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ON THE
STUDY AND DIFFICULTIES OF
MATHEMATICS
BY
AUGUSTUS DE MORGAN
THIRD RKPRINT EDITION
CHICAGO
THE OPEN COURT PUBLISHING COMPANY
LONDON
KEGAN PAUL. TRENCH, TRUBNER & Co., LTD.
1910
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EDITOR'S NOTE.
TV TO apology is needed for the publication of the present new
•L ^ edition of The Study and Difficulties of Mathematics, — a
characteristic production of one of the most eminent and lumi
nous 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 in
sufficiently 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. Ele
mentary instructors and students cannot fail to profit by the gen
eral 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
600772
iv ON THE STUDY OF MATHEMATICS.
out of print have either been omitted or supplemented by the men
tion of more modern works. The few notes which have been
added are mainly bibliographical in character, and refer, for in
stance, 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
Brockport, N. Y.
THOMAS J. MCCORMACK
LA SALLE, 111., Nov. i, 1898.
AUTHOR'S PREFACE.
TN 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 work
ing 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 recom
mence 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 dog
mas. 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,
VI ON THE STUDY OF MATHEMATICS.
however, responsible for the contents of this treatise, further than
for the manner in which they are presented, as most of the opin
ions 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 a priori reasoning can give to a beginner.
In treating, therefore, on the negative sign, on impossible quanti
ties, and on fractions of the form g, etc., I have followed the
method adopted by several of the most esteemed continental writ
ers, 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, \/ — a, g, 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 differ
ent misconceptions, which give rise to the various expressions
above alluded to. To this view of the question I have leaned,
rinding 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 con
fided to more skilful hands. AUGUSTUS DE MORGAN.
CONTENTS.
CHAPTER PAGE
Editor's Note iii
Author's Preface ......... ^, . . . v
I. Introductory Remarks on the Nature and Objects of
Mathematics .............. I
II. On Arithmetical Notation . ....... n
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
CHAPTER I.
INTRODUCTORY REMARKS ON THE NATURE AND
OBJECTS OF MATHEMATICS.
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 difficul
ties and which are apt to be misunderstood by begin
ners, 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 be
fore entering on the sciences of algebra and geometry,
are such as languages and history, etc. Of these,
neither appears to have any affinity with mathemat-
2 ON THE STUDY OF MATHEMATICS.
ics; yet, in order to see the difference which exists be
tween these studies,— for instance, history and geom
etry, — it will be useful to ask how we come by knowl
edge in each. Suppose, for example, we feel certain
of a fact related in history, such as the murder of
Ca3sar, whence did we derive the certainty? how came
we to feel sure of the general truth of the circum
stances 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 his
torians, men of credit, who lived and published their
accounts in the very time of which they write ; that
succeeding ages have icceived 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 ob
jection 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 Caesar 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 be
cause we can show that it must necessarily have been
that, at a particular day, at a particular place, a sue-
NATURE AND OBJECTS OF MATHEMATICS. 3
cessful 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 entertain
ing, 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 in
volve absolute absurdity, but only extreme improb
ability.
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 neces
sity, 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 them
selves, 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
4 ON THE STUDY OF MATHEMATICS.
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 dem
onstration 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 im
probable, it proves it at once to be absurd and impos
sible. This is done by showing that the contrary of
the proposition which is asserted is in direct contra
diction to some extremely evident fact, of the truth of
which our eyes and hands convince us. In geometry,
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 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., 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
NATURE AND OBJECTS OF MATHEMATICS. 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 imme
diately ; 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 demonstra
tion, 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 meta
physics. In reality, our senses are our first mathe
matical 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
6 ON THE STUDY OF MATHEMATICS.
by these names, in the shape of incontrovertible prop
ositions, such as have been already cited, on which,
if any remark is made by the beginner in mathemat
ics, it will probably be, that from such absurd truisms
as "the whole is greater than its part," no useful re
sult 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, neverthe
less ; 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 un
derstanding, 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 pro
ceed 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 im
pediments in the way of his progress appear. This
necessarily follows in a science which consists of rea-
NATURE AND OBJECTS OF MATHEMATICS. 7
soning from the very commencement, for it is evident
that every student will feel a claim to have his objec
tions 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 occa
sional 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 advan
tages 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 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, mathe
matics, or natural history, may be chosen for this pur
pose. 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,
8 ON THE STUDY OF MATHEMATICS.
whether the results are true or not. When the guid
ing property of the loadstone was first ascertained,
and it was necessary to learn how to use this new dis
covery, and to find out how far it might he 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 fol
lowing grounds :
1. Every term is distinctly explained, and has but
one meaning, and it is rarely that two words are em
ployed 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 prin
ciples, 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 arith
metical calculation. This gives confidence, and is
NATURE AND OBJECTS OF MATHEMATICS. 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 de
grees 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 asser
tion will prove its conclusion generally, that is, for all
cases in which A is greater than B, whether the dif
ference be great or little. Locke mentions the dis
tinctness of mathematical terms, and says in illustra
tion : "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 pow
ers. 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 concentrat-
IO ON THE STUDY OF MATHEMATICS.
ing the ideas which a successful study of them in
creases 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 atten
tion and forces the united faculties to use their utmost
exertions. The habit of mind thus formed soon ex
tends 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 edu
cation 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, geom
etry, and trigonometry. We have used these four di
visions 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.
CHAPTER II.
ON ARITHMETICAL NOTATION.
'TVHE first ideas of arithmetic, as well as those of
-*- other sciences, are derived from early observa
tion. 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 com
pletely 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
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
12 ON THE STUDY OF MATHEMATICS.
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 pre
cision with which the terms are defined, it is most es
sential 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
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 " ; a straight line is that which
"lies evenly between its extreme points." Now, let
any one ask himself whether he could have guessed
ON ARITHMETICAL NOTATION. 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 diffi
culty? 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 absurd
ity of attempting to make the simplest ideas yet more
simple.
A knowledge of our method of reckoning, and "of
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
a part of it ; and that we are not much indebted to
instruction for so simple a gift. It has been well ob
served, that if the whole earth spoke the same lan
guage, we should think that the name of any object
was not a mere sign 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
14 ON THE STUDY OF MATHEMATICS.
this error, because, as it is, we happen to know that
what we call by the sound "horse," the Romans dis
tinguished as well by that of "equus," but we commit
a similar mistake with regard to our system of nume
ration, 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 mysti
cism to numbers which has prevailed so extensively
in all times.
We know that we have nine signs to stand for the
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 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 reck
oning 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
ON ARITHMETICAL NOTATION. 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 one flock, twenty shillings
are one pound. Call each lot a 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 : — Sup
pose, 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 re
maining. 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
number in a short and convenient manner? It is plain
that the way to do this is a matter of choice. Suppose
then that we distinguish the tens by marking their
number with one accent, the hundreds with two ac
cents, and the thousands with three. We may then
represent this number in any of the following ways : —
l6 ON THE STUDY OF MATHEMATICS.
76'5"4"', 6'75"4'", 6'4'"5"7, 4"'5"6'7, the whole num
ber of ways being 24. But this is more than we want ;
one certain method of representing a number is suffi
cient. 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
be apparent that the accents are unnecessary. Since
the singly accented figure will always be the second
from the right, and so on, the 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 diffi
culty 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 cy
phers so as to bring each number into the place al
lotted 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 art of cypher, or 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
ON ARITHMETICAL NOTATION. IJ
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 rep
resented 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 bun
dles 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 nine, by 1" a nine of nines, and
so on, the number which we write 4567, must be writ
ten G'"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 num
bers in the common system are written, and in the
one which we have just explained :
COUNTING BY
Tens...l 2 3 4 5 6 7 8 9 10 11 12 13
Nines.. 1 2 3 4 5 6 7 8 I'O I'l 1'2 1'3 1'4
l8 ON THE STUDY OF MATHEMATICS.
COUNTING BV
Tens 14 15 16 17 18 19 20 30 40 50
Nines 1'5 1'6 VI 1'8 2'0 2'1 2'2 3'3 4'4 5'5
Tens 60 70 80 90 100
Nines 6'6 7'7 8'8 1"1'0 1"2'1
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 :
9)4567
9) 507 — rem. 4.
9) 56 — rem. 3.
9) 6 — rem. 2.
0 — rem. 6. Representation required, 6'" 2" 3' 4.
The processes of arithmetic are the same in prin
ciple 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.
ADDITION.
636 7" T 6
987 1"'3"1'6
403 4" 8' 7
2026 2"'7"0'1
SUBTRACTION.
1384 1"'8"0'7
797 l'"0" 7' 5
587 7" 2' 2
ON ARITHMETICAL NOTATION.
MULTIPLICATION.
297 3" 6' 0
136 1" 6' 1
1782 360
891 2400
297 360
40392 &'"!'" 3" 6' 0
DIVISION.
633) 79125 (125 7" 7' 3) lv 3^0-4" 71 6* (1"4' 8
633 773
4217
3423
6846
6846
0
The student should accustom himself to work
questions in different systems of numeration, which
will give him a clearer insight into the nature of arith
metical 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
of figures supposed, twelve will be represented by I'O ;
there must, therefore, be a distinct sign for ten and
eleven : a nine and six reversed, thus 9 and 6, might
be used for these.
*To avoid too great a number of accents, Roman numerals are put in
stead of them; also, to avoid confusion, the accents are omitted after the
first line.
CHAPTER III.
ELEMENTARY RULES OF ARITHMETIC,
AS SOON as the beginner has mastered the notion
-E~- *- of arithmetic, he may be made acquainted with
the meaning of the algebraical signs -J-, — , X? => and
also with that for division, or the common way of rep
resenting a fraction. There is no difficulty in these
signs or in their use. Five minutes' consideration will
make the symbol 5 -j- 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 gener
ally 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 ope
rations commences. The beginner should exercise
ELEMENTARY RULES OF ARITHMETIC. 21
himself in putting the simplest truths of arithmetic in
this new shape, and should write such sentences as
the following frequently :
2 + 7-9,
6-4 = 2,
2X2 + 12X 12 = 14x 10 + 2 X2X 2-
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 con
necting each operation with its sign, and should avoid
confounding operations together and considering them
as the same, because they produce the same result.
Thus 4x7 does not denote the same operation as
7 X 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 4x7 = 7x4 is a proposition to be
proved, not to be taken for granted. Again, $ X -A
and ^ 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
22 ON THE STUDY OF MATHEMATICS.
is following. Now, at the commencement of his ca
reer, a principle is not received and understood by
the student as quickly as it is explained by the in
structor. 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 all numbers unless by
having it shown that it holds good for some numbers,
and that for those "some numbers he may substitute
others, and use the same demonstration. Until he
can do this himself he does not understand the prin
ciple, 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 dic
tionary 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
ELEMENTARY RULES OF ARITHMETIC. 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 subtrac
tion ; 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
same number to itself — twelve times seven is twelve
sevens added together. Division is a continued sub
traction 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, there
fore, the result of additions and subtractions only.
The following principles, which occur so continu
ally 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.
24 ON THE STUDY OF MATHEMATICS.
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 :
(45 -f 7) — (22 + 7) = 45 — 22.
(45 — 8) — (22 — 8) = 45 — 22.
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 X 29 = 100 X 29 + 50 X 29 + 6 X 29-
IV. The same thing may be done with the multi
plier instead of the multiplicand. Thus, 29 is made
up of 18, 6, and 5. Then
156 X 29 = 156 X 18 + 156 x 6 + 156X 5.
V. If any two or more numbers be multiplied to
gether, it is indifferent in what order they are multi
plied, the result is the same. Thus,
10X6X4X3 = 3X10X4XG = 6X10X-±X3, etc.
VI. In dividing one number by another, for ex
ample 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 :
156 _ 72 60 24
T2~==12 + 12 + 12'
ELEMENTARY RULES OF ARITHMETIC. 25
The same thing cannot be done with the divisor. It
is not true that
156 _ 156 156 156
~12~z 4 ~3~ ~5~*
The student should discover the reasotn for himself.
A prime number is one which is not divisible by
any other number except 1. When the process of di
vision 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 reduced to its prime
factors* as it is called, that is, it may either be shown
to be a prime number itself or made up by multiply
ing several prime numbers together. Thus, 306 is
34 X 9, or 2x1^X9, or 2x17x3x3, 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
case, we can then find what is called their greatest
common measure or divisor; that is, the number made
* The factors of a number are those numbers by the multiplication of
Which it is made.
26 ON THE STUDY OF MATHEMATICS.
by multiplying all their common factors. It is an evi
dent 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 num
ber 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 X 2 or 12. It is also true that, if two numbers
are prime, their product is divisible by no numbers
except themselves. Thus, 17 X H 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 (meas
ure 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 2x5x3x3 and
2x2x^x5, have the common factors 2 and 5, and
are divisible by 2 X 5, or 10. The quotients are 3x3
and 2x5, or 9 and 10, which have no common factor
remaining, and 2x5, 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
ELEMENTARY RULES OF ARITHMETIC. 27
have mentioned of resolving numbers into their prime
factors, being troublesome to apply when the num
bers 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 pro
duct. 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 X 6 -f 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 meas
ures 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 di
visor and dividend is also the greatest common meas
ure of the remainder and divisor. Now take the com
mon process for finding the greatest common measure
of two numbers ; for example, 360 and 420, which is
as follows, and abbreviate the words greatest common
measure into their initials g. c. m. :
•28 ON THE STUDY OF MATHEMATICS.
360)420(1
360
~~60)360(6
360
0
From the theorem above enunciated it appears
that
g. c. m. of 420 and 360 is g. c. m. of 60 and 360 ;
g. c. m. of 60 and 360 is 60 ;
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
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 meas
ure except 1, the least common multiple is their pro
duct. 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 x 5 and
the second 2x4. The product is 2 X 2 X 4 X 5, but
ELEMENTARY RULES OF ARITHMETIC. 2Q
2 X 4 X 5 is also a common multiple, since it is divis
ible by 2 x ^, or 8, and by 2x5, or 10. To find this
common multiple we must, therefore, divide the pro
duct 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 X 7
or 56.
CHAPTER IV.
ARITHMETICAL FRACTIONS.
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 distinc
tions 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 an
other, as 8, the process is written thus : -6B6-. 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
ARITHMETICAL FRACTIONS. 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 -5g7-, and this
quotient is therefore called a fraction, or broken num
ber. 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
J, and -5^- is 7 -(- i, which is usually written 7J. Both
of these quantities -5^, and 7|, 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, |. This means
that 5 is to be divided into 8 parts, and | 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
J X 5. To prove this let each of the lines drawn be
low represent | of an inch ; repeat J five times, and
repeat the same line eight times.
In each column is Jth of an inch repeated 8 times ;
that is one inch. There are, then, 5 inches in all,
32 ON THE STUDY OF MATHEMATICS.
since there are five columns. But since there are 8
Jines, each line is the eighth of 5 inches, or f, but
each line is also Jth of an inch repeated 5 times, or
JX 5- Therefore, f = JX 5 J that is, in order to find
| inches, we may either divide five inches into 8 parts,
and take one of them, or divide one inch into 8 parts,
and takeySW of them. The symbol | is made to stand
for both these operations, since they lead to the same
result.
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, ^ is the same as J-J, 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 altera
tion in its value. Thus, \ — f — f = | — •£$, etc.;
? = T44=A=288> 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 diffi
culty in reducing fractions to a common denomina
tor, in reducing a fraction to its lowest terms ; neither
ARITHMETICAL FRACTIONS. 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 pe
culiar 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 be
ginner 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 gen-
34 ON THE STUDY OF MATHEMATICS.
eral, which applies to all cases contained under the
first, and to others besides. We have only the alter
native of inventing a new name, or of extending the
meaning of the former one so as to merge the particu
lar 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 f, J, J, etc. Now there is no num
ber which may not be considered as a fraction in as
many different ways as we please. Thus 7 is *•£• or
%£-, etc.; 12 is ±ffi, -7^2-, etc. Our new notion of frac
tion 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 ^, 3 and |. The rule of which we have spoken
is another instance. It is called the multiplication of
fractional numbers. Now, if we return to our mean
ing 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 ne-
ARITHMETICAL FRACTIONS.
35
cessary to use for fractions, in all cases where multi
plication would have been used had they been whole
numbers. Of this we shall now give a simple exam
ple. Take an oblong figure (which is called a rect
angle in geometry), such as A BCD, and find the mag
nitudes of the sides AB and BC in inches. Draw the
B
A D
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 rect
angle ABCD contains an exact number of squares,
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 X 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 exam
ple, that A' B' is | of an inch, and that B ' C is f of an
B' C
A'
36 ON THE STUDY OF MATHEMATICS.
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 f and f
for its numerator, and the product of their denomina
tors for its denominator; that is, that A' B' C' D' con
tains J£ of G. Here then appears a connexion be
tween the multiplication of whole numbers, and the
formation of a fraction, whose numerator is the pro
duct of two numerators, and its denominator the pro
duct of the corresponding denominators. These ope
rations 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 neces
sary, in the same case, to multiply the numerator by
the numerator, and the denominator by the denomina
tor, and form the result into a fraction, as above.
