ATHEMMICAL SERIES
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CALIFORNIA
. CALIFORNIAI
LONGMANS MODERN MATHEMATICAL SERIES
General Editors
P. Abbott, B.A., C. S. Jackson, *M.A.
F. S. Macaulav, M.A., D.Sc.
THE TEACHING OF ALGEBRA
BY THE SAME AUTHOR
Uniform with this Volume.
EXERCISES IN ALGEBRA (INCLUDING TRIG-
ONOMETRY). Parti. Without Answers, 3s.6d.;
with Answers, 4s.
Contents. — Section I, "Non-directed Numbers";
Section II, "Directed Numbers"; Section III, "Log-
arithms " ; Supplementary Exercises.
EXERCISES IN ALGEBRA (INCLUDING TRIG-
ONOMETRY). Part II. Without Answers, 6s. ;
with Answers, 6s. 6d.
Contents. — Section IV, " Mainly Revision " ; Sec-
tion V, " The Trigonometry of the Sphere"; Section
VI, "Complex Numbers"; Section VII, "Periodic
Functions"; Section VIII, "Limits"; Section IX,
" Statistics ".
LONGMANS, GREEN AND CO.
LONDON, NEW YORK, BOMBAY, CALCUTTA, AND MADRAS
Xonomang' /B^o^ern /IDatbemattcal Series
THE TEACHING OF
ALGEBRA
(INCLUDING TRIGONOMETRY)
T. PERCY _^^UNN, M.A., D.Sc
PROFESSOR OF EDUCATION IN THE UNIVERSITY OF LONDON
VICE-PRINCIPAL OF THE L.C.C. LONDON DAY TRAINING COLLEGE (UNIVERSITY OF
LONDON) ; FORMERLY SENIOR MATHEMATICAL AND SCIENCE MASTER
WILLIAM ELLIS SCHOOL
WITH DIAGRAMS
LONGMANS, GREEN AND CO.
39 PATERNOSTER ROW, LONDON
FOURTH AVENUE & 30th STREET, NEW YORK
BOMBAY, CALCUTTA, AND MADRAS
I9I4
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PREFACE.
In 1909 and 1910 the author had the honour of giving
courses of lectures on the teaching of Algebra addressed
respectively to masters and mistresses in Secondary-
Schools. The present volume is a practical handbook
based upon those lectures, and containing what seem
to the author the most useful things he has learnt
during the fifteen years of his work as a mathematical
master and the ten years in which it has been his duty
to discuss with teachers, actual and prospective, the
problems of their craft. It is accompanied by two col-
lections of examples — ''Exercises in Algebra, Part I,"
and " Exercises in Algebra, Part II " — which are in-
tended together to cover all stages of school instruction
in the subject. Thus the three volumes constitute a
single work. In the view of the author the term
"Algebra" should include in its reference all the
Trigonometry, plane and spherical, which it is desirable
to teach in schools, together with an exposition of the
fundamentals of the Calculus. He has sought, there-
fore, to present these subjects, both in this book and in
the " Exercises," as a unified whole.
" Exercises, Part I," is intended to supply materials
6c22r>7
vi PREFACE
for a course which every boy or girl who remains at a
secondary school until the age of sixteen or seventeen
may reasonably be expected to cover. Thus it meets
and in some directions exceeds the present requirements
of University Entrance and similar examinations. An
important note upon this point will be found on page 60
of the present book. " Exercises, Part II," presents a
continuation of this universal course, to be taken, as a
whole or in part, by boys and girls who remain at
school until the age of eighteen or nineteen, and by
those who, at an earlier age, forge ahead of their fel-
lows. It is hoped that it may also prove useful to
students preparing to enter the scientific professions
and to students in Training Colleges for Teachers.
In '* Exercises, Part I," explanatory matter has been
limited to a few brief notes — on the ground that in the
earlier stages of the pupil's progress exposition is of little
use unless given verbally by the teacher. Thus the
discussions and arguments presupposed in the exercises
have been relegated to the present volume, where they
could be given in a form which, it is hoped, the younger
teacher will find more helpful, and his experienced
colleague more suggestive and provocative. In " Exer-
cises, Part II," each set of examples is accompanied
by exposition intended for the student's reading. To
make it suitable for this purpose it has seemed neces-
sary to abandon the traditional reticence of the mathe-
matical textbook — a fact to be borne in mind in con-
nexion with the otherwise misleading length of the
book. The corresponding sections of the present book
consist of critical introductions, commentaries upon the
exposition and exercises, suggestions for illustrations
PREFACE vii
and other teaching devices, and solutions of typical
examples.
There is considerable evidence that teachers of the
present generation are no longer satisfied with either
the curriculum or the methods of instruction in Algebra
which they have inherited from their predecessors.
Moreover, there are signs that their dissatisfaction is
reflected in the public examining bodies which exercise
such immense influence upon mathematical teaching
in this country. These circumstances may be held to
justify an addition to the few works written expressly
for the teacher who wishes, while maintaining his
hold upon all that is sound in the traditional methods,
to orient himself in accordance with the present drift
of well-considered opinion. The author has worked
out in " Exercises, Part I," a curriculum congruent
on the whole with the recent Eeport of the Com-
mittee of the Mathematical Association, though in
some respects more radical and in some more conserva-
tive. He has noted with special satisfaction that, taking
the whole work into consideration, the programme of
studies which he has proposed is in close agreement
with the one outlined by Dr. A. N. Whitehead in the
very valuable address printed in the Mathematical
Gazette for March, 1913. With regard to method, the
author has sought to take due account both of the
pedagogical pragmatism of which Professor Perry has
long been our most influential advocate and also of the
modern critical movement represented so brilliantly in
this country by the authors of Principia Mathematica.
In addition, recognizing that mathematical ideas are
viii PREFACE
apt to become " flat " through long confinement in
textbooks, he has sought, by drawing them afresh from
their historical sources, to present them with as much
as possible of their original vigour.
Acknowledgment has been made in the text wherever
the author has consciously taken advantage of the work
of writers whose ideas have not yet been absorbed into
the common stock. He has the pleasure of recording
here many other obhgations, general and specific. From
his three Editors he has constantly received criticism
and advice of the most helpful kind. Mr. C. 0. Tuckey
of Charterhouse read a good deal of the earlier part of
the work in typoscript and made several important sug-
gestions, gratefully accepted. Professor C. Spearman of
University College very kindly read and criticised the
section on Statistics — a subject to which he has himself
made such important contributions. Similar service
was rendered, in connexion with other sections, by
Messrs. G. B. Jeffery of University College, B. M.
Neville of the William Ellis School, and E. Wasser-
man of Owen's School. Dr. L. Silberstein, from his
inexhaustible store of fertile mathematical ideas, has
kindly contributed those embodied in several important
examples. Intercourse with Mr. Benchara Branford
and Mr. David Mair has had an influence upon the
book which is not the less substantial because it is
not located in any particular pages. The author's
thanks are due also to Miss Doris Brookes, Miss Elsa
Nunn and Miss Clotilde von Wyss for assistance given
in the preparation of the diagrams, and to a number of
his present and former students for undertaking the un-
grateful task of providing the answers to the examples.
PREFACE IX
Lastly, he has to acknowledge the help received at every
stage of the work from his colleague, Miss Margaret
Punnett, who not only drew all the diagrams which
required special care or laborious calculations, corrected
all the proofs, and compiled the index of the present
volume, but also gave constant criticism and encourage-
ment of the greatest value.
It should be unnecessary to add that the author does
not seek by these acknowledgments to make his friends
share the responsibility for anything in the work that
may be judged unsatisfactory, or to suggest that they
accept his views upon all the disputed or disputable
questions to which he has offered answers.
London Day Training College
(University op London),
May, 1914.
CONTENTS.
General Introduction,
chaptbb page
I. The Nature op Algebra 1
II. Method and Curriculum 16
III. The Formula 26
IV. The Graph . 31
PAET I.
Alternative Schemes op Study .... 60
V. Introduction to Part I 61
SECTION I.
NON-DIRECTED NUMBERS.
The Exercises op Section I 62
VI. The Programme op Section I (Exercises I-XVI) . 63
VII. Factorization 82
A. Factorization ot ac + be (p. 82) ; B. Factoriza-
tion of a2 - 62 (p^ 87).
VIII. Square Root. Surds 90
A. The Calculation of Square Roots (p. 90) ; B.
The Radical Form (p. 93).
IX. Fractions 96
A. Fractions with Monomial Denominators (p. 96) ;
B. Fractions with Binomial Denominators
(p. 98).
X. Changing the Subject op a Formula . . . 104
XI. Programme op Section I (Exercises XVEI-XXVI) . 109
X
CONTENTS xi
CHAPTBR PAOE
XII. Direct Proportion 117
XIII. Trigonometrical Ratios (I) 121
A. The Tangent of an Angle (p. 121) ; B. The Sine
and Cosine, Vectors (p. 124).
XIV. Trigonometrical Ratios (II) 129
A. Circles of Latitude, Middle Latitude Sailing
(p. 129) ; B. Relations between the Sine, Cosine,
and Tangent (p. 132).
XV. The Combining op Formula 136
A. The Determination of Constants in a Formula
(p. 136) ; B. Common Values of Two Relations
(p. 139) ; C. Elimination (p. 142).
XVI. Further Types op Proportionality . . . 146
A. Inverse Proportion (p. 145) ; B. Direct Propor-
tion to the Square or Square Root (p. 149) ; C.
Inverse Proportion to the Square or Square
Root (p. 152) ; D. Combinations of Types of
Proportion (p. 154).
SECTION II.
DIRECTED NUMBERS.
The Exercises op Section II 157
XVII. The Programme op Section II (Exercises XXVII-
XXXVIII) 159
XVIII. Directed Numbers 181
A. The Uses of Directed Numbers (p. 181); B.
Algebraic Addition and Subtraction (p. 184);
C. The Multiplication and Division of Directed
Numbers (p. 193).
XIX. CONSTANT-DlPPERBNCE SbRIES 199
A. The Summation of Constant-Difference Series
(p. 199) ; B. The Calculation of Certain Areas
and Volumes (p. 203).
XX. Algebraic Multiplication 207
A. Algebraic Multiplication (p. 207); B. The
Binomial Expansion (p. 211).
rii CONTENTS
CHAPTER PAGE
XXI. Positive and Negative Indices .... 214
A. The Uses and Laws of Positive Indices (p. 214) ;
B. Negative Indices (p. 217).
XXII. Algebraic Division 222
A. Algebraic Division (p. 222) ; B. Geometric Series
(p. 224).
XXIII. The Complete Number-Scale 228
XXIV. The Programme op Section II (Exercises XXXIX-L) 235
XXV. Linear Functions. Extended Use of Sine, Cosine,
AND Tangent 258
A. Linear Functions (p. 258) ; B. Extension of
meaning of Sine and Cosine (p. 261).
XXVI. The Hyperbolic and Parabolic Functions . . 264
A. Hyperbolic Functions (p. 264) ; B. Parabolic
Functions (p. 266) ; C. Quadratic Equations
(p. 270) ; D. Inverse Functions (p. 274).
XXVII. Wallis's Law 279
A. Area Functions (p. 279) ; B. DiSerential For-
mulae (p. 282).
XXVIII. The Calculation op it and the Sine-Table . . 292
A. The Calculation of tt (p. 294) ; B. The Calcula
tion of the Sine-Table (p. 295).
SECTION III.
LOGARITHMS.
The Exercises op Section III .... 298
XXIX. The Programme op Section III ... . 299
XXX. The Graphic Solution op Growth-Problems . 312
XXXI. The Gunter Scale 319
XXXII. Logarithms 325
A. The Slide Rule (p. 325) ; B. Logarithms (p. 329).
XXXIII. Common Logarithms 333
A. Gunter's Scale and Logarithms Obtained by
Calculation (p. 333) ; B. Common Logarithms
(p. 335) ; C. The Use of Tables (p. 337).
CONTENTS xiii
CHAPTER PAGB;
XXXIV. The Logarithmic and Antilogarithmic Functions 341
XXXV. Nominal and Effective Growth-Factors . . 346
EXERCISES SUPPLEMENTARY TO SECTIONS II AND III.
Supplementary Exercises 354
XXXVI. The Programme op Exercises LX-LXV . . 355
XXXVII. The Programme op Exercises LXVI-LXIX . . 366
PAET II.
Alternative Schemes op Study .... 380
XXXVIII. Introduction to Part II 381
SECTION IV.
MAINLY REVISION.
The Exercises of Section IV 402
XXXIX. Number-Systems and Numerical Operations . 403
XL. Functions 422
XLI. The Exponential Function and Curve . . . 428
SECTION V.
THE TRIGONOMETRY OF THE SPHERE.
The Exercises op Section V 440
XLII. Projections 441
XLIII. The Trigonometry of Spherical Triangles . . 454
SECTION VI.
COMPLEX NUMBERS.
The Exercises op Section VI 468
XLIV. The Nature op Complex Numbers .... 469
XLV. Relations between a Real and a Complex
Variable 478
XLVI, Relations between two Complex Variables . 486
xiv CONTENTS
SECTION VII.
PERIODIC FUNCTIONS.
CHAPTER PAQB
The Exercises of Section VII .... 498
XLVII. The Circular Functions . . . . . .499
XLVIII, Wave-Motion 515
XLIX. Differential Formula of the Circular Functions 528
L. The Hyperbolic Functions 531
SECTION VIII.
LIMITS.
The Exercises of Section VIII
LI. Differentiation and Integration .
LII. Expansions. Supplementary Examples .
540
541
557
SECTION IX.
STATISTICS.
The Exercises of Section IX . . ... 568
Lin. Frequency-Distribution 569
LIV. The Calculation of Frequencies. Probability . 583
LV. Correlation 602
Index 611
COLOUEED DIAGRAMS.
Figs. 48, 49, 50
Fig. 51
Fig. 58
Fig. 59
Figs. 60, 61, 62
To face page 200
M .. 201
„ „ 208
„ 209
.. 210
GENEEAL INTEODUCTION.
CHAPTER I.
THE NATUEE OF ALGEBEA.
§ 1. A discussion of the proper definition of " Algebra " may
easily be carried to the point where its interest becomes
academic rather than practical. It will be limited here to a
review of the general nature of the topics traditionally studied
and taught under a name of ancient and obscure origin.^ In-
spection of these topics shows the presence of at least four
important elements : Analysis, the direct use of symbolism, the
extended use of symbolism, and the manipulation of symbolism.
§ 2. Analysis. — The most fundamental is analysis. The diffi-
culty of finding a precise boundary between arithmetic and alge-
bra (as these terms are commonly understood) is well known.
It is due to the fact that the distinction between them con-
'sists not so much in a difference of subject-matter as in a
difference of attitude towards the same subject-matter. A
simple example may make the difference clear. A small boy
has learnt that the " area " of a figure is the number of unit
squares (say, square inches) which would entirely cover it.
With this definition before him he is asked to determine the
area of a rectangle measuring 7 inches by 5. He soon ob-
serves that the unit squares into which the figure is to be
mapped out can be regarded as forming five rows each con-
1 It is unfortunate that a writer is compelled by convention to
place his introductory chapter at the threshold of his work. The
disadvantages of the arrangement are obvious. An introduction
often represents the author's attempt to meet difficulties and to
answer objections which the reader has not yet felt. It is therefore
liable to be misunderstood or misused. It is to be hoped that the
reader who finds in the present chapter little relevance to the
urgent problems of algebra teaching will by judicious skipping reach
the end without serious waste of time. He will find at many points
of the sequel references which will send him back, perhaps to a
more prohtable reading of these preliminary discussions.
T. 1
2 ALGEBRA
taining seven squares. This observation enables him to
shorten the process of finding the area ; for it is obvious that
the rectangle nxust contain 7 x 5 = 35 square inches. So
^ar arithmetic. But now let the boy's attention shift from
the actual manipulation of the figures to the process which
ilie maaipiila:ion follows ; and let him observe that the essence
of that process is the multiplication of the length of the
rectangle by its breadth. At this moment he has crossed the
frontier which separates arithmetic from algebra ; for it is an
important part of the business of algebra to disengage the
essential features of an arithmetical process of given type
from the numerical setting which a particular case presents.
The result of the boy's analysis of the process of area-cal-
culation will be expressed in a statement or "rule" that
makes no reference to the particular numbers, 7 and 5, and
would therefore hold good in any other area-calculation
of the same type. For this reason it is usually called a
" generalization ". Strictly speaking, however, the terms an-
alysis and generalization refer to two distinct mental move-
ments. In the former I bring to light the essential process
concealed in a particular or accidental numerical garb. In
the latter I recognize that this process may be followed
identically in solving all problems of the same type. The
distinction is not a trivial one, but demands the teacher's
serious attention. The neglect of it is largely responsible for
the common belief that the process of generalization must of
necessity build upon numerous examples of the truth to be
generalized, and that the security of the result depends upon
the number of instances upon which it is based. This belief
makes the schoolmaster regard "generalizing from a single
instance" as one of the most dangerous manifestations of
original intellectual sin.
The truth is that in some cases the certainty of a generali-
zation does depend upon the multiplicity of its data, while in
other cases the number of data is logically irrelevant ; a single
instance will establish the rule as securely as a hundred.
Thus, when John Wallis (1655) had shown that what we
now call the " integrals " of x, x^, x^, x^ are ^x^, ^x^, {x^, ix^,
he felt entitled to generalize the result and to assume that
the integral of re" "^ would be - x" for all integral values of n.
This was a generalization of the former kind and had probable
THE NATURE OF ALGEBRA 3
truth only. The degree of probability of such a generaliza-
tion clearly depends in the first instance upon the number of
data upon which it is based. Its credit will subsequently
rise if results deduced from it are found to agree with known
truths, or will collapse if a " negative instance " can be pro-
duced. Thus the numbers 41, 43, 47, 53, 61, 71, 83, 97,
113, 131, etc., are all prime and are all included in the
formula n^ + n + 4:1. But the hypothesis that this formula
always yields primes is at once discredited when it is seen that
it fails for n = 40.^ On the other hand there is no possibility
that the rule for the area of a rectangle should break down.^
Its certainty is due, not to verification in a large number of
instances, nor to its simplicity, but merely to the fact that it
is founded on analysis.
It may be objected that the two types of generalization
cannot be distinguished in this way ; that, for example, Wallis
had to analyse his individual results before he could include
them all in a single algebraic statement. This is, of course,
true, but it should be noted that in Wallis's case the analysis
was limited to the discovery that, in several instances, expres-
sions of the form re" ~ ^ actually have integrals of the form ^ x".
Analysis in the sense intended here would deal essentially
with the circumstances on which this relation depends. An
analysis in this sense of any one of Wallis's instances of
integration would establish for ever the certainty of all
instances covered by his formula.^ Thus the question
whether a generalization requires the evidence of a number of
examples is the question whether or not it is founded on
analysis. If it is not so founded the generalization has at best
only probability in its favour, a probability which increases
with the number of instances in which it is verified. If it is
so founded, though in practice several examples may be
needed to point the way to the generalization, yet in theory a
single instance is sufficient to render it certain. "^
1 Quoted from Jevons, Principles of Science.
^ It is assumed for the present that the dimensions are integral.
See § 3.
^ I.e. in which n is integral,
* Logicians (e.g. Bradley) have called the first kind of generaliza-
tion an "empirical universal," the second kind an "uncondi-
tional universal ". The philosophically minded reader may follow
1*
4 ALGEBRA
Each form of generalization has played an important part
in the history of mathematical discovery. Bach has its place
in school instruction. They have, however, very unequal
value. While a mathematical truth may for a time be ac-
cepted and used — as Newton accepted and used the binomial
theorem for a fractional index — on the guarantee of a number
of instances in which it is known to hold good, yet its position
is not felt to be satisfactory until it has been placed on the
firmer foundations of analysis. On the other hand, it must
not be forgotten that the power to reach by analysis a com-
plete guarantee of a mathematical truth is subject to the
weakness which limits all human powers. Thus Pythagoras
— probably as the result of the examination of a number of
special instances — first enunciated the proposition that the
sum of the squares on the sides of a right-angled triangle is
equal to the square on the hypothenuse. Euclid's " de-
monstration " was an attempt to guarantee the truth of the
proposition by analysis. Apparently the analysis seemed to
Euclid complete, and satisfied geometers down to modern
times. But it involves the proposition about congruent tri-
angles known to our youth as Prop. 4, and this truth is not
really guaranteed by Euclid's analysis of it. A recent
brilliant critic, generalizing illegitimately, like Macaulay, from
the single instance of his own amazing precocity, says that
Euclid's proof of Prop. 4 " strikes every intelligent child as a
juggle ". Euclid's proof of the theorem of Pythagoras thus
offers an instance of a piece of mathematical analysis, long
thought to be adequate, which the modern critical sense
requires to be corrected and made more complete. Similar
incidents are constantly occurring in various parts of mathe-
matical theory. They show us that mathematical reasonings
do not necessarily issue from the brains of mathematicians in
full-blown perfection, like Athene from the head of Zeus.
For them, as for all other human productions, perfection is a
goal to be reached, if at all, only after a long process of puri-
fication by criticism and reconstruction.
§ 3. Direct Use of Symbolism. — For the task of algebra as
described in the foregoing section only two tools are, in
up the question raised above in James's Principles of Psychology,
i., ch. XII., and Prof. Bosanquet's paper in the Proceedings of the
Aristotelian Society, 1910-11.
THE NATURE OF ALGEBRA 5
principle, necessary : the power (which every mind possesses
in some measure) of discerning the abstract essential process
in the concrete arithmetical case, and a sufficient command
of language to express it when discerned. But in practice
something more is needed. Neither the analysis of arith-
metical procedure nor the expression of the results of such
analysis can proceed very far without the help of symbolism.
Thus it is difficult to suppose that the generalization commonly
expressed in the form
{x + af = X'' + nx''-^ a + -^^^^ — -^■"~'^ot?' + ... + a"
could ever have been reached without the aid of symbols.
Moreover its expression in language would not only be in-
tolerably prolix but almost useless for practical purposes ;
only the conciseness of the symbolic expression makes it
possible for the student either to grasp the generalization or
to apply it. The development of a symbolism with the pro-
perties of making analysis easier and the expression of its
results more concise and available is, then, the second funda-
mental element in algebra.
This function of symbolism has an importance that reaches
much beyond the field of school mathematics. " The ideal of
mathematics," writes Dr. Whitehead, ^ " should be to erect a
calculus to facilitate reasoning in connexion with every pro-
vince of thought, or of external experience, in which the
succession of thoughts or of events can be definitely ascer-
tained and precisely stated. So that all serious thought
which is not philosophy, or inductive reasoning, or imaginative
literature, shall be mathematics developed by means of a cal-
culus." The definition of algebra suggested by this passage
is, of course, much too wide for our present purpose, but
it brings out a point of great importance. The algebra with
which we are all familiar is only one of an indefinite number
of possible algebras. Wherever there is a field for inquiry
of a certain type an algebra may be invented to facilitate that
inquiry.
The field of common algebra is that of numbers and their
relations, but the school curriculum itself yields an example
of an algebra concerned with an entirely different region.
For in chemical formulae and " equations " we have a system
^ Universal Algebra, p. viii.
6 ALGEBRA
of symbolism expressly designed to facilitate the expression
and investigation of truths in the field of chemical composition
and reaction. Optimistic philosophers have even hoped to
devise algebras that should impart to the treacherous ground
of theological and political controversy the solidity and cer-
tainty of mathematical inquiry. An invitation from one
theological algebraist to another to "sit down and calculate "
would then take the place of the denunciations of rival
champions inspired with odium theologicum.^
It is important to note that the aim of all these algebras
is the same : namely to correct the weaknesses and supple-
ment the deficiencies of language as an instrument of abstract
investigation and exact statement. Words and phrases as
the vehicles of ideas are replaced by symbols — with a con-
sequent gain in clearness and conciseness. A formula,
consisting of an arrangement of symbols, is free from the
ambiguity which often besets the arrangement of verbal units
into a sentence, and is, besides, a more effective vehicle of a
complicated meaning. It is easier to move forward in an
argument when the steps can be expressed in symbolic form,
and it is easier to check the correctness of the movements
afterwards.
The practical deduction from this general theory of algebra
is that in ordinary algebra the symbols are to be thought of
as substitutes, not immediately for numbers, but for words
as the vehicles of general ideas. The res, as and 6s of a
common algebraic statement refer to numbers only in the
same way as the verbal forms which they replace ; they do
not stand for numbers. To speak of a symbol as a " general-
ized number "is to employ a phrase which (with all deference
to the great algebraists who have used it) has no clear mean-
ing and is incompatible with modern logical ideas. In this
book, then, it will be assumed that such symbolisms as
(a + h)'^ = a^ + 2ab + b^, V = 7rr% (for the volume of a
cylinder), y = ax^ + bx + c, may always be regarded as
verbal statements about numbers expressed for a special
purpose in a conventional form, the letters and the graphic
symbols being immediately substitutes for words and " stand-
ing " for numbers only in the same sense as the verbal units
1 See the account of Leibniz' Universal Language in Venn's
Empirical Logic, ch. xxii.
THE NATURE OF ALGEBRA 7
to which they correspond stand for them. This practice will
be found to remove much of the difficulty which the beginner
has in understanding what algebra is " all about ". He can-
not easily see how a can mean a particular number without
meaning either this particular number or that one ; but he at
once appreciates the sport of expressing mathematical state-
ments in a new kind of " shorthand ". It is quite true that
the miracle of " ambiguous reference " is present in the words
" add any two numbers together " in exactly the same way as
in the symbolism "a + b". But it is so familiar a thing in
the former case that nobody but a philosopher finds anything
mysterious in it ; in the latter case the novelty of the ex-
pression brings the strangeness of the fact to our notice.
This point is so important as to be worth repeating in
another form. Every one knows that mathematics is essenti-
ally concerned with "variables". For instance, in the
formula V = Ah, which gives the rule for the volume of a
cylindrical solid, or in the " equation of the ellipse," x'^/a^ +
y'^jb'^ = 1, the symbols all represent variables. That is, while
the numerical connexion expressed by the symbolism V = Ah
holds good only between particular volumes, areas and
heights, the formula refers ambiguously to any set of the
volumes, areas and heights which could exist in combina-
tion. What is not generally noticed is that variables are
almost as common outside mathematics as within.^ Thus
in the statement " The King of England is a constitutional
monarch " the element " the King of England " is a variable
in exactly the same sense as V in the formula Y = Ah. The
sole difference is that while V refers ambiguously to one of
an indefinite collection of number?, " the King of England "
refers ambiguously to one of an indefinite collection of per-
sons. " Edward VII " and " George V " are particular values
which may be " substituted " for the latter variable, just as
the number of cubic centimetres of an actual cylinder may be
substituted for the former.
The invention of variables was, perhaps, the most import-
ant event in human evolution. The command of their use
remains the most significant achievement in the history
of the individual human being. Ordinary algebra simply
carries to a higher stage of usefulness in a special field the
^ See Russell, Principles of Mathematics (references in index).
8 ALGEBRA
device which common language employs over the w^hole
range of discourse. The prudent teacher will, therefore, in
the interests of clear understanding and economy of effort,
present the technical use of variables in mathematics not
as a new thing but as merely a modification of linguistic
uses which the pupil mastered, in principle, at his mother's
knee.
The use of the sign " = "is sometimes thought to be an
obstacle to the view here expounded. How can symbols be
anything but numbers, it is argued, when they are connected
by a sign of equality? The objection would have more
weight if there were not other algebras in which the sign
" = " is used to connect symbols which are certainly not
numbers : for example, M^ + H2O = M^O + H2. Con-
sideration shows two common elements in the meaning of
the sign " = "in all algebras, nB,me\y identity and equivalence.
For example, in the chemical equation just quoted, the sign
'* = " implies that the " matter " referred to by the symbolism
M^ + H2O and M^O + H^ is identically the same matter
manifesting itself in two different forms. The two sides of
the equation are therefore equivalent in the sense that one
gives us in a different shape all the matter that is contained
in the other. Similarly in {a + b)'^ = a'^ + 2ab + 6^ or in
y = ax'^ + bx + c Y7e have the ideas, first, that the two sides
of the equation have reference to identically the same number
and, second, that this number can be regarded in two
equivalent ways. Thus a chain of symbolical expressions
each linked to its predecessor by the sign " = " represents a
series of mental occurrences that can be likened to the " trans-
formations of energy " that take place when (say) the ex-
plosion of gas in an engine cylinder makes the fly-wheel turn
round and so generates in a dynamo an electric current which
in turn makes a lamp filament become incandescent. In
both cases we have the thought of something which remains
fundamentally identical in amount but is made to assume a
number of different forms. This is the reason why such a
chain of statements as 3-1-5 = 84-3 = 11 + 7 = 18 . . .
is inadmissible : it implies the introduction of new matter
where there ought to be identity of matter throughout.
The practical conclusion from this discussion is that, in
ordinary algebra, while the sign " = " always implies the
presence of numerical identity beneath changes of form, yet
THE NATURE OF ALGEBRA 9
it may appear in different contexts with varying force. Thus
in the chain of statements
d , ' V T . sin (x + h) - sin x
— (sma?) = Lit ^^ ^
ax
. h
sm^
= L^ cos (x + o) *
= cos X
sin|
h
2
the sign has an obviously different meaning each time it
occurs. In the first line it may be regarded as heralding
either a definition or a practical rule, in the second it asserts
the equivalence of two symbolic expressions, in the third it
claims validity for a certain regrouping of the symbols, in the
last its force is best given by a simple " is ". We may read it
''equals" in each case, but it is important, especially in the
earlier stages of the subject, to make the pupil realize the
varying colour of that convenient word.
The history of algebra throws interesting light on the
origin and functions of symbolism. After 230 years it may
still be read in the pages of John Wallis as profitably as any-
where. Wallis points out that Vieta (about 1590) greatly
improved -algebra by first denoting known numbers as well
as unknown by "Marks or Notes," and by exercising "all
the Operations of Arithmetick in such Notes and Marks as
were before exercised in the common Numerical Figures "
{Algebra, ch. xiv.). Suppose a problem to concern "any
three numbers ". Then the various numbers in so far as they
may be chosen to be the first, the second or the third of the
" any three numbers " may be regarded as forming three inde-
finitely numerous classes. In denoting any member of the first,
second or third class by the symbol A, B or C, Vieta followed
(says WaUis) the custom of lawyers who "put cases in the
name of John an-Oaks and John a-Stiles or John a-Down, and
the like, (by which names they mean any person indefinitely,
who may be so concerned;) and of later times (for brevity
10 ALGEBRA
sake) of J. O. and J. S. or J. D. ; (or yet more shortly) of A, B,
C, etc." This practice is (Uke Euclid's use of letters in referring
to lines and angles) to save the labour of describing the
members of each class ** by long periphrases or tedious De-
scriptions ".
In chapter xv. Wallis shows how the English Oughtred
{Clavis Mathematicae, 1631) " who affected brevity, and to
deliver what he taught as briefly as might be, and reduce all
to a short view," carried Vieta's " improvement " still farther.
" Thus what Vieta would have written
A Quadrate, into B Cube _ , _,^ ,
OPE s^ii^ Equal to FG plane,
would with him be thus expressed
ODE ~
A2B3
From this symbolism to the modern pryps = EG is but a short
though a very important step.^
§ 4. Extended Use of Symbolism. — A third distinct element
appears in algebra when a piece of symbolism originally in-
vented to express a simple arithmetical operation is found so.
convenient that our definition of the operation is deliberately
changed in order to bring other less simple operations within
the scope of the same symbolism.
Here again is a feature of such importance that some writers
have regarded it as the distinctive characteristic of the science.
It is, perhaps, more illuminating to think of it as simply a
further development of the power of symbolism to bring effici-
ency and the " short view ". It may be illustrated by continu-
ing a little further our observation of the small boy who was
studying areas in § 2. We left him at the point where he had
found that the area of a rectangle with integral dimensions is
obtained by multiplying the length by the breadth. Let us
follow him in his investigation of rectangles whose dimensions
are fractional.
There is, to begin with, no difficulty in a rectangle of frac-
tional length but integral breadth. For if the length is, say, 6 J
^Prof. U. G. Mitchell has contributed a useful review of the
growth of algebraic symbolism to Young's Fundamental Concepts
of Algebra and Geometry, p. 22b.
THE NATURE OF ALGEBRA 11
inches and the breadth 4 inches the rectangle can be divided
into 4 inch-strips, each containing 6 J square inches. The
area will, therefore, be found by taking four times 6J — an
operation which comes under the formula " multiply length by
breadth ". But if the breadth is also fractional — say 4:^ inches
— the matter is not so simple. The area cannot be divided
into an exact number of inch-strips, and cannot, therefore, be
calculated by multiplication — as multiplication has hitherto
been understood. We are driven to the more complicated
calculation : —
area = 6J x 4 + | of 6^
Now if the breadth were 5 inches, the calculation of the area
would again come under the simple formula " multiply length
and breadth together ". The formula holds good then, for
integral breadths, 3 inches, 4 inches, 5 inches, etc., but not
for intermediate breadths, 3 J inches, 4^ inches, 5f inches, etc.
But this is a very unsatisfactory state of affairs. If some of
these areas are determined by multiplication it seems reason-
able to regard them all as determined by that process. This
view is possible if we agree so to extend our notion of multi-
plication that an operation of the type 6;^ x 4 4- -^^ ^^ ^i shall
be defined as " multiplying 6 J by 4^ ". By this device the
useful formula, area = length x breadth, is made to include
every case in which the lengths of the sides of a rectangle
can be expressed in integral or fractional numbers.
The introduction of negative and fractional indices is
another typical instance of this influence of a useful symbolism
upon the original definition of an arithmetical process. Thus,
if a sum of money, say £250, increases at 3 per cent com-
pound interest for 5 years its arnount is given by the cal-
culation £250 X (1-03)^ — the symbolism implying (by defini-
tion) 5 successive multiplications by 1*03. Similarly, the
amount at the end of 6 years is obtained by 6 successive
multiplications, symbolized by £250 x (1*03)''. Now we
must not say that the amount after 5-^ years is found by 6^
successive multiplications, for the statement would be mean-
ingless. On the other hand it seems unreasonable that the
operation needed in the last case should not be regarded as
the same operation and be denoted by the same symbolism
as in the other cases. If we yield to this argument we shall
agree so to change our original definition of the symbolism
12 ALGEBRA
£250 X (l-03)-^ that the operation (whatever it may be) which
determines the amount after 5 "5 years shall be capable of
being described by the symbolism £250 x (l-03)^*^ In this
way all problems of this type can be brought under the
single formula
Amount = Pr"
where P is the principal, r the amount of £1 after 1 year and
n the time in years, integral or fractional.
The use of negative numbers is, of course, the capital in-
stance in elementary algebra of the characteristic considered
in this article.
§ 5. Manipulation of Symbols. — Wallis signalized two great
merits in Vieta's use of " species " or symbols. First, that
they made general arithmetical statements much more con-
cise ; second, that the symbols could be "exercised" like
numerals, in all the ordinary operations of arithmetic. The
second of these properties is the fourth fundamental character-
istic of algebra.
It has two chief uses. The first is exemplified by
" identities" such as (a + b) {a - b) = a^ - b^. It is profit-
able to examine carefully the meaning of this piece of
symbolism. It is, we have seen (§ 3), to be regarded as a
"shorthand" transcription of a verbal statement: "If any
two numbers be selected the product of their sum and differ-
ence will be the same as the difference between their squares ".
It can easily be seen how this result comes about in any
specific instance. Consider, for example, the product 15 x 3,
expressed as (9 + 6) (9 - 6). Simple arithmetical considera-
tions show that the product can be written in the form
(9 4- 6) (9 - 6) = 92 + 6 X 9
- 9 X 6 - 62
= 92-62
Now this analysis of the way in which the result comes
about shows clearly that it is quite independent of the fact
that we selected the numbers 9 and 6 for our experiment.
The process of multiplication would have taken the same
typical course if we had chosen any other numbers. We can,
that is, describe the process in words without using the names
of any particular numbers, and we can reduce our statement
THE NATURE OF ALGEBRA 13
to a '' short view " by expressing it in " species ". We then
have
(a + b) (a - b) = a^ + ba
- ab - b^
= a^ - ¥
But when we have gone through this process of analysing a
particular case and expressing its permanent or essential
features by the aid of a symbolic statement we may notice
that our work might have been much abbreviated. The
symbolism chosen is of such a character that, line by line,
it imitates the arrangement of the figures in the arithmetical
process. It follows that there was really no necessity to go
through the arithmetical process first and afterwards to de-
scribe its essential features in symbols. By manipulating the
letters as if they were figures we could with perfect certainty
have predicted the typical form of the arithmetical result.
The " simplification of an algebraic fraction " may be taken
as another example of the same property. Suppose that a
quantity t is such that 7 ^ Z "^ 7- Then we have
t 4 7
7 + 4
whence
4x7
4x7
7 + 4
There are certain features of this process which are obviously
independent of the particular numbers which here enter into
it. It is possible, therefore, by the employment of " species ''
to give a generalized account of its steps. Putting a and b
for the verbal units ** the first denominator," " the second de-
nominator," the three stages of the process can be analysed
as follows : —
1 = 1 + 1
tab
b + a
~~ ab
ab
t = ,
14 ALGEBRA
But it is obvious that here, as in the case previously
examined, the consideration of an arithmetical example is
really unnecessary. By treating the letters as if they were
figures we might have passed at once and with confidence
from the first symbolic statement to the third. That is, we
might be sure that if the typical form of an expression for
the reciprocal of a number is 1/a + 1/6, the typical form of
the number itself is ahjih + a).
It should now be easy to understand why "species" are
so often taken to be numbers. For on the one hand the
symbolic statement about a numerical relation imitates
exactly the arrangement of a particular numerical instance
of that relation — as, for example, — \- - imitates - + ~
ah 4 7
and, on the other hand, the general features of the result of
transforming a given arithmetical expression can always be
predicted by imitating the steps of the transformation with
the letters of the corresponding symbolic expression. ^ But
while it would be pedantic never to speak of "adding" or
" multiplying" symbolic expressions, yet it should be made
clear to the pupil that these are merely convenient forms of
speech; that, strictly speaking, we can add and multiply
nothing except numbers ; and that the operations which we
carry out upon symbols are merely a means of determining,
without considering a specific example, the general or typical
features of the result of manipulating an arithmetical ex-
pression in a given way.
We may summarize this discussion by saying that from
one statement expressed in algebraic symbols it is possible to
deduce, merely by manipulating the symbols in accordance
with the laws of arithmetic, an indefinite number of state-
ments each of which will be true if the original statement is
true. Thus if a number can truly be expressed in the form
(a + h) (a - h) it is certain that it can be expressed with
equal truth in the form a^ - 6^. But by far the most striking
instances of this property consist in the transformations
which are used in the solutions of equations. Suppose, for
example, that I am confronted with the formula
^ These remarks also make it clear why the Greeks, in the
absence of a convenient arithmetical notation, were unable to
develop algebra to any considerable extent.
THE NATURE OF ALGEBRA 15
C =
nr + R
taken from an electrician's note-book. Merely by treating
the letters as if they were figures I can obtain from this
statement a number of other symbolic statements, such as
r = E/C - B/n
R = w (E/C - r)
Now the important thing here is that I may be so grossly
ignorant of electricity that the original formula is meaning-
less to me. Yet I shall be quite certain that if the original
formula was valid the formulae I have derived from it are
equally valid. In this way it is possible for me to discover
electrical facts of which my friend the electrician (who though
an excellent practical man is, perhaps, but an indifferent
algebraist) was actually unaware.^ In such a way, too,
given a symbolic statement of one property of a curve and a
few geometrical ideas, it is possible to deduce statements of
an endless series of other properties of the curve — properties
which may in this way become known for the first time.
Point is given to earlier arguments of the chapter by the
observation that the property of yielding new truths as a
result of a merely mechanical rearrangement of symbols is not
confined to the algebra of number. It is to be found in some
form wherever a system of symbolism has been developed " to
facilitate reasoning " in a particular province of thought. The
late Prof. Jevons actually invented a " logical machine " in
which the exploration of the field of truth could be carried out
by pulling levers and turning handles. It would probably
not be impossible, if only it were worth while, to construct an
" algebra machine " which could in a similar way be made
to yield from a given formula other formulae which follow
from it.
^ Of course it will require his practical knowledge to give mean-
ing to my discoveries.
CHAPTEE II.
METHOD AND CURRICULUM.
§ 1. Our Aim in Teaching Mathematics. — Clear notions
about the nature of algebra are essential to a profitable dis-
cussion of teaching problems but do not themselves constitute
solutions of such problems. The purposes to be held in view
in teaching algebra to boys and girls, the proper selection of
topics, the best methods of presenting them : these questions
constitute a distinct subject of inquiry and are to be settled by
considerations of v^hich those adduced in the previous chapter
form only a part. For a full discussion of such matters the
reader must look elsewhere than in a practical handbook ;
but he may be asked to accept the following brief statement
of general principles as the author's confession of faith.
Mathematical truths always have two sides or aspects.
With the one they face and have contact with the world of
outer realities lying in time and space. With the other they
face and have relations with one another. Thus the fact that
equiangular triangles have proportional sides enables me to
determine by drawing or by calculation the height of an
unscaleable mountain peak twenty miles away. This is the
first or outer aspect of that particular mathematical truth.
On the other hand I can deduce the truth itself with complete
certainty from the assumed properties of congruent triangles.
This is its second or inner aspect. The history of mathe-
matics is a tale of ever-widening development on both these
sides. From its dim beginnings by the Euphrates and the
Nile mathematics has been on the one hand a means by
which man has constantly increased his understanding of
his environment and his power of manipulating it, and on the
other hand a body of pure ideas, slowly growing and con-
solidating into a noble rational structure. Progress has
brought about, and, indeed, has required, division of labour.
16
METHOD AND CURRICULUM 17
A Lagrange or a Clerk Maxwell is chiefly concerned to enlarge
the outer dominion of mathematics over matter ; a Gauss
or a Cantor seeks rather to perfect and extend the inner
realm of order among mathematical ideas themselves. But
these different currents of progress must not be thought of as
independent streams. One never has existed and probably
never will exist apart from the other. The view that they
represent wholly distinct forms of intellectual activity is
partial, unhistorical, and unphilosophical. A more serious
charge against it is that it has produced an infinite amount
of harm in the teaching of mathematics.
Our purpose in teaching mathematics in school should be
to enable the pupil to realize, at least in an elementary way,
this two-fold significance of mathematical progress. A person,
to be really "_educated," should have been taught the import-
ance of mathematics as an instrument of material conquests
and of social organization, and should be able to appreciate
the" value and significance of an ordered system of mathe-
matical ideas. There is no need to add that mathematical
instruction should also aim ai ^Ijdis ciplini ng his mind" or
giving him "mental training ". So far as t^e idealFintended
by these phrases are sound they are comprehended in the
wider purpose already stated. Nor should we add a clause
to safeguard the interests of those who are to enter the
mathematical professions. The treatment of the subject
prescribed by our principle is precisely the one which best
supplies their special needs.
§ 2. The Choice of Curriculum. — This principle adopted,
we have the practical problem of selecting for our curriculum
in algebra topics which shall illustrate adequately the two
aspects of mathematics. To the question whether the two
must be provided for separately we may reply with a confi-
dent negative. The theoretical questions which are of most
importance in an elementary course are just those which
arise naturally out of attempts to apply mathematical ideaa
and methods to practical purposes. We shall, therefore,
choose those subjects in which the practical value of algebra is-
most clearly exhibited, confident that this is the best means of
securing opportunities for fruitful theoretical discussions.
In accordance with this view algebra should be introduced
to the young pupil as a symbolic language specially adapted
for making concise statements of a numerical kind about
T, 2
18 ALGEBRA
matters with which he is already more or less familiar. In
other words, the earliest lessons in the subject should teach
the use of the formula, illustrations being drawn largely from
the "Pocket Book" of the engineer and similar formularies.
In these lessons the various algebraic notations should be in-
troduced as symbolic idioms needed for the transcription of
pieces of important or interesting fact. A little later comes
the study of those manipulations of a formula by which it
may be made to yield truths unknown or unperceived before
(ch. I., § 5). It is of the first importance that from the out-
set of his work the pupil should be made to perceive clearly
and feel constantly that both formulae and manipulations
always refer to realities beyond themselves. It is scarcely too
much to say that incompetence in, and distaste for, mathe-
matics nearly always spring from the neglect of this funda-
mental teaching principle. Even in the case of those who
have a natural fondness for the technique of mathematics the
same neglect often leads to an astonishing blindness to the
real significance of mathematical ideas and operations.
The gradual elaboration of the formula as an instrument of
description and investigation is, then, the first business of the
course in algebra. The pursuit of this task leads naturally
to two kinds of inquiries both of which belong to the
inner aspect of mathematical thought. In the first place,
the attempt to build up an effective system of symbolic ex-
pression leads to a number of problems of great importance
from the theoretical point of view. These necessarily include
questions about the meaning and use of negative numbers, the
interpretation of fractional indices, etc., and may range as far
as the fascinating inquiries into the ultimate nature of num-
bers and numerical operations which are so characteristic of
modern mathematics. In the second place, in trying to give
an account from the numerical standpoint of the concrete
things with which his formulae deal, the young algebraist
can hardly fail to notice and to become interested in the
fact that "variables" of widely differing character are yet
often bound to one another by identical quantitative laws.
From that moment onwards it is natural to give an increasing
amount of attention to these general forms of connexion be-
tween variables. Eventually — under the rather forbidding
name of " functions " — they may become the main object of
study.
METHOD AND CURRICULUM 19
§ 3. The Position of Trigonometry and the Calculus. — The
foregoing principles point directly to two important modifica-
tions in present teaching practice. The first concerns the
position of trigonometry, the second that of the differential
and integral calculus. At present it is usual to teach both
of these as " subjects," distinct from algebra and distinct from
one another. The reason is easily found. The professional
mathematician, interested chiefly in perfecting the technique
of his subject, finds it natural as well as most effective to take
a special group of allied methods or allied problems and
to develop them as far as he can without concerning himself
too greatly about the practical value of his work. Now this
systematic exploration of special parts of mathematics is, no
doubt, of vital importance for the continued growth of the
science. It does not, however, by any means follow that the
branches of mathematics should be presented to beginners with
the formal elaboration which is the inevitable mark of their
treatment as separate subjects. The absurdity to which this
practice may lead is sufficiently illustrated by the custom of
one very important public examination in which, until re-
cently, a paper was set requiring considerable technical know-
ledge of the differential calculus while assuming no knowledge
at all of integration. The mischief which follows from it is
still more evident from the fact — surely little less than a
scandal — that so far as the operation of University Leaving
Certificates is concerned a boy or a girl may at present [1913]
pass through a secondary school without making the smallest
acquaintance with the fascinating and powerful methods of
elementary trigonometry.
If we admit that the custom of teaching trigonometry and
the calculus as distinct subjects has no defensible founda-
tion and is responsible for very unsatisfactory consequences,
we shall be prepared to inquire what position is assigned to
them by the principles laid down in this chapter. Taking
first the case of trigonometry, it becomes at once evident that
the pupil's acquaintance with the tangent, sine and cosine
should begin in the region where arithmetic marches with
elementary geometry. As soon as the symbols of the trigono-
metrical ratios are recognized as capable of entering into
formulae and of being manipulated they should be regarded
as belonging to the vocabulary of algebra. There is, indeed,
no principle, except the invalid principle of formal segrega-
2*
20 ALGEBRA
tion, upon which we can include the study of a;" or of
a^ in the algebra course and exclude sin x or tan x. All
alike are pieces of symbolism invented for the description
and interpretation of facts of the external world. Each re-
presents a typical kind of " function ". To each corresponds
a specific form of curve which may be regarded as the
graphic symbol of the function. Both algebra and trigono-
metry would gain by fusion : the former through an added
variety and richness in the illustrations of its main themes ;
the latter by the removal of the excessive formalism which
at present obscures its value and interest for the begin-
ner. Fusion upon these lines is attempted in this book.
The formal work which constitutes so large a part of the
ordinary treatment of trigonometry is greatly reduced and
much more stress is laid in Part I upon the practical,
and in Part II upon the functional, aspect of the subject.
Among other advantages this plan permits the inclusion of a
brief section upon the trigonometry of the spherical surface.
The customary neglect of a branch of inquiry which is of such
importance and interest, in which also the essential results
are to be obtained with such ease, is, of course, only another
glaring instance of the mischief brought about by the over-
formal treatment of mathematics in school. There is ab-
solutely no reason why the main elements of the trigonometry
of the sphere should not be taught except the ridiculous one
that spherical trigonometry is a separate " subject " which
can be studied only after " plane trigonometry " has been
disposed of.
In arguing for the assimilation of trigonometry in the
algebra course we are arguing against the artificial separation
between problems which must in any case be attacked in a
similar spirit and by similar weapons. When we consider
the position of the differential and integral calculus we have
to protest against a tradition which forbids all but exceptional
pupils to become acquainted with the most powerful and
attractive of mathematical methods. As in the former case,
the mischief is the result of a technical elaboration which,
though essential to the historical development of the calculus,
has had the effect of making the really simple ideas upon
which it is built inaccessible to the ordinary boy or girl at
school. In this instance the history of the subject suggests
a remedy for a state of things which is generally regarded
METHOD AND CURRICULUM 21
as unsatisfactory. The calculus began, in the writings of
John Wallis and others, merely as a special kind of alge-
braic argument which might be introduced at any appropriate
point and without the apparatus of a technical notation.
The remedy suggested by this observation is adopted in the
present work. Notions which form part of the doctrine of
the calculus are introduced at an early stage and are developed
side by side with other algebraic ideas ; but only towards the
end of the work are the technical symbols introduced which
have been known to so many students only as hostile
standards floating above an impregnable citadel.
§ 4. Some Practical Suggestions. — It is probable that enough
has been said to indicate the point of view adopted in this
book, and there would be nothing to gain by anticipating here
what will be found in detail in later chapters. The whole
course treated in these chapters is divided into two main
stages upon a principle already explained in the Preface.
These stages are represented respectively by Parts I and II
of the Exercises in Algebra. The ground assigned to the two
stages, their subdivisions, and the order of treatment recom-
mended are dealt with in ch. v. as far as concerns Part I,
and in ch. xxxviii. as far as concerns Part II. The reader
who wishes to gain a complete view of the whole course may
with advantage turn immediately from this chapter to those.
Meanwhile it may be convenient to give some account of
the plan of the book and to suggest how it should be used.
The present or Introductory Section contains two more
chapters. These deal in some detail with the two instruments
of expression which the student of algebra has to use at
every stage of his progress — namely, the formula and the
graph. The conclusions reached are applicable all through
the course, but they have special reference to the first be-
ginnings of algebra. Thus they are illustrated largely by re-
ferences to Exercises I and II, which are intended to be regarded
as preliminary work to be covered before the systematic study
of the subject is begun.
The rest of the book — setting aside the general introduc-
tions to Part I and Part II — is divided into sections which
correspond to the ten sections of the exercises. In Part I
each of these divisions contains a chapter or chapters in
which the general ideas underlying the exposition of the
section are explained, and hints are given for the illustration
22 ALGEBRA
of the teacher's lessons, etc. These chapters also offer
remarks upon the examples of each exercise. Matters which
require emphasis are pointed out and the solutions recom-
mended for typical examples are given. The remaining
chapters indicate the substance of the lessons which the
teacher should give before setting his class to work upon the
various exercises. He should understand that these notes do
not always contain a complete discussion of the topic with
which they deal. They aim merely at carrying the discussion
up to the point at which the pupil can profitably attack the
exercise. In many cases the subject is developed farther in
the course of the exercise itself. For this reason alone it is
important that the teacher should take occasion, after an
exercise has been disposed of, to summarize afresh what has
been learnt both from the preliminary lesson and from the
subsequent examples. It may be added that sometimes the
examples, instead of developing farther a topic which has re-
ceived preliminary discussion in class, prepare the way for a
subsequent lesson. In such cases warning is always given
that certain examples are specially important, and what is to
be learnt from them is summarized in the chapter devoted to
the following lesson.
Part II is written upon a similar plan, with the important
difference that there are no systematic suggestions for lessons
preliminary to the exercises. These are unnecessary in view
of the detailed discussions which accompany the exercises.
With regard to the use of the exercises many points must
be left to the teacher's own discretion and experience. For
instance, it is hardly ever desirable that the whole class shall
at the first attack work through all the examples of an exercise.
Certain examples must be taken because, as we have said, the
theory of the subject is developed from them, while others
are essential as a preparation for instruction to come later.
The teacher must use his judgment as to the number of
examples, outside these, to be exacted from every pupil.
Some may be left for subsequent homework, some for re-
vision in class, some (especially in the harder sections of
the exercises) for the cleverer and quicker boys or girls.
The teacher should bear in mind that it is often a good
thing to run away from a difficulty in order to fight it
another day ; that of many a difficulty, especially in mathe-
matics, it may be said, solvitur ambulando ; and that a class
METHOD AND CURRICULUM 23
bored by long-continued study of a single topic is making
its minimum rate of progress and gaining the minimum profit
from its labours.
To these remarks the author may, perhaps, be permitted to
add for the benefit of his younger colleagues that it pays always
to exact from a boy or girl the best work of which he or she
is capable. The beginner, recognizing that his real business is
to keep his pupil's mind active, is apt to underestimate the
closeness of the connexion between orderly ways of thinking
and writing, and to be imposed upon by specious little rogues
who have developed a precocious talent for concealing lazi-
ness. He should be on his guard against this weakness, and
should constantly check any tendency to accept careless or
untidy work merely because it shows intelligence. When he
has acquired the diagnostic powers which only experience
can give he may trust his ability to determine the cases in
which he may safely relax. Even then these cases should
be rare.
References for Reading.
The present author has dealt more fully with some of the
points raised in chs. i. and ii. in the following papers : —
"On the Method of School Algebra." *' School," Sept. 1905.
(John Murray, 6d.)
"The Arithmetic of Infinites." "Mathematical Gazette," Dec.
1910 and Jan. 1911. (Bell & Co., Is. 6d. each.)
"The Aim and Methods of School Algebra." "Mathematical
Gazette," Dec. 1911 and Jan. 1912.
The following are among the most important and acces-
sible books upon the logic, pedagogy and history of mathe-
matics. Some of them contain bibliographies : —
A. N. Whitehead, "Introduction to Mathematics". (Home Uni-
versity Library, Williams & Norgate, Is.)
P. E. B. Jourdain, "The Nature of Mathematics ". (The People's
Books, T. C. & E. C. Jack, 6d.)
J. W. A. Young, "Lectures on Fundamental Concepts of Algebra
and Geometry". (The Macmillan Co., 7s.) An admirable
review of the subject, intended specifically for teachers.
Bertrand Russell, " The Principles of Mathematics ". (Cambridge
Univ. Press, 25s.) A book of the highest originality, impor-
tance and authority.
24 ALGEBRA
B. Branford, *' A Study of Mathematical Education". (Clarendon
Press, 4s. 6d.) The most important and original of recent
English contributions to the pedagogy of mathematics.
D. E. Smith, ''Teaching of Elementary Mathematics". (Mac-
millan Co., 4s. 6d. net.)
J. W. A. Young, "Teaching of Mathematics in the Elementary
and Secondary School ". (Longmans^ 6s.) The last two are
excellent American textbooks.
G. St. L. Carson, "Essays on Mathematical Education". (Ginn
& Co., 3s. net.)
J. Perry, "Report of a Discussion on the Teaching of Mathe-
matics," British Association, 1901. (Macmillan & Co., Is.)
A Report which has had a great influence upon the reform of
mathematical teaching in England.
A. Holier, "Didaktik und Methodik des Rechnen und der Mathe-
matik". (Leipzig, Teubner, 12s.)
W. W. Rouse Ball, "Short Account of the History of Mathe-
matics ". (Macmillan & Co., 10s.)
F. Cajori, "A History of Elementary Mathematics". (The
Macmillan Co., 6s. 6d. net.)
