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/ I 


A TO apology is needed for the publication of the present new 
-*-^ 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 


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. 

LA SALLE, 111., Nov. i, 1898. 


'N 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 
jmentary works on these sciences. There are two classes of men 

fho 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, 


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 

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 & priori reasoning can give to a beginner. 
In treating, therefore, on the negative sign, on impossible quanti- 
ties, and on fractions of the form , 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, 
finding no other so satisfactory to my own mind. 

The number of mathematical students, increased as it has 
been of late years, would be much augmented if those who hold 
the highest rank in science would condescend to give more effective 
assistance in clearing the elements of the difficulties which they 
present. If any one claiming that title should think my attempt 
obscure or erroneous, he must share the blame with me, since it is 
through his neglect that I have been enabled to avail myself of an 
opportunity to perform a task which I would gladly have seen con- 
fided to more skilful hands. AUGUSTUS DE MORGAN. 



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 



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- 


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 
Caesar, 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 received those accounts as true, 
and that succeeding historians have backed them with 
a mass of circumstantial evidence which makes it the 
most improbable thing in the world that the account, 
or any material part of it, should be false. This is 
perfectly correct, nor can there be the slightest 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- 


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- 

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 


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 


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 


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- 


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, 


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 be relied on, 
it would have been thought advisable to make many 
passages between ports that were well known before 
attempting a voyage of discovery. So it is with our 
reasoning faculties : it is desirable that their powers 
should be exerted upon objects of such a nature, that 
we can tell by other means whether the results which 
we obtain are true or false, and this before it is safe 
to trust entirely to reason. Now the mathematics are 
peculiarly well adapted for this purpose, on the 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. 


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- 


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. 



r I ^HE 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 


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 


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 

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 


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 


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 & 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 : 


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" p . 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 


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 6'" 2" 3' 4. In order to avoid confusion, we will 
suffer the accents to remain over all numbers which 
are not reckoned in tens, while those which are so 
reckoned shall be written in the common way. The 
following is a comparison of the way in which num- 
bers in the common system are written, and in the 
one which we have just explained : 


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 11 1'2 1'3 T4 



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) 507 rem. 4. 
9) 56 rem. 3. 
9) 6 rem. 2. 

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. 


636 7" 7' 6 

987 1"'3"1'6 

403 4" 8' 7 

2026 2"'7"0'1 


1384 r"8"0'7 

797 1"'0"7'5 

587 T 2' 2 


297 3" 6' 

136 1" 6' 1 

1782 360 

891 2400 

297 360 

40392 6""!'" 3" 6' 


633) 79125 (125 7" 7' 3) l v S^O 1 "^ 1 ? 1 6* (1"4' 8 
633 773 

1582 4217 

1266 3423 


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 TO ; 
there must, therefore, be a distinct sign for ten and 
eleven : a nine and six reversed, thus 9 and d, 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. 



AS SOON as the beginner has mastered the notion 
*"* of arithmetic, he may be made acquainted with 
the meaning of the algebraical signs -f, , 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 


himself in putting the simplest truths of arithmetic in 
this new shape, and should write such sentences as 
the following frequently : 

6 4 = 2, 

4 6 = 4 + 2 + 1, 
2 X 2 + 12 X 12 = 1 4 x 10 + 2 X 2 X 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 4 X ~ does not denote the same operation as 
7X4, 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 4 
and ^ are marks of distinct operations, though their 
result is the same, as we shall show in treating of 

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 


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 


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. 


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 + 7) (22 + 7)== 46 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 + 156 X 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 = 3X10X4X6 = 6X10X4X3, 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 


