI$; therefore, the approximate
fractions which represent $AC$~and~$AB$ may
be brought so near, that their product shall, as nearly
as we please, represent the number of square units in
their rectangle.
In precisely the same manner it may be proved,
that if the unit of content or solidity be the cube described
on the unit of length, the number of cubical
units in any rectangular parallelepiped, is the product
\PageSep{284}
of the number of linear units in its three sides, whether
these numbers be whole or fractional; and in the sense
just established, even if they be incommensurable with
the unit.
These algebraical relations between the sides and
content of a rectangle or parallelepiped were observed
by the Greek geometers; but as they had no distinct
science of algebra, and a very imperfect system of
arithmetic, while, with them, geometry was in an advanced
state; instead of applying algebra to geometry,
what they knew of the first was by deduction
from the last: hence the names which, to this day,
are given to $aa$, $aaa$, $ab$, which are called the \emph{square}
\index{Cube@{\emph{Cube}, the term}}%
\index{Square@{\emph{Square}, the term}}%
\index{Terms, geometrical and algebraical compared}%
of~$a$, the \emph{cube} of~$a$, the rectangle of $a$~and~$b$. The student
is thus led to imagine that he has proved that
square described on the line whose number of units
is~$a$, to contain $aa$~square units, because he calls the
latter the square of~$a$. He must, however, recollect,
that squares in algebra and geometry mean distinct
\index{Algebra!applied to the measurement of lines, angles, proportion of figures and surfaces|)}%
things. It would be much better if he would accustom
himself to call $aa$ and $aaa$ the second and third
powers of~$a$, by which means the confusion would be
avoided. It is, nevertheless, too much to expect that
a method of speaking, so commonly received, should
ever be changed; all that can be done is, to point out
the real connexion of the geometrical and algebraical
signification. This, if once thoroughly understood,
will prevent any future misconception.
\index{Measurement, of lines, angles, proportion of figures, and surfaces|)}%
\PageSep{285}
\BackMatter
\printindex
\iffalse
INDEX.
Addition 23, 67
Algebra
notation of|EtSeq 55 % et seq.;
elementary rules of|EtSeq 67 % et seq.;
advice@{advice on the study of} 53, 54, 62
advice@{advice on the study of|EtSeq} 175 % et seq.;
nature of the reasoning in 192
applied to the measurement of lines, angles, proportion of figures and surfaces 266-284
Algebraically@\emph{Algebraically greater} 144-145
Algebras@{\emph{Algebras}, bibliographical list of} 188-189
Analogy, in language of algebra 79
Angle
def@{definition of|EtSeq} 196 % et seq.
def@{definition of} 238
measure of|EtSeq 273 % et seq,
Angular units 275-276
Approximations 130, 267
Approximations@{Approximations|EtSeq} 48, 171, 242, 281 % et seq.
Arrangment of algebraical expressions 73
Arithmetic
elementary rules of|EtSeq 20 % et seq.;
compared with algebra 76
Arithmetical
notation|EtSeq 11 % et seq.;
notion of proportion|EtSeq 244 % et seq.
Assertions, logical|EtSeq 203 % et seq.
Assumptions 231, 232
Axioms 208
Axioms@{Axioms|EtSeq} 231 % et seq.
Babbage 168, 174
Bagay 168
Bain 212
Baltzer, R. 189
Beman, W. W. 188, 265, 276
Bertrand 239
Biermann, O. 189
Binomial theorem, exercises in|EtSeq 177 % et seq.
Bolyai 232
Bosanquet 212
Bourdon 188
Bourgeois gentilhomme, the 230
Bourlet, C.#Bourlet 188
Brackets 21
Bradley 212
Bremiker 168
Bruhns 168
Burnside, W. S. 189
Caesar@{Cæsar} 2, 229
Caillet 168
Callet 168
Carus, Paul 232
Change of algebraical form|EtSeq 105 % et seq.