This would lead us to suspect some connexion be
tween these two operations, and we shall accordingly
find that when whole numbers are formed into frac
tions, 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 -^5-, and
the second as f . Form a fraction from the product
of the numerators and denominators of these, which
will be !££-, 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 product of those two frac-
ARITHMETICAL FRACTIONS. 37
tions, and the process of finding the third fraction,
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 frac
tions are really whole numbers. The mathematics are
not the only branches of knowledge in which it is cus
tomary 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 carry
ing 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
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
any others, although his terms are deficient in analogy,
and his notation infinitely inferior to that which we
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
attention of the scientific world to the imperfect mech
anism of the science. A slight and almost casual im-
38 ON THE STUDY OF MATHEMATICS.
provement, made by Hariot in algebraical language,
has been the foundation of most important branches
of the science.* 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 f by f is equivalent to divid
ing ^ into 2 parts, and taking three such parts. Be
cause | being the same as -if, or 1 divided into 12
parts and 10 of them taken, the half of || is 5 of those
parts, or f^. Three times this quantity will be 15 of
those parts, or -J-|-, which is by our rule the same as
what we have called, | multiplied by f . But the same
result arises from multiplying f by |-, or dividing |
into 6 parts and taking 5 of them. Therefore, we find
that | multiplied by f is the same as f multiplied by
f, or | X f = % X f- This proposition is usually con
sidered 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 |- of an
apple divided into 2 parts and 3 of them taken, is the
same as f 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,
*The mathematician will be aware that I allude to writing an equation
in the form
jr2 -\- ax — b = o ; instead of
.r2.f ax=b.
ARITHMETICAL FRACTIONS. 39
for example, 12 by 2, we endeavor to find how many
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 exam
ple | and f , it will be obser.ved that in place of find
ing how many twos make 12, we shall have to find
into how many parts J must be divided, and how many
ot them must be taken, so as to give |. If we reduce
these fractions to a common denominator, in which
case they will be ^J and ^ ; and if we divide the sec
ond into 8 equal parts, each of which will be ^, and
take 15 of these parts, we shall get £J, or |. The
fraction whose numerator is 15, and whose denomina
tor is 8, or -Jg5-, 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 pro
cess is equivalent to the division of one whole number
by another, whenever the fractions are really whole
numbers ; for example, 3 is \2-, and 15 is -3Y°-. If this
process be applied to -^ and -L2-, the result is J^0-,
which is 5, or the same as 15 divided by 3. This pro
cess is then, by extension, called division : -Jg5- is called
the quotient of f divided by |, and is found by multi
plying 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 sec
ond for the denominator of the result. That this pro
cess does give the same result as ordinary division in
all cases where ordinary division is applicable, we can
4-O ON THE STUDY OF MATHEMATICS.
easily show from any two whole numbers, for exam
ple, 12 and 2, whose quotient is (>. Now 12 is -8^6-, and
2 is -^Q., and the rule for what we have called division
of fractions will give as the quotient -f^-, 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 f
is 1 divided into 6 parts and five of them taken, or the
sixth part of 5, and he should reason upon these sup
positions, neglecting all rules until he has established
them in his own mind by reflexion on particular in
stances. 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.
I I ! I
£ F G H
He must then recollect that each of these parts is ^
of a foot ; that any two of them together are ^ of a
foot ; any 3, -|, 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.
ARITHMETICAL FRACTIONS. 41
What is J of f of a foot? What is f of J of £ of a
foot? Into how many parts must f of a foot be di
vided, and how many of them must be taken to pro
duce if of a foot? What is £ -f ^ of a foot? and so on.
CHAPTER V.
DECIMAL FRACTIONS.
TT is a disadvantage attending rules received without
•*- 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 multiplica
tion ; — 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 con
spicuous 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
DECIMAL FRACTIONS. 43
the subject it is continually necessary to reduce frac
tions to a common denominator : he will see, there
fore, the advantage of always using either the same
denominator, or a set of denominators, so closely con
nected 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
with the number ten. All fractions, such as -fifa,
T3o3o%> ~ff ~> which have a decimal number for the
denominator, are called decimal fractions. Now a
denominator of this sort is known whenever the num
ber of cyphers in it are known ; thus a decimal num
ber with 4 cyphers can only be 10,000, or ten thou
sand. 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 figures as there are
cyphers in the denominator. Thus \7o3o3o4 ^s represented
by 17.334; ^^ thus, .229. We might, had we so
pleased, have represented them thus, 173343, 2293 ;
or thus, 173343, 2293, or in any way by which we
might choose to agree to recollect that the denomina
tor 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
44 ON THE STUDY OF MATHEMATICS.
we represent T^8oVo ? We must here extend our lan
guage 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 denom
inator. Something of the sort has already been done
in representing a number of tens, hundreds, or thou
sands, 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 de
nominator, and place a point before the first cypher,
the fraction To8o8o
or 3.625
2125 • 2.125
1500 1.5
3625 3.625
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 to
gether. Thus 100 X 1000 = 100000, 10 X 100 = 1000,
etc. Therefore the denominator of the product, which
is the product of the denominators, has as many cy
phers 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:
13 12 156
loo" x To = looo' or ' i '
4 6 24
1000 ^ 100 100000'
or . 004 X • 06 = . 00024, etc.
DECIMAL FRACTIONS. 47
It is a general rule, that wherever the number of fig
ures falls short of what we know ought to be the num
ber 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 demon
strated 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 X 10, nor 10 X 10
X 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 pro
ducts, either of 5 or 2, or of both combined, such as
5x2, 5x5x2, 5x5x5, 2x2, etc. To show that
this is the case, take any fraction with such a denomi
nator ; for example, = -. Multiply the numera-
«* X ** X «*
tor and denominator by 2, once for every 5, which is
contained in the denominator, and the fraction will
then become
13X2X2X2 2x 2X 2X 13
5X5X5X2X2X2'' 10x10x10'
which is yVVo' or -l^. 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 de
nominator, produce a decimal number, and then mul-
48 ON THE STUDY OF MATHEMATICS.
tiply both the numerator and denominator by that
number.
No fraction which has any other factor in its de
nominator can be reduced to a decimal fraction ex
actly. 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 com
mitted, it would not make a difference which would
be considered as of any consequence, as in all prob
ability 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 sur
vey, 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 in
struments hitherto made would show such an error:
and secondly, because if they could, no material dif-
DECIMAL FRACTIONS. 49
ference would be made in the result by a correction of
it. Again, we know that almost all bodies are length
ened 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, percept
ible 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 meas
urement of the base of a survey, which is several miles
in length and takes many days to perform, it is neces
sary 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 ef
fect, it would be useless to avoid even were ft pos
sible. We see, therefore, that we may, instead of a
fraction, which cannot be reduced to a decimal, sub
stitute a decimal fraction, if we can find one so near
to the former, that the error committed by the substi
tution 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 ^T. If we divide
the series of numbers 70, 700, 7000, etc., by 11, we
shall obtain the following results :
50 ON THE STUDY OF MATHEMATICS.
-7-0- gives the quotient 6, and the remainder 4, and is 6T4-j
a 63 " 7
t< 636 « 4
63 " 7 63T7T
io_o_o_o « 6363 " 7 6363T7T
etc. etc. etc.
Now observe that if two numbers do not differ by
so much as 1, their tenth parts do not differ by so
much as T^, their hundredth parts by so much as TJ^,
their thousandth parts by so much as T^Vo~> an<^ so on >
and also remember that T7T is the tenth part of Jf, the
hundredth part of -7T°T°-, and so on. The two following
tables will now be apparent :
iQ- does not differ from 6 by so much as 1
iff. « 63 " 1 *
vooo « 636 " 1
ip_o_oo tt 6363 " 1
etc. etc. etc.
Therefore
T7T does not differ from T6^ or . 6, by so much as T^ or . 1
r7T " TfW"-63 " rW"-01
T7T "
etc. etc. etc.
We have then a series of decimal fractions, viz., .6,
.63, .636, .6363, .63636, etc., which continually ap
proach more and more near to T7T, and therefore in
any calculation in which the fraction T\ appears, any
one of these may be substituted for it, which is suffi
ciently near to suit the purpose for which the calcula
tion is intended. For some purposes .636 would be a
DECIMAL FRACTIONS. 51
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 f J 2, and the sec
ond |JJJ. The quotient of these by the ordinary rule
is f 420 a ^ or 6420. 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 approxi
mation, according to the nature of the factors in the
denominator.
When the decimal fraction corresponding to a com
mon fraction cannot be exactly found, it always hap
pens that the series of decimals which approximates
to it, contains the same number repeated again and
again. Thus, in the example which we chose, T7T 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 con
sisting entirely of repetitions of the figures 63 after the
decimal point. Thus, in finding ^, 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 thor
oughly understood without some knowledge of alge-
52 ON THE STUDY OF MATHEMATICS.
bra. It is sufficient for the student to know that he
can always either reduce a common fraction to a deci
mal, or find a decimal near enough to it for his pur
pose, though the calculation in which he is engaged
requires a degree of accuracy which the finest micro
scope will not appreciate. But in using approximate
decimals there is one remark of importance, the ne
cessity for which occurs continually.
Suppose that the fraction 2.143876 has been ob
tained, and that it is more than sufficiently accurate
for the calculation in which it is to be employed. Sup
pose 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 con
cerned, 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, be
cause 900 is nearer to 876 than 800. In the same
manner, it will be nearer the truth to write 2.144 in-
DECIMAL FRACTIONS. 53
stead 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
of arithmetical calculation, considered as a part of
general Mathematics. With respect to the commer
cial 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 be
come familiar with algebraical operations, when he
will find no difficulty in understanding the principles
or practice of any of them. He should, before com
mencing the study of algebra, carefully review what
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 arith
metic 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 estab
lishing the rules of arithmetic. His progress in the
54 ON THE STUDY OF MATHEMATICS.
former science depends most materially, if not alto
gether, 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 prin
ciples and practice of arithmetic, and not before, he
should commence the study of algebra. It is usual
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 ele
ments of geometry should not precede those of alge
bra, and be studied together with arithmetic. In this
case the student should read some treatise which re
lates 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 begin
ner. We suppose him to be well acquainted with
what has been previously laid down in this treatise,
particularly with the meaning of the signs -f-, — , X>
and the sign of division.
CHAPTER VI.
ALGEBRAICAL NOTATION AND PRINCIPLES.
TTTHENEVER any idea is constantly recurring,
* * the best thing which can be done for the per
fection of language, and consequent advancement ol
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 rep
resent numbers, and the operations which are per
formed upon them. The first step was to write simple
signs for the first numbers, instead of words at full
length, such as 8 and 7, instead of eight and seven.
The next was to give these signs an additional mean
ing, according to the manner in which they were con
nected with one another; thus 187 was made to rep
resent 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, mul
tiplication, and division ; thus 5 + 8 was made to rep
resent 8 added to 5, and so on. With these signs
56 ON THE STUDY OF MATHEMATICS.
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 differ
ence, gives the greater of the two numbers. For ex
ample, 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 our
selves 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 follow
ing truths :
16 10 16-10 _,«
2 2
27 + 8 27-
~~~ ~
15+9 15—9
~T~ + ~2~ =15'
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 :
ALGEBRAICAL NOTATION AND PRINCIPLES. 57
First No. -f- Second No. First No. — Second No.
~^2~ ~^~
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 propo
sition, 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 asser
tion, which we should find to be always true :
(First No. -f Second No.) X (First No.— Second No.)
= First No. X First No. — Second No. X Second No.
When any sentence expresses that two numbers or
collections of numbers are equal to one another, it is
called an equation* thus 7-f- 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 fre
quently? This is done by putting letters of the alpha
bet to stand for these numbers. Suppose, e. g., we
let x stand for the first number, and y for the second,
the two assertions already made will then be written :
i x—y — x
~~~
(* +}') X (x —}'} = x X x —y X y.
By the use of letters we are thus enabled to write
sentences which say something of all numbers, with a
* As now usually defined an equation always contains an unknown quan
tity. See also p. 91. — Ed.
58 ON THE STUDY OF MATHEMATICS.
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 arithme
tic. Thus x -\- z is the sum of the numbers represented
by x and z. The following equations are sufficiently
evident :
* +J + *==#4- * +> =y -\-z-\-x.
If a = b, then a + c = b-\- c, a-\- c + d=b + c + d,
etc.
3. The sign — is used for subtraction, as in arith
metic. The following equations will show its use :
x -\- a — b — t-\- e = x-\- a-\- e — b — c
= a — c -\- e — b-\- x.
\{a = bt a — c = b — c, a — c + d=b — c -\- d, etc.
4. The sign X is used for multiplication as in
arithmetic, but when two numbers represented by let
ters are multiplied together it is useless, since aX b
can be represented by putting a and b together thus,
ab. Also a X b X c is represented by a be; aX#X a,
for the present we represent thus; a a a. When two
numbers are multiplied together, it is necessary to
ALGEBRAICAL NOTATION AND PRINCIPLES. 59
keep the sign X ; otherwise 7 X 5 or 35 would be mis
taken for 75. It is, however, usual to place a point
between two numbers which are to be multiplied to
gether; thus 7x5x3 is written 7.5.3. But this
point may sometimes be mistaken for the decimal
point : this will, however, be avoided by always writ
ing the decimal point at the head of the figure, viz.,
by writing ^-f^fJ- thus, 234*61.
5. Division is written as in arithmetic : thus, —
b
signifies that the number represented by a is to be di
vided by the number represented by b.
6. All collections of numbers are called expres
sions ; thus, a + b, a + b — c, aa-\-bb — d, are alge
braical expressions.
7. When two expressions are to be multiplied to
gether, it is indicated by placing them side by side,
and inclosing each of them in brackets. Thus, if
a -|- b -f c is to be multiplied by d-\- e -f /, the product
is written in any of the following ways :
.d+e+f,
8. That a is greater than b is written thus, 0>£.
9. That a is less than b is written thus, a * =
ab -j- a c — de-\- a-\- b -\- c — e — f — d.
When the first term changes its place, as in the fourth
of these expressions, the sign -f- is put before it, and
should, properly speaking, be written wherever there
is no sign, to indicate that the term in question in
creases 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 re
collected 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 under
stood that the expression in brackets is to be consid
ered 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 difference of b
64 ON THE STUDY OF MATHEMATICS.
and c which is to be subtracted from a. If a = 12,
b = $y and ab.
Such are the mistakes which beginners almost uni
versally make, mostly for want of a moment's consid
eration. They attempt to reduce quantities which
cannot be reduced, which they do by adding the ex
ponents 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 can
not 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 ac
cording to the sign, and affixing the common letters
with their indices. When there is no coefficient, as
66 ON THE STUDY OF MATHEMATICS.
in the expression a2 b, the quantity represented bya2^
being only taken once, 1 is called the coefficient.
Thus,
ab-\-§ab — ab — 1 ab =
-j- 3 xy^ — 5 xy* -j- xy* =
The student must also recollect that he is not at lib
erty to change an index from one letter to another, as
by so doing he changes the quantity represented.
Thus cfib and ab^ 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 pleas
ure instead of letters in the expressions, and calculat
ing their values ; but, in conclusion, the following re
mark must be attended to. If it were asserted that the
(P + P. 2ab
expression — is the same as a -4- b — •= — — -. , and
a-\- b 2a — b
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 3i. 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 2^ + 8, if x stand
for 12, these are the same, but if it stands for any
other number they are not the same.
CHAPTER VII.
ELEMENTARY RULES OF ALGEBRA.
THE student should be very well acquainted with
the principles and notation hitherto laid down
before he proceeds to the algebraical rules for addi
tion and subtraction. He should then take some sim
ple 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 :
a — b
c — d
Add"
or more properly thus : Find (a — £) -f (c — ^).
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 -y this quantity must therefore be de
creased 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 -j- c — b — d or
68 ON THE STUDY OF MATHEMATICS.
a — b -j- c — d. From a few reasonings of this sort the
rule may be deduced ; and not till then should an ex
ample be chosen so complicated as to make the stu
dent lose sight for one moment of his demonstration.
The process of subtraction we have already performed.
and from a few examples of this method the rule may
be deduced.
The multiplication of a by c — d is performed thus :
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 6x5 is
the same as 6 X 14 — 6 X 9, or as 84 — 54. Also that
f(t — T\)> °r *XTt* is the same as f X* — f XA,
or as ^2T — ¥\. Upon similar reasoning the following
equations may be proved :
a(b -\- c — d} = ab -f- ac — ad.
— &r)xz=.pxz -\-pqxz — arxz.
or aa
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
ELEMENTARY RULES OF ALGEBRA. 69
ab — ac -\- 02,
or ab — ac-\-aa,
we see that in its formation every term has been mul
tiplied by a ; that is, it has been made by multiplying
b — c -\- a by a,
or a by b — c -f- a.
There will now be no difficulty in perceiving that
a c -f a d -f b c -f b d = a ( c -f- d ) -f b ( c -f d ) =
— - — ps— (qr — qs}
By reasoning in the same way we may prove that
(/ — 4)(r + *)=pr+ps — qr — qs
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 -f- 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 -f- multiplied by -f- gives -f; — multiplied
by -)- gives — ; and so on. He must recollect that
the signs -)- and — are not quantities, but 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 multi
plied by take away gives add, as that — multiplied by
— gives +.*
The only way in which the student should accus
tom himself to state this rule is the following : "In
*Frend, Principles of Algebra. The author of this treatise is far from
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.
72 ON THE STUDY OF MATHEMATICS.
multiplying two algebraical expressions, multiply each
term of the one by each term of the other, and wher
ever two terms are preceded by the same sign put -f-
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 them
selves, such as
let him, for the present, reject the example in which
it occurs, and defer the consideration of such equa
tions 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 noth
ing;* 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 -f- 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.
*For a full critical and historical discussion of this point, see Duhamel.
Des mtthodes dans les sciences de raisonnement, 2me partie, chap. XIX. (third
edition, Paris, Gauthier-Villars, 1896).— Editor.