The same thing cannot be done with the divisor. It 
is not true that 

156 _ 156 156 156 
~~~~ ~~ ~~~ ~~' 

The student should discover the reason 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 2 X 17 X 9, 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. 


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 2x^X3x3 and 
2 X 2 X 5 X 5> have the common factors 2 and 5, and 
are divisible by 2 X 5, or 10. The quotients are 3 X 3 
and 2x5, or 9 and 10, which have no common factor 
remaining, and 2 X 5, or 10, is the greatest common 
measure of 90 and 100. The same may be shown in 
the case of any other numbers. But the method we 


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 -J- 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. : 




From the theorem above enunciated it appears 

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 


2 X 4 X ^ is also a common multiple, since it is divis- 
ible by 2 x 4, or 8, and by 2 X 5, 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. 



"TT7HEN 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 : - 5 B 6 -. 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 


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 $-, 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 -j- -J, which is usually written 7 J. Both 
of these quantities $-, and 7i, 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, f . 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 
i X 5. To prove this let each of the lines drawn be- 
low represent J of an inch ; repeat J five times, and 
repeat the same line eight times. 

In each column is th of an inch repeated 8 times ; 
that is one inch. There are, then, 5 inches in all, 


since there are five columns. But since there are 8 
lines, each line is the eighth of 5 inches, or f, but 
each line is also Jth of an inch repeated 5 times, or 
J X 5- Therefore, f = JX 5 ; 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 take^z^ of them. The symbol f is made to stand 
for both these operations, since they lead to the same 

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, f is the same as 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, i = J ^ = | = T 5 ^, etc. ; 

=A=A=A. 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 


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- 


cral, 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 |, 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 -2^, - 7 ^, 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 j. 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- 



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 


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'JD', 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 



inch. We may show, by reasoning, that we can find 
how much A'JS'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 ^ 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 -V 5 -, and 
the second as f . Form a fraction from the product 
of the numerators and denominators of these, which 
will be \2_0-, 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- 


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- 


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 f into 2 parts, and taking three such parts. Be- 
cause f being the same as -if, or 1 divided into 12 
parts and 10 of them taken, the half of -i 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, f multiplied by f . But the same 
result arises from multiplying J by f, or dividing | 
into 6 parts and taking 5 of them. Therefore, we find 
that | multiplied by f is the same as -| multiplied by 
f, or | X f 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 

x%-\-ax b = o; instead of 


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 J and J, it will be observed that in place of find- 
ing how many twos make 12, we shall have to find 
into how many parts f must be divided, and how many 
of them must be taken, so as to give |. If we reduce 
these fractions to a common denominator, in which 
case they will be JJ- and ^ ; and if we divide the sec- 
ond into 8 equal parts, each of which will be ^L, and 
take 15 of these parts, we shall get |f, or f. The 
fraction whose numerator is 15, and whose denomina- 
tor is 8, or Jg 5 -, 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 J- 2 -, and 15 is $-. If this 
process be applied to - 3 2- and -, the result is J^ -, 
which is 5, or the same as 15 divided by 3. This pro- 
cess is then, by extension, called division : -ig 5 - is called 
the quotient of f divided by f , 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 


easily show from any two whole numbers, for exam- 
ple, 12 and 2, whose quotient is 6. Now 12 is -\ 6 -, and 
2 is -^, and the rule for what we have called division 
of fractions will give as the quotient fi, 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 I ! 

C D E F G 

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 


What is J of f of a foot? What is f of | of f of a 
foot? Into how many parts must ^ of a foot be di- 
vided, and how many of them must be taken to pro- 
duce of a foot? What is -f- ^ a f ot ? an ^ so on- 



IT 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 


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 c'an 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 T 7 ^, 
ffifo, izff^, 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 ^Vo 4 * s represented 
by 17.334; fifrfc thus, .229. We might, had we so 
pleased, have represented them thus, 17334 3 , 229 3 ; 
or thus, 17334 3 , 229 3 , or in any way by which we 
might choose to agree to recollect that the 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 


we represent -j^ffa ? 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 T ^ 8 o 8 ^ will be represented thus, .0088 ; by 
which we understand a fraction w r hose numerator is 
88, and whose denominator is a decimal number con- 
taining four cyphers. 

There is a close connexion between the manner of 
representing decimal fractions, and the decimal nota- 
tion for numbers. Take, for example, the fraction 
217.3426 or H^t^F- You wil1 recollect that 2173426 
is made up of 2000000 -f 100000 -f 70000 -f- 3000 -f 
400 + 20 + 6. If each of these parts be divided by 
10000, and the quotient obtained or the fraction re- 
duced to its lowest terms, the result is as follows : 

2173426 -200 I 10 I 7 I 3 -1 _ 2 _ 6 
10000 MO^ 100^1000~MOOOO 

We see, then, that in the fraction 217.3426 the first 
figure 2 counts two hundred ; the second figure, 1, 


ten, and the third 7 units. It appears, then, that all 
figures on the left of the decimal point are reckoned 
as ordinary numbers. But on the right of that point 
we find the figure 3, which counts for T 3 ^ ; 4, which 
counts for ^j 2, for ^^; and 6, for T of^- It ap- 
pears therefore, that numbers on the right of the de- 
cimal point decrease as they move towards the right, 
each number being one-tenth of what it would have 
been had it come one place nearer to the decimal 
point. The first figure on the right hand of that point 
is so many tenths of a unit, the second figure so many 
hundredths of a unit, and so on. 

The learner should go through the same investiga- 
tion with other fractions, and should demonstrate by 
means of the principles of fractions, generally, such 
exercises as the following, until he is thoroughly ac- 
customed to this new method of writing fractions : 
.68342 = .6 + .08 + .003 + .0004 + .00002 
68342 6 8 3 4 2 

__ __ 

' r ~~ ~*~ + "*" 

__ __ 

100000 ~~ To ~ Too 1000 loooo 100000 

.00012 = .0001 + .00002 = 

10000 ' 100000 

iooo L6S 1000 = ~TF + Tooo 

16342 . 9 

100 r 1000' 

The rules of addition, subtraction, and multiplica- 
tion may now be understood. In addition and sub- 
traction, the keeping the decimal points under one 


another is equivalent to reducing the fractions to a 
common denominator, as we may show thus : Take 
two fractions, 1.5 and 2.125, or -J-JJ. and f if g, which, 
reducing the first to the denominator of the second, 
may be written ijgg and f if g. If we add the nume- 
rators together, we find the sum of the fractions f f g, 

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 100x1000 = 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: 

or . 004 x 06 = . 00024, etc. 

1000 X 100 ~~~ 100000' 


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- 

o X o X 
tor and denominator by 2, once for every 5, which is 

contained in the denominator, and the fraction will 
then become 

13X2X2X2 2X2X2X13 

5X5X5X2X2X2' 10x10x10' 

which is yVW or -104. In a similar way, any fraction 
whose denominator has no other factors than 2 or 5, 
can be reduced to a decimal fraction. We first search 
for such a number as will, when multiplied by the de- 
nominator, produce a decimal number, and then mul- 


tiply both the numerator and denominator by that 

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- 


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 it 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 7 T . If we divide 
the series of numbers 70, 700, 7000, etc., by 11, we 
shall obtain the following results : 


f gives the quotient 6, and the remainder 4, and is 6^ 
TjOfi " 63 " 7 63 T 7 T 

7000 636 <t 4 636^ 

? QflQ o 6363 " 7 6363 T 7 T 

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 -f^, their hundredth parts by so much as T J , 
their thousandth parts by so much as T ^o> and so on ; 
and also remember that T 7 T is the tenth part of -, the 
hundredth part of - 7 T T -, and so on. The two following 
tables will now be apparent : 

1Q- does not differ from 6 by so much as 1 

7_o_o (t 63 i 

oo 636 " 1 

_op_o. tt 6363 " 1 

etc. etc. etc. 


T 7 T does not differ from -f^ or . 6, by so much as y 1 ^ or . 1 

T 7 T " T 6 0%"' 63 " T*IF"- 01 

r 7 T " 

T 7 T " 

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 T 7 T , and therefore in 
any calculation in which the fraction T 7 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 w r ould be a 


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, an( } t h e sec- 
ond -if ij . The quotient of these by the ordinary rule 
is ffftoo* or ffft- 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 

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, T 7 T 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 1, 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- 


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- 


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 


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 

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 -|-, , X, 
and the sign of division. 



WHENEVER any idea is constantly recurring, 
the best thing which can be done for the per- 
fection of language, and consequent advancement of 
knowledge, is to shorten as much as possible the sign 
which is used to stand for that idea. All that we have 
accomplished hitherto has been owing to the short 
and expressive language which we have used to 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 -f- 8 was made to rep- 
resent 8 added to 5, and so on. With these signs 


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 + 8 278 _ 27 

15+9 15-9 

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 : 


First No. 4- 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+ 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 : 

. *y x 


(x _|-j)x(.* y) x X x 

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, 


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 

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 -j- 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 : 

If a = b, then a + c = b + c, a-\- c + d=b + c-\- d, 

3. The sign is used for subtraction, as in arith- 
metic. The following equations will show its use : 

x -\- a b c -\- e = x -j- a-\- e b c 
= a c-\- e b-}- x. 

If a = b, 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 a X b 
can be represented by putting a and b together thus, 
ab. Also a X b X c is represented by a be; # X # X #> 
for the present we represent thus, a a a. When two 
numbers are multiplied together, it is necessary to 


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 JL thus, 234' 61. 

5. Division is written as in arithmetic ; thus, -=- 


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-\-c is to be multiplied by d-}-e-\-f, the product 
is written in any of the following ways : 



8. That a is greater than b is written thus, 0> b. 

9. That a is less than b is written thus, a<^b, 

10. When there is a product in which all the fac- 
tors are the same, such as xxxxx, which means that 


five equal numbers, each of which is represented by 
x, are multiplied together, the letter is only written 
once, and above it is written the number of times 
which it occurs, thus xxxxx is written x 5 . The fol- 
lowing table should be carefully studied by the stu- 

x X x or xx is written x 2 , 

and is called the square, or second power of x. 

xy^xXxorxxx is written x 9 , 

and is called the cube or third power of x. 

xX^X^X^orxxxxis written x*, 
and is called the fourth power of x. 

.^X^X-^X^X^ or xxxxx is written x 5 , 
and is called the fifth power of x, 

etc., etc., etc. 

There is no point which is so likely to create con- 
fusion in the ideas of a beginner as the likeness be- 
tween such expressions as 4x and x*. On this account 
it would be better for him to omit using the latter ex- 
pression, and to put xxxx in its place until he has 
acquired some little facility in the operations of alge- 
bra. If he does not pursue this course, he must re- 
collect that the 4, in these two expressions, has differ- 
ent names and meanings. In 4x it is called a coeffi- 
cient, in x* an exponent or index. 

The difference of meaning will be apparent from 
the following tables : 


2x = x -- x x* = xx = xx 

x = x-- x-- x x=xy^xy^x 

4x = x-\-x-}-x-\-x, x* = xX x X x X-* or xxxx, 
etc., etc. 

If* = 3 2x= 6 x*= 9, 
3*:= 9 x* = 2T, 
4x = l2 * 4 = 81, 

The beginner should frequently write for himself 
such expressions as the following : 

= aaa ax -\-aaaax-\- a a a ax -\-aaaax -\-aaaax. 

aa bb aa bb' 
3 aaa 

a 2 & 2 aa bb a + b 

With many such expressions every book on algebra 
will furnish him, and he should then satisfy himself 
of their truth by putting some numbers at pleasure 
instead of the letters, and making the results agree 
with one another. Thus, to try the expression 


a b ~ 

or, which is the same, 
aaa bbb 

= aa-\- ab -\- bb. 

a b 

Let a stand for 6 and b stand for 4, then, if this ex- 
pression be true, 


which is correct, since each of these expressions is 
found, by calculation, to be 76. 

The student should then exercise himself in the 
solution of such questions as the following : What is 

* + * -- '* + - a '--a, 
a -f- b a b 

I. when a stands for 6, and b for 5, II. when a stands 
for 13, and b for 2, and so on. He should stop here 
until he has, by these means, made the signs familiar 
to his eye and their meaning to his mind ; nor should 
he proceed to any further algebraical operations until 
he can readily find the value of any algebraical ex- 
pression when he knows the numbers which the letters 
stand for. He cannot, at this* period of his course, 
write too many algebraical expressions, and he must 
particularly avoid slurring over the sense of what he 
has before him, and must write and rewrite each ex- 
pression until the meaning of the several parts forces 
itself upon his memory at first sight, without even 
the necessity of putting it in words. It is the neglect- 
ing to do this which renders the operations of algebra 
so tedious to the beginner. He usually proceeds to 
the addition, subtraction, etc., of symbols, of the 
meaning of which he has but an imperfect idea, and 
which have been newly introduced to him in such 
numbers that perpetual confusion is the consequence. 
We cannot, therefore, use too many arguments to in- 


duce him not to mind the drudgery of reducing alge- 
braical expressions into figures. This is the connect- 
ing link between the new science and arithmetic, and, 
unless that link be well fastened, the knowledge which 
he has previously acquired in arithmetic will help him 
but little in acquiring algebra. 

The order of the terms of any algebraical expres- 
sion may be changed without changing the value of 
that expression. This needs no proof, and the follow- 
ing are examples of the change : 

a -\- b -\- a b -\- c -\- ac d e de f= 
a </-(- b e-\- ab de-}- c f-\- ac = 
a -\- b d e de f-\- a c -f- c -f- a b = 
ab A- ac d 'e -{- a -f- b -f- c e f d. 
When the first term changes its place, as in the fourth 
of these expressions, the sign -|- is put before it, and 
should, properly speaking, be written wherever there 
is no sign, to indicate that the term in question 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 <r), 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 


and c which is to be subtracted from a. If = 12, 
^ = 6, and c = , this is 10. In the expression a b 
made by subtracting b from a, too much has been sub- 
tracted by the quantity c, since it is not b, but b c, 
which must be subtracted from a. In order, therefore, 
to make a (^ t), c must be added to a b, which 
gives a b-{-c. Therefore, a (^ c~} = a b-\-c. 

(c + d e f} = a + b c d-\- e+f, 

When the positive sign is written before an ex- 
pression in brackets, the brackets may be omitted 
altogether, unless they serve to show that the expres- 
sion in question is multiplied by some other. Thus, 
instead of (* + + *)+;&/+*+/), we may write 
a -{- b -j- c -j- d-\- e -\-f, which is, in fact, only saying 
that two wholes may be added together by adding to- 
gether all the parts of which they are composed. But 
the expression a-{-(b-\-c) (d-\- e) must not be written 
thus: a -\- b -{- c (d -\- e) , since the first expresses that 
(-f c) must be multiplied by (d-\- e) and the product 
added to a, and the second that c must be multiplied 
by (d-\- e) and the product added to a-\- b. If a, b, c, 
d, and e, stand for 1, 2, 3, 4, and 5, the first is 46 and 
the second 30. 

When two or more quantities have exactly the 
same letters repeated the same number of times, such 
as 40 2 ^ 3 , and 60 2 ^ 3 , they may be reduced into one by 


merely adding the coefficients and retaining the same 
letters. Thus, 2 a + 3 a is 50, bc kbc is 2 be, 
%(x+y)-\-%(x-\-y) is 5(>+_y). These things are 
evident, but beginners are very liable to carry this 
farther than they ought, and to attempt to reduce ex- 
pressions which do not admit of reduction. For ex- 
ample, they will say that 3/-f~^ 2 is kb or 4^ 2 , neither 
of which is true, except when b stands for 1. The ex- 
pression 3^-f-^ 2 , or %b-\-bb, cannot be made more 
simple until we know what b stands for. The follow- 
ing table will, perhaps, be of service : 

is not 9a 5 ^ 5 
is not 2a 
2ba-\-Sb is not 50 b. 

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 


in the expression cPb, the quantity represented by a 2 b 
being only taken once, 1 is called the coefficient. 



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 aft 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 

. . , 

expression is the same as 04- b ^ ,, and 

a -\- b La 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 3J. 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 3* 4 and 2x-{- 8, if x stand 
for 12, these are the same, but if it stands for any 
other number they are not the same. 



'T^HE 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 

or more properly thus : Find (a V)-\-(c d}. 

If we add c to a, or find a -f- c, we have too much : 
first, because it is not a which is to be increased by 
c d but a b ; this quantity must therefore be 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 -|- c b d or 


a b-{- 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 6y5 is 
the same as 6 X 14 6 X 9, or as 84 54. Also that 

*(* &)> or *X T ** is the same as fX} f X T 2 i>> 
or as / T 4 F . Upon similar reasoning the following 
equations may be proved : 

#(-j- c d} = ab -f- ac ad. 
(J>-\-pq ar)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 


ab ac-\- # 2 , 
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-\-a. 
There will now be no difficulty in perceiving that 


It is proved in arithmetic that if numbers, whether 
whole or fractional, are multiplied together, the pro- 
duct remains the same when the order in which they 
are multiplied is changed. Thus 6x4x3 = 3x6x4 = 
4x^x3, etc., and | xf = |X|, etc. Also, that a 
part of the multiplication may be made, and the par- 
tial product substituted instead of the factors which 
produced it, thus, 3x4X^X6 is 12X&X 6 , or 15x4x6, 
or 90x4. From these rules two complicated single 
terms may be multiplied together, and the product 
represented in the most simple manner which the case 
admits of. Thus if it be required to multiply 

6tf 3 ^ 4 *:, which is aaabbbbc 
by 12a 2 ^ 8 ^//, which is \Zaabbbcccd, 

the product is written thus : 

aaabbbbc YZaabbbccc d, 


which multiplication may be performed in the follow- 
ing order 


which is represented by 72 a 5 ft 1 c* d. A few examples 
of this sort will establish the rule for the multiplica- 
tion of such quantities which is usually given in the 
treatises on Algebra. 

It is to be recollected that in every algebraical 
formula which is true of all numbers, any algebraical 
expression may be substituted for one of the letters, 
provided care is taken to make the substitution wher- 
ever that letter occurs. Thus from the formula : 

we may deduce the following by making substitutions 
for a. If this formula be always true, it is true when 
a is equal to p-\-g, that is, it is true if p-\-q be put 
instead of a wherever that letter occurs in the form- 
ula. Therefore, 

Similarly, (b -f mf P = (26 + m)m t 

= xy, and so on. 

We have already established the formula, 
(/ q)a = ap aq. 

Instead of a let us put r s, and this formula be- 

(/ ?)(> s) = (r s)p (r. f)?. 



(r s)p = pr ps, and (r s^q = qr qs. 

(/ ?)O s^^pr ps (qr qs} 
=prps qr-\-qs. 

By reasoning in the same way we may prove that 

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 -j- 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 -j- gives -)-; multiplied 
by -f- 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 


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 -|- 
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 

These remarks do not apply to such an expression 
as b + a, which we sometimes write instead of a 
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 fall critical and historical diseass* 
Da mJOults Jmns la sdtmas Je rminmmttmmt 

edition, Paris, Gauthier-Villars, 1896). **r. 


In writing algebraical expressions, we have seen 
that various arrangements may be adopted. Thus 
ax 2 bx-\- c may be written as c-\- ax* bx, or bx 
-\-c-\-ax 2 . Of these three the first is generally chosen, 
because the highest power of x is written first ; the 
highest but one comes next ; and last of all the term 
which contains no power of x. When written in this 
way the expression is said to be arranged in descend- 
ing powers of x; had it been written thus, c bx -}- ax 2 , 
it would have been arranged in ascending powers of 
x; in either case it is said to be arranged in powers 
of x, which is called the principal letter. It is usual 
to arrange all expressions which occur in the same 
question in powers of the same letter, and practice 
must dictate the most convenient arrangement. Time 
and trouble is saved by this operation, as will be 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 : 
















bd cr. 


+ + 

fcO CO 





/ v 






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 yfrj/ 
power in one by the first power in the other. Also, 
that the lowest power of x, or should it so happen, 


the term in which there is no power of x t 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 

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 I 7 

the number which T represents is always an arith- 
a b 

metical fraction ; the contrary may be shown. Let a 

stand for 12, and b for 6, then -- - 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 

7 6 


it may, let a = % and l> = 2, then a*-\-ab = \\ 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 

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- 

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 

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- 

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. 


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- 


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 \ab-\-\ 
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 4 ft c divided by 6 a 2 b cl that is, what quantity must 
be multiplied by cPbc y in order to produce 420 4 3 <r. 
This last expression written at length, \s4aaaabbbc, 
and 42 is 6x7. We can then separate this expression 
into the product of two others, one of which shall be 
6 a 2 b c, or 6 a a b c ; it will then be Qaabcy^l aabb, 
and it is 7 aabb which must be multiplied by Saabc 
in order to produce 42 # 4 b* c. A few examples worked 
in this way, will lead the student to the rule usually 
given in all cases but one, to which we now come. 
We have represented cc, ccc, cccc, etc., by c 2 , <?, c*, 
etc., and have called them the second, third, fourth, 
etc., powers of c. The extension of this rule would 
lead us to represent c by c 1 , and call it the first power 
of c. Again, we have represented c -j- c, c -\- c -\- c, 
c -j- c -(- c -|- c, etc. by 2^-, 3<r, 4^-, and have called 2, 3, 
4, etc., the coefficients of c. The extension of this 


rule would lead us to write c thus, lc, or, rather, if we 
attend to the last remark, l^ 1 . This instance leads us 
to observe the gradual progress of our language. We 
begin with the quantity c by itself ; we proceed in our 
course, shortening by new signs the more complicated 
combinations of c, and the original quantity c forces 
itself anew upon our attention as a part of the series, 

c, 2 <r, 3 c, 4 <r, etc. , and <r, <r 2 , <r 3 , <r 4 , etc. , 
in each of which, except the first, there is a distinct 
figure, which is called a coefficient or exponent, ac- 
cording to its situation. We then deduce rules in 
which the terms coefficient or exponent occur, but 
which, of course, cannot apply to the first term in 
each series, because, as yet, it has neither coefficient 
nor exponent. Among such rules are the following : 
I. To add two terms of the first series, add the co- 
efficients, and affix to the sum the letter c. Thus 
c-\-Z<; = f l c. II. To multiply two terms of the sec- 
ond series, add the exponents, and make this sum the 
exponent of c. Thus ^X^ = ^- HI. To divide a term 
of the second series by one which comes before it, sub- 
tract the exponent of the divisor from the exponent 
of the dividend, and make this difference the exponent 
of c. Thus, c 1 


These rules are intelligible for all terms of the 
series except the first, to which, nevertheless, they 
will apply if we agree that lc l shall represent c, as 
will be evident by applying either of them to find 



j or . We therefore agree that I*: 1 shall 
stand for c, and although c is not written thus, it must 
be remembered that c is to be considered as having 
the coefficient 1 and the exponent 1, which is an 
amendment and enlargement of our algebraical lan- 
guage, derived from experience. It may be said that 
this is all superfluous, because, if c^ stand for cc, and 
fl for ccc, what can c 1 stand for but <r? But it must 
be recollected that, since the symbol c 1 has not yet re- 
ceived a meaning, we are at liberty to make it stand 
for anything which we please, for example, for - , 
or c r 2 , or any other. If we did this, there would, 
indeed, be a great violation of analogy, that is, what 
c 1 stands for would not be as like that which c 2 has 
been made to stand for, as the meaning of c* is to 
that of c^j but, nevertheless, we should not be led to 
any incorrect results as long as we remembered to 
make c 1 always stand for the same thing. These re- 
marks are here introduced in order to show the man- 
ner in which analogy is followed in extending the lan- 
guage of algebra, and to prove that, after a certain 
period, we may rather be said to discover new symbols 
than to make them. The immense importance of this 
branch of the subject makes it necessary that it should 
be fully and early understood by all who intend to 
pursue their mathematical studies to any depth. To 
illustrate it still further, we subjoin another instance, 
which has not been noticed in its proper place. 


The signs -f and were first used to connect one 
quantity with others, and to show what arithmetical 
operations were performed on other quantities by 
means of the first. But the first quantity on which 
we begin the operation is not preceded by any sign, 
not being considered as added to or subtracted from 
any previous one. Rules were afterwards deduced for 
the addition and subtraction of the total result of sev- 
eral expressions in which these signs occur, as follows : 

To add two expressions, form a third, which has 
all the quantities in the first two, with the same signs. 

To subtract one expression from another, change 
the sign of each term of the subtrahend, and proceed 
as in the last rule. 

The only terms in which these rules do not apply 
are those which have no sign, viz., the first of each. 
But they will apply to those terms, and will produce 
correct results, if we place the sign -f before each of 
them. We are thus led to see that an algebraical 
term which has no sign is equivalent in all operations 
to one which is preceded by the sign -f-. We, there- 
fore, consider this sign as prefixed, though it is not 
always written, and thus we are furnished with a 
method of containing under one rule that which would 
otherwise require two. 

From these considerations the following appears 
to be the best and most natural course of proceeding 
in the invention of additional symbols. When a rule 
has been discovered which is not quite general, and 


which only fails in its application to a few instances, 
annex such additional symbols to those already in use, 
or change and modify these so as to make the rule 
applicable in all cases, provided always this can be 
done without making the same symbol stand for two 
different things, and without any violation of analogy. 
If the rule itself, by its application to any case, should 
produce a new symbol hitherto unexplained, it is a 
sign that the rule has been applied to a case which 
was never intended to fall under it when it was made. 
For the solution of this case we must have recourse 
to first principles, but when, by these means, the re- 
sult has been found, it will be best to agree that the 
new symbol furnished by the rule shall stand for the 
result furnished by the principle, by which means the 
generality of the rule will be attained and the analogy 
of language will not be injured. Of this the following 
is a remarkable instance : 

To divide c 8 by <r 5 the rule tells us to subtract 5 
from 8, and make the result the exponent of c, which 
gives the quotient <?. If we apply the same rule to di- 
vide c* by <r 6 , since 6 subtracted from 6 leaves 0, the 
result is <r, a new symbol, to which we have attached 
no meaning. The fact is that the rule was formed 
from observation of different powers of <r, and was 
never intended to apply to the division of a power of 
c by the same power. If we apply the common prin- 
ciples to the division of <: 6 by c 6 , the result is 1. We, 
therefore, agree that <r shall stand for 1, and the least 


inspection will show that this agreement does not af- 
fect the truth of any result derived from the rule. If, 
in the solution of any problem, the symbol <r should 
appear, we must consider it is a sign that we have, in 
the course of the investigation, divided a power of c 
by itself by the common rule, without remarking that 
the quotient is 1. We must, therefore, replace <r by 
1-, but it is entirely indifferent at what stage of the 
process this is done. 

Several extensions might be noticed, which are 
made almost intuitively, to which these observations 
will apply. Such, for example, is the multiplication 
and division of any number by 1, which is not con- 
templated in the definition of these operations. Such 
is also the continual use of as a quantity, the addi- 
tion and subtraction of it from other quantities, and 
the multiplication of it by others, neither of which 
were contemplated when these operations were first 
thought of. 

We now proceed to the principles on which more 
complicated divisions are performed. The question 
proposed in division, and the manner of answering it, 
may be explained in the following manner. Let A be 
an expression which is to be divided by If, and let Q 
be the quotient of the two. By the meaning of divi- 
sion, if there be no remainder A = Qff, since the quo- 
tient is the expression which must multiply the di- 
visor, in order to produce the dividend. Now let the 


quotient be made up of different terms, a, b, c, etc., 
let it be a + b c + d. That is, let 

A = QH (1) 

Q = a + b c + d. (2) 

By putting, instead of Q in (1), that which is equal 
to it in (2), we find 

A = (a+b c + d}H=aH+bHcH+dH (3) 
Now suppose that we can by any method find the 
term a of the quotient, that is, that we can by trial or 
otherwise find one term of the quotient. In (3), when 
the term a is found, since H is known, the term aH 
is found. Now if two quantities are equal, and from 
them we subtract the same quantity, the remainders 
will be equal. Subtract affirom the equal quantities 
A and aH+ bHcH+ dH, and we shall find 

A aH= bH cH+ dH= (b c + d}H. (4) 
If, then, we multiply the term of the quotient found 
by the divisor, and subtract the product from the divi- 
dend, and call the remainder B; then 

B=(b c + d}H. (5) 

That is, if B be made a dividend, and H still continue 
the divisor, the quotient is b c -j- d, or all the first 
quotient, except the part of it which we have found. 
We then proceed in the same manner with this new 
dividend, that is, we find b and also bH, and subtract 
it from B, and let B bHbe represented by C, which 
gives by the process which has just been explained 

C=( c + d}H=cH+dH. (6) 

We now come to a negative term of the quotient. 


Let us suppose that we have found c, and that its sign 
in the quotient is . If two quantities are equal, and 
we add the same quantity to both, the sums are equal. 
Let us therefore add cH to both the equal quantities 
in (6), and the equation will become 

C+cH=dH; (7) 

or if we denote C-\- cH by Z>, this is 

There is only one term of the quotient remaining, and 
if that can be found the process is finished. But as 
we cannot know when we have come to the last term, 
we must continue the same process, that is, subtract 
dH from D, in doing which we shall find that dH is 
equal to D, or that the remainder is nothing. This 
indicates that the quotient is now exhausted and that 
the process is finished. 

We will now apply this to an example in which 
the quotient is of the same form as that in the last 
process, namely, consisting of four terms, the third of 
which has the negative sign. This is the division of 

# 4 / 3 x^y 2 + x*y+2 xy* by x y. 
Arrange the first quantity in descending powers of x 
which will make it stand thus : 

x * _|_ y* y 3 * 2 / -j- 2 xy* y* (A) 

One term of the quotient can be found immediately, 
for since it has been shown that the term containing 
the highest power of x in a product is made up of 
nothing but the product of the terms containing the 
highest powers of x which occur in the multiplier and 


multiplicand, and considering that the expression (A) 
is the product of x y and the quotient, we shall re- 
cover the highest power of x in the quotient by divid- 
ing x*, the highest power of x in (A), by x, its highest 
power in x y. This division gives x 3 as the first 
term of the quotient. The following is the common 
process, and with each line is put the corresponding 
step of the process above explained, of which this is 
an example : 





hrj H C/5 


P- p 

IT. C C/) D, V) ^ C/) 

C Cu C ""i* p. pi 

cr cr 2. cr 5 - 

2. a; a ft 3. fT 

rj O- 1 Q3 CD 3 fK pj 

o 2 o P Q. n> o 

! fc 


to to 

H Ho 

to cc 

L> L> 

to w 








Ve *~s 


The second and following terms of the quotient 
are determined in exactly the same manner as the 
first. In fact, this process is not the finding of a quo- 
tient directly from the divisor and dividend, but one 
term is first found, and by means -of that term another 
dividend is obtained, which only differs from the first 
in having one term less in the quotient, viz., that 
which was first found. From this second dividend 
one term of its quotient is found, and so on until we 
obtain a dividend whose quotient has only one term, 
the finding of which finishes the process. It is usual 
also to neglect all the terms of the first dividend, 
except those which are immediately wanted, taking 
down the others one by one as they become necessary. 
This is a very good method in practice but should be 
avoided in explaining the principle, since the first 
subtraction is made from the whole dividend, though 
the operation may only affect the form of some part 
of it. 

If the student will now read attentively what has 
been said on the greatest common measure of two 
numbers, and then examine the connexion of whole 
numbers in arithmetic and simple expressions in alge- 
bra with which we commenced the subject of division, 
he will see that the greatest algebraical common meas- 
ure of two expressions may be found in exactly the 
same manner as the same operation is performed in 
arithmetic. He must also recollect that the greatest 
common measure of two expressions A and B is not 


altered by multiplying or dividing either of them, A, 
for example, by any quantity, provided that quantity 
has no measure in common with B. For example, 
the greatest common measure of a 2 x 2 and bcfi bx* 
is the same with that of 2 a? 2x 2 and a B x 3 , since 
though a new measure is now introduced into the first 
and taken away from the second, nothing is introduced 
or taken away which is common to both. The same 
observation applies to arithmetic also. For example, 
take the numbers 162 and 180. We may, without 
altering their greatest common measure, multiply the 
first by 7 and the second by 11, etc. The rule for 
finding the greatest common measure should be prac- 
tised with great attention by all who intend to proceed 
beyond the usual stage in algebra. To others it is not 
of the same importance, as the necessity for it never 
occurs in the lower branches of the science. 

In proceeding to the subject of fractions, it must 
be observed that, in the same manner as in arithmetic, 
when there is a remainder which cannot be further 
divided by the divisor, that is, where the dividend is 
so reduced that no simple term multiplied by the first 
term of the divisor will give the first term of the re- 
mainder, as in the case where the divisor is a 2 x -\-bx 2 
and the remainder ax-\- b; in this case a fraction 
must be added to the quotient, whose numerator is 
this remainder, and whose denominator is the divisor. 
Thus, in dividing a*-\-fi* by a-\-b, the quotient is 
a 3 a?b-\-ab 2 b*, and the remainder 2 4 , whence 


The arithmetical rules for the addition, etc., of frac- 
tions hold equally good when the numerators and de- 

3 1 

nominators are themselves fractions. Thus -| and -| 

T ^ 

are added, etc., exactly in the same way as | and fy, 
the sum of the second being 


and that of the first 



The rules for the addition, etc., of algebraic fractions 
are exactly the same as in arithmetic ; for both the 
numerator and denominator of every algebraic frac- 
tion stands either for a whole number or a fraction, 
and therefore the fraction itself is either of the same 


form as f or-|. Nevertheless the student should at- 
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 formulae which it is highly desirable that the 
student should commit to memory before proceeding 
to any other part of the subject. 

(a + *) + (* *) = 2* (1) 

(a + t) (a f) = 26 (2) 

a(a V) = b (3) 


(2* # + ) 2 =4tf 2 * 2 -f abx + P 


(a-\-b}(a &) = <& & 


( ( :i:x ( :_?=J-E:+?:+:'} 


a ma 
b mb 


! <T #</-{-<: ^ 0d? C 


d d ' d d 

a ( c ad-\-bc a c ad be 


b d ~ bd ' b d ~ bd 

a ac a a c ac 



a a 

b a c 


a a 

b ad c 
c be ~b 


~d ~d 

1 _ b 

a ~~ a 




WE have already defined an equation, and have 
come to many equations of different sorts. But 
all of them had this character, that they did not 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, 2, etc., or any other number or fraction 
whatever. The second side of this equation is, in 
fact, the result of the operation pointed out on the 
first side. On the first side you are directed to divide 
a 2 1 by a-\- 1 ; the second side shows you the result 
of that division. An equation of this description is 
called an identical equation, because, in fact, its two 
sides are but different ways of writing down the same 


number. This will be more clearly seen in the iden- 

tical equations 

+ = 20, 70 30 + ^ = 40 


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 0+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 


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- 

Let x stand for the number which is to be found. 

Then * + 56 = 200 2x. 

If, instead of 56, 200, and 2, any other given num- 


bers, a, 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 x -J- 4 
hours, and will have travelled (^ + 4)2, or 2x-\-S 
miles. The second has been gone x hours, and will 
have travelled 3 x miles. And, when the second over- 
takes the first, they have travelled exactly the same 
distance, and, therefore, 

3x = 2x + 8. 

If, instead of these numbers, the first goes a miles 
an hour, the second b, and c hours elapse before the 
second is sent after the first, 

bx = ax -f- 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 2x . . . 

C paid x -f 2 x or 3 x . . . 
D paid 2x-\- 3x or x . . . 


All together paid x + 2x-\-3x+5x, or 11 x, there- 

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 

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 T 2 ^, etc., and in x hours -f^. Similarly, in 
x hours, the second cock fills / F of the cistern. When 
the two have exactly filled the cistern, the sum of 
these fractions must represent a whole or 1, and, 

- + --1 
12 ^ 15 ~ 

If the times in which the two can fill the cistern are a 
and b hours, the equation becomes 

f+f- 1 - 

A person bought 8 yards of cloth for ^"3 2s., giving 
9s. a yard for some of it and 7s. a yard for the rest ; 
how much of each sort did he buy ? 

Let x be the number of yards at 7s. Then 7 x is 
the number of shillings they cost. Also 8 x is the 
number of yards at 9s., and (8 #)g, or 72 9*, is 
the number of shillings they cost. And the sum of 


these, or 7jc-j-72 9#, is the whole price, which is 
;3 2s., 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 


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 t a-\- c = b-\-c, a <r 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 = d v 
add c to both quantities, then 

a-\-b <r-f c = d-}-c or a-\-b = 

Again subtract b from both quantities, then a-\-b 
c b = d b, or a c = d b. Without always re- 


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 T 5 . If we 
multiply them all by the product of the denominators 
4x^X14, 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- 

3 V 392 
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 -7-7^, will become simple 
b ae bdj 


expressions, if they are multiplied by b X <^X bdf y or 
ft d** ef. But the most simple common multiple of b, 
de, and bdf, is bdef, which should be used in pref- 
erence to t> 2 d* 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""F : ~To~ ~~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*) 

~3~ ~~5~ : ~TO~~ ~~6~~ 

30*. 30 60*. 60 

But -g-is-g-X*, or 10*, -g- is-jr-X*, or 12*, etc.; 

so that we have 

10*4-12*^21 5(3 2*), (2) 

or 10*4-12* = 21 (15 10*), (3) 

or 10*-f 12* = 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 


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 : 

+- f f ,.. 

a ^ = 7. 

a + b r+ 
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 x from both sides, which 
gives 10* + 12* 10^ = 21 15, or 12^ = 6: divide 

these equal quantities by 12, which gives -r-^- =^-^, or 

1 1 U 

x = 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 ^, 
we shall find that 


j.^ ~ (r __ . 

3 5 ~10 6 6 ^ 5 ~10 ~ 6' 

this we find to be true, since 

1 1 11 , 7 2 22 , 11 22 
+5 1S 30' and !0- 6- == 66' and 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 : 

L _i_ __ -^H 

a H 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 : 

acehx abcehx acdeh aceh(f goc) 
~~~ ~~ ~~~ ~~ ~' 

or, cehx-\- ab ehx = acdh ace(f gx), 
or, cehx-\-abehx = acdh acef-\- acegx. 
Subtract acegx from both sides, and it becomes 

cehx -j- abehx acegx=acdh acef, 
or, (ceh-\- ab eh aceg}x = acdh ac ef. 
Divide both sides byceh + abeh aceg, which gives 
ac dh acef 