Chrystal, Prof. 189
Cipher 16
Circulating decimals 51
Clifford 232
Coefficient 60
Collin, J. 188
Commercial arithmetic 53
Comparison of quantities|EtSeq 244 % et seq,
Computation 180
Comte 187
Condillac 187
Continued fractions 267-273
Contradictory 205
Contraries 205
Convergent fractions|EtSeq 269 % et seq.
Converse 205
Copula 203
Counting|EtSeq 13 % et seq.
Courier, problem of the two|EtSeq 112 % et seq.
Cube@{\emph{Cube}, the term} 284
Dalembert@{D'Alembert} 187
Dauge, F. 187
%\PageSep{286}
Decimal
system of numeration|EtSeq 14 % et seq.;
point 43
fractions 42-54
Definition 11, 207
Delboeuf@{Delb\oe{}uf, J.} 232
Demonstration
mathematical@{mathematical|EtSeq} 4 % et seq.
mathematical 184
inductive 179
Demorgan@{De Morgan} 187
Descartes 37
Differential calculus 265
Diminution, not necessarily without limit 251
Diophantus 186
Direction 196
Direct reasoning 226
Discovery, progress of, dependent on language 37
Division@{Division|EtSeq} 23 % et seq.
Division 38, 75, 165-167
Duhamel 72, 187, 232
Duhring@{Dühring} 187
Duodecimal system 19
English@{\Title{English Cyclopedia}} 174, 187
Equations
first@{of the first degree} 90-102
second@{of the second degree|EtSeq} 129 % et seq.;
identical 90
condition@{of condition} 91
condition@{of condition|EtSeq} 96 % et seq.;
reducing problems to|EtSeq 92 % et seq.
Errors
in mathematical computations|EtSeq 48 % et seq.;
in algebraical suppositions, corrected by a change of signs|EtSeq 106 % et seq.
Euclid 4, 37, 181, 234-235, 265
theory@{his theory of proportion} 240
theory@{his theory of proportion|EtSeq} 258 % et seq.
Exhaustions, method of 263-265
Exponents.|See Indices. 0
Experience, mathematical 104
Expressions, algebraical 59
Extension@{Extension of rules and meanings of terms|EtSeq} 33 % et seq.
Extension@{Extension of rules and meanings of terms} 80-82, 143-145, 163
Factoring@{Factoring|EtSeq} 132 % et seq.
Factoring 160
Figures, logical|EtSeq 216 % et seq.
Fine, H. B. 189
Fisher and Schwatt 189
Fluxions 265
Form, change of in algebraical expressions|EtSeq 105, 117 % et seq.
Formulæ, important 88-89, 99, 141-142, 163-167
Fowler, T. 212
Fractions
arithmetical@{arithmetical|EtSeq} 30 % et seq.
arithmetical 75
decimal 42
continued 267-273
singular values of|EtSeq 123 % et seq.;
evanescent 126-128
algebraical 75, 87-89, 97-99
Fractional exponents@{Fractional exponents|EtSeq} 163 % et seq.
Fractional exponents 185
French language 188
Frend|FN 71 %, foot-note.
Freycinet 187
Geometry
study of|EtSeq 4 % et seq.;
definitions and study of|EtSeq 191 % et seq.;
elementary ideas of|EtSeq 193 % et seq.
Geometrical reasoning and proof|EtSeq 203, 220 % et seq.
German language 188
Grassmann 232
Greatest common measure@{Greatest common measure|EtSeq} 25, 267 % et seq.
Greatest common measure 86
Greater@{\emph{Greater} and \emph{less}}, the meaning of 144
Greatness and smallness 170
Halsted 232
Hariot 38
Haskell 168
Hassler 168
Helmholtz 232
Hindu algebra 186
Hirsch 188
Holzmüller, G. 189
Hutton 168
Hyde 232
Hypothesis 208
Identical equations 90
Imaginary quantities|EtSeq 151 % et seq.
Impossible quantities|EtSeq 149 % et seq.
Incommensurables|EtSeq 246, 281 % et seq.
Increment 169
Indirect reasoning 226
Indeterminate problems 101
Indices, theory of 60, 166, 185
Indices, theory of@{Indices, theory of|EtSeq} 158 % et seq.