J. Tropfke, " Geschichte der Elementar-Mathematik, " 2 vols.
(Leipzig, Veit & Co., 9s. each.)
CHAPTEE III.
THE FORMULA.
-Algebra regarded as " general-
ized arithmetic " should have no formal beginning. As soon
as the child who sees the teacher write upon the blackboard
area = length x breadth
can translate this into the words: "To find the area of the
floor I must multiply its length by its breadth," ^ he has,
without knowing it, already begun his study of the subject.
What the teacher has set before him has the two character-
istics of a "formula": {a) it is a statement of a general
" rule " applicable to any one of a definite class of problems ;
and (6) the statement is expressed in a conventional form
chosen for its properties of conciseness and ready compre-
hensibility (ch. i., §§ 2, 3). By his twelfth year lessons
in arithmetic and science should have afforded the pupil
abundant opportunity of learning to write down and use
simple formulae of this kind.
Formulae must, of course, never be used in arithmetic un-
less the pupil clearly understands the processes which they
prescribe. In other words he is entitled to use a formula
only if it represents genuine results of his own thinking. He
may then with advantage write it at the head of his calcula-
tion as a memorandum of the process which he intends to
employ. Used in this way, the formula makes for greater
clearness both of the pupil's thinking and of his written state-
ments.
These preliminary exercises will give occasion for the use
^ The purist may make him say : " the number which measures its
length by the number which measures its breadth ". This greater
scrupulosity of diction need not affect the formula.
25
26 ALGEBRA
in verbal formulae of all the ordinary symbols of arithmetic :
area
e.g. , ,. , or area/length, (length)^, (length)^. The teacher
must also seek in them his opportunity for introducing his
pupils to the extremely important device of replacing the
constituent phrases of the verbal formula by single letters.
The use of words in an abbreviated form supplies a natural
transition to the stage in which this practice is definitely
adopted.
§ 2. Use of Literal Symbols. — The use of single letters can
best be explained (and made attractive) by teaching the class
to regard formulae as "shorthand" memoranda of the rules
which they have established in the course of their work and
are constantly needing. The principles of this " short-
hand" are {a) to represent certain constantly recurring
words (such as "multiply," "divide," "square") by conven-
tional symbols or " grammalogues " ; and (6) to reduce other
words or verbal expressions in the full statement of the rule
to single letters, chosen so as to suggest those words or ex-
pressions as readily as possible to the reader of the memo-
randum. Thus " circumference of the circle " may be
reduced to C, " rate of interest per cent per annum," to r, "the
number of passengers " to either n or p, according to con-
venience or the choice of the writer. Moreover, it will be in
accordance with the notion that we are developing a " short-
hand " to replace always by the same letters any words or
verbal expressions which frequently occur in the problems in
which our formulae are employed. Thus A can generally be
taken without special explanation to mean " area," the parti-
cular figure whose area is in question being known from the
context. This method of procedure follows so obviously from
the function of the formula as a labour-saving device, that it
is, perhaps, unnecessary to point out how it accords with the
practice of all persons who employ formulae for serious pur-
poses. To the electrical engineer or the actuary it is a matter
of no small importance that his text-books and formularies
should employ consistently the same symbols, and that these
should readily suggest the verbal units for which they stand. ^
1 The Institute of Actuaries have published an official set of
symbols which are used, without explanation, in all the papers and
discussions of their members.
THE FORMULA 27
When the verbal statement of the rule contains a numerical
constant the practice must be taught of placing it before the
literal symbols. Thus the rule that the volume of a pyra-
midal solid is obtained by multiplying the area of the base by
one-third of the height is, in accordance with this convention,
to be written neither in the form V = AJ/i, nor in the form
V = Ah^, but in the form V = ^Ah.
The use of the symbol -n- appears to contradict the statement
that symbols are to be taught as representing not numbers
but words or verbal expressions. This is not really the case.
There would be no point in using the symbol tt in a formula,
rather than a concrete number, if it was not understood by
the class that the ratio signified is one that in different calcula-
tions is taken to have different values according to the degree
of approximation required. Thus tt does not immediately
represent a number but is the " shorthand " rendering of the
phrase : " the ratio of the circumference of a circle to its
diameter, taken to the degree of approximation which the
problem requires ".
§ 3. Example of Method. — It is a matter of little import-
ance at what precise point the class should be taught to
adopt definitely the device of representing by a single letter
a word or larger verbal unit. Whenever the step is taken the
teacher will find it convenient to proceed much as in the
following example.
The topic under discussion is supposed to be the mode of
calculating the volume of a solid of uniform cross- section, e.g.
a cylinder. The teacher is provided with a cylindrical tin
over the bottom of which he has pasted a piece of paper
divided into centimetre squares. He proposes to the class
the problem of finding the number of cubic centimetres in the
content of the tin and conducts with them a colloquy to the
following effect : —
On counting up the number of square centimetres that
cover the base of the tin (making due allowance for incom-
plete squares) I find that there are (we will say) exactly 32.
If now I placed in the tin a layer of clay 1 cm. thick and just
large enough to cover the base I should evidently have a centi-
metre cube of clay standing on each of the squares. There
would be, therefore, 32 c.cm. in the layer altogether. If
the tin is (say) 15 cm. high, I can pack 15 of such layers of
clay on top of one another, so that the total amount of clay
28 ALGEBRA
that the tin would hold must be 32 x 15 c.cm. Now if
there had been on the base of the tin 17 or 148 or any other
number of square centimetres and 82 or 2003 or any other
number of linear centimetres in its height it is evident that I
could calculate the amount of clay the tin would hold in
exactly the same way. Thus we have the rule that the
number of cubic centimetres which the tin would hold (the
"volume" of the tin) is found by multiplying the number of
square centimetres in the area of the base by the number of
linear centimetres in the height — or, expressed more shortly,
by multiplying the area of the base by the height. If the area
of the base had been fractional — say 32-7 sq. cm. — each of
the slabs of clay would have contained 32*7 c.cm. instead of
32 c.cm. If, moreover, the height of the tin had been 15-3
cm., instead of exactly 15 cm., then it is clear that I could
have packed into the tin 15 slabs and a thinner slice of clay
3/10 of a centimetre thick, and therefore containing 3/10 of
32*7 c.cm. Thus the total volume of the clay would be given
by the product 32-7 x 15*3. The rule that the volume of the
tin is obtained by multiplying the area of the base by the
height evidently holds good, then, when the area or the height
is measured by a fractional number as well as when the
measures are whole numbers.
We may now proceed to write this rule upon the black-
board. It is unnecessary to write every word in full, for
you will have no difficulty in knowing what I mean if I
shorten it down to the following : —
vol. of cyl. = base x height
or if I make it briefer still : —
vol. of cyl. = base x ht.
Now there are a great many persons who have constantly
to make use of notes or memoranda of this kind. They
are such people as engineers, who have to keep notes of all
sorts of rules in regard to the weights which their materials
will bear, etc., ship-builders, electricians, architects, military
officers, sailors, etc., etc. Some of their rules are so compli-
cated that their notes would be very cumbersome even if they
shortened the words down as we have, and employed symbols
such as " = " and " x ". They find it necessary to use, there-
fore, a kind of shorthand in which they can express their
memoranda much more briefly even than we have expressed
THE FORMULA 29
the rule for finding the volume of the cylinder. The principle
is to use one letter only to represent a word such as " height "
or a phrase such as "area of the base". As far as possible
letters are chosen which readily suggest the words for which
the letters stand. Thus if we were to write our formula in
this '' shorthand " way we could choose the letter V to stand
for the words " volume of a cylinder," the letter B to stand
for the words " area of the base " and the letter h to stand
for the word "height". The rule would then take the ex-
tremely short form : —
V = B X /i
But if we make up our minds never to use more than one
letter to represent a word or group of words, the formula
may be shorter still. We can agree to indicate that two
numbers are to be multiplied together simply by writing the
letters which are the shorthand descriptions of them side by
side. Upon this plan our formula becomes : —
V = B/t
Such a formula is, remember, merely a shorthand way of
writing down the sentence " The volume of the cylinder is
obtained by multiplying the area of the base by the height ".
The sign " = " can be read " is obtained by " (or equivalent
words), while the word " multiply " is supplied by the fact that
the letters are side by side. If it is necessary to make clear
in your note-book what words the various letters stand for it
is best to write as follows : —
V ^ " volume of a cylinder "
Thus the symbol " = " is to be read " is the symbol for the
words ".
From this point onwards the teacher should take occasion
whenever a rule in connexion with mensuration, arithmetic,
or elementary science has been formulated in words to dis-
cuss with the class how to express it in " shorthand " form.
In this way the class will, without receiving special lessons
in algebra, acquire facility in handling the simpler forms of
symbolical expression.
Exercise I is intended to indicate the range of algebraic
expression which should be covered in these incidental dis-
cussions. It will, therefore, be most conveniently used as a
30 ALGEBRA
means of revising and assuring the pupil's preliminary know-
ledge before he begins, in Ex. Ill, the systematic study of
the art of formulation. If the pupil has no knowledge of the
uses of the formula, Ex. I must be worked through with
considerable care. In either case it is of much importance
that the method recommended in this chapter should be
followed ; that is, that the beginner should be taught to
regard a formula as nothing more than a " shorthand "
transcription of a verbal rule or other statement.
§ 4. Setting Down of Work. — By writing his formulae at
the head of arithmetical calculations the young student will
also learn the rudiments of the art of "substitution". In
connexion with this topic it is hardly possible to lay too
much stress upon the importance of cultivating a neat and
orderly way of setting down the steps in an arithmetical or
algebraical argument. A piece of algebraic symbolism should
be as capable of straightforward and continuous reading as a
passage from a newspaper. To achieve this end the teacher
will find it a sound rule never to permit a line to contain
more than two expressions connected by the sign of equality,
and to insist upon the pupil's setting the signs of equality,
in successive lines of the argument, directly underneath one
another. Thus such expressions as
Y = BA, = 32-7 X 12-4 = 405-48 c.cm.
should be excluded both from the exercise book and the black-
board in favour of the arrangement : — ^
Y = Bh
= 32-7 X 12-4
= 405-48 c.cm.
§ 5. No Manipulation of Symbols. — It should be noted
that in this preliminary work no manipulations of the symbols
will be taught and no question of the " sign " of the numbers
that are substituted in the formulae will be raised. Both these
matters belong distinctly to the formal study of algebra.
^ The need of economizing space unfortunately compels the author
to break this rule in his book. The reader should attend to the
precept and ignore the examples.
CHAPTER IV.
THE GRAPH.
§ 1. The Graph and the Formula. — Throughout algebra
the graph, alone or in conjunction with the formula, plays
an important part as an instrument of analysis and generaliza-
tion. It is obvious that it shares many of the properties of
the formula. Like the formula it can be used to bring out
and express the *' law " or identity which underlies the diver-
sity of a number of concrete numerical facts. Like the
formula it delivers its message in a form readily taken in by
the eye, and so ministers to the " short view ". It may often
be regarded as a general statement from which, as by sub-
stitution in a formula, an endless number of new particulars
can be deduced. Lastly, it can in some cases be manipulated
like a formula so as to yield new and unsuspected generaliza-
tions. On the other hand, it is inferior to the symbolic
formula in many important respects. Its accuracy depends
largely upon mechanical or non-intellectual conditions, such
as the skill of the draughtsman and the exactness of the
squared paper. It is less compact and less easily reproduced.
Its message is frequently inarticulate and obscure. For these
and similar reasons it should be regarded as a subsidiary
algebraic instrument which fulfils its best office when it either
leads up to a formula by which it may itself be superseded,
or serves to unfold more fully the implications of a formula
whose properties have been only partially explored.
As contrasted with the symbolic formula the usefulness
and the limitations of the graph both rest upon the same
circumstance — its relatively concrete character. Thus the
curve which represents the relations between the values of
two variables is abstract enough to be a means of concentrat-
ing attention upon the law of connexion as distinguished from
the variables connected, and is at the same time concrete
31
32 ALGEBRA
enough to make a vivid appeal to intuition. On the other
hand, though its concreteness makes it a more impressive
form of statement than the formula, it also makes it a much
less flexible instrument of investigation.
In view of these characteristics it is not surprising to find
that graphic methods were used as an effective instrument of
mathematical thinking before algebraic symbolism had de-
veloped beyond the rudiments. Thus the Greeks, who never
succeeded in producing a satisfactory algebraic method, yet
performed analytic feats of high importance with the aid of
graphic forms. The propositions of Euclid's second, fifth and
most of the later books exemplify this statement ; a simpler,
but very striking illustration, is afforded by the argument in
proof of the rule for extracting a square root which is given
in ch. viii. This method (as the present author found after
publishing it as his own !) was used, in practically the same
form, by the Alexandrian astronomer, Ptolemy, about a.d.
120.1
The superior vividness and intuitability of graphic modes
of expression suggests the conclusion that the young pupil
should be taught their simpler uses before he makes acquaint-
ance with the abstruser though more powerful instrument, the
formula. The historical circumstances just adduced will seem
to many to add support to this conclusion. It is true that it
is opposed to the prevailing practice, but no one is likely to
contest it who has observed the readiness with which a child
will express himself in pictures long before he has learnt to
command the more abstract medium of written words. Thus
the gradual penetration of graphic methods into elementary
instruction in mathematics and science may be welcomed as
one of the most significant features of present pedagogical
tendencies.
§ 2. First Lessons in Graphic Bepresentation (Ex. II, A.).
— In accordance with the observations of the last article the
second of the two preliminary exercises (Ex^ II) consists of
a number of examples illustrating those uses of the graph
which foreshadow the more effective use of the formula to be
learned at a later stage. Like those of Ex. I these examples
are best used as a means of revising and extending a know-
^ See the School World, Feb. 1911 ; also Heath, Archimedes^
p. Ixxvi.
THE GRAPH 33
ledge of graphic representation which the pupil acquired
before he began the systematic study of algebra. They sum-
marize, therefore, a course of instruction which will be all the
more useful if it has been spread over two or three years
of school life. If the pupil has not been practised in graphic
representation in his earlier lessons in arithmetic and geometry
this exercise will require special attention. It is,|however,
neither necessary nor desirable to work through all the
examples before going on to Ex. III. A few of the examples
in divisions A and B should be worked in class. The rest
can be set from week to week as homework, or worked in
class alternately with the first few examples of Ex. III.
The only thing essential is that the main principles of graphic
representation (i.e. those exemplified in divisions A and B)
should be well understood before the class attacks Ex. IV.
The best way to teach the graph is to let it grow out of the
use of the picture. This principle is illustrated in the examples
of Ex. II. Thus division A begins with graphs which are
very little removed from pictures, and goes on to others in
which the pictorial element is constantly less prominent, until,
in the examples of division B, the pupil is prepared to dis-
pense with it almost entirely.
Nos. 1, 2 and 4 are exercises which form part of a course
of lessons in Nature study given to children of about nine
years old. (The author owes them to the kindness of his
colleague, Miss C. von Wyss.) In No. I a base line is divided
into equal parts and graduated to represent minutes. In the
middle of each interval a perpendicular line is drawn of the
same length as the path traced out by the snail during the
corresponding minute. This length is to be determined by
laying a piece of cotton thread along the line in the diagram.
Eig. 1 1 shows the solution of No. 2. Vertical lines are drawn
through the minute graduations of the base, and the perpen-
diculars drawn in No. 1 are placed end to end across the
spaces between them. The firm sloping lines represent in
this way the movements of the snail which traced the path
AB ; the broken lines represent those of his competitor. The
variations in the average speed of the snails are, of course,
^ I.e. in this book. Figures in Exercises in Algebra, Part I and
Part II, are numbered consecutively among themselves. A refer-
ence to one of them will in this book always be prefaced by
" Exercises " for Part I, and by " Exercises II" for Part II.
T. 3
34
ALGEBRA
Fig. 1.
expressed by the variations in the slope of the lines. Thus
the pupil acquires in his first graphical
exercise the germ of the idea of measuring
a rate of change by the " gradient " of a
graph. On the whole, however, these
two examples illustrate the lowest grade
of usefulness of a graph. The diagrams
add practically nothing to the information
contained in the original figures ; they
merely present it in a more effective and
easily assimilable shape.
From this point of view No. 4 repre-
sents an important step forward, for it
introduces the process of interpolation.
Having graduated his base-line as before,
the pupil sets up at the end of each seg-
ment a vertical equal to the length of the
tulip as shown in the corresponding
drawing in Exercises, fig. 2. A vacant
place must, of course, be left at the end
of the third segment. The pupil is then to judge how long the
tulip would have been found to be if this particular measure-
ment had not been omitted. After (rather than before) he has
expressed his judgment the curve through the tops of the
ordinates is to be drawn, and it is to be made clear that,
consciously or unconsciously, the course of this curve really
determines the judgment.
The significance of the process of interpolation consists in
the draughtsman's analysis, out of the given lengths of the
tulip, of the law of succession which they suggest. He may
be quite unable to give a precise account of this law — he may
even not understand what is meant by calling it a law — but
the graphic presentation of the data forces it upon his atten-
tion as an actual fact. Nos. 5 and 6, taken together, are
meant to fortify the incipient notion of a law by contrast-
ing a case where law exists with one from which it is
absent. In No. 5 the " lawfulness " in the growth of the tulip
leads the pupil to expect a corresponding " lawfulness " in
its weekly increase. We may permit him at this stage to act
upon the expectation, even though he cannot give an adequate
defence of it. Thus when the growth during the fourth week
is determined by interpolation he will expect it to be the same
THE GRAPH 36
as the ditference between the length of the plant at the end
of the third week and the length inserted by interpolation in
No. 4. On the other hand it is evident that there is no law in
No. 3 and, therefore, that nothing can be deduced by inter-
polation.
An important technical detail must not be forgotten. The
vertical ^ lines in No. 4 represent the height of the plant at the
end of each week ; they should be drawn, therefore, at the
end of the corresponding segments of the base line. On the
other hand the verticals in No. 5 represent growth during the
interval and should, accordingly, be erected at the mid-points
of the time-intervals.
Of the remaining examples in division A nothing need be
said except that No. 7 describes an elementary type of
astronomical observations which may with great advantage
be carried out practically. Such observations lead to the
recognition (i) that the sun reaches its highest point in the
sky at a time which varies from about a quarter to twelve to
about a quarter past by local time, and (ii) that when at its
highest point it is always exactly in the same direction — the
direction called " south ".
§ 3. Ex. II, B. — In division B the pictorial element
present in all the examples of division A disappears. Instead
of thinking chiefly of his representations of the data the pupil
is now to attend directly to the graphic expression of the law
which governs their variation. The **note " before No. II is
intended to prepare him for this shifting of the centre of
interest. The teacher will see that there is an important
gradation in the abstractness of the examples. In the earlier
ones the ordinates, if they were drawn, would represent the
data pictorially, for the data are all lengths. In the later
ones the ordinates do not represent lengths, and, therefore,
cease to be even implicitly pictorial. Much of the difficulty
which beginners experience in understanding the representa-
tive character of a graph is met with at this point. The
teacher does not always make allowance for the gap which
the child feels between the case in which a length represents
^ The terms " horizontal" and " vertical " will be used through-
out this book to denote the directions upon a sheet of paper in which
the axes of x and y are respectively drawn according to the common
usaw-e. They may be taken to refer originally to lines drawn upon
a blackboard arranged with its plane vertical.
3*
36 ALGEBRA
a length and the case in which it represents something totally
different from itself. The note prefacing No. 15 suggests a
way of bridging the gap. In this example the ordinates are
to represent sums of money, and the pupil is told to think of
each of them as showing the height of a certain pile of
shillings or pence. Such devices facilitate the transition to
the stage in which the abscissae and ordinates become purely re-
presentative and the graph an expression of an entirely abstract
law of connexion between the numerical values of variables.
When a graph ceases to be pictorial we need a principle to
decide which of the two variables shall be represented by
horizontal measurements and which by vertical. There is a
perfectly clear rule upon this point which the pupil should
be taught to apply unaided. In every case in which a graph
is drawn we can regard one set of measurements as having
been chosen, and the question is how the other set of measure-
ments depends upon these. Thus in No. 15 we select from
the dealer's catalogue statements about the monthly payment
demanded for furniture of a certain total value, and the
question is how does the former sum depend upon the latter.
The former may be called the dependent variable, the
latter by contrast the independent variable. Then the
rule in question states that the independent variable is to
be represented by horizontal, and the dependent variable
by vertical measurements. The teacher may choose to add
the information that the former measurements are called
abscissce and the latter ordinates, but these technical terms
are probably better reserved until a later point.
§ 4. The Column-graph (Ex. II, C.).— So far the variables
whose connexion is the object of inquiry have been in each
case represented in the graph by measured lengths. There
have been, however, certain cases in which this mode of re-
presentation is not completely satisfactory. No. 5 offers a
typical instance. Here the thing to be represented is certainly
a length, but it is a length acquired by the growing tulip, not
at any particular moment of time, but gradually during the
course of the week. There is therefore something arbitrary
in connecting the representative line with the middle or any
other particular point of the base-segment. It is obvious
that a more satisfactory mode of representation would con-
nect the weekly growth with the whole of the segment repre-
senting the time in which it accrued.
THE GRAPH
37
t Zmi f -
IIIIIIIIIIIII?! IIIII IIIIIIIIIIIIIIZ
::::;:::::fi?f:::::::E:::;:::;:::::
J .._
::::::::::::::::::::!:?":::::::::::
:::::|:::::::::::::::::::|:::::::
t ^
"!""!::i:i::i::::::::::::::5"ii
mil mil IIIII IIIII mil IIIII ii_ii
a
3 4
Fig. 2.
The best way to meet this objection is to represent each
weekly increase by an area erected upon the corresponding
segment of the base. But the representation of a length by
an area is more difficult to understand even than the repre-
sentation of a sum of money by a length. It is important,
therefore, to introduce
this new form of gra-
phic symbolism by
examples in which it
is readily intelligible.
This condition is ful-
filled by Nos. 21 and
22, the solution of
which is shown in fig.
'2. The horizontal axis
is here divided into
equal segments to re-
present intervals of
one day. Upon each
segment is set a rect-
angle whose area measures the amount of fresh ground
swallowed up by the flood during the corresponding day.
A larger square of the paper is taken to represent an acre.
When completed the series of rectangles constitutes a " histo-
graph " or (as we shall prefer to call it) a " column-graph "
representing pictorially what we are told about the history
of the flood. It is obvious that the total area flooded at the
end of the first, second, third, . . . days is represented by
the combined area of the first, the first two, the first three,
. . . rectangles.
The data must in the first instance be represented by rect-
angles, for the reason that they consist merely of state-
ments of the total increase of the area of the flood during
each day. We are supplied with no direct information about
the way in which the flood grew from hour to hour. Yet as
contemplation of the pictures given in Exercises, fig. 2, in-
evitably suggests dimensions for the missing member of the
series, so contemplation of the gross results of the successive
days' flooding inevitably suggests probable details of its history.
We may assume, to begin with, that the growth of the flood
was continuous, and we know how much was added to its
surface during the course of each day. The assumption com-
38 ALGEBRA
bined with the knowledge suffice to suggest the interpolations
represented by the broken curve which is added in fig. 2 to
the original column-graph. The principle followed is to draw
a continuous curve across the rectangles in such a way that
the total area above each of the original segments of the base
remains unchanged. It is possible, of course, to draw many
curves which fulfil this condition, but if we qualify it by the
further condition that the curve shall be as smooth as possible
— that is, if we assume the changes in the rate of flooding to
have been as little violent as the data permit us to suppose —
the results obtained by different draughtsmen will show a
satisfactory agreement. To answer such a question as No.
22 (i) we shall, of course, refer to the column-graph as cor-
rected by the added curve. The total area under water at
the end of 1^ days is represented with much probability by
the area under the curve from the ordinate at to the
ordinate at 1'5.
The column-graph is the means of representation which
should be used whenever the data state, not the values of one
variable corresponding to definite values of the other, but the
gross amount of the change in the first variable corresponding
to a given change in the second. As in the case of the linear
graph, the representation varies with the nature of the
variables from a pictorial to a purely symbolic stage. The
earlier stages in the gradation are illustrated by the examples
of division C. It is unnecessary to delay the progress of the
class in order to work through these examples. It will be
sufficient, and probably best, to take them from time to time
side by side with Exs. III-XV.* They will be found of great
importance in later stages of the course, for they are not only
constantly used by modern statisticians but are also made in
this book to play an essential part in the exposition of the
main ideas of the differential and integral calculus. Thus
the method is one which the pupil will often be required
to use.
§ 5. Ex. II, D. — The last division of Ex. II is made up of
examples of varying interest and importance. They are given
here chiefly in order to prevent the pupil from acquiring an
unduly narrow conception of the scope of graphic methods.
The chief value of Nos. 26 and 30 is that they introduce the
pupil to curves, very different from the circle, yet having just
as definite an individuality. Nos. 27 and 28 show how the
THE GRAPH
39
course of a curve may be defined, either partially or entirely,
by angular " co-ordinates ". The solution of No. 27 is given
in fig. 3, that of No. 28 in fig. 4. No. 29 (solved in fig. 5) is
an example of a type which always proves very interesting to
beginners and has an obvious value as foreshadowing some
ideas of importance in the calculus. It is suggested in A
Bhythmic Approach to Mathematics, the little book in
which Miss Somervell has illustrated some of Mrs. Mary
Boole's interesting ideas. Let AB be the path of the cyclist
and the points on it marked 1, 2, 3, . . . the positions which
he occupies at the end of the first, second, third, . . . inter-
vals of time. Let D be the point from which the dog starts.
40 ALGEBRA
Then we may suppose that during the first interval the dog
runs in the direction DA and covers the distance Dl which
is, by hypothesis, half as long again as Al. At the end of
the first interval we may suppose the dog to observe his
master's change of position and to amend his own direction
accordingly. Thus during the second interval he will run
along 11 and will cover the distance 12. x\gain he changes
his direction and during the third interval runs along the line
22, reaching the point 3 at the end of it. The process is
continually repeated until the dog is found to have caught
the man up. Lastly we argue that our solution of this
artificial problem needs only slight amendment to fit the
given case. The polygonal figure D123 . . . indicates a con-
tinuous curve which may easily be accepted as representing
the actual path of the dog if he adjusted his direction at every
moment to the varying position of his master. Such a curve
is a " Curve of Pursuit ".
§ 6. Certain Principles of Method. — The examples of Ex.
II all conform to at least one principle which should
never be contravened. It is that a graph should not be
drawn unless there is a clear purpose to be served by it — a
purpose that the pupil can understand and accept. " A train
is going at 30 miles an hour. Determine by a graph how
far it will go in 3 hours." This is a type of " graphical work "
which offends the common- sense of the pupil and prejudices
him against its legitimate uses. He gets to regard it as merely
a capricious, inconvenient, and uncertain way of treating
problems which can be solved with ease and certainty by ordin-
ary arithmetic. He entertains the same sound objection to
solving by graphs equations which yield without difi&culty
to algebraic methods. If there is nothing to be gained by it
he sees no reason why the thing should be done.
The ends to be achieved by drawing a graph will, of course,
vary greatly from case to case. From the logical point of
view the simplest cases are those in which it serves as a
" ready reckoner " — either by presenting a number of numeri-
cal facts in a form convenient for inspection or by solving
troublesome calculations by a mechanical device. In most
other cases the special service it renders is to disengage from
a set of numerical data knowledge which is not obtainable
from the numbers considered separately. The extent and
value of this knowledge also vary considerably. The simplest
THE GRAPH 41
case is typified by the records of the snails' movements in
Ex. II, No. 1, or the column-graph of No. 21 before the ad-
dition of the continuous curve. Such records are useful be-
cause the facts can be " taken in " from them as a whole
more readily than from a column of figures. They give a
clearer impression of how the snails' speed varied or the area
of the flood increased — and one more easily retained in
memory. The graph performs a much higher function when,
as in No. 4, No. 22, and in most of the other examples of the
exercise, it is used to bring to light the mathematical law
underlying a set of data. It is important that clear thinking
— graduated in " rigour " to the age and experience of the pupil
— should accompany this use of the graph, otherwise there is
some danger that a fine heuristic instrument may be perverted
to the encouragement of slip-shod intellectual habits and the
blunting of the logical sensibility. We have seen that graphs
can in these cases be regarded as generalizations (§ 1). It
must be noted, however, that they are not generalizations based
upon analysis, but belong to the inferior type which requires a
number of instances (ch. i., § 2) ; it is impossible to construct
a graph upon one result. The trustworthiness of a graph de-
pends, therefore, in the first instance upon the number and
variety of the data originally plotted. Thus in Ex. II, No. 7,
if the sun had been hidden by clouds from 10.20 to 12.30 the
drawing of the middle part of the graph would have been at-
tended with much uncertainty. It would have been unsafe
to adopt any very definite conclusion about either the time of
noon or the minimum length of the shadow.
But while the risk attending these graphic generalizations
should always be clearly presented it should not be exagger-
ated. After all it is of precisely the same character and de-
gree as the risk run by a physicist or a chemist who publishes
a new law after carefully examining a number of well-chosen
instances. He predicts that all other instances will be found
to follow the law which he detects in his data. The confidence
of the mathematician in a generalization not based upon
analysis rests on the same foundation as that of the man of
science — a belief in the prevalence of simplicity and continuity
among natural phenomena of all kinds. " Simplex sigillum
veri " ; " natura nihil per saltumfacit ". Return in this con-
nexion to Ex. II, No. 4, and let a, 6, c, e, f be the terminal
points of the lines which represent the recorded lengths of
42 ALGEBRA
the tulip. Then the problem before the pupil is not merely
to join these points by a smooth curve — a problem soluble, of
course, in an infinite number of ways — but to find the con-
tinuous curve which in addition to passing through a, b, c,
etc., satisfies certain other definite conditions. Some of these
conditions represent actual knowledge. For example, it may
be taken as certain that the height of the tulip will not rise
and fall rhythmically, so the graph must rise continuously
from left to right. Other conditions are the expressions of
more or less reasonable assumptions. Thus, although a plant
may show rhythmic variations in its rate of growth, yet in
the case of observations of a plant which cover a short period
at the beginning of its career it may safely be assumed that
the acceleration which is a characteristic of the successive
weeks' growth when recorded was also a feature of the growth
during the vacant fortnight. This assumption, translated
into graphical terms, implies that the curve must show no
" waves ". Thus in the end ambiguity in the solution of the
problem is reduced within very moderate limits. It has
already been shown how similar assumptions limit the solu-
tion of problems hke that of No. 22. From the point of view
of the logical training of the pupil the important thing is that
he should recognize that he is making certain assumptions and
should understand how they work out in graphical terms.
Fortunately the investigation of such matters adds not difii-
culty but interest to the lesson.
The same principle applies in a modified form in instances
of which Ex. IV, No. 9, may be taken as typical. Here the
pupil is to draw a graph in order to find by interpolation the
square roots of numbers which cannot be determined by in-
spection. He plots the square roots of 1, 4, 16, etc., against
the numbers themselves and draws a smooth curve through
the points thus defined. This curve he uses to find the
square roots of intermediate numbers. It should be noted
that in this case there is no knowledge or plausible assumption
from which we can deduce that the curve must be " smooth ".
The choice of the smooth curve is determined merely by its
simplicity and uniqueness. Of all possible continuous curves
through the given points this is the one which it is reasonable
to try first. But in this case — as distinguished from those
hitherto considered — we already know the law of which the
graph is to be the expression. We can therefore use it to
THE GRAPH 43
test the success of our venture. Each pupil chooses at random
one or two numbers within the range of the graph, and reads
off the numbers which the graph asserts to be their square
roots. The truth of this pretension is verified by multiplica-
tion. It is now impossible to doubt that though our initial
choice of a smooth curve might have proved erroneous yet as
a matter of fact it has been justified at least to the extent that
square roots deduced from it may be expected as a rule to be
accurate within the degree obtainable by inspection of the
graph.
§ 7. Graphs in Practical Work. — When the points which
^re to determine the course of a graph represent the results
of practical measurement the problem of drawing the appro-
priate curve is complicated by other considerations. Meas-
urement is always subject to error, and the uncertainty
about the correct position of the graphic points may, for this
reason, be great enough to make it doubtful how the curve
should be drawn even when its general form is already known.
When the general form of the graph is itself unknown the
draughtsman will attach much weight to considerations of
simplicity. Thus if the assumption that the graph is " meant
to be " a straight line is not grossly discordant with the actual
position of the points, he will adopt it in preference to the as-
sumption of a more complicated curve. But when he has
learnt or has assumed that the graph has a certain general
form he has still to determine the particular instance. In
advanced work it is customary to apply in this connexion the
" method of least squares " which is illustrated in the last
section of this volume. In elementary work no systematic
method of dealing with the difficulty is generally used. The
pupil is told to select the straight line or curve which his eye
judges to fit the points best. In Ex. XXVI, D, a simple
method is discussed for which there is a good deal to be said
on statistical grounds, while it offers at least a definite principle
for dealing with the kind of situation which is now in view.
It may be illustrated by No. 17 {Exercises, p. 144). The
solution is shown in fig. 6 in this book. Since it is known
that P is proportional to Q^ the eleven given values of P are
plotted (in the left-hand portion of the figure) against numbers
which represent the squares of the values of Q. Each of the
graphic points corresponds, of course, to a definite possible
value of the ratio P/Q^. This value would be the tangent of
44
ALGEBRA
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THE GRAPH 45
the angle between the horizontal axis and the line joining the
graphic point to the origin. If all the points were joined up
with the origin the resultant lines would constitute a fan of
11 rays. In the figure the middle of these rays (the " median ")
is shown in a firm line, while the third from each end (the
" quartiles ") are shown in dotted lines. The others are
omitted for clearness. Now let us suppose that the correct
value of the ratio is k and let us, assuming some numerical
value for k, find the deviations from it of the ratios represented
by the points in the graph, and add them together. Then it
can be proved with little difficulty ^ that the sum of the devia-
tions can never be less than when k is assumed equal to the
ratio represented by the median line. In this very definite
sense, it can, therefore, be said that the median represents best
the general effect of the observations. But it is clear that, on
another occasion, while the median line might fall in the
same position as in the present instance the fan as a whole
might exhibit important differences. It might be more com-
pact or more extended. The former case would imply greater,
the latter less concordance among the measurements than in
the present instance. Thus it is evident that the median
should be supplemented by some indication of the degree of
" scatter " among the individual observations. But this is a
function which the quartile lines are well suited to perform.
For it is evident that if, speaking generally, the graphic points
deviate widely from the median there will be a wide angle
between the quartiles ; while at the same time their position
will not be affected by the accident of a single " wild "
observation. Thus the lines presented in the left-hand part
of fig. 6 may be taken as summarizing in a simple and useful
way the information conveyed by the original eleven observa-
tions. On the right of the figure this summary is translated
into the graphic form appropriate to the description of a case
in which one variable is proportionate to the square of another.
The abscissas are now the values, not of Q^ but of Q ; the
curve drawn with a firm line is the semi-parabola correspond-
ing to the median value of the ratio, while the dotted curves
correspond in the same way to the quartiles.
§ 8. The Graph in Algebraic Theory. — Our discussion has
as yet hardly touched upon what is after all the main function
1 See Exercises, II, Ex. CXXI, No. 17.
46 ALGEBRA
of the graph in algebra. This, as described in § 1, is either to
point the way to a formula by which it may itself be super-
seded or to unfold more fully the implications of a formula
whose properties have been only partially explored. On
the other hand this function is illustrated so constantly
throughout the book that it will suffice to give here a very
brief review of the different uses which are included in it.
The first of these is well illustrated by the lessons and ex-
amples which deal with the familiar identities a^ - h"^ =
(a + b){a - b), {a + by = a^ f 2ab + b^, etc. (chs. vii., viii.,
IX., Exs. V-XI). In these the figures composed of squares
and rectangles are used, much as the Greeks used them, to
reach certain analytical results of importance. The only
important difference between the older usage and the present
one is that in these lessons the manipulation of the graphic
forms serves a purely temporary purpose. It is used to sug-
gest algebraic identities which are afterwards established upon
an independent basis. The earlier lessons in Section III
(Logarithms) illustrate a similar use of the graph. Here the
" growth-curves " of Exercises, fig. 50 (p. 273), are employed
temporally for the solution of problems which are afterwards
to be solved by logarithms, and the properties of the curve
are made to suggest the algebraic method which eventually
makes a graphic method unnecessary.
The second use of the graph in algebraic theory is exempli-
fied every time the student draws the graph of a " function ".
The mere statement that two variables are connected by the
relation xy = a conveys comparatively little of its implicit
contents to a person who has not explored them by means of
graphic analysis. Even students of considerable mathe-
matical knowledge may often experience the truth of this
observation ; for example, when they try to realize the figure
which corresponds to a simple function involving three
variables. In this kind of application the graph not only
serves to bring out particular features of the function which
might otherwise not be noticed or not be realized ; it also
serves as a kind of challenge to the student to complete or
supplement his view of the function. Thus at the lowest the
drawing of a graph gives a useful occasion for a systematic
and orderly examination of the properties of the function.
But it may often do much more. Thus the exercises of
Section VII (Complex Numbers) illustrate important cases in
THE GRAPH 47
which the attempt to give a graphical representation of a
function leads to the discovery of a whole field of values which
might otherwise have remained unknown.
Lastly it must be observed that a graph is more than a
means by which a given function can be conveniently explored.
It is also by far the best means by which, in the earliest
stages, the function is taught and symbolized. In fact, long
before the student is ripe enough to discuss linear, parabolic
or other '' functions " under those abstract names his mind
has been prepared for the reception of the notions they con-
note by the repeated emergence of straight lines and parabolas
in his graphic exercises. And as the graph is the forerunner
of the idea of an algebraic function so it remains the chief
support of that idea and the chief means of maintaining in
the student's mind the distinct individuality of the different
functional relations. For this reason each of the more im-
portant functions is taught in these lessons in close con-
nexion with the curve which serves as its graphic symbol,
preserving the outlines of the function distinct in the memory
and ensuring the certainty of its recall.
§ 9. The Graph and Co-ordinate Geometry. — The discussion
of this chapter should have made clear the essential distinction
between the use of the graph in algebra and the formal study
of co-ordinate geometry. Briefly summarized, the difference
is that in algebra graphs are studied only for the light they
throw upon the properties of functions, while in co-ordinate
geometry the algebraic properties of functions are of interest
only in so far as they contribute to the exploration of the
properties of curves. This distinction between the two points
of view is important and should be used to define the range
of the geometrical ideas and knowledge to be embodied in the
algebra course. Although it would be pedantic to insist upon
a too scrupulous observance of the frontier, yet there is much
to be gained in definiteness of aim and unity of method
by restricting the attention of the student in this course to
the graphical work which has direct relevance to algebraic
investigation. It is well to add that these observations apply
to graphs of functions involving three variables as directly
as to those involving two variables. The surfaces which cor-
respond to the former should be treated — as they are in Section
IV, Ex. LXXVII — simply as three-dimensional graphs, and
should be studied in the algebra course for the same reasons
48 ALGEBRA
and to the same extent as graphs which can be represented
upon squared paper.
§ 10. Some Practical Suggestions. — This chapter may be
brought to an end by a few practical suggestions which may
be of use, at least to younger teachers. The pupil should be
taught to recognize that a graphic record is useless unless its
object is clearly stated and unless the scales of representation
are properly labelled and graduated. Beginners are prone to
label the vertical scale in such a way that the feet of the
letters are (like those of the horizontal scale) towards the
edge instead of towards the middle of the paper. A graph
gives excellent opportunities for insisting upon the " great
school virtues " of neatness, carefulness and accuracy. The
finest and cleanest of lines drawn with the sharpest of pencils
should be demanded, and no obscurity or untidiness of figures
or lettering allowed. The pupil should be taught to choose
the scales of representation so as, on the one hand, to make
the work of plotting and reading points easy, and on the
other hand, to utilize the sheet of paper as completely
as possible. The natural boy and girl tend to produce a
graph which clings closely to one edge of the paper. They
should be shown that such graphs are wasteful both of paper
and of the possibilities of accuracy.
The great extension of graphic work in schools has led to
the production of quantities of inferior squared paper. The
cheaper kinds are badly inaccurate. They should be carefully
avoided, for their use militates directly against the teacher's
attempt to insinuate ideals of painstaking and accurate work.
PAET I.
SCHEME OF ALTEENATIVE OEDERS OF STUDY.
(Of. p. 59.)
Section I.
Exs. m.-iv., Exs. xiv.-xv.
1
Exs. v.-xin.
1
Ex. XVI.
1
1
Exs. xvn., xxn.-xxv,
1
Exs. X VIII. -XXI.
Ex. XXVI.
1
Section II., Exs. xxvTr.-xxxvin.
1
1
Exs. XXXIX. -L.
1
Section III.
1
1
SUPPLBMBNTAEY EXERCISES.
B.
Section I, Exs. in.-xvi.
1
Exs. xvn. -XXVI. Exs.
1
Exs. XXXIX. -L.
1
Section II.
XXVII. -xxxvin.
1
Section III.
1
Supplementaby Exercises.
CHAPTER V.
INTEODUCTION TO PAKT I.
§ 1. The Essentials of a Course in Algebra. — The exercises
of Part I are intended to cover, and to be confined to, those
portions of the subject which are of such fundamental import-
ance that their study should form part of every scheme of
secondary instruction. When the details of such a curriculum
are worked out there is necessarily found to be scope for
much difference of opinion among those qualified to speak.
Any particular set of proposals will almost certainly exhibit
— by inclusion, rejection and emphasis — features for which
general acceptance can hardly be expected. Nevertheless
there is among thoughtful and experienced teachers a growing
agreement about the essentials of a general course in algebra,
and unanimity of opinion that the prevailing tradition fails in
some important respects to embody them. It is, no doubt, an
advantage that within the broad lines of this concensus there
should be many different individual presentations of the
subject.
The presentation offered in this book postulates that the
course to be taken by all boys (and possibly by all girls) in a
secondary school shall consist of the following items : —
(a) The use of the formula as a means of making and of
expressing arithmetical generalizations, and of describing the
quantitative regularities which characterize physical, social
and other phenomena. The making of formulae, including
practice in the simplest forms of algebraic symbolism. The
interpretation of formulae and the determination of particular
results by substitution.
{b) The art of graphic representation. The uses of the
graph as a subsidiary instrument of analysis and generaliza-
tion.
51 4*
52 ALGEBRA
(c) The manipulation of formulaB in order to bring out the
further relations which a given generalization may imply.
The application of these processes to the solution of problems
of real interest and of practical importance.
{d) In particular, the use in formulae of the symbols of the
trigonometrical ratios, the manipulation of such formulae, and
their application to simple practical problems.
(e) The extension of arithmetical ideas to include the com-
plete scale of positive and negative numbers. The use of
these positive and negative numbers in formulae (including
trigonometrical formulae) and the rules to be observed in
manipulating them.
(f) The use and theory of logarithms.
(g) A simple introduction to the ideas and methods of the
integral and differential calculus.
(h) The idea of a "function " as a generalization from con-
crete instances of numerical dependence. The simpler func-
tions of a single variable and their characteristic graphic
.symbols.
The exercises in which this programme is developed are
grouped into three sections. Speaking roughly, Section I
(" Non- directed Numbers") covers (a), (b), (c), and (d) ; the
items in (e), (g) and (h) fall mainly into Section II ("Directed
Numbers "), while those in (/) are dealt with in Section III.
In addition a group of " Supplementary Exercises" reviews
the whole course and completes the development assigned to
the various topics.
§2. Section I. Non-directed Numbers. — The course outlined
in Section I, Exs. I-XXVI, is intended to give the pupil clear
ideas about the fundamental processes of algebra together
with sufficient technical facility to ensure his appreciation
of their value as instruments of mathematical statement
and inquiry. The fundamental processes are taken to be :
(i) the making and use of formulae ; (ii) factorization ; (iii)
the converse process of expanding a product; (iv) the sim-
plification of easy algebraic fractions ; (v) the process here
called " changing the subject of a formula " and commonly
known as " the solution of simple equations ". The pro-
gramme includes in addition : (vi) the fundamental ideas of
functionality, illustrated by a simple study of direct and in-
verse proportion. The use of the trigonometrical ratios is
taught in connexion with this study.
INTRODUCTION TO PART I 53
The title " Non-directed Numbers," is meant to imply that
in this section algebraic symbols always stand for the ordinary,
signless numbers of arithmetic. Thus if, in a given instance,
the symbol a represents the number 8, it means neither + 8
nor - 8, but simply the cardinal number 8 used in the sense
intended when I say that I have eight coins in my pocket or
that there are eight books upon the table. We might have
said, reviving an ancient term, that the subject of Section I
is " specious arithmetic ". Not until Section II is reached
are symbols to be regarded as standing for numbers accom-
panied by signs, such as + 8 or - 8. There are two good
reasons for this procedure. The first is that the expression
of generalized statements by formulae and the use and
manipulation of numbers accompanied by signs are two
distinct processes which have no necessary connexion with
one another. Each has its own difficulties for the beginner,
difficulties which are best overcome if faced separately. The
second reason is that the two processes are not only different
in kind ; they depend upon the presence of characters in the
child's mind which begin to ripen at different ages. It is easy
enough to make a boy or a girl apply mechanically the rule
that " like signs produce plus, unlike signs minus " ; but if the
rule is to be used with intelligence the teacher must be able
to appeal to logical powers and interests which have rarely
emerged at the age when lessons in algebra begin. Obser-
vation will, in fact, show that the pupil who has been early
taught the properties of positive and negative numbers rarely
uses them spontaneously in his thinking. His mind works
freely only among signless numbers with their familiar pro-
perties. This fact is itself a strong indication that numbers
accompanied by signs have been taught prematurely.
It is of great importance to understand the bearing of these
observations upon the "rule of signs". There are, strictly
speaking, four distinct types of problems covered by this
rule. The first is the type discussed in Ex. X, note to No. 5,
and in ch. ix. B, § 3. It is there shown (for example) that
any number -5x6 = the same number - (9 - 4) x 6
= the same number -9x6-1-4x6.
The change of sign that occurs here is easily justified by
purely arithmetical considerations: taking away 6 fives is
equivalent to taking away 6 nines and adding 6 fours. The
next two types are those represented by such symbolisms as : —
54 ALGEBRA
(-3)- (-7)= -3 + 7
and (-3) X (-7) = +21
discussed respectively in ch. xvni. B and C. The fourth
type is exemphfied by the process of " multiplying " two factors,
such as a + 6 and a - b, when the literal symbols represent
positive or negative numbers. In none of the last three
cases does plus mean "add" or minus "subtract" in the
primary senses of those terms. The appropriate "rule of
signs " must therefore be discussed afresh in each of them.
The fact that as the result of this discussion the same rule
is found to hold good in all four types does not make it a
less serious crime against logic to assume that the rule which
obtains in the first case must be valid in the others. Yet this
practice, it is to be feared, is far from unusual.
In Section I, then, symbols always imply numbers without
sign ; plus and minus have their direct arithmetical signi-
ficance ; a number is never " subtracted " except from a
number larger than itself ; and the rule of signs is considered
only in relation to the first of the four cases just distinguished.
The exercises fall naturally into two groups — Exs. I-XVI,
and Exs. XVII-XXVI. Those of the first group teach the
fundamental uses of formulae and the main types of algebraic
manipulation — factorization, the simplification of fractions and
the processes by which the " subject " of a formula may be
changed. In the second group the formula describing the
numerical relations between a particular pair of variables
ceases to be the centre of interest, the pupil's attention being
now directed to the resemblances which are so often exhibited
in the numerical relations of pairs of variables of widely
differing character. Only the simplest and most important
of these resemblances are studied, namely, those to which the
mathematician applies the terms " direct " and "inverse pro-
portion ". It has already been indicated that the beginnings
of " numerical trigonometry " have their place here, the pro-
perties of the tangent, sine and cosine being treated as
especially important instances of direct proportion.
§3. Section II. Directed Numbers. — In the strictest sense
all algebra is " generalized arithmetic " ; that is to say, all
algebraic processes are based ultimately upon the properties of
numbers, and the results of algebraic investigation have their
meaning and value solely in the fact that they are discoveries
or demonstrations of further general or specific properties
INTRODUCTION TO PART I 55
of numbers. Nevertheless there are important differences
in nature between the ground covered in Section I and that
of Section II. In the first place the numbers considered in
Section I are, as we have seen, the signless numbers of ele-
mentary arithmetic — the numbers which begin with zero and
are continued through 1, 2, 3, etc., in an endless series. The
generalizations of Section II are based upon a different series —
the series of " directed numbers," or numbers with signs, which
has neither beginning nor end but from any starting-point (such
as zero) can be continued forwards and backwards without
end. In the second place this new series differs from the
numbers of ordinary arithmetic in its origin. The signless
numbers were known and used long before algebra was in-
vented ; the use of numbers with signs is itself a product of
the development of algebra. It is, in fact, the most important
instance of the characteristic explained in ch. i., § 4. Consider
a simple example : I am travelling with velocity v towards a
point from which at the present moment I am at distance
d^ ; what will be my distance from it (d) after time t ? The
formula d = d„-vt sums up the solutions of all particular
problems of this form which it is possible to state. But, re-
garded as a " shorthand " prescription of certain operations to
be carried out with given numbers it has a strictly limited
field. The instructions which it gives can be fulfilled only if
vt is not greater than do. Now the use of numbers with signs
enables us to set this limitation aside and to obtain results
from the prescribed operations even when vt is greater than
d„. Moreover it is easily shown that such results are solu-
tions of actual problems akin to, if not identical with, the
original ones. (In the present case a negative result is the
answer to the question: "How far shall I be beyond the
point in time t ? ") Thus the introduction of the new number
series kills two birds with one stone. It removes our natural
dissatisfaction with a generalization whose scope is limited
differently by the conditions of every different problem ; and
it greatly increases the range of problems to which the single
formula applies.
The study of the consequences and advantages of adopting
numbers with signs as the basis of algebraic generalizations
constitutes, then, the main subject of Section II and gives unity
to the topics considered. First the new numbers are intro-
duced and the laws of their combinations are determined
66 ALGEBRA
(Exs. XXVII-XXIX). Subsequent exercises illustrate the
extraordinary increase in scope and power which accrues to
the notations, notions and processes of Section I — formulae,
identities, " change of the subject," trigonometrical ratios,
specific functional relations — when the symbols are taken
to describe numbers with signs instead of the numbers without
signs of ordinary arithmetic.
Upon this plan an expression such as a + 6 may refer to
very diverse numerical combinations : for example, +8+3
(if a=+8 and 6= +3), +8-3 (if 6= -3), -8+3, etc.
Similarly the expression x 10" may mean a number either of
successive multiplications or of successive divisions by 10 ac-
cording as n is positive or negative. It follows that throughout
Section II much more attention must be paid to algebraic /orm
than in Section I. This statement does not mean that the
practical and heuristic aims of the former section are abandoned.
It means merely that a scientific study of algebraic form is
now recognized as necessary to their successful pursuit. In
other words, the idea of a function receives considerable em-
phasis, especially in the latter half of the work.
Like those of Section I the exercises of this section fall
naturally into two groups — Exs. XXVII-XXXVIII and
Exs. XXXIX-L. There is indeed a rough correspondence
between the divisions in the two sections ; for the aim of
each group of exercises in Section II may be regarded as that
of working out the results which follow when " directed " are
substituted for " non-directed " numbers in the arguments of
the corresponding group in Section I. This general state-
ment applies to one particularly important instance — namely
to the development of the ideas and methods which are
generally regarded as belonging to the differential and integral
calculus. The starting-point of this development will be
found in some of the earlier exercises of Section I. Thus in
Ex. IX the notion is introduced of an " approximation-
formula," i.e. of a formula which gives results true to a certain
degree of accuracy when one of the magnitudes is small rela-
tively to others. In the last exercises of Section II the notion
of an approximation- formula is generalized into the idea of a
" differential formula " which may be applied in accordance
with definite rules to functions of a definite form. Again in
the first division of Section II (Ex. XXX) the young student
learns to solve a certain group of problems (arising out of the
INTRODUCTION TO PART I 57
study of arithmetical series) by a method which is, in effect,
integration. At a later point (Ex. XLVII) this simple method
is itself generalized, and the student learns that a large
number of important problems can be solved by a knowledge
of the rule which connects a function of given form with its
"integral".
§4. Section III. Logarithms. — The best way of approach-
ing logarithms is a subject upon which teachers of mathe-
matics hold very different opinions. The quarrel revolves
about the question whether logarithms should be taught before
fractional indices or fractional indices before logarithms. The
former method follows the course of history, the latter com-
plies with the tradition of the text-books. The present writer
for some time dallied with a method which sought to combine
the advantages of both modes of attack. After further experi-
ment and reflexion he has become convinced that a treatment
which keeps close to the historical evolution of the subject is
on the whole the simplest and the most satisfactory from the
logical point of view. The arguments adduced against a too
early introduction of directed numbers weigh also against a
treatment of logarithms based upon the theory of indices. It
is at least doubtful whether the somewhat subtle logic needed
in this case is really appreciated by the young pupil. On the
other hand a treatment which follows the reasoning of Napier,
Briggs, Mercator and Gunter is not only much simpler but
also means much honester thinking on the part of the student.
In Section III, then, the reader will find the usual relation
between logarithms and fractional indices inverted. The
theory of the latter is based upon the theory of the former.
Though this order of presentation follows the order of
emergence in history of the ideas of logarithms and fractional
indices it does not represent exactly the manner in which the
latter notation found its way into algebraic practice and
theory. Fractional indices were actually invented by John
Wallis in the course of his investigations into the " arithmetic
of infinites ".^ The method of treatment illustrated in Ex.
LVIII is a kind of idealization of the course of history ; it
represents what might have happened rather than what
actually took place. It may perhaps claim to be more in
^ See the present author's articles on Wallis in the Mathematical
Gazette, Dec. 1910 and Jan. 1911.
58 ALGEBRA
accordance with the modern tendency to " arithmetize
mathematics " than the traditional method of the text-books.
Nevertheless, as the latter illustrates important points in the
general theory of algebra it is offered to the matm-er student
in the first section of Part II.
The exercises of Section III depart from history in another
important respect. Napier invented logarithms in order to
lighten the labour involved in the long multiplications and
divisions of spherical trigonometry. The disadvantage of
adopting this starting-point in teaching is that for some time
it is easier for the student to perform multiplications and
divisions by arithmetic than to find the results by logarithms.
If the numbers are large enough to make the arithmetical
operations troublesome they are too large to be dealt with
satisfactorily by the four-figure logarithms with which the
student usually begins. It seems better, therefore, to com-
mend logarithms to the beginner by showing their application
to problems which could not be solved by ordinary computa-
tion. In accordance with this view the earlier exercises of
Section III are given to the study of " growth problems " in
which a magnitude is contemplated as increasing or decreasing
in accordance with the geometric or " compound interest "
law. Problems of this kind, involving fractional periods
of time, cannot well be solved except by logarithms or by
graphic methods which lead directly to the conception of
logarithms. Such a method of procedure has the further
advantage of directing attention naturally to important
financial and social phenomena in which the logarithmic and
" exponential " functions are exemplified. Thus it becomes
possible, in the last exercise of the section, to introduce the
student to that important entity " e " in circumstances which
are calculated to give him sound ideas about its real
significance.
§ 5. Supplementary Exercises. — The supplementary exer-
cises may be regarded as an appendix to Sections II and III,
developing stiU further the topics of those sections and, in
some cases, bringing them into relation with one another.
Their position in the book is meant also to imply that their
contents have not the same fundamental importance as those
of the earlier sections. They may without great harm be
reserved, wholly or in part, for brighter or older pupils. In
Ex. LXII the "sum and difference " formulas of the trigono-
INTRODUCTION TO PART I 59
metrical ratios are investigated. In Exs. LXIV and LXV
these are used to complete the earlier study of functions of the
second degree and their graphic forms. Exs. LXVI and
LXVII supplement certain statistical notions acquired as far
back as Ex. XXVI, and (taken together with Ex. LXIX)
complete the elementary treatment of the calculus by ex-
emplifying some important applications of its methods. Ex.