*r - _ 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- 


tablishes them both, and the same errors are 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 d, 10 for <?, 3 for /, 2 for 
g, and 6 for h, which gives for the value of x, 
3X5X7X6 3X5X10X3 
3X2X10X6 3X5 
3x5x12 180 

' 3X2X10X6' ' 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, 3x 2y = 4', 
this second equation by itself has an infinite number 

of solutions: to find them, y may be taken at pleasure, 

and x = ~- -. Of all the solutions of the second 


equation, one only is a solution of the first ; thus there 
is only one value of x and y which satisfies both the 
equations, and the finding of these values is the 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- 


pens whenever one of the equations can be deduced 
from the other. For example, when x-\-y = 8, and 
4x 29 = 3 4j, the second of these is the same as 
4^ + 47 = 3 + 29, or 4* + 4jF = 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. 



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 b where b 
is the greater of two quantities, instead of b a. Had 
we done this with the sum of two quantities, it would 
have made no difference, because a -f b and b + a are 
the same, but this is not the case with a b and b a. 
For example, 8 3 is easily understood; 3 can be 
taken from 8 and the remainder is 5 ; but 3 8 is an 


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 


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 + 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 -j- 3 8. This may be written 
56 + 3 (3 + 5), or 56 + 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 


= 0, then b = c. We can now understand the follow- 
ing equations : 

a b -f- c d= e, or b-\-d a c = e, 
2ab a? b'* = de, or a 2 + P 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 y and then d ( r) = 
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 ( <r) 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 ( r), 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 


which we can correct as certainly at the end as at the 

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 & = c, and it follows that d (b a) = d c, 
or d b-\-a = d c, or d -f- a b = d c, or d -\- 
(a b} = d c. This is d-\- ( *:) */ c, another 
case in which the rule is verified. Again, multiply 
together a b and m n, the product is am an 
bm-\-bn. This is the same product as arises from 
multiplying b a by n m, written in a different or- 
der. If, then, b a = c, and n m=p, or a 6 = 
c, and m ;* /, we find that ( c) X ( P) = 
cp. By which result we mean that a mistake, in the 
form of both a b and m n, will not produce a mis- 
take in the form of their product, which remains what 
it would have been had the mistake not been made. 

(n ni)(b a~] = bn bm an -{-am 
(n m}(a b} = an am b n -f- b m. 

If the first product be real and equal to P, the second 
is represented by P. The first is cp, the second is 
( <OXA which gives 


That is, a mistake in the form of one factor only alters 
the form of the product. To distinguish the right 
form from the wrong one, we may prefix -f- to the 
first, and to the second, and we may then recapit- 
ulate the results, and add others, which the student 
will now be able to verify. 

The sign -f- placed before single quantities shows 
that the form of the quantity is correct ; the sign - 
shows that it has been mistaken or changed. 

(+)X (+) = + * (+*)X( ) = ab 
( *)X( *) = + * = (+ ) 

. ^ 
~ h b 

-}- a 

etc. etc. 

We see, then, that a change in the form of any 
quantity changes the form of those powers whose ex- 
ponent is an odd number, but not of those whose ex- 
ponent is an even number. By these rules we shall 


be able to tell what changes would be made in an ex- 
pression by altering the forms of any of its letters. It 
may be fairly asked whether we are not changing the 
meaning of the signs -f and , in making -f a stand 
for an expression in which we do not alter the signs, 
and a for one in which the signs are altered. The 
change is only in name, for since the rule of addition 
is, " annex the expressions which are to be added 
without altering the signs of either," or " annex the 
expressions without altering the form of either ; " the 
quantity a -j- b, which is the sum of the two expres- 
sions a and b, stands for the same as -\-a-\-b, in 
which the new notion of the sign -j- is used, viz., the 
expressions a and b are annexed with unaltered forms, 
which is denoted by writing together -f a and -f b. 
Again, the rule for subtraction is, " change the sign 
of the subtrahend or expression which is to be sub- 
tracted, and annex the result to the other expression," 
or "change the form of the subtrahend and annex it 
to the other, which, the expressions being a and b, is 
written a b, which answers equally well to the sec- 
ond notion of the sign , since -\-a b indicates that 
a and b are to be annexed, the first without, the sec- 
ond with a change of form. These ideas of the signs 
-f and give, therefore, in practice, the same results 
as the former ones, and, in future, the two meanings 
may be used indiscriminately. But when a single 
term is used, such as -\- a or a, the last acquired 
notions of -f- and are always understood. 


This much being premised, we can see, by num- 
berless instances, that, if the form of a quantity is to 
be changed, it matters nothing whether it is changed 
at the beginning of the process, or whether we wait 
till the end, and then follow the rules above men- 
tioned. This is evident to the more advanced stu- 
dent, from the nature of the rules themselves, but the 
beginner should satisfy himself of this fact from expe- 
rience. We now give a proof of this, as far as one 
expression can prove it, in the solution of the equa- 

. a 2 a? x 

and - -- ax = - -- a 

which two equations only differ in the form in which 
a appears. For, if the form of a in the first equation 

be altered, that of and is unaltered, -{-ax be- 
b b 

comes ax, and -\- a becomes a. We now solve 
the two equations in opposite columns. 

J+ax= -_ +a _t, __*;=__,,_, 

a 2 -\-atx = a <2 x + ab ti* a 2 atx = a' 2 x ab 

a* ab P = a> x abx 

_ _ 

The only difference between these expressions 


arises from the different form of a in the two. If, in 
either of them, a be put instead of -f 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 -f b into 1>, and this equation will become 

which last, common division will show to be true. 

Again, suppose than when a, b, and c are in a 
given form, which we denote by -f- a, -f- b, and -f- c, 
the solution of a problem is, 


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. 


+ a,+b,+c ^T**^ 

a-\- c b 

+ .+*.-' a-c-t 

-\-a, b t c 7- 

a c-\-b 

a, b t c 

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 : 

a + c b' b a c' a-\-c b 1 b a c' 

We now proceed to apply these principles to the 
solution of the following problems : 

q - 1 --- 1 -- j - \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 


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 D, and that m is greater than 
n. They will then meet at some point between B and 
D. Let that point be H, and let AH be called x. 
Then A travels through AH, or x, in the time during 
which B travels through BH, or x a. But, since A 
goes m miles an hour, he travels the distance x in 

hours. Again, B travels the distance x a in - 
m n 

hours. These times are the same, and, therefore, 

x x a ma 

or x - - = AH 

m n m n 

m n 
The time which elapses before they meet is 

x a 
or . 

in m 


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. 

q - ^ -- J. - j --- \D i 

Then since A travels through HA or x in the time 
during which B travels through HB, or x -\- a, in the 
same manner as in the last case, we show that 

x x -\- a ma 

-=- -or* - - AH 
m n n m 

- =BH. 


The time elapsed is ... 


Case third. If they are moving from D to C, and 
if B moves faster than A, the point His 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 


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 JB, they must 
now meet somewhere between A and B\ as before let 
them meet in H 9 and let Aff=x. 

C\ +_, J. 1* 

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 


x = 

a x = 

The time elapsed is . . . 



m -\- n 

m-\- 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 t 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 



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. 

of AH. 

of BH. 

Time of meeting 

j Both move from Cto D, 



n a 

a -iftrr 

t A moves faster than B. 
( Both move from Cto D, 

B and D. 


m n 
n a 

m n 
a hrfnrr 

i A moves slower than B. 
j Both move from D to C, 

A and C. 

n m 

n m 

n- m bCfOTC - 


5 ' ' A moves slower than B. 
( Both move from D to C, 

A and C. 

n m 

n m 
n a 

n m 

< A moves faster than B. 
j A moves towards D and 

B and D. 

m n 

m n 

m n 
a tt 

5 ' 1 B towards C 
j A moves towards C and 

A and B. 

m + n 
m a 

m + n 
n a 

m + n 


J< ' B towards D. 

A and B. 

m -\- n 

m -\- n 

m + n 



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 





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 



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 , and writing 
-(- n t 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\n the fifth, but in 

m -f 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 AH be- 

ma ma ma 

come , or 7 - - , 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 



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 

m-\- n 

Also the time in 

Circumstances of the case. 

Direction of 
the point H. 

of AH. 

of BH. 

Time of meeting 

j Both move from C to D, 



n a 

a iftpr 

1 A moves faster than B. 
j Both move from C to Z>, 

B and D, 

m n 

m n 

n a 

m n 

' A moves slower than B. 
j Both move from D to C, 

A and C. 

n tn 

n m 

n m 

a ft 

3 ' 1 A moves slower than B. 
j Both move from D to C, 

A and C. 

n m 

n m 

n m a tCr ' 

(A moves faster than B. 
j A moves towards D and 

B and D. 


m n 

m n 
a after 

5< 1 B towards C. 
j A moves towards C and 

A and B. 

m + n 

m + n 

m + 

'* 1 B towards D. 

A and B. 

m -\- n 

m + n 

m + n 


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 


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 , 

m n 

which may be written thus , it is a sign that 

n m 

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 


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 bm shillings, and since his gain is the differ- 
ence between his receipts and expenditure : 
ax bm = c 

bm-}- c 

or x= 


Now suppose that he had worked so little as to lose c 
shillings instead of gaining anything. The equation 
from which x is derived is now 

bm a x = Cj 
which, when its form is changed, becomes 

ax 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 -|- 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- 


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 ; 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 quanti- 


ties as -|- c and c to a later stage, when he will be 
better prepared to understand the difficulties of the 
subject. We now proceed to another class of 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- 
ample, ^ is greater than ^r or the twelfth part of 5 
14 ZU 2i 

is greater than its twentieth part. Similarly, - \ is 

2i 4 i 

greater than g, 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 

a a m 

-r- or is m. Now since b may diminish so as to be 


equal to any part of a, however small, that is, so as 

to make m any number, however great, which is 


= m may be any number however great. This dimi- 
nution of , 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 

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 



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 

^1 TTT J^1 , 7-7- ma ma 

that m = n. We find that Aff= - or - or 

ma m n n n 

-j-. On looking at the equation which produced this 

xx a 

result we find that it becomes = , or x=x a, 

m m 

which is impossible. On looking at the manner in 
which this equation was formed, we find that it was 
made on the supposition that A and B are together at 
some point, which in this case is also impossible, since 
if they move at the same rate, the same distance which 
separated them at one moment will separate them at 
any other, and they will never be together, nor will 
they ever have been together on the other side of A. 
The conclusion to be drawn is, that such an equation 
as x = -^ indicates that the supposition from which x 
was deduced can never hold good. Nevertheless in 
the common language of algebra it is said that they 
meet at an infinite distance, and that -~- is infinite. 
This phrase is one which in its literal meaning is an 
absurdity, since there is no such thing as an infinite 
number, that is a number which is greater than any 
other, because the mind can set no bounds to the 
magnitude of the numbers which it can conceive, and 


whatever number it can imagine, however great, it 
can imagine the next to it. But as the use of the 
phrase is very general, the only method is to attach a 
meaning which shall not involve absurdity or con- 
fusion of ideas. The phrase used is this : When 
c = b, - = and is infinitely great. The student 
should always recollect that this is an abbreviation of 

the following sentence. "The fraction becomes 

c o 

greater and greater as c approaches more and more 
near to b ; and if c, setting out from a certain value, 
should change gradually until it becomes equal to b, 

the fraction -j- setting out also from a certain value, 

will attain any magnitude however great, before c be- 
comes equal to b." That is, before a fraction can as- 
sume the form -^ , it must increase without limit. The 
symbol oo is used to denote such a fraction, or in gen- 
eral any quantity which increases without limit. The 
following equation will tend to elucidate the use of 

this symbol. In the problem of the two couriers, the 

... ., .. ma x x a 

equation which gave the result ^ was = , or 

m m 

x = x a, which is evidently impossible. Neverthe- 
less, the larger x is taken the more near is this equa- 
tion to the truth, as may be proved by dividing both 

sides by x, when it becomes 1 = 1 , which is never 

d ^ 
exactly true. But the fraction decreases as x in- 


creases, and by taking x sufficiently great may be re- 
duced to any degree of smallness. For example, if it 


is required that should be as small as ^ of a 

unit, take x as great as 10000000 a, and the fraction 

bec mes 10000000 a' r WOOO- But "^becomes 

smaller and smaller, the equation 1 = 1 -be- 


comes nearer and nearer the truth, which is expressed 
by saying that when 1=1 , or x=x a, the so- 


lution is x = oo. In the solution of the problem of 
the two couriers this does not appear to hold good, 
since when m = n and x=-- the same distance a 
always separates them, and no travelling will bring 
them nearer together. To show what is meant by 
saying that the greater x is, the nearer will it be a so- 
lution of the problem, suppose them to have travelled 
at the same rate to a great distance from C. They 

C\ 1 \B 


can never come together unless CA becomes equal to 
CB t or A coincides with B, which never happens, 
since the distance AB is always the same. But if we 
suppose that they have met, though an error always 
will arise from this false supposition, it will become 
less and less as they travel farther and farther from 
C. For example, let (7.4 = 10000000 AB, then the 
supposing that they have met, or that B and A co- 
incide, or that BA = 0, is an error which involves no 
more than of A C; and though AB is always 

of the same numerical magnitude, it grows smaller 


and smaller in comparison with AC, as the latter 
grows greater and greater. 

Let us suppose now that in the problem of the two 
couriers they move in the same direction at the same 
rate, as in the case we have just considered, but that 
moreover they set out from the same point, that is, 
let 0. It is now evident that they. will always be 
together, that is, that any value of x whatever is an 
answer to the question. On looking at the value of 

AH, or , we find the numerator and denomina- 

m n 

tor both equal to 0, and the value of AH appears in 
the form -^. But from the problem we have found 
that one value cannot be assigned to AH, since every 
point of their course is a point where they are to- 
gether. The solution of the following equation will 
further elucidate this. Let 

ax-\- t>y = c 

dx + cy^f, 
from which, by the common method of solution, we 


ce bf af cd 

\ V 

~i~j ' y ~i~j 
ae bd ae bd 

Now, let us suppose that fe = ^fand ae = bd. Divid- 
ing the first of these by the second, we find 

ce bf c f 
- = -^-, or == ^r, or cd=af. 

ae bd a d 

The values both of x and y in this case assume the 
form - to find the cause of this we must return to 


the equations. If we divide the first of these by c, 
and the second by/, we find that 

a . b 

But the equations <r^ = /and r</=0/give us = -, 
and = -j, that is, these two are, in fact, one and 
the same equation repeated, from which, as has been 
explained before, an infinite number of values of x 
and y can be found ; in fact, any value may be given 
to x provided y be then found from the equation. We 
see that in these instances, when the value of any 
quantity appears in the form -^-, that quantity admits 
of an infinite number of values, and this indicates that 
the conditions given to determine that quantity are 
not sufficient. But this is not the only cause of the 
appearance of a fraction in the form -^-. Take the 
identical equation 

When a approaches towards b, a -\- b approaches to- 
wards 2 , and a 2 2 and a b approach more and 
more nearly towards 0. If a = b the equation assumes 
this form : 


This may be explained thus : if we multiply the nu- 

merator and denominator of the fraction by a b 

J) * 47 

(which does not alter its value) it becomes ^7. 

Ha no 

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 
A > >a J3 o 

in the form 7-. But since the result would have been 


had that multiplication not been performed, this 


last fraction must be used instead of the unmeaning 

. a 2 P (a + b)(a b) 

form -JT-. Thus the fraction or - ' , - 

a + b c(a b^ c(ab} 

is the fraction - - after its numerator and denomi- 

nator have been multiplied by a b, 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. 



T^VERY 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 -f- b b is 
0, 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 a 2 , is called the square of a ; from 
which a is called the square root of a 2 . As 169 is 
called the square of 13, 13 is called the square root of 
169. The following table will show how this phrase- 
ology is carried on. 

a is called the square root of a 2 , . . denoted by 
a " " " cube root of 3 , . . " " ^~cP 
a " " " fourth root of 4 , . . " " \/~a* '. 
a" " " fifth root of a 5 , .. " " v'~cP 
etc. etc. etc. 


If b stand for a 5 , \/~b stands for a, and the foregoing 
table may be represented thus : 

\ia* = b,a = Vb\ 

if a* = b,a = $ / ~b, 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 J^ 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 ^^^^^ could 
be found a fraction which has a square root, and the 


fraction in the last might be decreased to any extent 
whatever, so that though we cannot find a fraction 
whose square is 7, we may nevertheless find one whose 
square is as near to 7 as we please. To take another 
example, if we multiply 1-4142 by itself the product 
is 1-99996164, which only differs from 2 by the very 
small fraction -00003836, so that the square of 1-4142 
is very nearly 2, and fractions might be found whose 
squares are still nearer to 2. Let us now suppose the 
following problem. A man buys a certain number of 
yards of stuff for two shillings, and the number of 
yards which he gets is exactly the number of shillings 

which he gives for a yard. How many yards does he 

buy? Let x be this number, then is the price of 

2 x 

one yard, and x= or # 2 = 2. This, from what we 

have said, is impossible, that is, there is no exact 
number of yards, or parts of yards, which will satisfy 
the conditions; nevertheless, 1-4142 yards will nearly 
do it, 1-4142136 still more nearly, and if the problem 
were ever proposed in practice, there would be no 
difficulty in solving it with sufficient nearness for any 
purpose. A problem, therefore, whose solution con- 
tains a square root which cannot be extracted, may be 
rendered useful by approximation to the square root. 
Equations of the second degree, commonly called 
quadratic equations, are those in which there is the 
second power, or square of an unknown quantity : 
such as x* 3 = 4* 2 15, .* 2 -f 3# = 2* 2 x 1, etc. 