Induction, mathematical 104, 179, 183
Inductive reasoning 219
Infinite quantity, meaning of|EtSeq 123 % et seq.
Infinite spaces, compared|EtSeq 235 % et seq.
Instruction
principles of natural|EtSeq 21 % et seq.;
faulty 182
books on mathematical 187
Interpolation 169-174
%\PageSep{287}
James, W. 232
Jevons 212
Jodl, F. 232
Jones 168
Keynes 212
Lacroix 187
Lagrange 187
Laisant 187
Lallande 168
Language 13, 37, 79
Laplace 185
Laurent, H. 188
Least common multiple 28
Leibnitz 37
Line 193, 242
Linear unit 267
Literal notation|EtSeq 57 % et seq.
Lobachévski 232
Locke 9
Logarithms|EtSeq 167 % et seq.
Logic of mathematics 203-230
Logics@{\emph{Logics}, bibliographical list of} 212
Mach, E. 232
Mathematics
nature, object, and utility of the study of|EtSeq 1 % et seq.;
language of|EtSeq 37 % et seq.;
advice on study of 175
philosophy of 187
Matthiessen 189
Measures 198
Measures@{Measures|EtSeq} 266 % et seq.
Measurement, of lines, angles, proportion of figures, and surfaces 266-284
Measuring|EtSeq 241 % et seq.
Mill, J. S. 212
Minus quantities 72
Mistaken suppositions|EtSeq 106 % et seq.
Moods, logical|EtSeq 212 % et seq.
Multiplication@{Multiplication|EtSeq} 23, 34, 68
Multiplication 164
Mysticism in numbers 14
Negative
quantities 72
sign@{sign, isolated|EtSeq} 103 % et seq.
sign@{sign, isolated} 181
squares 149, 151
indices 166, 185
Netto, E. 189
Newton 37, 185
Notation
arithmetical, decimal|EtSeq 11 % et seq.;
general principle of|EtSeq 15 % et seq.;
algebraical@{algebraical|EtSeq} 55 % et seq.
algebraical 79, 159
extension of 33, 80, 143, 163
Numbers, representation of|EtSeq 15 % et seq.
Numeration, systems of|EtSeq 14 % et seq.
Numerically@{\emph{Numerically greater}} 144
Oliver, Waite, and Jones 189
Panton, A. W. 189
Parallels, theory of 181, 231-237
Particular affirmative and negative 203
Perfect square 138
Petersen 189
Pi@{$\pi$} 277
Plane surface 195
Plato 239
Poincaré, H. 232
Point, geometrical 194-195
Postulate 210
Powers, theory of|EtSeq 158 % et seq.
Predicate 203
Premisses|EtSeq 211 % et seq.
Prime numbers and factors 25
Problems
reducing of, to equations|EtSeq 92 % et seq.;
general disciplinary utility of 95
of loss and gain as illustrating changes of sign 119
of the two couriers|EtSeq 112 % et seq.
Proportions 170
theory of 240-265
Proportional parts 173
Propositions|EtSeq 203 % et seq.
Pythagorean proposition|EtSeq 221 % et seq,
Quadratic
equations|EtSeq 129 % et seq.;
roots, discussion of the character of|EtSeq 137 % et seq.
Radian 276
Read, Carveth 212
Reasoning
geometrical|EtSeq 203 % et seq.;
direct and indirect 226
Reckoning|EtSeq 13 % et seq.
Riemann 232
Roots|EtSeq 129, 137, 158 % et seq.
Rules 42
mechanical 184
extension of meaning of 33, 80, 143, 163
Russell, B. A. W. 232
%\PageSep{288}
Schlömilch, O. 189
Schrön 168
Schubert, H. 189
Schüller, W. J. 189
Self-evidence 209
Serret, J. A. 188
Sexagesimal system of angular measurement 273
Shorthand symbols 55
Sidgwick 212
Signs
arithmetical and algebraical@{arithmetical and algebraical|EtSeq} 20 % et seq.
arithmetical and algebraical 55
rule of 96, 186
Sigwart 212
Simple expression 76, 77
Singular values|EtSeq 122 % et seq.