LXVIII collects a number of results scattered through previ-
ous exercises and generalizes them into the binomial theorem.
§ 6. Order of Study. — The teacher may welcome some
suggestions about the order in which the exercises should be
taken. The schemes printed on p. 50 suggest two alternative
orders each of which has certain advantages. It is assu^med
in each that two weekly lessons are given to algebra. The
first scheme is probably the more suitable for a class which
begins the book at a fairly early age and with no previous
knowledge of the subject. It subdivides the exercises in the
first group of Section I upon a plan which carries the pupil
quickly from the first lessons upon the formula to the lessons
on " changing the subject " — i.e. to simple equations. Fac-
torization, approximation-formulae and the simplification of
easy fractions are, upon this plan, studied in a parallel series
of lessons. A similar subdivision is made in the second
group of Section I. This has the effect of bringing together,
on the one hand, the exercises which deal with the various types
of proportionality, and, on the other hand, the exercises which
cover the first year's programme in " numerical trigonometry ".
Scheme B is more suitable for those who begin the study
of algebra at a later age or with previous knowledge of the
earlier parts of the subject, and may also be followed by
teachers who wish to reach as quickly as possible the doctrine
of positive and negative numbers. It differs from the former
scheme in prescribing the first group of exercises in Section II
to be taken side by side with the second group of Section I.
The first group of Section I may be taken either seriatim or
upon the plan indicated in Scheme A. It will be observed
that according to both arrangements Section III (Logarithms)
is to be worked simultaneously with the second group of
Section II. Reference to the table of exercises on p. 158 will
show that with this sequence the study of logarithms and
fractional indices follows shortly after the consideration of
positive and negative indices in Exs. XXXIII and XXXIV.
00 ALGEBRA
§ 7. Public Examinations. — Speaking generally the course
of Part I covers, and indeed exceeds, the requirements of
school-leaving examinations and tests of similar standard.
The teacher should, hov^ever, note that, for reasons assigned,
certain topics, often included in the syllabuses of these ex-
aminations, are omitted. There is no treatment of the
"imaginary" roots of equations nor of permutations and
combinations in algebra, nor of circular measure and the
ratios of angles of unlimited magnitude in trigonometry. The
author hopes and believes that in these exclusions he has
merely anticipated the action of the examining authorities.
Meanwhile the teacher who deems it necessary to deal with
these topics in the general course may easily supplement
Part I by the few exercises in Part II in which they are
treated.^
^ This is, perhaps, a suitable place for a note of acknowledgment.
The author has long recognized the importance of distinguishing
clearly between the algebras of "non-directed" and "directed"
numbers. In pubhshing a scheme of work based largely upon this
distinction he has, however, been anticipated by Messrs. Barnard
and Child in their New Algebra. The pedagogical ideas embodied
in the New Algebra appear in some respects to be very different
from those of the present author ; he is the more glad, therefore,
to confess his admiration for a work which exhibits so much ability
and sincerity.
SECTION I.
NON-DIEEOTBD NUMBERS.
THE EXERCISES OF SECTION I.
*^* The numbers in ordinary type refer to the pages of Exercises
in Algebra, Part I ; the numbers in heavy type to the pages of this
book.
EXBRCISB
I. Thb " Shoethand " OF Algebra
II. Graphic Representation
III. The Writing of Formula
IV. The Reading and Use op Formula
V. Factorization (I) .
YI. Factorization (II)
VII. Square Root .
VIII. '' Surds "
IX. Approximation- Formula (I) .
X. Approximation-Formula (II)
XI. Approximation-Formula (III)
XII. Fractions (I) ....
XIII. Fractions (II)
XIV. Changing the Subject op a Formula (I)
XV. Changing the Subject of a Formula (II)
XVI. Supplementary Examples
XVII. Direct Proportion ....
XVIII. The Use of the Tangent-Table .
XIX. The Use of the Sine- and Cosine -Tables
XX. Some Navigation Problems .
XXI. Relation of Sine, Cosine and Tangent
XXII. Linear Relations .....
XXIII. Inverse Proportion ....
XXIV. Proportion to Squares and Cubes
XXV. Joint Variation
XXVI. Supplementary Examples
pages
1,
25
4,
31
13.
63
23,
67
32,
82
39,
87
44,
90
46,
93
48,
72
53,
72
55,
72
51,
96
63,
98
70,
104
74,
io6
81,
79
103,
117
108,
121
112,
124
118,
129
120,
132
122,
136
126,
145
129,
149
132,
I.S4
135,
114
CHAPTER VI.
THE PROGEAMME OF SECTION I (EXS. I-XIV).
§ 1. The Cultivation of the Formula. — Formal work in
algebra — as distinguished from the incidental use of symbol-
ism in arithmetic and elementary science (ch. iii.) — is here
planned to begin with lessons intended to cultivate the formula
as an instrument of mathematical statement and investigation.
When it is considered how essential is their use in a vast
range of trades and professions — from plumbing to Dread-
nought building — it is hardly extravagant to say that facility
in the working, interpretation and application of formulae is
one of the most important objects at which early mathematical
studies can aim. A beginning at this point secures, therefore,
the tactical advantage of giving the pupil his first view of the
subject on its most obviously useful side.
The cultivation of the formula involves four distinct ele-
ments : {a) practice in analysing arithmetical processes and
rules of procedure ; (6) practice in symbolizing the results of
analysis ; (c) practice in interpreting given pieces of symbol-
ism ; (d) practice in " substitution ". The first two constitute
the art of formulation ; the second two the art of using formulae.
It is advantageous to give separate study to these two sides
of the work. For this reason Exs. I and III are devoted
almost entirely to formulation and Ex. IV to the reading of
formulae and to substitution. It is not intended, however, that
the whole of Ex. Ill should be worked before Ex. IV is begun.
After a good start in formulation has been made the two
exercises should be carried on concurrently.
§ 2. Formulation (Exs. I, III). — Ex. I contains very simple
examples of formulation. We have seen that it is to be re-
garded as representing the ground covered by the preliminary
work described in ch. in. Where the plan there recom-
mended has been followed, Ex. I will serve as revision ex-
63
64 ALGEBRA
amples preparatory to the formal study of the subject. In
this case the exercise may well be taken orally or set for
homework, with or without preparation in school. If the
simple uses of formulae have not yet been taught Ex. I
should be worked carefully according to the methods of
ch. in.
These methods hold good equally for Ex. III. That is, the
object in each of the examples is (i) to formulate in words the
mathematical relation or the rule of procedure which the ex-
ample illustrates,^ and (ii) to reduce this verbal expression to
a symbolic or "shorthand " expression. The quantity repre-
sented in symbolism on the left-hand side of the sign " = ''
may conveniently be called the " subject " of the formula.
Thus, in the first of the formulae in the footnote '* the tangent
of the angle a " is the subject ; in the second formula, " the
volume of a cylinder ". The term is useful for at least two
reasons. It reminds the pupil that his formula is always a sym-
bolic rendering of a verbal statement ; and it helps the teacher
to exact precision in verbal formulation before the symbolic
transcription is made. The teacher should never neglect
this opportunity of cultivating accuracy and directness of
statement.
In some examples (e.g. Nos. 21, 22) before a formula is
demanded a numerical instance is given. This is, of course,
intended to help the pupil to formulate the essential rule. In
other cases where he cannot formulate the rule without such
an instance he should invent one for himself. The numerical
example should not be worked out to its result but should
be set down in such a way that all the steps involved in ob-
taining the result are clearly exhibited. As a rule these steps
will be taken one by one without any clear consciousness of
the plan of procedure as a whole. The pupil should then
analyse the working so as to make himself clearly cognisant
of the details of the general plan, apart from the particular
numbers in which, in this case, the plan is realized. The
^ A formula can usually be regarded as stating both a mathe-
matical relation (i.e. a numerical identity underlying diverse
equivalent forms) and a rule of procedure. In most cases, how-
ever, one of these ways of looking at it is more natural than the
other. Thus the formula tan a = sin a/cos a suggests most readily
a fact of relationship, the formula V = irr% a practical rule. Cf .
the remarks on the sign of equality, ch. i. § 3.
THE PROGRAMME OF SECTION I 65
formula is, as we have already seen, nothing more than a
statement of this plan in accordance with a conventional
system of symbolism.
The formula obtained by analysis is, of course, not a de-
scription of the special case as such (ch. i., § 2). The special
case is used merely as a challenge to the student to formulate
the procedure which he would be bound to adopt in any case.
As he gains experience it should become less necessary to
begin with a numerical instance. At each step in the evolu-
tion of the formula he should determine what he would do
next if he were concerned with a particular case, and should
write down, symbol by symbol, his statement of procedure
without needing to have the special case as a whole before
him. Finally he should cease to be conscious that he is
making any appeal at all to numerical instances and should
handle his symbols exactly as if they were figures ^ (ch. i,, § 5).
But it cannot be urged too emphatically that the best way to
give the student this degree of mastery over symbolism is to
allow him first to obtain the full value of the lower degrees.
The examples of Ex. Ill have been selected to give
occasion for introducing, one by one, the simpler forms of
algebraic symbolism. Thus they necessarily illustrate some
of the most important types of numerical fact which the
world presents to the mathematician for analysis. Our con-
ception of algebra dictates that the pupil should have some
real acquaintance with these facts in their concrete settings
before he is asked to bring his analytical symbolism to bear
upon them. It follows that the order of the examples is to
some extent arbitrary ; for it must depend partly upon the
extent of the pupil's familiarity with their subject-matter.
Thus in a school where practical mensuration and simple
physical measurements have an early place many of the
examples placed later will seem easier than their predecessors,
because the class is familiar with the facts and processes to
be analysed. The teacher should, therefore, consider the
examples as a whole and should take them in the order most
appropriate in the circumstances.
That the pupil shall always in the algebra lesson feel that
1 Example : I buy an article for £a and sell it for £6. My gain
is, therefore, ^ — ^^-^ per cent.
T. 6
66 ALGEBRA
he is face to face with something in the real external world
and that his business is to give in symbolism an account of
its behaviour in its numerical aspect — this principle is of
quite fundamental importance. The teacher should keep it
prominently before him at every stage. Thus whenever the
facts underlying an example in Ex. Ill are unfamiliar some-
thing should be done to make them real to the class before
the analysis is attempted. In some cases the exhibition of a
model or a piece of apparatus will be possible and is the best
means of producing understanding. In other cases the same
result may be reached by putting the bare facts in a pictorial
setting. For example, finding his boys bafiled by the
relatively abstract statement of No. 3, the teacher may speak
to the following effect : You are on the side of a hill and you
notice a number of policemen, one at the top and others at
different points in the hedge along the road. Suddenly a
motor car appears on the summit of the hill. The policeman
stationed there looks at the car and at once writes something
in his note-book. The car descends the hill, of course with
increasing speed. Exactly a minute after its first appearance
it passes the second policeman, after another minute the
third, and so on. Each policeman makes an entry in his
note-book. Afterwards they gather together and you hear
what they have noted down. The first one says, " He passed
me at 14 miles an hour " ; the second, " He passed me at 17
miles an hour " ; the third, " He was going 20 miles an hour
when he passed me ". What will you expect the fourth man
to say if the rule shown by the first three statements con-
tinues to hold good? . . . the fifth? . . . the tenth? etc.,
etc.i
In conclusion the reader is reminded that an expression of
the form a - b implies throughout Section I that a smaller
number is to be subtracted arithmetically from a larger. If
the number intended by b becomes larger than the number
intended by a the symbolism must be held to be no longer
applicable. If the difference is still to be taken the operation
of taking it must now be represented by the symbolism b - a.
Ex. Ill, Nos. 28 and 32 give instances of formulae in which
a reversal of the symbolism is required by a change in the
^ This story is not offered as a veracious ' ' word-picture " of a
police-trap !
THE PROGRAMME OF SECTION I 67
values of the numbers. In Section II (Exs. XXVIII, XXIX)
it is shown that the device of using numbers with signs en-
ables us to include these different cases in a single symbolic
expression.
§ 3. The Beading of FormulcB. Substitution (Ex. IV). —
The correlative of the power to express a verbal statement
in algebraic symbolism is the power to retranslate such
symbolism into verbal terms. Most of the examples of Ex.
IV lend themselves to the cultivation of this power ; some are
specially intended to exercise it. It is a good plan to have a
number of the formulae " translated " round in class as the
sentences of a Latin or French exercise are translated — the
same care being taken to secure a ready and exact rendering.
But the chief object of Ex. IV is to cultivate the art of sub-
stitution. The principles of this art are simple and have been
sufficiently discussed in ch. iii., § 5. We pass on, therefore,
to note that, as far as possible, the formulae have been chosen
for the interest of their subject-matter as well as for their
value as exercises in substitution. It is not difl&cult to har-
monize the two conditions, for the symbolic forms which have
most frequent practical use naturally offer the most profitable
field for exercise in substitution. Many of the examples in
substitution have, therefore, been drawn from Molesworth's
Engineer's Pocket- Book and similar formularies. In some
cases (e.g. Nos. 3, 6, 15) the formulae admit of simple practi-
cal applications. A few such applications will do more to
illustrate the value of algebra than many formal lessons. The
teacher should, therefore, make as much use of them as
possible. Many more examples of this type could, of course,
be given, but it is advisable to avoid anticipating formulae
which the pupil may encounter in his later studies in science,
etc. This principle limits the selection to empirical formulae
and rational formulae which lie outside the ordinary scope of
school work.
Graphic methods enter in two distinct ways into the
solutions of these examples. In some cases (e.g. Nos. 13, 2l)
the object of the graph is to give a conspectus of the particular
numerical facts covered by the formula. Such a graphic
presentment of the results of successive substitutions conveys
a fuller meaning than can be gathered easily from the numbers
themselves. Moreover, a graph based upon a sufficient num-
ber of results obtained by direct substitution can be used to
6*
68 ALGEBRA
save the labour of further substitutions. This function of
the graph has still more obvious usefulness when, except in
certain cases, direct substitution is impossible. Thus in Nos.
6, 7, 8, it is easy to obtain results by substituting under the
radical sign numbers, such as 16, 81, etc., which are per-
fect squares; but for boys who have not learnt the square
root process other substitutions are far from easy. It is pos-
sible by trial to hit upon a number which when squared
comes (for example) approximately to 75, but the process is
inconvenient and uncertain. The class should, therefore,
draw a careful graph in which the abscissae are the numbers
from to 100 and the ordinates are their square roots.
Points representing the square roots of 1, 4, 16, 25, etc.,
should first be plotted and a smooth curve drawn carefully
through them. (Cf. ch. iv., § 6.) It is obvious that the
square root of a number 100 times as large as a given num-
ber will be 10 times the root of the latter. For example, the
square root of 2600 is 10 times the square root of 26. This
principle will be needed again in Ex. VII.
The teacher should take occasion to point out that finding
a square root is, like finding a quotient or guessing a riddle,
an inverse process.
The arguments of ch. iv., § 6, apply to both the uses of the
graph described in this section, for in each case we start out
with a formula and are seeking its graphic expression. The
curve based on a few calculations must be tested by confirm-
ing the results of random interpolations. If each member
of the class applies such a test successfully the whole volume
of the evidence may clearly be accepted as establishing the
claim of the " smooth curve " to be the graphic representative
of the formula.
§ 4. Factorization (Exs. V, VI). — With Ex. V begins the
study of a number of topics that may be regarded as de-
velopments of the art of substitution. The first is factoriza-
tion. As usually taught factorization is not a very elevating
exercise. The beginner can hardly appreciate its value as an
introduction to the study of " algebraic form ". It can be
justified to him only as a means of doing very simple multi-
plications and divisions in the head. But, since he has
probably never seen any reason for doing these multiplications
or divisions, factorization has, in effect, to be accepted as one
more " rule " — a trick, amusing or depressing according to the
THE PROGRAMME OF SECTION I 69
skill and temperament of the student. The optimist may, in
addition, believe vaguely that it is a trick which some day
may prove to have a use.
On the principles of this book, the usefulness of factoriza-
tion should be made clear at the outset. Nothing is easier,
for " identities " such as ac ± be = {a ±h) c and a^ - 6'^ =
{a + b) {a - b) have this obvious value : they can be used to
reduce the labour of arithmetical computation. Here, then,
is a clear reason for inviting the pupil to study them. There
need be no fear — here or elsewhere — that the grossness of the
utilitarian motive will destroy his sensitiveness to " algebraic
form ". On the contrary, in accordance with a universal
psychological law, he will arrive all the sooner at a genuine
interest in form if he sees how significant it may be from the
point of view of economy of thought and labour.
The factorization of an algebraic expression is, then, to be
taught, in the first instance, as a device which enables us to
throw a formula into the shape most suitable for substitution.
This principle decides both the range and the mode of treat-
ment of the identities to be studied. They will be limited to
the forms ac ±bc = {a ±b) c and a'^ - b^ = {a + b) {a - b)
together with forms readily derived from these. A kindred
principle justifies the addition of the expansions of (a±bY
and {a±bY to the young algebraist's armoury. All other
identities are best postponed to a later period.
Detailed suggestions for teaching these identities are given
in ch. VII. Here we need note only the following points.
(a) The identities are introduced by instances in which their
truth and their labour-saving virtue are obvious. This ac-
counts (for example) for choosing the problem of Exercises,
fig. 19, as the starting-point of ch. vii., B, instead of that
of Exercises, fig. 24. {b) The process called " multiplication "
is subsequently employed to show that the identities hold good
universally (ch. i., § 5). (c) The formal elaboration of the
identities is itself guided by the requirements of practical
problems. This point is illustrated in Ex. V, Nos. 5, 14,
and Ex. VI, Nos. 6, 7, 24. {d) It is an excellent plan to
have copies of the figures of Exs. V and VI cut from card-
board or metal and to circulate them or use them with the
class as a whole. Models of the solids of which these figures
are cross- sections are still more useful, and to construct them
in any suitable material is a valuable exercise.
70 ALGEBRA
The general principle followed in ch. vii. is to start with
certain obvious geometrical relations and to base upon them
an algebraic identity. This method is a reversal of the usual
plan in which the identity is used to " prove " the truth of
the geometrical propositions. Nevertheless it has historical
justification. In the absence of a convenient numerical
notation the Greeks were unable to develop an effective
symbolic algebra (ch. i., § 5). A good deal of what we call
their "geometry" was, as we have seen, really intended
to supply this defect. Thus in Euclid, Bks. V and VII-X,
the central interest is not in geometrical but in arithmetical
analysis. Strictly speaking, lines and figures are there
employed as the most convenient medium for expressing
general arithmetical truths; in other words, these books
contain largely a graphic algebra. Now, though the graphic
presentation of a law or a relation is in the long run not so
effective as its presentation in a formula (ch. iv., § 1), yet
it has great merits, especially in the earlier stages of the
subject. It presents the law in a less abstract medium than
the symbolism, so that the untrained mind can more easily
grasp it. This advantage of a graphic presentation is especi-
ally marked in cases like those of Exs. V and VI where the
geometrical relations can be demonstrated by actually moving
parts of a figure constructed of paper or cardboard. The
teacher may be surprised to find how strikingly superior
this method is to the mere contemplation of a blackboard
drawing — superior as leading both to a much more ready
discovery of the geometrical truth and a much more vivid
appreciation of it. These psychological facts underlie the
treatment of ch. vii. The algebraic presentation of the
identity is based upon the graphic presentation, and each
step in the manipulation of the symbols is simply a record of
a corresponding manipulation of a tangible figure.
§ 5. Square Boot (Ex. VII). — The calculation of square roots
may be regarded as a process subsidiary to substitution. It
enables the boy to find by a simple computation results that had
to be found in Ex. IV by graphic interpolation. Ch. viii. sug-
gests a method of evaluation based on the principles of the
preceding section. It is practically identical with that de-
scribed about A.D. 365 by Theon of Alexandria.^ It has two
^ See an article and letters in the School World for January,
February, and April, 1911.
THE PROGRAMME OF SECTION I 71
great advantages, (i) It is thoroughly heuristic; i.e. the
class may be expected to suggest spontaneously all the essential
stages of the process, (ii) It is easily remembered or recon-
structed if forgotten — a fact which illustrates the superior
teaching value of what the psychologist calls " visual
imagery ".1
5^ 6. The Radical Notation. Surds (Ex. VIII).— It is
convenient at this point to consider the devices by which
calculations involving the taking of square roots can be made
with the least labour. Under this head we may limit our
consideration to (i) the use of such a form as ^12 as a
symbol to be translated into a numerical value only at the
end of the calculation ; (ii) the obtaining of equivalences such
as
Vl2 = 2 73;
(iii) the advantage of "rationalizing the denominator" in
the case of such a fraction as 3/^7. These operations should
be taught and exemplified in connexion with Ex. VIII. The
more complicated cases in which the denominator is a
binomial surd are reserved to Ex. XXVI, E.
In dealing with this topic the teacher should remember
that a " surd '' such as J2 does not in elementary work stand
for any definite number. It means a number which when
squared will give as close an approximation to 2 as the nature
of a particular problem demands.
If our present inquiry had a purely scientific instead of
a practical aim we should find it necessary to go beyond
this point and to find a definition of J2 that should distin-
guish it from any of the approximations which the arithmetical
process of " finding the square root " can yield. The necessity
for doing so arises from the connexion between number and
geometrical magnitudes. Consider a right-angled triangle
with two sides each of unit length. The length of the hy-
potenuse, AB, of such a triangle can be shown to be J 2.
For practical purposes this may be taken to mean that the
length is 1-41 or 1-414 or 1-414235 . . . according to the
degree of accuracy of measurement possible. Strictly speak-
ing, however, each of these numbers represents a point situated
^ Cf. Report of L.C.C. Conference of Teachers, January, 1911,
discussion on '^Memory".
72 ALGEBRA
at a different distance from A, and however far the process
is continued it will never give the distance from A of the point
B itself. Thus we must either abandon the idea that the
distance AB can be exactly measured at all or else we must
invent a special number to measure it — a number, that is,
which has no place in the decimal scheme. Mathematicians
have proposed different definitions to meet this need. That
of Mr. Bertrand Russell ^ is the latest, simplest and most at-
tractive. Such a symbol as 8 may be taken to represent two
distinct numbers : (1) the ordinary " natural " or cardinal
number eight; and also (2) the whole class of "rational"
numbers, integral or fractional, which are less than eight.
We can then define ^2 as representing the class of rational
numbers which when squared give rational numbers less
than two. It is clear, on the one hand, that the *' irrational
number" thus defined is not a member of the ordinary decimal
scheme, and, on the other hand, that it may be used quite
unambiguously as the numerical label of the point B at the
end of our hypotenuse.^
It is equally clear that this attempt at philosophical pre-
cision would be entirely out of place at the present stage of
the course, and that the pupil should be taught to regard
^2 as the symbol for a number which yields, when squared,
a number differing from two by an unimportant amount.
§ 7. The expansions of (a ± b)^, (a ± b)^. Approximation-
formulcR (Exs. IX-XI). — The square root process of ch. viii. is
based directly upon a study of fig. 17. In Ex. IX, No. 2, the
pupil deduces from this figure the identity (a 4- 6)^, and in No.
l6 shows that it holds good universally. In the other examples
the identity is used to reach an approximate solution of many
problems in which an exact solution is either unnecessary or
out of place because it would give a result too " fine "to be
tested by measurement. The principle involved is very simple.
If BX is (l/n)th of AB (fig. 17) the square Q (fig. 18) must
be (l/w2)th of the whole square BD. For example,
if BX = AB X 0-01, Q = BD X 00001.
Thus if the calculation took account of only two decimal
places the square Q could be neglected. This principle may be
1 Principles of Mathematics, Vol. I, ch. xxxiii.
^ The whole question raised here is discussed in Exercises, Part
II, Ex. LXXII, and in ch. xxxix. of this book.
THE PROGRAMME OF SECTION I 73
simply illustrated by drawing several figures like fig. 16, the
side AB being the same in each while ED is progressively
smaller. The eye witnesses to the still more rapid progress
towards insignificance of the area FD.
In Ex. IX, B, this principle is applied to the approximate
evaluation of square roots. Thus in No. 25 we have
V 18 = 742 + 2.
Taking the area of XY (fig. 17) as 16 the area of the strip
of fig. 19 is 2, and the double base of RE' is 8. If Q is
ignored we can say that the area of R + R' = 2 and the
height BX = 2/8 = 0-25. Thus, ignoring Q, ^18 = 4-25.
In general, if p is small compared with a^ we have
J^F^p = a + ^
as the approximate value of the square root.
Ex. X illustrates the same principles in connexion with the
identity {a - hy = d^ - 2ab + 6^. The only fresh point of
interest lies in the manipulation by which fig. 17 can be
made to exhibit the identity (No. l). Putting
a = AB and 6 = BX
we have that ab is the rectangle composed of R + Q or that
composed of R' + Q. To reduce the square
BD = a2 to XY = (a - by
the following operations are therefore necessary. Take away
R + Q, replace Q, take away R' + Q. In symbols : —
{a - 6)2 = a^ - ab + b^ - ab
= a2 _ 2ab + b^
In Ex. XI the same principles are illustrated in connexion
with the identities (a + b)^. Some critics of authority have
argued against the retention of these identities in elemen-
tary syllabuses, but the demonstration of the former by a
model is so easy and attractive that there seems no sufficient
reason for excluding it. There is another much weightier
reason against exclusion. The natural field of application of the
expansion of {a + by is to the properties of volumes. It
plays the same part here as the expansion of {a + by plays
in connexion with areas. At a time when it is generally
recognized that elementary mathematics should give more
and not less time to tridimensional problems it seems unwise
74
ALGEBRA
to dispense with an effective and simple instrument of spatial
inquiry. Any valid objections to the identity in question are
probably avoided in Ex. XI by limiting the examples almost
entirely to problems on approximation.
The expansion of {a + b)'^ is most easily taught by means
of the model ^ shown in fig. 7. The cube {a + b)^ is to be
built up from the cube a^ by the following additions : (i) three
slabs of dimensions a x a x b ; (ii) three prisms measuring
a X 6 X 6 to fill up the spaces marked p, q, r ; (iii) a cube,
b^, to fill up the space still left empty. Thus we have
(a + bf = a^ + 3a26 + 3a62 + ^,3
The additions are exhibited as a flat slab in fig. 8. (The
letters refer to the problem described on p. 75.) It is evident
that as bl{a + b) decreases the cube b'^ soon becomes insignifi-
7a h
Fig. 7.
Fig. 8.
cant even when compared with 3a&2, the volume of the prisms.
As the fraction continues to decrease, the row of prisms itself
becomes insignificant compared with the three square slabs
^a^b. Thus if b is small enough we have : —
(a + bf = a^ + ^a%
This result makes it easy to calculate a cube root approxi-
mately (No. lo). Let the number be written as a^ + ^ ; then
p = 3a26 and b = p/3a2. Thus
approximately.
Ija'
+ p = a + p/3a-
The same model may be used to exhibit the expansion of
(a - by, (No. 8), by operations similar to those used in con-
nexion with {a - by. It must be possible to remove together
any one of the square blocks with the adjacent prisms and the
small cube, and also to replace the prisms and cube separately.
' The model is most easily made of cardboard or stitf paper.
THE PROGRAMME OF SECTION I 75
Call the whole cube a^ and the depth of the parts to be re-
moved h. Then a^ can be turned into {a - hy by operations
performed in the following sequence : —
(a - hf = a3 _ ci'b + ah'' - a^h + aW- - f' + a¥ - a'b
= a^ - 3a'b + Sab^ - ¥
The model may also be used to discover the identity
a^ - b^ = {a - b) (a-' + ab + b'^ given as No. 89 of Ex. XVI.
Taking the larger cube of fig. 7 as a^ and the smaller cube as
6^, the difference a^ - b^ may be exhibited in the form of
the uniform slab of fig. 8. The thickness of this slab is
a - b, its area Sab + {a - by = a' -\- ab + b'^. Hence its
volume = (a - b) (a^ + ah + b^).
Lastly it should be noted that Exs. IX-XI, important in
themselves as illuminating the meaning and usefulness of
identities which the pupil is apt to regard as barren and rather
irritating abstractions, have a further importance as aiding the
development of the notions upon which the later study of the
calculus must be based. For this reason the teacher will do
well to emphasize the question of the relative importance of
the first, second and third powers of a small number and to
illustrate it, when possible, by models. A series of models
like fig. 7 with varjnng proportions of i to a is very useful for
this purpose.
§ 8. Fractions (Exs. XII, XIII). — The proper significance
of the process called simplifying an algebraic fraction has been
pointed out in ch. i., § 5. Its aim is to predict by the mani-
pulation of symbols the result that would be reached by
simplifying an arithmetical expression containing fractions of
a given type. Thus an " algebraic fraction " is, strictly speak-
ing, no fraction at all. It is only a shorthand description of
a certain class of actual (i.e. arithmetical) fractions. For
instance aj {a + 5) is a shorthand description of the class of
fractions in which the denominator is greater by 5 than the
numerator.
The discussions of ch. ix. place the study of algebraic
fractions upon the same basis as that of factorization. It is
" worth while " because by " simplifying " a fractional formula
we can generally turn it into a form more suitable than the
original for purposes of computation.
In Section I the range of fractions studied is limited to
those with simple binomial denominators and monomial
76 ALGEBRA
numerators. Skill in manipulating such fractions suffices foi-
the solution of most ordinary problems of real interest. On
the other hand occasion is taken to complete the doctrine of
approximations built up in Exs. IX-XI. Thus in Ex. XIII,
No. 15, we have : —
1 + -gL- ^ (1 - g) + « ^^j „ ^ a' _ {a- a') + a'
1 - a 1 - a 1 - a 1 - a
1 a
1 - a 1 - a
whence _, = 1 + ... (i)
I - a 1 - a
= 1 + a+ -^ . . . (ii)
1 - a
When a^ is negligibly small this relation becomes
1/(1 - a) = 1 + a.
Under the same conditions
1/(1 + a) ^ 1 - a
(No. 16). To obtain a " second approximation " (No. 31) we
use the relation a'^ + «^/(l - a) = «V(1 - «)• Substituting
for a7(l - a) in (li) we have : —
— A— = 1 + a + a- + ^^ — . . . (iii)
1 - a 1 - a
Thus to a second approximation 1/(1 - a) = 1 + a + a^.
The process could evidently be continued indefinitely.
Fig. 9. Fig. 10.
Fig. 11.
The examples of Ex. XIII, B, give a number of instances
of the usefulness of these interesting approximations. Figs.
9-11 show a simple method of exhibiting to the eye the degree
of approximation reached in a given case. Fig. 9 represents
THE PROGRAMME OF SECTION I 77
the case of 1/(1 - a) when a = |. The whole strip measures
the full value of the fraction, i.e. 2. The shaded part
measures the value of the approximation 1 + a, i.e. 1-^. The
case when a = | is dealt with similarly in fig. 11. Fig. 10
carries the case a = ^ to a second approximation. The
teacher will recognize in the whole treatment a useful pre-
paration for the study of geometrical progressions (Ex.
XXXVI).
§ 9. Changing the Subject of a Formula. — Exs. XIV, XV
bring us to the important subject usually described as the
solution of simple equations. The procedure advocated in
oh. X. differs materially, both in spirit and method, from the
current treatment of this central topic of elementary algebra.
In history equations began as conundrums, and the school
tradition has not lifted them to a much higher level of
intellectual dignity. The pupil may become skilful in com-
pelling " ic " to reveal the value hidden in a symbolic state-
ment of baffling complexity ; he may become acute in thread-
ing the intricate mazes called " problems " which the ingenuity
of the text-book writer has set in his path. Yet in the end
he may still be only an expert solver of conundrums. He
may have gained but an imperfect idea either of the practical
or of the scientific importance of processes which he has
learnt to handle for merely artificial purposes. We may well
agree with Dr. Whitehead ^ that " one of the causes of the
apparent triviality of much of elementary algebra is the pre-
occupation of the text-books with the solution of equations ".
Nevertheless it would be bad tactics to ignore the peda-
gogical value of the conundrum. In all its varieties — from the
riddle to the tragic " mystery " — it may be a powerful
stimulus to intellectual activity. Thus in ch. x. the treat-
ment is openly based upon the attraction of the conundrum,
but this attraction is used to beguile the pupil into a study of
processes which are immediately put to a more serious use.
The nature of that use was explained in ch. i., I^ 5, where the
practical value of the rules established in ch. x. was shown
to consist in their power of leading us from old truths to new
by an infallible mechanical process. By these rules any
^ Introduction to Mathematics (Home University Library),
p. 18.
78 ALGEBRA
variable which enters into a formula can be made the subject
of the formula. For this reason the process may appropri-
ately be called " changing the subject of a formula ".^
Two points in the exposition of ch. x. should receive
special attention. The first is that the rules for removing a
number from one side of the formula to the other are not
based upon any axioms of equality. This departure from
traditional procedure needs but little justification. A boy is
told that when 7 is added to a certain number the sum is 12,
and at once states that the unnamed number was 5. It will
not be pretended that he reaches this result by reflecting that
" if equals be taken from equals the remainders are equal,"
nor that if he could not reach it unaided an appeal to that
axiom would help him to conviction. Children can, in fact,
solve such concrete riddles years before they can appreciate
the abstract axiom. ^ Moreover the pupil will see with per-
fect clearness that the mode of solution of this problem —
" take 7 from 12 and you have the other number " — is
perfectly general , that is, that its validity as a process does
not depend upon the specific numbers involved in it (ch. i.,
§2).
These psychological considerations point the teacher to the
plan followed in ch. x. The pupil first solves mentally a
series of simple arithmetical problems. He is then led to
analyse his solutions in symbols in order to make clear the
principles underlying them. Finally he formulates the results
of this analysis in "rules" of universal application (ch. i.,
The mode of transition from solving a numerical riddle to
changing the subject of a formula is the second point of im-
portance. The reader will see that it is a particular applica-
tion of the doctrine that symbols do not stand for numbers
^ Since this phrase has already obtained a certain amount of
currency, the author may be permitted to claim here the modest
credit of its paternity. He believes that it was used for the first
time in his lectures to teachers of mathematics in 1909. It was
subsequently adopted in the Report on the Teaching of Algebra by
the Committee of the Mathematical Association.
"Cf. the Board of Education's Circular on the Teaching of
Geometry and Graphic Algebra. On the significance of axioms in
algebra see the present writer's article in the Mathematical
Gazette, for January, 1912.
THE PROGRAMME OF SECTION I 79
but describe them (ch. i., ^ 3). There is, perhaps, no point
at which the practical value of that doctrine is more clearly-
shown. The class whose teacher adopts the method given
in ch. X., § 3, will find little difficulty in " literal equations "
or in the much more interesting and important exercise of
changing the subject.
Numerical results obtained as answers of the problems in
Exs. XIV, XV should, of course, always be tested by sub-
stitution. In Ex. XIV, B, Nos. 32-7, the pupil is instructed
to test also any formula derived from a given formula by
changing the subject. Thus in No. 32 he is, from the formula
W = 6 + mn, to derive the formula n = (W - 6)/m by mechan-
ical application of the rules of ch. x. He is then to think out
afresh the problem of finding the number of marbles in a bag,
given W, b and m, and to observe that he obtains the same
formula as before. The object here is not to confirm the
original answer so much as to convince the pupil of the labour-
saving virtue of the rules. It is, in fact, another signal advan-
tage of these exercises that they illustrate so clearly the
" economy of thought " at which mathematics constantly aims.
" Civilization advances by exteading the number of important
operations which we can perform without thinking about
them."i
Finally it should be noted that the word "equation" is
avoided throughout Section I. There seem good reasons for
withholding the term until Section II, where, with the introduc-
tion of directed numbers, it becomes appropriate to use the
typical form / (a;) = and to associate a new technical name
with it (Ex. XXXVII).
§ 10. Bevision (Ex. XVI). — Changing the subject of a
formula completes the tale of the fundamental algebraic
operations. Ex. XVI is, therefore, given to the revision and
extension of the results gained during this first stage of the
pupil's progress. The examples are classified into seven
groups, A to G. A and B are rather harder examples of
formulation and substitution and need no comment. In C
the pupil is confronted with a number of the numerical
puzzles and odd relationships which invariably tickle the
interest of boys and girls both in and out of school, and is
given the task of finding out how they " work ". The skilful
^Whitehead, Introduction, p. 61. See also Mach, Science of
Mechanics, passim.
80 ALGEBRA
teacher will find it possible, by a wise use of the gaiety ap-
propriate to the theme, to make valuable use of these examples
in exercising the analytical powers of his class. The method
described in ch. vi., § 1, is the one generally to be followed :
the pupil should work out a particular case of the puzzle and
should then analyse his solution in symbols. On the other
hand, if he is able to write down his analysis without previous
consideration of a particular case he should, of course, be
encouraged to do so.
Examples D give further exercise in graphic methods
already studied in Ex. I. An account of all these methods
will be found in ch. iv. No. 6o is particularly useful be-
cause it introduces the third dimension. The teacher is
recommended to take some pains, if necessary, to secure'
careful solutions of this example.
There are no novelties in E until No. 87 is reached where
the pupil is introduced to the method of " completing the
square " which is afterwards to be used so frequently in the
solution of quadratic equations and in other problems. The
note before No. 87 gives sufficiently fully the argument which
is intended to elucidate the process. It will be found a good
plan to cut out large copies of the figures and to make
members of the class actually carry out the " completion of
the square " upon the blackboard. In No. 88 the process is
to be applied in order to calculate the length of the side of the
square in given circumstances.
Examples E may be passed over as involving no new prin-
ciples, and we may turn at once to the important argument
developed in examples G, Nos. IO5-II8. From the stand-
point of the usual classification these are examples of
"quadratic equations". The teacher is, however, strongly
recommended to resist the temptation to regard them as in-
stances of a new " rule," but to treat them simply as cases in
which the ordinary rules for changing the subject of a formula
require supplementing by a little additional manipulation.
The stages in the evolution of this manipulation are marked
in Nos. 105, 109 and 110. In No. 105 the first step is to
change the subject of the formula to v^ ; the second, to obtain
a formula for v by simply taking the square root of the right-
hand side. These steps are so obvious that the pupil will
take them without prompting. No. 109 requires a slight
variation in the method ; the necessary steps are
THE PROGRAMME OF SECTION I 81
(s4-a)2=.A
a= ^A-s
Finally, in No. 1 10 the process of " completing the square "
must be invoked in aid before the method of the last example
can be applied. Thus we have
a2+12a + 36 = 100
(a + 6)2 = 100
a + 6 = 10
It will be noticed that there is no indication in these so-
lutions of the ** ambiguity of sign " of a square root. The
explanation is that as long as we are confined to non-directed
numbers there is no such ambiguity, for the simple reason that
the numbers have no signs. It will, indeed, be readily seen
that there cannot be two answers to the question, "What is the
length of the side of a square of given area ? and the suggestion
that there are two possible values for the square root would
in this case be only misleading. Nevertheless it is easy to
find cases in which a problem of this type has two answers,
though both are non-directed numbers. The note before No.
Il8 explains the circumstances in which double answers are
possible. It will be noted that the ambiguity which gives rise
to them concerns, not the sign of a square root, but the order
of the two terms of a binomial when the sign connecting them
is a minus. It is extremely desirable to defer the considera-
tion of the former type of ambiguity to the stage when the
pupil is ready to study the " parabolic function " ax'^ + bx + c
in which the variables are assumed capable of positive and
negative values, and to reserve the name " quadratic equation "
for the problem represented typically by the symbolism
ax'^ + bx+c =
Problems of these kinds will come before us in Exs. LXII
and LXIII.
T.
CHAPTEE VII.
FACTORIZATION.
A, Factorization of ac + be (ch. vi., § 4 ; Ex. Y)}
^ 1. The Identity ac + be = (a + b)e discovered. — The
E F
Fig. 13.
Fig. 14.
figure AF (fig. 12) is a plan of two rooms whieh have different
lengths but the same breadth. Suppose AB = 36 feet, BC =
^ A piece of cardboard shaped like fig. 14 should be prepared
and should, before the lesson, be cut half-way through along the
line 6E so that C6 may be readily detached and placed in the
position of the dotted rectangle in fig. 13.
82
FACTORIZATION 83
22 feet and AD = 19 feet. Then the joint area of the two
rooms would be found by the calculation
A = 36 X 19 + 22 X 19
That is, it would be found by doing two multiplication sums
and adding the results.
But we can calculate the joint area in a shorter way than
this. Suppose the partition BE to be removed. Then instead
of two rooms we shall have one room, 58 feet long, whose
area must be the same as the sum of the areas of the original
two rooms. (The partition may be supposed so thin that no
allowance need be made for the area it stands on.) Thus the
calculation is reduced to a single multiplication. The steps
by which the reduction has been brought about can be written
down as follows : —
A = 36 X 19 + 22 X 19
= (36 + 22) X 19
= 58 X 19
The working of such a problem will always be shortened by
this device, and the shortening may sometimes be very great.
Thus if we wanted to know the total area of two rooms whose
lengths were 63 feet and 37 feet respectively while they were
both 29 feet wide, we should have : —
A = 63 X 29 + 37 X 29
= (63 + 37) X 29
= 100 X 29
= 2900 square feet.
In this example the working is so much simplified that it can
easily be done in the head.
The same plan could be used for shortening the work if we
had to find the area of two rooms arranged as in fig. 14, or
of a single room shaped like fig. 13. For since GE is of the
same length as BE the rectangle FE can be supposed cut off
at GE and placed in the position shown by the dotted rectangle.
The area of the whole rectangle AF' can then be calculated
as before.
The rule we have just found can be used, of course, in
calculations concerning any two rooms or other rectangular
areas so long as they have one dimension in common. If a
and h are used as symbols of the two unequal dimensions and
6*
84
ALGEBRA
c as the symbol of the common dimension, the rule can be
written thus : —
A = ac + be
= (a + b)c
[Ex. V, Nos. 1 to 5, may now be taken.]
§ 2. The Identity ac - be = (a - b)c discovered. — Let us
turn to a different kind of area-problem. The rectangle AD
(fig. 15) represents the wall of a passage 32 feet long and 12
feet high. The passage is to be wainscoted up to EF, a dis-
tance of 4 feet 6 inches, and to be painted above that height.
How many square feet are there to be covered with paint ?
There are evidently two ways in which this calculation can
be made. One way is to calculate the whole area AD, which
A
C
E
4'S
r
?'
32'
Fig. 15.
is 32 feet by 12 feet, and deduct the area of the wainscot ED
which is 32 feet x 4| feet. Thus : —
A = 32 X 12 - 32 X 4|
But it would obviously be quicker to calculate straight away
the area of AF, whose length is also 32 feet and whose height
is 1\ feet, that is 12 feet - 4-| feet. Setting down the steps
by which the calculation is thus reduced to its simplest form,
we have : —
A = 32 X 12 - 32 X 4i
= 32 X (12 - 4i)
= 32 X 7i
It is clear that this rule can be applied to find the area of any
rectangle that can be considered as the differeace of two
rectangles which have one dimension in common. We
have : —
A = ca - c6
= c{a - h)
FACTORIZATION 85
In the formulae which we have written the letters have
been placed in exactly the same position as the numbers to
which they correspond in the calculations. This is generally
speaking the best thing to do, since it shows clearly the con-
nexion between the formula and the arithmetical working
which the formula describes. But in the present problem there
is no reason why we should take such pains to make our
formula imitate the arithmetic, for we should not always write
the arithmetic down in the same way. For example, the
last calculation might on another occasion have been written —
A = 12 X 32 - 4i X 32
= (13 - 4i) X 32
= 7^ X 32
Corresponding to this the algebraic description of the working
would be
k = ac - he
= {a - b)c
Thus the same calculation can sometimes be set down in
two or more ways each of which can be described in a
formula. Since, however, one of the things which a formula
is meant to do is to serve as a convenient memorandum of a
rule, there is an advantage in writing it always in the same
way. It is usual, therefore, when the symbols can be written
in different orders to choose some definite order such as the
order of the alphabet. Thus we shall generally write ac - be
rather than ca - c6, and {a - b)G rather than c{a - b).
Sometimes, however, it may be convenient to reverse the usual
order and, of course, it can never be actually wrong to do so.
§ 3. When one of the Rectangles is a Square. — Suppose in
fig. 12 that BF was a square. Then the common dimension of
the two rooms would in this case be the same as one of the
two unequal dimensions — namely the one denoted by the
symbol b. Thus the simplest way to represent the area of
BF is by the symbol b^, that is bb. It is unnecessary, there-
fore, to have another symbol, c, for the common width of the
two rooms, since b will suffice. The rule for shortening the
calculation now becomes : —
A = ab + h'
= {a + b)b
Similarly if we take away a rectangle whose dimensions are
86 ALGEBRA
represented by the symbols a and b from a square the length
of whose sides is also represented by a, we have
A = a^ - ab
= a (a - b)
[Ex. V, Nos. 6-14, may now be taken.]
§ 4. The Identities can be used in all, Calculations. — These
rules are so useful in simplifying calculations of areas that
we shall naturally inquire whether they cannot be used to
simplify other calculations also. It will easily be seen that
they can be so used.
Taking the first rule first, we note that it applies to cal-
culations in which we have to find the sum of two products
that possess one factor in common — for example 13 x 7 +
13 X 3. Now whatever these numbers measure — or even if
they are not meant to be measurements of anything at all — it
is clear that 3 thir teens added to 7 thirteens must give 10
thirteens. That is
13 X 7 + 13 X 3 = 13 X (7 + 3)
Similarly if we had to work out the calculation
13 X 7 - 13 X 3
it is plain that 3 thirteens taken from 7 thirteens will leave
4 thirteens, or that
13 X 7 - 13 X 3 = 13 X (7 - 3)
Moreover we can obviously argue in the same way what-
ever numbers are substituted for the 13, 7 and 3, so long as
one factor in the two products is the same. Thus using
the symbol c to mean now, not necessarily a common length,
but simply a number which is the common factor of two
products, while a and b represent the unequal factors of the
same products, we have
ac + be = {a + b)c
and ac - be = {a - b)c
For greater brevity these two results can be given together in
the form
ac ± be = {a ± b)c
which is to be read " ac 2:)lus or minus be,'' etc.
The expressions which are linked here by the sign = simply
represent different ways of carrying out the same calculation.
For this reason they are often called identities.
[The remaining examples of Ex. V may now be worked.]
FACTORIZATION 87
B. Factorization of z? - b^ (ch. vi., § 4 ; Ex. VI). ^
§ 1. The Discovery of the Identity.— Fig. 16 may be sup-
posed to be the plan of a courtyard. The shape would be a
complete square if there did
not project into it a building
which cuts off from the corner
a smaller square, FD.
Let a = the length of the side
of the larger square and b ^ the
length of the side of the square
FD ; then the formula for the
area of the courtyard is obvi-
ously
A = a^ - h' Fig. 16.
Can we turn this into a form
that would make calculation of the area easier ? Our experi-
ence with the examples of the previous lesson suggests the
inquiry whether it is possible to cut off the rectangle EC and
place it so that with AG it forms one large rectangle. In
order that this may be done either EG or ED must fit on
toBG.
Let us first examine EG. Since FG = a and FE = b,
EG = a - 6. But since BC = a and GC = ED = b, BG
— a - b also. Thus EC can be cut off from the plan and
placed in the position BD'.
We have now one rectangle AD' whose area is the same
as that of the plan. But AC = a -{■ b (since BC is GC)
and CD' = BG = a - ^)
Hence A = a^ _ 52
= (a -{■ b) (a - b)
Thus instead of squaring the two given lengths and subtract-
ing one result from the other we can find the area in question
by multiplying together the sum and difference of the given
lengths. Since in the second method there is only one
multiplication instead of two it will generally be an easier
way of carrying out the calculation. Sometimes it will be
a very much easier way. Thus if the side of the larger
1 A large cardboard copy of ABCDEF (fig. 16) should be prepared
beforehand and half cut through along the line EG, so that at
the proper moment the rectangle EC may be detached and pinned
on the blackboard in the position BD'.
88 ALGEBRA
square is 186 feet long and that of the smaller 86 feet long, we
have : —
A = (186)2 - (86)2
= (186 + 86) (186 - 86)
= 272 X 100
= 27200 square feet.
In this example the calculation is reduced to one that can
be done in the head because one of the resulting factors is
100. It is obvious that there will also be great advantage
in replacing a^ - 6^ ^y (^ _j_ ^^ (^ _ ^^ whenever a and b
represent numbers which are nearly equal. For example, if
a = 97 feet and 6 = 92 feet, we have
A = (97)2 - (92)2
= 189 X 5
= 945 square feet.
[Ex. VI, Nos. 1 to 10, may now be worked.]
§ 2. The Identity can be used in any Calculation. — We
found in the last lesson that the expression ac + be may be
replaced by {a + b)c not only in area-formulae but also in
any other formula in which it appears. For this reason (it
was said) ac + be = {a + b)c is called an identity. It is
evidently important to find whether a'^ - b^ can also be re-
placed by {a + b) {a - b) in any formula.
As before we must examine cases in which the numbers
are not intended as measurements of lengths or of anything
else in particular. For example let a be supposed to stand
for the number 9 and b for the number 6. Then
92 - 62 = 81 - 36
= 45
while (9 + 6) (9 - 6) = 15 X 3
= 45
So that in this case it is true that
92 - 62 = (9 + 6) (9 - 6)
whatever the 9 and 6 measure.
Each member of the class should verify the rule in another
case chosen by himself.
Let us next try to find out how this result comes about.
The best way to do so is to start with (9 + 6) (9 - 6)
and try to show that the multiplication leads to the result
92 - 62.
The expression (9 + 6) (9 - 6) means, of course, that
FACTORIZATION 89
(9 + 6) or 15 is to be multiplied by (9 - 6) or 3. We can
calculate the result by taking 6 times (9 + 6) from 9 times
(9 + 6), for the residue will obviously be 3 times (9 + 6).
That is
(9 + 6) (9 - 6) = (9 + 6) X 9 - (9 + 6) X 6
But (9 + 6) X 9 (i.e. 9 fifteens) is evidently the same as 9
nines together with 9 sixes ; and (9 4- 6) x 6 is the same as
6 nines and 6 sixes. We have, therefore, to take 6 nines
and 6 sixes away from 9 nines and 9 sixes. Let us put this
down in figures : —
(9 + 6) (9 - 6) = 9 X 9 + 6 X 9
-9x6-6x6
= 9^ - 6'^
For 6x9 = 9x6, so that the addition of 6 x 9 and the
subtraction of 9 x 6 cancel one another.
We can deal similarly with any other difference of squares.
For example : —
(13 + 5) (13 - 5) = 13 X 13 + 5 X 13
-13x5-5x5
= 13-^ - 52
Each member of the class should at this point analyse his
own chosen example in the same way. Note that it is con-
venient so to arrange the figures that the cancelling products
come one underneath the other.
It is evident that whatever numbers we take the same re-
sult will always follow. We can, therefore, describe the way
in which the numbers behave by means of symbols. Putting
a and h for the two numbers we shall have : —
{a + b) {a - b) = a^ + ba
- ab - 62
= «2 _ ^2
For whatever number ba represents ab must stand for the
same number. Thus the addition of the number described
by ba is always cancelled by the subtraction of the number
described by ab, so that we are left with the difference be-
tween the squares of the two numbers.
[Ex. VI, Nos. 11 to 25, may now be worked.]
CHAPTEE VIII.
SQUAEE BOOT. SUEDS.
A. The Calculation of Square Boots (ch. vi., § 5 ; Ex. YII).
§ 1. The Simplest Case. — If we know that the area of the
square AC (fig. 17) is 16 square inches, then AB = 4 inches ;
if 25 square inches, AB = 5 inches, etc. But if we are in-
formed that the area is 14 -44 square inches, we cannot tell
at a glance the length of AB. All that we know, at first,
is that AB is between 3 and 4 inches. Mark off AX = 3
B
i
D ''^\
w
VI
J"
J-
3"
R'
£
Y
Pk
Fig.
B
17.
Fig. 18.
08
R 1 R'
1q
oa"
X
V
3" 3"
^0-6
"
6'
Fig. 19.
inches, draw the square XY of area 9 square inches, and sup-
pose it cut out from the square AC. The residue (fig. 18)
can be divided into two equal rectangles, K and R', and a
square, Q, having a total area of 14'44 - 9 = 5*44 square
inches. This residue can be re-arranged as in fig. 19 to
form a long rectangle. The problem is to find XB, the height of
the rectangle. If we knew the base of the rectangle we should
9U
SQUARE ROOT. SURDS 91
divide it into 5*44, the area of the whole strip, and so obtain
the height exactly. We know that the base of
R+R'=3x2 = 6,
so that 6 divided into 5*44 will give us the height very roughly.
The quotient is 0*9, but since this must be too big we will
suppose that the height is really 0*8. On this assumption
the base of the rectangle would be 6*8 inches and the height
0'8 inch. The area would be
6*8 X 0*8 = 5*44 square inches.
But this is exactly the required area. Therefore XB = 0*8
and ^14-44 = 3-8. The various steps are best set down in
the form : —
14-44 (3 + 0-8
^•^^444
5-44
.-. Jl¥U = 3-8
In a similar manner the square roots of 32*49 (57) ; 51"84
(7-2) ; 68-89 (8-3) should be reasoned out.
§ 2. Non- terminating Boots. — Suppose the area of the square
to be 40 square inches. What is the length of AB ? As before
mark off AX = 6 inches, and remove the square XY so as to
leave a residue R + R' + Q = 40 - 36 = 4 square inches.
Since the base ofR+R' = 6x2 = 12, the height XB is ap-
proximately 4/12, i.e. about 0-3 inch. Taking it as exactly
0-3 we have that the area of the strip
R + R' + Q = 12-3 X 0-3 = 3-69 square inches.
This area is less than the required area by
4 - 3-69 = 0-31 square inch.
The state of affairs we have reached is represented by fig. 20.
W^e have removed from the whole
area of 40 square inches the area
of a square whose side is 6-3
inches, and we have a residue
r + q + r' whose area is 0-31
square inch. This residue can in
turn be arranged as in fig. 19, ex-
cept that the three sections of the
rectangle must be labelled r, r'
and q. We know that the base of
r + r' is 6 3 X 2 = 12*6 inches, so that the height of the
Fig. 20.
92 ALGEBRA
rectangle is rather less than 0-31/12'6, i.e. about 0'02 inch.
This value would give for the whole rectangle r + r' + q
the area 12-62 x 0*02 = 0*2524 square inch, leaving a
deficiency of 0-31 - 0*25 = 0*06 square inch. That is,
the area of a square 6 + 0*3 + 0*02 = 6*32 inches in
the side would be only 0*06 square inch less than 40 square
inches. We could carry the approximation still farther in
exactly the same way, but as it is already so close that the
error cannot be exhibited in the figure the labour would be
superfluous. The only further step we need take is to find
whether a square of 6-32 inches, or one of 6*33 inches in the
side would give the nearer approximation to an area of 40
square inches. The residue 0*06 divided by 6*32 x 2 gives
a number less than 0*005. It is clear, therefore, that 6*32
inches is the value to be adopted. The working will be set
down thus : —
40 (6 + 0-3 + 0*02 +
36
12-3 1 400
3*69
12-62 \ 0*3100
0-2524
12*64 1 00576
.*. 740 = 6*32
§ 3. Numbers greater- than 100 or less than 1. — Suppose
our plans all have to be made on a scale of 1 foot to 10 feet.
Every linear dimension would be 10 times as small as in
reality, every area 100 times as small. Thus in the various
cases considered the areas would really be 1444 square feet,
3249 square feet, 5184 square feet, 6889 square feet, and
4000 square feet, while the sides of the squares would be 38
feet, 57 feet, 72 feet, 83 feet, and 63 2 feet. This considera-
tion gives us a rule for finding the square root of a number
greater than 100. Divide the number by 100, find the square
root and multiply it by 10. (This is of course only a ques-
tion of the position of the decimal point.) If it is greater
than 100 X 100, divide it by this number, take the square root
and multiply the latter by 10 x 10, i.e. 100.