By transposition of their terms, they may always be 
reduced to one of the following forms : 

ax* -\-bx-\- <r = 

For example, the two equations given above, are 
equivalentto 3* 2 12 = 0, and # 2 4* 10, which 
agree in form with the second and last. In order to 
proceed to each of these equations, first take the equa- 
tion x* = a 2 . This equation is the same as x 2 a 2 = Q, 
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 : 

xa = Q 
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 + a) (a* -f x s ) = (** -f a x) (a 2 + a x -f 2 * 2 ) 
develop this equation, and transpose all its terms on 
one side, when it becomes 

or x a 

or O 2 a *)(x* 2ax 2 ) = 0. 
This last equation is true when A 2 2 = 0, or when 


x 2 = a 2 , 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 4a X 2a s = 2a 2 X 4# 2 ; if a be 
substituted instead of x, the result is = 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 x 2 2 ax 
# 2 = 0, or x 2 2ax = a 2 . Every equation of the sec- 
ond degree, if it has one value of x, has a second, of 
which x 2 a 2 is an instance, where x = a, in which 
by the double sign is meant, that either of them 
may be used at pleasure. We now proceed to the so- 
lution of ax 2 ^.#-f-<r 0. In order to understand 
the nature of this equation, let us suppose that we 
take for x such a value, that ax 2 bx-\- c, instead of 
being equal to 0, is equal to j, that is 

y = ax ei 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 ax 2 bx -\- c equal to nothing, in which case m 
is called a root of the equation, 

ax 2 bx + c = (2) 

and it follows that 

= Q (3) 

*In the investigations which follow, a, b, and c are considered as having 
the sign which is marked before them, and no change of form is supposed to 
take place. 


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 -\-rn~) 

or n-\- m= (4) 

In (3) substitute for b its value derived from (4), from 
which b = a(n-\-m}\ it then becomes 

am 1 a m (n -f- m~) -\-c = 0, or c amn Q, 

which gives m n = . (5) 

Substitute in (1) the values of b and c derived from 
(4) and (5), which gives 

y ax 2 a(m-\-n)x-\- amn 

= a(x 2 m-\-n x-\-mri). 

Now the second factor of this expression arises from 
multiplying together x m and x n ; therefore, 

y = a(x m) (x ) (6) 

To take an example of this, let y = 4x 2 5#-fl. 
Here when x = l, y = 4: 5-fl = 0, and therefore 
m = \. If we divide 4 x 2 5^-j-l by x 1, the quo- 
tient (which is without remainder) is 4x 1, and 


y = (x 1)(4* 1). 

This is also nothing when 4x 1 = 0, or when x is J. 
Therefore n = , and y = 4 (x 1 ) (x J) , a result 


coinciding with that of (6). If, therefore, we can find 
one of the values of x which satisfy the equation 
a x 2 b x -f- c = 0, we can find the other and can divide 
ax* bx-\-c into the factors a, x m and x n, or 

ax 2 bx-\- c = a(x m) (x ). 

If we multiply x -f- m by x -J- #, the only difference be- 
tween (x -j- m) (x -f- ) and (x m)(x ) is in the 
sign of the term which contains the first power of x. 
If, therefore, 

ax 2 bx-{- c = a(x /) (x ), 
it follows that 

ax 2 -f- b x -f- c = a(x-\-m) (x -|- ) . 
We now take the expression <z x 2 b x c. If there 
is one value of x which will make this quantity equal 
to 0, let this be m, and 

Let y = ax 2 bx c 
Then Q=am* bm c, 
from which y = a (x 2 ;/z 2 ) b (x m) 
= (x m) (a x-\-m b) 
= (x m) (a x -\- a m ^) . 

Let be called , or let am b = an ', then 


y'= (x m)(ax-\-a n) 
= a(x m) (x -(-). 

As an example, it may be shown that 

3* 2 * 2 = 3(> l)O-ff). 

Again, with regard to ax 2 -\-bx c, since (x-\-m) 
(x n) only differs from (x m) (^ + )in the sign 


of the term which contains the first power of x, it is 
evident that 

if a x 2 b x c = a (x m) (x -f- ri) 
a x 2 -j- b x c = a (x -\- m) (x n). 

Results similar to those of the first case may be ob- 
tained for all the others, and these results may be ar- 
ranged in the following way. In the first and third, 
m is a quantity, which, when substituted for x, makes 
y = 0, and in the second and fourth m and n are the 
same as in the first and third. 

1st. . . y = ax 2 bx-}-c = a(x m)(x ri) 

b c 

m-\-n= mn= . 
a a 

2d . . . = 

b c 

m 4- n = mn= . 
a a 

3d . . . y = ax 2 bx c = a(x 

b c 

m n mn=. 
a a 

4th . . y = a x 2 -\- b x c a (x -\- vi) (x n) 

b c 

m n = m n = . 
a a 

We must now inquire in what cases a value can be 
found for x } which will make jy = in these different 
expressions, and in this consists the solution of equa- 
tions of the second degree. 



and observe that (2ax 6) 2 = 4a 2 x 2 4abx-\- P. 
Multiply both sides of (1) by 4 a, which gives 

4ay = 4a'> x* 4abx + 4ac (2) 

Add ft to the first two terms of the second side of (2), 
and subtract it from the third, which will not alter the 
whole, and this gives 

b* (3) 

Now it must be recollected that the square of any 
quantity is positive whether that quantity is positive 
or negative. This has been already sufficiently ex- 
plained in saying that a change of the form of any ex- 
pression does not change the form of its square. Com- 
mon multiplication shows that (c ^/) 2 and (d r) 2 
are the same thing ; and, since one of these must be 
positive, the other must be also positive. Whenever, 
therefore, we wish to say that a quantity is positive, it 
can be done by supposing it equal to the square of an 
algebraical quantity. In equation (3) there are three 
distinct cases to be considered. 

I. When ft 2 is greater than 4ac, that is, when 
b^ 4ac is positive, let ft 4ac & 2 , which expresses 
the condition. 

Then 4ay = (2ax ) 2 & (4) 

and we determine those values of x, which makejF=0, 
from the equation, 

(2ax ) 2 & = Q. 

We have already solved such an equation, and we 
find that 


where either sign may be taken. This shows that y 
or ax 2 bx-\- c is equal to nothing either when 

instead of x is put -^ = 


Li d U d 

the second values being formed by putting, instead of 

k its value l 2 kac. They are both positive quan- 

tities, because & being equal to 2 ac is less than 

b-\- k 

ft, and therefore k is less than b, and therefore - 
^ _ ^ & a 

and - are both positive. These are the quantities 


which we have called m and n in the former investiga- 

tions, and, therefore, 

ax* bx-\-c = a(x m)(x ri) 

= a\x 

2a )' 

Actual multiplication of the factors will show that this 
is an identical equation. 

II. When / 2 , instead of being greater than 4tac, is 
equal to it, or when P 40<r and /& = 0. In this 

case the values of m and n are equal, each being -= 



/ b \ 2 

y = ax^ bx-\- c = a(x m)(x n) = a\x ^1 . 

In this case y is said, in algebra, to be a perfect 
square, since its square root can be extracted, and is 

V a\x j. Arithmetically speaking, this would 


not be a perfect square unless a was a number whose 
square root could be extracted, but in algebra it is 
usual to call any quantity a perfect square with re- 
spect to any letter, which, when reduced, does not 
contain that letter under the sign i/. This result is 
one which often occurs, and it must be recollected 
that when P lac = Q, ax* bx-\-c is a perfect 

III. When b* is less than lac, or when P lac 
is negative and lac & 2 positive, let lac & = &, 
and equation (3) becomes 

In this case no value of x can ever make y = 0, for 
the equation z; 2 + ^ 2 = indicates that v 2 is equal to 
a/ 2 with a contrary sign, which cannot be, since all 
squares have the same sign. The values of x are said, 
in this case, to be impossible, and it indicates that 
there is something absurd or contradictory in the con- 
ditions of a problem which leads to such a result. 

Having found that whenever 
ax 1 

it follows that a x 2 -f b x -f c = a (x -f- m) {x -\- ) , we 
know that 

(1) when b* is greater than lac, 

o o 

la 2a 

(2) when fl = 



and y is a perfect square ; 

(3) when 2 is less than 4#r, ax^ + bx + c cannot 
be divided into factors. 

Now, let 

y = ax* bxc (1) 

As before, 


Let P-\-ac = &. Then 

4ay = (2ax ) 2 P. (3) 

Therefore y is when (2 ax />) 2 = / 2 , or when 


Now, because ft is less than ^ 2 -j-4^r, b is less 
than Vb* -\-kac, therefore n is a negative quantity. 
Leaving, for the present, the consideration of the 
negative quantity, we may decompose (3) into factors 
by means of the general formula 

/ 2 _ 2 = (/_^)(/-f-^), which gives 
= (2ax-b k} (2ax 


from which y or 


ax* bx c = 

V P + ac +b\( V P + a~c b\ 



Therefore, from what has been proved before, 
x c = 

The following are some examples, of the truth of 
which the student should satisfy himself, both by ref- 
erence to the ones just established, and by actual 
multiplication : 

1/49 24 \ f 7 1/49 

X ; 





10 y v 10 

If we collect together the different results at which 
we have arrived, to which species of tabulation the 
student should take care to accustom himself, we have 
the following : 

* Recollect that > 7 ^ = 1/6X1 = v 6~X ^T= 2 ^67 


ax* bx-\-c = 

a ( X b VV 9 l '~^ 1 ' } {X*""* "~~ ) ( B ) 

\ / 


bx = 

) (D) 

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 
ax* -\-bx-\-c can be made equal to nothing by two 
values of x. For example, in (1), when 

*u . 

either #4 - = 

Z a 

L- <7 

If we call the values of x derived from the equations 
m and n, we find that 

1/> ac b VP lac 


In the cases marked (B), (C), and (D), the results 


m= n= ^ (B 

2a 2a 

m= n= 5 (D) 

2a 2a 

and in all the four cases the form of ax 2 -}- bx-\-c 
which is used, is the same as the corresponding form 

a(x m~) (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* kac is negative, the quantity ax 2 -f bx-\- c 
cannot be divided into factors at all, since l/ 2 ac 
is then no algebraical quantity, either positive or neg- 

In (B') both, if they exist at all, are positive. 

In (C') there are always real values for m and , 
since ^ 2 -j- kac is always positive; one of these values 
is positive, and the other negative, and the negative 
one is numerically the greatest. 

In (D') there are also real values of m and , one 
positive and the other negative, of which the positive 
one is numerically the greatest. Before proceeding 
any further, we must notice an extension of a phrase 


which is usually adopted. The words greater and less, 
as applied to numbers, offer no difficult}' 1 , and from 
them we deduce, that if a be greater than b, a c is 
greater than b c, as long as these subtractions are 
possible, that is, as long as c can be taken both from 
a and b. This is the only case which was considered 
when the rule was made, but in extending the mean- 
ing of the word subtraction, and using the symbol 
3 to stand for 5 8, the principle that if a be greater 
than b, a c is greater than b c, leads to the follow- 
ing results. Since 6 is greater than 4, 6 12 is greater 
than 4 12, or 6 is greater than 8; again 6 6 
is greater than 4 6, or is greater than 2. These 
results, particularly the last, are absurd, as has been 
noticed, if we continue to mean by the terms greater 
and less, nothing more than is usually meant by them 
in arithmetic ; but in extending the meaning of one 
term, we must extend the meaning of all which are 
connected with it, and we are obliged to apply the 
terms greater and less in the following way. Of two 
algebraical quantities with the same or different signs, 
that one is the greater which, when both are connected 
with a number numerically greater than either of them, 
gives the greater result. Thus 6 is said to be greater 
than 8, because 20 6 is greater than 20 8, is 
greater than 4, because 6 -f is greater than 6 4 ; 
+ 12 is greater than 30, because 40 -f 12 is greater 
than 40 30. Nevertheless 30 is said to be numeri- 
cally greater than -f 12, because the number contained 


in the first is greater than that in the second. For this 
reason it was said, that in (C), the negative quantity 
was numerically greater than the positive, because any 
positive quantity is in algebra called greater than any 
negative one, even though the number contained in 
the first should be less than that in the second. In 
the same way 14 is said to lie between + 3 and 
20, being less than the first and greater than the 
second. The advantage of these extensions is the 
same as that of others ; the disadvantage attached to 
them, which it is not fair to disguise, is that, if used 
without proper caution, they lead the student into 
erroneous notions, which some elementary works, far 
from destroying, confirm, and even render necessary, 
by adopting these very notions as definitions ; as for 
example, when they say that a negative quantity is 
one which is less than nothing ; as if there could be 
such a thing, the usual meaning of the word less being 
considered, and as if the student had an idea of a 
quantity less than nothing already in his mind, to 
which it was only necessary to give a name. 

The product (x ni} (x ) is positive when 
(x m) and (x n) have the same, and negative when 
they have different signs. This last can never happen 
except when x lies between m and n, that is, when x 
is algebraically greater than the one, and less than 
the other. The following table will exhibit this, where 
different products are taken with various signs of m 
and , and three values are given to x one after the 


other, the first of which is less than both m and , 
the second between both, and the third greater than 





4) (x 7) 

+ 1 

+ 18 

in = 

+ 4 

+ & 


n = - 


+ 10 

+ 18 

(x + 



+ 30 

m = 




n = 

+ 3 

+ 4 

+ 14 

(x + 

2)O + 12) 


+ 11 

m = 







+ 11 

The student will see the reason of this, and per- 
form a useful exercise in making two or three tables 
of this description for himself. The result is that 
(# m)(x n) is negative when x lies between m and 
n, is nothing when x is either equal to m or to n, and 
positive when x is greater than both, or less than 
both. Consequently, a(x m}(x n) has the same 
sign as a when x is greater than both m and n, or less 
than both, and a different sign from a when x lies be- 
tween both. But whatever may be the signs of a, b, 
and c, if there are two quantities m arid n, which make 

ax* + *+ c = a(x m) (x ), 

that is, if the equation ax 2 -\- bx-\- c = Q has real roots, 
the expression ax 2 + bx + c always has the same sign 


as a for all values of x, except when x lies between 
these roots. 

It only remains to consider those cases in which 
ax 2 -\-bx-\-c cannot be decomposed into different fac- 
tors, which happens whenever b 2 4ac is 0, or nega- 
tive. In the first case when b 2 4ac = Q, we have 




and as these expressions are composed of factors, one 
of which is a square, and therefore positive, they have 
always the same sign as the other factor, which is a. 
When b 2 4ac is negative, we have proved that if 
y = a x 2 b x -j- *"> 4ay = (2axd=. ) 2 -f- k 2 , where k 2 = 
kac b 2 , and therefore 4 ay being the sum of two 
squares is always positive, that is, ax 2 bx-\- c has 
the same sign as a, whatever may be the value of x. 
When = 0, the expression becomes ax 2 -\-frx, or 
which is nothing either when x = Q, or 

when ax-\-b-= and x = -- ; the general expres- 
sions for m and n become in this case - 5 -- and 
, which give the same results. 


When ^ = 0, the expression is reduced to ax^-\- 
= 0, which is nothing when x= */ , which is 
not possible, except when c and a have different signs. 
In this case, that is, when the expression assumes the 
form ax 2 c, it is the same as 



The same result might be deduced by making /> = 
in the general expressions for m and n. 

When 0, the expression is reduced to bx-\-c, 
which is made equal to nothing by one value of x only, 
that is - - . If we take the general expressions 
for m and n, and make a Q in them, that is, in 

, and - , we find as 

Za La 

the results -y- and ^ . These have been already ex- 
plained. The first may either indicate that any value 
of x will solve the problem which produced the equa- 
tion ax" 2 -f bx + <: = 0, or that we have applied a rule 
to a case which was not contemplated in its forma- 
tion, and have thereby created a factor in the numera- 
tor and denominator of x, which, in attempting to 
apply the rule, becomes equal to nothing. The stu- 
dent is referred to the problem of the two couriers, 
solved in the preceding part of this treatise. The 
latter is evidently the case here, because in returning 
to the original equation, we find it reduced to bx-\- 
<r = 0, which gives a rational value for x t namely, -- -. 

2 h 

The second value, or - -^-, which in algebraical lan- 
guage is called infinite, may indicate, that though 
there is no other value of x, except - - , which 
solves the equation, still that the greater the number 
which is taken for x, the more nearly is a second so- 


lution obtained. The use of these expressions is to 
point out the cases in which there is anything remark- 
able in the general problem ; to the problem itself we 
must resort for further explanation. 

The importance of the investigations connected 
with the expression a x 2 -j- b x -j- c, can hardly be over- 
rated, at least to those students who pursue mathe- 
matics to any extent. In the higher branches, great 
familiarity with these results is indispensable. The 
student is therefore recommended not to proceed until 
he has completely mastered the details here given, 
which have been hitherto too much neglected in Eng- 
lish works on algebra. 

In solving equations of the second degree, we have 
obtained a new species of result, which indicates that 
the problem cannot be solved at all. We refer to 
those results which contain the square root of a nega- 
tive quantity. We find that by multiplication the 
squares of c d and of d c are the same, both being 
c* 2cd-\- d*. Now either c d or d c is positive, 
and since they both have the same square, it appears 
that the squares of all quantities, whether positive or 
negative, are positive. It is therefore absurd to sup- 
pose that there is any quantity which x can represent, 
and which satisfies the equation x* = a 2 , since that 
would be supposing that x 2 , a positive quantity, is 
equal to the negative quantity a 2 . The solution is 
then said to be impossible, and it will be easy to show 
an instance in which such a result is obtained, and 


also to show that it arises from the absurdity of the 

Let a number a be divided into any two parts, one 
of which is greater than the half, and the other less. 

Call the first of these -5- -f x, then the second must be 

-^ x, since the sum of both parts must be a. Mul- 


tiply these parts together, which gives 

As x diminishes, this product increases, and is great- 
est of all when x = Q, that is, when the two parts, into 
which a is divided, are -^ and -^-, or when the number 

Z L 

a is halved. In this case the product of the parts is 

a a a 2 

o- X -K-, or , and a number a can never be divided 

* z #2 

into two parts whose product is greater than -j. This 

being premised, suppose that we attempt to divide 
the number a into two parts, whose product is b. Let 
x be one of these parts, then a x is the other, and 
their product is ax x 2 . 
We have, therefore, 

ax x* = b 
or x 2 ax-\-b = Q. 