ELEMENTARY RULES OF ALGEBRA. 73
In writing algebraical expressions, we have seen
that various arrangements may be adopted. Thus
ax* — bxA^ c may be written as c-^-ax* — bx, or — bx
-\-c-\-a x-. 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 descend
ing powers of x; had it been written thus, c — bx -\- ax2,
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 evi
dent from multiplying two unarranged expressions to
gether, 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 :
ON THE STUDY OF MATHEMATICS.
td £
«v
<\
4
g
c
td £
*» • ."
0*5*. 0 ^
0, • ^ 0- .
C ^
o
CO* ' w
.
ft
5X ft ^
K^
H 5-
h*
^o Ho S
0
1
4- £
4- 4- w
c^ ^^ ^
1
CO
HP
to c
^ V
M
^ ft.
^ Ho *
4-
4- H
SL ^
LN° Ho
fc 5-
44-4-
+ 4" Ho
tv
00
ft ^ ^
^5 ~r
4- 4-
« ^ ^
•51
^ Oi
M CO CO
H
"* ^
"t t"
1 1
Sv, ^ c^
CO tO
H *L ^
^ ^
*^ Od
O5 Oi
4 +
4
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 first
power in one by the first power in the other. Also,
that the lowest power of x, or should it so happen,
ELEMENTARY RULES OF ALGEBRA. 75
the term in which there is no power of x, is made by
multiplying the last terms in each factor. These be
ing the highest and lowest, there can be no other such
power, consequently neither of these terms can co
alesce with any other, as is the case in the intermedi
ate products. This remark will be of most convenient
application in division, to which we now come.
Division is in all respects the reverse of multipli
cation. 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 — -, and all that remains is to see whether that
b
fraction can be represented by a simple algebraical
expression without fractions or not; just as in arith
metic the division of 200 by 26 is the reduction of the
fraction 2££. to a whole number, if possible. But we
must here observe that a distinction must be drawn
between algebraical and arithmetical fractions. For
example, is an algebraical fraction, that is, there
is no expression without fractions which is always
equal to . But it does not follow from this that
a — b a + b
the number which - - represents is always an arith-
a — o
metical fraction ; the contrary may be shown. Let a
•stand for 12, and b for 6, then —^--J is 3. Again,
a — b
a*-{-ab is a quantity which does not contain algebrai
cal fractions, but it by no means follows that it may
not represent an arithmetical fraction. To show that
76
ON THE STUDY OF MATHEMATICS.
it may, let a = % and 6 — 2, then a2-\-afr = li or J.
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,
which only differ in this, that "whole number" in the
arithmetical proposition is replaced by "simple ex
pression"* 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 :
The sum, difference, or pro
duct of two whole numbers, is a
whole number.
One number is said to be a
measure of another when the
quotient of the two is a whole
number.
The greatest common meas
ure of two whole numbers is the
greatest whole number which
measures both, and is the pro
duct of all the prime numbers
which will measure both.
When one number measures
two others, it measures their
sum, difference, and product.
In the division of one number
by another, the remainder is
measured by any number which
measures the dividend and di
visor.
The sum, difference, or pro
duct of two simple expressions
is a simple expression.
One expression is said to be a
measure of another when the
quotient of the two is a simple
expression
The greatest common meas
ure of two expressions is the
common measure which has the
highest • exponents and coeffi
cients, and is the product of all
prime simple expressions which
measure both.
When one expression meas
ures two others, it measures
their sum, difference, and pro
duct.
In the division of one expres
sion by another, the remainder
is measured by any expression
which measures the dividend
and divisor.
*By a simple expression is meant one which does not contain the princi
pal letter in the denominator of any fraction.
ELEMENTARY RULES OF ALGEBRA. 77
A fraction is not altered by A fractional expression is not
multiplying or dividing both its altered by multiplying or divid-
numerator and denominator by ing both its numerator and de-
the same quantity. nominator by the same expres
sion.
In the term simple expression are included those
quantities which contain arithmetical fractions, pro
vided there is no algebraical quantity, or quantity rep
resented by letters in the denominator; thus ±a £ -f J
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 con
tains more than one term. For example, what is
42 fl4 ft c divided by 6 a2 b t? that is, what quantity must
be multiplied by §cPbc, in order to produce 42 #4 b* c.
This last expression written at length, is ¥Z a a a abb be,
and 42 is GX?. We can then separate this expression
into the product of two others, one of which shall be
bcPbc, or §aabc\ it will then be Saab <: X ? a abb,
and it is 7 aabb which must be multiplied by §aabc
in order to produce 42 cfib* 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 c1, <:*, c*,
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 r1, and call it the first power
of c. Again, we have represented c-\-c, c -\-c-\-c,
c -j- c -\- c -j- c, etc. by 2c, 3r, 4
tend to some examples of each operation upon alge
braic fractions, by way of practice in the previous
operations. As the subject is not one which presents
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.
-K«-~*)=fci (i)
) — (a — b}=2b (2)
(a — b} = b (3)
ELEMENTARY RULES OF ALGEBRA.
89
fr+V-*+*J+J
(*)
(2,, + ^ = 4^ + 4^, + ^
(6)
(a+f)(a^-l,)=J—P
(7)
(.r -\- a) (x -{- b~) = x1* -{- (a -\- b} x -}- a b \
(x — a) (x — <£) == x'2 — {a -\- b} x -{- a b )
(8)
a ma
0>)
b m b
c ad-\-c c ad — c
ST \ • /7
(10)
(ii)
d d d d
a c ad-}- be a c ad — be
~b d~ bd ' b d~ bd
a ac a a c ac
(12)
c
a a
b a c
(13)
a a
b ad c
c be b
(14)
d d
1 b
a a
~b
(15)
CHAPTER VIII.
EQUATIONS OF THE FIRST DEGREE.
TT 7E 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 de
pend 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
the truth of the assertion made in this algebraical sen
tence is the same, whether a be considered as repre
senting 1, 2, 2J, 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
a2 — 1 by a-\- 1 ; the second side shows you the result
of that division. An equation of this description is
called an identical equation, because, in fact, its two
sides are but different ways of writing down the same
EQUATIONS OF THE FIRST DEGREE. QI
number. This will be more clearly seen in the iden
tical equations
a-\- a = 2a, 7a — 30 -j- b = ±a — 3£ -f 4£, and ^- X ^ = a.
b
The whole of the formulae 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, pro
vided 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 « -(- 1 =8, we
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 repre
sent consistently with this equation is 7, as any other
supposition involves absurdity. This is a new spe
cies of equation, which can only exist in some partic
ular 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 distin
guish them from identical equations, they are called
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
Q2 ON THE STUDY OF MATHEMATICS.
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 suppos
ing 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
.general rule can be given. The problem is some prop
erty 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 fol
lowing 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 num
ber?
Let x stand for the number which is to be found.
Then x-\- 56 = 200 — 2jc.
If, instead of 56, 200, and 2, any other given num-
EQUATIONS OF THE FIRST DEGREE. 93
bers, <7, b, and c, are made use of in the same man
ner, the equation which determines x is
x -\- a = b — ex.
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 be
fore 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 #-j-4
hours, and will have travelled (^-f 4)2, or 2x-\-8
miles. The second has been gone x hours, and will
have travelled 3x miles. And, when the second over
takes the first, they have travelled exactly the same
distance, and, therefore,
3*=2* + 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,
b x = ax -j- 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?
Suppose that A paid x pounds,
then B paid 2^- ...
C paid x -f 2 x or 3 x . . .
D paid 2x + 3x or 5# . . .
94 ON THE STUDY OF MATHEMATICS.
All together paid x-\-2x + 3x + 5x, or 11 *, there
fore
There are two cocks, from the first of which a cis
tern 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 T^ of it, in
two hours T2^, etc., and in x hours y^. Similarly, in
x hours, the second cock fills T^ of the cistern. When
the two have exactly filled the cistern, the sum of
these fractions must represent a whole or 1, and,
therefore,
- + --1
12 + 15"
If the times in which the two can fill the cistern are a
and b hours, the equation becomes
A person bought 8 yards of cloth for £& 2s., giving
9s. a yard for some of it and 7j. a yard for the rest ;
how much of each sort did he buy?
Let x be the number of yards at Is. Then 1 x is
the number of shillings they cost. Also 8 — x is the
number of yards at 9s. , and (8 — x")g, or 72 — 9#, is
the number of shillings they cost. And the sum of
EQUATIONS OF THE FIRST DEGREE. 95
these, or 7 x + 72 — 9jr, is the whole price, which is
^3 2.$-., or 62 shillings, and, therefore,
7*4-72 — 9# = 62.
These examples will be sufficient to show the
method of reducing a problem to an equation. As
suming 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
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 conun
drums with which such books are usually filled, that,
therefore, he is not made for the study of mathemat
ics, and should give it up. His future progress de
pends 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 con
siderable step for himself, but, without doing this, he
may derive all the benefits which the study of mathe
matics 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
g ON THE STUDY OF MATHEMATICS.
this. We do not wish to depreciate its utility as an
exercise for the mind, or to hinder all from attempt
ing to conquer the difficulties which present them
selves ; but to remind every one that, if he can read
and understand all that is set before him, the essen
tial 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 con
dition. 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 un
known 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 dimin
ished, 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
— = — . 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 = dt
add c to both quantities, then
a-\-b — c+ c = d-\- c or a-\-b
Again subtract b from both quantities, then a -f- b —
c — b = d — bt or a — c = d — b. Without always re-
EQUATIONS OF THE FIRST DEGREE. Q7
peating 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 abridg
ment 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 "sub
tract 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 f, ^, and -f±. If we
multiply them all by the product of the denominators
4x7x1*4, or 392, all the products will be whole num-
TU -11 u 3x392 1X392 , 5x392
bers. They will be -^ , --^= , and — ^- — ,
and since 392 is measured by 4, 3 X 392 is also meas
ured by 4, and - - 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 - , — , and -r-j-,, will become simple
b ae oaf
9 ON THE STUDY OF MATHEMATICS.
expressions, if they are multiplied by b X deX bdf, or
tPd** ef. But the most simple common multiple of b,
de, and ^//, is bdefy which should be used in pref
erence to lPd?ef.
This being premised, we can now reduce any equa
tion which contains fractions to one which does not.
For example, take the equation
x_ 2* _ 7_ _ 3 — 2*
3" 5" :~10~ 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 mul
tiple of the denominators 3, 5, 10, and 6, which is 30.
If we multiply both equal quantities by 30, the equa
tion becomes
30* 60*__210 30(3 — 2*)
IT ~5~ :~HT " ~~6~~
30*. 30 60*. 60
But -=— is — X x> or 10*, — — is -=- X x, or 12*, etc.;
o O 00
so that we have
10*-fl2* = 21 — 5(3 — 2*), (2)
or 10*+12* = 21— (15 — 10*), (3)
or 10*+ 12*r=21 — 15 + 10*. (4)
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
EQUATIONS OF THE FIRST DEGREE. 99
signify this by putting the numerator between brack
ets, 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 subtrac
tion has been performed, as is done in equation (4).
The following operations will afford exercise to the
student, sufficient, perhaps, to enable him to avoid
this error :
b — c-\- d — e af-\- b — c -J- d — e
~T~ ~T~
a -f = — f
a + b a + b'
We can now proceed with the solution of the equa
tion. Taking up the equation (4) which we have de
duced from it, subtract 10* from both sides, which
gives 10# + 12*— 10# = 21 — 15, or 12# = 6: divide
12 v 6
these equal quantities by 12, which gives — ^- =^, or
A: = J. This is the only value which x can have so as
to make the given equation true, or, as it is called, to
satisfy the equation. If instead of x we substitute J,
we shall find that
100 ON THE STUDY OF MATHEMATICS.
j 2XI= 7 3-2Xj fr 1 1 ^ _2
3 5 "10 6 6 "*" 5 ~~ 10 ~ T1
this we find to be true, since
I 1 ' U 7 2 _22 ,1 n _22
6" "" 5 1S 30' S d 10 ~~ IT "~ 60' S d 30 ~ 60'
In these equations of the first degree there is one un
known 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, put
ting letters in place of numbers :
— _|_ ^ — — __ f^gJ?L
» '~l c ~ ~ e " h
The solution of this equation is as follows : multi
ply both quantities by aceh, the most simple multiple
of the denominators, it then becomes :
ace hoc abcehx acdeh aceh(f — gx)
a ~7~ ~~e~ ~~JT '
or, c ehx -\- ab ehx = acdh — ace(f — gx},
or, c ehx -f- abehx — acdh — acef-\- ac egx.
Subtract acegx from both sides, and it becomes
cehx-\-abehx — ac egx = acdh — acef,
or, (c e h -j- a b e h — a c eg) x = acdh — a c ef.
Divide both sides by ceh^abeh — aceg, which gives
ac dh — ac ef
£ 1 ^
ceh-\-abeh — ac eg'
The steps of the process in the second case are ex
actly the same as in the first ; the same reasoning es-
EQUATIONS OF THE FIRST DBGKSifc . lOt
tablishes them both, and the same tfrfo.rs, aye to^be
avoided in each. If from this we wish to find the so
lution of the equation first given, we must substitute
3 for a, 2 for b, 5 for c, 7 for //, 10 for e, 3 for /, 2 for
g, and 6 for h, which gives for the value of x,
5X10X6 + 3X2X10X6 — 3X5X10X2
3X5X12 180
' 3X2X10 X 6' ' ' 360'
which is ^, the same as before.
If in one equation there are two unknown quanti
ties, 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 = S admits of an in
finite number of solutions, for take x to represent any
whole number or fraction less than 8, and let y repre
sent what x wants of 8, and this equation is satisfied.
If we have another equation of condition existing be
tween the same quantities, for example, 3jc — 2y = 4;
this second equation by itself has an infinite number
of solutions: to find them, y may be taken at pleasure,
4 _i_ 2y
and x = — ~— . Of all the solutions of the second
o
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 solu
tion 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 hap-
102 ONT THE STUDY OF MATHEMATICS.
pens .whenever oiie of the equations can be deduced
from the other. For example, when ,r-|-j — 8, and
4x — 29 = 3 — 4y, the second of these is the same as
4# + 4y = 3 + 29, or 4^-f 4^ = 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 pe
culiar difficulty ; we shall therefore proceed to the
consideration of the isolated negative sign.
CHAPTER IX.
ON THE NEGATIVE SIGN, ETC.
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 let
ters 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 — I) 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 -j- 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
104 ON THE STUDY OF MATHEMATICS.
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 mis
takes give rise to them, and in what manner they af
fect the process of investigation.
We would recommend to the beginner to make
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 ob
jection. We shall observe the effect of this error in
denoting a subtraction on every species of investiga
tion to which we have hitherto come, and shall de
duce rules which the student will recollect are the re
sults of experience, not of abstract reasoning. The
extensions to which he will be led have rendered Al
gebra 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
ON THE NEGATIVE SIGN, ETC. IO5
part of the groundwork of modern Algebra and should
be considered by the student who is desirous of mak
ing 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.
Where the signs of every term in an expression are
changed, it is said to have changed its form. Thus
-\- a — b and -j- 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 -f 3 — 8 is connected with any
other number thus, 56 -f- 3 — 8. This may be written
56 -|- 3 — (3 + 5), or 56 -f 3 — 3 — 5, or 56 — 5. It ap
pears, 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 writ
ing 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
IO6 ON THE STUDY OF MATHEMATICS.
a = 0, then b = c. We can now understand the follow
ing equations :
a — b-\-c — d= — e, or b-\- d — a — c = e,
— ai — b'2 = — d—e, or a2 -f £2 — 2ab = d-{- e.
We must not commence any operations upon such
an equation as a — b = — c, until we have satisfied our
selves of the manner in which they should be per
formed, by reference to the correct form of the equa
tion. This correct form is b — a = c. This gives
d-\- b — a = d-\-c, or d — (a — b} = d-\-c. Write in
stead 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 — ( — <:) 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 — ( — ^), 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
ON THE NEGATIVE SIGN, ETC. lO'J
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 nega
tive sign. Thus the real form of d — ( — c~} is d-\-c.
Again, take the real form b — a = c of the equation
a — 1> = — c, and it follows that d — (/> — a) = d — x
. a2 a2 x
and — -- ax= — - -- a — o
which two equations only differ in the form in which
a appears. For, if the form of a in the first equation
CL CL 3C
be altered, that of— and — — is unaltered, -4- ax be-
b b
comes — ax, and -}- a becomes — a. We now solve
the two equations in opposite columns.
a1 a? x a2 a2 x
-=~ -4- ax = — — -4-a — b - -- a x = — - -- a — b
b b b b
cP + abx^cPx + ab — fi a^ — abx — a^ x — ab — P
= a*x -f abx
The only difference between these expressions
ON THE NEGATIVE SIGN, ETC. Ill
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 pro
duced. 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 so
lution 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 iden
tical equation, the form of one or more of its quanti
ties may be altered throughout, and the equation will
still remain identically true. For example,
a — o
Change -j- b into — />, and this equation will become
which last, common division will show to be true.
Again, suppose than when a, />, and c are in a
given form, which we denote by -\- a, -\- 1), and -}- c,
the solution of a problem is,
112 ON THE STUDY OF MATHEMATICS.
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.
FORMS OF a, b, AND C. VALUES OF X.