Similarly if the number is a decimal less than unity
SQUARE ROOT. SURDS 93
multiply it by 100 or 100 x 100, etc., as the case requires,
find the square root and divide by 10, or by 10 x 10, etc.
[Ex. VII may now be worked.]
B. The Radical Form. Surds (ch. vi., § 6 ; Ex. VIII).
§ 1. Belations between Square Boots. — We have seen (A,
§ 3) that if we know the square root of a number it is un-
necessary to calculate the square root of a number 100 times
as large or as small. The root of the second number is obtained
by multiplying or dividing the root of the former by 10. This
is an instance of a principle that is often useful. Suppose
that we know that ^40 = 6*32 approximately and want to
find the square root of 360. Then we have : —
360 = 9 X 40
= 3 X 3 X 6-32 X 6-32
= (3 X 6-32) X (3 X 6-32)
whence ^360 = 3 x 6-32
= 18*96 approximately
Similarly, since 1-6 = 40/25
JYE = 6-32/5
= 1*26 approximately
We conclude that when a number is the product or the
quotient of two numbers whose roots are known its root is
the product or the quotient of their roots.
§ 2. Use of the Badical Notation. — While there are some
numbers which have an exact square root — like 16, 178929
and 13-1044 of which the roots are 4, 423 and 3-62 respectively
— the great majority of numbers really have no square root at
all. These are called " surds ". Thus it is evident from A, § 2,
that the process of finding the square root of 40 would never
come to an end. In other words, there is no decimal fraction
which when squared yields exactly 40. All we can say is that
by prolonging the process of A, § 2, we can get numbers which,
when squared, come constantly nearer to 40. Thus the first
stage of the process consists in observing that 7^ > 40 > 6^.
Since the length of the strip of fig. 19 is between 12 and
13 its height is between 4/13 and 4/12, that is, between
0-307 ... and 0-333 . . . , or, say,' between 0-3 and
0-4. We have, then, (6-4)2 > 40 > (6-3)2. ^y examining the
successive strips r + r' + q, etc., we obtain the following series
94 ALGEBRA
of pairs of numbers such that the square of one is greater and
the square of the other less than 40. The one whose square
is nearer to 40 is in heavy type.
72 =
49
>40>
36
= 62
(6-4)2 =
40-96
>40>
39*69
= (6-3)2
(6-33)2 =
40-0689
>40>
399424
= (6-32)2
(6-325)2 =
40-005625
>40>
39-992976
= (6-324)2
(6-3246)2 =
4000056516
>40>
39-99930025
= (6-3245)2
etc
:.
etc.
Although
40 has really no square root it is
customary
to say that its square root is one of the numbers 6, 6-3,
6-32, 6-325, 6-3246, etc., according to the degree of exactness
required. Thus ^40 occurring in a calculation must be
taken to mean one of this series. It is generally convenient
not to decide until the end of the calculation which of the
numbers is to be taken. For this reason the expression
^40 is often treated in calculations as if it were a symbol
like a, the value of which is to be filled in after factorizing,
etc. The following is a good example of the advantage of
this procedure : —
§ 3. Bationalizing the Denominator. — In the formula of
Ex. IV, No. 31, let D = 40. Then we_have P = 18/ /ia
Now if at this point we substitute for ^40 one of the series
obtained in § 2 we shall have to carry out a decimal division.
But by treating ^40 as a symbol it is possible to avoid this
inconvenient operation. We proceed thus : —
P = ii
18
V40 _
18^ ^ V40 . . . (i)
V40 V40
isyiQ
40
9740
20
9 X 6-32 .... (ii)
20
2-844 lb.
SQUARE ROOT. SURDS 95
In (i) the denominator is " rationalized " by multiplying
the fraction by J 4:0/ J 4:0 — an operation which leaves its
value unchanged. The substitution of a value of ^40 occurs
only at the end of the calculation in (ii). Note that if we
had known ^10 instead of ^40 our method would have
been P = 18/2 ^10 = 9/ ^lO = 9 jTO/10.
[Ex. VIII may now be worked.]
CHAPTEE IX.
FRACTIONS.
A. Fractions with Monomial Denominators (ch. i., § 5 ;
ch. VI., § 8 ; Ex. XII).
§ 1. Reduction of Formulce containing Fractions. — A cis-
tern is being fed by a ball-tap which would fill it in forty-
eight minutes. At the same time water is being drawn out
by a pipe which would empty it in fifty-three minutes. If it
is empty to begin with, in how many minutes will it be full ?
For the solution we have : —
Fraction of cistern filled in 1 minute
1 1 ...
= 48-53 • • «
48 X 53 ^ ^
48 X 53 r-'\
.'. time taken = ^^ _ aq • • V^^)
It is obvious that we should have followed the same
method whatever numbers had been given. The process can,
therefore, be described in words and expressed in symbols.
Put ^1 = " the number of minutes in which the tap would fill
the cistern," and use t^ and T similarly as the other symbols
required. Then we have the following " shorthand " descrip-
tion of steps (i), (ii), (iii) : —
^ = t-' . . . . (iy
T ^1 ^2
= ^2 - h .... (ii)'
T = -iii- . . (iii)'
96
FRACTIONS 97
Now it is clear that formula (iii)' is more convenient for cal-
culating T than formula (i)', for its use enables us to dispense
with two steps of arithmetic. Moreover, if the. arithmetical
calculation goes through steps like (i), (ii) and (iii) the sym-
bolic description of those steps must always proceed by corre-
sponding steps like (i) ', (ii)' and (iii)'. Thus if we are given
a formula like (i)' we can always work through a stage corre-
sponding to (ii)' to a formula corresponding to (iii)' which is
in the form most suitable for calculation.
§ 2. Bules for Manipulating Formulm. — There is no need
of special rules — like the rule a^ - b^ = {a + b) {a - b) — for
carrying out these changes in a formula. The one sufficient
rule is, obviously, the following : Whenever a formula contains
symbolism which describes a fractional expression it may be
simplified for purposes of calculation by manipulating the
symbols in exactly the same ways as the figures would be
manipulated in the arithmetic which the symbolism describes.
Thus, suppose we are given a formula 1/R = ajb + cjd and
are asked to simplify it for calculation. The fact that we do
not know what words the symbols stand for does not hinder
us in the least, for we know that the right-hand symbolism
describes the sum of two fractions ; a and c being the descrip-
tion of the numerators, b and d of the denominators. The
steps involved in the arithmetical calculation must, therefore,
be those described by the steps : —
lac
B,^ b'^~d
ad -\- be
R =
bd
bd
ad + be
§ 3. The Least Common Denominator. — If the denomi-
nators of two fractions contain a common factor that factor
occurs only once, not twice, in the denominator used to
express the sum or difference of the fractions. For example
o + ifTi = — Y 7i w- ' If we know that the denominators
8 12 4x2x3
of the fractions described in a formula have a common factor
it is easy to arrange the statement in symbols so as to show
T. 7
ALGEBRA
the efifect of this circumstance upon the calculation. Thus,
putting p for the common factor we should have : —
a c _ ad + be
pb pd ~ pbd
[Ex. XII may now be worked. No. 1 may be taken orally.]
B. Fractions with Binomial Denominators (ch. i., § 5 ;
ch. VI., § 8 ; Ex. XIII).
§ 1. Algebraic Fractions. — It would be tiresome always to
a
say that " r represents or describes a fraction," that " a de-
scribes the numerator," and that "b describes the denomi-
a
nator ". For the sake of brevity we can say that / ^s a fraction,
that a is its numerator and b its denominator. This is merely
repeating what we did when we agreed to use the form " let
the length of the room be I feet," instead of the longer form,
"let I be the symbol for the length of the room in feet ".
In both cases for the sake of brevity and convenience we
speak of symbols which stand for the descriptions of numbers
just as if they were themselves numbers. When we want to
make it clear that we are speaking of these " fractions " made
up of symbols and not of the real fractions of arithmetic, we
will call them " algebraic fractions ". It should be noticed
that the number described by the algebraic fraction a/b may
not even be an arithmetical fraction at all, but a whole num-
ber. This will be so, for instance, if, in a given case, we
have a = 12, 6 = 3.
§ 2. One Binomial Denominator. — In previous examples of
algebraic fractions the numerators and denominators have
been either single symbols such as c or products of single
symbols ^ such as pq or p^. There is, however, no reason
why the denominator of a fraction should not sometimes be
more conveniently expressed as the sum or difference of two
symbols.
For an example suppose that the pipe of A, § 1, takes five
minutes longer to empty the cistern than the ball-tap takes to
^ Note that pq is really a symbol for the product of two numbers
represented by p and q. It is called a product of the symbols p
and q for brevity.
FRACTIONS 99
fill it. In these circumstances if we represent the latter time
by t the former time is best represented by ^ + 5. The original
formula now becomes : —
1 _ 1 _ 1
T~ t t + 5
This fractional expression can be simplified in exactly the
same way as before. For although t + 6 is a more compli-
cated symbol, yet, after all, it represents a single number just
as the symbol t^ does in A, § 1, The only difference between
what we must do now and what we did then is that when we
manipulate i + 5 we should enclose it in brackets. This
practice will remind us that ^ + 5 really represents a single
number which, if we were doing arithmetic, would be moved
as a whole from one place in the expression to another. We
have, then,
1 _ 1 ]^
T " t~ t + 5
_ {t+ 5) - t
tit + 5)
But the numerator of the new fraction can obviously be
simplified. For if we take t away from t -h 5, whatever t
may represent, the residue must always be 5. Thus, the
next line of the working will be
5
while, finally : —
T
t{t + 5)
t{t + 6)
5
In general,
minutes, is
as follows :
if the difference of time, which
represented by the symbol d,
111
in this case is five
the work will read
T ~
t ~ t + d
.-.T =
(t + d) - t
t{t + d)
d
t{t + d)
t{t + d)
d
7*
100 ALGEBRA
Instead of taking t to represent the time in which the tap
fills the cistern let it represent the time taken by the pipe to
empty the cistern. Then the former time must now be re-
presented by t - 5, and the formula becomes : —
1
f
1
~ t - 5
1
t
t- (t
-5)
- t(t-
5
5)
t(t - 5)
"^ ~ 5
We have seen in Ex. VI, Note to No. 15, how to simplify the
numerator of the fraction in the second line. We have to
take t - 6 away from t, that is, to take from t the whole of t
except 5. Whatever number t represents the residue must,
of course, be 5. We can, if you prefer it, argue in another
way. We have to take away i - 5, that is, a number 5 less
than t. If we take away t we shall have taken away 5 too
much and must give back 5 to make the account square. But
if we take t from t we shall have nothing left, so that when
we restore the 5 this number will be the total. By both argu-
ments, then, t - {t - 5) = 5.
The conclusion must hold good for all numbers. Thus
whatever numbers a and b represent we have : —
_1 1 _ a- (a - b)
a - b a ~ a{a - b)
b
a(a - b)
[Ex. XIII, Nos. 1-3 and 7-33, can now be worked.]
§ 3. Two Binomial Denominators. — We may now go on
to consider other kinds of fractional expressions that are likely
to occur in formulae. We shall then be prepared to deal with
them when they present themselves in actual problems.
It is very likely, for example, that we shall sometimes meet
with a fractional expression slightly more complicated than
the last, such as ^ -I q- Ijg* us begin by finding an
a — I a -\- o
expression more suitable for calculation by which this could
FRACTIONS 101
be replaced. Remembering that a - 7 and a + 3 are really
symbols for single numbers such as 20 and 30 (i.e. 27-7 and
27 + 3) we have : —
1 1 ^ (a + 3) + (g - 7)
a - 7 a + 3 {a - 7) {a + 3)
^-J
~ {a - 7) {a + 3)
There is no difficulty about simplifying the numerator.
We have to add to the number a + 3 a number which is 7
less than a. This can be done by adding a and taking away
7. Thus
(a + 3) + (a-7) = a + 3 + a-7
= 2a - 4
Consider next the expression in which the fractions of the
last expression are connected by a minus instead of by a plus
sign •— i ^f^': ^'
1 _ 1 ^ (g + 3) - ( a --^ 7) ■
a- 7 a + 3 (« - 7) (^ + 3) ■ iil':'\iJl
{a - 7) (a + 3)
To find the simplified numerator we have to take the
number a - 1 from the number a + 3. We can do this in
two ways.
We can take a - 1 from a and then add 3. But if we
take a -1 from a the result is, as we know, 7. Thus, the
numerator would be 10. This argument written in symbols
would be
{ a - (a - 7) } + 3 = 7 + 3
= 10
Note that we put a pair of curled brackets round the
expression a - (a - 7) to show that we are thinking of it as
representing a single number.
Or we can say that instead of taking away a - 7, that is,
a number 7 less than a, we shall reach the same result by
taking away a and adding 7. Expressed in symbols this
argument reads : —
(a+3)-(a-7) = a + 3-a+7
= 10
By similar arguments the numerator (a - 3) - (a - 7)
could be simplified in either of the following ways : —
102 ALGEBRA
(a - 3) - {a - 7) ={ a - {a - 7)} - S
= 7-3
= 4
(a-3)-(a-7) = a-3-a + 7
= 7-3
= 4
What should we do if the numerator were 13 - (a - 7)?
In this case the first argument would be rather round-about.
There is no a from which to take a - 1. It we want one we
must add it to the expression and at the end take it away
again. Thus we may write : —
13 - (a - 7) = a + 13 - (a - 7) - a
= { a - (a - 7) } + 13 - a
= 7 + 13 - a
= 20 - a
The otht^r ' d;rgument is, this time, much simpler. We
take away d from 13 and then add 7. Thus
j^Ir/:?; ; .13^^ (a - 7) = 13 - a + ?
'' ' ' = 7 + 13 - a
= 20 - a
The only difficulty about the second argument is that a
might be greater than 13, for example 15. We could not
take 15 away from 13 and then add 7. The first argument,
however, always holds good. It shows us that even if a is
greater than 13 it is still correct to write
13 - (a - 7) = 13 - a + 7
The reason why it is correct is, of course, that to take
away a and then add 7 ought to produce the same result as
adding 7 and then taking away a. If a is greater than 13
we cannot do the first thing but we can do the second.
§ 4. Numerators not Unity. — The fractions whose com-
binations we have studied in this lesson have all had the
same numerator, namely unity. There will, of course, be
many occasions when this is not the case. Another " cistern
problem " will serve to exemplify this fact. Suppose the
cistern to be fed by four taps, each capable of filling it in
i - 5 minutes, while water is being drawn off by three pipes,
each of which would empty it in ^ + 2 minutes. Then the
time taken to fill the cistern when all the taps and pipes are
at work is given by the formula : —
FRACTIONS 103
14 3
T ^. - 5 ^ + 2
4:(t + 2) - 3(^ - 5)
" (t- 6){t + 2)
Before we proceed further we must know how to simplify
the numerator 4: {t + 2) - 3 {t - 5).
To beo;in with, 4 (i + 2) means 4 times a number made up
of t and 2. This product must, of course, be the same as four
ts together with four 2s, or 4( + 8. Similarly the product
represented by 3 (i - 5) must be the same as the number
3t - 15. Thus we have
4 (^ + 2) - 3 (^ - 5) = (4^ + 8) - (3^ - 15)
The further simplification of this expression can be carried
out by either of the two methods which we have used before.
Thus we may argue : —
{U + 8) - (3^ - 15) = I (3^ - (3^ _ 15) I + ^ + 8
= 15 + ^ + 8
= ^ + 23
Or we may produce the result of taking away the number
{3t - 15) by first taking away St and then adding 15 : —
(4i + 8) - (3^ - 15) = 4i + 8 - 3£ + 15
= t + 23
As before, the second is the better practical method. It
has a simple rule : remove the brackets and change the sign
connecting the terms within them.
The simplification can now be completed. We have : —
1^ 4 3_
T~ t - t + 2
_ 4:(t + 2) - 3(t - 5)
~ (t- 5){t + 2)
_ t+ 23
~ {t - 5){t + 2)
{t - 5){t + 2)
t + 23
[Ex. XIII, Nos. 4-6 and 34-6, may now be taken.]
.•.T =
CHAPTER X.
CHANGING THE SUBJECT OF A FORMULA.i
§ 1. " Think of a Number " Problems. — " I am thinking of
a nunaber. I multiply it by 2 and subtract 7 from the product.
The result is 11. What is the number ? " The whole class see
at once that the number is 9. One member is then asked to
describe the steps by which he reached this conclusion. *'If
after taking away 7 from twice the number 11 is left, twice the
number must be 18. If twice the number is 18, the number
itself must be 9." These steps are now, at the dictation of
the class, to be recorded on the blackboard ^ : —
(i) 27t - 7 = 11
2n= 11 + 7
= 18
18
= 9
" Think of a number " questions of each of the following forms
should be asked and the modes of solution analysed in the
same way. That is, the answer should be obtained mentally,
the steps by which it is reached described verbally, and the
description transcribed in symbols. In addition to the former
example the blackboard should now exhibit the following or
similar matter : —
(ii) 3% + 6 = 30 (iii) 3{n - 4) = 18
3w=30-6 n - 4: = 18/3
= 24 = 6
n = 24/3 ?z = 6 + 4
= 8 =10
1 See ch. i., § 5 ; ch. vi., § 9 ; Exs. XIV, XV.
^ It is vital to note that what is written on the blackboard is
vierely a shorthand transcription of the answer just given. It is
not " doing it by algebra instead of by arithmetic ".
104
CHANGING THE SUBJECT OF A FORMULA 105
(iv) i(n + 7) = 5
w + 7 = 5 X 2
= 10
w = 10 - 7
= 3
§ 2. Analysis and Generalization of the Method. — A study
of these four examples shows that they are solved by the use
of two rules : (A) If the expression on the left-hand side of
the sign *' = " is written as a sum (or difference) the number
to be added (or subtracted) is removed and is subtracted from
(or added to) the number on the right of the sign. (B) If the
expression on the left is written as a product (or a quotient)
the numerical factor (or divisor) is removed and becomes a
divisor (or a multiplier) of the number on the right. Rule A
is exemplified in the second line of (i) and (ii) and the fourth
line of (iii) and (iv) ; Rule B in the fourth Une of (i) and (ii)
and the second line of (iii) and (iv).
Care must be taken to apply each rule at the proper point.
Thus in (iii) it would be wrong to begin by adding 4 to 18,
because 3(n - 4) is written as a product not as a difference.
We could, however, easily turn it into a difference thus : — ^
3(n - 4) = 18
3n - 12 = 18
3n = dO
n = 10
reaching the same result as before.
Again, it would be wrong to begin in (ii) by dividing the
30 by 3 (i.e. to write w + 6 = 10) because 3w + 6 is written
as a sum not as a product. We can, however, turn it into a
product thus : — ^
3w + 6 = 30
d{n + 2) = 30
n + 2 = 10
n= 8
Thus, although these problems can often be solved mentally
in more than one way, Rules A and B hold good whichever
way is taken.
Lastly it is clear that the rules hold good in all cases.
Thus in (i) it is not because the number to be subtracted on
1 Note that the algebra is, once more, only a transcription of the
verbal explanation.
106 ALGEBRA
the left is 7 that it may be removed and added on the right.
We should begin solving the problem in the same way what-
ever number stood in the place of the 7. Similarly the divi-
sion in the fourth line is not performed because the multiplier
on the left is 2 ; we should deal in the same way with any
number standing in the place of the 2.
Now suppose the following question is asked (Ex. XIV, No.
1) : "I am thinking of a number. I multiply it by 3*6 and add
14-7. The result is 23-18. What is the number ? " This
problem is too hard for most of us to do in our heads, but that
fact need not prevent us from solving it. We can write the
statement down in symbols and can find the number required
simply by applying to it Eules A and B.
[Ex. XIV, Nos. 1-6, may be taken here.]
§ 3. Description of Method in Symbols. — Since a " Think of
a number " problem could be solved in the same way even if
you changed all the numbers in the statement, it is possible to
describe in symbols the methods followed in solving each kind.
Consider, for example, problems like (i) in § 1. Let a = " the
number multiplying n," & = "the number to be subtracted on
the left," c = " the number on the right". Then the method
of solution can be described, line by line, as follows : —
an - b = c
an — c + b
c + b
n =
a
The actual working of No. 6 and the description of the
method are here set side by side.
~ -f 7-35 = 13-6 - + ft = c
4-4 a
^= 6-25 " = c-6
4-4 a
n = 6*25 X 4-4 w = a{c - b)
= 27-5
[Ex. XIV, Nos. 7-31, may now be taken.]
§ 4. Changing the Subject of a Formula. — In Ex. Ill, No.
2 (ii) we found a formula for the salary of a clerk after so
many years of service. Suppose now that we are asked for
a formula for the number of years before his salary will reach
a given amount. We can, of course, obtain the new formula
CHANGING THE SUBJECT OF A FORMULA 107
in the same way as we obtained the old one, by thinking out
the rule and then writing it down in symbols. But by
means of Rules A and B of § 2 we can obtain it in a more
convenient way directly from the formula S = S^ + ni.
To begin with, the statement
S = S„ + wt . . . . (i)
may also be written
S, + m = S . , . . (ii)
for (i) says, " The salary after n years is obtained by add-
ing 7ii to the original salary," while (ii) says, " Add ni to the
original salary and you will obtain the salary after n years ".
Thus the two statements differ only in the order of the words.
But in the formula which we are seeking S^, i and S will be
descriptions of known numbers — just as a, b, c, etc., were in § 3
— while n is here, just as it was there, a symbol for a number
which is not given but has to be calculated. It follows that
we may apply Rules A and B to t and S„ in (ii) : —
m = S - S„ . . . . (iii)
n = (S - S„)A- . . . (iv)
in order to obtain a formula, (iv), with 7i as subject instead
of S. This operation may be called changing the subject of
the formula. It is easy to satisfy oneself that the new
formula is true.
§ 5. The Meaning of the Symbols need not be known. — We
have, then, the following rules for changing the subject of
a formula. First manipulate the formula so that the new
subject appears on the left. (This manipulation will often
consist in interchanging the two sides of the formula — a
process which may be called Rule C.) Next apply Rules
A and B to the other symbols until they are all on the right-
hand side of the sign of equality. The result is the desired
formula.
The most interesting thing about this process is that you
do not have to know what a formula means in order to
change its subject. If the first formula is true and Rules A,
B and C are performed correctly, the resulting formula must
also be true. Take as examples the unexplained formulae
p = and P = TT^r , and change the subject to Q in
n Ov^ — t
each case. For the first we have : —
108 ALGEBRA
P = (a + bQ)ln
{a + bQ)ln = P .
. RuleC
a + 6Q = nP
. RuleB
bQ = nV -a.
. Rule A
Q = (tiP - a)/b
. Rule B
For the second we have :—
P = a/{bQ - t)
{bQ - t)-p = a.
. Rule B
bQ - t = alV .
. Rule B
bQ = ajV + t
. Rule A
«-"V'
. Rule B
[Ex. XIV, B, and Ex. XV may now
be taken.]
CHAPTEE XI.
THE PROGRAMME OF SECTION I (EXS. XVII-XXVI).
§ 1. Direct and Inverse Proportion, — The general course of
the argument running through Exs. XVII-XXVI has been
described in ch. v., § 2. Ex. XVII introduces the idea of
"functionality," which is further developed in Exs. XXIf-
XXV. Strictly speaking, the use of a formula always implies
this idea ; for a formula always exhibits the value of one vari-
able as depending upon the values of other variables. The
aim of these exercises is to make the implicit idea explicit.
In previous exercises the pupil has been chiefly interested in
the matter of a formula — that is, in the information which
it gives about the subject. He is now to see that formulas
which deal with a most heterogeneous collection of subjects
may yet have precisely the same /orm; and this text is to be
illustrated by a detailed study of a few specific functional
relations.
Two features of the exposition are of special importance.
In the first place the more abstract notion of algebraic form
is approached by way of the more concrete graphical form.
For example, the straight line and the hyperbola which are
respectively the expressions of direct and inverse proportion
are literally " forms ". The student who has stored them in
his memory has ever at hand a means of keeping hold of
ideas which are apt to evade the grip of an algebraic expression.
Their remembered shapes exhibit in a flash to the mental
eye the typical features of each relation. For these reasons
emphasis is laid in this part of the subject upon the corre-
spondence between certain graphic and certain algebraic
forms [cf. ch. iv., § 8].
In the second place those familiar denizens of the algebraic
page — X and y — now come into view for the first time.
109
110 ALGEBRA
Letters have hitherto been the shorthand representatives of
verbal descriptions of specific or concrete variables. For this
reason they have always been chosen so as to suggest the
variables to which they refer. The best way to record the
fact that the weight of a piece cut from a sheet of cardboard
is proportional to its area is to use the form W = kA. But
if we wish to forget the specificity of the variables and to
attend only to their form of connexion it is well to use letters
which will not suggest any particular variables but shall mean
only " variables in connexion with one another ". For this
purpose X, y and z will be reserved. It follows that, as our
students come to deal more and more with variables in
general instead of specific variables, xs and ^s will the more
abound. The teacher is strongly advised not to throw away
the great advantage of this special use of x and y by using
them as symbols for unknown values of a concrete variable.
The forms of relation (or " functions ") ^ considered in Exs.
XVII and XXII- XXV are (with one exception) limited to
direct and inverse proportion. Ex. XVII deals with y = kx,
Ex. XXIII with y = k/x — each function being approached
by way of its graphic symbol. In Ex. XXIV the study of direct
and inverse proportion is extended to the functions y = kx^
and y = k Jx, in the one connexion, and to the functions
y = k/x^ and y = k/ Jx in the other. Attention should be given
to the way in which the relation between the primitive and
the derived form of proportion is brought out. Each form
of direct proportion is in the first instance referred to the
straight line through the origin ; each form of inverse propor-
tion to the hyperbola. The graphs of the more complex forms
are then derived from the straight line or hyperbola by a
simple method of transformation. In this way the pupil is
taught to realize the unity underlying the various forms in
which direct proportion and inverse proportion may be ex-
hibited. Ex. XXV is given to a simple study of mixed forms,
such as 2r = kxy, of such common occurrence that they could
not be omitted even in the first stage.
The exception referred to above is the " linear relation "
which grows so naturally out of the relation of direct propor-
tion that its study could hardly be excluded. It appears in
Ex. XVIII, Nos. 19-24.
iThe Urm '* function " is reserved for Section II.
THE PROGRAMME OF SECTION I 111
§ 2. The Trigonometrical Batios. — The reasons for includ-
ing in the algebra course the fundamental notions of trigono-
metry have been given in ch. ii., § 3. Methods of imparting
them are suggested in ch. xiii. It will be seen that the sub-
ject is here treated as a department of the doctrine of direct
proportion. The study of the sine, cosine and tangent as
functions which have a " field " of their own apart from their
relations to triangles is postponed to Part II, Section VII.
At first the treatment is confined to these three fundamental
functions ; the secant, cosecant and cotangent are probably
not worth the confusion they introduce.
All three notions are taught in connexion with practical prob-
lems, in which they appear as factors which when multiplied
into the length of one side of a right-angled triangle give the
length of another. The average pupil finds these definitions
of tangent, sine and cosine much easier to understand and
to apply than their definitions as ratios. The tangent is
taught first (ch. xiii., A) since the practical problems in which
it can be applied are the most obvious and simple. The sine
and cosine are taught (ch. xiii., B) in connexion with navigation
problems for two reasons besides the intrinsic attractiveness
of such problems. First, the calculation of easting and
northing from a given course and distance exhibits the sine and
cosine in the role in which they play their most characteristic
part in all branches of pure and applied mathematics — namely
as factors for determining the "projections," "components"
or "resolved parts'' of some directed magnitude. Next, in
navigation problems the right-angled triangle is in varied
positions, so that the pupil learns from the outset to think of
the sine and cosine as connecting the hypotenuse with the
sides opposite and adjacent to a given angle. Taught in
this way these notions do not contract the haziness which
persistently clings round them when they are defined in terms
of the " perpendicular " and " base " of a triangle.^
In connexion with the aim of illustrating the special use
1 In order to eliminate a fruitful source of confusion the Greek
letters a, /3, y will be used as symbols for the rneasure of an angle
in degrees. The Roman letters A, B, C, etc., P, Q, R, etc., will
be used (as in geometry) merely to name the angle. Thus if in
fig. 32, p. 128, the angle P contains 34° we have a = 34°. (Note
that we do not write a°.) Later in the work the symbols 6, ^ will
be used for the measure of an angle in radians.
112
ALGEBRA
of the sine and cosine the idea of a vector is introduced
(Ex. XIX, Note to No. 18). The reader is no doubt familiar
with this notion. In fig. 21, which illustrates Ex. XIX, No.
20, the vectors b and a represent two movements carrying a
point from A to B. The vector c represents the single move-
ment which would have carried the point from the same
origin to the same terminus. Then b and a are component
vectors, c their resultant. It is important to note that the
angle between b and a is the external angle at C. It is the
change of direction which one would make at B in walking
Fig. 21.
tiG. 22.
along the route ACB. The angle A = 69° - 20° = 49°.
Hence
c = b cos 49° + a cos 20°.
By the aid of the vector notion all cases ^ of the "solution
of triangles " can easily be solved without the aid of
special formulae. Thus if, in fig. 21, the information that
ABC = 20° had been withheld (No. 22), the length of c could
still have been calculated by the method shown in fig. 22.
Suppose the point to travel from A to B by the rectangular
route APB instead of by ACB. Then we have
AP = 10-6 + 23-5 cos 69° = 19, and PB = 23-5 sin 69°
= 22 (nearly).
But tan a = PB/AP = 22/19 = 1-16 = tan 49°.
Hence a = 49° and ft = 20°,
while c can be calculated by two or three methods already
studied.
^ Except that of determining the three angles from the three
sides.
THE PROGRAMME OF SECTION I 113
Pig. 23 shows the solution of No. 28 :
PB = 800 sin 53" = 639 ; a = PB/sin 78° = 639/sin 78"
= 653.
Hence h = 800 cos 53° + 653 cos 78°.
Ch. XIV., A, continues the subject of navigation problems in
order to exhibit the
application of the
trigonometrical func-
tions to the simple geo-
metry of the sphere.
The neglect of this
geometry in schools
where globes and
atlases are in constant
use is little short of
a scandal. Ex. XX
is the beginning of Fig. 23.
an attempt continued
throughout the book to give clear though strictly elementary
ideas upon a subject which — apart from its special claims upon
the interest of a seafaring people — offers the best possible op-
portunities for cultivating the knowledge of tridimensional
space.
In ch. XIV., B, and Ex. XXI, the pupil is made acquainted
with the fundamental relations between the three trigono-
metrical ratios of the same angle. It will be noted that they
are taught not as abstract propositions but as practical tests
of the accuracy of the ratios supposed to be determined by
measurement. Any cultivation of " identities " beyond this
point would be quite out of harmony with the general treat-
ment.
§ 3. The Combining of Formulce (Exercise XXII). — Be-
tween the earlier and the later exercises on proportionality an
exercise dealing with "simple equations of two unknowns"
is interpolated. The traditional treatment which takes this
topic immediately or shortly after equations of one unknown
simply because two follows after one is based upon the in-
fertile principle of " logical " arrangement. In ch. xv. an
attempt is made to find a more natural and fruitful mode of
attack. There are three outstanding uses of systems of
equations. The first is to determine the constants in a re-
lation, such as y = a -i- bx, the form of which is already
T. 8
114 ALGEBRA
known. The second is to determine whether any pair (or
triad) of values of the variables will satisfy each of a
given set of relations ; and, if so, what is the pair or triad.
The third use is elimination — the process of " editing " a set
of symbolic statements so as to derive from them a statement
in which all they have to say about the connexion between
certain variables is included while all the irrelevant things
they say about other variables are deleted. These three topics
form the subjects of ch. xv. and Ex. XXII. It should be
noted that the discussion is in each case based upon operations
which the pupil has learnt in earlier exercises. The aim is
to systematize these operations into technical methods.
§ 4. Revision. — Ex. XXVI and last of Section I is given
to general revision. Eevision of this kind is of great import-
ance, especially in order to secure technical facility. But it
is still more important that the pupil shall constantly be
called upon to apply his knowledge outside the formal mathe-
matics lesson. The making and manipulating of formulae in
physics — including change of the subject — statistical work and
field-surveying in geography offer opportunities for the kind
of application which makes mathematical knowledge at once
a reality and a delight. A simple plan of co-ordination be-
tween the school departments will provide many such oppor-
tunities to the profit of all concerned and will secure the unity
of method which is the essence of fruitful co-operation.
The bulk of the examples are arranged in six " test papers,"
printed in two blocks — A,B,C, and r,G,H. An important
feature consists in the problems in which the pupil is called
to solve simple problems of solid geometry by means of
trigonometrical formulas. Simple models constructed in
paper or thin card are of great assistance in building up the
power of dealing intelligently with problems of this kind.
Between the two sets of test papers are two groups of ex-
amples whose aim is to extend rather than to revise the
earlier work. The group headed " E " deals with the manipu-
lation of surds, and may be regarded as an appendix to
Ex. VIII. Its main purpose is to familiarize the pupil with
the process of " rationalizing the denominator ." of a surd frac-
tion when the denominator is a binomial. The group headed
" D " is composed of examples of a more novel character.
These aim at introducing certain simple methods which
statisticians use in summarizing the information to be derived
THE PROGRAMME OF SECTION I 115
from a given group of data. A little consideration will show
that any such method involves the use of at least two numbers
or " co-ordinates ". Take as a simple example the attempts
of a marksman to hit a certain point on a target. If we
suppose that his shots have left visible marks upon the target
it is clear that they may, in general, be regarded as constitut-
ing a swarm or constellation of points. With regard to this
swarm we may ask two questions : (i) What is its general
position on the target ? and (ii) What is the degree of closeness
or " scatter " of the individual shots? If by an astonishing
fortuity the marks happened to lie in a perfect circle it would
be easy to give a definite numerical answer to both those
questions : the '* general position " of the swarm would be
defined by the position of the centre of the circle, the degree
of " scatter " by its radius. The problem of the statistician
is to find similarly definite replies in less simple cases. The
examples before us illustrate two methods, either of which
may appropriately be used . when the data are non-directed
numbers. The method most suitable for directed numbers
(the method of " root- mean- square deviation ") is studied in
Ex. LVII. Both these incursions into the province of statisti-
cal science are to be regarded as preliminary to the formal
treatment of that important and typically modern subject in
the last section of the whole work.
In both of the methods here to be considered the " general
position " of the group of data is regarded as determined by
the position of the "median," or middle term of the series.
There are two alternative ways of measuring the degree of
dispersion or " scatter ". The first is to find by calculation
the arithmetical mean or average of the differences between
the various data and their median. This number is called
the " mean deviation " of the group of data. The other
method is to divide the series into four compartments, each
containing one quarter of the whole number of data arranged
in order, and to find the semi-distance between the " quartiles "
or points of division which lie on either side of the median
and include half the series between them. It is obvious that
this " quartile deviation " or " semi-interquartile range " indi-
cates in a general way the degree of condensation or disper-
sion of the data about the median.
Of these two methods the latter has the advantage of
requiring practically no calculation. It is also directly ap
8*
116 ALGEBRA
pJicable when the data are represented graphically instead
of numerically. Nos. 13-20 illustrate this important point
in connexion with the problem of evaluating the evidence
afforded by a series of practical measurements. The way in
which the application is made was described in ch. iv., § 7.
CHAPTEE XII.
DIBECT PROPORTION.
(Gh. IV., § 6 ; oh. XL, § 1 ; Ex. XVII.)i
§ 1. Direct Proportion : its Graphic Symbol. — The idea of
direct proportion is familiar from arithmetic. We say, for
example, that the cost of a length of stair carpet is directly-
proportional to the number of yards bought ; meaning that if
on one occasion we buy 2, 3, 4 . . . times, or -|^, J, f , y, • • •
as much carpet as on another occasion the cost will be 2, 3, 4
. . . times or -|, J, f , 4^ . . . as much as before.
Suppose a man who sells stair-carpets at different prices to
need unexpectedly a " ready reckoner ". It could be made
most expeditiously upon the following plan. On squared
paper graduate a base-line to represent the number of yards
of carpet sold (fig. 24), and a vertical line to represent the
cost. Above the graduation 20 insert dots marking the
price of 20 yards of each kind of carpet. We may suppose
that there are four kinds, priced respectively at 28. 3d.,
3s. l|d., 3s. 9d. and 4s. 6d. a yard. (To use the prices of 1
yard would be to risk inaccuracy in drawing ; the cost of
20 yards is easily calculated mentally.) Through each dot
and the origin draw the straight lines OA, OB, etc. These
lines constitute the required " ready reckoner ". For example,
the cost of 48 yards of the cheapest carpet is given by the
point P and is £5 8s.
§ 2. Geometrical Proof. — It is easy to show why the
ready reckoner " works ". Suppose OQ to be divided into
any number of equal parts of which Oq^^ is the first, and
J^i?2» P2^3> ®*^-' *^® equal to the others. Then all the triangles
^^iPv Pi^2P2y ®*®-> *^^ congruent. It follows that the line
^ The teacher should also consult the chapter on Proportion in
Abbott's Teaching of Arithmetic, published in this series.
117
118
ALGEBRA
rises from O to P by equal steps, p^q^, p^q.^^ p^q^, etc. Thus
if there are n steps, PQ = p^q^ x n. Suppose q^ to be taken
at the point representing 1 yard. Then there would be 48
steps and PQ = p^q^ x 48. Thus if p^q^^ represents correctly
the price of 1 yard, PQ will represent correctly the price of
Jf7
^6
jts
1
,
/
//
/ /
'A
1
//
/
1
///
//
^--
1
//.>
Xr
\
/// /
/ p^-
'
A
"93
<f2
j£2
jef
O <frlO 20 30 4-0 Q' q5(?
JVP of yards of carpet
Fig. 24.
48 yards. But there would be 20 equal steps from O to
A, so that, since the vertical at A represents correctly the
price of 20 yards, p-^q^ does represent correctly the price of
1 yard.
We are certain, then, that the points on the line OAP give
correctly the cost of any exact number of yards. It is easy
DIRECT PROPORTION 119
to make sure that the reckoner will also " work " for fractions
of a yard — for example, that the point P' gives correctly the
price of 42 yards 2 feet, or 42f yards = i|-. For this result
the line can be supposed to rise from O to P' by 128 equal
steps, of which the first should represent the cost of ^ yard.
But since 60 of these equal steps will carry the line to A, the
first step does represent correctly the cost of ^ yard. Thus
P'Q' represents correctly the cost of 42| yards. It is obvious
that the argument could be repeated with any other fractional
number of yards.
i^ 3. Correspondence betiveen Graphs and Formulce. — The
cost in pounds (C) of a length of L yards of the cheapest
carpet is given by the formula
C = 0-1125L
Since by the straight line OA the same problems can be solved
as by this formula, the line may be called the graph of the
formula G = 0-1125L. Similarly the line OD is the graph of
the formula C = 0225L, etc. In general, if p be the price of
a single yard of carpet, then the formula G = ph will give
the cost of a given number of yards, and to this formula will
correspond some straight line through the origin — the slope
of the line depending on the value of p.
§ 4. Generalization of Formula. — It is clear that we should
have reached results like the foregoing if we had started with
any other example of direct proportion. That is, no matter
what the things are of which one is directly proportional to
the other, the graph of the formula connecting them would
always be a straight line through the origin. Such a straight
line may, therefore, be regarded as an universal symbol of
direct proportion.
It is convenient to have a formula which can also be used
as an " universal symbol " of direct proportion. For this pur-
pose we must first replace the letters C and L in the formula
G = ph by letters that shall not suggest any definite ' ' vari-
ables ". The letters x and y are very suitable for this use.
It is usual to let x represent the "independent variable" —
that is, the variable whose values are supposed to be given.
The letter y will then represent the " dependent variable " —
that is, the variable which would be the subject of the corre-
sponding formula. In accordance with the rule followed in
graph-drawing (ch. iv., § 3), x, the unnamed independent
variable, will always be measured along the horizontal axis.
120 ALGEBRA
For this reason this axis is often called the axis of x or the
ic-axis. The vertical axis is then called the axis of y or the
2/-axis.^
Lastly p must be replaced by a letter that will not suggest
any definite example of proportion. The letter k will generally
be used. It will suggest that in the formula corresponding to
any straight line through the origin each value of y^ is obtained
by multiplying the proper value of a; by a constant number.
With given scales of graduation for x and y, the value of k
depends on the slope of the graph, being greater as the latter
is steeper.
[Ex. XVII may now be worked.]
^ The distance of a point P from the axis of x is called the
ordinate of P ; its distance from the axis of y the abscissa of P.
CHAPTER XIII.
THE TRIGONOMETRICAL RATIOS (I).
A. The Tangent of an Angle (ch. xi., § 2 ; Ex. XVIII).
§ 1. " Height Problems " by Calculation. — Standing 150
feet from the wall of a building I note that the angle of eleva-
tion of its top is 22°. How high is the top of the wall above
my eye-level? This problem could, of course, be answered
by means of a drawing (fig. 25). Draw OA representing 150
feet to any scale you please, set off OB so that the angle
AOB = 22°. The perpendicular AB read off on the scale
used in drawing OA gives the height of the wall. It will be
found to be 60 feet.
It is easy to see why differences of scale make no differ-
ence to the result. (The method would, of course, be useless
if they did.) Let OAjBi, OAgB^, etc. (fig. 26), represent
different figures obtained by taking different scales and placed
so that they share the angle AOB. Then, by the last
lesson we have : —
AjBi = k . OAj, A2B2 = k . OA2, etc.,
k being a constant number. If, therefore, in any one draw-
ing the perpendicular is (say) f of the base, it will be so in
all possible drawings representing the same data,
121
122
ALGEBRA
Now the value of k depends, as we have seen, upon the
angle at O. When this angle is 22° it appears that ^ = | or
0'4. Supposing this result to be accurate it would be possible
to solve any other similar problem involving an angle of 22°
by calculation and so to avoid the tedium and risk of drawing.
For example : I stand 80 yards from the point immediately
under the spire of a church and the weathercock has an
elevation of 22°. How high is it above my eye-level?
Suppose OA (fig. 25) on any scale to represent 80 yards ; then
AB represents the height of the spire. But we have
AB = OA X t
= 80 X I
= 32 yards.
§ 2. Extension to other Angles. The Table of Tangents. —
It is easily seen that to each angle there corresponds a
definite value of h by means of which all " height problems "
involving that angle could be solved by calculation instead of
measurement. It would clearly be worth while to make
a table of these values by means of very careful drawings and
to preserve ic for future use. Such a table is called a table
of tangents.
It is not necessary to make a drawing for every degree.
It will be sufficient to measure the " tangents " (i.e. the
values of h) for every tenth degree. The tangent for any
other angle can be determined from these by a graph. The
work should be divided among the class. Bach member
should draw a right-angled triangle containing one of the
angles in the following table. To ensure accuracy the
triangle should be large, but no attempt should be made
THE TRIGONOMETRICAL RATIOS
123
to ensure uniformity. It should be pointed out that each
triangle supplies materials for calculating two tangents. For
example, the person who starts out to determine the tangent
of 40° can also calculate from his figure the tangent of 50°.
The conclusion is drawn that the tangent of an angle is the
reciprocal of the tangent of its complement. By this plan
at least four independent measurements of each tangent will
be obtained. The results should be collected upon the black-
board, averaged, and set out in the following table. The
tangents of 0° and 90° must be discussed separately. In
the case of the latter it must be shown that as the angle
increases the tangent also increases in such a way that by
bringing the angle near enough to 90° we can make the
tangent as large as we please. Thus there is no number so
large that we cannot find an angle whose tangent is larger
still. Moreover, the tangents continue to grow larger with
the angle until the latter becomes a right angle. Neverthe-
less, there is no tangent corresponding to the right angle
itself, for at the moment when the angle becomes a right
angle the figure ceases to be a triangle. (A triangle whose
base is zero would be an absurdity.) To indicate (i) that
there is no tangent corresponding to 90° and (ii) that the
tangent increases without end as the angle approaches 90°
it is usual to make use of the symbolism
tan 90° = 00 . ■
This statement is usually read " the tangent of 90° is in-
finity ". The pupil must remember that this does not mean
that 90° has a tangent like the other angles, but that the
words are a convenient brief way of expressing what was
said above.
Angle
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
Tangent
0-00
018
0-36
0-58
0-84
1-19
1-73
2-75
5-67
00
From these data the graph of fig. 27 should now be drawn.
Individuals should be asked to read from their graphs the
tangents of given angles. The accuracy of the results should
be challenged by comparison with those of other individuals
124
ALGEBRA
and with the numbers in the Table of Tangents (Exercises,
p. 107).
[Ex. XVIII may now be worked.]
C. The Sine and Cosine. Vectors (ch. xi., § 2 ; Ex. XIX).
§ 1. The Navigator's Problem. — The master of a ship which
leaves one port for another, perhaps at a great distance, must
be able at any moment to mark his position upon his chart
or map of the seas. He can do this very exactly from time
to time by observations of the sun from which he can calcu-
T
\
\
1
'^i *
/
K
y
r
/^
^
O" iO° 20 30 40 SO 60 70' 80 90
■T
Angle
Fig. 27.
late his latitude and longitude. But such observations can
be made only in favourable circumstances — e.g. in clear
weather. They must be supplemented by other calculations
which will fix the position of the ship at times between the
observations on the sun.
These calculations — which the sailor calls his " dead reckon-
ing " — are made as follows. When he has left a known
point he can fix his position at a later moment if he knows
two things : the direction in which he has sailed and the
distance the ship has travelled in that direction. The former
THE TRIGONOMETRICAL RATIOS
125
is called the course and is determined by the compass. On
steamers, at any rate, the course is usually reckoned by the
number of degrees between the direction in which the ship is
being steered and the north and south line. Thus if the ship
is being steered W.S.W. the course is " 67^° W. of S." The
distance run on a given course is determined by the log. In
modern ships this is an instrument which can be compared
with a cyclometer, since it records at any moment how far the
ship has gone. (A brief description of the patent log may
Fig. 28.
well be given, prefaced by an account of the older method of
"heaving the log". The term "ten knots" which means
"ten sea-miles an hour" receives its explanation here.)
Whenever the direction of the ship is changed the new course
is recorded in the "log book " together with the distance run
upon the last course.
§ 2. Graphic Solution. — Knowing the distance run on
each course the sailor could now fix his position on the
chart. Let O (fig. 28) be the port of departure. Let the ship
begin by steaming 10 miles on a course 50° E. of N. This
movement can be represented by the line OP drawn to scale
126 0<*7 ALGEBRA
and at an angle of 50° with the north and south Une Op.
The hnes PQ, QR may similarly represent movements of the
ship in which it runs 38° E. of N., 8 miles, and 56° E. of N.,
12 miles, successively. The point R now gives the position
of the ship according to the "dead reckoning". But there
is a more convenient method than this. The movement of
10 miles along OP will carry the ship a certain distance to
the north of O and also a certain distance to the east. The
same thing is true of the other movements. If we knew the
amount of the " northings " represented by Op, Pg, Qr, and
of the " eastings " represented by_pP, qQ, rR we could fix the
position of R by marking off Or' = the total northing and
r'B, = the total easting. In this way the trouble and risk
involved in drawing would be avoided. It is evident after the
last chapter {a) that, given the distance run, the northing or
southing and the easting or westing depend entirely upon the
angle of the course ; and (b) that, given the course, they are
directly proportional to the distance run. Thus if Op is
(say) 6*4: miles when OP is 10 miles, it would be 12*8 miles if
the distance run were 20 miles, etc.
To apply this idea it is necessary to know the northing (or
southing) and the easting (or westing) produced by a given
distance for each possible course. Knowing these, the north-
ings, etc., for other distances can be calculated by proportion.
The most convenient distance to take is 10 miles. The
method of A, § 2, should be followed. Let the class determine
by careful drawing the northing and easting when a ship sails
10 miles on courses 10°, 20°, 30°, . . . 80'^ E. of N.— each
member being responsible for one angle or more. Let the
class then be divided into two sections — one section to graph
the eastings (fig. 29), the other the northings (fig. 30).
§ 3. Sine and Cosine. — From these graphs, taken together,
we can read off the northing (or southing) and the easting (or
westing) corresponding to a run of 10 miles on any given
course. For example if a ship runs 37° W. of S. she will be
carried 8 miles south and 6 miles west of the starting point.
Suppose that, the course being the same, the distance run
was 17 miles, we should argue that each mile run along that
course carries the ship 0*8 miles south and 0'6 miles west.
Thus for a run of 17 miles : —
Southing = 17 X 0-8 and Westing = 17 x 0-6
= 13-6 miles = 102 miles
THE TRIGONOMETRICAL RATIOS
127
The numbers 0-8 and 0*6 and the numbers corresponding
in the same way to the other possible courses can, then, be re-
garded as factors or coefficients by which the distance run is
to be multiplied to give the northing (or southing) and the
easting (or westing). They will obviously all be fractions
ranging between and 1.
miles
10
o" lo" 20" 30" ^o" so" 60° 70" ao" 9o°'E.ofN
Course
Fig. 29.
miles
10
^
"-N
\
N,
N
\
30' ^o" so'
Course
Fig. 30.
<w'
ao" so°E. ofN.
It is convenient to give these coefficients definite names.
The coefficient used to calculate the easting or westing is
called the sine of the angle to which it belongs, the other co-
efficient the cosine. They may, of course, be used to calcu-
late the sides of any right-angled triangle when the hypoten-
128
ALGEBRA
use and one of the angles are given. The sine is the factor
by which the hypotenuse must be multiplied to give the side
opposite to this angle ; the cosine is the factor for calculating
the side adjacent to this angle (j5g. 31).^
§ 4. Complementary Angles. — It will be noticed that the
sine and cosine of an angle a are respectively the same as the
cosine and sine of 90° - a. Fig. 32 shows the reason of this
relation : the angle R = 90" - a and it is evident that the
side adjacent to P is the side opposite to R.
The angle 90° - a is called the complement of the angle a.
We have, then, that the sine of an angle is equal to the cosine
of its complement, and conversely.
[Ex. XIX, Nos. 1-17, may now be taken.]
§ 5. Vectors. — The term vector, if not already familiar, may,
at the teacher's discretion, be introduced during the foregoing
argument to describe the lines OP, PQ, QR of fig. 28. The
definition of a vector is given in the note preceding Ex. XIX,
No. 18, at which point it may, if the teacher prefers, be con-
sidered. The rest of Ex. XIX can then be worked.
^ The teacher who does not despise these things may be glad of
the following mnemonic device : " When you were told that the
sine is the name of the fraction used in calculating the length of the
side (O) opposite the given angle from the length of the hypotenuse
(H), and that the word is spelt, not sign, but sine, you might well
have expressed your surprise by saying OH I When you were told
that the length of the adjacent side (A) is calculated from that of
the hypotenuse (H) by means of another fraction called the co-sine
you might have made the milder remark AH ! Now ' OH ' comes
necessarily before ' AH ' and ' sine ' necessarily before ' co-sine ' ;
thus you can always remember which stands for which."
CHAPTBE XIV.
THE TRIGONOMETRICAL RATIOS (II).
A. Circles of Latitude. Middle Latitude Sailing (ch. xi.,
§ 2 ; Ex. XX).
[The teacher will need a globe — preferably one with a surface
upon which lines may be drawn with chalk.]
i^ 1. Changes of Latitude : the Nautical Mile. — In ch. xiii.,
B, the northing (or southing) and the easting (or westing) were
calculated in miles. The sailor does not actually leave his
results in this form, for, as every one knows, positions on a
map or chart are fixed not by measurements representing
miles, but by measurements representing latitude and longi-
tude. Thus, before he can use the results of his calculation
the sailor must express his northing or southing in degrees
and minutes of latitude and his easting or westing in degrees
and minutes of longitude.
There is no difficulty with regard to the northing or southing.
Reference to the globe shows that the meridians or circles of
longitude are all " great circles " whose circumferences are
simply the circumference of the earth. When we know the
length of this circumference a simple calculation will give the
change of latitude produced by a given northing or southing.
The polar circumference of the earth is 24,856 miles, whence
the length of a degree of latitude is 24856/360 = 69 miles.
Now if this number were 60 the sailor's calculations would
be much simplified ; every mile run north or south along a
meridian would mean a difference of latitude of exactly one
minute. To secure the advantage of so simple a relation the
sailor abandons the ordinary or statute mile and uses a
' nautical " or sea-mile which contains 6080 feet instead of
5280. Sixty of these miles are of the same length as 69
statute miles. In other words, 60 sea-miles are equivalent
to a degree of latitude and 1 sea-mile to a minute of latitude.
T. 129 9
130
ALGEBRA
Thus a northing of 72 sea-miles implies a difference of latitude
of 1° 12' ; a northing of 354 sea-miles a difference of latitude
of 5° 54'.
v^ 2. Changes of Longitude. — The question of the difference
of longitude produced by a given easting or westing is not so
simple. The parallels of latitude are ** small circles " whose
circumferences decrease from the equator towards the poles.
Only along the equator itself will 1 sea- mile imply 1' differ-
ence of longitude. The readiest way to see this is to note how
the meridians on the globe approach one another towards the
poles.
What is the law which the length of a degree of longitude
follows? It will probably be suggested that the length is
proportional to the distance from the pole. This suggestion
is easily found to be inadequate ; measurement on the globe
shows that the greatest distance between two meridians is
halved not at latitude 45° but at latitude 60°. The explana-
tion of this fact must be sought in
the law which fixes the relative
lengths of the circles of latitude.
Why has the parallel of 60° half
the circumference of the equator?
Fig. 33 answers the question.
It represents a section through
the centre of the globe (C) and the
poles (P, P'). CE is a radius join-
ing the centre to a point E on the
equator; A is a point in latitude
60°. Let the figure rotate about
PK. The circle would trace out
the surface of the globe, E the equator, A the parallel of 60°.
It is evident that CE would be the radius of the equator and
AB the radius of the 60th parallel. Let CE = CA = R, then
AB = AC sin ACB
= R cos 60°
= 0-5 R, by the table p. 111.
But if the radius of the 60th parallel is one-half of the radius
of the equator, its circumference will be one-half of the length
of the equator. That is, the length of a degree in 60° latitude
is, as we found by measurement, one-half of the length of a
degree along the equator. Conversely a voyage of a given
number of sea-miles along the 60° parallel implies a change of
THE TRIGONOMETRICAL RATIOS 131
longitude twice as great as if it had been taken along the
equator.
The rule is easily generalized. If the latitude be called X.
the radius of the parallel is R cos A.. To find the change of
longitude corresponding to a given easting or westing we have
the rule : —
. , . n . . iio- of sea-miles
change of longitude in minutes = t — —
Thus if a ship sails 420 miles along the 53rd parallel the
change of longitude is 420/cos 53° = 700 minutes = 11° 40'.
§ 3. Middle Latitude Sailing. — A new difficulty now comes
into view. Suppose a ship to start in longitude 42° 18' W.
and latitude 60° N. and to run 20 miles on a course 37°
E. of N., then we should have : —
northing = 20 cos 37° easting = 20 sin 37°
= 16 miles = 12 miles
The 16 sea-miles to the north imply a difference of latitude
of 16 minutes. The ship's latitude is, therefore, now
60° 16' N. But when we seek to convert the easting into
minutes of longitude what cosine are we to take as divisor ?
If the ship had sailed 12 miles east in latitude 60° the divisor
would have been cos 60° ; if in latitude 60° 16' it would have
been cos 60° 16'. But as a matter of fact it made its easting
on neither of these parallels but on its way from one to the
other. A strictly accurate result cannot be obtained, there-
fore, by using either of the cosines as divisor.
This objection is sound, but in the case before us is not
serious. There is so little difference between the two cosines
that it hardly matters which we take as divisor. We can
without serious error assume that all the easting took place
in latitude 60°. Then we have : —
diff. of long. = 12/cos 60°
= 24'
so that the new longitude is 42° 18' - 24' = 41° 54' W.