If we solve this equation, the two roots are the two 
parts required, since from what we have proved of 
the expression x 2 ax-}- b the sum of the roots is a 
and their product b. These roots are 


which are impossible when - -- b is negative, or when 

b is greater than ^-, which agrees with what has just 

been proved, that no number is capable of being di- 


vided into two parts whose product is greater than -j. 
We have shown the symbol V a to be void of 
meaning, or rather self-contradictory and absurd. 
Nevertheless, by means of such symbols, a part of 
algebra is established which is of great utility. It 
depends upon the fact, which must be verified by ex- 
perience, that the common rules of algebra may be 
applied to these expressions without leading to any 
false results. An appeal to experience of this nature 
appears to be contrary to the first principles laid down 
at the beginning of this work. We cannot deny that 
it is so in reality, but it must be recollected that this 
is but a small and isolated part of an immense sub- 
ject, to all other branches of which these principles 
apply in their fullest extent. There have not been 
wanting some to assert that these symbols may be 
used as rationally as any others, and that the results 
derived from them are as conclusive as any reasoning 
could make them. I leave the student to discuss this 
question as soon as he has acquired sufficient knowl- 
edge to understand the various arguments: at present 

*The general expressions for in and give a ~ a * as the roots of 



let him proceed with the subject as a part of the 
mechanism of algebra, on the assurance that by care- 
ful attention to the rules laid down he can never be 
led to any incorrect result. The simple rule is, apply 
all those rules to such expressions as V a, a -f V b, 
etc., which have been proved to hold good for such 
quantities as \/a, a -f- 1/^, etc. Such expressions as 
the first of these are called imaginary, to distinguish 
them from the second, which are called real; and it 
must always be recollected that there is no quantity, 
either positive or negative, which an imaginary ex- 
pression can represent. 

It is usual to write such symbols as V b in a dif- 
ferent form. To the equation = ^x( 1) a pply 
the rule derived from the equation V ' xy = V 'xX V y t 
which gives V b = Vb X V- 1, of which the first 
factor is real and the second imaginary. Let J/=*i 
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 : 

V '-^24 = 1/24 1/ 1 = 2 1/6V1 

The following tables exhibit other applications of 
the rules : 


c =aV--l ci^ aty 

^ 2 a 2 c s = a s , etc. 

= a 

The powers of such an expression as a\/ 1 are 
therefore alternately real and imaginary, and are posi- 
tive and negative in pairs. 


Let the roots of the equation ax* -\- frx-\-c = Q be im- 
possible, that is, let 2 ac be negative and equal 
to 2 . Its roots, as derived from the rules estab- 
lished when 2 ac was positive, are 

and - = -- , or 


Take either of these instead of x ; for example, let 
b k 


P bk / T 2 

Then a x 2 = -. - V 1 . 

4a 2a 40 

ffi bk 

-35 + H"- 

Therefore, ax"*-}-bx-\-c=-- - \- c, in 

4a 4a &a 

which, if ac & 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 -f b V - - 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=0. Now, if b is not 
equal to d y let b d=e; then a c -\- eV 1 = 0, and 

/ T c a 
V 1 = -- ; that is, an imaginary expression is 

equal to a real one, which is absurd. Therefore, b = d 


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- 
ity in V 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 

The imaginary expression I/ 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 
a is as inconceivable as V a. What, then, is the 
difference of signification? The following problems 
will elucidate this. A father is fifty-six, and his son 
twenty-nine years old : when will the father be twice 
as old as the son? Let this happen x years from the 
present time ; then the age of the father will be 56 -|- x, 
and that of the son 29 -f#; and therefore, 56-f-# = 
2 (29 -f #) = 58 -j- 2 x, 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 


terms o! 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 


or x= -=- 

the roots of which are imaginary when b is greater 
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 


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. 



'T^HE 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 : 


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 a 2 # 3 # 4 a 5 . . a m . . a m+n , etc. 
Roots f/^* ^a Va V^ Va m+ t/^, etc. 

which series are connected together by the following 
equation, (v / a) n = a. 

*The z is usually omitted, and the square root is written thus iTa. 


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 20 2 ^ 3 ; that 
is, to find 2 a? ^ X 2 a 2 P X 2 a 2 &. The common rules 
of multiplication give, as the result, 80 6 ^ 9 , which is 
expressed in the following equation, 


2 a] 64 a Q ' 
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 , and multiply the index of 
every letter by n, that is, 

(a? b q c r y a np b nq c nr . 

In extracting the root of any simple term, we are 
guided by the manner in which the corresponding 
power is found. The rule is : Extract the required root 
of the coefficient, and divide the index of each letter 
by the index of the root. Where these divisions do 
not give whole numbers as the quotients, the expres- 


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 
16a 12 M, or find f/160 12 ^M. The expression here 
given is the same as the following : 

or (2# 3 < 2 ^) 4 , the fourth root of which is 20 3 ^ 2 ^, 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, T^ZZ? = ^ ~A f. For, raise ^A V~B to the 
third power, the result of which is, 


or iTZ f/A- V~A x f^ 

or AB. 

In the same way it may be proved generally, that 
VABC=v r Av r BV^ 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 fl6 a^b 1 c we must 
observe that 16 is 8 X 2, at is a s X ^, b 1 is b* X ^ and 
the expression is 8 a* b* X 2 a b c, 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, 'Zabc maybe found either 


exactly, or, when that is not possible, by approxima- 

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 O 2 ) 4 or a 2 X <$ X # 2 X 2 is a 8 , or 4X2 . 
This is the same as (# 4 ) 2 , which is cfi X # 4 > or a s . 
Therefore, generally (a m ) n = (a M ) m = a mn . In the same 
manner, a root of a root is the root whose index is the 
product of the indices of the two roots. Thus 

For since a = %/a Va l/a X V~a V~a i/a, the square 
root of a is I/a i/a v/fl, the cube root of which is Va. 
This is the same as v i/a, and generally 

m / n / n / nt /^ mn / 

V y a = v v a = v a. 

Again, when a power is raised and a root extracted, 
it is indifferent which is done first. Thus f 7 ^ 2 is the 
same thing as (^ a ) 2 . For since a 2 = a^a, the cube 
root may be found by taking the cube root of each of 
these factors, that is it / a? = f / aX f / ^=(f / a) 2 ) and 

In the expression v 7 ^, n and m may both be mul- 
tiplied by any number, without altering the expres- 
sion, that is 

To prove this, recollect that 


But a" 1 * is (a 1 *)', and by definition \/(a m y = a m . There- 
fore "\/a m * = \/~cr. This multiplication is equivalent 
to raising a power of 1/0"*, and afterwards reducing 

the result to its former value, by extracting the corre- 

mp . 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 aid 
principles hitherto laid down. 

z/7 7T j 

1 -f- b} n ~ m X V^( ^)"+ WI = 
(cP P) 

n \ab __ v^ab __ {/a yHJ _ n \a_ *\b 

\^ = " V^ = " 'VTVd = : \^ ' ' W 

The quantity \/a m is a simple expression when m 
can be divided by n, without remainder, for example 
i/a 12 = a*, f/0 20 = a 4 , and in general, whenever m can 


be divided by n without remainder, i/ 7 ^" 1 = a n . 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, 


and not in reality, that is, when m is divisible by 


n, and the quotient is /, a n shall stand for a p , or 


aaa (/)* It will be convenient to let a n always 

stand for i/0, in which case the condition alluded to 


is fulfilled, since when /, cF or {X5" = a*. This 

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 : 

a% stands for i/a 1 or I/a 

a% stands for ^ a 

a* stands for \/a 
a% stands for f 7 ^ or (f/^ ) 2 
al stands for I/a* or (Va ) 7 

(P + ^) 2 stands for V (p + q} 

stands for 

(-i\\ . 
dy stands for 

The results at which we have arrived in this chap- 
ter, translated into this new language, are as follows : 

^') "=(*")"=* (!) 

BC)" = A" " C" (2) 

*This is a notation in common use, and means that aaa is to be 

continued until it has been repeated/ times. Thus 

a+a + a+ (/)=/, 

a X a X a X (?! = <**. 


f l\- 1 

(a* )*,= * (3) 

/ \1 / i\* m 

{a m ) n = \a n ) =a n (4) 

" = (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 a m , hold good also for performing 
the same operations upon such quantities as a", in 
which the exponents are fractional. The truth of this 
last assertion we proceed to establish. 

Suppose it required to multiply together a" and 

a n , or Va and Va l . From (2) this is V a" 1 X *', or 

1/a m+i , or a"*". Suppose it now required to multiply 

w p_ 

a n and a q '. From (5) the first of these is the same as 

*n? ftf 

a nq , and the second is the same as a nq . The product 


of these by the last case is a n , or T / l ** + "->. But 

m g -|- np . m p 

- - is --- h an( i therefore 


a n X <* q =a 


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 


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. 

Suppose it required to find \a") . It is evident 

tn_ tn i zm f fn \ ^ 2 " z 

that a n X a n = a" * = """, or \a* ) = a*~. Similarly 

/ \3 yn / y rnp 

\a*J = a n , and so on. Therefore \a n ) a n , 
Again to find (/**)*, or V a" . Let this be a? . 

Then a* = V a" , or (a*) = a", or a~ -.= a". There- 

xq m x m ( ^\j ZL 

fore - - = , or = , and \a n ) = a nq 
y n y nq 

( \2- */~7 -V 
Again to find \a n ) or ]/ \a n ) . Apply the last 

/ y m 

two rules, and it appears that \a n ) = a n , and 

g / mp mp / \2- mp ^ X 

V a n = a"* . Therefore {a* ) q = a n * = a n * . 

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 


or fractional numbers, = a m ~ n . The only cases 

,, * 


which have been considered in forming this rule are 


those in which m is greater than #, 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- 

pie, suppose it required to find . If we apply the 

rule, we find the result # 3 ~ 7 , or a-*, 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, cr* may be 

used in future to stand for this result. Now the frac- 
as 3 1 

tion -= is the same as -7, which is obtained by divid- 
a 1 a* 

ing both its numerator and denominator by a B . There- 
fore r is the rational result, for which we have ob- 

tained a" 4 by applying a rule in too extensive a manner. 

Nevertheless, if or* be made to stand for -r, and 

1 a 

a- 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, *x<r~ is X , or -^, which is 
^, or tf-^+'O, or ar m - H . Similarly 

a nt a nt a n 

IF*' or T' is ^' or an ~ m > or *"" ~ n) 


(^)" W is^,or-L, orir-, 

and so on. 


It has before been shown that a stands for 1 when- 
ever it occurs in the solution of a problem. We can 
now, therefore, assign a meaning to the expression a, 
whether m be whole or fractional, positive, negative, 
or nothing, and in all these cases the following rules 
hold good : 


= a m ~" = a" 1 a 
a n 

The student can now understand the meaning of such 
an expression as 10 - 301 , where the index or exponent 
is a decimal fraction. Since .301 is $}$, this stands 
for lcx j/ / (10) 3 1 , 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. 


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- 

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 Encyclopedia Bri- 
tannica, Article "Tables," Vol. XXIIL 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. 


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 x 2 -\- x upon the letter 
or letters which it contains, if a very small addition be 
made to the value of one of these letters, in 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. 


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 

x 2 = . 0001 x 2 = . 0000000001 

x* = . 000001 x s = . 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 x 2 and X B are both small fractions whem com- 
pared with unity ; nevertheless, x 2 is very great when 
compared with x s , 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 


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 -f- BW -j- Ch 3 , we may, if h is 
very small, reject BW -f- Ch B , 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 
x 2 -{- x, and suppose that x is increased by a very small 
quantity h. The expression then becomes ( < #-f~^) 2 -|~ 
(x + h), or x 2 + x + (2x+l)A + tf. But it was x* + x; 
therefore, in consequence of x receiving the increment 
h, x 2 -f- x has received the increment (2x-{- \}h-\-h 2 t 
for which (2^-J-l)^ may be written, since h is very 
small. This is proportional to h, since, if h were 
doubled, (2#-|-l)^ 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 -f- h, y is changed into a quantity of 
the form y -f Ah + Bh* -f Ch* + etc.; that is, y re- 
ceives an increment of the form Ah -j- Bh^ -j- Ch* -j- 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 


all the instances which occur in elementary treatises 
on arithmetic and algebra. 
For example : 

= x m -f m x m ~ l h + m ^ x m ~ 2 A 2 + etc. 

Here A = m x m-1 , B = m - x m ~' 2 , etc.; and if h be 


very small, (x -\- hy = x m -\- mx m ~ 1 A, nearly. 

Again, ^5 1 + * + * + '^ + etc. Therefore, 

e* X e* or *+* = e x + e*h+ W + etc. And if h be 


very small, c*+ h = e x -f e*h, nearly. 

Again, log. (1 + ') = M(n' 1' 2 -f J' 3 etc.). 
To each side add log.*, recollecting that 
log.A: + log. (1 + ') = log. *(1 + ') = log. (x + xtt), 

and let 

, x 

xn =h or n = . 

Making these substitutions, the equation becomes 


If h is very small, log. (.# -f- /f) = log. x-\- " - h. 


We can now apply this to the logarithmic example 
with which we commenced this subject. It appears 

Log. 41713 = 4 . 6202714 

Log. (41713 + 1) = 4 . 6202714 + . 0000104 
Log. (41713 + 2) = 4 . 6202714 + . 0000104 X 2. 
From which, and the considerations above-men- 


Log. (41713 -M) = log. 41713 -f . 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 /$ = .27. 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 
. 00001 04 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- 


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. 



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 


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 


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- 

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 
development of (a-\-x)"$ will exercise him in the use 
of the binomial theorem ; when he has obtained the 


series which is equivalent to (#-|-#) 3 , let him, in the 


same way, develop (a-\-x^' } the product of these, 
since |-|-|=r3, ought to be the same as the develop- 
ment of (a + x)*, or as a 9 -f 3a* x -\-3ax* + x*. 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 -|- A\ 
Again, when any development is obtained, it should 
be seen whether the development *p ssesses a U the 
properties of the expression from which it has been 
derived. For example, - is proved to be equiv- 

-1 OC 

alent to the series 

1 -j- x -\- x 2 -\- x s +, etc. , ad infinitum. 

This, when multiplied by 1 x, should give 1 ; when 
multiplied by 1 x 2 , should give \-\-x, because 

-JX(1 *)=!, r - X(l x^ = l + x, etc. 

J- - OC -L - OC 


3 +...admf. 

etc . 

Now, since a* X a- y = a* fj ', 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, 


is often doubtful, on account of the generality of the 

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-^c) 3 , or any other case, which is there applied 
to (1 -|- #)*". Again, the general form of the product 
(x + a), (#-(-), (x-\-c), 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 

+ 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 


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 


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 


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 -f- a X = # 2 > he considers the multipli- 
cation of a m -\- a n by b m -f b n 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- 


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- 


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 (#-|-^) 2 > (#-M) 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 


objection to this appears to have been the necessity 
which arose for departing from the appearance 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. 

"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- 

have observed that beginners are not disposed 


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 
+ #X b = 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 

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 


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 ax 2 -\- bx + = assume, according to the signs 
of a, b, and <:, etc. Both the preceptor and the pupil, 
but especially the former, will derive great advantage 
from the perusal of Lacroix, Essais sur PEnseignement 
'en general et sur celui des Mathe'matiques 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 Mathtmatique. Philosophie- 
Enseignement. Paris, 1898, 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 Methodologie MathSmatique of Felix Dauge (Deuxieme edition, revue 
et augmented. 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 M&thodes dans les Sciences de Raisonne- 
ment, (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 der Prinzipien der Mechanik and his Neue 
Grundmittel 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 
Cyclopcedia, 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. 


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 Traitf 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' Algtbre 
Elementaire, Paris, Colin, 1896. A standard and exhaustive work on higher 
algebra is the Cours d'Algebre 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 Afgebra 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 excellent recent text-book for beginners, and one that skil- 
fully introduces modern notions, is the Elements of Algebra of W. W. Beman 
and P. E. Smith (Boston, 1900). Fisher and Schwatt's elementary text-books 


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 Algeb ra (London : Mac- 
millan), and Oliver, Wait, and Jones's Treatise on Algebra (Ithaca, N. Y., 
1887), also Fine's Number 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 Recreations, Chicago, 
1898). Professor Schubert has recently begun the editing of a new and larger 
series of mathematical text-books called the Sammlung Schubert (Leipsic : 
Goschen), which contains three works treating of algebra. In this connexion 
maybe mentioned also Matthiessen's admirable Grundziige 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. Schiiller's Arithmetik und Algebra fur hb'here 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 uber 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. 


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- 



IN this treatise on the difficulties of Geometry and 
Trigonometry, we propose, as in the former part 
of the work, to touch on those points only which, from 
novelty in their principle, are found to present 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 


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 


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 3 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 


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 ends of which, A and B, draw two lines, AF 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 F in AF, could be thus connected with the same 
point of AB, it is evident that two straight lines would 
enclose a space, viz., the lines CE and CF, which 

C B 

Fig. x. 

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 


such, indeed, #s 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, 


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 <- 

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 


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 

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 



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 



Fig. 3- 

right line, but on opposite sides of the point O, evi- 
dently make an opening or angle which is equal to 
the sum of the angles AOC and COB, or equal to two 
right angles. A line may also be said to make with 
itself an opening equal to four right angles, since 
after revolving through four right angles, the moving 
line reassumes its original position. We may even 
carry this notion farther : for if the moving line be in 


the position OE when P hours have elapsed, it will 
recover that position after every twenty-four hours, 
that is, for every additional four right angles de- 
scribed; so that the angle AOE is equally well repre- 
sented by any of the following angles : 
4 right angles -f A OE 
8 right angles -f A OE 
12 right angles -f AOE, etc. 

These formulae which suppose an opening greater 
than any apparent opening, and which take in and 
represent the fact that the moving line has attained 
its position for the second, third, fourth, etc., time, 
since the commencement of the motion, are not of 
any use in elementary geometry ; but as they play an 
important part in the application of algebra to the 
theory of angles, we have thought it right to mention 
them here. 

It is plain also that we may conceive the line OE 
to make two openings or angles with the original po- 
sition OA : (1) that through which it has moved to re- 
cede from OA ; (2) that through which it must move 
to reach OA again. The first (in the position in which 
we have placed OA} is what is called in geometry the 
angle A OE ; the second is more simply described as 
composed of the openings or angles EOC, COB, 
BOD, DO A, and is not used except in the application 
of algebra above mentioned.* Of the two angles just 

*But use is made of it in some modern text-books of elementary geome- 


alluded to, one must be less than two right angles, 
and the second greater ; the first is the one usually 
referred to. 

It is plain that the angle or opening made by two 
lines does not depend upon their length but upon 
their position ; if either be shortened or lengthened, 
the angle still remains the same ; and if while the an- 
gle increases or decreases one of the straight lines 
containing it is diminished, the angle so contained 
may have a definite and given magnitude at the mo- 

Fig- 4 

ment when the straight line disappears altogether and 
becomes nothing. For example, take two points of 
any curve A, and join A and B by a straight line. 
Let the point B move towards A \ it is evident that 
the angle made by the moving line with AB increases 
continually, while as much of one of the lines contain- 
ing it as is intercepted by the curve, diminishes with- 
out limit. When this intercepted part disappears en- 
tirely, the line in which it would have lain had it had 
any length, has reached the line AG, which is called 
the tangent of the curve. 


In elementary geometry two equal angles lying on 
different sides of a line, such as A OJS, A OH (Fig. 3), 
would be considered as the same. In the application 
of algebra, they would be considered as having differ- 
ent signs, for reasons stated at length in pages 112 et 
seq., of the first part of this Treatise. It is also com- 
mon in the latter branch of the science to measure 
angles in one direction only ; for example, in Figure 3 
the angles made by OE, OF, OG, and OH, if measured 
upwards from OA y would be the openings through 
which a line must move in the same direction from OA, 
to attain those positions; and the second, third, and 
fourth angles would be greater than one, two, and 
three right angles respectively. 

We proceed to the method of reasoning in geom- 
etry, or rather to the method of reasoning in general, 
since there is, or ought to be, no essential difference 
between the manner of deducing results from first 
principles, in any science. 