P — lac
+*,
c — b
a — c — b
-\-a, — by — c —
— a, — by —
b — a — c
Also, the expression for x may be written in the
following different ways, the forms of a, b, and c re
maining the same :
— b' b — a — c' a-\-c — 1>' b — a — c '
We now proceed to apply these principles to the
solution of the following problems :
q -- 1 --- 1 -- 1 -- \D
A B H
Two couriers, A and B, in the course of a journey
between the towns C and D, are at the same moment
ON THE NEGATIVE SIGN, ETC. 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 direc
tions, 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 dis
tance AB be called a,
Case first. — Suppose that they are going in the
same direction from C to Z>, 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 ;;/ miles an hour, he travels the distance x in
*£
— hours. Again, B travels the distance x — a in
m n
hours. These times are the same, and, therefore,
x x — a ma
— = - - or * = - - = AH
m n m — n
and x — a= - =BH.
m — n
The time which elapses before they meet is
x a
— or - .
m m-
114 ON THE STUDY OF MATHEMATICS.
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.
c\
Then since A travels through HA or x in the time
during which B travels through HB, or x -j- a, in the
same manner as in the last case, we show that
x x -- a ma
or x=
m n n — m
, na
and x 4- a = — - = BH.
n — m
The time elapsed is ...
-m
Case third. — If they are moving from D to C, and
if B moves faster than A, the point Zfis the same as
in the last case, since, if having in the last case ar
rived 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.
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
ON THE NEGATIVE SIGN, ETC. 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.
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 Aff=x.
cl 4_| lg p
Then A moves through AH, or x, in the same time as
B moves through BH, or a — x. Therefore
x a — x
— = , or
m n
ma
x =
a — x =
The time elapsed is . . .
m-\-n
na
m-\- n
a
m -4- n
Case sixth. — Let them be moving in contrary direc
tions, 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
n6
ON THE STUDY OF MATHEMATICS.
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.
Circumstances of the case.
Direction of
the point H.
Value
of AH.
Value
of BH.
Time of meeting
j Both move from C to D,
Between
m a
n a
a
' A moves faster than B.
j Both move from C to Z>,
B and D.
Between
m — n
m a
in — n
n a
ttt — n
' A moves slower than B.
j Both move from D to C,
A and C.
Between
n — m
tna
n — m
n a
n — m
a.
5> 1 A moves slower than B.
j Both move from D to C,
A and C.
Between
n — m
ma
n — m
na
n — m
i A moves faster than B.
j A moves towards D and
B and D.
Between
m — »
vn a
m — n
n a
m — n
CL
after
5' ' B towards C.
j A moves towards C and
A and B.
Between
m + n
m a
m -f- n
n a
m -f- n
1 B towards D.
A and B.
m + n
in -f- n
tn -\- n
Now
and
are the same quantity written in
m — n n — m
different forms, for n — m is — (m — «); and accord
ing to the rules
a a
m
Similarly
ma
ma
n — m 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 in the first, and
6 m — n m — n
ON THE NEGATIVE SIGN, ETC.
in the second. Again, in the same cases, in the first
of which the time of meeting is after, and in the sec
ond 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 , and in the second , or .
m — n m — n n — m
The same remarks apply to the third and fourth ex
amples. 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
-f n, instead of — n. The expression for BH in the
first, if the form of n be likewise changed, becomes
— , which is the value of BH'm the fifth, but in
m-\- n
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 Affbe-
— ma — ma ma
come- — , or — -. , or . If we alter the
— m — n — (m-\-n) m-\-n
value of the time in the first, in the same manner, it
becomes , or — , which is of a different
— m — n m -\- n
form from that in the sixth ; but it must also be ob
served 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
n8
ON THE STUDY OF MATHEMATICS.
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
-, or — .
n m -j- n
Also the time in
Circumstances of the case.
Direction of
the point H.
Value
of AH.
Value
of BH.
Time of meeting-
j Both move from Cto D,
Between
m a
na
a
' A moves faster than B.
f Both move from Cto D,
B and D.
Between
tn — n
•m a
m — n
n a
tn — n
' A moves slower than B.
J Both move from D to C,
A and C.
Between
n — m
tn a
n — m
n a
n — m
a
1 A moves slower than B.
j Both move from D to C,
A and C.
Between
n — m
m a
n — m
n a
«_^aftcr-
i A moves faster than B.
j A moves towards D and
B and D.
Between
m — n
m a
tn — n
na
tn — n
a
'• \ B towards C.
j A moves towards C and
A and B.
Between
m + n
ma
m + n
n a
m -f- n
' ' B towards D.
A and B.
m -f- n
m -f- n
tn + n
(TABLE OF PAGE 116 REPEATED.)
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
ON THE NEGATIVE SIGN, ETC.
second case from those of the first, this is done by
changing the sign of all those quantities whose direc
tions are opposite in the second case to what they are
in the first, and if any answer should appear in a neg
ative form, such as , when m is less than n.
m — n ma
which may be written thus , it is a sign that
n — ni
the quantity which it represents is different in direc
tion 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 fol
lowing problem :
A workman receives a shillings a day for his labor
or a proportion of a shillings for any part of a day
which he works. His expenses are b shillings every
120 ON THE STUDY OF MATHEMATICS.
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 re
ceives ax shillings, and during the m days his expen
diture is Inn shillings, and since his gain is the differ
ence between his receipts and expenditure:
a x — bm=c
b m 4- c
or x= ! —
a
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
b m — a x = c,
which, when its form is changed, becomes
a x — bm=. — c,
an equation which only differs from the former in hav
ing — c written instead of c. The solution of the equa
tion is
bm — c
~~^~>
which only differs from the former in having — c in
stead of -f c. It appears then that we may alter the
solution of a problem which proceeds upon the sup
position of a gain into the solution of one which sup
poses an equal loss, by changing the form of the ex
pression 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 nega-
ON THE NEGATIVE SIGN, ETC. 121
tive, 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 ex
ample, 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 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 re
sult from observation only, which taught us first to
deduce rules for making that alteration in the result
which arises from altering -f c into — c at the com
mencement ; 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, re
serving all metaphysical discussion upon such quanty
122 ON THE STUDY OF MATHEMATICS.
ties as -f 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 difficul
ties, which are generally, if possible, as much miscon
ceived by the beginner as the use of the negative sign.
Take any fraction — . Suppose its numerator to
remain the same, but its denominator to decrease, by
which means the fraction itself is increased. For ex-
5 5
ample, ^ is greater than ^— or the twelfth part of 5
is greater than its twentieth part. Similarly, — f is
2i 4i
greater than — |, etc. If, then, b be diminished more
and more, the fraction - - 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 = — . Then
— or — is m. Now since b may diminish so as to be
•m
equal to any part of a, however small, that is, so as
to make m any number, however great, — which is
= m may be any number however great. This dimi
nution of b, and the consequent increase of—, 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 — increases, and in the interval in
b
which b passes from its first magnitude to 0, the frac
tion — - passes from its first value through every pos
sible greater number. Now, suppose that the solution
of a problem in its most general form is — , but that
ON THE NEGATIVE SIGN, ETC. 123
in one particular case of that problem b is =0. We
have then instead of a solution — , a symbol to which
we have not hitherto given a meaning.
To take an instance : return to the problem of the
two couriers, and suppose that they move in the same
direction from C to D (Case first} at the same rate, or
i T -r T r- i i Mia m (I
that m = n. We find that Aff= — or — - or
ma m — n n — n
-=-. On looking at the equation which produced this
*Y* y = — •
ae — bd ae — bd
Now, let us suppose that — P a-\-b
When a approaches towards b, a -f b approaches to
wards 2£, and a2 — ft and a — b approach more and
more nearly towards 0. If a = b the equation assumes
this form :
0 _2£
"o ~=T
128 ON THE STUDY OF MATHEMATICS.
This may be explained thus : if we multiply the nu-
^
merator and denominator of the fraction — by a — b
(which does not alter its value) it becomes — — .
J3a — Bb
If in the course of an investigation this has been
done when the two quantities a and b are equal to
... . A Aa — Ab ...
one another, the fraction — or — — will appear
Q ±> i) a — no
in the form -=-. But since the result would have been
A
— had that multiplication not been performed, this
£>
last fraction must be used instead of the unmeaning
t ° TU *u r *• <*2 — 6* (a + &)(* — 6)
form — . Thus the fraction — or - — , ,
0 a,b c(a — b~} c(a — b-)
18 the fraction after its numerator and denomi-
c
nator have been multiplied by a — ^, and may be used
in all cases except that in which a = b. When the
form -^ occurs, the problem must be carefully ex
amined in order to ascertain the reason.
CHAPTER X.
EQUATIONS OF THE SECOND DEGREE.
T7VERY operation of algebra is connected with an-.
*— ' other which is exactly opposite to it in its effects.
Thus addition and subtraction, multiplication and di
vision, are reverse operations, that is, what is done
by the one is undone by the other. Thus a-\-b — b is
a, and — is a. Now in connexion with the raising of
powers is a contrary operation called the extraction
of roots. The term root is thus explained : We have
seen that a a, or #2, is called the square of a\ from
which a is called the square root of a1. As 169 is
called the square of 13, 13 is called the square root of
169. The following table will show how this phrase
ology is carried on.
a is called the square root of a2, . . denoted by V a2
a " " " cube root of 03, . . " " f/a?
a" " " fourth root of 04, .. " " i/a*
a " " " fifth root of a5, . . " " v~a*
etc. etc. etc.
130 ON THE STUDY OF MATHEMATICS.
If b stand for #5, \/ b stands for a, and the foregoing
table may be represented thus :
If a* = b,a=VH\
if a2 = Z>,a = it/J, etc.
The usual method of proceeding is to teach the
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 solu
tion 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 ex
tracted. For example, 7 is not made by the multipli
cation 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 | X f or -^6- 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 T-jytnTFo^'iro o~ cou^
be found a fraction which has a square root, and the
EQUATIONS OF THE SECOND DEGREE. 13!
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
2
buy? Let x be this number, then — is the price of
2 x
one yard, and x= — or x1 = 2. This, from what we
oc
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, !• 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 con
tains 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 *2 — 3 = 4#2 — 15, *2-f 3.* = 2.*:2 — *— 1, etc.
132 ON THE STUDY OF MATHEMATICS.
By transposition of their terms, they may always be
reduced to one of the following forms :
ax2 — b x — c — 0.
For example, the two equations given above, are
equivalent to 3 *2 — 12 = Q, and *2 — 4# — 1 =0, which
agree in form with the second and last. In order to
proceed to each of these equations, first take the equa
tion x'2 = a2. This equation is the same as x2 — #2 — 0,
or (x-\-a)(x — 0) = 0. Now, in order that the pro
duct of two or more quantities may be equal to noth
ing, it is sufficient that one of those quantities be noth
ing, and therefore a value of x may be derived from
either of the following equations:
or x-{-a = Q
the first of which gives x = a, and the second x = — a.
To elucidate this, find x from the following equation :
(3 x -f a) (a3 -f A:8) = (x2 -f a x] (a* -f a x -f 2 *2)
develop this equation, and transpose all its terms on
one side, when it becomes
or x — #
or (x2 — tf2) — ax — ^ = .
This last equation is true when x2 — #2 = 0, or when
EQUATIONS OF THE SECOND DEGREE. 133
x2 = a2, which is true either when x=-\-a, or x = — a.
If in the original equation -f a is substituted instead
of x, the result is 4# X 2 a* = 2 a* X 4^2 5 if — « be
substituted instead of #, the result is 0 — 0, which
show 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 x2 — 2 ax —
a2 = Q, or x2 — 2ax = a2. Every equation of the sec
ond degree, if it has one value of x, has a second, of
which x2 — a2 is an instance, where x=±a, in which
by the double sign d= is meant, that either of them
may be used at pleasure. We now proceed to the so
lution of ax1 — bx-\- c—§. In order to understand
the nature of this equation, let us suppose that we
take for x such a value, that ax2 — bx-\-ct instead of
being equal to 0, is equal to j, that is
y = ax* — bx+c* (1)
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 ax2 — bx-\- c equal to nothing, in which case m
is called a root of the equation,
ax2 — bx-\- c = b (2)
and it follows that
am2 — bm + c-=§ (3)
*In the investigations which follow, a, 6, and c are considered as having
the sipn which is marked before them, and no change of form is sujjoosed to
tane i«ut.
134 ON THE STUDY OF MATHEMATICS.
Subtract (3) from (1), the result of which is
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") — <£ — 0; there
is, therefore, another value of x, for which j = 0 ; if
we call this n we find it from the equation a (n -\-nt) —
£ = 0,
or n -f- m = - (4)
In (3) substitute for b its value derived from (4), from
which &.= a(n-}-m')'> it then becomes
am2 — am(n-\- m}-\- <4 = V 6~x ^4"= 2 ^
142 ON THE STUDY OF MATHEMATICS.
ax? — bx-\-c —
ax2 -j- b x — c =
ax2 — b x — c =
a\x * " ' """- i " 1*4- ' (D)
\ 2a J \ 2a J
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 (C) is made from that of (A) by changing the sign
of c; the second side of (C) is made from that of (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
a x1 -\- b x -\- c can be made equal to nothing by two
values of x. For example, in (1), when
. , ,
either x -\
_
Z a
A a
If we call the values of x derived from the equations
m and n, we find that
EQUATIONS OF THE SECOND DEGREE. 143
In the cases marked (B), (C^ and (D), the results
are
—
~~
iac
m=~ *=- -
and in all the four cases the form of a x* -\- b x -\- c
which is used, is the same as the corresponding form
of
a (x — ni) (x — «)
and the following results may be easily obtained. In
(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^ — 4ac is negative, the quantity a x2 -f b x -}- c
cannot be divided into factors at all, since l/^2 — ±ac
is then no algebraical quantity, either positive or neg
ative.
In (B') both, if they exist at all, are positive.
In (C') there are always real values for ;// and n,
since <$2-{-40
which gives I/— ^ = 1/^X V— 1, of which the first
factor is real and the second imaginary. Let i/^ = c,
then I/ — b = cV/ — 1. In this way all expressions
may be so arranged that j/ — 1 shall be the only im
aginary quantity which appears in them. Of this re
duction the following are examples :
1/IT24 = 1/24 I/— T = 2 1/6VZ.1
V — a X V — a= — a
V~a? x V— ~P = aV--\ X b = — ab.
The following tables exhibit other applications of
the rules :
EQUATIONS OF THE SECOND DEGREE. 153
c = aV- ,7 = _07|/;
The powers of such an expression as aV — 1 are
therefore alternately real and imaginary, and are posi
tive and negative in pairs.
(a -f b l/-^l) 2 =r
(a + b l/'-^l) (a — b i/ITI ) = ^2 _|_
a + bV'-^ = a? — P
' ~
Let the roots of the equation ax2 -f- ^jc-j-r — 0 be im
possible, that is, let P — ±ac be negative and equal
to — £2. Its roots, as derived from the rules estab
lished when ft — \ac was positive, are
Take either of these instead of x ; for example, let
b k —
154 ON THE STUDY OF MATHEMATICS.
/;2 h k &
Then a x* = --.- - ~ V— 1 - \
4a 2a 4a
Therefore, # x1 -4- b x -\- c = -- — - --- ^ — \- c, in
4a 4a Za
which, if 4ac — ^2 be substituted instead of £2, the re
sult is 0. It appears, then, that.the imaginary expres
sions which take the place of the roots when ft — 4ac
is negative, will, if the ordinary rules be applied, pro
duce the same results as the roots. They are thence
called imaginary roots, and we say that every equa
tion 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 re
sults will follow from the application of these expres
sions 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 -j- b y — 1 =
c-\- dV — 1, when all the terms are transferred on one
side, the part which is real and that which is imagin
ary must each of them be equal to nothing. The
equation just given when its left side is transposed
becomes a — c-\-(b — ^)1// — 1 = 0. Now, if b is not
equal to d, let b — d=e; then a — c + eV — 1 = 0, and
T/ — 1=— — ; that is, an imaginary expression is
equal to a real one, which is absurd. Therefore, b = d
EQUATIONS OF THE SECOND DEGREE. 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 absurd-
, _ /• _ Sf
ity in y — 1 = — — . 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 -f b V ^—l =
The imaginary expression V — a and the negative ex
pression • — b have this resemblance, that either of
them occurring as the solution of a problem indicates
some inconsistency or absurdity. As far as real mean
ing is concerned, both are equally imaginary, since
0 — a is as inconceivable as y — 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 -j- x,
and that of the son 29 -f- x ; and therefore, 56-f-.r =
2 (29 + *) = 58+ 2.Y, or x = — 2. The result is ab
surd ; nevertheless, if in the equation we change the
sign of x throughout it becomes 56 — .# = 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 was the father twice as old as the son ? the an
swer to which is, two years ago. In this case the
negative sign arises from too great a limitation in the
156 ON THE STUDY OF MATHEMATICS.
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 nega
tive form of the result. In this case the negative re
sult 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 ques
tion 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 re
sulting equation is
a
or x= -=-
the roots of which are imaginary when b is greater
a1
than — . If we change the sign of x in the equation
it becomes
and the roots of the second are imaginary, if those of
the first are so. There is, then, this distinct difference
EQUATIONS OF THE SECOND DEGREE. 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.
CHAPTER XI.
ON ROOTS IN GENERAL, AND LOGARITHMS.
THE meaning of the terms square root, cube root,
fourth root, etc., has already been defined. We
now proceed to the difficulties attending the connex
ion of the roots of a with the powers of a. The fol
lowing table will refresh the memory of the student
with respect to the meaning of the terms :
NAME OF JT. NAME OF X.
Square of a ..... x=aa Square Root of a - - xx=a
Cube of a ...... x=aaa Cube Root of a - - - xxx=a
Fourth Power of a - x=aaaa Fourth Root of a - - xxxx=a
Fifth Power of a - - x=aaaaa Fifth Root of a - - - xxxxx=a
The different powers and roots of a have hitherto
been expressed in the following way :
Powers tf2 a3 a* a5 . . am . . am + H, etc.