But now suppose the distance run to have been 200 miles,
so that the northing would be 160 miles and the easting 120
miles. In the first place we have : —
change of lat. = 160'
= 2° 40'
.-. new latitude = 62° 40'
9'
132 ALGEBRA
The cosine graph shows that cos 62° 40' = 0*46. The dififer-
ence between
120/cos 62° 40' and 120/cos 60° is 261' - 240' = 21'.
This is too serious a difference to be ignored ; so the
sailor compromises and assumes that the easting was made
on a parallel half-way between 60° and 62° 40', i.e. 61° 20'.
The graph shows that cos 61° 20' = 0'48. So we have : —
change of long. = 120/0-48
--= 250'
= 4° 10'
.-. new longitude = 42° 18' - 4° 10'
= 38° 8' W.
This method is called the rule of middle latitud-e sailing.
The question may be asked — Would the rule apply if the
distance run were much greater still — say 2000 miles ? The
answer is obviously, No. It can only be used for moderate
runs. But as a single day's run is never anything like so great
as 2000 miles the failure of the rule is unimportant. The
final longitude after several days' running can always be
determined by adding the daily changes together.^
[Ex. XX may now be taken.]
B. The Belations between the Sine, Cosine and Tangent of
an Angle. The Calculation of the Batios of Certain Angles
(ch. XI., g 2 ; Ex. XXI).
§ 1. Tan a = sin a/cos a. — The tables on pages 107 and 111
give the values of the tangents, sines and cosines as they
could be determined by careful drawing and measurement.
But when numbers are obtained by a graphic process it is
always well to check them by calculation if it is possible to
do so. There are several ways in which the accuracy of our
tables can be tested. Here is one way. In fig. 31 we have
tan a = 0/A, sin a = 0/H, cos a = A/H
^"* A = H^H'
that is, tan a = sin a/cos a.
If, therefore, the numbers in the tables are correct the value
^ The complete theory of Mercator sailing will be found in Part
II of this work.
THE TRIGONOMETRICAL RATIOS
133
given in the first as the tangent of an angle should be equal
to the quotient of the sine of the same angle by the cosine,
as these are given in the second table.
§ 2. Si'n? a + Gos^ a = 1. — A second test applies only to the
sine and cosine of the same angle. Let the triangle ABC
(fig. 34) be right angled at C, and let the perpendicular CD
divide the base into two parts of lengthy and q. Then we
have : b = c cos a, a = c cos yS = c sin a, and : —
c = jp + g
= h cos a 4- a cos y8
= h cos a + a sin a
= c cos^ a + c sin'^ a
= c (cos^ a + sin'^ a)
Hence (i) sin^ a + cos^ a = 1, (ii) cos a = ^1 - sin*^ a, and
(iii) sin a = ^1 - cos^ a.
These formulae are proved still more easily if the class knows
the theorem of Pythagoras ^
have : —
Fia. 34.
(Euclid
I, 47), for we then
^2 + 62 = c2
a^jc^ + 62/c^ = 1
that is, sin^ a + cos^ a = 1.
§ 3. The Identities Combined. — The identities of §§ 1, 2
can be combined to yield tests by which, given the sine or
cosine, the tangent can be calculated, and vice versa. Thus
we have : —
iThis theorem can itself be proved most easily by fig. 34. We
have : —
cos a = bjc — pjb ; sin a = cos /3
.•. b^ = pc .-. a^
whence a^ + 6^ = (^ + q)c
ajc = qla
134
ALGEBRA
tan a = sin a/cos a
= sin a/ ^(1 - sin-^ a) by 5^ 2, (ii)
= ^(1 - cos- a)/cos tt by § 2, (iii)
Again
whence
Finally
tan^ a + 1 =
cos^ a
+ 1
by the preceding result.
sin'^ a + cos^ a
~~ COS-^ a
= 1/cOS^ a
COS a = 1/ ^(tan^ a + 1)
sin a
Sin a = COS a
COS a
= COS a tan a
= tan a/ ^(tan'"' a + 1)
^
/^
y^o°
^^ A=
^3 1
I
Fig. 36.
Fig. 37.
§ 4. The Ratios of 45°, 30°, 60°.— The preceding tests apply
to all angles. In the case of certain special angles still better
tests are available. Thus if a = 45° and we put A = 1 (fig.
35), it follows (since the triangle is isosceles) that 0=1.
By Pythagoras' Theorem (§ 2 above)
H = jK^+a' = V2 = 1-414.
Hence tan 45° = 1, sin 45° = cos 45° = 1/^2 = 0-707.
All three results are in agreement with the tables.
In fig. 36 the whole triangle is an equilateral triangle
which has been divided into two identical right-angled triangles
by a perpendicular from the vertex to the base. Here, if we
put H = 2, we have A = 1 and O = JK'^ - A'' = ^3:
THE TRIGONOMETRICAL RATIOS 135
Hence tan 60° = V 3 = 1-732, sin 60° ^ ^3/2 = 0866,
cos 60° = 1/2 = 0-500. Fig. 37 shows the equilateral triangle
differently arranged, and gives us tan 30° =1^3 = 0*577,
sin 30° = 0-500, cos 30° = 0-866. All these six results agree
with those in the tables.
[Ex. XXI should now be taken.]
CHAPTER XV.
THE COMBINING OF FORMULAE.
A, The Determination of Co7istants in a Formula (ch. xi.,
§ 3 ; Ex. XXII, A).
§ 1. Determination by a Graph. — When a weight of 20
grm. is suspended by a rubber cord the length of the cord is
25-1 cm. When the weight is 30 grm. the length is 28-5
cm. Assuming a linear relation between I and w find the
formula.
Fig. 38.
This problem is easily solved by the methods of Ex. XVII,
Nos. 18, 19, 20. Let AB (fig. 38) be the graph, P and Q
the points representing the data. Then Q/i = 28*5 - 25-1
= 3*4 cm. is the increase of length for the addition of
Vn - 30 - 20 = 10 grm.
Thus the increase of length per gramme added is
Q?i/Pn = 0-34
136
THE COMBINING OF FORMULA 137
To find the original length AO we must subtract from PM
0-34 X 20 - 6-8 cm.
AO = 25-1 - 6-8 = 18-3 cm.
Thus the formula is
I = 18-3 + 0-34:W
The method can, of course, be generalized. Let the linear
relation between any variables x and y have the form
y = a + bx
where a and b are constant numbers. To fix the exact
form of the relation in a given case a and b must be known.
They can be determined if two pairs of corresponding values
of X and y are given. For example, let the pairs be a? = 8,
y = 41, and x = 13, y = 56. Then Qw = 56 - 41 = 15,
Pti = 13 - 8 = 5 and Qn/Vn = 15/5 = 3. Also
AO = PM - 3 X 8 = 41 - 24 = 17.
Hence
2/ = 17 + Sx,
§ 2. Algebraic Methods : (i) by Composition. — Let us now
seek rules for determining a and b without drawing a figure.
We have y = a + bx
(QN) 56 = a + 136 . . ' . (i)
(PM) 41 = a + 86 . . . (ii)
(Qn) 15 = 56 1 . . . . (iii)
whence 6=3
Again by (ii) 41 = a + 86
= a + 24 (since 6=3)
(AO) a = 41 - 24
= 17
Thus the relation is y = 17 + 3x.
The essence of the method is that by subtracting the two
sides of (ii) from the corresponding sides of (i) we obtain a new
relation (iii) from which the value of one constant is at once
determined. This known, the value of the other can be ob-
tained from one of the original relations. We may call this the
method of composition, since relation (iii) is obtained by com-
pounding (i) and (ii).
§ 3. Algebraic Methods : (ii) by Substitution. — Another
method is readily suggested. In (i) change the subject to
a : —
a = 56 - 136 . . . (iv)
1 Note that 5 = Fn.
138 ALGEBRA
In (ii) replace a by its equivalent in (iv). Thus
41 = (56 - 136) + 86
whence 6=3. The value of a naay be determined as be-
fore.
This method is called the substitution method.
§ 4. The Relation y = a - bx. — A linear relation is not
always of the form y = a + bx. li y decreases as x increases
a graph shows that the form must he y = a - bx. For ex-
ample, let (5, 33) and (12, 5) be the two pairs of values of x
and y. Then, by the composition method : —
y == a - bx
33 = a - 56 (i)
5 = a - 126 . . . . (ii)
33 - 5 = (a - 56) - (a - 126)
or 28 = 76 (ill)
6=4
The line between (ii) and (iii) may be omitted when its
effect is understood.
The substitution method gives the same result for b ; a can
be determined as in § 2.
,§ 5. The Relation y = bx - a. — Consider now the relation
y — 4:X - 12. Its graph must be a straight line, for if x increases
by equal steps y increases also by equal steps. But it is clear
that 4:X can never be less than 12, i.e. x cannot be less than
3. When x = 3, y = 0. Thus i/ = 4a? - 12 is a linear re-
lation whose graph begins, like A'B' (fig. 38), at some point
on the axis of x instead of beginning, like AB, at some point
on the axis of y. Relations corresponding to lines in such
positions must be of the form y = bx - a.
Is "there any way of finding, without drawing the graph,
which form the linear relation has in a given case ? Consider
the relations : —
y = bx, y = bx + a, y = bx - a.
These may be written : —
y/x = 6, y/x = b + a/x, y/x = 6 - a/x.
In the first the quotient y/x is the same for all values of x ; in
the second it decreases as x increases, in the third it increases
as x increases. 1
^ For the fraction a/x grows smaller as its denominator x grows
larger. Thus as x increases there is a smaller amount to add in
the second case and to subtract in the third case.
THE COMBINING OF FORMULA 139
We can, then, decide the relation appropriate to a given pair
of values of x and y by noting the value of ylx for each pair.
If this value decreases with a greater value of x the relation
\s,y = a + hx ; if it increases the relation i^ y = hx - a. For
example, find by the composition method a linear relation
such that vsrhen a^ = 7, 1/ = 33, and when x = 11, y — 57.
Here 57/11 > 33/7 so that the relation is
y = bx - a
57 = lib - a . . . . (i)
33 = 76 - a . . . . (ii)
57 - 33 = (116 - a) - (76 - a)
24 = 46 (iii)
6 = 6
From (i) or (ii) a = 9. Hence y = 6x - 9.
[Ex. XXII, A, may now be worked.]
B. Common Values of Two Relations (ch. xi., § 3 ;
Ex. XXII, B).
§ 1. Determination of Common Values by Graph. — Two
rubber cords hang side by side and are loaded with weights
as in A, j^ 1. The formula giving the length of one cord is
I =14-2 + l-2w;
the formula for the other cord is
I = 18-7 + 0-7w.
The units of measurement are inches and ounces. Is there
any weight which will make the cords assume the same length ?
This problem is similar to Ex. XVII, Nos. 21-24. Let
AB (fig. 39) be the graph of the first formula, A'B' that of the
second. Then C clearly gives a weight OM which will
stretch each cord to the same length, CM. To find this
weight by calculation we have : —
CM = AO + Cw and also CM = A'O + Cm'
= 14-2 + l'2w = 18-7 + 0'7w
Hence we can put
14-2 + l-2w; = 18-7 + 0'7w
w = 9 ounces.
From either of the original relations we now find I = 25
inches.
140
ALGEBRA
§ 2. Algebraic Methods. — The method just employed can
obviously be regarded as the " substitution method " employed
to find the value of a variable instead of the value of a con-
stant. For example, let the question be proposed whether
the two linear relations y = 14: + 6x and y = Sx - 7 have
a pair of values of x and y in common. By the method of
substitution we can replace y in the second relation by 14 +
6x. We then have
14 +
6aj = 8a; - 7
X = 10-5
From the first relation it now follows that y = 77.
the pair (10-5, 77) is common to both relations.
Thus
inches
4-0
B/
y
/
/ y^
B'
a
^
"/
'-■'{
I
\^'o
20
Fig. 39.
30
4-0 oz.
Have the relations Sx - 4:y = 7 and 5x + 2y = 16 any
common pairs of values of x and y ?
3 7 6
The relations can be written y = ix - ^ and y = 8 - xX
respectively. These transformations show that they are
linear so that they may have one pair of values in common.
Using the substitution method we have : —
THE COMBINING OF FORMULA 141
3 7^5
13 _ 39
4 '^ ~ 4
From y = -X - j we now have V = q-
The transformation that necessarily precedes substitution
in such examples as this could be avoided by a " composition "
method. We have
From (ii) we have
if(i)
have
3a; - 42/ = 7 . . • W
5x + 2y = 16 . . . (ii)
10.1; + 4?/ = 32 . . . (iii)
If (i) and (iii) be now added y will disappear and we shall
13a; = 39 . . . (iv)
X = S
From (i) 9 - 4?/ = 7
1
[The class should at this point turn to Ex. XXII, No. 13, and
should (without working out the examples) consider how the
composition method is to be applied to them.]
§ 3. Common Values not always Possible. — It is evident
that if two graphs do not intersect the corresponding relations
have no pair of common values of x and y. A little con-
sideration will determine without the aid of graphs whether
common values do or do not exist in a given case.
Have the relations 2x - 3?/ = 4 and Sy - 5x = 7 a
common pair of values of x and y ? Here, using the method
of composition, we have : —
2x - dy = i . . . • (i)
Sy - 6x = 7 . . . . (ii)
2x - 5x= 4: + 7 . . . (iii)
But, as 5x cannot be taken from 2x, (iii) cannot be formed. ^
Thus no single pair of values of x and y satisfies both re-
lations, and the graphs will be found not to cross.
^ Note that 2£c - 5x = 4 - 7 would give no difficulty for it could
be transformed into 7 - 4 = 5x - 2x.
142 ALGEBRA
Consider next the relations Sx - 4:y = 5 and ix - Sy = 7.
Multiplying these by 4 and 3 respectively we obtain : —
12x - 16?/ = 20 . . . (i)
12x - 9y = 21 . . . (ii)
Since 21 > 20 we subtract downwards and obtain
16y - 9y = 1
and from Zx - ^y = 6 obtain x = 1^.
But if the former relation had been 3a; - 4t/ = 6 we should
have had
12a; - 161/ = 24
12a; - % = 21
Here subtraction of the lower line from the upper leads to
% - 16?/ = 24-21
which is obviously impossible. There is, then, no common
pair of values of x and y.
Finally take the pair y = ^x + 6 and y = I - ^x. The
substitution method gives : —
p+6 = i - ix
leading to f a; + fa; = |^ - 6
which is impossible.
On the other hand, suppose the relations to have been
given in the equivalent forms 2^/ - 3a:; = 12and5x + Sy = 14.
Applying the composition method we should have, after
multiplying the former by 5 and the latter by 3 : —
lOy - 15a; = 60 . . . (i)
15a; + 242/ = 42 • • • (")
and, by addition, 34i/ = 102
y=3
But on putting ^ = 3 in 5a7 + 8^/ = 14 we have
5a; + 24 = 14
which is again impossible. Thus we cannot, by either
method, find a pair of values of x and y common to the two
relations.
[Ex. XXII, B, may now be worked.]
C. Elimination ^ (ch. xi., § 3 ; Ex. XXII, C).
§ 1. An Example of Elimination. — Turn back to Ex. XVIII,
No. 13 ; let h he the height of the flagstafif and d the distance
^ This section may be omitted and taken in revision,
t
THE COMBINING OF FORMULA 143
of the observer from the building. Then we have
{h+ 110)/d = ta>n4:6° . • • (i)
Now this relation, as it stands, does not enable us to calcu-
late h because of the presence of d. We must find, therefore,
some means of eliminating or getting rid of d. This can be
done by the help of the relation 110/5 = tan 37°, using either
the method of substitution or the method of composition. By
the former method we have d = 110/ tan 37°, which, substi-
tuted in (i), leads to
{h + 110) tan 377110 = tan 46° . . (ii)
Composition must in this case take the form not of adding or
subtracting but of dividing the first relation by the second.
Thus we have : —
h + 110 d , ,ao 1
= tan 46 ■
d 110 tan 37°
or [h + 110)/110 = tan 46°/tan 37° . . (iii)
— a relation obviously equivalent to (ii). From either of them
it is easy to calculate h.
§ 2. Elimination in General. — The foregoing problem
affords an example of the process called eliminatioji. Elimi-
nation, in general, aims at reducing the number of variables
or unknown numbers in a given set of relations. Thus in
§ 1 our aim was to obtain a relation involving h only from
two relations involving hand d. The following is an example
in which the number of variables, originally three, is reduced
to two.
Eliminate z from the relations
^x - ^z = 13 and ^y + ^z = 3.
Upon multiplying the second relation by 2 and compound-
ing, z disappears, and we have
3x + IQy = 19
as the required relation containing only x and y.
§ 3. Trigonometrical Examples. — The term composition is
to be taken to mean any manipulation of two given relations
which leads to a third relation different from either of them.
The variety of manipulations sometimes required in elimina-
tion may be illustrated by two examples.
(i) Eliminate a from the three relations
xja = sin a, hjy = cos a, cjxy = tan a.
Here we have, by compounding the first and second rela-
144 ALGEBRA
tions, xyjah = sin a/cos a = tan a. Substitution for tan a in
the third relation leads to
xy/ab = c/xy or xhj^ == abc.
(ii) Eliminate a from the relations
X = a cos a, 2/ = 6 sin a
Here we argue as follows : —
X = a cos a .'. COS a = x/a
y = b sin a .'. sin a = ?//6
But cos^ a + sin^ a = 1 ; .-. x'^/a^ + y^jb^ = 1.
[Ex. XXII, C, may now be worked.]
GHAPTEE XVI.
FURTHER TYPES OF PROPORTIONALITY.
A. Inverse Proportion (ch. iv., § 8 ; ch. xi., § 1 ; Ex.
XXIII).
§ 1. Inverse Proportion ; its Graphic Symbol. — Two
places, P and Q, are 16 miles apart. How long will it take
10 travel with constant speed from P to Q? The answer
obviously depends upon the rate of movement. A walker
going at 4 miles/hour will take four hours ; a donkey-cart going
twice as fast will take half as long ; a cyclist going three
times as fast one-third as long; an express train going 40
miles/hour, one-tenth as long, and so on. These facts are
expressed by saying that the time taken is inversely pro-
portiojial to the speed.
We saw that direct proportion has a definite graphic sym-
bol — the straight line through the origin. Let us see what
results from graphing a series of pairs of numbers in inverse
proportion. The problem just considered will serve as an ex-
ample. The speed of the movement must be taken as the
independent variable. Any number of points may readily be
determined by noting that the product of speed and time must
always be 16. Some small speeds should be considered (e.g.
a creature crawling ^ ml./hr.) as well as large speeds. The
points obtained determine the smooth curve A (fig. 40).
Now if A can really be taken as a graphic symbol of this
case of inverse proportion then the product of the co-ordinates
of any point on the curve should be 16, just as the ratio of
any pair of co-ordinates in the straight line symbol of direct
proportion is h. In the case of the straight line we were able
to prove the required property by geometry. Since we do not
know any properties of the curve A from which to start a proof
by reasoning we must be contented with the results of measure-
T. 145 10
146
ALGEBRA
ment. Let each pupil choose, at random, a couple of points
on the curve and i find that the product of the co-ordinates is,
in every case, as nearly 16 as we could expect. We may then
conclude that the curve A is truly the graphic symbol of the
formula st = 16, and, when once drawn, may be used (as the
straight lines were used in ch. xii., A) as a ready-reckoner.
[Examples of interpolation should be given.]
§ 2. Generalization of Formula and Graphic Symbol. — As
in ch. XII., we may now note that the curve A would be the
graphic symbol of any case of inverse proportion between two
variables whose constant product is 16. That is, we may
regard the curve as the graph corresponding to the relation
xy = 1^ or 2/ = l^jx
where x and y represent any variables whatever.
«ll
\
\
\
\
/
/
\
/
/
\
/
/
\
•^
\
\
/
/
/
/
^
\^
^
\^
/
/
^^
o
^
V
1.
~r
D
I A
0^
IS 20 25
Fig. 40.
30 35 ^O
It is evident that the 16 here plays a part similar to the
constant. A;, in the formula y = Izx. Now when Iz received
different values we found that the corresponding graphs
were all straight lines, and that they differed only in their
inclination to the axis of x. It is important, therefore, to
inquire how the graphic symbol of inverse proportion will
change if we substitute another constant for 16. In other
FURTHER TYPES OF PROPORTIONALITY 147
words, what are the graphs of the relation y = hjx corre-
sponding to different values of A; ?
The investigation is best divided among the class, each
taking a different value for k. Some of the curves should be
transferred with rough accuracy to the blackboard. It will
then be seen that we have a number of curves which all bear
a family resemblance to A, For example, B (fig. 40) is the
curve answering to A; = 1, C, D and E those obtained when
A; = 36, k = 100 and k = 400. The '' family resemblance "
consists in the following facts, (i) The straight line, OV,
drawn at an inclination of 45° to OX, divides each curve
symmetrically ; for every point on one side of this line there
is a corresponding point on the other. The line OV is for
this reason called the axis of the curves, (ii) The point where
this line cuts each curve is the point nearest to the origin.
It is called the vertex or head of the curve, (iii) At any
vertex, V, x and y are equal ; and, since xy = k, we have
X — y = J k. Hence the distance
OV = V {x^ + 2/^) = J^k
(iv) Each curve approaches constantly nearer to the axes
but never reaches them. However large x i^in y — kjx, y
can never be zero ; and, however small x is, it will always
be possible to calculate a value of y though it will become
endlessly large. Thus neither x nor y can be ever ab-
solutely zero. These facts are expressed by saying that the
axes are asymptotes of the curves.
It is evident, then, that we can regard all the curves of fig.
40 as simply different specimens of the same kind of curve,
just as the curves we draw with compasses are all specimens
of the circle. We will call it the inverse proportion curve}
§ 3. Mechanical Constructions for the Curve.'^ — When we
need the graph corresponding to y = kx it is unnecessary to
plot a number of points. Knowing that it will be a straight
line through the origin we determine a single point and join
this point up with the origin by means of the ruler. It may
be asked whether there is any means of drawing the curve
corresponding to 2/ = kjx without plotting points, just as a
line is drawn with a ruler or a circle with compasses.
^The term "rectangular hyperbola" is, perhaps, best reserved
until ch. XXVI. is reached.
'^ This section may be omitted or taken in revision .
10*
148
ALGEBRA
Such means exist though they are not so simple as the use
of ruler or compasses. The following is, perhaps, the simplest.
Begin by drawing the
axis FOF (fig. 41).
On OX take A where
OA = J k, and so de-
termine the vertex V.
Mark off OF and OF'
each of length 2 J k.
Also determine a point /
where V/ = VF. Now
take a piece of string,
tape or other unstretch-
able material (F'B) and
fasten to one end of it a
length of elastic cord.
Lay the combination
along FF so that (B)
the join of string and
elastic cord is at /. Fix
the string by a drawing-pin at F' and the elastic cord by
another at F. Fasten a piece of string, SP, to the elastic cord
at V and pull the string out gradually, taking care that it
always makes equal angles with the two parts of the cord.
The point of a pencil following the movement of the point on
the elastic cord corresponding to the point P will trace out the
inverse proportion curve. ^
A more accurate but less simple method '^ is to hold a ruler
so that it rotates about F'. At the other end fix on its edge
at K a piece of string which is shorter than KF' by a length
F'/. Fasten the free end at F. As the ruler rotates hold the
string tight against its edge by the pencil point. The curve
will in this way be obtained.
[Ex. XXIII may now be worked.]
Fig. 41.
^ In the rectangular hyperbola the eccentricity = y/2. Thus if
F is a focus, OF = J 2, OV = 2 Jk. Also, if P is any point on
the curve, and F' the other focus, PF' - PF = 2 OV = FJ. Since
PF = PB and since F'B does not stretch the last condition is ful-
filled.
2 Described by Milne and Davis, Geometrical Conies, p. 82.
FURTHER TYPES OF PROPORTIONALITY
149
B. Direct Proportion to the Square or the Square Boot (oh.
IV., § 8 ; ch. XL, § 1 ; Ex. XXIV).
§ 1. Graphic Symbol of y = kx^. — Suppose a number of
squares to be cut from a uniform sheet of cardboard or metal
and to be weighed. The weight (W) would be directly pro-
portional to the area (A). The formula expressing the rela-
tion is W = A;A and its graphic symbol a straight line through
the origin. In fig. 42, the line OP corresponds to the for-
mula in the case where k = ^.
Now, if I is the length of the side of a square, A = P and
W = kP. The weight of a square is, then, directly pro-
portional not only to its area but also to the square of its
side. If we regard OP (fig. 42) as the graph of the formula
W = ^P then the numbers along the horizontal axis are values
not of I but of the square of I. We may ask what form the
graph would take if we regarded these numbers as values of I
itself.
Consider the point P, which indicates the weight of a square
for which A = 16. For this square Z = 4 ; consequently if
P ^
IP
h , .9--'
/ ^
00
10 15
Fig. 42.
20 X
the base numbers are to be regarded as values of I instead of
values of A the point P must be moved until it stands in the
position p above 4. Similarly Q must be moved to q above
3-46 (= 7 12), R above 3 (= ^ 9), etc. The point T above
1 will remain undisturbed, but the point above \ must be
moved to the right till it stands above \.
Joining the points by a smooth curve we have O^ suggested
as the graphic symbol of W = \P-. As in the last lesson the
validity of the suggestion must be tested by taking points at
150
ALGEBRA
random and seeing whether their co-ordinates satisfy the re-
lation.
Generalizing as before we may now regard Op as the graph
corresponding to the formula y = ^x^ in which reference is
made to no special pair of variables. Further, we may regard
this formula as a special case of the formula y = kx^ and in-
quire how the form of the graph changes with the values of k.
The work is, as before, best divided among the class, the
results being represented
with rough accuracy
upon the blackboard.
The points should be
obtained by direct substi-
tution in the formula.
In fig. 43 the curves A,
B, C, D represent the
results of putting k = yg >
J, 1, 3. It is obvious
that the curves form a
family with the pro-
perties (i) of touching
OX at O, and (ii) of
receding from OX with-
out end as x is in-
creased without end.
The point O is called the vertex of each curve. Seeking a
name for the family, we note that C, the graph oi y = x^ is
a " curve of squares ". We may conveniently extend this
name ^ to all graphs corresponding to y = kx^.
§ 2. Mechanical Construction for the Curve.'^ — The curve
of squares corresponding to any value of k can be drawn
mechanically as follows. On OY (fig. 44) take F so that
OF = 1/4A;. Draw DD' parallel to OX and 1/4A; below it.
Take a thick set-square or any right-angled board, A. Pin
at B a piece of string of length BC. Pin the free end at F.
Draw the string tight by a pencil at P and slide A along DD'.
The curve corresponding to y = kx'^ will be traced.^
^ Reserving the name parabola for the complete curve studied in
ch. XXVI., B.
2 See note on A, § 3. The method described is taken from Milne
and Davis, op. cit.
^ Putting OF = a, we have x^ = 4ai/ ; whence a = 1/4A;.
Fig. 43.
FURTHER TYPES OF PROPORTIONALITY
151
§ 3. The Graph of y = k Jx. — The distance visible from
the top of a cliff is given (Ex. IV, No. 6) by the formula
d = 1'22 Jh. This is a case of direct proportion to the square
root of a variable. We may obtain the corresponding graph
as a straight line through the origin if we take the square root
of h as the independent variable. But if we now wish to
exhibit h itself as the independent variable each point in the
straight line must be moved horizontally until it stands over
the square of the original abscissa. For if the abscissa of
a point P was 4 when the abscissa represented Jh it must
be taken as 16 when it represents h itself.
Fig. 44.
The transformation from one set of abscissae to the other
is shown in fig. 45. The points P, Q of the straight line which
stand over the numbers 3 and 2 are removed horizontally to
p, q where they stand over 9 and 4. The point R standing
over 1 is undisturbed. The point S standing over | is moved
to the left and stands at s over J.
The curve Op can now be regarded as merely a special
example of the graph oi y = kjx in which k = 1'22. If
desired we can, as before, explore the results that follow from
assigning different values to k. The different curves can be
drawn in three ways, (i) The co-ordinates of the points can
be obtained by direct substitution in the formula, (ii) Draw
the line y — kx and shift the points horizontally until they
162
ALGEBRA
stand above the squares of the original numbers, (iii) Take
the graph oi y = x^/k^, turn the paper in its own plane
through a" right angle in the clockwise direction ; turn it
upside down, transferring the graph to the back of the paper
by tracing against a window or by pricking points.
The graph ot y = kjx may be called the square root
curve.
[Ex. XXIV, A, may now be worked.]
C. Inverse Proportioji to the Square or the Square Boot of
a Variable (ch. xi., § 1 ; Ex. XXIV, B).
§ 1. The Graph of y == k/x^. — A number of tins are to be
made of J' different sizes but each is to contain the same
Y
6
£
/'
ft
p
J-
-^-
3
2
/'
q
^
^^
/
Q
}
;>
f%
0^
^56
Fig. 45.
'OX
quantity of liquid. The height h of any tin is given by
h = &/A, where k is the constant volume and A the area of
the bottom. That is, the height is inversely proportional to
the area of the bottom. The graph, when k = ^\ cu. feet and
h and A are measured in linear and square feet, is the curve
PTU (fig. 46).
Suppose the bottom of each tin to be a square. Then the
formula becomes h = kjP, that is, the height is inversely pro-
portional to the square of the side of the base. In order that
the graph may express the relation in this form, the abscissae
must be taken to represent I instead of A and the points P, Q, R,
FURTHER TYPES OF PROPORTIONALITY
153
etc., must be removed horizontally until they stand at p, q, r,
etc., above the square roots of the original numbers. (Cf. B,
§1.)
Generalizing, we may say that pHu is the graph of the
relation y = kjx^ when k = 4-5. Similar curves would be
obtained by assigning other values to k.
§ 2. The Graph of y = kj Jx. — In exactly the same
way we could obtain the graph corresponding to the relation
y = kj Jx — i.e. the connexion between two variables one of
I
10
9
a
7
6
5
uiiu
' T '
i
\ i
\\
vr
3
v\
N^
y ^
s"^
s'
/
x
"v.
R
2_
"
~
r^
■
^
^
b:=rr:
.P..
■
Fig. 46.
which is inversely proportional to the square root of the
other. The case when k = 4*5 is obtained by moving the
points P, Q, E, S, etc., of fig. 46 horizontally to the positions
p\ q, r\ s\ etc., where they stand above numbers which are
the squares of the original abscissae. Ex, IV, No. 31, gives
a simple instance of this kind of proportion.
These methods of obtaining the graphs oi y = k/x'^ and
y = kj Jx are instructive as indicating their relations to the
inverse proportion curve. The graphs could, of course, also
154 ALGEBRA
be plotted as the result of direct substitution in the formulae.
Moreover the graph oi y = hj Jx could be derived from that
oiy = 1/^V^ in the manaer described in B, i^ 3.
[Ex. XXIV, B, may now be worked.]
D. Combinations of Types of Proportion (ch. xi. , § 1 ;
Ex. XXV).
§ 1. Conjoint Direct Variation. — A suite of rooms has the
same breadth throughout, but the lengths of the rooms differ.
If they had to be carpeted we should say that the cost of each
room would be directly proportional to the length. Similarly
if a number of rooms had the same length but differed
in breadth the cost of carpeting them would be directly pro-
portional to their breadth. Finally, if we are considering
rooms in which both length and breadth differ the cost will be
proportional to the area, that is, to the product of the length
and breadth.
These facts can be expressed concisely in the following
form : —
G cc I {b constant)
C Gc 6 (Z constant)
G a: bl (when b and I vary)
The symbol oc = "is directly proportional to ". It is con-
venient to use it in cases like this since each of the three state-
ments, if expressed in the usual form, would require a different
constant. The symbol is often read " varies . . . with ".
Thus y cc X would be read "y varies directly with x," while
y cc - reads " y varies inversely with x ".
X
§ 2. Generalization. — The result just obtained can easily be
generalized. Let P, Q and E be any three variables whose
values are at any moment x, y, z. Suppose also that : —
zee X when y is constant
zee y when x is constant
then shall z a: xy when both vary.
Let k be the value of R when P and Q both have unit value.
Let Q remain of unit value while P's value changes from 1 to x.
At the same time let R's value become z. Then, by the first
relation we have z' = hx. Now that P has reached its final
value let it remain constant and let Q begin to change from
FURTHER TYPES OF PROPORTIONALITY 155
1 to y. Then R which has the value z = kx when Q begins
to change will (by the second relation) reach a value z such
that
z = z.y
= kx.y
when Q has reached its final value y. This argument holds
good whatever be the final values, x and y, of the variables
P and Q. Thus
z = k.xy or z cc xy.
§ 3. More Complicated Cases. — This weight of liquid (W)
that can be poured into a cylindrical tin of given cross- sec-
tional area (A) varies directly with the height (h). That
is : —
yV cc h (A constant)
Similarly W oc A {h constant)
and, in accordance with § 2, W cc A^ when both vary.
But we have also Ace r'^ where r is the radius of the base.
Hence we may write W cc hr'^.
This result when generalized takes the form : —
If z cc x (y constant)
and z cc y^ (x constant)
then z cc xy^ (if both vary).
The proof will be conducted precisely as in § 2.
If in the above problem we wished to calculate the height
of the tin required to hold a given weight of liquid when
the base had a given area we should have : —
h cc W (A constant)
h cc 1/A (W constant)
and, when both vary, h cc W/A.
Here we have an example in which a case of direct pro-
portion or variation is combined with a case of inverse pro-
portion or variation. Moreover since A oc r^ we can transform
it into a still more complicated case of " conjoint variation " : —
h cc W (r constant)
h cc Ijr^ ( W constant)
h cc V^ jr^ (when both vary).
These results can both be generalized and established in
their generalized form as in § 2.
[Ex. XXV may now be worked.]
SECTION II.
DIRECTED NUMBEBS.
THE EXERCISES OF SECTION II.
*^'^ The numbers in ordinary type refer to the pages of Exercises
in Algebra, Part I ; the numbers in heavy type to the pages of
this book.
BXERCISE
XXVII, The Use of Directed Numbers
XXVIII. Algebraic Addition and Subtraction
XXIX. Directed Products
XXX. Summation of Arithmetic Series
XXXI. Algebraic Multiplication
XXXII. The Index Notation .
XXXIII. Negative Indices
XXXIV. Factorization ....
XXXV. Algebraic Division
XXXVI. Geometric Series
XXXVII. \ The Complete Number-Scale
XXXVIII. Further Examples on Directed Numbers
XXXIX. Linear Functions
XL. Directed Trigonometrical Ratios
XL I. Surveying Problems .
XLII. Hyperbolic and Parabolic Functions
XLIII. Quadratic Equations .
XLIV. Further Equations
XLV. Inverse Parabolic Functions (I)
XLVI. Inverse Parabolic Functions (II)
XL VII. Area Functions ....
XL VIII. Differential Formul/E
XLIX. Gradients
L. The Calculation of it and the Sinb-Tablb
PAGES
155, i8i
161, 184
168, 193
175, 199
183, 207
187, 214
191, 217
193, 176
196, 222
201, 224
208, 228
213, 230
217, 258
221, 261
224, 256
229, 264
235, 270
239, 240
242, 274
247, 243
250, 279
255, 282
258, 252
264, 292
CHAPTEE XVII.
THE PROGRAMME OF SECTION II (EXS. XXVII-XXXVIII).
^ 1. The Use of Directed Numbers (Ex. XXVII).— It was
explained in ch. v., § 3, that the distinctive task of Section II
is the study of the consequences which follow when algebraic
symbols are understood to represent numbers with signs
instead of the signless numbers of arithmetic. The object of
Ex. XXVII is to give the pupil his first introduction to these
new numbers. Simple as the lesson is, it is one of critical
importance.
Historians of mathematics have remarked upon the slow-
ness with which ideas about the nature of positive and nega-
tive numbers developed into clearness in the early days of
algebra. It is, perhaps, not too much to say that the diffi-
culties which the older thinkers found or raised remain, like
an undispersed fog, to obscure the path of the beginner to-day.
These difficulties all have the same origin — the failure to per-
ceive that the difference between positive and negative is not
synonymous with the difference between greater and less.
This confusion of two distinct ideas vitiates, for instance, the
example with which beginners have made first acquaintance
with negative numbers in every generation since the end of
the sixteenth century. If a person (we are told) who
possesses £5 is to estimate his wealth as " +5," then a
person who has no money in his purse but owes £5 must write
his wealth down as " - 5," for he has £5 less than nothing.
Now the difficulty which the beginner has to face here lies
not in any abstruseness of the new idea but in its absurdity ;
for he is asked, as the price of admission to the new subject, to
give up the conviction of common sense that there cannot be
anything less than nothing. So unfortunate a result might
well have suggested a careful scrutiny of positive and negative
numbers with the object of determining whether it is inevitable,
159
160 ALGEBRA
or whether the true nature of the numbers had been mis-
apprehended. It is, therefore, a rather strange fact that it
remained for the mathematical philosophers of the last cen-
tury to put the theory of these important entities upon a
more satisfactory basis.
The upshot of their investigations is the discovery that
over a large part of the field of mathematics the fundamental
idea is not magnitude but order. Some of the more salient
consequences of this discovery are given in the exercises of
Part II, Section IV, to which the reader who is unacquainted
with these modern developments may turn for information in
the first instance.^ Their application to the question of nega-
tive wealth is easily illustrated. Suppose a company of
persons of varied financial position to be gathered together,
and let a humorous tyrant insist upon examining their bank
pass-books and arranging them in a single row in order of
their financial status. We will suppose his edict to be that
every man must stand so that his left-hand neighbour is in
a better, and his right-hand neighbour in a worse, financial
position than himself. Then common sense will itself insist
that the man who owes £5 shall be to the right of the man
who has nothing and owes nothing, and that a man who owes
£6 shall stand on his right and one who owes £4 on his left.
Now let the tyrant proceed to label his victims with numbers
indicative of their relative status. He may, perhaps, start by
assigning to the possessor of £5 the numerical label "5," and,
proceeding on this plan, will mark the man who has neither
possessions nor debts with the label " ". He has now to
deal with the unfortunates who have to a greater or less ex-
tent overdrawn their banking accounts. In labelling these
he may reasonably adopt the view that the man who owes
his banker £5 is, in respect of his financial position, just as
far removed in one direction from the person marked zero
as the possessor of a balance of £5 is in the other direction.
To signify at once the resemblance and the difference in their
situations he may adopt the device of prefixing to the common
number "5" the distinguishing signs + and -. If these
signs are distributed in a similar way to all the participants
in this one-sided game, we shall have the following results.
1 Hq will find the treatment moat suitable for his purpose in
Young's Fundamental Concepts of Algebra and Geometry.
THE PROGRAMME OF SECTION II 161
All persons on one side of the zero-person will bear numbers
prefixed with the sign plus, the numbers increasing as the eye
moves along the row to the right. All on the other side of the
zero-person will have numbers prefixed with the sign minus,
and these numbers will increase towards the left. Thus the
numerical label, together with the sign, serves to fix the
position of each person in a perfectly definite way, so that
they could instantly be rearranged as before if the row were
to be temporarily broken. Moreover the device chosen by
the tyrant would obviously enable him to assign a definite
place and label to an endless number of fresh victims.
This example, in spite of its triviality, brings out the real
significance of positive and negative numbers. Their essential
function is not to measure magnitude but to register position
in a series in which the terms are arranged upon some prin-
ciple of order. They are most appropriately used when the
series is either actually or potentially extensible without end
on both sides of any given term, and they serve to indicate
the (real or metaphorical) distance and the (real or meta-
phorical) direction of the various terms from a term selected
as the point of reference and marked zero.
The reader will now see why the name ** directed numbers "
is given in this book to numbers accompanied by plus or
minus signs. Such a number always refers, expHcitly or
implicitly, to a term of an ordered series, and the sign always
indicates the direction in which the term is to be sought,
starting from a certain point of reference or zero. In many
cases — and those among the most important — the word " direc-
tion "is to be understood literally ; in other cases it is only
a readily intelligible metaphor. But, in every case in which
positive and negative numbers are applicable, direction in one
of its senses must be present and constitutes the essence of
the conditions which make their use appropriate.
From these considerations an important practical conclu-
sion follows. If positive and negative numbers are used only
when the idea of direction is present, then it will obviously
be best to teach their use first in cases where the element of
direction is explicit and has the familiar spatial character.
This is the procedure followed, not only in Ex. XXVII, but
also in the subsequent exercises in which the class are to ex-
plore the laws of manipulation of directed numbers.
The exposition suggested in ch. xviii.. A, needs no further
T. 11
162 ALGEBRA
infcroiuction. Nevertheless it may be well to emphasise the
recommendation made in the lesson, that plus and minus
signs should consistently be prefixed to directed numbers.^
The adoption of this simple custom proves vsronderfully
effective in maintaining clearness of ideas, and, therefore,
well repays the trifling trouble and delay which it involves.
In particular it will serve to keep the young algebraist on
his guard against the notion — generally harmless enough,
but occasionally a source of mischief — that if a number has
no sign it is necessarily positive.
§ 2. The Manipulation of Directed Numbers (Exs.
XXVIII, XXIX). — The next two exercises deal with subjects
which, by general consent, are among the most difi&cult in the
teaching of elementary algebra. These are the rules which
are to be observed when directed numbers are added and
subtracted, multiplied and divided. As was pointed out in
ch. v., § 2, the problem of the ** rule of signs " is met with at
four distinct points in the theory of algebra. It arises first
when we are called upon to evaluate such an expression as
46 - 3(15 - 8).
Here, as we saw in Ex. XIII and ch. ix., B, the problem is
an arithmetical one and can be solved by arithmetical con-
siderations. We have merely to take from 46 three times
the difference between 15 and 8, that is, three times 7 or 21.
But if the problem is to find the value oia + h ox a - h when
a.= + 8, 6 = - 4, the case is very different. The root of
the difficulty in which we find ourselves is that the signs
connecting the a and h can no longer mean "add" and
" subtract " in the common meaning of those terms. This is
evident from the fact that when we " add '' the - 4 to the
+ 8 the arithmetical operation which actually has to be carried
out is subtraction, and is addition when we " subtract " it.
Thus our first task is to determine what the signs plus and
minus mean when they connect symbols which stand for
directed numbers. It is evident that a similar difficulty will
face us if we seek to evaluate the forms
ah or afh
when the numbers represented by the symbols are directed.
^ Mr. G. St. L. Carson has suggested writing 5 instead of - 5,
etc., when the minus belongs to the number. The one objection
to this proposal is that it would be inconvenient, at any rate in
print, to deal similarly with a plus.
THE PROGRAMME OF SECTION II 163
For what can possibly be the meaning of an instruction to
take ( + 8) ( - '4) times ? Thus here again the first task is to
interpret a familiar piece of symbolism in circumstances in
which it no longer bears its usual meaning.
It is worth while to remark that the difficulties here in view
affect the manipulation of positive equally with negative
numbers. It is generally supposed, for example, that although
the determination of the value of the * ' product " of - 4 by
- 8 constitutes a real problem, the evaluation of ( + 4) x
( + 8) does not. It is assumed that the answer in the latter
case is obviously +32. But consideration will show that
this easy triumph is a hollow one, being gained by the totally
illegitimate identification of + 4 and + 8 with the signless
numbers 4 and 8. The truth is that, without some definite
convention or definition, " nmltiplication " in the case of two
positive numbers is just as mysterious, and, indeed, meaning-
less an operation as in the case of two negatives. Both cases
are equally in need of examination.
There are two distinct ways in which these problems may
be attacked. The first is by the " high priori road " of arbi-
trary definition and postulate. We may, for example, say
that the operation denoted by the symbol x is to mean ordin-
ary multiplication when it connects two non-directed numbers,
and when it connects directed numbers is to be taken as
defined by the following laws : —
{+p)x(+q)=+pq, {-P)x(-q)= +M
{+p)x{-q)=^ -pq, ^ {-p)^{ + q)= -pq
Of this method it is sufficient to say here that although
theoretically unimpeachable it is utterly unsuitable as an
introduction to the subject. It is impossible for the young
student to appreciate or to profit by a mode of treatment
which presupposes a trained power of logical detachment and
abstract reasoning. In practice it can lead only to " symbol
juggling " of the most unsatisfactory kind, in which the
student carries out manipulations in accordance with given
rules but has only the most shadowy idea of their real
meaning.
The general principle that all forms of algebraic symbolism
should be taught as natural ways of describing certain facts
suggests a method of dealing with this question much more
suitable to the needs of the beginner. The essence of the
method is the rule that the results of a given manipulation
11*
164 ALGEBRA
of the algebraic symbols shall always accord with the knoivn
behaviour of the things to tvhich they correspond. Thus the
formula d = vt ia an obvious way of describing the position
at any moment of a point moving with uniform velocity v.
Moreover the three elements to which the three symbols
correspond are all capable of description by directed numbers.
Thus the study of the relation between this formula and the
behaviour which it is intended to describe gives a good oppor-
tunity for determining the rules to be followed in evaluating
a piece of symbolism of the form vt when the symbols repre-
sent directed numbers. Now it is easily shown (as in ch.
XVIII., C) that when both the velocity of the point and the
moment of time under consideration are to be denoted by
negative numbers then the distance of the point from its zero
will always be denoted by a positive number. From this
known fact about the things referred to we draw a definite
rule for the manipulation of the symbols which refer to them :
namely, that the product of two negative numbers must
always be considered to be positive.
The reader may object that the conclusion goes beyond the
evidence, and that this rule for the product of two negative
numbers has been proved to hold only when they represent
respectively a velocity and a time. This objection is removed
by the argument of ch. xviii., C, which shows that the rule
must hold good for the products of all pairs of numbers which
can be represented graphically upon the same principle as
velocity and time in the case of a moving point. The reader
may readily satisfy himself that this condition covers all cases
in which the ordinary laws of algebraic multiplication apply,
and that it fails to cover only those cases, such as the product
of vectors, which as a matter of fact lie outside the scope of
the ordinary rules of algebra.
§ 3. Algebraic Addition and Subtraction (Ex. XXVIII,
ch. XVIII., B). — Before products and quotients of directed
numbers are considered the case of sums and differences is
discussed in ch. xviii., B. We have said that the root diffi-
culty here is that in speaking of a "sum" or "difference"
of directed numbers we cannot be using these words in their
ordinary arithmetical sense, and that we have, therefore, to
make clear the new meanings which are to be attributed to
the familiar terms. But closer examination shows that this
statement does not express very accurately the nature of our
THE PROGRAMME OF SECTION II 165
task ; it will be found that we have not so much to discover
new meanings for the words " sum " and " difference " as to
determine what parts of their common meaning do and what
parts do not apply to directed numbers. Perhaps the best
way to discover the common element is to examine a case of
" addition " in which the addenda are not numbers at all.
A simple case of this kind is the addition of " vectors " illus-
trated by fig. 21, p. 112. In this figure the lines AC and CB
represent two vectors and AB represents a third vector which
is called their " sum ". Thus if the three vectors are denoted
in order by the letters a, h and c, the relation between them
is expressed by the formula
c = a -\- h.
Now what is the meaning of this formula ? Remembering
that the vectors represent certain movements whose directions
are parallel to those of the lines and whose magnitudes are
represented to scale, we see that the formula is equivalent to
the statement : If you take the movement a and take also the
movement h your ultimate position will be the same as if you
had taken the movement c. It is clear that the one element
common to the use here of the sign *< + " and to its use in
arithmetic is the idea represented by the words " and take
also". If I take 4 books "and take also" 6 books the
result is the same as if I had taken 10 books. This is the
meaning of the statement that 4 + 6 = 10, or that 10 is the
" sum " of 4 and 6. Thus, if some one asks me what is the
" sum " of 4 books and 6 books and also what is the " sum "
of a movement of 4 yards in one direction and a movement of
6 yards in another direction, I recognize that the element of
identity in the two problems which justifies the use of the
same word in enunciating them is the fact that I have to con-
sider in each case the result of taking two components together.
But the identity between the problems stops at this point.
To find the resultant (as it may be called) of the two numbers
of books I have to follow one rule — the rule of arithmetical
addition. To find the resultant of the two movements I have
to follow another rule — the rule of " vector addition ".
Similar considerations bring out the common meaning con-
tained in different uses of the terms " subtraction " and
"difference". Thus, in fig. 21, if the vectors h and c are
given and the vector a is required, the process by which a is
166 ALGEBRA
determined is called " subtracting " the vector b from the
vector c and is represented by the formula
a = c - b
To understand how this process can be called subtraction
we must note that c is here regarded as the resultant of two
components, a and b, of which b is known and a unknown.
To find the unknown a we draw the combination a + b which
is equivalent to c and then discard the b. Thus we have
a = (a + b) - b
= c -b
as above. Similarly, if I want to know the result of sub-
tracting 4 from 10 I remember that a whole composed of
10 books, or other countable objects, can be regarded as made
up of two parts composed of 4 and 6 respectively. Hence if I
remove or ignore the 4 I am left with 6 as the result of my
subtraction. Thus the common element in all problems of
subtraction is that we are given a resultant of two components
and also one of the components, and are required to determine
the other component. But, as in the case of addition so here,
rules for making this determination in any given case will
depend upon the known properties of the things combined.
The foregoing considerations underlie the treatment of
algebraic addition and subtraction in ch. xviii., C. The argu-
ment is, in brief, as follows. A piece of symbolism such as
- 6 + 2 = - 4
may represent the solution of two distinct problems. In one
of those problems we may have been told that two movements
of - 6 and + 2 have been made in succession, and the question
is what is the single movement to which they are equivalent.
In the other we may have been told that two successive
movements have resulted in a displacement of - 6, one of
these movements being - 2, and the question is what was the
other component movement. Now our uncertainty as to
what the problem was comes from the fact that the numbers
are directed. Similar problems involving numbers without
signs could be discriminated at once : if the numbers were
connected by a plus the problem would be one of the first
kind, if by a minus one of the second kind. The question
arises, therefore, whether the signs phis and minus cannot
be used to discriminate the problems symbolically when the
numbers are directed as well as when they are undirected.
THE PROGRAMME OF SECTION II 167
The result of this inquiry is the decision that the first problem
may be written in the form
(_6)+( + 2)=-4
and the second in the form
(_6)-(-2)= -4
The result of this part of the discussion is, therefore, to
bring out the fact that, in pieces of symbolism like those just
quoted, the signs + and - are each used for two distinct
purposes, neither of which is the same as its ordinary arith-
metical use. The signs within the brackets are attached to,
and form part of, the numbers, and are indicative of direction.
The signs between the brackets have a totally different func-
tion — they are there simply to show to which of two types
the given problem belongs.
The first stage in the actual solution must be the removal
of these latter signs, which have done their work when they
have informed us of the nature of the problem under con-
sideration. The rules for manipulating the signs attached
to the numbers when the enclosing brackets are removed form
a very important detail of the argument. The method fol-
lowed is the one explained in § 2. By considering the actual
and known results of typical problems of each kind we de-
termine what rules of signs must be adopted in order that
the manipulations of our formulae may agree with the known
behaviour of the things to which they refer.
Ex. XXVIII, A, consists of simple examples intended to
illustrate and to *' drive home " the distinctions and the rules
reached in the course of the lesson. No. I may be taken
orally. Nos. 2 to 9 have a flavour of the puzzle about them
which makes them surprisingly attractive to the young
student. In division B the newly acquired ideas are applied
to the making of formulae in which the symbols are to stand
for directed numbers. These formulae illustrate the striking
gain in generalizing power which comes with the adoption of
these modes of expression. Cases which formerly could be ex-
hausted only by using several different formulae are now found
to come within the scope of a single one. After working
the first few of the examples the student should be convinced
that to obtain a formula applicable to all possible cases it is
sufficient to consider a case in which the numbers either are
non-directed or are all positive. The formula which describes
this case will be equally applicable to all others. The ex-
168 ALGEBRA
amples of division C may be considered as optional. They
are inserted rather as a tribute to convention than because
they are of any particular use at the present stage of the
pupil's progress. But if done at all they should certainly be
done with the careful attention to their proper significance
which is illustrated in ch. xviii., B, §§ 6, 7.
§ 4. Products and Quotients of Directed Numbers (Ex.
XXIX ; ch. XVIII., C). — The method followed in dealing with
the rules of signs of products and quotients of directed numbers
has been sufficiently explained in § 2. In Ex. XXIX, A, the
student is shown how these rules make it possible to widen
still further the range of cases included in the reference of a
single formula. Thus these examples supplement those of
Ex. XXVIII, B, and complete our study of the primary con-
sequences of permitting the symbols of a formula to represent
directed instead of non -directed numbers.
The examples of division B illustrate the power of the new
methods of algebraic statement by applying them to the de-
scription of the terms of an arithmetic sequence — that is, an
endless series of numbers exhibiting from term to term a con-
stant difference. The objections which good authorities have
urged against the conventional treatment of arithmetic pro-
gressions would probably not be maintained here. The con-
cept of the doubly- endless sequence of directed numbers is
not only most attractive to young students but is also of
fundamental importance throughout algebra. If a practical
justification for its introduction is required we may point to
the theory of logarithms in which the central idea (at least
as it is treated in this book) is the correlation, term by term,
of a doubly-endless arithmetic sequence with a doubly- endless
geometric sequence. For the present we rely upon its intrin-
sic interest and theoretical importance and its value as a
means of demonstrating the powers of directed numbers.
In division C the examples deal with " progressions," that
is with sections of a complete sequence counting forwards
from a given term. Here again the subject is introduced
episodically and as a further study in the properties of directed
numbers, and not as a new " rule ". The examples have,
however, special importance in connexion with the summa-
tions which are to be studied in the next exercise.
§ 5. The Summation of Constant-difference Series (Ex.
XXX, A ; ch. XIX., A). — The summation of arithmetic series
THE PROGRAMME OF SECTION II 169
is a rather trivial occupation regarded as an end in itself.
The indubitable attraction which it has for the young mathe-
matician may, however, as in the exposition of ch. xix., A, be
made a convenient means of introducing a concept — that of
positive and negative areas — which will be found of great use
in developing the theory of algebraic multiplication in ch. xx.
This consideration alone would suffice to justify the attention
given to the subject. But as will shortly be seen, this process
of summation is preliminary to an enterprise of considerable
moment — namely, the student's first excursion into the pro-
vince of the integral calculus. Thus it has much practical as
well as theoretical significance.
The exposition of ch. xix., A, needs no supplementary ex-
planation. The teacher is advised to adopt the device there
described, and to distinguish between positive and negative
areas by the use either of coloured paper or of coloured chalk.
The pupils should use the same device in their answers to
Ex. XXX, No. 1. Time may be saved by distributing long
strips of red and blue paper of a convenient width which
have previously been coated with gum upon the reverse side.
From these it is easy to cut columns of the required length
to be stuck in the exercise book.
§ 6. The Calculation of certain Areas and Volumes (Ex.
XXX, B ; ch. XIX., B). — The grounds for including an ele-
mentary treatment of the ideas of the calculus in the algebra
course have been stated in ch. ii., § 3, where the general spirit
in which it should be undertaken was also discussed. We
have now reached the point at which the systematic develop-
ment of the ideas there defended is to begin. It will be
observed that we start, not with differentiation, but with
integration. Many reasons may be given for this order of
procedure, but the teacher whose class has worked Ex. XXX
will probably agree that the simplicity of the argument and
the power which it puts at once into his pupils' hands are a
sufficient justification even for so radical a departure from
tradition. If a further defence is required we point to the
fact that the exposition of ch. xix., B, follows closely the track
which led our famous countryman, John Wallis (1616-1703),
to what was practically the invention of the integral calculus.
Wallis built, of course, upon the ideas of his predecessors —
notably Archimedes among the ancients and Cavalieri (1598-
1647) among the moderns — but he was the first mathematician
170 ALGEBRA
to attempt a systematic " arithmetic of infinites ". His work
was the starting-point of Newton, Leibniz, and the other
mathematicians who gradually gave to the calculus the shape
in which we know it and use it to-day. But, as in so many
cases, the later form of the science, although technically
more perfect and a more effective instrument of mathematical
investigation, is not so suitable for the purposes of the teacher
as the earlier form. The beginner gets a more rapid insight
into the meaning of the new ideas and a more complete
mastery over them if they are presented to him as they oc-
curred to the pioneer and in connexion with attempts to solve
the same kind of problems. For this reason the method and
subject-matter of ch. xix., B, are borrowed, almost as they
stand, from the first chapter of Wallis's Arithmetica In-
finitorum (Latin folio edition, Oxford, 1699).