IT is evident that all reasoning, of what form soever, 
can be reduced at last to a number of simple prop- 
ositions or assertions ; each of which, if it be not self- 
evident, depends upon those which have preceded it. 
Every assertion can be divided into three distinct 
parts. Thus the phrase, "all right angles are equal," 
consists of : (1) the subject spoken of, viz. , right an- 
gles, which is here spoken of universally, since every 
right angle is a part of the subject ; (2) the copula, or 
manner in which the two are joined together, which 
is generally the verb is, or is equal to, and can always 
be reduced to one or the other : in this case the co- 
pula is affirmative ; (3) the predicate, or thing asserted 
of the subject, viz., equal angles. The phrase, thus 
divided, stands as written below under 1, and is called 
a universal affirmative. The second is called a particu- 
lar affirmative proposition ; the third a universal nega- 
tive; the fourth a particular negative: 

1. All right angles are equal (magnitudes). 


2. Some triangles are equilateral (figures). 

3. No circle is convex to its diameter. 

4. Some triangles are not equilateral (figures). 
Many assertions appear in a form which, at first 

sight, cannot be reduced to one of the preceding ; the 
following are instances of the change which it is nec- 
essary to make in them : 

1. Parallel lines never meet, or parallel lines are 
lines which never meet. 

2. The angles at the base of an isosceles triangle 
are equal, or an isosceles triangle is a triangle having 
the angles at the base equal. 

The different species of assertions, and the argu- 
ments which are compounded of them, may be dis- 
tinctly conceived by referring them all to one species 
of subject and predicate. Since every assertion, gen- 
erally speaking, includes a number of individual cases 
in its subject, let the points of a circle be the subject 
and those of a triangle the predicate. These figures 
being drawn, the four species of assertions just alluded 
to are as follows : 

1. Every point of the circle is a point of the tri- 
angle, or the circle is contained in the triangle. 

2. Some points of the circle are points of the tri- 
angle, or part of the circle is contained in the tri- 

3. No point of the circle is a point of the triangle, 
or the circle is entirely without the triangle. 


4. Some points of the circle are not points of the 
triangle, or part of the circle is outside the triangle. 

On these we observe that the second follows from 
the first, as also the fourth from the third, since that 
which is true of all is true of some or any ; while the 
first and third do not follow from the second and 
fourth, necessarily, since that which is true of some 
only need not be true of all. Again, the second and 
fourth are not necessarily inconsistent with each other 
for the same reason. Also two of these assertions 
must be true and the others untrue. The first and 
the third are called contraries, while the first and 
fourth, and the second and third are contradictory. 
The converse of a proposition is made by changing the 
predicate into the subject, and the subject into the 
predicate. No mistake is more common than con- 
founding together a proposition and its converse, the 
tendency to which is rather increased in those who 
begin geometry, by the number of propositions which 
they find, the converses of which are true. Thus all 
the definitions are necessarily conversely true, since 
the identity of the subject and predicate is not merely 
asserted, but the subject is declared to be a name 
given to all those magnitudes which have the proper- 
ties laid down in the predicate, and to no others. 
Thus a square is a four-sided figure having equal 
sides and one right angle, that is, let every four-sided 
figure having, etc., be called a square, and let no other 
figure be called by that name, whence the truth of the 


converse is evident. Also many of the facts proved 
in geometry are conversely true. Thus all equilateral 
triangles are equiangular, from which it is proved that 
all equiangular triangles are equilateral. Of the first 
species of assertion, the universal affirmative, the con- 
verse is not necessarily true. Thus " every point in 
figure A is a point of B," does not imply that "every 
point of B is a point of A," although this may be the 
case, and is, if the two figures coincide entirely. The 
second species, the particular affirmative, is conversely 
true, since if some points of A are points of B, some 
points of B are also points of A. The first species of 
assertion is conversely true, if the converse be made 
to take the form of the second species : thus from 
"all right angles are equal/' it may be inferred that 
"seme equal magnitudes are right angles." The third 
species, the universal negative, is conversely true, 
since if "no point of B is a point of A," it may be in- 
ferred that "no point of A is a point of B." The 
fourth species, the particular negative, is not neces- 
sarily conversely true. From "some points of A are 
not points of B" or A is not entirely contained within 
B, we can infer nothing as to whether B is or is not 
entirely contained in A. It is plain that the converse 
of a proposition is not necessarily true, if it says more 
either of the subject or predicate than was said before. 
Now "every equilateral triangle is equiangular," does 
not speak of all equiangular triangles, but asserts that 
among all possible equiangular triangles are to be 


found all the equilateral ones. There may then, for 
anything to the contrary to be discovered in our as- 
sertion, be classes of equiangular triangles not in- 
cluded under this assertion, of which we can therefore 
say nothing. But in saying "no right angles are un- 
equal," that which we exclude, we exclude from all 
unequal angles, and therefore "no unequal angles are 
right angles " is not more general than the first. 

The various assertions brought forward in a geo- 
metrical demonstration must be derived in one of the 
following ways : 

I. From definition. This is merely substituting, 
instead of a description, the name which it has been 
agreed to give to whatever bears that description. No 
definition ought to be introduced until it is certain 
that the thing defined is really possible. Thus though 
parallel lines are defined to be "lines which are in 
the same plane, and which being ever so far produced 
never meet," the mere agreement to call such lines, 
should they exist, by the name of parallels, is no suffi- 
cient ground to assume that they do exist. The defi- 
nition is therefore inadmissible until it is really shown 
that there are such things as lines which being in the 
same plane never meet. Again, before applying the 
name, care must be taken that all the circumstances 
connected with the definition have been attended to. 
Thus, though in plane geometry, where all lines are 
in one plane, it is sufficient that two lines would never 
meet though ever so far produced, to call them par- 


allel, yet in solid geometry the first circumstance must 
be attended to, and it must be shown that lines are in 
the same plane before the name can be applied. Some 
of the axioms come so near to definitions in their na- 
ture, that their place may be considered as doubtful. 
Such are, " the whole is greater than its part," and 
"magnitudes which entirely coincide are equal to one 

II. From hypothesis. In the statement of every 
proposition, certain connexions are supposed to exist 
from which it is asserted that certain consequences 
will follow. Thus "in an isosceles triangle the angles 
at the base are equal," or, " if a triangle be isosceles 
the angles at the base will be equal." Here the hy- 
pothesis or supposition is that the triangle has two 
equal sides, the consequence asserted is that the an- 
gles at the base or third side will be equal. The con- 
sequence being only asserted to be true when the an- 
gle is isosceles, such a triangle is supposed to be taken 
as the basis of the reasonings, and the condition that 
its two sides are equal, when introduced in the proof, 
is said to be introduced by hypothesis. 

In order to establish the result it may be necessary 
to draw other lines, etc., which are not mentioned in 
the first hypothesis. These, when introduced, form 
what is called the construction. 

There is another species of hypothesis much in 
use, principally when it is required to deduce the con- 
verse of a theorem from the theorem itself. Instead 


of proving the consequence directly, the contradictory 
of the consequence is assumed to hold good, and if 
from this new hypothesis, supposed to exist together 
with the old one, any evidently absurd result can be 
derived, such as that the whole is greater than its 
part, this shows that the two hypotheses are not con- 
sistent, and that if the first be true, the second cannot 
be so. But if the second be not true, its contradic- 
tory is true, which is what was required to be proved. 
III. From the evidence of the assertions themselves. 
The propositions thus introduced without proof are 
only such as are in their nature too simple to ad- 
mit of it. They are called axioms. But it is neces- 
sary to observe, that the claim of an assertion to be 
called an axiom does not depend only on its being 
self-evident. Were this the case many propositions 
which are always proved might be assumed ; for ex- 
ample, that two sides of a triangle are greater than 
the third, or that a straight line is the shortest dis- 
tance between two points. In addition to being self- 
evident, it must be incapable of proof by any other 
means, and it is one of the objects of geometry to re- 
duce the demonstrations to the least possible number 
of axioms. There are only two axioms which are dis- 
tinctly geometrical in their nature, viz., "two straight 
lines cannot enclose a space, " and " through each 
point outside a line, not more than one parallel to 
that line can be drawn." All the rest of the proposi- 
tions commonly given as axioms are either arithmet- 


ical in their nature; such as "the whole is greater 
than its part," "the doubles of equals are equals," 
etc.; or mere definitions, such as "magnitudes which 
entirely coincide are equal "j or theorems admitting 
of proof, such as "all right angles are equal." There 
is however one more species of self-evident proposi- 
tion, the postulate or self-evident problem, such as 
the possibility of drawing a right line, etc. 

IV. From proof already given. What has been 
proved once may be always taken for granted after- 
wards. It is evident that this is merely for the sake 
of brevity, since it would be possible to begin from 
the axioms and proceed direct to the proof of any one 
proposition, however far removed from them ; and 
this is an exercise which we recommend to the stu- 
dent. Thus much for the legitimate use of any single 
assertion or proposition. We proceed to the manner 
of deducing a third proposition from two others. 

It is evident that no assertion can be the direct 
and necessary consequence of two others, unless those 
two contain something in common, or which is spoken 
of in both. In many, nay most, cases of ordinary con- 
versation and writing, we leave out one of the asser- 
tions, which is, usually speaking, very evident, and 
make the other assertion followed by the consequence 
of both. Thus, "Geometry is useful, and therefore 
ought to be studied," contains not only what is ex- 
pressed, but also the following, "That which is useful 
ought to be studied ; " for were this not admitted, the 


former assertion would not be necessarily true. This 
may be written thus : 

Every thing useful is what ought to be stud- 
Geometry is useful, therefore geometry is 

what ought to be studied. 

This, in its present state, is called a syllogism, and 
may be compared with the following, from which it 
only differs in the things spoken of, and not in the 
manner in which they are spoken of. 

Every point of the circle is a point of the tri- 

The point B is a point of the circle. 
Therefore the point B is a point of the tri- 

Here a connexion is established between the point 
B and the points of the triangle (viz., that the first is 
one of the second) by comparing them with the points 
of the circle ; that which is asserted of every point of 
the circle in the first can be asserted of the point B, 
because from the second B is one of these points. 
Again, in the former argument, whatever is asserted 
of every thing useful is true of geometry, because ge- 
ometry is useful. 

The common term of the two propositions is called 
the middle term, while the predicate and subject of the 
conclusion are called the major and minor terms, re- 
spectively. The two first assertions are called the 
major and minor premisses, and the last the conclusion. 


Suppose now the two premisses and conclusion of the 
syllogism just quoted to be varied in every possible 
way from affirmative to negative, from universal to 
particular, and vice versa, where the number of changes 
will be 4x^X4, or 64 (called moods); since each 
proposition may receive four different forms, and each 
form of one may be compounded with any of the other 
two. And these may be still further varied, if instead 
of the middle term being the subject of the first, and 
the predicate of the second, this order be reversed, or 
if the middle term be the subject of both, or the pred- 
icate of both, which will give four different figures, as 
they are called, to each of the sixty-four moods above 
mentioned. But of these very few are correct deduc- 
tions, and without entering into every case we will 
state some general rules, being the methods which 
common reason would take to ascertain the truth or 
falsehood of any one of them, collected and general- 

I. The middle term must be the same in both 

*Whately's Logic, page 76, third edition. A work which should be read 
by all mathematical students. [Whately's Logic is procurable in modern edi- 
tions, many of which were, until recently, widely read in our academies and 
colleges. The following works in which the same material is presented in a 
shape more comforming to modern methods may be mentioned : T. Fowler's 
Elements of Deductive Logic ; Bain's Logic; Venn's Empirical Logic and Sym- 
bolical Logic; Keynes's Formal Logic; Carveth Read's Logic, Deductive and 
Inductive; Mill's System of Logic (a. discussion rather than a presentation). 
Strictly contemporary logic will be found represented in the following works 
in English: Jevons's Principles of Science and Studies in Deductive Logic; 
Bradley's Principles of Logic ; Sidgwick's Process of Argument; Bosanquet's 
Logic: or, the Morphology of Knowledge; and the same author's Essentials of 
Logic: Sigwart's Logic, recently translated from the German; and Ueber- 
weg's System of Logic and History of Logical Doctrines. Ed.] 


premisses, by what has just been observed ; since in 
the comparison of two things with one and the same 
third thing, in order to ascertain their connexion or 
discrepancy, consists the whole of reasoning. Thus, 
the deduction without further process of the equation 
a 2 -}-^ = f 2 from the proposition, which proves that 
the sum of the squares described on the sides of a 
right-angled triangle is equal to the square on its hy- 
pothenuse, a, b, and c being the number of linear units 
in the sides and the hypothenuse, is incorrect, since 
syllogistically stated the argument would stand thus : 
The sum of the squares of the 

lines a and 6 are j tities> 


the square of the line c 

# 2 + P } [ the sum of the squares of a and , 
and \ are \ and 

f 2 J I the square of c. 

a 2 + #M 

Therefore and \ are equal quantities. 
S J 

Here the term square in the major premiss has its 
geometrical, and in the minor its algebraical sense, 
being in the first a geometrical figure, and in the sec- 
ond an arithmetical operation. The term of compari- 
son is not therefore the same in both, and the conclu- 
sion does not therefore follow from the premisses. 

The same error is committed if all that can be con- 
tained under the middle term be not spoken of either 
in the major or minor premiss. For if each premiss 


mentions only a part of the middle term, these parts 
may be different, and the term of comparison really 
different in the two, though passing under the same 
name in both. Thus, 

All the triangle is in the circle, 

All the square is in the circle, 

proves nothing, since the square may, consistently 
with these conditions, be either wholly, partly, or not 
at all contained in the triangle. In fact, as we have 
before shown, each of these assertions speaks of a part 
of the circle only. The following is of the same kind : 

Some of the triangle is in the circle. 

Some of the circle is not in the square, etc. 

II. If both premisses are negative, no conclusion 
can be drawn. For it can evidently be no proof either 
of agreement or disagreement that two things both 
disagree with a third. Thus the following is incon- 
clusive : 

None of the circle is in the triangle. 
None of the square is in the circle. 

III. If both premisses are particular, no conclusion 
can be drawn, as will appear from every instance that 
can be taken, thus : 

Some of the circle is in the triangle. 
Some of the square is not in the circle, 
proves nothing. 

IV. In forming a conclusion, where a conclusion 
can be formed, nothing must be asserted more gener- 


ally in the conclusion than in the premisses. Thus, if 
from the following, 

All the triangle is in the circle, 

All the circle is in the square, 

we would draw a conclusion in which the square 
should be the subject, since the whole square is not 
mentioned in the minor premiss, but only part of it, 
the conclusion must be, 

Part of the square is in the triangle. 

V. If either of the premisses be negative, the con- 
clusion must be negative. For as both premisses can- 
not be negative, there is asserted in one premiss an 
agreement between the term of the conclusion and 
the middle term, and in the other premiss a disagree- 
ment between the other term of the conclusion, and 
the same middle term. From these nothing can be 
inferred but a disagreement or negative conclusion. 
Thus, from 

None of the circle is in the triangle, 
All the circle is in the square, 
can only be inferred, 

Some of the square is not in the triangle. 

VI. If either premiss be particular, the conclusion 
must be particular. For example, from 

None of the circle is in the triangle, 
Some of the circle is in the square, 
we deduce, 

Some of the square is not in the triangle. 
If the student now applies these rules, he will find 


that of the sixty-four moods eleven only are admis- 
sible in any case ; and in applying these eleven moods 
to the different figures he will also find that some of 
them are not admissible in every figure, and some not 
necessary, on account of the conclusion, though true, 
not being as general as from the premisses it might be. 
This he may do either by reasoning or by actual in- 
spection of the figures, drawn and arranged according 
to the premisses. The admissible moods are nineteen 
in number, and are as follows, where A at the begin- 
ning of a proposition signifies that it is a universal 
affirmative, E a universal negative, / a particular 
affirmative, O a particular negative. 

Figure I. The middle term is the subject of the 
major, and the predicate of the minor premiss. 

1.* A All the O is in the A 

A All the n is in the O 

. . A All the n is in the A 

2. E None of the O is in the A 
A All the n is in the Q 

. . E None of the n is in the A 

3. A All the O is in the A 
/ Some of the n is in the O 

. *. / Some of the n is in the A 

4. E None of the O is in the A 
/ Some of the n is in the O 

.. O Some of the n is not in A 

*This, and 3, are the most simple of all the combinations, and the most 
frequently used, especially in geometry. 


Figure II. The middle term is the predicate of 
both premisses. 

1. E None of the A is in the O 
A All the n is in the Q 

. . E None of the n is in the A 

2. A All the A is in the Q 
E None of the n is in the O 

. . E None of the n is in the A 

3. E None of the A is in the Q 
/ Some of the n is in the O 

. . O Some of the n is not in A 

4. A All the A is in the O 

Some of the n is not in O 
. . O Some of the D is not in A 

Figure III. The middle term is the subject of both 

1. A All the O is in the A 
A All the O is in the n 

. . 7 Some of the D is in the A 

2. / Some of the O is in the A 
A All the O is in the n 

. . / Some of the n is in the A 

3. A All the O is in the A 

1 Some of the O is in the n 
. . / Some of the n is in the A 

4. E None of the O is in the A 
A All the O is in the n 

. . O Some of the n is not in A 

5. O Some of the O is not in A 


A All the O is in the n 

. . O Some of the n is not in A 
6. E None of the O is in the A 
/ Some of the O is in the n 
. . O Some of the D is not in A 
Figure IV. The middle term is the predicate of 
the major, and the subject of the minor premiss. 

1. A All the A is in the Q 
A All the O is in the n 

. . / Some of the n is in the A 

2. A All the A is in the O 
E None of the O is in the n 

. . E None of the n is in the A 

3. / Some of the A is in the O 
A All the O is in the n 

. . / Some of the n is in the A 

4. E None of the A is in the O 
A All the O is in the n 

. . O Some of the n is not in A 

5. E None of the A is in the O 
/ Some of the O is in the n 

. . O Some of the n is not in A 
We may observe that it is sometimes possible to 
condense two or more syllogisms into one argument, 
thus : Every A is B (1), 

Every B is C (2), 
Every C is D (3), 
Every D is E (4), 
Therefore Every A is E (5), 


is equivalent to three distinct syllogisms of the form 
Fig. l.j these syllogisms at length being (1), (2), a\ 
a, (3), b; b, (4), (5). 

The student, when he has well considered each of 
these, and satisfied himself, first by the rules, and 
afterwards by inspection, that each of them is legiti- 
mate ; and also that all other moods, not contained 
in the above, are not allowable, or at least do not give 
the most general conclusion, should form for himself 
examples of each case, for instance of Fig. Ill, 3 : 

The axioms constitute part of the basis of 


Some of the axioms are grounded on the evi- 
dence of the senses. 
. . Some evidence derived from the senses is 

part of the basis of geometry. 

He should also exercise himself in the first princi- 
ples of reasoning by reducing arguments as found in 
books to the syllogistic form. Any controversial or 
argumentative work will furnish him with a sufficient 
number of instances. 

Inductive reasoning is that in which a universal 
proposition is proved by proving separately every one 
of its particular cases. As where, for example, a 
figure, A BCD, is proved to be a rectangle by proving 
each of its angles separately to be a right angle, or 
proving all the premisses of the following, from which 
the conclusion follows necessarily : 


The angles at A, B, C, and D are all the an- 
gles of the figure A BCD. 
A is a right angle, 
B is a right angle, 
C is a right angle, 
Z? is a right angle, 
Therefore all the angles of the figure A BCD 

are right angles. 

This may be considered as one syllogism of which 
the minor premiss is, 

A, B, C, and D are right angles, 
where each part is to be separately proved. 

Reasoning a fortiori, is that contained in Fig. I. 1. 
in a different form, thus : A is greater than B, B is 
greater than C ; a fortiori A is greater than C ; which 
may be also stated as follows : 

The whole of B is contained in A, 
The whole of C is contained in B y 
Therefore C is contained in A. 
The premisses of the second do not necessarily im- 
ply as much as those of the first ; the complete reduc- 
tion we leave to the student. 