Roots W * Pa ya Va Va °+fa, etc.
which series are connected together by the following
equation,
*The 2 is usually omitted, and the square root is written thus Va.
ROOTS IN GENERAL AND LOGARITHMS. 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 202^3 ; that
is, to find 2 a2 fi X 2 a1 P X 2 a1 fi. The common rules
of multiplication give, as the result, 8#6^9, which is
expressed in the following equation,
Similarly (
1 ^
~ 64 "^;
and the general rule by which any single term may be
raised to the power whose index is n, is : Raise the co
efficient to the power n, and multiply the index of
every letter by n, that is,
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 expres-
l6o ON THE STUDY OF MATHEMATICS.
sion 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
16 a12 £*<:*, or find V\&d*W?. The expression here
given is the same as the following :
or (203^V)4, the fourth root of which is 2#3^, con
formably to the rule.
Any root of a product, such as AB, may be ex
tracted by extracting the root of each of its factors.
Thus, I^Ztf = i^A -&~B. For, raise ^~A -fit to the
third power, the result of which is,
or -A A i A X
or AB.
In the same way it may be proved generally, that
'{/ABC—V^dVB \/C. The most simple way of rep
resenting 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 IQa^b1? we must
observe that 16 is 8 X 2, a* is a* X a, b1 is b* X £, and
the expression is 8 a3 b* X 2 a b cy the cube root of which,
found by extracting the cube root of each factor, is
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, ^^abc maybe found either
ROOTS IN GENERAL AND LOGARITHMS. l6l
exactly, or, when that is not possible, by approxima
tion.
We find that a power of a power is found by affix
ing, as an index, the product of the indices of the two
powers. Thus O2)4 or a? X a1 X a2 X is (a™)', and by definition {/(amy = a'H. There
fore \/cT* = i/a™. This multiplication is equivalent
to raising a power of v' a™, and afterwards reducing
the result to its former value, by extracting the corre-
m P . m
spending root, in the same way as — signifies that -
np n
has been multiplied by/, and the result has been re
stored to its former value by dividing it by/.
The following equations should be established by
the student to familiarise him with the notation ai d
principles hitherto laid down.
a -f- by~m X
, 2
(a1 — £
n\ab v'ab \/ a i/7^ n\a n\b
W ^= VTct ^ VTVd ***& ^\d
b v a"~l b
The quantity v ' am is a simple expression when m
can be divided by «, without remainder, for example
v/fl12 = aQ, \/a2Q = a*, and in general, whenever m can
M
be divided by « without remainder, jX^=dP. This
symbol, viz., a letter which has an exponent appear
ing in a fractional form, has not hitherto been used.
We may give it any meaning which we please, pro
vided it be such that when — is fractional in form only,
n
and not in reality, that is, when m is divisible by
ROOTS IN GENERAL AND LOGARITHMS. 163
n, and the quotient is /, an shall stand for ap, or
rn
a a a (/)*• It will be convenient to let a" always
stand for i/ar, in which case the condition alluded to
is fulfilled, since when — =/, a" or i/a"1 = ap. This
n
extension of a rule, the advantages of which will soon
be apparent, is exemplified in the following table,
which will familiarise the student with the different
cases of this new notation :
c& stands for v'a1 or Va
a% stands for va
a* stands for v^
a% stands for V~cP or (f/a)*
a* stands for I/a7 or (I/a )7
ytt-\-n
a"1'" stands for n!~i/aM+ft
stands for !/(/ +
stands for
\an j
stands for V \/~a
The results at which we have arrived in this chap
ter, translated into this new language, are as follows :
(*•)"=(*•)"=* a)
\A£C)" = A" £" C" (2)
*This is a notation in common use, and means that a a a ..... is to be
continued until it has been repeated/ times. Thus
a +a+a+ ..... (/}=/«,
a X a X a X ..... (/;=«*.
164 ON THE STUDY OF MATHEMATICS.
(3)
a
=a» (4)
an =
(5)
The advantages resulting from the adoption of this
notation, are, (1) that time is saved in writing alge
braical expressions ; (2) all rules which have been
shown to hold good for performing operations upon
such quantities as am, hold good also for performing
•m
the same operations upon such quantities as an , in
which the exponents are fractional. The truth of this
last assertion we proceed to establish.
Suppose it required to multiply together an and
_£
an , or v/0™ and \/al. From (2) this is V ani X «'» or
m+l
'i/am+l , or a n . Suppose it now required to multiply
m p_
an and aq '. From (5) the first of these is the same as
ing ttp
anq , and the second is the same as anq. The product
mq-\-np
of these by the last case is a ng , or *{/am9+nf. But
m a -4- n p . m p
y^ ' is -- h — , and therefore
nq n q
an X aq =a
t ™ + L re-.
This is the same result as was obtained when the
indices were whole numbers. The rule is : To multi
ply together two powers of the same quantity, add
the indices, and make the sum the index of the pro
duct. It follows in the same way that
ROOTS IN GENERAL AND LOGARITHMS. 165
or, to divide one power of a quantity by another, sub
tract the index of the divisor from that of the divi
dend, and make the difference the index of the result.
( -Y
Suppose it required to find \an ) . It is evident
m m mm 2»t / tn\ 2 2>«
that a" X a* = a"+" = «""", or (a"J = a*. Similarly
/ ,«\3 yn / ,n\P ' mfi
\an ) = a" , and so on. Therefore \an ) = a " .
( ™\— i/~™. !L
Again to find \an )q , or V an . Let this be a? .
£. i/~™ ( *Y '» ZL ™
Then ay = V a" , or (a*) = a", or ay --=an. There-
xq m x m ( f\~
fore --- — — , or - = — , and (a" ) = anq .
y n y nq
( -\~ q/ ( — V
Again to find \f.j or ]/ \an ) . Apply the last
/ m-y mf
two rules, and it appears that \an j = a H , and
„ / nip rnp / m\$- tn£ —X —
V an — anq . Therefore \an ) q = anq — an 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 re
sult. 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 ex
tension, which will complete this subject.
We have proved that whether m and n be whole
a"1
or fractional numbers, — = am~n. The only cases
which have been considered in forming this rule are
1 66 ON THE STUDY OF MATHEMATICS.
those in which ;// 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 exam
ple, suppose it required to find—. If we apply the
rule, we find the result aB~7, or «~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, #~4 may be
used in future to stand for this result. Now the frac-
a3 1
tion —7 is the same as -^, which is obtained by divid
ing both its numerator and denominator by #3. There
fore -^ is the rational result, for which we have ob
tained <2~4 by applying a rule in too extensive a manner.
Nevertheless, if a~^ be made to stand for —., and
1 a
a~m for — , 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 X a~n is — X — > or , which is
-j a a a a
— ^-, or a~(m+n\ or a-m~n. Similarly
——, or — , is — , or a"~m, or
a
Again
, or -, or ^ls stands
for 1°1//(10J301, an expression of which it would be im
possible 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 under
stand 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
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.
l68 ON THE STUDY OF MATHEMATICS.
In order to make use of logarithms, he must pro
vide himself with a table. Either of the following
works may be recommended to him :
[1. Bruhns, A New Manual of Logarithms to Seven
Places of Decimals (English preface, Leipsic).
2. Schron, Seven-Figure Logarithms (English edi
tion, London).
3. Bremiker's various editions of Vega's Logarith
mic Tables (Weidmann, Berlin). With English pref
ace.]
4. Callet, Tables portatives de Logarithmes. (Last
impression, Paris, 1890).
5. V. Caillet, Tables des Logarithmes et Co-Loga-
rithmes des nombres (Paris).
6. Hutton's Mathematical Tables (London).
7. Chambers's Mathematical Tables (Edinburgh).
8. The American six-figure Tables of Jones, of
Wells, and of Haskell.
For fuller bibliographical information on the sub
ject of tables of logarithms, see the Encyclopaedia Bri-
tannica, Article "Tables," Vol. XXIII. — Ed.~\*
The limits of this treatise will not allow us to enter
*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: (i) Taylor's Logarithms. (2) Hutton's Logarithms.
(3) Babbage, Logarithms of Numbers; Callet, Logarithms of Sines, Cosines,
etc. (4) Bagay, Tables Astronomiques et Hydrographiques. The first and last
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
in the United States."— Ed.
ROOTS IN GENERAL AND LOGARITHMS. 169
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 Trig
onometry. There is, however, one consideration con
nected with the tables, which, as it involves a princi
ple 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 :
Log. 41713 = 4.6202714
Log. 41714 = 4.6202818
Log. 41715 = 4.6202922
The difference of these successive logarithms and of
almost all others in the same page is .0000104. There
fore 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 oc1 -J- x upon the letter
or letters which it contains, if a very small addition be
made to trie-value of one of these letters, in conse
quence of which the expression itself is increased or
diminished; generally speaking, the increment* of the
* When any quantity is increased, the quantity by which it is increased is
called its increment.
I7O ON THE STUDY OF MATHEMATICS.
expression will be very nearly proportional to the in
crement of the letter whose value is increased, and the
more nearly so the smaller is the increment of the let
ter. 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 de
crease, as the following examples will show :
Let x = .01 Let x =.00001
#2 = .0001 x* = .0000000001
x* = . 000001 x* = . 000000000000001
etc. etc.
Now quantities are compared, not by the actual
difference which exists between them, but by the num
ber 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 ex
ample x2 and x9 are both small fractions whem com
pared with unity ; nevertheless, x* is very great when
compared with ^c3, being 100,000 times its magnitude.
This use of the words small and great sometimes em
barrasses the beginner; nevertheless, on considera
tion, 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
ROOTS IN GENERAL AND LOGARITHMS. IJl
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 ut
terly insignificant in point of numbers. Hence, in
such an expression as Ah -j- BW -{- Chz, we may, if h is
very small, reject £h?-\- Ch*, as being very small com
pared with Ah. An error will thus be committed, but
a very small one only, and which becomes smaller as
h becomes smaller.
Let us take any algebraical expression, such as
#2 -f x, and suppose that x is increased by a very small
quantity h. The expression then becomes C#-f^)2-f
(x + h), or #a + #-f (2* +!)£ + #. But it was x2 + x\
therefore, in consequence of x receiving the increment
h, x*-\-x has received the increment (2x-\-V)ti-\--X*9
for which (2x-{-l)A may be written, since h is very
small. This is proportional to h, since, if h were
doubled, (2x-\-\}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 -f Ah -f BW -f Ch* -f etc. ; that is, y re
ceives an increment of the form Ah -f BK* -j- Ch* + etc.
If h be very small, this may, without sensible error,
be reduced to its first term, viz., Ah, which is propor
tional to h. The general proof of this proposition be
longs to a higher department of mathematics ; never
theless, the student may observe that it holds good in
172 ON THE STUDY OF MATHEMATICS.
all the instances which occur in elementary treatises
on arithmetic and algebra.
For example :
O -f- h}m = xm -+- m xm~l h + m M~ xm~2 A2 -f etc.
a
Here A=mxm~l, B = m m~ xm~2, etc.; and if h be
very small, (x -|- h}™ = xm-\- mxm~lh, nearly,
}{i ft*
Again, eh = 1 -f h + -=- -f --^+ etc. Therefore,
Lt A.O
ex X eh or e*+h = e* -j- e* h -}- ^ ti* + etc. And if h be
a
very small, ^*+A = ^*'-j- e* h, nearly.
Again, log. (1 -f- n') = M(n' — %n'2 + ^»'»_ etc<^
To each side add log.jc, recollecting that
log. * + log. (l4-»') = log. x(l + »') = log. (^ + ^«'),
and let
xn' = h or »'==—.
^
Making these substitutions, the equation becomes
M
If ^ is very small, log. (x-\- fc) = log. ^r -f " - h.
We can now apply this to the logarithmic example
with which we commenced this subject. It appears
that
Log. 41713 =4.6202714
Log. (41713 + 1) = 4. 6202714 + .0000104
Log. (41713 -f-2) = 4. 6202714 -f-. 0000104 X 2.
From which, and the considerations above-men
tioned,
ROOTS IN GENERAL AND LOGARITHMS. 173
Log. (41713 -f h-) = log. 41713 + . 0000104 X h,
which is extremely near the truth, even when h is a
much larger number, as the tables will show. Sup
pose, then, that the logarithm of 41713.27 is required.
Here /i = .21. It therefore only remains to calculate
.0000104 X-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 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, ob
tained in the following manner :
Log. 41713 =4.6202714
. 0000104 X- 2 = .0000021
. 0000104 x -07= .0000007
Log. 41713.27 =4.6202742
This, when the useless cyphers and parts of the opera
tion are omitted, is the process given in all the books
of logarithms. If the logarithm of a number contain-
174 ON THE STUDY OF MATHEMATICS.
ing 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
consult the preface to Babbage's Table of Logarithms *
• Copies of Babbage's Table of Logarithms are now scarce, and the reader
may accordingly be referred to the prefaces of the treatises mentioned no
page 168. The article on " Logarithms, Use of" in the English Cyclopedia,
may also be consulted with profit.— Ed,
CHAPTER XII.
ON THE STUDY OF ALGEBRA.
IN this chapter we shall give the student some ad
vice as to the manner in which he should prose
cute 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 them
selves. 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 con
tinual applications of the preceding rules and devel
opments 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 under
standing of the different processes there exhibited,
and of their connexion with one another. This can
not be attained by a mere reading of the book, how
ever great the attention which may be given. It is
176 ON THE STUDY OF MATHEMATICS.
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 ad
ditions, 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 neces
sary 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 con
vinced that it is by far the shortest method of pro
ceeding, since the deliberate consideration which the
act of writing forces us to give, will prevent the con
fusion 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 oc
casioned the embarrassment will not stand the trial to
which it is thus brought. Should there be still any
THE STUDY OF ALGEBRA. 177
matter of doubt which is not removed by attentive re
consideration, the student should proceed, first mak
ing 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 geo
metrical, there is less necessity for committing to writ
ing the whole detail of the arguments, since the sym
bolical language is more quickly understood, and the
subject is in a great measure independent of the mech
anism 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 in
sist.
II. On arriving at any new rule or process, the
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 pre
vious knowledge will suggest some method of proving
whether his result is true or not. For example, the
7
development of (tf-f--*)3 will exercise him in the use
of the binomial theorem ; when he has obtained the
series which is equivalent to (dt-j-^c)¥, let him, in the
178 ON THE STUDY OF MATHEMATICS.
same way, develop (a-\-x)^ ; the product of these,
since -|-f f — 3, ought to be the same as the develop
ment of (0-j-.*)3, or as 03-f 3rt2*-|-3tf.x2-J-.x3. He
may also try whether the development of {a-\-x)^ by
the binomial theorem, gives the same result as is ob
tained 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, •= - is proved to be equiv
alent to the series
1 -f- x -f x2 -f xz -J-, . etc. , ad infinitum.
This, when multiplied by 1 — x, should give 1 ; when
multiplied by 1 — x2, should give !-}-.#, because
etc.
Again,
etc.
Now, since a* X ay = a*+y9 the product of the two
first series should give the third. Many other in
stances of the same sort will suggest themselves, and
a careful attention to them will confirm the demon
stration of the several theorems, which, to a beginner,
THE STUDY OF ALGEBRA. 179
is often doubtful, on account of the generality of the
reasoning.
III. Whenever a demonstration appears perplexed,
on account of the number and generality of the sym
bols, 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 -f *)?, or any other case, which is there applied
trt
to (1 -f •*)"• Again, the general form of the product
C*-f-0), (x^- <£), (#-}-<-), etc., . . . containing n factors,
will be made apparent by taking first two, then three,
and four factors, before attempting to apply the rea
soning which establishes the form of the general pro
duct. 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
first, instead of taking the general equation of the
form
x" + Ax»~l + Bx"-* -f + 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
l8o ON THE STUDY OF MATHEMATICS.
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 demon
stration. 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 ad
vantage of being able to give to each part of the sub
ject 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 prom
inent parts, and will waste time upon unimportant
details, from which even the best books are not free.
The necessity for this continual reference is particu
larly felt in the Elements of Geometry, where allusion
is constantly made to preceding propositions, and
where many theorems are of no importance, consid
ered 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 prac
tical utility is in direct proportion to the facility which
he possesses in computation. Though it is in plane
THE STUDY OF ALGEBRA. l8l
and spherical trigonometry that the most direct nu
merical 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 de
duced from them. From the imperfect manner in
which arithmetic is usually taught, the student ought
in most cases to recommence this study before pro
ceeding to the practice of logarithms.
VI. The greatest difficulty, in fact almost the only
one of any importance which algebra offers to the rea
son, is the use of the isolated negative sign in such
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 sub
jects of discussion for centuries. In geometry, for
example, the theory of parallel lines has never yet
been freed from the difficulty which presented itself to
Euclid, and obliged him to assume, instead of proving,
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
l82 ON THE STUDY OF MATHEMATICS.
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 them
selves. These are not such as would suggest them
selves 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 re
sult 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 X — « = — #2> he considers the multipli
cation of am -f a" by bm -{- b" as one of the difficulties
of algebra. This arises, in our opinion, from the man
ner in which his previous studies are usually con
ducted. 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 arith
metic, 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 alge
bra 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 recom-
THE STUDY OF ALGEBRA. 183
mended. It is no wonder, then, that he considers the
difficulty of a process as proportional to that of re
membering 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 incon
clusive 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 con
sideration 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,
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 pecu
liar 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 rig-
184 ON THE STUDY OF MATHEMATICS.
orous 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 com
prehend. 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
to see it so early in his career), or from observation
of its truth in the particular cases of the development
of (a -|- £)2, (a -f- <£)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 rea
soning which prevailed so much in English works on
elementary mathematics. Rules which the student
could follow in practice supplied the place of argu
ments 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
THE STUDY OF ALGEBRA. 185
objection to this appears to have been the necessity
which arose for departing from the a] pearance of rig
orous 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.