Smith, D. E. IV, 187, 265, 276
Solutions, general algebraical 111
Square@{\emph{Square}, the term} 282, 284
Stäckel, Paul 232
Stallo, J. B. 232
Straight line 12, 193
Subject 203
Subtraction 23
Subtractions, impossible 103-104
Surfaces, measurement of incommensurable|EtSeq 280 % et seq.
Syllogisms|EtSeq 210 % et seq.
Symbols@{Symbols, invention of} 80-81
Symbols@{Symbols|See Signs}. 0
Syllabi, mathematical 186
Tables, mathematical, recommended 168
Tannery, P. 232
Taylor 168
Terms, geometrical and algebraical compared 284
Theory of equations@{Theory of equations|EtSeq} 132 % et seq.
Theory of equations 179, 190
Todhunter 189
Triangles, measurement of proportions of|EtSeq 277 % et seq.
Trigonometrical ratios|EtSeq 278 % et seq.
Ueberweg 212
Universal affirmative and negative 203
Vega 168
Venn 212
Weber, H. 189
Wells 168
Whately 212
Whole number 76
Zero
as a figure 16
its varying significance as an algebraical result|EtSeq 122 % et seq.;
exponents 81, 166
\fi %** End of index text
\iffalse
%[** TN: Very lightly proofread OCR from the Internet Archive follows]
THE OPEN COURT MATHEMATICAL SERIES
Essays on the Theory of Numbers.
(1) Continuity and Irrational Numbers, (2) The Nature
and Meaning of Numbers. By RICHARD DEDEKIND. From
the German by W. W. BEMAN. Pages, 115. Cloth, 75
cents net. (3s. 6d. net.)
These essays mark one of the distinct stages in the devel
opment of the theory of numbers. They give the founda
tion upon which the whole science of numbers may be es
tablished. The first can be read without any technical,
philosophical or mathematical knowledge; the second re
quires more power of abstraction for its perusal, but power
of a logical nature only.
``A model of clear and beautiful reasoning.''
Journal of Physical Chemistry.
``The work of Dedekind is very fundamental, and I am glad to have it
in this carefully wrought English version. I think the book should be
of much service to American mathematicians and teachers.''
Prof. E. H. Moore, University of Chicago.
``It is to be hoped that the translation will make the essays better
known to English mathematicians; they are of the very first importance,
and rank with the work of Weierstrass, Kronecker, and Cantor in the
same field.'' Nature.
Elementary Illustrations of the Differential
and Integral Calculus.
By AUGUSTUS DE MORGAN. New reprint edition. With
subheadings and bibliography of English and foreign works
on the Calculus. Price, cloth, \$1.00 net. (4s. 6d net.)
``It aims not at helping students to cram for examinations, but to give
a scientific explanation of the rationale of these branches of mathe
matics. Like all that De Morgan wrote, it is accurate, clear and
philosophic.'' Literary World, London.
On the Study and Difficulties of Mathe
matics.
By AUGUSTUS DE MORGAN. With portrait of De Morgan,
Index, and Bibliographies of Modern Works on Algebra,
the Philosophy of Mathematics, Pangeometry, etc. Pages,
viii, 288. Cloth, \$1.25 net. (5s. net.)
``The point of view is unusual; we are confronted by a genius, who,
like his kind, shows little heed for customary conventions. The shaking
up which this little work will give to the young teacher, the stim
ulus and implied criticism it can furnish to the more experienced, make
its possession most desirable.'' Michigan Alumnus.
THE OPEN COURT MATHEMATICAL SERIES
The Foundations of Geometry.
By DAVID HILBERT, Ph. D., Professor of Mathematics in
the University of Gottingen. With many new additions
still unpublished in German. Translated by E. J. TOWN-
SEND, Ph. D., Associate Professor of Mathematics in the
University of Illinois. Pages, viii, 132. Cloth, \$1.00 net.
(4s. 6d net)
``Professor Hilbert has become so well known to the mathematical
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for them to obtain in English such an important discussion of these
points by such an authority.'' Journal of Pedagogy.