The elements of geometry present a collection of 
such reasonings as we have just described, though in 
a more condensed form. It is true that, for the con- 
venience of the learner, it is broken up into distinct 
propositions, as a journey is divided into stages ; but 
nevertheless, from the very commencement, there is 
nothing which is not of the nature just described. We 



present the following as a specimen of a geometrical 
proposition reduced nearly to a syllogistic form. To 
avoid multiplying petty syllogisms, we have omitted 
some few which the student can easily supply. 

Hypothesis. ABC is a right-angled triangle the 
right angle being at A. 

Consequence. The squares on AB and -^Care to- 
gether equal to the square on BC. 


and BA describe squares, 
produce DB to meet EF t 
produced, if necessary, in 
G, and through A draw 
HAK parallel to BD. 


I. Conterminous sides 
of a square are at right 
angles to one another. 

EB and BA are conter- 
minous sides of a square. 

. . EB and BA are at right angles. 

II. A similar syllogism to prove that DB and BC 
are at right angles, and another to prove that GB and 
BC are at right angles. 

III. Two right lines drawn perpendicular to two 
other right lines make the same angle as those others 


(already proved) ; EB and BG and AB and BC are 
two right lines, etc., (I. II.). 

. . The angle EBG is equal to ABC. 

IV. All sides of a square are equal. (Definition.) 
AB and BE are sides of a square. (Construction.) 
. . AB and BE are equal. 

V. All right angles are equal. (Already proved.) 
BEG and BAG are right angles. (Hypothesis and 


. . BEG and BAC are equal angles. 

VI. Two triangles having two angles of one equal 
to two angles of the other, and the interjacent sides 
equal, are equal in all respects. (Proved.) 

BEG and BAC are two triangles having BEG and 
EBG respectively equal to BAC and ABC and the 
sides EB and BA equal. (III. IV. V.) 

.-. The triangles BEG, BAC are equal in all re- 

VII. BG is equal to BC. (VI.) 

BC is equal to BD. (Proved as IV.) 
.-. BGis equal to BD. 

VIII. A four-sided figure whose opposite sides 
are parallel is a parallelogram. (Definition.) BGHA 
and BPKD are four-sided figures, etc, (Construc- 

. . BGHA and BPKD are parallelograms. 

IX. Parallelograms upon the same base and be- 
tween the same parallels are equal. (Proved.) EBAF 
and BGHA, are parallelograms, etc. (Construction.) 


. . EBAFand. BGHA are equal. 

X. Parallelograms on equal bases and between the 
same parallels, are equal. (Proved.) 

BGHA and BDKP are parallelograms, etc. (Con- 

. . BGHA and BDKP are equal. 

XI. EBAF'is equal to BGHA. (IX.) 
BGHA is equal to BDKP. (X.) 

. . EBAF (that is the square on AB) is equal to 

XII. A similar argument from the commencement 
to prove that the square on A C is equal to the rectan- 
gle CPK. 

XIII. The rectangles BK and CK are together 
equal to the square on BC. (Self-evident from the 

The squares on BA and A C are together equal to 
the rectangles BK and CK. (Self-evident from XI 
and XII.) 

. . The squares on BA and A C are together equal 
to the square on BC. 

Such is an outline of the process, every step of 
which the student must pass through before he has 
understood the demonstration. Many of these steps 
are not contained in the book, because the most ordi- 
nary intelligence is sufficient to suggest them, but the 
least is as necessary to the process as the greatest. 
Instead of writing the propositions at this length, the 

22 4 


student is recommended to adopt the plan which we 
now lay before him. 

Hyp. 1 ABC is a triangle, right-an- 

gled at A. 

Constr. 2 a On BA describe a square 


3 a On BC describe a square. 

4 Produce BD to meet EF, pro- 

duced if necessary, in G. 

5 b Through A draw HAK par- 

allel to BD. 
Demonst. 6 2, Def. EBA is a right angle. 

7 3 .#C is a right angle. 

8 6, 7, c LEBG is equal to /_ABC. 

9 2, 1, </ /_BEG is equal to /_BAC. 

10 2 jET? is equal to AB. 

11 8, 9, 10, e The triangles .# and ABC 

are equal. 

12 11, 3 BG is equal to BD. 

13 5, 2, Def. AHGB is a parallelogram. 

14 5, 3, Def. BPDK\s a parallelogram. 

15 13, 2, y AHGB and ABEF are equal. 

16 13,14,^ ^ZT# and ^Z^ST are equal. 

17 15, 16 BPDK and the square on 

are equal. 

f By 
18{ similar 

and the S 1 uare on 


19 17, 18 The square on BC is equal to 
the squares on BA and A C. 
a, b Here refer to the necessary problems. 
c If two lines be drawn at right angles to 
two others, the angles made by the 
first and second pair are equal. 
d All right angles are equal. 
e Two triangles which have two angles of 
one equal to two angles of the other, 
and the interjacent sides equal, are 
equal in all respects. 

/, g Parallelograms on the same or equal 
bases, and between the same paral- 
lels, are equal. 

The explanation of this is as follows : the whole 
proposition is divided into distinct assertions, which 
are placed in separate consecutive paragraphs, which 
paragraphs are numbered in the first column on the 
left ; in the second column on the left we state the 
reasons for each paragraph, either by referring to the 
preceding paragraphs from which they follow, or the 
preceding propositions in which they have been 
proved. In the latter case a letter is placed in the 
column, and at the end, the enunciation of the propo- 
sition there used is written opposite to the letter. By 
this method, the proposition is much shortened, its 
more prominent parts are brought immediately under 
notice, and the beginner, if he recollect the preceding 
propositions perfectly well, is not troubled by the 


repetition of prolix enunciations, while in the contrary 
case he has them at hand for reference. 

In all that has been said, we have taken instances 
only of direct reasoning, that is, where the required 
result is immediately obtained without any reference 
to what might have happened if the result to be proved 
had not been true. But there are many propositions 
in which the only possible result is one of two things 
which cannot be true at the same time, and it is more 
easy to show that one is not the truth, than that the 
other is. This is called indirect reasoning ; not that 
it is less satisfactory than the first species, but be- 
cause, as its name imports, the method does not ap- 
pear so direct and natural. There are two proposi- 
tions of which it is required to show that whenever 
the first is true the second is true ; that is, the first 
being the hypothesis the second is a necessary conclu- 
sion from it, whence the hypothesis in question, and 
anything contradictory to, or inconsistent with, the 
conclusion cannot exist together. In indirect reason- 
ing, we suppose that, the original hypothesis existing 
and being true, something inconsistent with or con- 
tradictory to the conclusion is true also. If from com- 
bining the consequences of these two suppositions, 
something evidently erroneous or absurd is deduced, 
it is plain that there is something wrong in the as- 
sumptions. Now care is taken that the only doubtful 
point shall be the one just alluded to, namely, the 
supposition that one proposition and the contradictory 


of the other are true together. This then is incorrect, 
that is, the first proposition cannot exist with anything 
contradictory to the second, or the second must exist 
wherever the first exists, since if any proposition be 
not true its contradictory must be true, and vice versa. 
This is rather embarrassing to the beginner, who finds 
that he is required to admit, for argument's sake, a 
proposition which the argument itself goes to destroy. 
But the difficulty would be materially lessened, if in- 
stead of assuming the contradictory of the second 
proposition positively, it were hypothetically stated, 
and the consequences of it asserted with the verb 
"would be," instead of "is." For example : suppose 
it to be known that if A is , then C must be D, and 
it is required to show indirectly that when C is not D, 
A is not B. This put into the form in which such a 
proposition would appear in most elementary works, 
is as follows. 

It being granted that if A is B, C is D, it is 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 y which is absurd ; that is, it is absurd to suppose 
that C is not D and A is B, consequently when C is 
not D, A is not B. The following, which is exactly 
the same thing, is plainer in its language. Let C be 
not D. Then if A were B, C would be D by the prop- 
osition granted. But by hypothesis C is not D, etc. 


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 


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 


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. 



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 


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 Mlthodes 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 Euklidbis auf Gauss 
(Leipsic, 1895). Of interest are the essays of Prof. J. Delboeuf on The Old 
and the New 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 gin f 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' s Ausdehnungslehre (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 
Geometry," and Hyde's Directional Calculus, Ginn & Co., Boston). Dr. Paul 
Cams 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. 



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 BAC and BAD. 


Fig. 6. 

III. Whatever may be the length of BC and BD, 
the angles BA C and BAD are each less than a right 


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 


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 



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 

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 a O. Take the angles tOv, vOw y 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 


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.} 



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. 


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 



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. 8. 


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- 


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 T 2 T and ^ 
of a foot, are better compared by observing that ^ 
and f 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 B, which contain 
the same line C five and six times respectively, are 
said to have to one another the ratio of five to six, or 
to be in the proportion of five to six. If then we de- 


note the first by A,* and the second by B, and the 
common measure by C, we have 

A = 5C, or QA=3QC, 
B = 6C, or 5^ = 30C, 
whence A = B, or 6A 5 = Q. 
Generally, when mA nJ? = Q, the lines, or what- 
ever they are, represented by A and B, are said to be 
in the proportion of n to m, or to have the ratio of n 
to m. 

Let there be two other magnitudes P and Q, of 
the same kind with one another, either differing from 
the first in kind or not, (thus A and B may be lines, 
and P and Q surfaces, etc.,) and let them contain a 
common measure R, just as A and B contain C, viz. : 
Let P contain R five times, and let Q contain R six 
times, we have by the same reasoning 
6^5(2 = 0, 

and P and Q, being also in the ratio of five to six, as 
well as A and B, are said to be proportional to A and 
B, which is denoted thus 

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, mA nB, 

etc., are the only ones admissible. AB, , A 2 , etc., are unmeaning, while 

> m 

is the line which is contained m times in A, or the ;th part of A. The capital 
letters throughout stand for concrete quantities, not for their representations 
in abstract numbers, 

2 4 6 


some two whole numbers m and n such that, at the 

same time 

mA n = Q, 

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 

Fig 9. 

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 E, m X m or m* 
times. Similarly the square described on BC contains 
E n X n or n 2 times. But, because AB is an isosce- 


les right-angled triangle, the square on AB is double 
that on BC, whence m X m = 2 (n X ) or m 2 = 2n 2 . To 
prove the impossibility of this equation (when m and 
n are whole numbers), observe that m 2 must be an 
even number, since it is twice the number n 2 . But 
my^m cannot be an even number unless m is an even 
number, since an odd number multiplied by itself 
produces an odd number.* Let m (which has been 
shown to be even) be double m' or m = 2m'. Then 
2m' X 2m' = 2n 2 or 4m' 2 = 2n 2 or n 2 = 2m' 2 . By repeat- 
ing the same reasoning we show that n is even. Let 
it be 2n'. Then 2n' X 2' = 2m' 2 or m' 2 = 2n' 2 . By the 
same reasoning m' and n' are both even, and so on ad 
infinitum. This reasoning shows that the whole num- 
bers which satisfy the equation n 2 = 2m 2 (if such there 
be) are divisible by 2 without remainder, ad infinitum. 
The absurdity of such a supposition is manifest: there 
are then no such whole numbers, and consequently no 
common measure to BA and BC. 

Before proceeding any further, it will be necessary 
to establish the following proposition. 

If the greater of two lines A and B be divided into 
m equal parts, and one of these parts be taken away ; 
if the remainder be then divided into m equal parts, 
and one of them be taken away, and so on, the re- 

* Every odd number, when divided by 2, gives a remainder r, and is there- 
fore of the form 2/ + i where/ is a whole number. Multiply <zp + i by itself, 
which gives 4/ 2 + 4/ + i, or 2 (zp2 + zp}-\-i, which is an odd number, since, 
when divided by 2, it gives the quotient 2/2 -(- 2/, a whole number, and the 
remainder i. 


mainder of the line A shall in time become less than 
the line B, how small soever the line B may be. 

Take a line which is less than B, and call it C. It 
is evident that, by a continual addition of the same 
quantity to C, this last will come in time to exceed A\ 
and still moire will it do so if the quantity added to C 
be increased at each step. To simplify the proof we 
suppose that 20 is the number of equal parts into 
which A and its remainders are successively divided, 
so that 19 out of the 20 parts remain after subtraction. 

Divide C into 19 equal parts and add to C a. line 
equal to one of these parts. Let the length of C, so 
increased, be C'. Divide C' into 19 equal parts and 
let C', increased by its 19th part, be C". Now, since 
we add more and more each time to C, in forming C', 
C", etc, we shall in time exceed A. Let this have 
been done, and let D be the line so obtained, which 
is greater than A. Observe now that C' contains 19, 
and C", 20 of the same parts, whence C' is made by 
dividing C" into 20 parts and removing one of them. 
The same of all the rest. Therefore we may return 
from D to C by dividing D into 20 parts, removing 
one of them, and repeating the process continually. 
But C is less than B by hypothesis. If then we can, 
by this process, reduce D below B, still more can we 
do so with A, which is less than D, by the same 

This depends on the obvious truth, that if, at the 
end of any number of subtractions (Z> being taken), 


we have left Z>, at the end of the same number of 

9 P 

subtractions (A being taken), we shall have A, since 

the method pursued in both cases is the same. But 

since A is less than D, A is less than ->, 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 E, 2E, 3E, etc., we shall arrive at last at 
two consecutive terms, pE and (/+ 1)^ 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 (/-f- 1)2? are commen- 

* Algebraically, let a be the given line, and let th part of the remainder 

191 a 

be removed at every subtraction. The first quantity taken away is and the 
ft ( i \ ^* 

remainder a or all ), whence the second quantity removed is 

at \\ m ^ m/ t a\ f i \ / i \ 2 

1 i ) , and the remainder [a II (i I or a I i I . 

m ^ m' V / A m) \ m> 

Similarly, the nth remainder is a (i -) . Now, since i is less 

^ in ' 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. 


surable with A, that is with mE, 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 #2-^ nB = Q', nevertheless, 
we can prove that values of m 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 m and n,* we find 
mA nB = E, and let the first multiple of E, which 
is greater than B, be^E, so that pE = B + E' where 
E is less than E, for were it greater, (/ !)-> or 
pE E, which is B + (E' E), would be greater 
than B, which is against the supposition. 

The equation mA nB = E gives 

/ mA p nB pE = B + E', 

*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 5< 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 

mpAlnp + i) B 
zmA ^mB, etc., etc. 



pm=m' and p n -\- 1 = ', 


m'A riB = E'. 

We have therefore found a difference of multiples 
which is less than E. Let 'p'E' be the first multiple 
of E' which is greater than B, where p' must be at 
least as great as /, since E being greater than E', it 
cannot take more* of E than of E' to exceed B. Let 

then, as before, 

m'p'A (rip' + 1) B = .", 


w"^ n" = "-, 

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. 


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. 


D C" C' C 

Fig. 10. 

To compare the quantities E, E', etc., we have 

the equations 

pE = B + E' 

/' = B + 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/, p' , etc., can only remain the same 
for a finite number of steps, and must then increase, 
and after the increase can only remain the same for 
another finite number of steps, and then must increase 
again, and so on, we show that the process can be 
continued, until one of them is as great as we please ; 


let this be/ ( * } , 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 pE^ is the first multiple of E (z \ which 
exceeds the given quantity B, if p w can be as great as 
we please, E (z) can be as small as we please. To show 
that/^ 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 =/' =/', etc. The equations be- 

etc. etc. 
Then by subtraction, 

E' E" =p(E E'} 

E" E'" =p(E' E'")=pp(EE') 

E"'E""=p (E" E'"} =ppp (E E"} 

etc. etc. 


E-E" =E-E'+E'-E" =(^-^')(l+/) 

E-E" r =E-E'+E'-E"+E"-E'"=(E-E'}(\+p+p^ 

etc. etc. etc. 

E _ E ( ^ = E _ E' + E' E" -\- 


which is derived from w steps of the process. Now, 
if this can go on ad infinitum, it can go on until 1 + 

/+/ 2 + +/ 7 "" 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 -\-p +/ 2 -f- . . . -\-p' w ~ l becomes greater than the num- 
ber of times which E E' is contained in E ; for, 
from the above equation, E E 1 is contained in 
EE^\ 1 + /+/2 _j_ _ e _|_ pv- 1 t i mes . an d i t ; s 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 E' = 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 + 2 2 + 23+, etc., greater than 54; 
that is, it cannot be the same for six steps. And we 
find, on actually performing the operations, 

2x54^55'+ 53' 

2x53'= 55'+ 51' 

2x51'= 55'+ 47' 

2x47'= 55'+ 39' 

2x39'= 55'+ 23' 

3x23'= 55'+ 14' 
We do not say that /, /', etc., will remain the 


same until 1+/H-/ 2 -j- . . . 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 Cby 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 (/ -f- !)', 
neither of which can differ from C by so much as E' t 
and therefore not by so much as E. Then since 

therefore pmA pnB=pE', 

and (p+l)mA (p + \}n = (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 


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 , 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 

Resume the equations 

nB = Q, or A=-B 

nQ = b, or P= - Q 


If we take any other expression of the same sort 
; ,B and ,Q, it is plain that, according as the arith- 
metical fraction is greater than, equal to, or less 



than ; so will B be greater than, equal to. or less 
m m 

than -,B, and the same of Q and ,Q. Let the 
m mm 


be the abbreviation of the following sentence : "when 
x is greater than y, z is greater than w ; when x is 
equal to y, z is equal to w ; when x is less than y, z is 
less than /. " The following conclusions will be evi- 
dent : 

* i > :=< (^ an ') >=< 1/ 


'l^ ^\ d 

M > < 1/ W 

And from the first of these alone it follows that 




We have just noticed the following : 


n' n 



> = < 

^ and n 

> = < 


m . 

~ri B ~m@ 

. m' 

Therefore (1) 



n ' 


>> = < 

* or A p }> = <- 


Therefore (2) 

m A 



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 
mA n = Q 

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 m and , they are either both 
positive or both negative, for every other value of m and n. 


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 which B is of the same kind as A, and Q 
as P, are said to be proportional, if magnitudes B-\- C 
and Q-\- R can be found as near as we please to B and 
Q, so that A, B + C, P and Q + R, are proportional 
according to the common notion, that is, if whole 
numbers m and n can satisfy the equations 

mA n(B-\- C)=0 

mPn( Q + R)=. 

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- 


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 

This being true, it will follow that we can take m and 
n, so as not only to make mA nB less than a given 
magnitude E, which may be as small as we please, 
but also so that mP nQ shall at the same time be 
less than a given magnitude F, however small this 
last may be. For if not, while m and n are so taken 
as to make mA nB less than E (which it has been 
proved can be done, however small E may be) sup- 
pose, if possible, that the same values of m and n will 
never make mP nQ less than some certain quantity 
F, 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 pFis greater 
than Q, and mP nQ is by hypothesis greater than 
F t 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, 

mpPnpQ is greater than Q, 


mpA is less than (np-\- V)B, 

mpP is greater than (np + !)(?, 


which is contrary to our first hypothesis respecting 
A t By Py said Qy that hypothesis being Euclid's defi- 
nition of proportion, from which if 

mpA is less than (np -f- V)B 

mpP is less than (np + \)Q. 

We must therefore conclude that if the four quantities 
Ay By Py &&& Q satisfy Euclid's definition of propor- 
tion, then m and n may be so taken that mA nB and 
mP nQ shall be as small as we please. 


mA nB = E and = nC 

mP nQ = Fa.nd F=nR. 
Then mA n(B -f C) = 

and since E and F can, by properly assuming m and 
Hy be made as small as we please, much more can the 
same be done with C and R, consequently we can pro- 
duce B -f- C and Q -f- R as near as we please to B and 
Qy and proportional to A and P t according to the 
common arithmetical notion. In the same way it may 
be proved, that on the same hypothesis B C and 
Q R can be found as near to B and Q as we please, 
and so that A, B C, P and Q R are proportional 
according to the ordinary notion. It only remains to 
show that if the last-mentioned property be assumed, 
Euclid's definition of proportion will follow from it. 
That is, if quantities can be exhibited as near to P 
and Q as we please, which are proportional to A and 
By according to the ordinary notion, it follows that 


For let B-\- C and Q-\- be two quantities, such that 

in which, by the hypothesis, /and g can be so taken 
that C and ^ are as small as we please. We have al- 
ready shown that in this case (m and n being any 
numbers whatever) mA is never greater or less than 
n(B -f- C), without mP being at the same time the 
same with regard to n(Q-\- 7?). That is, if 

mA is greater than nB-\-nC, 

mP is greater than nQ-\- nR. 