11 Newton extended to fractional and negative
powers the analytical expression which he had found
for whole and positive ones. You see in this exten
sion 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 multi
tude of new truths out of formulae which were founded
upon very limited suppositions. At first, people were
afraid to admit the general consequences with which
analytical formulae furnished them ; 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 formulae a greater exten
sion than they really admit, and that it is always well
to demonstrate rigorously the results which are ob
tained."
We have observed that beginners are not disposed
1 86 ON THE STUDY OF MATHEMATICS.
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 al
gebra, we find a principle equivalent to the equations
-f-#X — 1> = — ab, and — #X — b — -\-ab, admitted
as an axiom, without proof or difficulty. In the Hin
doo 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, pos
sessed of many formulae whose developments pre
sented striking examples of an universality in alge
braic 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 sylla
bus of results only, unaccompanied by any demonstra
tion. It is essential to acquire a correct memory for
algebraical formulae, 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 formulae. Whenever that
THE STUDY OF ALGEBRA. 187
can be done, the student should have recourse to the
system of tabulation, of which he will have seen sev
eral examples in this treatise. In this way he should
write the various forms which the roots of the equa
tion ax2 -{- b x -f c = Q 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
from the perusal of Lacroix, Essais sur rEnseignement
en general et sur celui des Mathtmatiques en particulier, *
Condillac, La Langue des Calculs, and the various ar
ticles on the elements of algebra in the French En
cyclopedia, which are for the most part written by
D'Alembert. The reader will here find the first prin-
*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 Philosophy and Teach'
ing of Mathematics is that of C. A. Laisant (La Mathhnatique. Philosophie-
Enseignement, Paris, 1^98, Georges Carre et C. Naud, publishers.) Perhaps
the most accessible and useful work in English for the elements is David
Eugene Smith's new book The Teaching of Elementary Mathematics. (New
York : The Macmillan Company, 1900). Mention might be made also of W. M.
Gillespie's translation from Comte's Cours de Philosophie Positive, under the
title of The Philosophy of Mathematics (New York : Harpers, 1851), and of the
Cours de Mlthodologie Mathtrnptique of Felix Dauge (Deuxieme edition, revue
etaugmentee. Gand, Ad. Hoste ; Paris, Gauthier-Villars, 1896). The recent
work of Freycinet on the Philosophy of the Sciences (Paris, 1896, Gauthier-Vil
lars) will be found valuable. One of the best and most comprehensive of the
modern works is that of Duhamel, Des Methodcs dans les Sciences de Raisonne-
ntent, (5 parts, Paris, Gauthier-Villars), a work giving a comprehensive expo
sition of the foundations of all the mathematical sciences. The chapters in
Diihring's Kritische Geschichte d?r Prinzipien der Mechanik and his Neue
Crundmittel on the study of mathematics and mechanics is replete with orig
inal, but hazardous, advice, and may be consulted as a counter-irritant to
the traditional professional views of the subject. The articles in the English
Cyclopedia, by DeMorgan himself, contain refreshing hints on this subject.
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
Calculus of 'Probabilities, or from the historical and philosophical reflexions
that uniformly accompany the later works of Lagrange. The same remark
applies to the later mathematicians of note — Ed.
1 88 ON THE STUDY OF MATHEMATICS.
ciples of algebra, developed and elucidated in a mas
terly manner. A great collection of examples will be
found in most elementary works, but particularly in
Hirsch, Sammlung von Beispielen, etc., translated into
English under the title of Self -Examinations in Algebra,
etc., London: Black, Young and Young, 1825.* The
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,
should be acquainted with the French language, f A
knowledge of this, sufficient to enable him to read the
simple and easy style in which the writers of that na
tion treat the first principles of every subject, may be
acquired in a short time. When that is done, we re
commend to the student the algebra of M. Bourdon, J
* Hirsch s Collection, enlarged and modernised, can be obtained in vari
ous recent German editions. The old English translations of the original
are not easily procured. — Ed.
t German is now of as much importance as French. But the French text
books still retain their high standard. — Ed.
\ Bourdon's 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
complete treatise which begins with the elements and extends to the higher
branches of the subject is the Traitt d'Algtbre of H. Laurent, in four small
volumes (Gauthier-Villars, Paris). This work contains a large collection of
examples. Another elementary work is that of C. Bourlet, Lecons d' Alglbre
Elementatre, Paris, Colin, 1896. A standard and exhaustive work on higher
algebra is the Cours d' Alglbre Suptrieure, of J. A. Serret, two large volumes
(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 Algebra and Theory of Equations
(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 exce'lent recent text-book for beginners, and one that skil
fully introduces modern notions, is the Elements of Algebra of W. W. Beman
and D. E. Smith (Boston, 1900). Fisher and Schwatt's elementary text-books
THE STUDY OF ALGEBRA. 189
a work of eminent merit, though of some difficulty to
the English student, and requiring some previous
habits of algebraical reasoning.
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 pur
poses of practical mathematics, algebra is principally
useful as the guide to trigonometry, logarithms, and
the solution of equations. Much and profound study
of algebra are also recommendable from both a practical and theoretical
point of view, Valuable are C. Smith's Treatise on Algebra (London : Mac-
millan), and Oliver, Wait, and Jones's Treatise on Algebra (Ithaca, N. Y.,
1887), also Fine's Nitmber System of Algebra (Boston : Leach). The best Eng
lish work on the theory of equations is Burnside and Panton's (Longmans).
A very exhaustive presentation of the subject from the modern point
of view is the Algebra of Professor George Chrystal (Edinburgh : Adam and
Charles Black, publishers), in two large volumes of nearly six hundred pages
each. Recently Professor Chrystal has published a more elementary work
entitled 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 Schu
bert has, in various forms, given systematic expositions of the elementary
principles of arithmetic, (e. g., see his Arithmetik und Algebra, Sammlung
Goschen, Leipsic,— an extremely cheap series containing several other ele
mentary mathematical works of high standard; also, for a statement of
Schubert's views in English consult his Mathematical Recreatio?is, Chicago,
1898). Professor Schubert has recently begun the editing of a new and larger
series of mathematical text-books called the Sammhing Schubert (Leipsic :
Goschen), which contains three works treating of algebra. In this connexion
maybe mentioned also Matthiessen's admirable Grundzuge der antiken und
modernen Algebra (Leipsic: Teubner) for literal equations. The following
are all excellent: (i) Otto Biermann's Elemente der hb'heren Mathematik
(Leipsic, 1895); (2) Petersen's Theorie der algebraischen Gleichungen (Copen
hagen: Host; also in French, Paris: Gauthier-Villars); (3) Richard Baltzer's
Elemente der Mathematik (2 vols., Leipsic: Hirzel); (4) Gustav Holzmiiller's
Methodisches Lehrbuch der Elementarmathematik (3 parts, Leipsic: Teubner);
(5) Werner Jos. Schuller's Arithmetik und Algebra fur hohere Schulen und
Lehrerseminare, besonders zum Selbstunterricht, etc. (Leipsic, 1891, Teubner; ;
(6) Oskar Schlomilch's Handbuch der algebraischen Analysis (Frommann,
Stuttgart); (7) Eugen Netto's Vorlesungen iiber Algebra (Leipsic : Teubner, 2
vols.); (8) Heinrich Weber's Lehrbuch der Algebra (Braunschweig: Vieweg, 2
vols ). This last work is the most advanced treatise that has yet appeared.
A French translation has been announced.— Ed.-— April, 1902.
I QO ON THE STUDY OF MATHEMATICS.
is not therefore requisite ; the student should pay
great attention to all numerical processes and particu
larly 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 knowl
edge 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, par
ticularly in the theory of equations of higher degrees
than the second. Instead of occupying himself upon
these, he should proceed to the application of algebra
to geometry, and afterwards to the differential cal
culus. 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 at
tention according to his own opinion of the use which
he is likely to have for them. This applies particu
larly 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 diffi
culties which arise in the study of Geometry and Tri
gonometry.
CHAPTER XIII.
ON THE DEFINITIONS OF GEOMETRY.
TN 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 diffi
culties 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 tem
porary embarrassment which always attends the com
mencement of a new study ; that, for example, there
is nothing in the elements of pure geometry compar
able, in point of complexity, to the theory of the nega
tive sign, of fractional indices, or of the decomposi
tion 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
IQ2 ON THE STUDY OF MATHEMATICS.
thought it necessary to call the attention of our read
ers to the points which we shall proceed to place be
fore 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 compre
hend, to future extensions of what is before him.
The reason why geometry is not so difficult as al
gebra, 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 con
clusive, unless the symbols are as general as the argu
ments. 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 gen
eral 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
THE DEFINITIONS OF GEOMETRY. IQ3
proposition may be safely demonstrated by reasonings
on any one particular example. For though in prov
ing 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 gen
eral, that is, true for all instances of the same kind.
It also affords some facility that the results of elemen
tary geometry are in many cases sufficiently evident
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 evi
dence from the senses; for example, that a* — b* is
always divisible without remainder by a — b.
The definitions of the simple terms point, line, and
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
straight line, which cannot be mistaken for one mo
ment, 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 magni
tude, 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 di
vided into two parts by a line, the dividing line is not
194 ON THE STUDY OF MATHEMATICS.
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 dimin
ished, which is what we mean when we say that the
line is not a part of the surface. The same considera
tions apply to a point, considered as the boundary of
the divisions of a line.
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 representa
tives of lines and points, have in reality three dimen
sions, 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 suffi
ciently 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 fol
lowing manner. Let AB be any straight line, from
THE DEFINITIONS OF GEOMETRY.
195
the ends of which, A and B, draw two lines, AT7 and
CB, parallel to one another. Consider AF as pro
duced 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 -Fin 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 CFt which
C B
Fig. i.
both pass through C, and would, were our supposi
tion 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.
The next definition which we shall consider is that
of a plane surface. The word plane or flat is as hard
to define, without reference to any thing but the idea
we have of it, as it is easy to understand. Neverthe
less the practical method of ascertaining whether or
no a surface is plane, will furnish a definition, not
196 ON THE STUDY OF MATHEMATICS.
such, indeed, as to render the nature of a plane sur
face more evident, but which will serve, in a mathe
matical 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 sur
face throughout its whole length, in whatever direc
tion 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 com
monly 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 directions. By
the direction of the needle we ascertain the direction in
which to proceed at sea, and by the direction in which
the hands of a clock are placed we tell the hour. It
remains to reduce this common notion to a more pre
cise form.
Suppose a straight line OA to be given in magni
tude 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 com
passes 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,
THE DEFINITIONS OF GEOMETRY. 1 97
and this opening is a species of magnitude, since one
opening may be compared with another, so as to as
certain 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 open
ing A OB, 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
O A P C
Fig. 2.
is less than the opening A OB, and if PD does not
fall between OA and OB, or on OB, the opening
CPD is greater than the opening BOA. To this spe
cies 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 magni
tude being one, the measure of which has seldom fal
len 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
198 ON THE STUDY OF MATHEMATICS.
been accustomed to the consideration, that one open
ing 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 spe
cies of magnitude either by means of time, length, or
number.
One magnitude is said to be a measure of another,
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) meas
ures 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 28th part, we know that
the atmospheric pressure of yesterday is increased by
its 28th part to-day. Again, in a watch, the number
of hours elapsed since twelve o'clock is measured by
the angle which a hand makes with the position it oc
cupied at twelve o'clock. In the spring balances a
weight is measured by an angle, and many other sim
ilar 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
THE DEFINITIONS OF GEOMETRY.
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
C
D
Fig. 3-
right line, but on opposite sides of the point ,
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 re
quired 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 hy
pothesis C is not D, therefore both C is D and is not
D, which is absurd ; that is, it is absurd to suppose
that C is not D and A is B, consequently when C is
not Dt A is not B. The following, which is exactly
the same thing, is plainer in its language. Let Cbe
not D. Then if A were B, C would be D by the prop
osition granted. But by hypothesis C is not D, etc.
228 ON THE STUDY OF MATHEMATICS.
This sort of indirect reasoning frequently goes by the
name of 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 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 de
pended 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 cor
rection 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 accus
tomed to talk of mathematical reasoning as above all
other, in point of accuracy and soundness. This, if
by the term reasoning we mean the comparing together
of different ideas and producing other ideas from the
comparison, is not correct, for in this view mathemat
ical reasonings and all other reasonings correspond
exactly. For the real difference between mathematics
GEOMETRICAL REASONING.
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 ad
vantage 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 meas
urement, 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 Caesar justifi
able or not? The actors in that deed justified them
selves by saying, that a tyrant and usurper, who med
itated the destruction of his country's liberty, made it
the duty of every citizen to put him to death, and that
Caesar was a tyrant and usurper, etc. Their 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 gen
eral 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 de
struction of civilisation, etc. Now in these reasonings
and all others, with the exception of those which oc
cur in mathematics, it must be observed that there
230 ON THE STUDY OF MATHEMATICS.
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 mathe
matics, 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 pro
cess to which they used to attach such ideas of diffi
culty. There may, perhaps, be many who imagine
that reasoning is for the mathematician, the logician,
etc., and who, like the Bourgeois Gentilhomme, may
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 an
other. 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, differ
ing 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.
CHAPTER XV.
ON AXIOMS-
EOMETRY, then, is the application of strict logic
to those properties of space and figure which
are self-evident, and which therefore cannot be dis
puted. 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 demon
strate all geometrical truths with the smallest possible
number of assumptions. These assumptions are called
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 ref
erence 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
232 ON THE STUDY OF MATHEMATICS.
always passed as perfectly self-evident.* 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
*But see J. B. Stallo, Concepts and Theories of Modern Physics, New York,
1884, p. 242, p. 208 et seq., and p. 248 et seq. For popular philosophical dis
cussions of the subject of Axioms generally, in the light of modern psychol
ogy and pangeometry, the reader may consult the following works : Helm-
holtz's "Origin and Meaning of Geometrical Axioms," Mind, Vol. III., p. 215,
and the article in the same author's Popular Lectures on Scientific Subjects,
Second Series, London, 1881, pp. 27-71; W. K. Clifford's Lectures and Essays,
Vol. I., p. 297, p. 317; Duhamel, Des M&thodes dans les Sciences de Raisonne-
ment, Part 2; and the articles "Axiom" and "Measurement" in the Encyclo
pedia Britannica, Vol. XV. See also Riemann's Essay on the Hypotheses
Which Lie at the Basis of Geometry, a translation of which is published in
Clifford's Works, pp. 55-69. For part of the enormous technical literature of
this subject cf. Halsted's Bibliography of Hyper-Space and Non-Euclidean
Geometry, 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 Lobachevski and Bo-
lyai, for instance, may be had in the Neomonic Series of Dr. G. B. Halsted
(Austin, Texas). A full history of the theory of parallels till recent times
is given in Paul Stackel's Theorie der Parallellinien von Euklid bis auf Gauss
(Leipsic, 1895). Of interest are the essays of Prof. J. Delboeuf on The Old
and the Neiv Geometries (Revue Philosophique , 1893-1895), and those of Profes
sor Poincare and of other controversialists in the recent volumes of the
Revue de Mltaphysique et de Morale, where valuable bibliographical refer
ences will be found to literature not mentioned in this note. See also P. Tan
nery in the recent volumes of the Revue g£n£rale and the Revue philosophique,
Poincare in The Monist for October, 1898, and B. A. W. Russell's Foundations
of Geometry (Cambridge, 1897). In Grassmann'sAusJe/tnung-slehre (1844), " as
sumptions" and "axioms" are replaced by purely formal (logical) "predica
tions," which presuppose merely the consistency of mental operations. (See
The Open Court, Vol. II. p. 1464, Grassmann, "A Flaw in the Foundation of
Geometr-y," and Hyde's Directional Calculus, Ginn & Co., Boston). Dr. Paul
Carus in his Primer of Philosophy (Chicago), p. 51 et seq., has treated the sub
ject of Axioms at length, from a similar point of view. On the psychological
side, consult Mach's Analysis of the Sensations (Chicago, 1897), and the biblio
graphical references and related discussions in such works as James's Psy
chology and Jodl's Psychology (Stuttgart, 1896). — Ed.
AXIOMS.
233
therefore been called the defect and disgrace of geom
etry. 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 BAG and BAD.
F
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
234 ON THE STUDY OF MATHEMATICS.
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 any line AD cutting CD and make
the angle DAE equal to ADB." But this our senses
immediately assure us is impossible.
It appears also a proposition to which no degree
of doubt can attach, that if the straight line AB, pro
duced indefinitely both ways, set out from the posi
tion 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 one
such position only. This is in reality the assumption of
Euclid ; for having proved that AE and BF are par
allel 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 re
maining 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
AXIOMS. 235
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
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 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 some
thing equivalent to the axiom of Euclid. Of this last,
indeed, a proof has been given, but involving consid
erations 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
would have been received without difficulty. The
Greek geometer confines his notion of equal magni
tudes 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
236
ON THE STUDY OF MATHEMATICS.
spaces contained between two equal angles whose
sides are produced without end, which by the defini
tion 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
O
Fig. 7-
lines Oa and Ob, making any angle with one another,
and produced ad infinitum* On Oa take off the equal
spaces OP, PQ, QR, etc., 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 in-
* Every line in this figure must be produced ad infinitum, from that ex
tremity at which the small letter is placed.