Euclid's Parallel Postulate: Its Nature, Validity
and Place in Geometrical Systems.
By JOHN WILLIAM WITHERS, Ph. D. Pages vii, 192. Cloth,
net \$1.25. (4s. 6d. net.)
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parallel postulate is empirical, and this may be admitted in the sense
that his argument requires; at any rate, he shows the absurdity of
some statements of the a priori school.'' Nature.
Mathematical Essays and Recreations.
By HERMANN SCHUBERT, Professor of Mathematics in
Hamburg. Contents: Notion and Definition of Number;
Monism in Arithmetic; On the Nature of Mathematical
Knowledge; The Magic Square; The Fourth Dimension;
The Squaring of the Circle. From the German by T. J.
McCormack. Pages, 149. Cuts, 37. Cloth, 75 cents net.
(3s. 6d. net.)
``Professor Schubert's essays make delightful as well as instructive
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ligibility. Even the lay mind can understand and take a deep interest
in what the German professor has to say on the history of magic
squares, the fourth dimension and squaring of the circle.''
Saturday Review.
THE OPEN COURT MATHEMATICAL SERIES
A Brief History oi Mathematics.
By the late DR. KARL FINK, Tubingen, Germany. Trans
lated by Wooster Woodruff Beman, Professor of Math
ematics in the University of Michigan, and David Eugene
Smith, Professor of Mathematics in Teachers College,
Columbia University, New York City. With biographical
notes and full index. Second revised edition. Pages,
xii, 333. Cloth, \$1.50 net. (5s. 6d. net.)
``Dr. Fink's work is the most systematic attempt yet made to present a
compendious history of mathematics.'' The Outlook.
``This book is the best that has appeared in English. It should find a
place in the library of every teacher of mathematics.''
The Inland Educator.
Lectures on Elementary Mathematics.
By JOSEPH Louis LAGRANGE. With portrait and biography
of Lagrange. Translated from the French by T. J. Mc-
Cormack. Pages, 172. Cloth, \$1.00 net. (4s. 6d. net.)
``Historical and methodological remarks abound, and are so woven together
with the mathematical material proper, and the whole is so
vivified by the clear and almost chatty style of the author as to give
the lectures a charm for the readers not often to be found in mathe
matical works.'' Bulletin American Mathematical Society.
A Scrapbook of Elementary Mathematics.
By WM. F. WHITE, State Normal School, New Paltz, N.
Y. Cloth. Pages, 248. \$1.00 net. (5s. net.)
A collection of Accounts, Essays, Recreations and Notes,
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book is supplied with Bibliographic Notes, Bibliographic
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The Educator-Journal.
THE OPEN COURT MATHEMATICAL SERIES
Geometric Exercises in Paper-Folding.
By T. SUNDARA Row. Edited and revised by W. W. BE-
MAN and D. E. SMITH. With half-tone engravings from
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``The book is simply a revelation in paper folding. All sorts of things
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Teachers Institute.
Magic Squares and Cubes.
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The first two chapters consist of a general discussion of the
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on Magic Squares'' by Dr. Carus, in which he brings out
the intrinsic harmony and symmetry which exists in the
laws governing the construction of these apparently mag
ical groups of numbers. Mr. Frierson's ``Mathematical
Study of Magic Squares,'' which forms the fifth chapter,
states the laws in algebraic formulas. Mr. Browne con
tributes a chapter on ``Magic Squares and Pythagorean
Numbers,'' in which he shows the importance laid by the
ancients on strange and mystical combinations of figures.
The book closes with three chapters of generalizations in
which Mr. Andrews discusses ``Some Curious Magic
Squares and Combinations,'' ``Notes on Various Con
structive Plans by Which Magic Squares May Be Classi
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``The examples are numerous; the laws and rules, some of them
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The Foundations of Mathematics.
A Contribution to The Philosophy of Geometry. BY DR.
PAUL CARUS. 140 pages. Cloth. Gilt top. 75 cents net.
(3s. 6d. net.)
The Open Court Publishing Co.
378-388 Wabash Avenue Chicago
\fi %[** TN: End of catalog text]
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