The teacher will find it useful to prepare for exhibition to
his class the models which correspond to figs. 54 and 55.
It is convenient to have in duplicate the one represented in
fig. 55 and to fix the grey rectangles of one of the two speci-
mens in the positions shown in fig. 56. It is interesting to
note that the argument used in connexion with these figures
and Exercises, fig. 34, is taken from Cavalieri, Geometria
Indivisibilibus Gontinuorum, a work from which Wallis
admits drawing much inspiration.
The examples of Ex. XXX, B, are of great importance and
should be worked very thoroughly, for they illustrate in a
simple and concrete manner the central problem of integra-
tion and the way in which it may be solved. Briefly, the
problem is : Given the law followed by the rate of change of
a thing's magnitude, what is the law governing the magnitudes
themselves? The contribution of the present examples to
the solution of this general problem is that if the rate of
change [v) of the magnitude of a variable x follows the law
V = a + ht
then the magnitude of x is itself given by the formula
X = at ■\- ^bP
The examples contain various applications of this interest-
ing result, but the most important are those in which the
variable is a distance and the rate of change of its magnitude
the velocity of a moving point. These examples are not to be
regarded as a trespass into the province of mechanics. It is
THE PROGRAMME OF SECTION II 171
true that a text-book of mechanics always contains a section in
which the formulae v = at, s = ^at^, etc., are proved and applied
to kinematical problems, but this circumstance is to be taken
rather as evidence that the teacher of mathematics leaves part
of his proper work undone than as an argument against the
inclusion of these same problems in this book. The business
of the student of mechanics is to determine what forms of
motion are actually exhibited by material bodies, to define
them quantitatively, and to determine the physical conditions
which govern their appearance. The investigation of the
mathematical consequences of laws of motion suggested by
common observation and experience is no more an invasion of
his territory than is the investigation of the trigonometrical
relations of which he also constantly makes use. Motion is
simply " geometry _pZws time," and any reason which justifies
the study of geometry as a branch of mathematics must
justify equally the inclusion of kinematics.
In Nos. 10-14 it is important to note that the constancy
in the final rate in question is represented in the graph by the
constant height of the last column. This is true however
narrow the column may be — that is, however short a time is
under consideration. Thus when the increase in the rate is
continuous the rate at any moment is measured by the height
of the corresponding ordinate. This argument gives the
auswer to No. 14. It also prepares the pupil for the con-
sideration of a rate as a " limit " — an idea which will demand
careful attention at a later stage of our work (see ch. li.).
In solving problems (such as Nos. 19, 21, 22) in which the
units employed in measuring the rate are not identical with
the unit represented along the horizontal axis the teacher is
strongly advised not to deal with the " change of unit " by
any formula or similar manipulation of the numbers, but to
work directly on the principle that the area of the diagram
represents the space covered in the given time. Thus in
No. 19 the abscissae measure minutes and the ordinates miles
per hour. The triangle whose base is 12 and height 60 has
an area of ^ of 60 x 12, that is 360 units, and represents the
distance travelled in 12 minutes, which will be 6 miles.
Thus each unit of area, of the figure represents a distance of
6/360, or 1/60 of a mile. (Note that the unit of area will not
be a square unless the horizontal and vertical scales of the
diagram are identical ; in general it will be a rectangle.) Now
172 ALGEBRA
the law for the velocity in miles per hour measured after t
minutes is evidently
V =^ 5t
so that the area of the triangle whose base is t will be ^. of
6t X t. Applying the condition that each unit of area repre-
sents 1/60 of a mile we have the formula
s = ^ ot 5t X t X ^^
No. 29 represents the type of calculation in which the argu-
ment reaches its culmination. Given the relation
V = 7-2 + 6'4:t
the student should now be able to proceed at once to the
conclusion
s = 7'2t + 3-2^2
The argument here dififers from an integration only in the
absence of the traditional notation in which integration-
processes are commonly expressed.
§ 7. Algebraic Multiplication (Ex. XXXI, ch. xx.). — The
problem studied in ch. xviii., C, was to find the single
directed number represented by the symbolism ab when both
a and b represent given directed numbers. Thus it is a
problem which concerns not the manipulation of a formula
but its arithmetical evaluation. The problem of algebraic
multiplication is quite different from this^ for it concerns the
validity of algebraic identities when the factors of one of the
equivalent expressions are themselves algebraic sums or dif-
ferences. Consider, for example, the familiar identity
(a + b){a - b)=^a^ - b^
When a and b are symbols for signless numbers and -f and
- mean "add "and "subtract" in the ordinary senses of
the words there is no difficulty in proving the validity of this
general statement by the process which we learnt in ch. vii.
to call " multiplication ". But when a and b become symbols
for directed numbers -f and - lose their ordinary signifi-
cance and acquire the new meanings explained in ch. xviii.,
B, and in § 3 of this chapter. It would, therefore, be quite
illegitimate to assume that the argument which established the
equivalence in the former case suffices to establish it also in
the new. It is evident that the new case must be examined
upon its own merits.
The task of ch. xx., A, is to prove that the method of
THE PROGRAMME OF SECTION II 173
algebraic multiplication which may be used to find and to
prove identities, when the symbols stand for non-directed
numbers and the connecting signs imply arithmetical addition
and subtraction, does as a matter of fact hold good when the
symbols stand for directed numbers and the connecting signs
imply algebraic multiplication and subtraction. When this
point has been established we shall have proved that the
symbolism which describes the relations and manipulations
of non- directed numbers holds good in every respect for
directed numbers. Thus any result which has been shown
to be true for the one kind of numbers may henceforward be
assumed to be true also for the other kind. In other words
we shall have proved that a single rule governs the combina-
tions of plus and minus signs over the whole range of their
significance.
A word may be added about the method of investigation
adopted in ch. xx., A. It would, no doubt, be possible to find
a general method which would prove at once that the rule of
signs must hold good when plus and minus signify algebraic
addition and subtraction. A general proof has, however, the
disadvantage that it does not bring the student's mind into
that close relation with particular cases which. is the necessary
foundation of full and exact knowledge. For this reason the
opposite plan is adopted and the student is led to establish
his hypothesis by examining all the standard cases. It will
also be observed that the method of examination followed in
ch. XVIII. is again followed here. That is, the rule which
governs a particular kind of manipulation is determined by
observing the actual behaviour of the things which the
formulae describe. The details of the lesson — including the
use of the coloured rectangles — are given too fully to need
further elucidation. It will be noted that one set of coloured
diagrams is given as the frontispiece of Exercises, Part I,
in order that the pupil may have it before him in working
Ex. XXXI, A.
Of the examples in divisions B and C it is sufficient to say
that they are chosen to illustrate the real significance of
algebraic multiplication — namely, that it is a process by which
we can predict the typical form of a product when we know
the typical forms of its factors. The danger in this part of
algebra is always that the pupil may forget that his Results
are valuable only for their analytical significance, and may
174 ALGEBRA
come to suppose that the power of manipulating symbols
skilfully is itself to be counted as a virtue.
The fact that the process of algebraic multiplication is
always the determination of an identity is best kept before
the student's mind by insistence upon the mode of setting
down illustrated in a simple case in ch. xx., A, 5, and previ-
ously in ch. VII., B, § 2. Thus Ex. XXXI, No. 32 (ii) should
be worked as follows : —
(3^2 _ 2i+ i)(i2 _ 2^+ 3) = 3i* - 2^3+ t2
-6t"+ 4:t^-2t
One of the advantages of postponing algebraic multiplica-
tion to the point here assigned to it is the possibility of illus-
trating the power of the process by setting the pupil to solve
simple problems of the type which will be recognized later as
covered by the formula of the binomial theorem. A selection
of these is given in Ex. XXXI, D. The coefficients of the
various expansions are to be determined by the empirical
method which was the only one known until Newton dis-
covered the general rule by which any binomial expansion
can be written down without reference to the coefficients of
any other. This generalization is reserved to Ex. LXVIII.^
§ 8. The Theory of Integral Indices (Exs. XXXII,
XXXIII ; ch. xxi., A, B). — One of the first algebraic idioms
which the pupil learnt was the use of the index to symbolize
the square, cube and occasionally the fourth power of a
variable. Many of the manipulative processes studied — in
particular those of the last exercise — have also involved raising
the index of a variable by multiplication. Nevertheless there
has been no point at which there was anything to be gained
by a systematic examination of the principles which underlie
the use of the integral power-index, and the pupil has been
left to deal with the particular cases as they arose without
being called upon to formulate the rules of his procedure.
The point has now been reached at which such a formulation
would be useful ; it is made, therefore, the subject of ch. xxi.,
A, and is illustrated by Ex. XXXII. In accordance with our
1 The author is indebted to Mr. 0. O. Tuckey of Charterhouse
for the idea of introducing at this point some preliminary exer-
cises in binomial expansions.
THE PROGRAMME OF SECTION II 175
usual plan, the theoretical question of the " laws of indices "
is raised in connexion with a practical problem — namely the
problem of representing concisely and manipulating con-
veniently the large numbers which so frequently occur in
physics and in the statistics of demography and economics.
The indices of ch. xxi., A, and Ex. XXXII are non -directed
or signless numbers ; ch. xxi., B, and Ex. XXXIII introduce
the highly important idea of indices as directed numbers.
Once more the argument has its origin in practical considera-
tions. It is shown that the representation and manipulation
of small numbers would be facilitated if we had a method of
symbolizing repeated division similar to the index method of
symbolizing repeated multiplication. The principle of con-
tinuity suggests the negative index as the most convenient
solution of this problem. There then arise the questions :
(i) what rules must be followed when negative indices are to
be combined with one another, and (ii) what rules must be
followed when negative are combined with positive indices.
The inquiry takes the course made familiar by our investiga-
tions, of the rules of signs, and leads to the conclusion that the
laws of indices established for non-direcfced indices also hold
good when the indices are directed.
Attention may be called to a detail of the method. In both
divisions of the chapter the index is studied not so much as a
notation for expressing numbers as for expressing operations
upon numbers. Thus the question raised does not directly
concern the meaning of the symbolism a" but that of the
symbolism x a". The distinction is not one of vital import-
ance, but the adoption of the point of view here recommended
will certainly be found to add clearness and interest to the
discussion. As the result of it the student should possess the
perfectly simple and definite notion that x a+^ means p
successive multiplications by the number a, and x a~^ p
successive divisions. The number a'^^ can be regarded as
equivalent to either 1 x a+^ or a x a+^~i, and the number a~^
as equivalent to either l-i-a"''^ or a-^a+^+^. It is supposed
here, of course, that p is non-directed.
The term " standard form " used in Ex. XXXIII, though
becoming widespread, cannot, perhaps, be regarded as univer-
sally known. A number is expressed in the standard form
when it is written as a decimal with a single digit before the
decimal point multiplied by some positive or negative power
176 ALGEBRA
of ten. Thus the standard forms of the numbers 18574
and 0-0005937 are respectively 1-8574 x 10* and 5-937 x 10"*.
Facility in expressing numbers in the standard form will be
found very serviceable in dealing with logarithms.
It is usual to teach fractional positive indices before nega-
tive indices are introduced at all. The teacher may possibly
think that the simplicity and directness of the exposition
suggested in ch. xxi., B, justify a departure from this custom ;
nevertheless, it may be worth while to add a brief note upon
the point. In the first place it should be observed that until
the pupil comes to the study either of logarithms or of
" Wallis's law " of integration he has no occasion to feel the
need of fractional indices ; upon the general principle which
should govern an elementary treatment of the subject there
is, therefore, no justification for introducing them. In the
second place it is obvious that the step from integral to frac-
tional indices is, from the theoretical standpoint, much
greater than the step from positive to negative integral in-
dices. As we have seen, nothing is involved in the latter step
except the extension of the scope of the symbol x a" to in-
clude successive division as well as successive multiplication.
But, as the argument of ch. xxxiv. shows, the interpretation
of a", when n is fractional, involves considerations so much
more complicated than those of the present chapter that they
should certainly be left until a later stage. Many weighty
authorities would exclude them altogether from the elemen-
tary course ; but if included there they should unquestionably
be taken in connexion with the theory of logarithms (cf.
ch. v., 6).
§ 9. Algebraic Division. Constant-ratio Series (Exs.
XXXIV-XXXVI, C; ch. xxii.).— The process of algebraic
division is, of course, the inverse of algebraic multiplication.
There are, however, two ways of regarding its results suffi-
ciently different to demand separate attention. According to
the first the problem of division is simply to retrace the steps
taken in multiplication, that is, to find the two factors of a
given product, or, given one factor, to find the other. This
point of view is represented in the examples of Ex. XXXIV,
including the special case — not of any particular practical
importance — in which the two factors are identical (division
B). The second point of view is presented in Ex. XXXV.
Here the typical problem is to find an integral expression
THE PROGRAMME OF SECTION II 177
equivalent to a given algebraic fraction, or, if no such exact
equivalent exists, to find an equivalent consisting of an
integral expression together with the simplest possible
" fractional complement ".
The practical importance of this process appears in division
C where it is employed to yield " expansions " of fractional
expressions, such as 1/(1 - a), which may be carried as far
as any desired power of the variable. When | a \ <V
these expansions can also be regarded as " approximation -
formulae " representing the values of the fractions to any
desired degree of accuracy. The reasoning upon which this
statement is based deserves careful attention in view of its
application in the theory of endless geometric, or constant -
ratio, series. After a given number of the terms of the ex-
pansion have been computed the difference between their sum
and the real value of the fraction is given exactly by the
fractional complement. Thus the complement is always an
exact measure of the degree of error involved in the adoption
of a given approximation-formula. Moreover, by taking a
sufficient number of terms it seems evident that the comple-
ment (and therefore the error) can be made smaller than any
number that can be named. An exhaustive analysis of this
conclusion need not necessarily be taken at this stage. The
student should, however, see that it involves the assumption
that if the product of a number a by a factor r produces a
number b which is less than a, then a sufficient number of
repetitions of the multiplication will reduce the product below
any given value however small. On the other hand it may prove
desirable to submit this assumption to further examination in
order to make it more acceptable to the critically-minded stu-
dent who occasionally appears in our classes even at the age of
thirteen or fourteen. If Z is a non-directed number >1 it is cer-
tain that 111 is <1 and that by making I large enough Ijl may
be made less than any number that can be named. Now let
any non-directed number a be multiplied by a number r=l + i,
both r and i being non -directed, and let the product be b.
Then it is clear that b^a, for it may be written as a -h ia.
Similarly, when b = ar is multiplied hy r = l + i, the product
ar^ = b + ib must be greater than b. Moreover the excess, ib,
^ The symbol | a | means '^ the non-directed number contained
in a ".
T, 12
178 ALGEBRA
is greater than the former excess ia, for h is greater than a.
The argument may evidently be repeated in connexion with
each of an endless number of successive multiplications by
r. Let I be the number reached as the result of n such
multiplications — that is, let I — ar^. Then we have evidently
proved that I could also be regarded as reached from a by a
series of n constantly increasing steps of which the smallest
is the first, ia. Now every one will admit that, however
small ia may be, a sufficient number of steps of that magni-
tude will carry us from a past any other finite number
however large. ^ If the steps (with the exception of the
first) are all larger than ia the only difference will be that
fewer will be required. Thus it becomes certain that when
r is ^1 we may, by taking n sujQ&ciently large, make ar'^ = l
larger than any number that any one may choose to name.
It follows by the preceding argument that 1/ar" may be made
smaller than any number that can be named. But since this
conclusion is quite independent of the value of a it is equi-
valent to the conclusion that, when r<Cl, ar"^ may, by taking a
sufficiently large value of n, be made as small as we please.
This was the assumption to be justified.
The argument underlying Ex. XXXV, C, enables us, in ch.
XXII. and Ex. XXXVI, to proceed to summations of constant-
ratio series without resorting to the undesirable course of
teaching them as a separate " rule ". (Compare the treat-
ment of constant -difference series in Exs. XXIX and XXX.)
The practical problems presented in Ex. XXXVI are to be
regarded simply as cases for the immediate application of the
equivalences established in Ex. XXXV, Nos. 24-6. In every
case it is important to realize the significance of the " com-
plementary fraction " which measures the degree of approxi-
mation of the calculation. This is well brought out by the
graphic exercises included in some of the examples. It
would probably be inconvenient to avoid altogether the
traditional term " sum to infinity " but the teacher should see
that no mystical idea is allowed to attach to it. The so-called
" sum to infinity " is not the sum of any number of terms at
all; it is merely the value of the series <i(l-fr -hr^-fr^-f r*
-f . . .) increased by the value of the appropriate fractional
complement. Its usefulness consists simply in the fact that
^ This assumption is known as the axiom of Archimedes.
THE PROGRAMME OF SECTION II 179
the sum of any actual number of terms of the series constantly
approaches nearer to it as the number increases, and may be
made to become and subsequently to remain as near to it as
we please.
The examples on annuities, etc., with which the exercise
closes are an important justification for the introduction of
constant-ratio series but hardly require special comment.
They are further developed in the later exercises of Section
III and are made, in Section IV (Part II), to lead to an
elementary theory of life insurance and other actuarial topics.
Finally it may be observed that the whole treatment of con-
stant-ratio series is a natural preparation for, and reaches its
culmination in, the study of logarithms (cf. § 4 of this
chapter).
§ 10. The Complete Number-scale (Exs. XXXVII,
XXXVIII ; ch. XXIII.). — In the preceding exercises of this
section the student has examined the consequences of sup-
posing his symbols to represent directed numbers in the case
of all the fundamental processes of algebra except two — the
simplification of fractions and the manipulations used in
changing the subject of a formula. The consideration of
these two groups of processes is undertaken in ch. xxiii. and
illustrated in the last two exercises of the present group.
The essence of the treatment of these topics in Section I
consisted in studying the ways in which we actually manipu-
late numbers in arithmetic and in basing on those ways our
rules for manipulating the symbols of numbers. Thus if we
are told that a certain number, less 6, is 12 we know that the
number in question must be 12 -f 6 or 18. Since it is clear
that we should proceed to deal with any other similar case in
the same way we have the rule that, given
n- b = a
we may at once write
n = a + b
But it is evident that this rule has been proved only for the
symbols of non-directed numbers, and that (as in the case
of identities) a new investigation is necessary before we may
use it when our symbols stand for directed numbers.
As a preliminary to this investigation it is suggested in ch.
xxiii. that the teacher shall review the properties of directed
numbers, making explicit the various matters that have been
implicitly assumed at earlier stages of the work. Thus the
12*
180 ALGEBRA
pupil is taught to see that the dominant notion underlying
the scheme of directed numbers is the notion of order, and
that the rules for changing the subject of a formula or simpli-
fying fractions must be established by considering not what
happens when numbers are added, subtracted, etc., as in
arithmetic, but what happens when a point moves backwards
and forwards along an endless line representative of the
complete scale of positive and negative numbers.
The examples of Ex. XXXVII are all important but con-
tain no technical difficulties. The positions of the points P
and Q on the parallel scales of fig. 63 (p. 230) illustrate the
following successive stages in the solution of No. II.
i(2a;4-17)-7 = 10-i(l-3a;). . (OP = OQ = -10)
^(2a;-hl7) + i(l-3a;) = 17 . . (OPj = OQi = 4- 17)
|x-|a;=+17-V-i
-fa^=-MOJ . . . (OP2 = OQ.= -hlOf)
a^=-13 . . . (OP3 = OQ3= -13)^
The teacher should note that the technical term " equation "
is introduced for the first time in connexion with these ex-
amples. No doubt it would be inconvenient never to apply
it except when a relation is presented in the form f{x) = but
there is much to be said for restricting it, as a rule, to such
cases.
Ex. XXXVIII is the last of the first group of Section II.
The examples present rather greater technical difficulty than
those of the earlier exercises but do not involve any new
principles. Nos. 9 and lO will be referred to in Ex. XLVII,
and Nos. II-16 are important as foreshadowing the applica-
tion of the binomial theorem to the case of a negative
integral exponent.
CHAPTER XVIII.
DIEECTED NUMBEBS.
A. The Use of Directed Numbers'^ (ch. xvii., § 1;
Ex. XXVII).
§ 1. The Minus Sign as an Index of Direction. — A boy
comes out of a class-room at having been dispatched on an
errand upstairs. When he has reached the 14th stair he
drops his pencil and has to go down 5 stairs to pick it
up. He is now 9 stairs above the landing. You did not
need, of course, to count the steps to find this result. You
merely performed in your head the calculation which we can
write down thus : —
Number of stairs from = 14 - 5
= 9
We should generally say that the minus sign here shows that
the 5 has to be taken away from the 14. It is clear, how-
ever, that the 14 may be regarded as indicating the distance
the boy goes upwards, and the 5 the distance he goes down-
wards. Thus the minus sign is in this problem an indication
not only of subtraction but also of reversal of direction. This
argument may be repeated with other numbers less than 14,
including finally the case in which the pencil rolls on to the
landing. In this case 5 = 14 - 14 = ; the boy has gone
up 14 steps and come down as many.
But if the pencil rolls down the upper flight, across the
landing and stops only on the 8th stair below, then the boy
must descend 22 stairs to reach it. In other words, if he
first ascends 14 stairs and then descends 22 he will eventu-
ally be 8 stairs below the starting-point. Can a subtraction
1 A blackboard diagram is required representing a staircase with
a landing in the middle marked " O ".
181
182 ALGEBRA
sum, arranged upon the same plan as before, give this result ?
Setting down first the movement upstairs and then the move-
ment downstairs we should have : —
s = 14 - 22.
Now it is clear that if minus is taken to mean " subtract "
here the operation is impossible ; you cannot take 22 from
14. But if the minus means that the boy takes 22 stairs
downwards after taking 14 upwards, there is, of course,
nothing absurd in the expression. The only question is, can
it show us that after his second movement the boy will be 8
stairs below O just as the former calculation showed us that
he was 9 stairs above 0? Let us write it down again
thus : —
s = 14 - 22
= 14-14-8
= - 8
If - 22 means "Take 22 steps downwards" we may, of
course, break it up into two stages: "Take 14 steps down-
wards on to the landing and then 8 more on the lower flight ".
These instructions are represented by - 14 - 8. But we
can take 14 away from 14, so the part of the expression
which reads " 14 - 14 " can be dropped out altogether. We
are left with the expression " - 8" as the answer to our
problem, and that answer can evidently be taken to mean
" 8 stairs below O ".
§ 2. Positive and Negative Numbers. — We now see that
when the numbers in a calculation measure distances up or
down a scale the sign minus can be taken to mean ' ' Move
downwards " as well as " Subtract," and the former meaning
remains even when the latter becomes impossible. That is,
we can obtain an answer to our problem when the number
with the minus in front of it is greater than the other just as
well as when it is smaller. We have seen, further, that the
answer in the former case will be a number with a minus
before it, and that this kind of answer always means a
position below the " origin " instead of above it.
In such cases we can think of the minus sign as at-
tached to the numbers that represent movements or measure-
ments downwards. Thus - 5 will be a number which re-
presents a movement downwards on our scale, the minus
being regarded as a part of the number, just as the 5 is.
We can also attach mi^ius signs to the numbers on our
DIRECTED NUMBERS 183
scale below the origin to distinguish them from the numbers
attached to distances above the origin. Numbers which in
this way indicate a movement downwards or a position below
the origin may be called negative numbers.
If our boy ran up two flights of stairs, the first containing
24 and the second 17 steps, we should find his position at the
end by the calculation : —
s = 24 + 17
= 41
It is clear that just as the minus in the former problems
could be taken to mean " Go down " as well as " Subtract,"
so in this problem the plus sign can be taken to mean not
only " Add " but also "Go up ". The plus sign is, then, a
sign that may be attached to numbers to indicate upward
movement or measurement.
In the former cases the fact that the final position of the
boy was below the origin was shown by the minus in front of
the result of our calculation. It will be well to put the plus
sign in front of the result here to show that the final position
of the boy is above the origin, as well as in front of the 24 to
show that this number also represents a movement upwards.
Thus our calculation will now read : —
s = + 24 + 17
= + 41
Similarly it will be convenient to attach plus signs to the
numbers in our scale that indicate positions above the origin.
Numbers which have a plus sign attached to them to indicate
upward movement or measurement may be called positive
numbers. Positive and negative numbers may conveniently
be called directed numbers.
Directed numbers may be used in calculations concerning
many other movements and measurements besides those
which are taken up or down : for example, movements to
right or left, the forward and backward movements of watch-
hands, the thermometer scale, etc. The arbitrary nature of
the choice of the positive direction should be brought out.
In the illustrative calculations every directed number should
have its sign attached as in the last example above.
§ 3. Evaluation of Combinations of Directed Numbers. —
If a boy, starting as before from the landing, goes down 6
stairs, then is called back and goes up 11 stairs on to the
184 ALGEBRA
upper flight, and finally goes down 18 stairs from that point,
where is he at the end of his movements ?
It is clear that he will be at the same place as if he had
taken the two downward movements one after the other and
then the upward movement of 11 steps, or had taken first
the upward movement and then the two downward ones in
succession. That is : —
s = -6+11-18
= -6-18+11
= -24+11
= - 13
That is, he will be 13 steps below the origin.
It seems, then, that we can alter as we please the order
of the numbers representing the movements without altering
the result. But it is easier to calculate the result if the plus
numbers are combined into a single number representing a
combination of the upward movements and if the minus
numbers are similarly combined.
At first the numbers themselves may be rearranged for this
purpose. As familiarity with the manipulations is gained
the numbers can be combined without rearrangement.
[Ex. XXVII may now be worked.]
B. Algebraic Addition and Subtraction (ch. xvii., §§2, 3;
Ex. XXVIII).
§ 1. Two Kinds of " Stair Problems ". — We will begin by
working on the board two " stair problems ". (i) Starting
from the landing I take 5 steps upstairs and then turn round
and descend 18 stairs. Where am I after the second move-
ment ? Answer : Thirteen stairs below the landing. Or, by
calculation
s = + 5 - 18
= - 13
(ii) Starting from the landing I mean to go to a room open-
ing on the staircase 5 stairs above. By mistake I go up 18
stairs. What must I do to reach the proper destination ?
Answer : Descend 13 stairs. To obtain this result by calcu-
lation the working must be arranged as follows : —
s = +5-18
= - 13
DIRECTED NUMBERS 185
That is to say, to answer the second question by calculation
requires exactly the sanae working as to answer the first. A
person coming into the room and ignorant of our discussion
would think that, for some reason, we had worked the same
problem twice over. Yet the questions were really entirely
distinct. In the first you were told two movements which
I made in succession and were asked to calculate where I
should be at the end of them ; while in the second you were
told where I should be after two movements and, being given
one of them, were asked to calculate the other.
We shall find it convenient to use here the terms component
and resultant already employed in connexion with vectors.
(Ex. XIX, B). If we represented movements up and down
stairs by straight lines those lines would, in fact, be vectors,
differing from the vectors of Ex. XIX merely in being re-
stricted to one direction along which they point either for-
wards or backwards. Thus in the first of our problems + 5
and - 18 were components and - 13 was a resultant ; in
the second + 18 and - 13 were components and + 5 was
the resultant.
We conclude that calculations involving directed numbers
may, although exactly alike, represent attempts to answer
two quite different kinds of questions. Expressed briefly
these questions are : — (A) Given two components to find
their resultant ; and (B) Given the resultant and one com-
ponent to find the other component.
§ 2. The Problems distinguished by Symbolism. — Is it
possible to set the calculations down in a way which will
show which of the two kinds is under consideration ?
Suppose the movements to be confined to a single direc-
tion — for example, up a ladder down which it is impossible
to return. Then the setting down of the calculations would
suffice by its form to show what questions have been asked.
Thus the form s = 30 -^ 17 could mean only that a person
has taken two successive upward movements of 30 and 17
steps and that the question is how far is he now up the
ladder. That is, the presence of the plus sign imphes a
problem of the first kind. Similarly the form s = 30 - 17
could mean only that he has set out to climb 30 steps and has
actually taken 17, the question being what second movement
is necessary to carry him to his destination.
It is natural, then, to inquire whether the signs plus and
186 ALGEBRA
minus cannot still be used to indicate the character of the
problem even when the movements involved may be down as
well as up, and require, therefore, other plus and minus signs
as labels to show which way they are taken. There is no
difficulty, provided that we are careful to distinguish the
latter signs from the former — which can be done by the
simple device of enclosing each number with its directive
label in brackets. Thus the two problems of § 1 may be
stated respectively in the forms
s = (+ 5) + (- 18) and s = (+ 5) - (+ 18)
which are no longer ambiguous. The plus between the
bracketed numbers in the former shows it to be a problem
in which we are given two component movements, + 5 and
- 18, and are asked to find their resultant. The minus
between the bracketed numbers in the latter shows that it is
a problem in which we are told a resultant movement, + 5,
and one of the (actually taken) component movements, + 18,
and are asked to find the other component.
§ 3. Bules of Procedure. — The only difficulty left is to
settle the practical rules by which we are to proceed from
the setting down of the problem in symbols to the calcula-
tion of the answer. Consider once more the problems of § 1.
" Common sense " shows that the answer to both problems
must be - 13. To obtain this answer from the numbers 5
and 18 we 7nust have + 5 - 18 in each case. Thus from
our knowledge of the correct answer we see that the calcula-
tions must be written : —
s = (+ 5) -1- ( - 18) and s = (+ 5) - (+ 18) . (i)
= +5-18 =+5-18 . . (ii)
= - 13 = - 13 . . . (iii)
An analysis of these cases suggests the following rules. The
plus and mijius signs between the bracketed numbers in (i)
are there simply to show the nature of the problem. When
we proceed to the actual calculation we drop them. If the
sign is phis the directed number following it (i.e. the second
component) is written down as it stands. If the sign is
minus the sign of the component is changed. We thus
obtain line (ii). The final results (iii) are obtained by com-
bining the directed numbers.
These suggested rules must be tested by applying them to
specimens of all possible cases. The following are instances.
DIRECTED NUMBERS 187
(1) A person goes down 7 stairs, then turns and comes up
18. Where is he now ? This is a problem of the first kind
to which the answer is obviously +11. The statement must
take the form s = ( - 7) + ( + 18) since the two components
are - 7 and + 18. The rule bids us drop the " + " between
the bracketed numbers, for it is there merely to show the
nature of the problem. It tells us to write the two com-
ponents, with their signs, just as they stand in the brackets
and to combine the directed numbers. Carrying out these
instructions we have
s=(-7) + (+18)
= -7 + 18
= +11
The answer is correct and the rule is, in this case also,
justified. (2) A person intended to go to a room 8 stairs
below the landing on which he stood, but, by mistake, went
down 20 stairs. What movement must he take to reach his
intended destination ? Here the problem is of the second
kind involving a resultant - 8 and a component - 20. The
answer to be expected is + 12. Following the rule we have
s = (-8)- (-20)
= -8 + 20
= + 12
That is, we dispense with the minus sign whose sole function
is to indicate the type of problem, and at the same time re-
verse the sign of the following component. The answer is
correct and the rule, therefore, justified.
Members of the class will themselves suggest further
problems involving different combinations of signs. They
will state the answers to be expected and verify that the
rules always give them. The result in a case in which (for
example) the first number is negative, the second positive
and numerically greater than the other may clearly be taken
as showing what must happen in all such cases. When the
various typical cases have been explored we may justly con-
clude that the rules have universal application.
^ § 4. Algebraic Addition and Subtraction. — The problem
represented by such symbolism as ( + 14) + ( - 9) may be
called " addition " if we remember that it is something very
different from ordinary or arithmetical addition which always
produces an increased total. We "add" here only in the
188 ALGEBRA
sense in which we may say that we " add " the two vectors
AC and CB (fig. 21) to obtain the resultant AB, and + 5 is the
" sum " in this case only in the same way that AB is the
"sum" of the two vectors AC and CB. In other words
" adding " means here combining components, and the " sum "
is simply the resultant of the combination. We can distin-
guish, in fact, three distinct kinds of addition: (1) ordinary
arithmetical addition, in which the numbers are non-directed
and the total is increased by each term added ; (2) " algebraic
addition " — the kind now before us — in which the numbers
indicate movements or distances backwards or forwards along
a line from a certain origin and the "sum " of two compon-
ents is a third movement or distance along the same line
which may be either backwards or forwards and either greater
or less than both components; (3) "vector addition," in
which we deal with lines representing movements inclined to
one another. Here the " sum " is a third line representing a
movement which is in a direction different from those of the
components and either greater or less than either of them.
These considerations show why the rule of procedure in
algebraic addition holds good. The plus between the bracketed
directed numbers implies that they are to be combined, and
the combination, as we saw in Ex. XXVII, is effected by
taking account of their signs. Thus to proceed with an
algebraic addition we simply drop the plus sign that separates
the components and deal with the components in accordance
with their signs.
Answering to the three forms of addition there are three
kinds of subtraction : (1) arithmetical subtraction, which al-
ways consists in reducing a non-directed total by taking a
number away from it ; (2) algebraic subtraction, in which we
are given a directed number representing a resultant move-
ment along a line from a certain origin together with another
directed number representing a component movement along
the same line, and are to find a third directed number which
will represent the second component — and may be numeri-
cally greater than the resultant; (3) vector subtraction, in
which we are given a line AB (fig. 21) representing a re-
sultant movement and a line AC representing a component
movement, and are to find the second component CB.
Algebraic and vector subtraction differ from arithmetical
subtraction in a further important respect. Every problem
DIRECTED NUMBERS 189
of algebraic or vector subtraction can be turned into a problem
of addition involving the same numbers or lengths. Thus
suppose we are given the vectors AB and CB (fig. 21) and wish
to find the vector AC. This is a problem of vector subtraction.
Draw AB, and from B draw, not CB but BC, that is, a vector
of the same length and direction as CB but with the arrow-
head reversed. The figure now represents a problem of
vector addition, but the answer, AC, is exactly the same as
the answer to the original problem of vector subtraction.
We can explain the result by saying that AB, being equivalent
to AC and CB, can be replaced by them. When we " add "
BC, the addition cancels CB and leaves us with the other
component AC.
Consider similarly the problem of algebraic subtraction,
s = ( - 8) - ( - 20). The resultant ( - 8) is equivalent to
the known component movement (- 20) together with the
unknown movement s and may therefore be replaced by them.
If we add to the combination of s and ( - 20) — that is, to
(- 8) — the component (+ 20), the effect of the movement
( - 20) will be cancelled ^ and the unknown component s
will be revealed. In other words the addition problem
s = (- 8) + (+ 20) must yield the same answer as the
subtractioni^rohlem s = (- 8) - (- 20).
We now see the justification of the rule of procedure in
algebraic subtraction. A problem of algebraic subtraction
can always be replaced by one of algebraic addition in which
the second component is the original component with sign
reversed. Since the " problem sign " in front of this com-
ponent is now plus, the number suffers no further change of
sign when the plus is suppressed and the calculation begins.
That is, the second term of the subtraction enters into the
calculation with sign reversed.
§ 5. The Use of Symbols. — Hitherto such an expression as
a + b has always meant " addition " in the ordinary sense of
the word ; a and b have stood for two numbers of which the
second was to be added to the first and so to increase it.
Similarly a - b has always meant the subtraction of one
number from another with a diminution of the total. But
henceforward if we are not told that our symbols stand for
non-directed numbers it must be assumed that they stand for
^ Going up 20 stairs cancels the eflFect of going down 20.
190 ALGEBRA
directed numbers and that the operations required are alge-
braic addition and subtraction.
If we are given the values of a and b in such expressions
we now know how to proceed to calculate the resultant or
the unknown component as the case may be. As a rule the
symbols will be accompanied by numerical coefficients. Here
is an example : —
Given that c = 2a - 36, find the value of c when a = - 3
and b = - 5.
c = 2a - 36
= 2 (-3) -3 (-5)
= (-6) -(-15)
= - 6 + 15
= + 9
The problem here is one of algebraic subtraction. The re-
sultant is given as 2a, that is, as equivalent to two movements
each to be denoted by a. In a similar way the known com-
ponent is given as 36, that is, as equivalent to three move-
ments each to be denoted by 6. In passing from the second
to the third line of the working we argue that two movements
of - 3 steps are equivalent to a single movement of - 6 steps
and that three movements each of - 5 steps are equivalent
to a single movement of - 15 steps. After this we apply
the rule for algebraic subtraction.
[Ex. XXVIII, Nos. 1-31, may now be worked.]
§ 6. Mare Complicated Cases.^ — Given that
c = (3a - 46) + (2a - 56)
find the value of c when a = - 4 and 6 = -f 3.
This is obviously a double problem. In the first place
the plus sign between the bracketed expressions shows that
the problem is one of algebraic addition ; c is to be regarded
as the resultant of two components represented respectively by
3a - 46 and 2a - 56. But to determine the value of these
components we have to solve two problems of algebraic sub-
traction. The working will begin with the subtraction with-
in the brackets and will proceed as follows : —
^ §§ 6 and 7 with Examples C, may be omitted at discretion or
taken in revision.
DIRECTED NUMBERS 191
c= {3a - 46) + (2a - 5b)
= {3(-4)-4( + -3)}+{2(-4)-5(+3)}
= {(-12)- (+12)} +{(-8) - (+15)}
= (- 12 - 12) + (- 8 - 15)
= (-24) + (-23)
= - 24 - 23
= - 47
Given that c = (3a + 2b) - {6a - 2b) find the value of c
when a = - 3 and 6 == - 4.
Here the problem is fundamentally one of algebraic sub-
traction : c is the unknown component of a resultant repre-
sented by (3a + 2b), the known component being represented
by (5a - 2b). The determination of the value of the result-
ant (3a + 2b) itself involves a subsidiary problem in algebraic
addition ; the determination of the value of the component
(5a - 2b) a subsidiary problem in algebraic subtraction. In
working we begin as before with these subsidiary computa-
tions.
c = (3a + 2b) - {5a - 2b)
= {3(-3) + 2(-4)}-{5(-3)-2(-4)}
= {(_9) + (-8)}-{(-15)-(-8)}
= (- 9- 8) - (-15 + 8)
= -17+7
= - 10
The meaning of each step in these calculations should be
elicited. In doing the exercises the teacher should insist
upon having the problems worked in full until the signi-
ficance of the successive operations is thoroughly mastered.
Those who are able to do so may then be allowed to shorten
the working by omitting (say) the second and third lines.
§ 7. Shorter Methods of Working. — The working can often
be greatly shortened by a preliminary treatment of the
formula (cf. the use of factorization and fractions). The
shortening depends on the fact that the algebraic subtraction
of a given component always produces the same effect as the
addition of a component numerically the same as the former
but of opposite sign. This rule may be expressed in symbols
by writing
a - b = a + {- b)
the symbol ( - b) being used to indicate the component that
has the same number as b but the opposite sign. Applying
the rule to the first example we have : —
(i)
(ii)
(iii)
(iv)
(V)
192 ALGEBRA
c = (3a - 46) + ,(2a - 5b) .
= {3a+ (- 46)}+ {2a+ {- 5b)} .
= 3a + (- ^b) + 2a + (- 5b)
= 5a + (-9b)
= 5a - 9b
= - 47
The problem has been turned [line (ii)] into one in which we
have first to find the resultant of a pair of components, then
the resultant of another pair, and, finally, the resultant of
these two resultants regarded as components. It is quite
obvious that these operations would have exactly the same
result as if we simply found the total resultant of the four
components [line (iii)] without first finding their partial re-
sultants in pairs. Line • (iv) fallows obviously. In line (v)
the problem is expressed as one of subtraction so that we
may use the Original component b instead of its opposite.
Thus by these manipulations the long expression with which
we started is reduced to the simple form 5a - 96 before cal-
culations begin. The same method may be applied in the
second example : —
c= {Sa+ 26) - (5a - 26) ... (i)
(3a + 26) + {(- 5a)- (- 26)}
= (3a. + 26)+ {(- 5a) + 26} .
= 3a + 26 + ( - 5a) + 26 .
= - 2a + 46
= - 10
(ii)
(iii)
(iv)
The subtraction of (5a - 26) in line (i) is to be replaced by
the addition of the opposite component. It is obvious that
if we change the signs of both members of (5a - 26) the
component represented will be the one required ; in this way
we obtain line (ii).
When the arguments are thoroughly understood it may be
pointed out that the expressions could be reduced in two
steps as follows : —
c = (3a - 46) + (2a - 56) . . . (i)
= 3a - 46 + 2a - 56 . . . (ii)
= 5a - 96
c = (3a + 26) - (5a - 26) . . (i)
= 3a + 26 - 5a + 26 . . . (ii)
= - 2a + 46
DIRECTED NUMBERS 193
The sign in front of a bracket is to be prefixed to the first
number within the bracket. The sign in the bracket remains
unchanged if the sign before the bracket is plus and is
changed if that sign is minus.
The rules exemplified by the transition from line (i) to line
(ii) are called the " law of association ". Compound quantities
may be resolved into their elements by the removal- of
brackets — accompanied by a change of sign whenever there
is a minus before the bracketed group of numbers.
[Ex. XXVIII may now be finished.]
C. The Multiplication and Division of Directed Numbers
(ch. XVII., §§ 2, 4 ; Ex. XXIX).
g 1. Train Problems ; Graphic Solution. — Suppose that we
were standing on the platform of the railway station at Don-
caster at the moment when a G.N.R. express passes through
on its way from London to the north. Let us suppose, also,
that the train keeps up a uniform speed of 42 miles an hour.
Then it would be easy to draw a graph which would represent
its position at any moment after it passed us. (The graph is
the line OP of fig. 47, the axes being OT and OD.)
Now this train had a history before it reached Doncaster,
and we might, of course, draw a second graph to represent it.
But it would clearly be better if we could extend the present
graph so as to include this history ; that is, so as to show the
position of the train at any moment of its journey, either be-
fore or after it reached Doncaster. This can be done very
simply. Produce the axis TO to T', and the axis DO to D'.
Graduate OT' from O to represent times before the train
reached Doncaster, and graduate OD', also from 0, to repre-
sent distances from Doncaster towards London. Then if we
calculate the positions of the train 1, 2, 3 . . . hours before
it reached Doncaster and insert points in the extended graphic
scheme to represent them, these points will mark out a straight
line, OQ, which is obviously a continuation of OP.
We have now a graph showing the position of the train at
all moments of its journey. To make it completely service-
able it will be well to indicate in some way the facts that
horizontal distances to the right measure times after and
distances to the left measure times before the train reached
T. 13
194
ALGEBRA
Doncaster ; also that distances above and below the line TT'
measure respectively distances of the train to the north or the
south of Doncaster. The simplest way to do this will be to
P'
mifes
D
p
\
\
r-
+14-0
-+120
-+I00
-+80
-y
/
/
r
^
- +60
c +40
\
/
V
T
_i
5 -A
1- <
\ -2
~
;
\*
+i
> +^
J +<
+i
> hrs.
±
t-4-O
-60
A
\
/
^
r
— ou
-100
—120
H40
\
\
. , .
D'
Fig. 47.
Q'
distinguish these times and distances by directed numbers.
It does not matter at all which are reckoned positive and
which negative, but it is usual, in drawing a graph, to mark
distances to the right or upwards, positive, to the left or
DIRECTED NUMBERS WB
downwards, negative. Thus we are going to count as posi-
tive the times after the train reached Doncaster and its dis-
tances from Doncaster after it has passed through the station.
The other times and distances will be reckoned negative.
§ 2. Graph of a Southward, Journey. — Next suppose that at
the moment when this train is passing through Doncaster on
its way north another train is also passing through and at
the same speed of 42 miles an hour, but towards London.
How will the graph representing its positions at various times
diifer from the one just constructed ? To answer this question
we will plot a few of the points that it must contain. Thus,
2 hours before it reached Doncaster it would be 84 miles to
the north ; the point representing this fact must be placed on
a level with the graduation + 84 on the axis DD' and directly
above the graduation - 2 on the axis TT'. Again, 3 hours after
passing us the train would be 126 miles to the south. The
representative point will this time be on a level with the
graduation - 126 in the distance scale and below the gradua-
tion + 3 on the time scale. One or two more points having
been fixed in the same way, the graph comes out as a straight
line, P'Q', making with the axes DD' and TT' exactly the
same angle as the former line but lying on opposite sides of
both of them.
From the two graphs thus drawn with the same axes and
the same scales it is possible to read off, without calculation,
the answers to a number of problems : e.g. How far apart
are the two trains 3 J hours before they reach Doncaster?
What length of time elapses between their passages through
York(+ 32 miles)?
§ 3. The Position of the Train by Calculation. — We should
have, of course, no difi&culty in obtaining answers to any of
these same problems by calculation. The familiar relation
distance covered by train = speed x time taken
would enable us to deal easily with all of them. But in this
formula, as you have hitherto used it, the distance, time and
speed have not been thought of as directed numbers. The
question may now be considered whether it is possible to
bring directed numbers under its operation. That is to say,
can the formula be made to contain and impart all the infor-
mation about the direction (as well as the magnitude) of the
speed, time and distance which is dyen ug by the lines upon,
the graph paper ?
13*
196 ALGEBRA
In the first place we will mark the fact that we wish
our calculation to show which way the train is going as well
as how fast it is moving by replacing the word " speed," which
refers only to the train's rate of movement, by the word " velo-
city," which is generally taken to refer also to the direction
of movement. When the train is moving northwards the
velocity will be reckoned positive, when southwards, negative.
Our formula may now be written in the form d = vt, and it
is to be understood that all three symbols stand for directed
numbers.
Let us now attempt to work out some examples by means
of the formula.
{a) Where will the northward moving train be 3 hours
after it leaves Doncaster? Here v — +42,^= +3, and
we see from the graph tha,t d = +126. Thus our calculation
must take the form : —
d = vt
= (+ 42) X (+ 3)
= + 126
That is, the multiplication of the two positive directed numbers
must be supposed to yield another positive number. It is
clear from the graph that this rule will hold for all cases in
which a positive speed is to be multiplied by a positive time ;
for the products are all represented by points on the line OP
— that is, by points which represent positive distances from
Doncaster.
(b) Where will the same train be 3 hours before it reaches
Doncaster? Here v = + 42, ^ = - 3, and we see from the
graph that the distance is - 126. Hence we must have : —
d = vt
= (+42)x (-3)
= - 126
That is, the multiplication of the positive by the negative
directed number yields a negative number. Again we see
that the rule always holds good. For the points which
represent all such products lie on the line OQ which repre-
sents places whose distances from Doncaster are negative.
(c) Where will the southward-moving train be 3 hours
after it reaches Doncaster? Here v = - 42, t = + S, and
we see from the graph that the distance is - 126. Thus we
must have : —
DIRECTED NUMBERS 197
d = vt
= (-42)x (+3)
= - 126
That is, the multiplication of the negative number by the
positive number yields a negative number. Since all such
products are on the line OQ', this rule always holds good.
It shows, too, that directed, like ordinary numbers, follow the
" law of commutation " in multiplication — that is, that the
multiplier may become the multiplicand and conversely with-
out change in the value of the product. We could not have
been sure, because 3x4 = 4x3, that ( - 3) x ( + 4) =
( + 4) X ( - 3) ; but we have found that it is so.
(d) Lastly, Where will the southward- moving train be
3 hours before it reaches Doncaster? Here v = - 42,
i = - 3, and we see from the graph that d = + 126. Thus
we must have : —
d = vt
= ( - 42) X ( - 3)
= + 126
That is, the multiplication of the negative number by another
negative number yields a positive number. As in the other
cases the fact that the points representing such products all
lie upon OP' shows that the rule is universally true.
§ 4. Comparison luith former Besults ; the " Bule of
Signs". — The rules just found for replacing two signs by a
single sign when two directed numbers are multiplied are
exactly the same as those which we discovered when we were
working with the formulae c = a + b and c = a - b. We
found that such combinations as + (+ 4), + (- 4), - (+ 4)
and - ( - 4) can be replaced by the single directed num-
bers + 4, - 4, - 4, -f 4, respectively ; the rule being that the
plus sign of the formula may be dropped without afifecting
the sign of the directed number, while the minus sign of the
formula could be dropped only if at the same time the sign of
the directed number was reversed. These rules, those of the
present lesson, and the rules of arithmetical addition and sub-
traction of ch. VIII., B, can all be summed up in one : *' Like
signs produce plus ; unlike signs produce minus ". This is
called '• the rule of signs ".
It is true that we have arrived at these rules from the
consideration of only one type of example in each case ; but
there can be no doubt that they will hold good whenever we
198 ALGEBRA
are dealing with magnitudes that can be represented graphic-
ally to the right and left or above and below a zero point.
We should not doubt that three sevens always made two tens
and a one simply because we had discovered the truth only in
counting up piles of pebbles. It is obvious that the result
must be true of any things that can be gathered into groups,
even though we have never tried it with them. For the same
reason we may be sure that the rule of signs can be trusted
in all cases in which a graph like that of the present lesson
can be used to represent the magnitudes in question.
i; 5. Division of Directed Numbers. — It can easily be
shown that the rule of signs will hold good for division of
directed numbers as well as for multiplication. When we say
12
that — = 4 we mean, of course, that 4 is the number by which
3 must be multiplied in order to produce 12. In the same
- 12
way, if I inquire the value of — — ^ I mean to ask what - 3
must be multiplied by in order to yield the product - 12. In
accordance with the rule of signs the answer is + 4 ; that
is, in division two minus signs will be replaced in the quotient
by a plus. The other cases of division can be treated in the
same way.
[Ex. XXIX may now be worked.]
CHAPTER XIX.
CONSTANT-DIFFERENCE SERIES.
A. The Summation of Constant- Difference Series (ch. xvii.,
^ 5 ; Ex. XXX, A).
i^ 1. The Problem. — Among the " events " of school athletic
sports the "Block Race" frequently finds a place. Each
competitor, as he stands on the starting line at 0, has in front
of him a number of wooden blocks, placed at equal distances
at the points A, B, C, etc. When the signal is given he
has to run from to A and return with the block to 0.
Leaving it there he runs to B, fetches the second block and,
returning again to 0, places it upon the first. He then runs
to C, returns to with the third block, and places it on the
top of the second. The race continues in this way until one
competitor has fetched in and piled up all his blocks. Suppose
that the distance OA is 4 yards and that the blocks are 3
yards apart, then the lengths of the various journeys of a
competitor, measured in yards, form the sequence : —
8, 14, 20, 26, 32, . . .
This is evidently an arithmetical progression with first term
8 and constant difference 6. To find the total distance run
in the race we could, of course, simply add the successive
terms of the series together. But if it was a very long race
this addition w^ould be a tedious business. We will try,
therefore, to find an easier way of arriving at the answer to
our problem.
§2. Summation Formula; (i) 2vhen the Terms are Non-
directed Nuinhers. — Fig. 48 is a column-graph representing the
distances covered in the first seven journeys. The area of the
shortest column represents 8 yards, that of the longest
8 + 6 X 6 = 44 yards.
The total distance run is represented by the total area of
the figure. Since we have chosen an odd number of strips
199
200 ALGEBRA
there must be one, marked P in the figure, which has an
equal number of others to right and left of it. Moreover,
since the strips increase uniformly in height, it is evident
that if the piece marked q were cut off from the strip Q it
would fit exactly into the space q' on the top of Q'. Simi-
larly r and s would fit into the spaces r and s. In this
way the figure could be turned into a rectangle, consisting of
seven strips of the same height (fig. 49). If we inquire what
that height is, we note that the two strips S and & of fig. 48
together make two of the equal strips of fig. 49. Each of these
strips is, therefore, one half of the sum of the first and last
strips of fig. 48 — that is -^ of (8 + 44), or 26. Since there
are seven such strips, the total area of the figure — the total
distance run — is 26 x 7, or 182 yards. Even with so small
a number as seven journeys this way of performing the cal-
culation is quicker than addition. But it is obvious that
the argument could be applied equally well to any odd
number of journeys. For every piece cut off from a strip to
the left of P there will be an equal space to be filled above
a strip on the right so as to convert the whole area into a
rectangle composed of equal strips. Moreover, each of these
equal strips must always be one half of the sum of the first
and last of the original strips. For example, suppose that
the race consisted in fifty-one journeys, so that the last would
be one of 8 + 50 x 6 = 308 yards. Then the total distance
would be represented by a rectangle composed of fifty-one
equal strips each of area -^(8 + 308), i.e. 158. Thus the
total length of the race would reach the formidable total of
158 X 51, or 8058 yards, that is about 4^ miles !
We conclude that when n is an odd number, a the first
and I the last of the terms, the sum S is given by the
formula
S = |(a + I).
Consider next the case of an even number of terms of the
series, say 8 (fig. 50). This time there is no middle strip, so
the 5th and 4th strips are marked P and P'. But if we draw
a line parallel to the base half-way between the top of P and
and the top of P' it is clear that the piece p will fit into the
space p, the piece q into the space q\ etc., as before. The
figure will once more be converted into a rectangle made up
of equal strips and once more each of these strips will be one
S' R' Q' P Q R S
Fig. 48.
Fig. 51.
T'
,^_t'_i.s'_i_r'_L_ci:
X
!x'
CONSTANT-DIFFERENCE SERIES
201
half the sum of the first and last of the original strips. Thus
the area of the rectangle, and consequently the total distance
run in the race, will in this case also be given by the formula
S = |(a + I).
§ 3. The Summation Formula ; [ii) when the Terms are
Directed Numbers. — It is natural to inquire whether this
formula holds good for constant-difference series composed
of directed numbers, some of them negative. Let us ex-
amine the case of the series
ll + 8-f-5 + 2-l-4-7-10-13
As before we will begin with an odd number of terms — say
9 — and will represent their values by the areas of strips of
constant width.
The first four (positive) terms will be represented as before
by the decreasing strips T', S', R', Q', of fig. 51. We then
come to the negative terms. It is quite natural to represent
these by strips P, Q, R, S, T, drawn below the base line XX.
The principle we are adopting is that if the area of a rect-
angle is taken as representing
a positive directed number the
area of the same rectangle
when inverted can be taken
to represent the correspond-
ing negative number. It will
be noticed that upon this plan
the horizontal ends of the
strips descend in height by
uniform steps along the dia-
gram. Since the bottom line
of a negative column corre-
sponds to the top line of a
positive column the column
must be supposed to become
negative not by slipping down
the face of the paper but by
revolving out of the plane of
the paper about XX (fig. 53).
If it is coloured blue on one
side and red on the other, the
change of sign will be signal-
ized by a change of colour. Suppose the strip to contain a
Fig. 53.
202 Al^GEBRA
transparent clock-dial, and that when it is in the positive
position we are standing behind the dial. Then when the
strip revolves about XX the dial will be facing us (though
upside down) and the hands will be going round in the
ordinary way. This idea may be used to distinguish between
areas that are to be measured by positive and negative numbers
when we cannot conveniently distinguish them by colours.
An area measured by a positive number may be distinguished
by a curved arrow (fig. 53) indicating movement in the
counter-clockwise direction ; the area measured by a negative
number being marked by an arrow indicating clockwise
movements.
Since P in fig. 51 is the middle strip we will, as in the former
case, draw the dotted line X'X' at the level of its end. In this
way we cut otf from each of the strips Q, R, S, T, a piece equal
to P. Moreover, it is clear that the residue from Q, marked q,
would, if laid upon Q' cover the whole of it and project below
XX to the line X'X'. Similar statements are true with re-
gard to the pieces of E, S, and T marked r, 5, t. But g, ;■, &•,
and t are negative areas. If they are laid upon positive areas
we must suppose that they obliterate or cancel any positive
surface beneath them, disappearing themselves to an equal
extent in the process. Thus q, r, s and t, entirely wipe out
the positive (or blue) areas Q', R', S', T', above the base XX
and are themselves destroyed to the same extent, but leave
behind them the negative (red) spaces q , r, s\ t\ outstand-
ing between XX and X'X'. As before the figure is trans-
formed into a rectangle (fig. 52) made up of equal strips.