AXIOMS. 237
finite 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 ad infini-
tum 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 an
gle ; 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 page 239.]
This demonstration involves the consideration of
a new species of magnitude, namely, the whole space
contained by the sides of an angle produced without
238 ON THE STUDY OF MATHEMATICS.
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 magni
tude can be instituted between it and any finite space
whatever, but that affords no reason against compar
ing this magnitude with others of the same kind.
Any thing may become the subject of mathemati
cal 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 geometri
cal reasoning, in all other cases at least, can be ap
plied as soon as a criterion of equality is discovered.
Thus the angle, to beginners, is a perfectly new spe
cies 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 neces
sary for them. All that can be said of the introduc
tion of the angle in geometry holds with some, (to us
it appears an equal force,) with regard to these unlim
ited 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 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
AXIOMS. 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 in
quirer into first principles. The theory to which we
are coming abounds in difficulties of both classes.
[NOTE TO PAGE 237. — The demonstration given on pp. 235-
237 is now regarded as fallacious by mathematicians; the consid
erations that apply to finite aggregates not being transferable to
infinite aggregates, — for example, it is not true for infinite aggre
gates that the part is always less than the whole Even Plato is
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.]
CHAPTER XVI.
ON PROPORTION.
TN 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 an
other, 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 greater
is the essence of the proposition ; no measure is given
of the excess, nor does anything follow from the theo
rem 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,* of the theory laid down in the
fifth book of Euclid, f Considered as a complete con-
* In England. t See Todhunter's Euclid (Macmillan, London).— Ed.
PROPORTION. 241
quest over a great and acknowledged difficulty of prin
ciple, this book of Euclid well deserves the immortal
ity of which its existence, at the present moment, is
the guarantee ; nay, had the speculations of the math
ematician been wholly confined to geometrical magni
tude, 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 propo
sitions of geometry are bound together, and as little
open to cavil and disputation.
The student is aware that before pronouncing upon
the connexion of two lines with one another, it is ne
cessary to 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
242
ON THE STUDY OF MATHEMATICS.
of measuring a line, the more strict geometrical method
is derived.
To measure the line AB, apply to it another line
(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 subdivi
sion 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, any
where but in geometry, or algebra, we
only mean accurate 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
Fig.
PROPORTION. 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 dimin
ished 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 diminu
tion 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 arbi
trary. 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 compari-
244 QN THE STUDY OF MATHEMATICS.
sons 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 T2T and |
of a foot, are better compared by observing that -j?r
and ^ being ^ and |^, the given lines contain one
77th 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.
Again, a line which is measured by another is called
a multiple of it, as in arithmetic.
The same definition, mutatis mutandis, applies to
surfaces, solids, and all other magnitudes ; and though
in our succeeding remarks we use lines as an illustra
tion, 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 com
parison 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 13, 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 de-
PROPORTION. 245
note the first by A,* and the second by B, and the
common measure by C, we have
A = 5C, or GA=3QC,
B = §C, or 5^ = 30C,
whence QA = 5B, or 6A— 5^ = 0.
Generally, when mA — nB = Q, the lines, or what
ever they are, represented by A and B, are said to be
in the proportion of n to ;//, or to have the ratio of n
to m.
Let there be two other magnitudes P and (?, 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 ft five times, and let Q contain R six
times, we have by the same reasoning
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
AiBiiP-.Q,
by which at present all we mean is this, that there are
* 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
units of some sort or other contained in the line A, stands for the line A itself,
and mA (the small letters throughout meaning whole numbers) stands for the
line made by taking A, m times. Thus such expressions as mA + B, tnA — nB,
etc., are the only ones admissible. AB, — , A2, etc., are unmeaning, while —
D tn
is the line which is contained m times in A, or the wth part of A. The capital
letters throughout stand for concrete quantities, not for their representations
in abstract numbers.
246
ON THE STUDY OF MATHEMATICS.
some two whole numbers ;;/ and n such that, at the
same time
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 as
sume that two lines A
and B, whatever may be
their lengths, both con
tain some other line an
exact number of times.
We can, moreover, pro
duce 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 meas
ure whatever. If possible let D be a common meas
ure of BC and AB ; let BC contain D, n times, and
let AB contain D, m times. Let E be the square de
scribed on D. Then since AB contains D, m times,
the square described on AB contains £, m X M or m2
times. Similarly the square described on BC contains
E n X n or n2 times. But, because AB is an isosce-
D
Fig 9.
PROPORTION. 247
les right-angled triangle, the square on AB is double
that on BC, whence m X w = 2 (, which be-
? p V
comes equal to C, therefore - A becomes less than C.*
?
We now resume the isosceles right-angled triangle.
• The lines BC and AB, which were there shown to
have no common measure, are called incommensurable
quantities, and to their existence the theory of pro
portion 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 Et 2E, 3E, etc., we shall arrive at last at
two consecutive terms, pE and (p-\-\")J& 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-\-l)E are commen-
* Algebraically, let a be the given line, and let — th part of the remainder
be removed at every subtraction. The first quantity taken away is — and the
a ( i \ rn
remainder a orali I, whence the second quantity removed is
a ( i \ m, , V "! , ( a \ / I \ / I \ 2
— I i I , and the remainder \a — — 1 li 1 or # I i I .
m V m' V m / V m ) V m*
Similarly, the «th remainder is-«(i I I . Now, since i — is less
> }>i ' m
than unity, its powers decrease, and a power of so great an index may be
taken as to be less than any given quantity.
250 ON THE STUDY OF MATHEMATICS.
surable with A, that is with m-E, since E is a common
measure of both. If therefore A and B are incommen
surable, 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 incommen
surable, there are no whole values of m and n, which
will satisfy the equation;;/^ — n£ = Q; nevertheless,
we can prove that values of ;;/ and n can be found
which will make mA — nB less than any given magni
tude C, of the same kind, how small soever it may be.
Suppose, that for certain values of ;;/ and «,* we find
mA — nB = E, and let the first multiple of E, which
is greater than B, beflE, so that pE = B -|- E' where
E' is less than E, for were it greater, (/ — !)-#» or
pE — E, which is B + (E' — J5), would be greater
than B, which is against the supposition.
The equation mA- — nB — E gives
p mA —p nB =pE = £-}-£',
whence
* 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 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 mul
tiple of A;
mA — nB
mpA — (np^ i) B
zmA — ^mB, etc., etc.
PROPORTION. 251
Let
p m = m and fin-\-~L=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 /' must be at
least as great as /, since E being greater than E ', it
cannot take more* of E than of E' to exceed B. Let
p'E' = B + E",
then, as before,
m'p'A — (rip' -f 1) B = E",
or
m"A — ri'B = E";
we have therefore still further diminished the differ
ence of the multiples ; and the process may be re
peated 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 ex
tracted, 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
diminution is not without limit. Again, from the point
* It may require as many. Thus it requires as many of 7 as of 8 to exceed
33, though 7 is less than 8.
252 ON THE STUDY OF MATHEMATICS.
D in the line AB draw DE, making an angle with
AB less than half a right angle. Draw BE perpen
dicular to AB, and take BC=BE. Draw CF perpen
dicular 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.
E
To compare the quantities E, E' , etc., we have
the equations
pE =
p'E'
etc. etc.
The numbers/, /', /", etc., do not diminish; the
lines E, E' , E" , etc., diminish at every step. If then
we can show that/, /', 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 ;
PROPORTION. 253
let this be/U), where z is not an exponent, but marks
the number which our notation will have reached, and
indicates the (z -f l)th step of the process. Let E(z} be
the corresponding remainder from the former step.
Then, since p™E™ is the first multiple of £(~~\ 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/U) can be as great as we please, observe, that/,
/',/", etc., must remain the same, or increase, since,
as appears from their method of formation, they can
not diminish. Let them remain the same for some
steps, that is, let p=p' =p", etc. The equations be
come
pE =
pE' =
etc. etc.
Then by subtraction,
E' —E" =p(E — E')
E" —E" =p(E' — £")=&(£—£')
E'" - E"" =p (E" — E'"} =ppp (E - E')
etc. etc.
Now,
E-E" =£-£'+£*—£" =(
etc. etc. etc.
Generally,
254 ON THE STUDY OF MATHEMATICS.
which is derived from w steps of the process. Now,
if this can go on ad infinitum, it can go on until 1 +
/ + /2 + • • • • +/?A'~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
given number. This is absurd, from the step at which
1 +/ +/2 -j- . . . -\-p™~1 becomes greater than the num
ber of times which E — E' is contained in E ; for,
from the above equation, E — E' is contained in
E — E(w), 1 + / +/2 -f- . . . +/7*"1 times ; and it is con
tradictory to suppose that E — E' should be contained
in E — E^ 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 x 54'= 55'+ 53',
where .£" = 53' and E — £' = !/, and is contained in
E 54 times. If, then, we continue the process, 2 can
not maintain its present place through so many steps
of the process as will, if the same number of terms be
taken, give l + 2 + 22 + 23+, etc., greater than 54;
that is, it cannot be the same for six steps. And we
find, on actually performing the operations,
2 x 54'= 55'+ 53'
2x53'= 55'+ 51'
2x47'= 55'+ 39'
2x39'= 55'+ 23'
3x23'= 55'+ 14'
We do not say that /, /', etc., will remain the
PROPORTION. 255
same until 1 -j-/-h/2 + • . . would be greater than the
number of times which E contains E — E ', but only
that they cannot remain the same longer. By repeti
tion 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 conse
quence 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 (/-{-!)£',
neither of which can differ from C by so much as E',
and therefore not by so much as E. Then since
therefore pmA — pnB=pE',
and (p+l)mA — (p+\)nB = (p
Both which last expressions differ from C by a. quan
tity less than E> the first being less and the second
greater than C, and both are of the 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 supposi
tion is practically correct, if there are any limits to
256 ON THE STUDY OF MATHEMATICS.
the senses, may be shown, for let any quantity be re
jected as imperceptible, then since a quantity can be
found as near to B as we please, which is commensur
able with A, the difference between B and its approx
imate commensurable magnitude, may be reduced be
low the limits of perceptible quantity. Nevertheless,
inaccuracy to some extent must infest all general con
clusions drawn from the supposition that A and B
being two magnitudes, whole numbers, m and ?z, can
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 pre
ceding reasonings in geometry. A definition of pro
portion should therefore be substituted, which, while
it reduces itself, in the case of commensurable quan
tities to the one already given, is equally applicable
to the case of incommensurables. We proceed to ex
amine the definition already given with a view to this
object.
Resume the equations
mA — n£ = Q, or A=~B
m
mP—nQ = Q, or P= — Q
m
If we take any other expression of the same sort
—.B and —Q, it is plain that, according as the arith-
m m* nt
metical fraction — is greater than, equal to, or less
PROPORTION.
257
than — , so will — B be greater than, equal to, or less
than —Bt and the same of — Q and — ,O. Let the
m m m
symbol
be the abbreviation of the following sentence : "when
x is greater than y, z is greater than w ; when x is
equal to jc, z is equal to w ; when x is less than yt z is
less than w." The following conclusions will be evi
dent :
•}>H$"'d ;}>=-<'?
Therefore (2)
m'A
m'P
n'B
n'Q
258 ON THE STUDY OF MATHEMATICS.
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 sec
ond and fourth, the multiple of the first is greater
than, equal to, or less than, that of the second, ac
cording as that of the third is greater than, equal to,
or less than, that of the fourth. This property* ne
cessarily follows from the equations
but it does not therefore follow that the equations are
necessary consequences of the property, since the lat
ter may possibly be true of incommensurable quanti
ties, of which, by definition, the former is not. The
existence of this property is Euclid's definition of pro
portion : he says, let four magnitudes, two and two,
of the same kind, be called proportional, when, if equi
multiples be taken of the first and third, etc., repeat
ing 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 at
tending 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
*It would be expressed algebraically by saying that if mA — nB and
mP — nQ are nothing for the same values of in and «, they are either both
positive or both negative, for every other value of in and n,
PROPORTION. 259
connexion between Euclid's definition and the ordi
nary and well-established ideas of proportion ; the
definition itself is made to involve the idea of infinity,
since all possible multiples of the four quantities enter
into it; and lastly, the very existence of the four
quantities, called proportional, is matter for subse
quent demonstration, since to a beginner it cannot
but appear very unlikely that there are any magni
tudes 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 \vhich 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 as near as we please to B and
Q, so that A, B + C, P and Q + Jt, are proportional
according to the common notion, that is, if whole
numbers m and n can satisfy the equations
mA — n(B-\- C)=0
mP— n(Q-{- J?)=0.
We have now to show that Euclid's definition fol
lows from the one just given, and also that the last
follows from Euclid's, that is, if there are four magni-
260 ON THE STUDY OF MATHEMATICS.
tudes which fall under either definition, they fall un
der the other also. Let us first suppose that Euclid's
definition is true of A, B, P, and Q, so that
mA\ (nB
mPy (nQ
This being true, it will follow that we can take ;;/ 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 Ft 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) sup
pose, if possible, that the same values of m and n will
never make mP — nQ less than some certain quantity
F9 and let pF 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\s 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 np, for which
mpA — npB is less than B,
while
mpP — npQ is greater than Q,
or
mpA is less than (np-\- 1)/?,
mpP is greater than (np -f !)(?,
PROPORTION. 26l
which is contrary to our first hypothesis respecting
A, B, P, and Q, that hypothesis being Euclid's defi
nition of proportion, from which if
mpA is less than (np -f- V)B
mpP is less than (»/ + !)nB + nC
which last equation is evidently impossible ; therefore
if mA>nB, mP>nQ. In the same way it may be
*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 /?, or rather the limits which we assign to them ; but m and*
n are given.
PROPORTION. 263
proved that if mA -, then
1 -L- 2.
therefore ' ™ is less than unity, and any fraction
multiplied by this is diminished. But
•4-n . m 1 -f-15
is — X T-Pr
and is therefore less than — , the greater of the two.
P
In the same way it may be proved to be greater than
— , the least of the two.
9
This being premised, since — = , „ — , it lies
a" a x -\-a
, b'x" , b , ? A*
between -7—7-, and — or between — and — .
a x a a a
Call the coefficients of A and B in the series of
equations, a\, 02, etc., b\, b^, etc., and form the series
of fractions — , — , — , etc. The two first of these
a\ #2 &8
will be v- and ^ . of which the second is the
1
ON THE APPLICATION OF ALGEBRA, ETC. 271
greater, since it is p-\ -- . Hence by what has been
proved — is less than — and greater than—; and
a 3 a? #1
every fraction is greater or less than the one which
comes before it, according as the number of its equa
tion is even or odd. Again, as the numerator of the
difference of two successive fractions -^ and —„ is the
i U U
same as that of-^ and -, whatever the numerator of
o b
the first difference is, the same must be that of the
second, third, etc., and of all the rest. But the nu
merator of the difference of ~- and **q is 1; there-
1 ? b'
fore either ab' — a'b, or a'b — ab' , is 1 according as — ;
b . a
or — is the greater of the two, that is according as «
is odd or even.* Now since the «th and (n -f l)th equa
tions, n being odd, are
and a'A=b'B — X'\
by eliminating A we have
or B=a'X+aX'
since ab' — a'b = 1 ; and since the remainders decrease
and the coefficients increase, a and X> X',
r>
whence 2 aX' < a'X+ aX', or 2aX' ***
and since X' < ^-, — 7 < ^ -- ,, or if B be taken as
£a a Z a a
if
the linear unit, — will express the line A with an error
1 a
less than ~ — -., which last may be made as small as
Aaa
we please by continuing the process.
It is also evident that — is too small, while —, is
a a
too great; and since X and X' are less than B,
aAb'B — B,
V—\ . a „ . b 0 X
or - — , is too small. Again, A -- B=— and
b' a X' a a
-7 B — A = —. - . Now X' < X and a' > a : whence
a a
•yi -y- it
— < — ; that is, — , B exceeds A by a less quantity
(Id Ct
than — B falls short of it, so that — r is a nearer repre-
* b a'
sentation of A than — , though on a different side of it.
ON THE APPLICATION OF ALGEBRA, ETC. 273
We have thus shown how to find the representa
tion of a line by means of a linear unit, which is in
commensurable with it, to any degree of nearness
which we please. This, though little used in prac
tice, 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.*
We now come to the measurement of an angle ;
and here it must be observed that there are two dis
tinct measures employed, one exclusively in theory,
and one in practice. The latter is the well-known di
vision 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 the
ory 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 nat
urally led by the theorems relating to the circle. As
sume any angle, AOBy as the angular unit, and any
other angle, AOC(Y\g. n). Let r be the numberf of
linear units contained in the radius OA, and / and s
the lengths, or number of units contained in the arcs
AB and AC. Then since the angles AOB and AOC
* See Lagrange's Elementary Mathematics (Chicago, 1898), p. 2 et seq.— Ed.
t It must be recollected that the word number means both whole and
fractional number.
274 ON THE STUDY OF MATHEMATICS.
are proportional to the arcs AB and AC, or to the
numbers / and s, we have
Angle AOC'is- of the angle AOB ;
and the angle AOB being the angular unit, the num
ber - is that which expresses the angle AOC. This
number is the same for the same angle, whatever
circle is chosen j in the circle FD the proportion of
A
Fig. ii.
the arcs DE and DF is the same as that of AB and
AC: for since similar arcs of different circles are pro
portional to their radii,
AB:DE::OA'.OD
Also AC\DF\\OA\OD
.-. AB-.DE\'.AC'.DF\
therefore the proportion of DF to DE is that of s to /,
and - is the measure of the angle DOF, — DOE being
the unit, as before. It only remains to choose the
angular unit AOB, and here that angle naturally pre
sents itself, whose arc is equal to the radius in length.