Take some given* values for m and n, fulfilling the 
first condition ; then, since C and R may be as small 
as we please, the same is true of nC and nR; if then 

mA is greater than nB 

mP is greater than nQ. 

For if not, let mA=nJ3-{-x, while mP=nQ y, x 
and y being some definite magnitudes. Then if 

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/" 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 


proved that if mA<^nB, mP<nQ, etc., so that Eu- 
clid's definition is shown to be a necessary consequence 
of the one proposed. 

The definition of proportion which we have here 
given, and the methods by which we have established 
its identity with the one in use, bear a close analogy 
to the process used by the ancients, and denominated 
by the moderns the method of exhaustions. We have 
seen that the common definition of proportion fails in 
certain cases where the magnitudes are what we have 
called incommensurable, but at the same time we 
have shown that though in this case we can never 
take m and n, so that mA = nB, or mA nB = 0, we 
can nevertheless find m and n, so that mA shall differ 
from nB by a quantity less than any which we please 
to assign. We therefore extend the definition of the 
word proportion, and make it embrace not only those 
magnitudes which fulfil a given condition, but also 
others, of which it is impossible that they should fulfil 
that condition, provided always, that whatever magni- 
tudes we call by the name of proportionals, they must 
be such as to admit of other magnitudes being taken 
as near as we please to the first, which are propor- 
tional, according to the common arithmetical notion. 
It is on the same principle that in algebra we admit 
the existence of such a quantity as 1/2, and use it in 
the same manner as a definite fraction, although there 
is no such fraction in reality as, multiplied by itself, 
will give 2 as the product. But, however small a 


quantity we may name, we can assign a fraction which, 
multiplied by itself, shall differ less from 2 than that 

Having established the properties of rectilinear 
figures, as far as their proportions are concerned, it is 
necessary to ascertain the properties of curvilinear 
figures in this respect. And here occurs a difficulty 
of the same kind as that which met us at the outset, 
for no rectilinear figure, how small soever its sides 
may be, or how great soever their number, can be 
called curvilinear. Nevertheless, it may be shown 
that in every curve a rectilinear figure may be in- 
scribed, whose area and perimeter shall differ from 
the area and perimeter of the curve by magnitudes 
less than any assigned magnitudes. The circle is the 
only curve whose properties are considered in elemen- 
tary geometry, and the proposition in question is dis- 
cussed in all standard treatises on geometry. Indeed, 
for this or any other curve the proposition is almost 
self-evident. This being granted, the properties of 
curvilinear figures are established by help of the fol- 
lowing theorem. 

If A y B, C, and D are always proportional, and of 
these, if C and D may be made as near as we please to 
P and Q, than which they are always both greater or 
both less, then A, B, P, and Q are proportional. 

Let C=P+P', and >=Q-\-Q f , where by hy- 
pothesis P and Q' may be made as small as we please, 
and A, , P+P, and Q+ Q' are proportionals. If 


A, B, P, and Q are not proportionals, let P and 
be proportional to A and B. Then, since A and B 
are proportional to P-\-P' and Q-\- Q', and also to P 
and <2 + ^?, therefore 

JP+P' : Q+ Qr.P-.Q + R 

in which all the magnitudes are of the same kind. 
Now let P' and Q be so taken that Q' is less than ^?, 
which may be done, since by hypothesis Q' can be as 
small as we please. Hence Q-{- @ is less than^-j-^, 
and therefore P-\-P' is less than P, which is absurd. 
In the same way it may be proved that P is not to 
O R in the proportion of A to B, and consequently 
P is to Q in the proportion of A to B. This theorem, 
with those which prove that the surfaces, solidities, 
areas, and lengths, of curve lines and surfaces, may 
be represented as nearly as we please by the surfaces, 
etc., of rectilinear figures and solids, form the method 
of exhaustions.* In this method are the first germs of 
that theory which, under the name of Fluxions, or the 
Differential Calculus, contains the principles of all 
the methods of investigation now employed, whether 
in pure or mixed mathematics. 

*For a classical example, see Prop. II. of the twelfth book of Euclid 
(Simson's edition). Consult also Beman and Smith's Plane and Solid Geom- 
etry (Ginn & Co., Boston), pp. 144-145, and igo.Ed. 





WE have already defined a measure, and have no- 
ticed several instances of magnitudes of one 
kind being measured by those of another. But the 
most useful measure, and that with which we are most 
familiar, is number. We express one line by the num- 
ber of times which another line is repeated in it, or if 
the second is not exactly contained in the first, by the 
greatest number of the second contained in the first, 
together with the fraction of the second, which will 
complete the first. Thus, suppose the line A contains 
B m times, with a remainder which can be formed by 
dividing B into q parts, and taking p of them. Then 
B is to A in the proportion of 1 to m -\- , or as q to 
mq-\-p, and if B be a fixed line, which is used for the 
comparison of all lines whatsoever, then the line A is 
m -f , or , if it be understood that for every 

* * 'h 

unit in m, B is to be taken, and also that for the 


same fraction of B is to be taken that is of unity. 
In this case B is called the linear unit. 

But here we suppose that a line B being taken, 
the ratio of any other line A to B can be expressed by 
that of the whole numbers mq-\- p to q, which we have 
shown in some cases to be impossible. If we take 
one of these cases, mA nB, though it can never be 
made equal to nothing, can be made as small as we 
please, by properly assuming m and n. Let mA nB 

= , then A = B -\ , and since can be made as 

m tn m 

small as we please, A can be represented as nearly as 

we please by a fraction , where B is the linear unit. 

Hence, in practice an approximation may be found to 
the value of A, sufficient for any purpose whatever, 
in the following manner, which will be easily under- 
stood by the student who has a tolerable facility in 
performing the operations of algebra. Let 

A contain B, p times with a remainder P, 
B contain P, q times with a remainder Q, 
P contain Q, r times with a remainder R, 
and so on. If the two magnitudes are commensur- 
able, this operation will end by one of the remainders 
becoming nothing. For, let A and B have a common 
measure E, then P has the same measure, for P is 
A pB, of which both A and pB contain E an exact 
number of times. Again, because B and P contain 
the common measure E, Q has the same measure, 
and so on. All the remainders are therefore multiples 


of E, and if E be the linear unit, are represented by 
whole numbers. Now, if a whole number be contin- 
ually diminished by a whole number, it must, if the 
operation can be continued without end, eventually 
become nothing. If, therefore, the remainder never 
disappears, it is a sign that the magnitudes A and B 
are incommensurable. Nevertheless, approximate 
whole numbers can be found whose ratio is as near as 
we please to the ratio of A and B. 

From the suppositions above mentioned it appears 






etc., etc. 

Substitute in () the value of P derived from (0), find 
Q from the result, and substitute the values of P and 
Q in (t) ; find a value of R from the result, and sub- 
stitute the values of Q and R in O/), an d so on, which 
give the following series of equations : 

* Throughout these investigations the capital letters represent the lines 


On inspection it will be found that the coefficients 
of A and B in these equations may be formed by a 
very simple law. In each a letter is introduced which 
was not in the preceding one, and every coefficient is 
formed from the two preceding, by multiplying the 
one immediately preceding by the new letter, and ad- 
ding to the product the one which comes before that. 
Thus the third coefficient of B is pqr-\-p-\-r; the 
new letter is r, and the two preceding coefficients are 
pq+l and /, zn& pqr+p + r = (pq-\- V)r-\-p. The 
remainders enter also with signs alternately positive 
and negative. Let x, x', and x" be the th , (n -(- 1)*, 
and (-f 2)* numbers of the series /, q, r, etc., and 
X, X' y and X" the corresponding remainders. Let 
the corresponding equations be 

a A=b B + X 


Here n must be supposed odd, since, were it even, 
the first equation would be aA = bB X, as will be 
seen by reference to the equations deduced. Hence, 
from the law of formation of the coefficients, x" being 
the new letter in the last equation, 

a" = a'x" + a 

b" = b'x" -\-b. 

Eliminate x" from these two, the result of which 
is a" b' a'b" = ab' a' b, the first side of which is 

themselves, and not the numbers of units, which represent them, while the 
small letters are whole numbers, as in the last chapter. 


the numerator of , r ., and the second of , . 

a a a a 

It appears then that , is either greater than both 

b , b" , a , . b' b" . b' b 

- and ^ or less than both, since w and 

a a a a a a 

will both have the same sign, the numerators being 
the same and the denominators positive. It may also 

7ff J II 

be proved that -,- lies between and . by means of 
a a a 

the following lemma. 

The fraction must lie between and ; for 

P+9 P q 

let be the greater of the two last, or > , then 
/ P 9 

tno Tip on <j n 

or - L > , or > , and 1 4- > 1 -\ ; 

mp mp p m p m 

1 -h- 
therefore 1 _! ^ is less than unity, and any fraction 

multiplied by this is diminished. But 
H is ^ X T^-, 

and is therefore less than , the greater of the two. 

In the same way it may be proved to be greater than 

-, the least of the two. 

h" h f v" I A 

This being premised, since r, = . r , , it lies 

a ax -\-a 

b'x" ,b. b' . b 

between -. Tl and or between ; and . 
a x a a a 

Call the coefficients of A and B in the series of 

equations, a\ t a?, etc., b\ t b^ etc., and form the series 

of fractions , , , etc. The two first of these 
a\ a<2 a$ 

be ^ an d , of which the second is the 




greater, since it is / H -- . Hence by what has been 
proved is less than and greater than; and 

#3 #2 #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 -777 and -77, is the 
r b b 

same as that of , and T , 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 4~ and is 1; there- 

1 ? b' 

fore either ab' a'b, or a'b ab', is 1 according as - t 

7 a 

or is the greater of the two, that is according as n 
is odd or even.* Now since the th and (n-\- l) th equa- 
tions, n being odd, are 

and a'A=b'BX'; 
by eliminating A we have 

(a V a'b~) B = a'X+ aX! 
or =a'X+aX' 

since ab' a'b = 1 ; and since the remainders decrease 
and the coefficients increase, a'^> a and X*> X', 

. Tt 

whence 2 aX' < a'X -f aX\ or 2aX'< and X'<-=-; 


the remainder therefore which comes in the (n -{- 1)* 
equation is less than the part of B arising from divid- 
ing it into twice as many equal parts as there are 

*We might say that ab' a'b is alternately + i and i; but we wish to 
avoid the use of the isolated negative sign. 


units in the th coefficient of A ; and as this number 
of units may increase to any amount whatever, by 


carrying the process far enough, - may be made as 


small as we please, and a fortiori, the remainders may 
be made as small as we please. 

The same theorem may be proved in a similar 
way, if we begin at an even step of the process. Re- 
suming the equations 

a A=b B + X 

a' A=b' BX' 

we obtain from the second, 

T> Vt T> 

and since Jf < , 7 < ^ -- ; , or if B be taken as 
&a a A a a 


the linear unit, will express the line A with an error 

1 a 

less than -^ -,, which last may be made as small as 
a a 

we please by continuing the process. 

7 Jt 

It is also evident that is too small, while is 
a a 

too great ; and since X and X' are less than B, 
aA<bB-\-B,oic - is too great, while a'A>b'B B, 

or - , is too small. Again, A -- B= and 
b' a X' a a 

-j B A=r. Now X' <X and tf'>0; whence 
a a 

yf y jf 

-j < ; that is, , B exceeds A by a less quantity 

than B falls short of it, so that r is a nearer repre- 

a b 

sentation of A than , though on a different side of it. 


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, A OB, as the angular unit, and any 
other angle, AOC(\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 A OB 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. 


are proportional to the arcs AB and AC, or to the 
numbers / and s, we have 

Angle A OC is - of the angle A OB ; 

and the angle A OB being the angular unit, the num- 
ber is that which expresses the angle A OC. This 
number is the same for the same angle, whatever 
circle is chosen ; in the circle FD the proportion of 

Fig. ii. 

the arcs DE and DF is the same as that of AB and 
ACi for since similar arcs of different circles are pro- 
portional to their radii, 

AB : DE : : OA : OD 
Also AC:DF::OA:OD 
.-. ABi>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 A OB, 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 


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 / = 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 s and 12 rare the numbers of inches in 

the same lines, and by the common theory of frac- 

s 12 s 
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 
= -j-. , 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 r, one-half of the circumference is 


r X 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 -p , or 1.57079633. And the number of seconds 

in a right angle is 90 X 60 X 60, or 324000. Hence if 
3- 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 

Hence A : 324000 : : S : 1.57079633 : 

Of A -^ e-rrnrro^oo X ' 

1. 57079633 
or A =206265 X $, very nearly. 

Suppose, for example, the number of seconds in the 
theoretical unit itself is required. Here $ = 1 and 
^==206265; similarly if A be 1, 5= , 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. 


thus, in the formula, for what is called in trigonom- 
etry the sine of x, viz. : 

-, etc. 

If x be an angle of one second, it is not 1 which must 
be substituted for x, but 2Q6265' 

The number 3.14159265, etc., is called ;r, and is 
the measure, in theoretical units, of two right angles. 
Also ^ is the measure of one right angle ; but it must 


not be confounded, as is frequently done, with 90. 
It is true that they stand for the same angle, but on 
different suppositions with respect to the unit ; the 
unit of the first being very nearly ^ ^ 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 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 

2 7 8 


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. 



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 

L BC side opposite to A . 

referred. Thus, =, or -. ^-. , is called 

AB* hypothenuse 

... AC side opposite to B 

the sine of A : while -^=- , 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 



of A. The following table expresses the names which 

, . . BC AC BC AC AB 
ra given to the six ratios, -jg and 

-=^, 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. 









sine of A 



cosine of B 


sin A 

cos B 


cosine of A 


sine of B 



cos A 

sin B 


tangent of A 


cotangent of B 


tan A 

cot B 


cotangent of A 


tangent of B 


cot A 

tan B 



secant of A 


cosecant of B 


sec A 



cosecant of A 



secant of B 




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 10. 


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 A. 
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 


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 n q 

is of the first, or is obtained by dividing the first 

unit into nq parts, and taking one of them. Then, 
by the proposition just quoted, the square described 
on the larger unit contains 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 

nqY^nq in the square on the longer unit ; and, there- 

, mqy^np mp Y.nq mp . 
fore, - , or - , or , is the number of 
nqy^nq nqy^nq nq 

the larger squares contained in the rectangle. But 
-- is the algebraical product of and . This prop- 
osition is true in the following sense, where the sides 
of the rectangle are incommensurable with the unit. 
Whatever the unit may be, we have shown that, for 



any incommensurable magnitude, we can go on finding 

b and a, two whole numbers, so that is too little, and 

b+ 1 a 

- too great : until a is as great as we please. Let 

AB and A C be the sides of a rectangle AK, and let 
them be incommensurable with the unit M. Let the 

lines AF and AG, containing and - - units, be 

a a 

respectively less and greater than AC; and let AD 
and AE, containing and units, be respectively 



Fig. 13. 

F C G 

less and greater than AB ; and complete the figure. 
The rectangles AH and AI contain, respectively, 

X T, an d X j square units,* and the 

ad a d 

first is less than the given rectangle, and the second 
greater ; consequently the given rectangle does not 
differ from either, so much as they differ from one 
another. But the difference of AH and AI is 
! be 


-,, or 



*" Square unit" is the abbreviation of "square described on the unit." 


6r 1 A + 1 .1 4. -L 

d a ad' ad' 

Proceed through two,* four, six, etc., steps of the 
approximation. The linear unit being M, the results 

will be such, that M will be always less than A C, 
a -1 z 

but continually approaching to it. Hence -= M is 

AC a 

always less than =- ; and since A C remains the same, 

and d is a number which may increase as much as we 


please, by carrying on the approximation, - r and 
I/ a 

a fortiori M may be made as small a line as we 

1 b 

please ; that is, may be made as small as we 
da j 

please, and so may T in the same manner. Also 

i ad 

j may be made as small as we please ; and there- 
ad -i i -i -i 

fore, also, the sum 1 r -I -j. But this num- 

d a ad ad 

ber, when the unit is the square unit, represents the 
difference of the rectangles AH and AI, and is greater 
than the difference of AK and AI ; therefore, the ap- 
proximate fractions which represent AC and AB may 
be brought so near, that their product shall, as nearly 
as we please, represent the number of square units in 
their rectangle. 

In precisely the same manner it may be proved, 
that if the unit of content or solidity be the cube de- 
scribed on the unit of length, the number of cubical 
units in any rectangular parallelepiped, is the product 

*This is done, because, by proceeding one step at a time, is alternately 
too little and too great to represent AC; whereas we wish the successive 
steps to give results always less than AC. 


of the number of linear units in its three sides, whether 
these numbers be whole or fractional ; and in the sense 
just established, even if they be incommensurable with 
the unit. 

These algebraical relations between the sides and 
content of a rectangle or parallelepiped were observed 
by the Greek geometers ; but as they had no distinct 
science of algebra, and a very imperfect system of 
arithmetic, while, with them, geometry was in an ad- 
vanced state ; instead of applying algebra to geom- 
etry, what they knew of the first was by deduction 
from the last : hence the names which, to this day, 
are given to aa, aaa, ab t which are called the square 
of a, 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. 


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- 

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 


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 


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 

Cube, the term, 284. 

D'Alembert, 187. 
Dauge, F., 187. 



Decimal, system of numeration, 14 et 

seq. ; point, 43 ; fractions, 42-54. 
Definition, u, 207. 
Delbceuf, 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. 
Duhring, 187. 
Duodecimal system, 19. 

E;:^iisJi 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. 

Formula?, important, 88-89, 99. 141- 
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. 
Harlot, 38. 
Haskell, 168. 
Hassler, 168. 
Helmholtz, 232. 
Hindu algebra, 186. 
Hirsch, 188. 
Holzmiiller, 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 

Increment, 169. 

Indirect reasoning, 226. 

Indeterminate problems, 101. 

Indices, theory of, 60, 158 et seq., 166, 

Induction, mathematical, 104, 179,183. 

Inductive reasoning, 219. 

Infinite quantity, meaning of, 123 et 

Infinite spaces, compared, 235 et seq. 

Instruction, principles of natural, 21 
et seq.; faulty, 182; books on math- 
ematical, 187. 

Interpolation, 169-174. 



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 suriaces, 

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 

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, 

Perfect square, 138. 

Petersen, 189. 

T, 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, 254 

Problems, reducing of, to equations, 
92 et seq.; 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 


Rules, 42; mechanical, 184. 
Rules, extension of meaning of, 33, 

80, 143, 163. 
Russell, B. A. W., 232. 



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 

Syllabi, mathematical, 186. 

Tables, mathematical, recommended, 

Tannery, P., 232. 

Taylor, 168. 

Terms, geometrical and algebraical 
compared, 284. 

Theory of equations, 132 et seq., 179, 

Todhunter, 189. 

Triangles, measurement of propor- 
tions of, 277 et seq. 

Trigonometrical ratios, 278 et seq. 

Ueberweg, 212. 

Universal affirmative and negative, 

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. 

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T.APR 16 1971 

QA De Morgan, Augustus 

11 On the study and difficulties 

D^6 of mathematics 2d reprint ed.