Moreover, out of T and T', the first and last of the original
strips, we have remaining, after the equal positive and negative
ones have cancelled one another, two of these equal strips, t'
and the corresponding piece above t at the other end of the
base. Thus each of the equal strips is one half of the sum
of the first and last of the original terms. The only differ-
ence between this case and the former cases is that the areas
represent directed numbers and that the " sum " is not their
arithmetical sum but the algebraic resultant of their combina-
tion. In the instance before us the areas of the first and last
of the original strips are + 11 and - 13 respectively. It
follows that the common area of the nine equal strips
constituting the rectangle XXX'X' is |{( -f 11) + ( - 13)},
i.e. - 1. The total resultant area and the sum of the series
CONSTANT-DIFFERENCE SERIES 203
will, therefore, be (- 1) x 9 = - 9. This result can be
verified by addition. We conclude that in this case also the
sum of the series is given by the formula
S = I (a + i)
a and I now being directed numbers.
^ 4. Proof by Symbols. — We could examine all other
possible cases in the same way but there can be no doubt
that the formula is universally valid. The essence of the
argument is to show that in each case we get a number of
equal rectangles — just as many rectangles as there are strips
or terms, that the first and last of the original strips together
make two of the equal rectangles, that the second and last
but one make two more, and so on. We can represent this
argument in symbols in such a way as to prove that it al-
ways holds good. In one line we write the series forwards and
in a second line we write it backwards, so that the last term
comes under the first, the last but one under the second, etc.
S = a + {a + d) + {a + 2d) + (a + Sd) + ...
+ {a + n - Id)
S = Z + (Z - d) + { I - 2d) + {I - Bd) + ...
+ {I - n - Id)
2^ = {a + I) + {a + I) + (a + I) + [a + I) + ...
+ {a+ I)
= n{a + I)
■: S = '^ (a + l).
On addition we obtain an {a + I) for each term of the series
— a result which corresponds, of course, to the n equal rect-
angles of our graphic method. The result S = x (a + Z)
follows at once.
The advantage of the proof by symbols is that a, I and ^
may be taken to represent directed as well as ordinary numbers
and the addition may be algebraic as well as ordinary addition.
Hence the one argument by symbols covers all the cases that
had to be treated separately by the graphic method.
B. The Calculation of certain Areas and Volumes (ch. xvii.,
§ 6 ; Ex. XXX, B, C).
^ 1. Fig. 54 represents a model consisting of six white card-
board rectangles all of the same size, and of a group of grey
204
AliGEBRA
rectangles resting on them. The grey rectangles increase in
size uniformly from b up
to c which is as large as
the white rectangle be-
low it. To the vulgar
eye there are only five
grey rectangles but the
mathematical eye will
discern upon a a sixth
whose size is zero ! What
^^^- ^^- ratio does the total area
of the grey rectangles bear to that of the large white
rectangle built up of the equal strips below ?
We can take the area of grey rectangle b as unity. The
area of the largest grey rectangle will then be 5, and this will
also be the area of each of the equal white rectangles. The
area of the imaginary grey rectangle (a) is, as we have al-
ready said, zero. The required ratio is given by the frac-
tion
0-t-l + 2-f-3 + 4: + 5 _ 15
~ 30
5+5+5+5+5+5
When we turn to fig. 55 we see a model just like that of fig.
54, with the difference that, although the white strips cover
the same total area they are thinner and more numerous.
There are 12 of them.
It will be convenient again to take the smallest of the visible
grey rectangles as the unit of area. (The change of unit does
no harm since we are seeking only a ratio of areas.) The
largest grey, and each of the equal white, rectangles will,
therefore, have an area of 11. The ratio of the grey area to
the white will be given by the fraction
+ 1 + 2 + . . . + 11 _ ^6
11 + 11 +Tl + . . . + 11 ~ 132
^ 1
~ 2
We need not trouble to set down all the figures of the
numerator. They form an a.p. of 12 terms and the sum of
the first and last is 11. Thus the sum of the series is
CONSTANT-DIFFERENCE SERIES 205
y^ X 12 = 66. The denominator is 11 x 12 = 132. Thus
we again find that the ratio is one half.
It is easy to show that, however numerous the rectangles,
the grey area will always be one half of the white area below.
Suppose the total white rectangle to remain of the same size
but to be built up of (p + 1) equal strips where p is any
number you please. There will be p visible grey strips and
in addition the invisible one of zero area. Calling the area
of the smallest visible grey strip 1, the area of the largest
will be p. This will also be the area of each of the white
strips. Thus for our ratio we have
+ 1 + 2+ ... + p ^ -kip + l)p
p + p + p + ... + p (p + l)p
_ 1
~ 2
§ 2. Application of Besult to Area of Triangle. — It appears,
then, that no matter how numerous the rectangular strips the
area covered by the grey ones is one half of the whole area
covered by the white ones. But if the strips, while still
covering the same area, became immensely numerous and
correspondingly thin, a time would come when it would be
impossible to distinguish their corners either by eye or touch.
To sight and feeling they would be indistinguishable from this
cardboard triangle. But since the ratio of areas has remained
one half all along, we conclude that the area of the triangle
must itself be one half of the area of the rectangle — or so
little different from one half that no one could ever estimate
the degree of difference.
§ 3. Other Applications. — Of course we already knew that
the area of a triangle is one half of that of the corresponding
rectangle. But this mode of discovering it is particularly
useful because it can be applied to the measurement of many
other areas. For example, by moving the grey rectangles of
fig. 55 they can be made to present fig. 56. By being moved
they are not, of course, changed in area. The total grey area
remains, therefore, one half of the big white rectangle and this
will be true however numerous the grey rectangles may be-
come. But if I make them numerous enough they will form
an area indistinguishable from Exercises, fig. 33. There are
no rectangles visible in this case, but if I measure the horizontal
206
ALGEBRA
distance across the figure at any equidistant intervals, the
heights obtained will be in a.p. I shall be measuring, as it
were, the heights of submerged rectangles picked out at equal
intervals along the line. Whatever intervals I choose, these
heights must form an a.p. if the whole of the series is in a.p.
(Ex. XXIX, No. 88). The conclusion follows that the area of
Exercises, fig. 33, is one half of the rectangle whose length is the
perpendicular distance from its apex to the bottom bound-
ary and whose width is equal to the length of that boundary.
Fig. 55.
Fig. 56.
Exercises, fig. 34, represents a shape cut out in paper and
containing a hole. If any number of equidistant lines are
drawn across the figure their lengths (missing out the parts
that bridge the hole) will be found to be in a.p. It follows
that the shape may be thought of as produced by placing an
immense number of rectangular strips in a.p. side by side
in certain positions. (Some of the strips must, of course,
have been cut and their segments separated so as to leave the
hole.) We conclude from this discovery that the area of the
shape may be calculated by the same process as that of fig. 33.
[Exercise XXX may now be worked.]
CHAPTEE XX.
ALGEBRAIC MULTIPLICATION.
A. Algebraic Multiplication (ch. xvii., ^:^ 7; Ex. XXXI,
A, B, C).
§ 1. Are the Identities of Chs. VIII and IX true of Di-
rected Numbers ? — In ch. vii. we saw that formulae could
often be simplified for the purpose of calculation by means of
the identities
ab ± ac == (a ± b)c and a- - b'^ = (a + b){a - b).
As there used the symbols a, b, c referred to non-directed num-
bers, and the signs plus and minus indicated ordinary arith-
metical addition or subtraction. That being the case it was
easy to show that the identities hold good for non -directed
numbers whatever measurements are intended by them. But
if a, b, c are symbols for directed numbers the plus and
minus signs in {a + b) and {a - b) imply algebraic addition
and subtraction, and the former proof of the identities is no
longer sufiQcient. To determine whether they may be used to
simplify formulae expressed in directed numbers requires a
fresh investigation.
§ 2. The Method of Proof; Use of Directed Areas.— ThQ
identities could be tested by substituting in them all kinds of
directed numbers, positive or negative, large or small, and
seeing if they worked out correctly or not. But this is not a
very satisfactory kind of proof, for it would not show lohy the
results are correct or incorrect. It will be better, therefore,
to seek magnitudes which can be represented by the identity
which we are testing and see whether the identity expresses
what we know to be the behaviour of the magnitudes repre-
sented. It was in this way that we found out the rule of signs
in multiplying directed numbers. We used the directed num-
bers to describe the behaviour of a moving train — a thing with
208
ALGEBRA
which we are quite familiar — and so found out what must be
the laws of their combination when multiplied together.
We first discovered the identities in question in connexion
with the calculation of areas. The use of directed areas in
the last chapter suggests that the old way of investigation,
suitably modified, may again give what we require.
When we thought of the product {a + h)c or the product
{a + h){a - h) as measuring areas, the two factors were re-
garded as measuring the lengths of the adjacent sides of a
rectangle. We can retain this idea with the difference that
a + h (for example) must be a directed length, the algebraic
sum of two directed lengths or measurements, a and h. Thus
if a = - 7 and 6 = + 3 then the side of the rectangle
measured hj a + b must be - 4, and our diagram must repre-
sent this fact. Similarly if c = +3, then the other side of the
rectangle must be marked so as to indicate the direction as well
as the magnitude, of this number. We will adopt the rule
always to draw the two directed sides /row one of the corners
of the rectangle. With this condition the product {a + b)c
or ( - 4) X (+ 3) would be represented by one or both of the
following figures : —
Fig. 57.
Now if these figures differ at all in what they represent,
one of them must stand for (+3) x (- 4), the other for
(- 4) X (-1- 3). But, as we have seen, these two products
are both - 12. We must, therefore, fix the sign of the areas
A and B by a rule which makes them both negative. The
following rule will be found to produce the required result.
Imagine the area to contain a clock-face (see p. 201) with the
Fig. 58.
(a + b)c = ac + be.
a and b both negative, c positive.
c
a + b
be
(a + b)e
a positive, b negative, e positive ; a numerically greater than b.
^^^^^^^^^ c
bl
a + b
be
(a + b)e ac
a negative^ b positive, e negative ; a numerically less than b.
^^^^■[4 a + b
(a + b)c
Fig. 59.
(a - b)c = ac - be.
and b both positive, c negative ; a numerically less than b.
b> a + (-b) = a-b
(a - b)c
■ i 't t, €,
- be
ALGEBRAIC MULTIPLICATION 209
hand pointing to the marked horizontal side. Place your
finger on the tip of the imaginary hand and move it in the
direction indicated by the arrow on the horizontal side.
As your finger approaches the vertical side it will turn upwards
or downwards according to the direction of the arrow on the
other vertical side. It will be seen that in both cases (as
indicated by the curved arrow) the hand has been turned the
same way and that that way is the "clock- wise" direction
which we agreed in ch. xv., A, to call negative. Thus each of
the figures A and B will equally well represent either of the
products (+ 3) X (- 4) or (- 4) X (+ 3). It will, how-
ever, add to clearness to restrict each figure to the repre-
sentation of one product. We will adopt the rule that the
base shall represent the multiplier and the vertical the multi-
plicand. Thus fig. A will represent ( + 3) x ( - 4).
The class should suggest and verify the rules of repre-
sentation in other cases.
^ 3. Examination of {a ± b)c = ab ± ac. — This idea may
now be applied to the examination of identities, beginning
with (a + h)c. Various cases are possible ; three of them
are worked out in the diagrams of fig. 58. As in the last
lesson, positive areas are coloured blue, negative areas red.
When a blue area is superimposed upon a red area or a
red area upon a blue one they destroy one another to the
extent of their coincidence. In each set of diagrams there is
first represented the area (a + h)c, a + h being represented
by the vertical and c, the multipher, by the horizontal side.
Next we have the areas ac and he separately. It is obvious
that when these areas are brought together their algebraic
sum is always identical, both in extent and in sign, with the
area (a + h)c.
In testing (a - h)c (fig. 59) we have to face the difficulty
of representing the algebraic difference a - i at the beginning
of the investigation, and ah - ac at the end. The difficulty
is overcome by remembering that the algebraic subtraction of
a component h is identical in result with the addition of a
component - h (p. 189). Thus we start by drawing the
rectangle one of whose sides is a+ (- h), and when we
have obtained the area he we take the area - be, equal to
the former but opposite in sign, and proceed to add it alge-
braically to the area ac. It will be seen that in each case
the identity is verified. Thus in fig. 59 the red ac will cancel
T. 14
210 ALGEBRA
the upper part of the blue - 6c, leaving a blue area identical
with the original rectangle.
The other possible cases could all be tested in the same
way, and it then becomes certain that we may in all circum-
stances assert the truth of the two identities (a ± h)c = ac ± he.
§ 4. Examination of {a + b){a - b) = a"^ - b^. — In figs.
60, 61 the identity {a + b){a - b) = a'"^.- b'^ is tested in a
similar way. We begin by drawing an area whose sides
represent the factors a + b and a - b respectively. We then
draw the area a^ and the area b'^. (Note that they must be
positive, whether a and b are themselves positive or negative.)
The latter area with sign reversed represents - b^. The
algebraic addition of - b'^ to d^ must in each case give a
figure like fig. 16, for a- is always positive and - 6^ always
negative. The only difference possible in different cases is
the " colour " of the residual area. The diagrams show that
it has always the same colour, i.e. the same sign, as the area
(a + b){a - b) while by the method of dissection familiar to
us from ch. vii., B, it can be shown to have the same extent.
Thus in every case (a + b){a - b) = d'^ - b^.
§ 5. Identities Proved by Multiplication. — In ch. vii. we
were not content until we had proved by multiplication that
the identities are universally true. It is important to find
whether the multiplication-process can be applied to com-
binations of symbols in which the letters stand for directed
numbers and the connecting signs imply algebraic addition
or subtraction. Fig. 62 illustrates the analysis by multi-
plication of the identity (a + b)(a - b) = d^ - b^. The first
line represents the argument
(a-\-b)(a- b)^ {a+b){a+ (- b)} . . (i)
= {a+b)a+ (a + b)(- b) . . (ii)
Below, the two terms of (ii) are analysed respectively into
(a + b)a = a^ + ba and (a + 6)( - 6) = - ab + (- b^). (iii)
The elements within the dotted rectangle are now '* collected ".
The terms ba and - ab cancel, and a'^ + ( - 6^) yields the
familiar figure representmg a' - 6^. The three horizontal
lines of figures answer, therefore, to the first three steps of
the process
(a + b)(a - b) = d^ + ba
- ab - b^
= a^ - b''
First there is the (mental) analysis of the multiplier a - b
Fig. 60.
(a + b)(a - b) = a2 - h\
a positive, b negative ; a numerically less than b.
a a + (-b) = a-b
b2
a2 + ( - 52) = a2 - b2
• • « • •»
r
Fig. 61.
(a + b)(a - b) = a2 - b'^
a negative, b positive ; a numerically greater than b.
a + (-b) = a-b
*a+ b
(a + b) (a - h)
■b"
a* + (-b«) = (a'^-b''')
Fig. 62.
(a + b)(a - b) = a2 - b^.
a positive^ b negative ; a numerically greater than b.
(a + b)(a - b) = a2 + ba
- ab - b2
= a2 - b2.
'^"tp
(a+b)(a-b) = (a+b)a +(a+b)(-b)
fai
a
(a + b)a =
k
-b
(a+b)(-b)
1
+ ;
1 a^
+ ba ' ;
; -ab
» J * it
a^-b-
o- » •>: •
ALGEBRAIC MULTIPLICATION 211
into two multipliers a and - b ; and then the successive ex-
pansion of the products {a + b)a and {a + b){- b). Finally
in each case comes the collection of terms.
The frontispiece of Exercises, I, gives an analysis of a special
case of {a + b){a + b) exhibiting exactly similar stages.
A study of these cases leaves no doubt that the multiplica-
tion process can be applied universally. The multiplier can
always be expressed as an algebraic sum by the device illus-
trated in line (i). The analysis into a series of partial pro-
ducts, of which two cases are represented by the top lines
of fig. 62 and Exercises, frontispiece, can then always be
carried out. If there are n terms gathered together in the
multiplier there will be n of these partial products. Analyses,
such as those of figs. 58 and 59, show that the partial pro-
ducts can then always be expanded into their elements by
applying the ordinary rule of signs. There remains the col-
lection of terms which is, of course, nothing more than a
counting up of the elementary terms of each sort which have
been produced by the expansion.
The whole process may be called " algebraic multiplication ".
This term is meant to remind us that the factors are algebraic
sums and differences, not arithmetical. The discovery that
algebraic multiplications can be carried out as if they were
multiplications of arithmetical sums and differences is a proof
that the signs plus and minus when they indicate the alge-
braic addition and subtraction of directed numbers are sub-
ject to the familiar rule of signs.
[Ex. XXXI, A, B, C, can now be worked.]
J5. The Binomial Expansion (ch. xvii., § 7 ; Ex. XXXI, D).
§ 1. StifeVs Table. — The results obtained by expanding
the products (a + b){a + b), {a + b){a + b)(a + b) or (a + bf,
{a + by, {a + by, etc., have special interest because each can
be derived from its predecessor in a very striking way.
There are two things to determine in these expansions'or
distributions : first the succession of literal terms and secondly
the numerical coefficients that go with them. As an ex-
ample consider the derivation of the expansion of (a + by
from that of '(a + by
{a + by = {a^ + 3a^b + 3ab^ + b^)(a + b).
The multiplication by the term a will produce a series of
14*
212 ALGEBRA
terms in which the literal eleraents are a^, a^b, a^h^, ab^ —
that is, the terms of {a + 6)^ with the a raised to the next
higher power in each. Multiplication by b produces the
series a'^b, a^b^, ab'^, b^ — that is the same series again, with
the exception that the first term a* is missing while an end
term b^ is added. On the whole, therefore, we shall have a
gamut of terms from a^ to b^ running through intermediate
terms in which the power of a constantly falls by unit steps
and the power of b constantly rises by unit steps. It
is obvious that if we proceeded from the expansion of
(a + by to that of {a + bf, from that of (a + bf to that of
(a + bY and so on indefinitely, this feature would constantly
be reproduced. Thus in the expansion of (a + bY we must
expect the terms a^, a~b, a^b^, . . . ab', b^.
Having considered the literal terms apart from the co-
efficients let us next consider the coefficients apart from the
literal terms. Take again the derivation of [a + by from
(a + by :
(1 + 3 + 3 + 1)(1 +1) =1 + 3+3+1
1+3+3+1
=1+4+6+4+1
This scheme of "detached coefficients" shows clearly that
the second coefficient (4) in the expansion of {a + by is
reached by adding to the second coefficient (3) in the expan-
sion of (a + by the first coefficient (1) ; that the third
coefficient (6) in (a + by is formed by adding the third
coefficient (3) of (a + 6)^ to the second (3), and so on. Simi-
larly to find the coefficients in (a + by we have
(1+4+6+4 + 1)(1 + l) = l+4+ 6+ 4+1
1+ 4+ 6 + 4+1
= 1 + 5+10+10+5+1
That is, any coefficient in (a + by is found by taking the
corresponding coefficient in (a + by and adding to it its
predecessor. It is obvious that the rule must hold good at
each successive multiplication. It is also clear that it will
always yield unity as the coefficient both of the first and of
the last term.
The law of derivation of the coefficients is best exhibited by
the following arrangement which has been known to the
Chinese since about 1300 but appears to have been first pub-
lished in Europe by the German algebraist Stifel (1544) : —
ALGEBRAIC MULTIPLICATION 213
1
1 1
12 1
13 3 1
14 6 4 1 ,
1 5 10 10 5 1 ^
etc. etc.
Starting with the second, the successive rows give the co-
efficients in the expansions of {a + b), {a + by, (a + by, etc.
The numbers in one row are to be obtained from those in the
row above by the rule just discovered. For example, the
second 10 in the last row is derived from 4, the correspond-
ing number in the row above, by adding to it its predecessor,
6. The table can, of course, be continued without limit.
§ 2. Derived Besults. — Stifel's numbers make it possible to
write down very easily the results of other expansions. Take
the expansion of {a - by as a first example. Here, since
a-b = a+{-b), we can derive the expansion we seek
from that of {a + by by simply substituting - b for b
wherever the latter occurs. Taking the coefficients from the
table we have
{a - by = a^ + 5a^{- b) + 10a»(- by + 10a2(- by
+ 5ft(- by + (- by
= a^ - 5a^b + lOa^b^ - lOa'b^ + 5ab^ - ¥
Take as a second expansion (2a + 36)*. Here 2a must be
put for a and 36 for b wherever they occur. Then
(2a + Sby = {2ay + 4(2a)3(36) + 6(2a)2(36)2 + 4(2a)(36)3
+ (36)4
= IGa* + 96^36 + 216^262 + 1i^a¥ + 816*
The labour-saving virtue of this method is obviously very
considerable.
[Ex. XXXI, D, can now be worked.]
CHAPTEE XXI.
POSITIVE AND NEGATIVE INDICES.
A. The Uses and Laws of Positive Indices (ch. xvii., § 8;
Ex. XXXII).
§ 1. The Index Notation for Numbers. — For many pur-
poses " round " numbers are more useful than exact numbers.
For instance, we are told that when the census was taken at
the stroke of midnight on 5th April, 1911, the population of
London was 4,521,685. Supposing this number to have been
correct as Big Ben began to strike the hour it is possible that
it was incorrect by the twelfth stroke. It was almost certainly
wrong by breakfast time and must be some thousands wrong
now. In two or three years' time it will be safe to say only
that the population is 4-^ millions — ignoring the lower denom-
inations. Statisticians in dealing with very large numbers —
such as the number of bushels of wheat imported in a year,
etc. — generally adopt this plan ; they give round numbers*
For example, a table of Imports in 1908. in the Dictionary of
Statistics gives against the entry '* Eggs " the number 2,185 ;
against " Iron and Steel " the number 1,119. But the former
is a number of millions, the latter a number of thousands of
tons. The column headed " value of imports " gives respec-
tively 7,183 and 12,235 against the same names, but each of
these is a number of thousands of pounds.
In all such cases the unit must of course be stated in some
form. In statistics the statement usually takes the form of
a note that the unit is 100,000 cwts. or £100,000, etc. Engi-
neers, physicists, astronomers and mathematicians generally
adopt another form — fearing the inconvenience and risk of
error involved in changing the units. They get rid of super-
fluous zeros by what is called the index notation. Thus the
present population of London can be expressed roundly as
4:'5 X 10^. The pages of a reference book of Physics or an
2U
POSITIVE AND NEGATIVE INDICES 215
Engineer's Pocket Book give many examples of the use of
this notation. For example, according to the latest deter-
mination, the velocity of light is 3 '002 X 10^^ — or. less exactly,
3 X 10^0 — centimetres per second, the elasticity of steel is
about 2 X 10^2. Here are two numbers, constantly written and
used, in which the compactness of the index notation is obvi-
ously very serviceable. Written in full the latter number
would be 2,000,000,000,000.
§ 2. Multiplication and Division in the Index Notation. —
The advantages of the index notation become still clearer
when large round numbers are to be multiplied or divided.
Here is an instance. We are told that the nearest fixed star
is so far away that light takes about two and a half years to
reach it. What is the distance in round numbers ? A day
has 24 X 60 X 60 = 8*6 x 10"^ seconds, roughly, and in two
and a half years there are roughly 9 x 10^ days. Hence
Distance = 3 x lO^o x 8-6 x 10^ x 9 x 10^
= 3 X 8-6 X 9 X 1010 X 10* X 10^
= 232 X 10^^ cms. roughly.
The rearrangement in the second line is easily justified: to
alter the order in which the multiplications are made cannot
alter the product (Law of Commutation). The step from
'' X 1010 X 10* X 102 " to " X 1016 » ig also obvious, for
both expressions mean that what precedes is to be multiplied
by 10 sixteen times.
The centimetre, though the common linear unit of science,
is out of place in measuring the distance of a star. If we
take the kilometre as unit we must divide by 1 x 100 x 1000
= 1 X 10^. Or, taking a mile as, roughly, 1-6 kilometres,
we can obtain the answer in miles by dividing by 1*6 x 10^
or 16 X 10*. The answer is
232 X 1016
Distance = -^r^ yf^t
16 X 10*
= 14-5 X 1012
= 145 X 1011 miles.
Here again, the procedure is clearly correct. For
232 X 1016 1016
= 14-5 X
16 X 10* ~ 10*
10^
10^
1016
Since the double operation " x "^- " means " multiply by
216 ALGEBRA
10 sixteen times and divide the result by 10 four times," it
may obviously be replaced by the single operation " x 10^^ ".
§ 3. The Law of Indices. — The instances of § 2 exemplify
strikingly the useful properties of the index symbolism, but
the properties are, of course, by no means peculiar to the
number 10. Let a, b, c, d, and p be any numbers, integral or
fractional, directed or non-directed, and let w, n, r, s be any
non-directed whole numbers. Let A = a x p^, B = 6 x p\
G = c X f, D = d X f and let M - A x B x C x D.
Then we have
M = ap"" X 6j9" X cp" X dp
= abed X p^ X p' X p" x p'
= abed X p>^+n+r+>
The justification is simply the former one generalized. The
second line follows from the first because the order of multi-
plication makes no difference to the product. The third follows
from the second because " x ^ "»+''+»•+*" merely sums up in
one command the successive multiplications by p which
a X _p"V' " X j3",'' etc., order, as it were, in batches.
Similarly if N = AB/CD we have
ap"' X bp"
op'' X dp*
ab p"" X p"
cd p^ X p^
ab
N =
-cd " ^
— r — *
If after multiplying (w -I- n) times by p we are to divide
( r + s) times by _p, we can obviously reach the result more
directly by multiplying (m -}- n) - (r + s) times by p. It is
here assumed that the number of multiplications exceeds the
number of divisions. If the reverse is the case we must write
N = — , X
+ «
cd p"
The argument can clearly be extended to any number of
numbers, always with the restriction that the indices — be-
cause they simply indicate the number of multiplications or
divisions — are non-directed whole numbers. Its results can,
however, be summed up in the three "laws of indices " : —
(i) X a'" X a" = x «"*+'»
ill) X a'" -^ a" = x a'"~** w>?i
(ill) X a"* -7- a" = -r ar~''' m<:^n
POSITIVE AND NEGATIVE INDICES 217
Lastly we may note that although the laws of indices deal
only with successive multiplication and division, yet indices
are commonly used to express single numbers. Thus a^ and
a^ are very familiar expressions and such symbolisms as
{a + by appeared in the last chapter. It is best to regard
these forms as convenient ways of expressing a x a, a x a^,
{a + b) X (a + by, etc. Thus we may write
(a + bf X {a + by = {a + b) x {a + by x {a + by
=^ {a+ b) X {a + by^
= {a+ bY'
or more briefly
{a + by X (a + by = (a + by\
[Ex. XXXII can now be worked.]
B. Negative Indices (ch. xvii., § 8 ; Ex. XXXIII).
§ 1. Small Numbers. — The index notation can also be used
for expressing compactly the very small numbers with which
the chemist, the physicist and the engineer often have to deal.
Thus the wave length of yellow light, which is said to be
0-000027 inch, could be written 27 -^ 10^.
But this notation, though a great improvement upon the
extended method of writing small numbers, is not so effective
in calculations as it was in the case of large numbers. Con-
sider the following instance. A plate of glass is O'OOSl inch
thick. What is the measure of its thickness if the wave length
of yellow light is taken as the unit ? We have
No. of wave lengths
81 4- 10*
"27-
10«
81 X
10«
27 X
10*
= 3 X
102
= 300
It would be convenient if the step represented by the second
line could be avoided.
§ 2. Negative Indices. — The following argument suggests a
method of doing so : —
218 ALGEBRA
2,700,000 = 27 X 10^
270,000 = 27 X 10*
27,000 = 27 X 103
2,700 = 27 X 10'^
270 = 27 X 101
27 = 27 X 100
2-7 = 27 X 10-1
0-27 = 27 X 10-2
0027 = 27 X 10-3
etc. etc.
We can regard the operation represented by x 10^ as hav-
ing the effect of shifting the digits 27 five places to the left ;
so that the 7 moves from the units' place to the place for
hundreds of thousands. Similarly the effect of x 10* is to
shift the figures four places to the left, and so on. When we
multiply by 10 the figures move one place to the left ; it is
evident, therefore, that 10 can be written lO^. What are we
to say about the next line where the figures are not shifted
at all? Obviously, we may indicate this fact by the sym-
bol x 10^. When we come to 2*7 the figures have moved not
to the left but to the right. How shall we indicate this fact ?
If X 10^ may be taken as meaning " shift the figures one
place to the left " then it is tempting to use x 10"! to imply
" shift them one place to the right ". In other words the
table suggests that directed numbers may usefully be used
instead of non-directed numbers as indices of powers of the
number 10 ; that positive indices may be taken to mean a move-
ment of the digits to the left and so to imply a correspond-
ing number of successive multiplications by 10; and that
negative indices may be taken to mean movements of the
digits to the right and so to imply a corresponding number of
successive divisions by 10. Upon this plan the form " x 10^"
would mean " leave the digits as they are " and so would be
equivalent to " x 1 ".
g 3. The Validity of Negative Indices. — Two questions at
once arise : (1) On what principle can negative powers of 10
be combined with one another and with positive powers ? —
and (2) Is the use of directed indices permissible with numbers
other than 10 ?
The first question is easily answered. Let P and Q be
small numbers such that P = j9 -^ 10" and Q = ^ -^ 10*, a
and b being non-directed. Also let M = PQ, N = P/Q.
POSITIVE AND NEGATIVE INDICES 219
Then we can write P = ^ x lO"** and Q = 3 x 10"*, and we
have
M=_pxlO-"xg'xlO-* N = (j9xlO- "')l{q x 10 - *)
=j9gxlO-»xlO-* =(pxlO-")x(10 + *-T-g) (i)
= j7gxlO (" + *) =;)/gxlO-"xlO + * . (ii)
=_p^x 10" -») + (-*» =2)/?xlO(-'^ + *) . . (iii)
=i?/5xl0«-'^)-(-*" . (iv)
Taking the product first the argument runs as follows : The
order of multiplications and divisions is indifferent ; hence
line (i), which signifies that the product pq is to be divided by
10 a times and then again h times. That is, pq is to be
divided by 10 (a + h) times, a fact expressed by line (ii).
But - (a + h) = {- a) + (- h) ', that is, when the total
number of divisions by 10 is expressed by a single index, that
index is the algebraic sum of the two original indices.
The argument concerning the quotient, N, takes a similar
course. Line (i) follows by the fundamental rule for division
by a fraction. Line (ii) merely gives the same operations in
a different order. Since the complex "x 10 ~" x 10+*"
means " divide pjq by 10 a times and then multiply by 10
h times," it may be replaced, as in line (iii) by " x 10 ""■*"* ".
For if a is greater than h this can be written " x 10 "<""''' "
and means " divide by 10 (a - b) times " ; while if a is less
than 6 it can be written **x lO*-«" and means "multiply
by 10 b - a times ". One of these alternative instructions
must be equivalent to the former pair. Lastly we have that
since -a+b={-a)-{-b) the single index is the
algebraic difference of the two original indices.
Precisely similar arguments may be followed in cases in
which one index is positive and the other negative or both
positive. They will all lead to the same result, namely that
the laws of indices discussed in A § 3 may be extended to
directed indices of 10 with the substitution of algebraic addi-
tion and subtraction for arithmetical addition and subtraction.
Further, the argument holds good for all numbers. Let
m and n be two directed numbers, positive or negative,
whose numerical values are a and b. That is let m = ± a
and n = ±b. Let p be any other number, directed or non-
directed. Then x f" = x jo ± ", and x p*^ = x p-\ the
symbols implying multiplications by p when the index is
220 ALGEBRA
positive, divisions when it is negative. Moreover it is clear
that
the same sign being taken with a or 6 on the right as is taken
on the left of the sign of equality. For instance
X J9+'' X p~^ = X ^+"-*
For the left-hand side means a multiplications followed by b
divisions ; that is, a - 6 multiplications if a > 6 and b - a
divisions if a<Cb. But either of these operations can be
represented by x ^ " ~ *. For if a > 6, a - 6 is positive and
the symbols mean a - b multiplications by p. On the other
hand iiaKb,a - 6 is negative and may be written - (b - a) ;
so that X _p"~* = X ^ "**""', which, by hypothesis, means
b - a divisions by p.
The other possible combinations of signs can be examined
in the same way and the equivalence
x^±"x_p±*= x_p±"±*
established in each case. But
± a±b = {± a) + {±b) = m + 11.
Hence in all cases
X p"^ X _p" = X p^-^""
By similar arguments we have that
For instance, x p'"" -=r p^^ means a divisions by p followed
by b more divisions, that is a + i divisions in all. This total
operation may be represented by -f- ^ " ^^ or x ^ ~"" *. The
other cases lead to similar results. But
±a + fe=(±a)- (±b) = m - n, and
±b + a={±b)-{ + a)=^n-m.
Hence we have in all cases
X p'" -^ J9" = X J9"' ~ " = -^ p"~ '"
We conclude that powers of any number may be expressed
by positive indices to indicate successive multiplications by
that number and by negative indices to indicate successive
divisions, and that such directed indices may be combined in
accordance with the laws followed by non-directed indices.
Finally we may inquire whether the assumption that j9^ = 1
will also harmonize with these laws. It p^ = 1 then it is
clear that
p" X p^ = p"" X 1 and p'' -^ p^ = p'' -^ 1
= ^" = |>"
POSITIVE AND NEGATIVE INDICES 221
But in the first case p^ may be written as_p''+^ and in the
second case as p" " ^ without error. Hence if p^ be taken to
be 1 the symbol may be used in combination with other
symbols without the need of any modification of the laws of
non -directed indices.
[Exs. XXXIII and XXXIV may now be worked.]
CHAPTEE XXII.
ALGEBEAIC DIVISION.
A. Algebraic Division (ch. xvii., § 9 ; Ex. XXXV).
§ 1. Nature of Algebraic Division. — As was seen in ch.
XVIII., C, § 5, to divide - 12 by - 3 is to ask by what num-
ber - 3 must be multiplied to yield - 12. Similarly to divide
(say) a^ + b^ by a + b is to inquire what is the other factor
P such that
a^ + b^ = (a+ 6)P
This other factor P is the algebraic quotient of a^ + b^ by
a + b. As we know, it is a^ - ab + 6^.
The matter can also be put in the following way. The
expression {a^ + 6^)/(a + b) is an algebraic fraction. Now
it happens in this case that the numerator can be factorized
into a + b and a'^ - ab + b^. Hence
a3_+&3 _ (g + 6)(a^ - ab + b^)
a + b ~ a + b
= a^ - ab+b''
That is, the algebraic fraction (a^ + b^)/{a + b) could be
replaced in any formula by the expression a^ - ab + 6^ which
is not fractional. For this reason a^ - ab + b^ is called the
integral equivalent of the algebraic fraction {a^ + b^)/{a + b).
If the numerator of an algebraic fraction does not contain
the denominator as one of its factors it has of course no in-
tegral equivalent. Thus (a'^ + l)/{a + 1) has no integral
equivalent. Nevertheless we can in this case write
g-^ + 1 _ (g -f l)(g - 1) -h 2
a + 1 a + 1
2
= g - 1 + — — -
g -i- 1
Here the equivalent of (g^ 4- l)/(g -f- 1) is partly integral and
partly a fraction. But the fraction 2/(g 4- 1) dififers from the
222
ALGEBRAIC DIVISION ^23
original fraction in an important respect. The numerator of
{a^ + l)/{a + 1) is of a higher degree than the denominator as
regards the variable a ; the numerator of 2/(a + 1) is of
lower degree than the denominator. An algebraic fraction in
which the degree of the numerator, N, is, with respect to a
variable in the denominator, of higher degree than the de-
nominator, D, may be called an improper algebraic fraction.
In such a case we can always write N = DP + Q where Q
is of a lower degree than D.
§ 2. The Division Process. — When the algebraic quotient
cannot be seen upon inspection it should be sought in the
following way : —
Example : Divide a^ - 5a^ + 13a^ - 29a + 24 by a - 3.
Solution : —
a^ - 5a^ + 13a'^ - 29a + 24 = (a - 3)(a» - 2a^ + la - 8).
Our object is to find the second factor or, failing a complete
quotient, to express the dividend in the form {a - 3)P + Q.
The first term of the factor must be a^. Multiplied by - 3
this would give a term - 3a^ ; the actual term in the divi-
dend involving d^ is - 5a.^, hence we must provide for a
term - 20-^. This is done by adding to the second factor the
term - 1d^ which gives - 2a^ when multiplied by a. But
when multiplied by - 3 the - 2a^ gives -1- Ga^ while the divi-
dend contains + l^d^. We need, therefore, a term + la^.
To produce this we must add to the new factor a term -\- la.
This term multiplied by - 3 gives - 21a, instead of the - 29a
of the dividend, and so on.
With a little practice division by a trinomial can be carried
out in the same way. At first, however, it is best to proceed
as follows : —
Divide Sa"^ + Id^ - ISa^ + 47a - 21 by 2a2 + 5a - 3
Solution : —
6a* + Id^ - 15a2 -i- 47a - 21 = (2a2 + oa- 3)(3a2) - 8a» - Ga^
+ 47a -21
= (2a2 -^6a- 3)(3a2 - 4a) + 14a^
-h 35a -21
= (2a2 4- 5a - 3)(3a2 - 4a + 7)
The first term of the quotient must be 3a^. But + 5a x Sa^
gives + 15a^ instead of + 7a^ and - 3 x 3d^ gives - 9a'^
instead of - 15a^. We set down, therefore, the complement
- Ba^ - Ga^ + 47a - 21 which is necessary to make the
right-hand expression equivalent to the left. The term - 8a^
224 ALGEBRA
may be removed from this complement by adding - 4a to the
Sa^. The complement now assumes the value 14a^ + 35a - 21.
The next stage removes it altogether.
B. Geometric Series (ch. xvii., § 9 ; Ex. XXXVI).
§ 1. ^ Fraction as the Sum of a Series. — A long defunct
London newspaper, The News, published in its issue of
January 10, 1813, the following piece of intelligence : " A
few days since a bargain was made at Oswestry market be-
tween a farmer whose name is Evans and the ostler at the
Crosskeys Inn, for a goose which weighed 11 lb. The ostler
agreed to give Mr. E. one halfpenny for the first pound, a
penny for the second and in like manner to double the sum
for every succeeding pound, which raised the price of the
goose to . . ." But instead of hearing at once the conclusion
of the matter it will be more interesting if — no doubt like the
astute Mr. Evans — we determine the cost of the goose by
calculation.
If P is the cost in halfpence we have
P=l+2+4+8+...
= 1 + 2 + 22 + 23 + . . . + 210
The calculation of the sum of this series can be greatly ab-
breviated by noting that it is 1 + a 4- a'^ -f- . . . + a"~ ^
with a = 2, n = 11. But by Ex. XXXV, No. o, we have
1 - a"
1 + a + a^ + . . . + a" ~ 1 = =
1 - a
a" - 1
a - 1
Hence in this case
P = (2" - 1) halfpence
= 2047 halfpence
= £4 5s. 3id.
We are told that the discovery of this answer to the problem
caused " great mortification to the purchaser ".^ No wonder !
Here is another problem. A tree grows 1 foot during the
first year of its life ; in each successive year its increase in
height is nine-tenths of the increase during the previous year.
How high will it become in a given number of years ?
* The story was reprinted in the London Observer of January
13, 1913, under the heading " A Hundred Years Ago".
ALGEBRAIC DIVISION 225
In this case we have for the height h after n years
/i = 1 + 0-9 + (0-9)2 + (0-9)3 + /o-9)4 + . . . + (0-9)" -1
_ 1 - (0-9)"
1 - 0-9
= 10{1 - (0-9)"}
Now the interesting thing about this answer is that the term
(0-9)" or {-^^Y becomes smaller as n increases. For what-
ever value it has for a given year its value for the next
year will be only nine-tenths of the previous value. Thus, in
time, (0"9)" will become too small to be measurable or
visible — smaller, in fact, than any minutest fraction of a
foot that you can name. In other words, the second factor
of the expression 10{1 - (0'9)"} can be made to differ from 1
by as little as we please. Hence the height of the tree will
in time differ from 10 feet by an entirely inappreciable amount.
For all practical purposes, then, 10 feet is the final height
which the tree would reach even if it grew for ever. This
fact is conveniently expressed by saying that the sum of the
series which gives us the height of the tree approaches 10
as the number of terms increases. The full meaning of this
statement is that, although the tree will never reach exactly
10 feet, yet if you name a height as little short of 10 feet as
you please the tree's height will in time become still nearer
to 10 feet, and will ever after remain still nearer. No such
statement could be made about a series of increasing terms.
§ 2. Geometric Series. — These examples suggest a method
that can be used for summing any series in which each term
is obtained by multiplying its predecessor by a constant
factor. Such a series of n terms can always be expressed in
the form
a + ar + ar'^ + ... + ar'"''^,
r being the constant factor or (as it is generally termed) con-
stant ratio. Then we have for the sum S : —
^ = a + ar + ar^ + ar^ + . . . + ar""^
= a(l + r + r2 + . . . + r""!)
1 - r"
a.
1 - r
r» - 1
r - 1
The last form of the sum is more appropriate when r is
T. 15
226 ALGEBRA
numerically greater than 1 and the earlier form when r is
numerically less than 1.
Series of this kind are called geometric series. (See Note
on Ex. XXXVI, No. 6.)
In general we have, by Ex. XXXV, No. 24, that
= a + ar -h ar^ + . .
Now if r is numerically less than 1 the fractional complement
:i becomes smaller and smaller as n increases and by
1 - r -^
taking n large enough may be made smaller than any number
that any one chooses to specify. Hence the fraction a/(l - r)
becomes ever more approximately equal to the sum of the
series the more numerous the terms. When the terms are
unlimited in number or are very numerous it may be taken,
therefore, as the sum of the series. Since, however, in strict-
ness the complement ar'Y(l - r) always retains some value,
though a negligible one, a/(l-r) is called the "sum to in-
finity ". This term simply means that by a convenient fiction
we may suppose that if the terms were " infinite " in number
the sum would be exactly a/{l - r). It should be remembered
that this is nothing more than a convenient fiction. The
actual facts are, as stated, that by making the terms numerous
enough the exact sum of the series may be brought as near to
a/(l - r) as we please, and that the addition of further terms
will bring it still nearer to this number.
If r is positive and < 1 the complement ar'*/(l - r) is always
positive. The fraction a/(l - r) marks, therefore, a limit which
the sum of the series never quite reaches. But if r is negative
odd powers of r are negative and the corresponding values
of the complement ar"/(l - r) will be negative, while those
corresponding to even values of n will as before be positive.
Thus the sum of the series will be alternately greater and less
than a/{l - r) but the successive values of the sum will swing
less and less above and below a/(l -r) as n increases. The
table below gives the values of the sum of
l-i+i-i + xV- . . . for 1, 2, 3, . . . 8 terms.
Here the " sum to infinity " is
ALGEBRAIC DIVISION 227
1 + i
= 0-6666 . . .
and it will be seen that the odd terms approach this value from
above and the even terms from below. However far the series
is continued no term will exactly reach it.
w=i 3 5 7
S = 1 0-75 0-6875 0-671875
S= 0-5 0-625 0-65625 0-6640625
w = 2 4 6 8
[Ex. XXXVI may now be worked.]
15
CHAPTEE XXIII.
THE COMPLETE NUMBEK-SCALE.
(Ch. XVII., § 10; Exs. XXXVII, XXXVIII.)
g 1. The Complete Number -Scale. — It will be well at this
point to summarize and complete the chief ideas which under-
lie the work of Exs. XXVII-XXXVI.
In Ex. XXVII we began to face problems the study of
which is greatly facilitated by the device of attaching plus and
minus signs to the numbers of arithmetic. In that way we
reached the notion of a scale of directed numbers (symbolized
by the points of an endless line AB) which may be supposed
to start at zero and be continued without end each way, so
that a mirror held across the series at the zero-point, O, and
facing A would give OB, the negative part of the scale, as a
reflection of OA, the positive part. One important advantage
of using this scale is that when we are dealing with any set of
things which naturally fall into a settled order with regard to
one another we can assign a special number to each of them
and need have no fear that the assignment will be upset by the
occurrence of new members of the series for which provision
was not originally made. Thus when Fahrenheit adopted
(1714) the thermometer graduation which is commonly used
in England, he chose 32° below freezing-point as the starting-
point of his scale, because he believed that this graduation
represented the greatest cold obtainable. It was, of course,
soon found that even the natural coldness of the air of the
Arctic regions is often more intense than the coldness which
brings the mercury to the Fahrenheit zero. But by giving
signs to the degrees and continuing the scale below zero as
a negative graduation a means was easily found of placing
records of Arctic cold in the same series as the records of our
own more moderate climate.
A greater advantage is that by the use of directed numbers
THE COMPLETE NUMBER-SCALE. 229
problems which would have to be considered separately if we
were restricted to the ordinary numbers of arithmetic can all
be studied together and solved by a single algebraic investi-
gation. Take as a simple instance the formula which gives
the distance (d) apart after time t of two motor cars which
move along the same road with different speeds {v^ v^) and
are originally at a given distance apart {do). If our sym-
bols stood only for non-directed numbers we should have
to give a whole set of formulae to suit the various cases — cars
going the same way, the faster in front ; going the same way
the slower in front ; going different ways and towards one
another ; different ways but away from one another, etc., etc.
But with the aid of directed numbers all possible cases are
included in the single formula d = do + {v-i^ - V2) t. This
advantage depends entirely upon the fact that all possible
velocities and distances both to the right and to the left, and
all possible times both in the future and in the past can be
treated respectively as members of a single series of velocities,
distances or times by assigning directed numbers to them.
Bxs. XXVIII and XXIX contain many instances of this
labour-saving virtue of directed numbers.
One more example. Suppose there are a number of persons
possessed of different sums of ready money — £1, £5, £10,
£100, etc., — and a number of others who possess nothing
but owe different sums — £1, £5, £20, etc. These can all be
arranged in one series in order of their financial position. If
we take zero to represent the financial position of the person
who has empty pockets but no debts then the position of A
who possesses £5 can be labelled -f- 5 and that of B who
owes £5 labelled - 5, etc. Upon this plan transactions
which carry a person from the category of possessors into
the category of debtors can be brought into line with those
v/hich affect merely the amount of possessions or debts.
Thus if A incurs a liability for £20 his financial position is
carried from 4- 5 to - 15.
It is often said of directed numbers that (for example)
while + 20 is greater than + 18, - 20 is less than - 18 and
that all negative numbers are less than nothing. The last
example shows the inadvisability of such a statement. A
person who owes £20 does not possess less than a person
who owes £18, for neither possesses anything, and, indeed, it
is obviously absurd to suppose anything to be less than
230 ALGEBRA
nothing. Directed numbers should be spoken of as differing
not in magnitude but in position with regard to the zero. Per-
haps the most convenient form of statement to use is that, of
two numbers on the scale, the first, represented in fig. 63 by a
point to the right of the point which represents the second,
is higher in the scale, and the second number lower in
the scale. Thus - 18 is higher than - 20 but +18 lower
than + 20, while all positive numbers are higher than all
negative numbers and all negative numbers are lower than
zero. When a and h are directed numbers the symbolisms
a^b, a<b should always be read "a is higher than b,"
"a is lower than b".
. "» P
,
?
,
. P. . P. .
•20
-15 -10
-5
+5
+10 +15 +20
. 95 9
9
. Q. . q. .
-20 -15 -10 -5 +5 +10 +15 +20
Fig. 63.
§ 2. Operations on Directed Numbers. — Let the points Pj
and Pg (fig. 63) represent two directed numbers a and b.
Then we may think of all operations performed on a and b as
having the effect of moving Pj and Pg higher or lower along
the scale. Thus a x n, if w is a non-directed number such
as 3, or 4-2, or 3/11, simply carries Pj to a point n times OPj
from but can never cause it to pass over 0. The result of
a + b could be shown by sliding the segment OPg to right or
left until the end now at coincides with P^. Pg then marks
the point to which Pj must be shifted if it is to represent
a -{• b. The operation may or may not carry Pj across 0.
The result of a -ft is shown by sliding OPg until P2 coincides
with Pp The other end of the segment now marks the point
to which Pj must be shifted to represent a - b. For the
movement from O to this point represents the combination
a + {-b) = a-b.
The effect of multiplying a by a directed number b is best
thought of in the first instance as the production of a rect-
angle on the positive (upper) or negative (lower) side of OPj.
But since the number of positive or negative units of area in
the rectangle will have a place at some point E on AB the
effect may finally be taken to be to shift Pj to R. Pj will thus
THE COMPLETE NUMBER-SCALE 231
be carried across O if b is negative and left farther from or
nearer to O according as Pg is without or within the range
from + 1 to - 1. If 6 is negative and - 1< 6 < + 1, suc-
cessive multiplication by b (i.e. ab'\ n positive) will carry Pj
backwards and forwards across O and bring it constantly
nearer to — but never quite up to it. The effects of single
or repeated division of a by 6 (i.e. afe", n negative) can be
investigated similarly.
There is only one kind of operation that gives rise to difi&-
culty. Wherever P^ is originally, the operation a^ always
leaves it on the positive side of 0. It follows that the inverse
operation ^a can be performed only if P^ starts on the posi-
tive side. In other words we cannot take the square root of
a negative number. This difficulty does not arise with an
odd power such as a^, for a x a'^ will leave P^ on the positive
side of if it starts on that side and on the negative side
if it starts there. Moreover, since a x a always leaves P^
on the positive side no matter on which side it started the
result of ^a is ambiguous. It may imply carrying P^ on to
the negative side of or leaving it on the positive side.
That is, every positive number has two square roots — one
positive and the other negative.
A further difficulty which may affect all root-operations is
that of " surds ". As we saw (ch. vii., B, § 2) there is really
no number on our scale which when squared gives (for ex-
ample) + 20. We could by a geometrical construction de-
fine a point P so distant from that the area of the square
upon OP is -1- 20, but we cannot name any whole or
fractional number which belongs to that point. We can only
find pairs of numbers, belonging to points that constantly
approach each other, between which the unnumbered point
lies. For purposes of calculation this solution of the diffi-
culty suffices. If we want anything better we must regard
" ^20 " as itself the label of this point in our graduated line.
§ 3. Fractio7is. — We have seen that combinations of directed
numbers may be treated according to the same rules as com-
binations of non-directed numbers connected by plus and
minus signs bearing the ordinary arithmetical senses of
"add" and "subtract". The manipulation of fractions in
which the numerators and denominators are directed num-
bers has not yet been formally examined.
232 ALGEBRA
In ch. IX. it was assumed for convenience that an alge-
braic fraction, such as a/6, always describes an arithmetical
fraction. This need not be the case. For example, ajh
may measure the ratio of the distance {a) that a point has
travelled to the time {h) that the journey has taken. In that
case if the distance (say - 12 feet) is an exact multiple of the
time (say + 3 seconds) the number represented by ajh will
be integral ( - 4). Thus the value of ajh when a and h are
directed may be represented by any point on our scale. That
is, although ajb is always fractional in form it need not be
fractional in value.
We must now inquire whether algebraic fractions whose
numerators and denominators are directed numbers can be
treated in the same way as the fractions of Exs. XII and
XIII. To begin with, can we say that r x t = lj ? Let
us suppose that afh (represented by P, fig. 63) is negative and
cjd (represented by P^) is positive. Then a and b must have
different signs and c and d the same sign. The various
possibilities, so far as they affect the sign of the product, may
be represented thus : —
(i) — X — = — (n) — X — = —
(m) — X — = — (iv) — X — = —
Thus, however the operations may be supposed to occur, the
final result will always be to bring the indicating point to the
same position (say R) on the negative side of O. Similar
investigations, assuming that a/6 and cjd are both positive or
both negative, obviously lead to the same result. We conclude
that we may write 7 x ~ = — in all cases, without inquiry
into the signs of the numbers.
Next, is the value of ajb affected by multiplying numerator
and denominator by the same (directed) number dl Ob-
viously not, for (by the preceding argument)
ad a d a , ^^
It follows that the operation of " reduction to a common
THE COMPLETE NUMBEE-SCALE 233
denominator " for the purpose of adding or subtracting frac-
tions may be carried out with directed as with non-directed
numbers. The only difference is that the addition and sub-
traction are themselves algebraic.
Finally, since we may treat bracketed groups of directed
numbers just as we may treat similar groups of non-directed
numbers, all cases of the " simplification " of algebraic frac-
tions may be performed with symbols representing directed
numbers exactly as with those representing non-directed
numbers.
§ 4. Equations. — A last question remains. Do the rules
for " changing the subject " hold good in the case of relations
between directed numbers? As an example consider the
problem of finding the value of n from
b
a H = c
q - pn
all the symbols standing for positive or negative numbers.
If we knew the value of n and substituted it on the left the
numbers on the two sides of the relation would, of course, be
identical. Let the point P moving along the upper scale
(fig. 63) record the values of the left-hand side and Q moving
along the lower scale those of the right-hand side. Then P
and Q occupy, at first, corresponding positions on their respec-
tive scales. To remove a from the left add the component
{-a) to both sides. This operation must have the same
effect both on P and Q, moving them (say) to Pj and Qj.
The relation becomes, therefore,
b
= c - a.
q - pn
Next multiply both sides hj q - pn. Here again since the
same operation is performed upon numbers occupying the same
position in the number scale the results must be identical.
We may suppose them to be represented by Pg and Qg. We
may write, therefore,
b ^ {c - a) {q - pn).
Now divide each side by (c - a). By the same argument the
results, represented by Pg and Qg, must be identical and we
may write
b ...
-—- = q-pn . . . (1)
By continuing this process — first adding - g to both sides
234 ALGEBRA
and then dividing by - p — we bring the representative
points to their final positions — still identical — and we have
n= {b/(c - a) - q]l{- p)
= {g - b/{c - a)]lp
In this example all the typical operations involved in
changing the subject have been involved, and the argument
shows that, although they must be justified on different grounds
from those of ch. x., yet the operations themselves may be
carried out in exactly the same way upon directed as upon
non-directed numbers.
Suppose that when the calculation had reached the stage
represented by (i) and by the points Pg and Q3, a component
-(q ~ pn) had been added to each side. The obvious effect
on Q would be to bring it to zero. P would therefore have
come to zero also. The relation would then have read
b/{c - a) - {q - pn) =
or p7i + b/{c - a) - q = . . (ii)
No matter what the original relation may be it can always be
reduced, by the algebraic addition to each side of a suitable com-
ponent, to the form in which the right-hand side is zero. This
is to be regarded, therefore, as a standard form of expression.
It plays so important a part in algebra that we shall give it a
special name — equation. It is true that this term is often
applied to relations in other forms, such as (i), but there is an
advantage in reserving it for the standard form and we shall
generally follow that practice. To find the value of a vari-
able from a relation expressed in this standard form is called
" solving the equation " and any value of the variable which
satisfies the relation is called a root of the equation.
[Exs. XXXVII and XXXVIII may now be worked.]
CHAPTER XXIV.
THE PEOGEAMME OF SECTION II (EXS. XXXIX-L)
§ 1. The Contents of the Group. — It has already been said
(ch. v., § 3) that the second group of exercises of Section II
corresponds in character to the second group of Section I.
Thus those exercises, Hke these, fall naturally into two sub-
divisions. In Exs. XXXIX and XLII-XLIX the simple study
of proportionality of the earlier section is, by the introduction
of directed numbers, expanded and generalized into an ele-
mentary doctrine of functions of one variable of the first and
second degrees, culminating in a study of the relations be-
tween these functions which are covered by the technical
terms " differentiation " and " integration ". In Exs. XL
and XLI directed numbers are used in a similar way to ex-
tend the range of the trigonometrical ratios to all angles up
to 360°. In Ex. L this part of the work is completed by a
simple inquiry into the methods by which tables of the
trigonometrical ratios may be calculated. It will be more
convenient to consider the exercises of the sub-divisions
separately than to discuss the contents of the group seriatim.