This, from what is proved in Geometry, will be the
ON THE APPLICATION OF ALGEBRA, ETC. 275
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
- 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 12.$- and 12 rare the numbers of inches in
the same lines, and by the common theory of frac-
s 12s
tions - = -— . Generally, the alteration of the unit
does not affect the number which expresses the ratio
of two magnitudes. When it is said that the angle
= —-T. — , it is only meant that, on one particular sup
position, (namely, that the angle 1 is that angle whose
arc is equal to the radius,) the number of these units
in any other angle is found by dividing the number of
linear units in its arc by the number of linear units in
the radius. It only remains to give a formula for find
ing 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 circum
ference to its radius, though it cannot be expressed
exactly, is between 3.14159265 and 3.14159266. Tak
ing the last of these, which will be more than a suffi
cient approximation for our purpose, it follows that
the radius being rt one-half of the circumference is
276 ON THE STUDY OF MATHEMATICS.
3.14159266 ; and one-fourth of the circumference,
or the arc of a right angle, is rX 1.57079633. Hence
the number of units above described, in a right angle,
is — -j-. — , or 1.57079633. And the number of seconds
radius
in a right angle is 90 X 60 X 60, or 324000. Hence if
$ be an angle expressed in units of the first kind, and
A the number of seconds in the same angle, the pro
portion of A to 324000 will also be that of 5 to
1.57079633. To understand this, recollect that the
proportion of any angle to the right angle is not al
tered by changing the units in which both are ex
pressed, so -that the numbers which express the two
for one unit, are proportional to the like numbers for
another.
Hence A : 324000 : : 3 : 1.57079633 :
324000
- 1.57079633 X^;
or A = 206265 X S, very nearly.
Suppose, for example, the number of seconds in the
theoretical unit itself is required. Here $ = 1 and
,4 = 206265 ; similarly if A be 1, 5=OA^ong, 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 theoretical
unit.* This unit is used without exception in analysis ;
*Also called a radian. See Beman and Smith's Geometry, p. igz.—EJ.
ON THE APPLICATION OF ALGEBRA, ETC. 277
thus, in the formula, for what is called in trigonom
etry the sine of x, viz.:
9 *
—' etc>
If x be an angle of one second, it is not 1 which must
be substituted for x, but
The number 3.14159265, etc., is called TT, and is
the measure, in theoretical units, of two right angles.
Also jj- is the measure of one right angle ; but it must
2
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 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 exam
ple, 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,
the proportions of the sides are the same. If, there
fore, any angle CAB be given, and from any points
278
ON THE STUDY OF MATHEMATICS.
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 BAG, B'AC, bAc, etc., having a
right angle in all, and the angle A common, are equi
angular ; 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, 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.
B"
b'
To these ratios names are given ; and as the ra
tios 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
. , „,, BC side opposite to A . . ,
referred. Thus, -—^, or — -. — ^-. , is called
AB hypothenuse
. AC side opposite to B ,
the sine of A ; while -r-^, or — : — =-£= , or the
AB hypothenuse
sine of B, the complement* of A> is called the cosine
*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
ON THE APPLICATION OF ALGEBRA, ETC
279
of A. The following table expresses the names which
, . BC AC BC AC AB
are given to the six ratios, _ _ _ _ _ and
AB
~BC
AB AB AC BC AC
, relatively to both angles, with the abbreviations
made use of. The terms opp., adj., and hyp., stand
for, opposite side, adjacent side, and hypothenuse, and
refer to the angle last mentioned in the table.
THE
RATIO
IS THE
BEING
OR
BEING
THESE ARE WRITTEN
BC
AB
sine of A
opp.
hyp.
cosine of B
adj.
hyp.
sin A
cos£
AC
AB
cosine of A
adj.
hyp.
sine of B
opp.
hyp.
cos A
sin B
BC
'AC
tangent of A
opp.
adj.
cotangent of B
adj.
opp.
tan A
cot B
AC
£C
cotangent of A
adj.
opp.
tangent of B
opp.
adj.
cot A
tan B
AB
AC
secant of A
hyp.
adj.
cosecant of B
hyp.
opp.
see A
cosec B
AB
~BC
cosecant of A
hyp.
opp.
secant of B
hyp.
adj.
cosec A
sec£
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 pro
portions of every right-angled triangle, one of whose
angles is an exact number of minutes, are registered.
part of a whole relatively to the rest. Thus, 10 being made of 7 and 3, 7 is
the complement of 3 to.io,
280 ON THE STUDY OF MATHEMATICS.
We say sines, cosines, and tangents only, because it
is evident, from the table above made, that the co
secant, secant, and cotangent of any angle, are the
reciprocals of its sine, cosine, and tangent, respec
tively. Again, the table need only include 45°, in
stead 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 pro
portions may be found from the tables by the method
explained in Chapter XI. of this treatise. It must be
observed, that the sine, cosine, etc., are not 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^.
The measurement of surfaces may be reduced to
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
ON THE APPLICATION OF ALGEBRA, ETC. 28l
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 de
scribed on the unit. This proposition is true, even
when a and b are fractional. Let the number of units
in the sides be — and — , and take another unit which
1 ..*...*
is — of the' first, or is obtained by dividing the first
nq ^
unit into nq parts, and taking one of them. Then,
by the proposition just quoted, the square described
on the larger unit contains nqy^nq of that described
on the smaller. Again, since — and — are the same
fractions as — and — , they are formed by dividing
nq nq
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 mqy^np of the square
described on the smaller unit. But of these there are
ngX n which are called the square
of #, the cube of a, the rectangle of a and b. The stu
dent 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
things. It would be much better if he would accus
tom himself to call a a 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.
Addition, 23, 67.
Algebra, notation of, 55 et seq.; ele
mentary rules of, 67 et seq.; advice
on the study of, 53, 54, 62, 175 et
seq.; nature of the reasoning in,
192 ; applied to the measurement
of lines, angles, proportion of fig
ures and surfaces, 266-284.
Algebraically greater, 144-145.
Algebras, bibliographical list of, 188-
189.
Analogy, in language of algebra, 79.
Angle, definition of, 196 et seq., 238;
measure of, 273 et seq,
Angular units, 275-276.
Approximations, 48 et seq., 130; 171
et seq., 242 et seq.; 267; 281 et seq.
Arrangment of algebraical expres
sions, 73.
Arithmetic, elementary rules of, 20
et seq.; compared with algebra, 76.
Arithmetical, notation, n et seq.; no
tion of proportion, 244 et seq.
Assertions, logical, 203 et seq.
Assumptions, 231 232.
Axioms, 208, 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, 177 et
seq.
Bolyai, 232.
Bosanquet, 212.
Bourdon, 188.
Bourgeois gentilhomme, the, 230.
Bourlet, C., 188.
Brackets, 21.
Bradley, 212.
Bremiker, 168.
Bruhns, 168.
Burnside, W. S., 189.
Caesar, 2, 229.
Caillet, 168.
Callet, 168.
Carus, Paul, 232.
Change of algebraical form, 105 et
seq.
Chrystal, Prof., 189.
Cipher, 16.
Circulating decimals, 51.
Clifford, 232.
Coefficient, 60.
Collin, J., 188.
Commercial arithmetic, 53.
Comparison of quantities, 244 et seq,
Computation, 180.
Comte, 187.
Condillac, 187.
Continued fractions, 267-273.
Contradictory, 205.
Contraries, 205.
Convergent fractions, 269 et seq.
Converse, 205.
Copula, 203.
Counting, 13 et seq.
Courier, problem of the two, 112 et
seq.
Cube, the term, 284.
D'Alembert, 187.
Dauge, F., 187.
286
ON THE STUDY OF MATHEMATICS.
Decimal, system of numeration, 14 et
seq.; point, 43; fractions, 42-54.
Definition, n, 207.
Delboeuf, J., 232.
Demonstration, mathematical, 4 et
seq., 184; inductive, 179.
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, 23 et seq., 38, 75, 165-167.
Duhamel, 72, 187, 232.
Diihring, 187.
Duodecimal system, 19.
English Cyclopedia, 174, 187.
Equations, of the first degree, 90-102;
of the second degree, 129 et seq. ;
identical, 90; of condition, 91, 96
et seq.; reducing problems to, 92
et seq.
Errors, in mathematical computa
tions, 48 et seq. ; in algebraical
suppositions, corrected by a change
of signs, 106 et seq.
Euclid, 4, 37, 181, 234-235, 265; his
theory of proportion, 240, 258 et seq.
Exhaustions, method of, 263-265.
Exponents. See Indices.
Experience, mathematical, 104.
Expressions, algebraical, 59.
Extension of rules and meanings of
terms, 33 et seq., 80-82, 143-145, 163.
Factoring, 132 et seq., 160.
Figures, logical, 216 ei seq.
Fine, H. B., 189.
Fisher and Schwatt, 189.
Fluxions, 265.
Form, change of in algebraical ex
pressions, 105 et seq., 117 et seq.
Formulae, important, 88-89, 99, J4i-
142, 163-167.
Fowler, T., 212.
Fractions, arithmetical, 30 et seq.,
75; decimal, 42; continued, 267-
273 ; singular values of, 123 et seq.;
evanescent, 126-128; algebraical,
75, 87-89, 97-99.
Fractional exponents, 163 et seq., 185.
French language, 188.
Frend, 71, foot-note.
Freycinet, 187.
Geometry, study of, 4 et seq.; defini
tions and study of, 191 et seq.; ele
mentary ideas of, 193 et seq.
Geometrical reasoning and proof, 203
et seq., 220 et seq.
German language, 188.
Grassmann, 232.
Greatest common measure, 25 et seq.,
86, 267 et seq.
Greater and less, the meaning of, 144,
Greatness and smallness, 170.
Halsted, 232.
Hariot, 38.
Haskell, 168.
Hassler, 168.
Helmholtz, 232.
Hindu algebra, 186.
Hirsch, 188.
Holzmuller, G., 189.
Hutton, 168.
Hyde, 232.
Hypothesis, 208.
Identical equations, 90.
Imaginary quantities, 151 et seq.
Impossible quantities, 149 et seq.
Incommensurables, 246 et seq., 281 et
seq.
Increment, 169.
Indirect reasoning, 226.
Indeterminate problems, 101.
Indices, theory of, 60, 158 et seq., 166,
185.
Induction, mathematical, 104, 179,183.
Inductive reasoning, 219.
Infinite quantity, meaning of, 123 et
seq.
Infinite spaces, compared, 235 et seq.
Instruction, principles of natural, 21
et seq.; faulty, 182; books on math
ematical, 187.
Interpolation, 169-174.
INDEX.
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, 57 et seq.
Lobachevski, 232.
Locke, 9.
Logarithms, 167 et seq.
Logic of mathematics, 203-230.
Logics, bibliographical list of, 212.
Mach, E., 232.
Mathematics, nature, object, and
utility of the study of, i et seq.;
language of, 37 et seq.; advice on
study of, 175 ; philosophy of, 187.
Matthiessen, 189.
Measures, 198, 266 et seq.
Measurement, of lines, angles, pro
portion of figures, and surfaces,
266-284.
Measuring, 241 et seq.
Mill, J. S.,212.
Minus quantities, 72.
Mistaken suppositions, 106 et seq.
Moods, logical, 212 et seq.
Multiplication, 23 et seq., 34 et seq.,
68 et seq., 164.
Mysticism in numbers, 14.
Negative, quantities, 72; sign, iso
lated, 103 et seq., 181 ; squares, 149,
151; indices, 166, 185.
Netto, E., 189.
Newton, 37, 185.
Notation, arithmetical, decimal, n
et seq.; general principle of, 15 et
seq.; algebraical, 55 et seq.; 79, 159,
extension of, 33, 80, 143, 163.
Numbers, representation of, 15 et
seq.
Numeration, systems of, 14 et seq.
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.
JT, 277.
Plane surface, 195.
Plato, 239.
Poincare, H., 232.
Point, geometrical, 194-195.
Postulate, 210.
Powers, theory of, 158 et seq.
Predicate, 203.
Premisses, 211 et seq.
Prime numbers and factors, 25.
Problems, reducing of, to equations,
92 etseq.; general disciplinary util
ity of, 95 ; of loss and gain as illus
trating changes of sign, 119 ; of the
two couriers, 112 et seq.
Proportions, 170; theory of, 240-265.
Proportional parts, 173.
Propositions, 203 et seq.
Pythagorean proposition, 221 et seq,
Quadratic, equations, 129 et seq.;
roots, discussion of the character
of, 137 et seq.
Radian, 276.
Read, Carveth, 212.
Reasoning, geometrical, 203 et seq.;
direct and indirect, 226.
Reckoning. 13 et seq.
Riemann, 232.
Roots, 129 et seq., 137 et seq., 158 et
seq.
Rules, 42; mechanical, 184.
Rules, extension of meaning of, 33,
80, 143, 163.
Russell, B. A. W., 232.
288
ON THE STUDY OF MATHEMATICS.
Schlomilch, O., 189.
Schron, 168.
Schubert, H., 189.
Schiiller, W. J., 189.
Self-evidence, 209.
Serret, J. A., 188.
Sexagesimal system of angular meas
urement, 273.
Shorthand symbols, 55.
Sidgwick, 212.
Signs, arithmetical and algebraical,
20 et seq.; 55 ; rule of, 96, 186.
Sigwart, 212.
Simple expression, 76, 77.
Singular values, 122 et seq.
Smith, D. E., IV., 187, 265, 276.
Solutions, general algebraical, in.
Square, the term, 282, 284.
Stackel, Paul, 232.
Stallo, J. B., 232.
Straight line, 12, 193.
Subject, 203.
Subtraction, 23.
Subtractions, impossible, 103-104.
Surfaces, measurement of incom
mensurable, 280 et seq.
Syllogisms, 210 et seq.
Symbols, invention of, 80-81. See
Signs.
Syllabi, mathematical, 186.
Tables.mathematical.recommended,
168.
Tannery, P., 232.
Taylor, 168.
Terms, geometrical and algebraical
compared, 284.
Theory of equations, 132 et seq., 179,
190.
Todhunter, 189.
Triangles, measurement of propor
tions of, 277 et seq.
Trigonometrical ratios, 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 sig
nificance as an algebraical icsult,
122 et seq.; exponents, 81, 166.
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."
— Pro/. 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.)
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On the Study and Difficulties of Mathe
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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.)
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its possession most desirable." — Michigan Alumnus.
THE OPEN COURT MATHEMATICAL SERIES
The Foundations oi Geometry*
By DAyiD 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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points by such an authority." — Journal of Pedagogy.
Euclid's Parallel Postulate : Its Nature, Val
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By JOHN WILLIAM WITHERS, Ph. D. Pages vii, 192. Cloth,
net $1.25. (4s. 6d. net.)
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finally there is a bibliography of fifteen pages. Mr. Withers's critique,
on the whole, is quite sound, although there are a few passages either
vague or disputable. Mr. Withers's main contention is that Euclid's
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
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The Squaring of the Circle. From the German by T. J.
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They are admirably lucid and simple and answer questions in which
every intelligent man is interested." — Chicago Evening Post.
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is too liable to drop into a mere rule of thumb system and forget the
scientific side of his work. Their chief merit is however their intel
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.
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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."
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Lectures on Elementary Mathematics.
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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 to
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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,
selected for their conspicuous interest from the domain of
mathematics, and calculated to reveal that domain as a
world in which invention and imagination are prodigiously
enabled, and in which the practice of generalization is car
ried to extents undreamed of by the ordinary thinker, who
has at his command only the resources of ordinary lan
guage. A few of the seventy sections of this attractive
book have the following suggestive titles : Familiar Tricks,
Algebraic Fallacies, Geometric Puzzles, Linkages, A Few
Surprising Facts, Labyrinths, The Nature of Mathematical
Reasoning, Alice in the Wonderland of Mathematics. The
book is supplied with Bibliographic Notes, Bibliographic
Index and a copious General Index.
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— *The Educator-Journal.
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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
photographs of actual exercises, and a package of papers
for folding. Pages, x, 148. Price, cloth, $1.00 net. (4s.
6d. net.)
"The book is simply a revelation in paper folding. All sorts of things
are done with the paper squares, and a large number of geometric
figures are constructed and explained in the simplest way."
— Teachers' Institute.
Magic Squares and Cubes.
By W. S. ANDREWS. With chapters by PAUL CARUS, L. S.
FRIERSON and C. A. BROWNE, JR., and Introduction by
PAUL CARUS. Price, $1.50 net. (7s. 6d. net.)
The first two chapters consist of a general discussion of the
general qualities and characteristics of odd and even magic
squares and cubes, and notes on their construction. The
third describes the squares of Benjamin Franklin and their
characteristics, while Dr. Carus adds a further analysis
of these squares. The fourth chapter contains "Reflections
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
fied," and "The Mathematical Value of Magic Squares."
"The examples are numerous ; the laws and rules, some of them
original, for making squares are well worked out. The volume is
attractive in appearance, and what is of the greatest importance in
such a work, the proof-reading has been careful." — The Nation.
The Foundations of Mathematics.
A Contribution to The Philosophy of Geometry. BY DR.
PAUL CARUS. 140 pages. Cloth. Gilt top. 75 cents net.
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