The first half of this chapter is, accordingly, given to the ex-
ercises in which the idea of a function is developed, and the
second half to the trigonometrical exercises.
§ 2. Linear Functions (Ex. XXXIX, ch. xxv., A). —
The discussion in ch. xviii., C, and the graphical examples
in Ex. XXIX have prepared the way for ch. xxv., A, and
the corresponding exercise. The relation of the present
argument to the work upon which it is based is that described
in ch. XI., § 1, and ch. xii., § 4. That is, from the study of
concrete cases in which the variables are connected by a linear
relation we are now to turn our attention to the relation
itself. As before, our study of the relation is to be guided
largely by consideration of the familiar properties of its
285
236 ALGEBRA
" graphic symbol " — the straight Hne — while y and x are to
be our algebraic syrabols for the words " dependent variable "
and " independent variable ".
The lesson is simple but the teacher should attend care-
fully to a point of logic which is sometimes slurred. The
original definition of the tangent of an angle (ch. xiii., A)
applies only to angles less than 90°, for it is based upon the
properties of right-angled triangles. If, then, angles between
90° and 180° are to be considered as possessing tangents these
must be defined in a new way. For the present the simplest
way is that of ch. xxv., A, § 2 ; % definition, the tangent
of an obtuse angle is to be considered equal to that of its
supplement but negative in sign.
In connexion with Ex. XXXIX the teacher should read
again ch. xi., § 1. All the examples must be considered im-
portant because of their direct bearing upon much subsequent
work. The teacher is advised to pay particular attention to
the device of " shifting the graph " which will be constantly
used throughout the rest of the book. This device is equiva-
lent analytically to the more usual " change of origin," but
is preferred here as a more vivid, attractive and intelligible
idea. In demonstrating it upon the squared blackboard the
teacher may conveniently make use of a piece of thin white
tape to be secured by a couple of drawing pins, or a metre
rule turned edgewise to the class.
In No. 3 the positions of the lines should be determined
by substituting first y = and then x = 0, and so finding
where they cross the axes.
In No. 4 (i) the answer is, of course, that when expressed
in the form
y = ax + b
a is the same for both lines. In No. 4 (ii) the coeflScients
of X are respectively + | and - ^. The angle whose tan-
gent is ^ is the complement of the angle whose tangent is |.
Calling the former a, the latter is 90° - a. It follows that
the angle whose tangent is - | is 180° - (90° - a), i.e.
90° + a ; that is, the two lines are perpendicular. The same
argument is generalized in No. 5.
§ 3. Hyperbolic and Parabolic Functions (Ex. XLII,
ch. xxvi., A, B). — The arguments of these lessons and ex-
amples are the natural extension of those of ch. xvi., A, B,
and Exs. XXIII, XXIV. The exploration of the " fields " of
THE PROGRAMME OF SECTION II 237
the two kinds of functions offers an excellent opportunity for
heuristic work on the part of the class, and an admirable
illustration of the power of a graphic method to stimulate and
guide mathematical thinking. The teacher should, in par-
ticular, refrain from anticipating the class's discovery of the
situation of the branches needed to complete the hyperbola
and the parabola, and should be ready to make good use of
inevitable first mistakes.
The examples showing how the position of the graph de-
pends upon the form of its algebraic expression are of great
importance and should be studied by means of a movable
graph drawn upon tracing paper (Nos. I-4, 9-II). A mov-
able parabola for blackboard demonstration is easily made by
shaping a. length of stiff iron or copper wire, and may be used
with much effect. The construction of a movable hyperbola
is not so simple since it must include some device for securing
to one another the two branches of the curve. In any case
Nos. 3, 4, 10, 11, 13 should be taken orally, the movements
of the graphs being made simultaneously by the whole class.
Nos. 15-18 should be emphasized since they contain the
essence of the method which is to be used in Ex. XLIII for
the solution of quadratic equations.
Parabolic functions involving positive and negative values
of the variable are of great importance in the physical world.
For example, the path followed by the centre of a cricket
ball thrown into the air is very nearly a parabola ; that is, its
vertical height is a parabolic function of its horizontal dis-
placement. The path of a bullet fired from a rifle is less
perfectly parabolic because it is distorted by a greater air-
resistance.
The practical study of one or two cases of parabolic motion
is a legitimate application of the methods of Ex. XLIII and
will do much to secure mastery of the theory of the function.
Mr. G. Goodwill {Elementary Mechanics, Clarendon Press,
1913) has described (pp. 34-6) some beautiful experiments
for tracing the path of a projectile. Nos. 29, 30 of Ex.
XLIII indicate another more easily performed. For the pur-
poses of the mathematical classroom the following extremely
simple method will be found quite satisfactory (No. 28).
Pin a sheet of squared or plain paper on a smooth drawing-
board. Fix the board in a sloping position with its lower
edge at the edge of the table. Take a smooth and uniform
238 ALGEBRA
ball (a motor-car ball-bearing or a billiard ball has the requisite
smoothness and uniformity) and smear it lightly with lubricat-
ing oil. After one or two trials (performed before the oiling)
it will be found easy to project the ball diagonally across the
drawing-board so that it leaves an oily parabolic track. A line
can be drawn to preserve the record which can be given out to
the class for study.
§ 4. Quadratic Equations (Ex. XLIII, ch. xxvi., C). —
Quadratic equations are undoubtedly important but they have
loomed too large in the elementary algebra course. They are
too often introduced without an adequate motive — generally,
perhaps, because after " doing " equations with one root it is
time to pass on to equations with two ! Moreover, as the re-
sult of this formal and unpedagogical mode of treatment, they
are too often nothing but formal exercises leading to " prob-
lems " more than usually unconvincing and infertile. All
these disadvantages are avoided by treating the study of quad-
ratics as merely an episode in the general elementary theory
of parabolic functions. This is the point of view adopted in
ch. XXVI., C, and Ex. XLIII. The pupil has already learnt
a good deal about the parabolic function and its graph — in-
cluding the determination of turning values and the turning
points which correspond to them. He knows that, in general,
a given value of the function is produced by two different values
oi x; it is natural, therefore, to inquire how these values may
be calculated. In the search for an universal method of deal-
ing with this problem he discovers that a particular case — the
case oiy = — can be solved (when it is soluble at all) by the
simple device of factorizing the function, and finds also that
all other cases can be reduced to this case. Thus the deter-
mination of the values of x for which the value of the function
vanishes is seen to be a process of considerable technical im-
portance and therefore worthy of detailed study.
This way of approaching the subject leads to a noteworthy
consequence, namely, that no attention is given to the so-
called " imaginary roots ". In the case of every parabolic
function the field of y has a certain definite upper or lower
limit. Each value of y within the field corresponds to two
different values of x with the exception of the value of y which
bounds the field. This last corresponds, strictly speaking, to
only one value of a;, though it is convenient to adopt the con-
vention that X has in this case two identical values. To ask
THE PROGRAMME OF SECTION II 239
what values of x correspond to a value of y outside the field
of y is to ask a self-contradictory and, therefore, absurd
question. It follows that some quadratic equations will have
two roots, others one root — or, if you prefer to put it so, two
identical roots — while others will have no roots at all.
There should be no need to defend this way of regarding
the matter from the standpoint either of teaching practice or
of mathematical theory. The "imaginary " root is certainly
a rock of stumbling to the thoughtful beginner whose common
sense is offended by the supposition that a negative number
may in any sense possess a square root. On the other hand,
he is not at all perturbed to find that some equations have no
solutions — especially when he sees that this circumstance is
correlated with an obvious property of the parabolic graph —
and is, as a rule, secretly of opinion that it is " silly " to insist
upon finding them. Again, the notion of "imaginary " num-
bers is equally offensive to modern mathematical theory. It
is, in fact, simply another remnant of the cloud of confusion
that shrouded the early history of many of the main ideas of
algebra (cf. ch. xvii., § 1). It is high time to delete the
term from elementary text-books and to get rid of its mislead-
ing associations. But this act of salutary purgation demands
as its correlative a suitable treatment of the doctrine of " com-
plex numbers " which replaces in modern theory the illegiti-
mate idea of imaginaries. No doubt it would be possible to
make the study of quadratics the occasion for teaching the
elements of this doctrine, but there is much more to be said
for postponing it to a later stage in mathematical instruction.
Postponement both avoids the risk of blurring the clear lines
of association between the quadratic equation and the para-
bolic function, and secures that the student shall attack the
question of complex numbers with a mind mature enough to
appreciate the logical subtleties involved in it. For these
reasons the suggestion that quadratic equations which are
apparently insoluble may after all be solved is reserved until
the student is well into Part II of this work. The inevitable
consequence that many boys and girls may leave school with-
out hearing that all quadratic equations have roots is one
which we ought, perhaps, to be able to face with equanimity.
Most of the examples in Ex. XLIII are of familiar types.
The solution of No. l6 is, of course,
{x - ma){x - mfi) =
or x^ - m(a + P)x + m^ayS = 0.
240 ALGEBRA
It shows that to produce an equation whose roots are m
times those of a given equation we need only multiply the
coefficient of xhy m and the constant term by rn?. Apply-
ing this principle to No. VJ (i) we have for the required
equation
a;2 - 8ic - 48 = 0.
No. l8 illustrates a rather useful application of this method.
In (i) we have
Ix^ + 4aj - 3 =
whence a?^ + |ic - f = 0.
The next step is to obtain from this another equation whose
roots are seven times as large. In accordance with the fore-
going result this equation must be
a?2 + 4a; - 21 = 0,
the roots of which are obviously - 7 and + 3. Hence the
roots of the original equation are - 1 and -f ^.
The same result is applied in Ex. XLIV, No. I, where we
have
ax'^ + hx + c =
x^ + ~x + ~ = 0,
a a
The equation whose roots are 2a times the roots of this
equation is
a;2 + ^hx + ^ac =
whence {x + by - (6^ - Aac) =
and X = - b + J{b^ - 4<xc).
From this result we see at once that the roots of the original
equation are { - 6 + J{b^ - 4ac)}/2a.
§ 5. Further Equations (Ex. XLIV). — The examples of
this exercise present more technical difficulty than those of
the last. They include simple illustrations of equations with
more than two roots. Nos. II-15 ^^^^ ^^ required in the
discussion of ch. xxvii., B.
The graph of No. 3 is shown in fig. 64. It is clear that
y = + 4: corresponds to no positive value of x. It is also
obvious that it can correspond to no negative value, for if ic is
negative each of the two terms of the function is negative.
The turning-point asked for may be read with approximate ac-
curacy from the graph. It may be determined by calculation
by the following argument : —
THE PROGRAMME OF SECTION II
241
y = X +
whe
nee X —
X -1
x" - {y + l)x + (2/ + 3) =0
y + 1 ± J{{y - If - 12}
Now this expression for x shows that there are no values of
y which make (2/ - 1)^ - 12 negative. The upper and lower
limits of y are given, therefore, by the roots of the equation
{y - If - 12 = 0,
that is, by + 1 + 2 ^3. In the right-hand branch of the
curve, therefore, the lowest value of 2/ is + 4-46.
Y
Y
+fi "X z
X " ■
: ± ~ : :
i--
4.fi _ i J
UtI ' +id
Jj>fl ! ]
---H
"- - - -■ :::_:___: ±b: : :
"_ -" - — i^iiEi : __:
^'0 +1 +2 +
5 +L Js M^ +U
_
■■[■ ' *2
\ - - - -
_^ !!::::::::::::::
::::::::::::::: x';::==2"=!=5 = = -i2""
-^
^ -- -- - - -- - V^ F -
-c - ~
-- — --- : _jd-.-- - I-
_A_-j-:-_::___:::
°Y'
Y'
Fig. 64.
Fig. 65.
The graph of No. 5 is fig. 65. There are evidently no
values of x for which y = 0. The asymptotic values corre-
sponding to ic = and a? = - 0-5 should be noted with
regard to both the upper and the lower branch of the curve.
In No. II if the graph is moved 1 unit to the right we
must substitute x - 1 for a; ; the corresponding function now
becomes
y = {X - If + 3{x - If - 4.{x - 1 - 1
= x^ - 7x - 6
T. 16
242 ALGEBRA
It is easy to see that by a similar substitution it is always
possible to reduce the function
y = ax^ + hx^ ■\- ex ■\- d
to the form
y = ax^ + c'x 4- d'
in which there is no term involving a;^. This property of a
cubic function has great importance in the algebraic solution
of equations of the third degree' ; it is also the basis of the
graphic method of solution exemplified in Nos. 12- 15. In
fig. 66 the curve is the graph oi y = x^, the line AB that of
Fig. 66.
y = 7ic + 6. The two graphs intersect where x has the
values + 3,-1,-2. That is,
aj8 = 7£C + 6
or x^ - 7x - 6 =
when X has any one of those three values. But the roots of
the original equation are lower by 1 than the roots of the
derived equation ; hence the roots of
x^ + 3a;2 - 4a; - 12 =
are + 2,-2, and - 3.
In No. 14 the graph when shifted 0*5 to the left corre-
sponds to
y =^ x^ - 3-25a; + 1-5.
THE PROGRAMME OF SECTION II 243
Thus we have to determine the intersection of the graphs of
y = x^ and y = 3 "250; - 1-5. The latter is the line CED in
fig. 66, and the points of intersection are where x has the
values - 2, + 0-5, and + 1-5. The roots of the original
equation are, therefore, - 1'5, + 1, and + 2.
§ 6. Inverse Functions (Exs. XLV, XLVI ; ch. xxvi., D).
— The algebraic process for finding the roots of a quadratic
equation may be looked at from a different point of view —
namely, as a means of changing the subject of a formula in
which one concrete variable is represented as connected with
another by a parabolic relation. This use of the process is
illustrated in division A of Ex. XLV. The natural sequel is
to regard the formulae obtained in this way as exemplifying a
new set of functions, each of which is the " inverse " of' the
parabolic function represented by the formula from which it
was derived. The formal study of the relations between
direct and inverse parabolic functions and their respective
graphs becomes, then, the subject of division B of the exer-
cise. Both these matters are treated fully in ch. xxvi., D,
but a few comments upon the examples may be of service.
The formula of No. 3 can be thrown into the form
h = + 72 - 0-02(d - 4)2
from which it is seen that the ball reaches its greatest height
(6 feet) when 4 feet past the bowling crease. The answers
to the other two questions are obtained by putting d = and
d = + 6Q. The foregoing formula leads directly to the one
required in No. 4 with d as subject : —
(d - 4)2 = (72 - h)IO'02
d= + 4: ± V(3600 - 50h).
The substitution oi h = 1'45 and h = 12 gives the answers
to the questions set in No. 4.
The two methods of solving No. II are those represented
by the formulae
S = ^ff^ - f (w - V-)'
and
_ + 41 + V (1681 - 24 S)
n- g .
Both show that S cannot rise above 1681/24 or TQJj- The
exact value of n corresponding to this value of S is 41/6 or
6J. Since, however, n must be integral it is clear that the
greatest sum must actually correspond either to n = 6 or
16*
244 ALGEBRA
n = 7. The second of these hypotheses makes {n - y)^less
than the former. We conclude that + 70, the value of S
when n = 7, is its maximum. If the conclusion is correct
the 8th and all subsequent terms must be negative so that
their inclusion in the series would lower its value. This
deduction is easily seen to be true.
The examples of No. l6 are to be solved by the method ex-
plained at the end of ch. xxvi., D, § 2. Following the same
rule in No. 17 we have
y ^ ax + h
y - b
X =
a
whence the inverse function must be
1 b
y = ~x .
^ a a
It will be found that the graphs of the original and inverse
functions intersect where x = y = bjil - a), that is, in a
point whose co-ordinates are always equal. Such points
can be found only on the line y — x.
Using the same method in No. 18 we find that the function
inverse to
y = ^^ + c or (a; + b)(y - c) = a
IS
y — b or (x - c)(y + b) = a.
Here, without actually solving the equations, we can appeal
to their *^* symmetry " as a proof that the values of x and y
must be identical.
The argument of No. 21 is of much importance in con-
nexion with No. 22 and many subsequent examples.
In No. 24 let a be any non-directed number less than \.
Then when both x = + a and when x = - a the numerator
of the function will be positive and the denominator negative.
Thus the graph is below the a?- axis both immediately to the
right and immediately to the left of the ^/-axis. Also it is
evident that it passes through the origin. It follows that the
value of the function when a; = 0, i.e. zero, must be an upper
turning value. Next substitute for x the values 1 + a and
THE PROGRAMME OF SECTION II 245
1- ain succession. The corresponding values of the function
are easily found to be
o + :j| — —-?r- and 3 +
1 + 2a "^ " ^ 1 - 2a'
But the second term of each of these expressions is positive so
long as I a I < i. Thus the graph is above the line y = + 3
on both sides of the line x = + 1. Also, when a = (or
X = + 1), 2/ = +3. We conclude, then, that when x = + 1
the value of the function ( + 3) is a lower turning value.
No. 26 can be solved by the method exemplified on p. 241.
Putting
we have
whence
("-')
{X - If
X + 1 =
(2 + Hy) ± J{{4.y + l)/^/^}
Since the number under the root sign must not be negative
its lowest value is zero. But the fraction (4^/ + 1)/^/^ is
clearly zero when ^^ = - J, positive for all higher and negative
for all lower values. Hence the function may have any value
which is not below - \ ; that is, - ;J is a lower turning value.
Substitution shows that the corresponding value of ic is - 1.
The problem may also be solved by the simpler argument
illustrated in the case of No. 24. Substituting - \ior y in
the original relation we find that x = - 1. For x substitute
- 1 + a in
X
y
(X - If
a being a small positive number ; then the value of the func-
tion changes from - J to
1 - a
{a - 2f
The numerator is now higher than before and the denomi-
nator lower ; the value of the function is, therefore, raised. It
follows that - -J is a lower turning value. Strictly speaking,
neither of the foregoing arguments is complete, for they prove
246 ALGEBRA
only that the .function has no values below - J. It is con-
ceivable that the graph should run out in a single curved line
to the point ( - 1, - J) and simply end there like an un-
finished railroad. To prove that the point is a turning-point
in the natural sense of the term, we must show that the graph
approaches it along one course and leaves it along another.
That this is the case follows from the consideration that to
every value of y above - J there correspond two values of x.
Nos. 27-30 are solved by similar arguments.
Ex. XLVI illustrates the application of the preceding
methods to formulae containing trigonometrical ratios. These
examples may, perhaps, be claimed as evidence of the sim-
plicity and economy which result from the assimilation of
trigonometry with algebra. Nos. I-5 are instances of the
" ambiguous case " in the solution of triangles. No. 9 is a
very simple proof of Euclid, III, 36. Since the constant term
of a quadratic equation is the product of the roots we see
that the rectangle contained by the two values of OP is d^ - r^.
Inspection of a figure shows that this is equal to the square
on the tangent. When is within the circle d is less than r
so that d^ - r^ becomes negative (No. lO). The interpreta-
tion of this result is that the two values of OP are the lengths
of the lines drawn from O in opposite directions to meet the
circle. That is, our result is now equivalent to Euclid, III, 35.
§ 7. Walliss Law (Exs. XLVII, XLVIII, XLIX, ch. xxvii.,
A, B). — In the last two exercises of the present subdivision two
important steps are taken towards the theory of the calculus.
Ex. XLVII carries the doctrine of " integration " a stage
forward from the simple introduction in Ex. XXX ; Ex.
XLVIII takes up the notion of an *' approximation-formula,"
which we have had before us at intervals from Ex. IX onwards,
and develops it into a more or less formal doctrine of *' differ-
entiation ". Ch. xxvir. gives a full outline of the necessary
exposition ; we may, therefore, confine ourselves here to
certain general considerations which the teacher should have
before him.
The first point to note is that integration and differentiation
are taught in these exercises as a " calculus of approxima-
tions ". That is to say, our investigations, though giving re-
sults which may be regarded as true to any required degree
of approximation, do not give, and must not be represented
as giving, absolute results. This is a point of great import-
THE PROGRAMME OF SECTION II 247
ance. There are two common views about the calculus, both
erroneous and equally apt to cause confusion of ideas. The
first is that the calculus is incapable of giving anything but
approximate results although the error involved in them may
be regarded as " infinitely " small. The other is that argu-
ments which actually prove only an approximate result of
this kind may be treated as if they had established an exact
truth. The former error is pardonable, for it has its roots in
the history of the subject. Until the mathematicians of the
nineteenth century worked out the theory of " limits " it was
not easy to see that, when properly stated, the arguments of
the calculus do as a matter of fact give results which are as
unequivocally exact as those obtained by multiplication or
any other arithmetical process. For the latter error there is
much less excuse, for it is due to a lack of precision in think-
ing which almost deserves to be called intellectual dishonesty.
The appearance in an argument of either of the phrases " in-
finitely small " and " infinitely great " should always put the
reader on his guard against an illegitimate deduction born
of this vicious thinking ; the occurrence of the word " ulti-
mately " should make him actively suspicious. Nor let it be
supposed that the logical lapses here in view are to be re-
garded as of trivial importance on the ground that " nobody
seems a penny the worse " for them. On the contrary, one
of the greatest hindrances to the beginner lies in the fact that
arguments are so often presented to him as sufficient which
he feels to be quite unconvincing. It is probable that the
mature reader himself once felt the same discomfort when
asked to assent to these specious reasonings. He has now
ceased to be troubled by them chiefly because experience has
given him so many opportunities of verifying the accuracy of
results which he had originally to take partly and, perhaps,
reluctantly on faith.
If the reader asks why the arguments which lead to exact
conclusions are not taught in this book from the outset, the
answer is that the doctrine of limits, though beautiful and
entirely satisfactory, cannot profitably be presented to the
beginner. It is better to reserve it, therefore, until familiarity
with inferior methods of investigation has made him feel the
need of something better. Moreover, it must be remembered
that the results obtained by the lower type of reasoning are
quite sufficient for all practical purposes. The demand for a
248 ALGEBRA
higher type commonly arises only when immediate practical
needs are satisfied and the impulse which seeks its satisfaction
in a completer theory begins to stir in the student's mind.
The important thing is not to prejudice the success of this
theoretical activity by encouraging or permitting an uncritical
acceptance of an argument's claim to prove what in fact it
does not and cannot prove. For these reasons care should
be taken to keep the student awake to the exact significance
of the conclusions he reaches.
In the second place the teacher will note that the traditional
or Leibnizian notation is used in neither of the lessons of ch.
xxvii. In the case of lesson A the reason is that the use
of a technical notation would be unnecessary and would tend
to distract the student's attention from the real business in
band. That business may be expressed as follows. The
essence of the idea conveyed by the term " function " is that
the value of one variable can be calculated from the value of
another by the uniform application of a definite rule expres-
sible in algebraic symbolism. In addition to this idea our
studies have made us familiar with another notion, namely, of
a function whose algebraic form can be calculated from the
form of another function by the uniform application of a
definite rule. As a simple example let y he & variable whose
value is calculable from the value of x by the function 2a? - 3,
and let it be required to find a function of x by which the
value of the variable y^ may be calculated. Then we know
that this function can be derived from the function 2ic - 3
by the application of a perfectly definite and universal rule :
it will, in fact, be 4a;2 - 12ic + 9. It is easy, if we wish to
do so, to give a graphic expression to the connexion between
these two functions. If any given value of the former is repre-
sented by a straight line of the proper length the correspond-
ing value of the latter will be represented by the area of the
square drawn upon it. Now there are many important
problems in which, if the successive values of the first of two
functions are represented by the ordinates of a certain curve,
the corresponding values of the second function will be repre-
sented by the area under the curve from the i/-axis up to the
successive ordinates. The question then arises whether it is
possible in these cases, as in the former example, to lay down a
definite rule from which, given the form of the " ordinate-
function," the form of the "area-function " can be at once
THE PROGRAMME OF SECTION II 249
determined. The contribution of ch. xxvii. to the solution
of this question is the proof that in certain cases, at any
rate, such a rule can be given. If the " ordinate-function "
is of the form hx, or kx^, or kx^^ then the " area-function "
has the form ^kx^, or ^kx^, or \kx^ as the case may be. Or,
in general, if the original function is of the form
y = a + hx + cx^ + dx^
then the second function is of the form
A = ax + ^bx^ + ^cx^ + idx^.
This important rule we call, in memory of its discoverer,
" Wallis's Law". In order that its true significance as a
rule by which in certain arguments we may pass from a func-
tion of one form to a function of another form may be kept
clearly before the student, it is well to avoid a technical
notation whose original meaning was very different from this.
A graphic presentation is by far the best means of keeping
in mind the conditions under which the rule is applicable.
In ch. XXVII., B, and Ex. XLVIII, the case is different.
Here the argument requires a definite notation to symbolize
a new idea — namely, the idea expressed verbally by the term
"differential" with its carefully defined connotation of ap-
proximate equality. The Leibnizian notation, dy/dx is not
used for this purpose because, when correctly understood, it
does 7iot symbolize an approximation, but expresses an exact
equality. It is reserved, therefore, for Section VIII where it
can be introduced without danger of confusion. The same
remark applies to the Leibnizian symbol for integration. We
may add that for the same reason the terms " integration "
and " differentiation " are not given to the pupil until he
reaches Section VIII. They are employed in these observa-
tions merely because they are familiar to the reader. It is de-
sirable that the student should be taught to use them only
in connexion with results based upon the theory of limits.
The teacher will recognize the application of these remarks
in the lessons of ch. xxvii. In lesson A Wallis's favourite
argument is used to show that if (m + 1) columns of equal
width whose heights are successively proportional to the
numbers 0'^, 1^, 2^, 3^, . . . m^ are laid upon the rectangle
AM (fig. 76) they will cover a fraction of its area which is
given by the expression
1 J^
3 "^ 6m
250 * ALGEBRA
It is certain that, as m increases, this fraction becomes con-
stantly nearer to 1/3, and that by taking m sufficiently large
the approximation may be made as close as we please. It also
seems obvious that the area covered by the columns approxi-
mates in much the same way to the area under the semi-
parabola OP. Thus ha,ymg proved that the. area under the
curve is, apparently to an unlimited degree of closeness, one-
third of the area of the rectangle AM, we are almost forced
to believe that the former is exactly one-third of the latter.
At any rate if the fraction is not truly one-third there is
practically no likelihood that the discrepancy could be de-
tected by measurement. Still the fact remains that what we
have proved is only an approximation. However numerous
the columns are made they will remain a set of columns and
their tops can never become a parabolic curve. It remains
an assumption that, because their united area approaches one-
third of AM, the area under the parabola is exactly one-third
of that area. Thus while we may legitimately emphasize the
practical certainty of our conclusion no attempt should be
made to disguise its theoretical imperfection.
Similarly, in ch. xxvii., B, and Ex. XLVIII, no attempt
should be made to slur over the fact that the " differential
formula " is always, from the logical standpoint, merely an
approximation -formula, though it is quite proper to emphasize
the other fact that no limit can be set to the closeness of
the approximation which it represents. In the interests of
lucidity and exact thinking it is equally important that
careful attention should also be given to the points brought
out in the Notes before Nos. 12, 13, 15 and 17.
It has already been remarked (p. 57) that in the investiga-
tion of his law Wallis was led to the invention of negative
and fractional indices. In the same place it was said that,
on the whole, it is better to introduce the theory of fractional
indices in connexion with logarithms. The class which
follows either of the schemes set out on p. 50 will probably
reach the theory of indices in Section III before it reaches
Ex. XLVII. Nevertheless, the index method of representing
a square root has been introduced in the argument of ch.
xxvii.. A, upon Wallis's lines and without reference to the
more formal treatment of ch. xxxiv. This procedure leaves
the teacher free to choose an order of treatment different from
either of those given on p. 50, and also enables the student to
THE PROGRAMME OF SECTION II 251
become acquainted with an argument of great intrinsic interest
and historical importance.
In Ex. XL VII, No. I (vi), the area-function is, by WalHs's
Law,
A = 3-4 ( - x)^
the negative sign implying that the whole of the curve is on
3.
the left of the ^/-axis. In No. 5 the area-function is A = x'^-
Hence the areas under the curve up to the ordinates ic = + 49
and X = + 4: are respectively
(4- 49)^ = + 343 and (+ 4)^ = + 8.
Thus the area between these ordinates is 343 - 8 = 335.
In solving No. 6 the easiest method is to apply directly
the theorem that the area within the curve is two-thirds of
that of the rectangle with the same height and base. Thus
the height of A above the ic-axis is 25 "6 and the distance
between the points where the curve cuts the ic-axis is 32 ;
hence the required area is
§ of 25-45 X 32 = 542-9.
In No. 7 it will, of course, be necessary to find two para-
bolic areas in this way and to take their difference.
Nos. 9- 1 1 are important as leading to the conclusion that
Wallis's Law can be used to find the area-function when the
ordinate-function is a complex of the form
y = a + bx + cx" + . . . .
in No. 9 we find that
pq = 0'04aj2
whence we deduce that the area between AB and AC {Exer-
cises, fig. 44) may be regarded as the area under the para-
bolic curve y = 0*04a?^ after every ordinate has been shifted
vertically upwards — just as the dark columns in fig. 55 (p.
206) have been shifted vertically to produce fig. 56. It follows
that the area-function of the surface between AB and AG is
the same as that of the space under y = \0-04:x'^, namely,
A = ^V x^
and that the total area is one-third of that of a rectangle
whose sides are equal to AD and BC. To answer the first
part of No. 10 we have merely to add the area of the triangle
ADC. But the ordinate-function for points on AC is easily
seen tohe y = O'Sa?; hence the ordinate-function for points
on AB (No. 11) must be
y = 0-8aj + 0-04aj2
252 ALGEBRA
In view of the preceding argument the area-function must be
A = K + rh^^
Substituting a; = 20 we obtain the same result as before for
the total area under the curve AB.
In No. 21 the successive fractions can be arranged in a
sequence in which the denominators show a constant difference
of 4:—
2^ 3^ 4 5 6
4' 8> 12' 1 6' ■2(J"» • • • •
It is obvious that they are all of the form
m + 1 _ 1 ^ ^
4w 4 4m'
Assuming in No. 22 that this law will hold good for all values
of w, it follows that the area under the curve will be one-
quarter of that of the underlying rectangle. In No. 30 this
assumption is to be justified by the method of recurrence
already illustrated in ch. xxvii., A, 3. The proof is as follows.
Assume that
0^ + 1^ + 2^ + . . , . +p^ _ p + 1
(j) + l)p^ 4tp
that is, that
03 + 13 + 23 + . . . . + ^3 _ 1 (^ + 1)2^2,
Add the term (p + 1)3 to each side. Thus we have
03 + 13 + 23 + — + {p + 1)^ = iip + lyy + {p + iy
= i{p + 2Y{p + ly
as in Ex. XXXVIII, No. 10. It follows that
0^ + 13 + 2^ + . . . +p^ + {p + iy ^ 1 1
(p + 2){p + 1)3 4 "^ 4(p -f ly
We conclude that if the result in question holds good when
m = _p it also holds good when m =- p + 1. But it is known
by trial to hold good when m = 1, 2, 3. . . . Therefore it
holds good universally.
Practically all the difficulties hkely to be met with in
Ex. XLVIII have been anticipated in the exposition of
ch. XXVII., B.
In Ex. XLIX, fig. 67 is the graph of Nos. 2 and 8. From
the formula
t/ = x3 - 3a: -1- 2
we deduce that
1 = 3x^-3 . . . A
THE PROGRAMME OF SECTION II
253
and that
hx^
6a;
The gradients at the points specified in No. I are found by
substituting the given values of x in A. For example, when
X = - 1, 8y/Sx = ; that is, the point is a turning point.
li X is a little below - 1, ^y/^x is positive, if a little above,
negative. It is clear, therefore, that this turning point is a
maximum. This fact is also proved (No. 8) by the circum-
stance that S'^y/Sx^ is negative when x = -1. At ic = 0,
Sy/Sx = - 3, and ^^yjhx^ = 0. If a^ is taken a little below
zero ^^yjSx^ is negative, if a little above, positive. It follows
(No. 8) that the intersection of the graph with the y-a.xis is a
point of inflexion.
?^
Y'
Fig. 67.
Fig. 68 is the graph of Nos. 4 and 9. Here we have
8y
8x
Qx'^ + 6a; - 36
A
8x^
= Ux+ 6
B
As before, the gradients at the points specified in No. 3 are
264:
ALGEBRA
to be found by substituting the given values of x in A, while
the character of the curvature at each point may be examined
either by considering how the gradient varies to right and
left of it or by substituting for x in B. For ic = - 0*5
^ylW = ; for a value a little below this, 8^?//8a;"^ is nega-
tive, for one a libtle above, positive. Thus the point is a
point of inflexion ; the curve is below the tangent on the left
of it and above the tangent on the right.
In No. 22 let a be the side of a square end and I the length
m
ts
sa
2£
"^^
Fig. 68.
of the box, both being measured in inches.
volume in cubic inches. Then we have
4a + Z = 72
V = aH
= 72a2 - 4a3
144a - 12a2
Also let V be the
8a
8a^
144 - 24a
A
B
From A we have that 8V/8a = when a = or a = 12.
JFrom B we see that a = makes S'^Y/Ba^ positive, while
THE PROGRAMME OF SECTION II 255
a = 12 makes it negative. Hence the latter value implies a
maximum. The corresponding value of I is 24 inches. Thus
the largest box is 2 feet long with an end 1 foot square.
§ 8. Directed Trigonometrical ^Ratios (Exs. XL, XLI;
ch. XXV., B). — We now turn to the consideration of the exer-
cises and lessons in which the idea of directed numbers is
applied, to the trigonometrical ratios. The case of the tangent
has been dealt with already (§2). The principle followed
there is applied again in the case of the sine and cosine. It
is shown that certain formulae would become simplified and
(so to speak) condensed if it could be supposed that angles
between 90° and 180° have, like acute angles, sines and cosines.
These new sines and cosines are then defined in such a way
that the desired simplifications are secured. Thus we decide
that the sine of an obtuse angle shall be considered identical
with the sine of its supplement, and the cosine equal to the
cosine of the supplement but opposite in sign. It is impor-
tant to observe that no proof is or can be offered of these
statements ; they are simply conventions adopted in order
to bring different cases of a rule (such as the rule for finding
the area of a triangle) under a single formula. The examples
of Ex. XL give practice in the application of these new sines
and cosines and also in the two important general properties
which are proved in the course of the lesson.
From the theoretical standpoint the importance of Ex.
XLI is that it carries still farther the principle just explained.
It is shown that the work of the surveyor is much simplified
by assigning sines and cosines to every possible angle from
0° to 360°. Once more the new sines and cosines must be
defined in such a way that they actually perform the service
which is required of them and for which they are called into
being. The guiding idea this time is that it shall be possible
to find correctly the northing or southing and the easting or
westing (ch. xiii., B) which correspond to any given vector
by multiplying its length, in the first place by the cosine, and
in the second case by the sine, of its bearing, bearings being
measured continuously round from the north through the
east. The values of the sines and cosines to be allotted to
the various angles are to be determined in such a way that
this convenient result follows. The exercises close with a few
examples (forming division C) to introduce the idea of a
negative angle which may have any value up to 360° and
256 ALGEBRA
may also be thought of as possessing a sine, cosine, and
tangent.
This extension of the field of the sine, cosine, and tangent
marks the limit reached in Part I. Any further extension
should be based, as these extensions have been, upon the re-
quirements of some practical problem. Such problems do
not arise until, in Part II, Section VII, we meet them in con-
nexion with the study of harmonic motion and waves.
Ex. XLI is also intended to illustrate with a certain amount
of actuality the methods used by the surveyor in mapping out
a country. It is necessary that the work in some examples
should be divided among the members of the class or it may
become undesirably burdensome. If possible, the examples
should be supplemented by simple field work carried out as
nearly as may be in accordance with the actual methods of
the surveyor. Inexpensive prismatic compasses and theo-
dolites are now to be obtained from several makers of scientific
apparatus. In boys' schools there should be little difficulty in
supplying efi&cient instruments out of the resources of the
manual training department.
§ 9. The Calculation of ir. Trigo^wmetrical Tables (Ex.
L ; ch. XXVIII., A, B). — Even an elementary course must be
considered incomplete if it includes no discussion of the
methods by which trigonometrical tables may be calculated.
This highly interesting and important subject is generally
omitted from elementary treatises because it is thought to
presuppose a knowledge of the expansions for the sine and
cosine of a given angle. Yet it is probable that all the tables
actually in use in our schools are lineally descended from
tables calculated long before the conception of these expan-
sions had entered into any mathematician's head. Archimedes
obtained his famous approximations for tt by considering the
perimeters of inscribed and circumscribed polygons. Hippar-
chus (c. 150 B.C.) probably, and Ptolemy (c. a.d. 100) cer-
tainly, calculated the lengths of the chords which correspond
to the various angles of the quadrant. These were used in
trigonometrical calculations right down to the later middle
ages when the use of semi-chords or sines filtered into Europe
from the Arabs. Tangents were invented by the celebrated
Regiomontanus about 1450 ; secants followed about a century
later, but the credit of their invention is disputed.
The first modem method of calculating tt is that of Ludolph
THE PROGRAMME OF SECTION II 257
van Ceulen ; our own John Wallis followed in 1655 with a
totally different method which gave the value as an infinite
product. Van Ceulen's method and the method of calculating
trigonometrical tables which goes with it are illustrated in
ch. XXXVIII. and Ex. L. The exposition of the chapter is too
full and the examples too straightforward to need comment.
Archimedes' famous evaluation of tt is given in full in
Sir T. L. Heath's translation of his collected works. Much
interesting information about the general history of the
trigonometrical ratios is given in the introduction to Hutton's
Mathematical Tables (1785). Wallis's account of his dis-
covery of the product-approximation for tt is given in his
Arithmetica Infinitorum (1655). The present writer has
given a summary of it in the Mathematical Gazette for Dec.
1910 and Jan. 1911. Van Ceulen's book, as far as he knows,
remains inaccessible except to readers of the original Dutch
or of Snell's Latin translation of 1619. A few of the
huge numbers with which it teems are given in ch. xxxviii.
to illustrate the immense labours which the founders of
modern mathematics were sometimes ready and even glad to
face. His longest value for tt — the " Ludolphian number "
as it was called — is said to have been inscribed upon his
tomb. He was modestly proud of his achievements, and
records with an obvious swelling of the soul the precise date
upon which his method occurred to him : " Anno reparatae
salutis 1586 mense septembri istam ad laterum polygonorum
circulo adscriptorum investigationem viam inveni " ; and relates
that he immediately proposed to himself the tremendous task
of calculating the perimeters of the inscribed and circum-
scribed polygons of 167,772,160 sides !
17
CHAPTEK XXV.
LINEAR FUNCTIONS.
EXTENDED USE OF SINE, COSINE AND
TANGENT.
A.
obvious
Linear Functio7is (ch. xxiv., § 2 ; Ex
The Belation y = a + bx alivays
the relation y = kx
that
, XXXIX).
Linear. — It is
describes a straight line
through the origin
even when k, as well
as X and y, is directed.
For example, y = +
0'9a; describes the line
AA' (fig. 69) and y
= - 0-9a; the line
BB'. The proof is
essentially that of ch.
XII., A, § 2. In y
= + 0-9ic let the
values of x rise by
equal steps from zero
to OM. Then the
value of y rises by
equal steps, 0-9 times those of x, to PM. On the other
hand if the value of x falls by equal steps from zero to OM',
y falls by equal steps, 0*9 times those of x, to P'M'. Thus
the graph is a continuous straight line lying in what may be
conveniently called the first and third quadrants of the graph-
paper. The argument may be repeated with regard to
y = - O'^x with the difference that upward steps in x imply
downward steps in y and vice versa, so that the corresponding
straight line, BB', lies in the second and fourth quadrants.
Similar results could obviously be obtained for all values of k,
positive or negative.
258
LINEAR FUNCTIONS 259
Consider next the relations
y = + 0-9rc + 5 and y = - 0*9aj + 5.
It is clear that the effect on the graph of the addition of + 5
to + 0*9a; in one case and - 0'9ir in the other is to raise each
ordinate through 5 units of the vertical scale. Thus P and
P' rise to p and p, Q and Q' to q and q\ We may say, then,
that the new relations describe the old lines raised, parallel
to themselves, through 5 units. Similarly y = + Sx - 7
describes the line through the origin, y = + Sx, after it has
been lowered 7 units.
We conclude that a relation of the form y = px + q is b,
linear relation for all values of p and q, and that it describes
the straight line through the origin, y = px, moved through q
units — upwards if q is positive, downwards if q is negative.
§ 2. Tangents as Directed Numbers. — AA' and BB' (fig. 69)
may be described as lines in which the ordinates bear a con-
stant ratio to the abscissae — the ratio being in the former
case + 0*9, in the latter, - 0*9. This description is an
obvious extension to directed variables of the notion of direct
proportion (ch. xii.. A). But the lines, as lines, can be de-
scribed in another manner. The table on Exercises, p. 107,
gives 0*9 = tan 42". AA' and BB' are, therefore, lines in-
clined at 42° to the x-a,xis, the difference between them being
only in the direction of the inclination. Both lines might be
supposed to have started from the position XX' and to have
reached their present position by rotating about O through
42°, AA' in the anticlockwise and BB' in the clockwise
direction. Thus, speaking generally, the relation y = kx
would describe a line through the origin having a certain
inclination to the a;-axis. The inclination would be the angle
whose tangent is numerically equal to k, and it would be
anticlockwise if k were positive and clockwise if k were
negative.
But there is a disadvantage in this way of describing the
line y — kx. Suppose a line to start from XX' and to be
rotated anticlockwise about O until it coincides with YY'.
Then k in y = kx will assume successively all possible posi-
tive values from zero upwards. According to the last para-
graph, if the line is now to assume the various negative
inclinations in due order it must go back to XX' and begin
to rotate again in the opposite direction. Now it seems
much more natural tQ suppose the Une to reach the position
17*
260 ALGEBRA
BB' by a continuation of the process which brought it through
A A' to YY' — that is, by a continuous anticlockwise rotation.
To secure this simpler way of looking at the matter all we
need do is (a) to measure the " inclination " of the line by
supposing in every case an anticlockwise rotation from XX' ;
and (6) to extend the idea of a tangent to include nega-
tive and positive values. Thus the tangents corresponding
to rotations of 100°, 110°, 120°, etc., must be taken to
be - 5*67, - 2-75, - 1-73, etc. ; that is to say, the tangents
of 80°, 70°, 60°, etc., with a negative sign prefixed. In general,
if a is an angle between 90° and 180° we must assume (or
rather define) that
tan a = - tan (180° - a).
Upon this understanding it becomes easy to lay down a single
rule for describing the line which corresponds to any given
linear relation. For example, y = + 0-7a? - 3*7 describes a
straight line, inclined 35° to the ic-axis (shortly, " of inclina-
tion 35° "), which has been lowered from the standard position
through 3*7 units; y = - O'lx + 12'8 describes a line, in-
clined 145° (= 180° - 35°) to the a;-axis, which has been
raised through 12*8 units.
§ 3. Linear Functions. — We have constantly spoken of a
relation between two variables, meaning that when one vari- .
able changes the other variable also changes in some definite
way. This kind of connexion between variables is often
described by saying that one is a function of the other. The
idea of a function as it is used in mathematics is perfectly
simple ; only the name is alarming. Thus the length of a
rod is said to be a function of its temperature because if you
change the temperature the length will change. The rent a
man pays for his house is (or should be) a function of two
variables — his income, and the size of his family. If he has
a larger or smaller income or a larger or smaller family he
will normally pay a larger or smaller rent. According to
uncompromising temperance advocates the amount of crime
in England is almost entirely a function of the amount of
alcohol consumed. As a last example we may take the
celebrated statement that the quantity of red clover on a
farm is a function of the number of its cats. For red clover
depends for fertilization upon humble-bees which are preyed
upon by field-mice which in their turn are the prey of the
catsl
LINEAR FUNCTIONS 261
It is obvious that one variable may be a function of any
number of variables. For example, the national income
obviously depends upon a vast number of changing factors,
each of which by rising or falling in amount affects the total
in some way. For the present, however, we shall consider
only functions of a single variable, and we shall denote that
variable by x. By speaking of a function of x we shall mean,
then, some expression whose value depends on the value of
X. Thus
dx + 7, j9-2x - 3-4, x^ - 2x+ 4, 1/(4 - dx + 2x^)
are all functions of x. The value of the function for a given
value of X will always be denoted by y. Thus the expression
y = ^/9-2x - 3-4 may, as we please, be regarded either as
stating that a certain relation exists between the values of x
and y, or as stating that y is the value of a particular function
of X, namely J9'2x - 3*4. The graph of a given relation
between x and y may also be described as the graph of the
corresponding function of x. Thus since the graph corre-
sponding to any relation of the form y = px + q is & straight
line we may say that any function of x which can be reduced
to the form px + g is a linear function.
[Ex. XXXIX may now be worked.]
B. Extension of Meaning of Sine and Cosine (ch. xxiv., § 8 ;
Exs. XL and XLI).
§ 1. The Problem. — In ch. xiii., B, the sine and cosine were
studied as factors by which the sides of a right-angled triangle
can be calculated, given the hypothenuse. Since the angles
of such a triangle cannot be greater than 90°, only angles be-
tween 0° and 90° were considered as having sines and cosines.
The advantage of assigning tangents to angles between 90°
and 180° now suggests the inquiry whether the range of sines
and cosines is capable of a similar useful extension.
§ 2. Sines of Angles between 90° and 180°. — Let us take
the sine first. Fig. 70 shows that, when the angles A and B
are both acute, we have : —
area of triangle = ^cp = ^ca sin/8 = ^cb sin a.
But, when the angle A is obtuse (fig. 71), while the equiva-
lence
area = ^ca sin (3
still remains, the second formula becomes
area = -i^bc sin (180° - a).
262
ALGEBRA
That is to say, we cannot express the area of a triangle in
terms of two sides and the included angle without first in-
quiring whether the angle is acute or obtuse ; the form of the
expression being different in the two cases.
This inconvenience could be removed by agreeing that
angles between 90° and 180° shall be considered to have sines,
the rule being that sin a = sin (180° - a) — or, in words,
that the sine of an angle shall be reckoned the same as the
sine of its stippiement. Upon this understanding the area of
a triangle can always be expressed as = | fee sin a whether a
is acute or obtuse.
Fig. 71.
and 180°. — The same
fig. 70
In
Fig. 70.
§ 3. Cosines of Angles behveen 90
figures suggest a similar extension of the cosine,
we have
c = b cos a + a cos /S
but in fig. 71 the formula must be written : —
c = a cos f3 - b cos (180° - a).
As before, the inconvenience of two rules — one for an acute
angled and another for an obtuse angled triangle — may be
avoided by supposing that every angle between 90° and 180°
has its own cosine as well as its own tangent and sine. But
the rule takes this time the form
cos a = - cos (180° - a)
or, in words, that the cosine of an angle is to be reckoned the
negative of the cosine of its supplement.
§ 4. Harmony of these Exte7isions. — For angles between
0° and 90° tan a = sin a/cos a. Does this result still hold
good when a is between 90° and 180° ? In such a case we
have
tan a = - tan (ISO*" - a)
sin a = sin (180° - a)
cos a = - cos (180° - a)
so that sin a/cos a == - tan (180° - a) = tan a as before.
LINEAR FUNCTIONS 263
It is easy to redefine the sine, cosine and tangent of any
angle between 0° and 180° in such a way as to secure all the
results of eh. xiii. and the present
chapter. Let the radius of the
semicircle of fig. 72 be taken as
unity and let the angle a be traced
by the radius OP starting from the
position OA. When it stops in any Blfe"
position such as OPj or OP2 drop a
perpendicular PjNj or PaNo. Then
PN is to be defined as sin a, ON as cos a and PN/ON as
tan a. It is clear that for a < 90° (as AOP^) sin a, cos a and
tan a are all positive, while for a > 90" sin a is positive, cos a
and tan a negative. Moreover it is obvious from the sym-
metry of the semicircle about 00 that the sine, cosine and
tangent of any angle are numerically the same as those of
the supplement of the angle.
§ 5. Two Important Theorems. — Two important properties
of the triangle can now be stated very concisely : —
(a) Whether a triangle be acute angled or obtuse angled we
have : —
area = ^bc sin a = ^ca sin /3 = ^ab sin y
Hence (multiplying by 2labc) : —
sin a/a = sin ^/b = sin yjc
or : each side is directly proportional to the sine of the opposite
angle.
(6) Whether the triangle be acute angled or obtuse angled ^
we have (figs. 70 and 71) : —
a^ = {b sin a)^ + {c - b cos a)^
= 62 sin2 a + c^ - 2bc cos a + b^ cos^ a
= b^ (sin2 a + cos^ a) + c^ - 2bc cos a
= 62 + c2 _ 26c cos u
The former theorem makes it possible to solve many prob-
lems more conveniently than could otherwise be done ; the
latter enables us for the first time to calculate any angle of a
triangle when the lengths of its sides are known.
[Exs. XL and XLI can now be taken.]
1 In fig. 7lp = h sin (180° - a) = 6 sin a ; AD = 6 cos (180° - a)
= 6 X (- cos a).
CHAPTBE XXVI.
HYPERBOLIC AND PARABOLIC FUNCTIONS.
A. Hyperbolic Functions (ch. xxiv., § 3 ; Ex. XLII, A).
§ 1. The Rectangular Hyperbola. — In ch. xxv., A, we saw
how the use of directed numbers leads to an important exten-
sion of the notions of direct proportion and of the hnear rela-
tion. We now inquire into the effect of admitting these
numbers into the relation called inverse proportion.
For this purpose take the relation xy = 24: and plot the
values of y corres-
ponding to negative
as well as positive
values of x. Fig. 73
shows the result. In
addition to the curve
AVB, familiar from
ch. XVI., A, we have,
in the opposite quad-
rant, the curve A'V'B'.
The second curve is a
precise reproduction
of the first, except
in position ; for every
point on AVB, such
as (4- 6, + 4), is
balanced by a point
in A'V'B', such as
(- 6, - 4). If a mirror were fixed along MM, at right
angles to the axis W and facing AVB, the image of AVB
would be A'V'B'.
Although AVB and A'V'B' are separated and lie in distinct
quadrants yet they must be regarded as simply two branches
264
Fig. 73.
HYPERBOLIC AND PARABOLIC FUNCTIONS 265
of one and the same curve, just as OA and OA' in fig. 69
are parts of one line. Suppose a point, Q, to approach O
along X'O from an endless distance beyond X'. Then the
point, P, defined by the relation y = + 24:/ x, will move along
B'V'A'. As the distance QO diminishes the length of the
ordinate increases without end. That is, it is never so long
that it does not become still longer as Q comes nearer still to
0. As Q passes from one side of O to the other the story of
the tangent is repeated (ch. xiii., A). We cannot say that the
curve has a point on the ^/-axis, for it is senseless to speak of
the quotient of 24 -=- 0. Yet no interval can be specified to
the left and right of so short that there are no correspond-
ing positions of P still nearer to the y-s^xis — on the left at
an endless depth along the y-scsde, on the right at an end-
less height. As Q proceeds along OX, P simply repeats
in reverse order along BVA its previous adventures along
B^rA'.
The one curve with its two branches is called a rectangular
hyperbola. V and V are the vertices, the line VV ( = 2 J^.k)
is the axis, XX' and YY' the asymptotes (ch. xvi., A, § 2),
O (because it bisects all chords such as CC) the centre.
§ 2. Movements of the Hyperbola. — If the hyperbola is
raised (say) through 7 units the corresponding relation must
become y = 24/ic + 1, or y - 7 = 24/rr. If it is lowered 7
units we have y = 24: /x - 1, or y + 7 = 24/ x. Moving the
figure to the left will not (as in Ex. XL, No. 12) produce
the same effect as raising it, but it will affect the algebraic
description of the curve in the same way as in the case of
the straight line. If (for example) the curve is moved 3 units
to the left the distance of a point from the vertical asymptote
is no longer x ; it becomes x + S. Similarly a movement of
3 units to the right changes the distance from the asymptote
from X into x - 3. Thus if the figure is moved 3 units to the
right and then 7 downwards the relation becomes : —
y + 7 = 24:l{x-3) or i/ = 24/(a; - 3) - 7.
§ 3. Negative Values of k. — If we had plotted y = - 24/a;
instead oi y = + 24/ic the result would evidently have been
to give the dotted curve of fig. 73. Two movements (for ex-
ample) which carry the centre to the point ( - 6, -H. 5) change
the corresponding relation to : —
2/-5= -24/(x-f6) ory= - 24/(a^-l-6) + 5.
§ 4. Hyperbolic Functions. — Just as ax + 6 is termed a
266 ALGEBRA
linear function of x because the graph oi y = ax + b is a,
straight line, so we may term any function that can be thrown
into the form kl{x + a) + & a hyperbolic function of x because
the graph oi y = kl{x +a) + b is a, rectangular hyperbola.
This function is by no means unusual in physics where
it occurs in cases that admit of negative and positive values
of x.^
[Ex. XLII, A, may now be worked.]
B. Parabolic Functions (ch. xxiv., § 3 ; Ex. XLII, B).
§ 1. The Parabola. — Let us now explore the wider range
included in the relation y = kx^
when X, y and k may be directed
numbers. The hard line of fig.
74 is the graph oi y = + \x'^.
Like the extended graph of
y = kjx it lies in two quadrants,
but they are adjacent instead
of opposite quadrants. The
reason is obvious. Whether
X = p or X = - p, in each
case \x'^ = \p^. Thus OF is
the image of OP in a mirror
set up along OY. The curve
extends upwards and outwards
both ways without limit but no
part of it lies below the ic-axis.
jijQ ^^ It is called a parabola. V (here
identical with the origin O)
is the vertex or head ; VA (here identical with the ?/-axis) is
the axis of the parabola about which the curve balances;
TVT' (here identical with the a? -axis) is clearly a tangent at
the vertex.
If we plot y = - Ix^ instead oi y = + {x^ we obtain the
dotted curve, identical with the former except that it is " head
up " instead of " head down ".
For example, the familiar formula - + -
which gives the
distances of object and image from a lens. Substituting x for u and
y for V we have Ijy + Ijx = 1//, becoming {x - f) {y - f) = P, or
y=Pl{x-})+f-
HYPERBOLIC AND PARABOLIC FUNCTIONS 267
§ 2. Movements of the Parabola. — Like the hyperbola the
parabola can be supposed moved so that the axis and
tangent at the vertex remain unchanged in direction al-
though altered in position. Thus y - 7 = l{x + 5)^ or
y = ^{x + 5)'^ + 7 describes the parabola y = + \x'^ moved 5
units to the left and 7 units upwards. So 2/ + 8 = - \(x - lOy^
ovy = - ^(x - 10)2 - 8 describes y = - \x^ moved 10 units
to the right and 8 downwards. The shortest way to express
the movements is to say that in the first case the vertex is
now at (- 5, + 7) and in the second case at (+ 10, - 8).
§ 3. Parabolic Functions. Turning Values. — In general,
if we start with a parabola y = ax^ and move it horizont-
ally through b units, and vertically through c units (a, 6, and
c being all directed), it corresponds in its final position to
y = a{x - by + c. For this reason any function that can
be thrown into the form a{x - b)"^ + c may be called a
parabolic function of x.
Consider the functions - 2x'^ - 12x + 4 and + 3x'^ - 12x - 3.
The first can be written - 2(ir -f 3)^ -t- 22 and the second
+ '^{x - 2)2 - 15. Both are, therefore, parabolic functions.
The graph of the first is the "head up " parabola y = - 'ix^
moved 3 units to the left and 22 units upwards ; that of the
second the " head down " parabola 2/ = + Sa?^ moved 2 units
to the right and 15 units downwards. Moreover it is clear
that this reshaping of the function could be carried out in
the same way whatever numbers, positive or negative, re-
placed the - 2, - 12, and + 4 of the former function. That
is, a function of the form px^ + qx + r \q always parabolic
whatever the values of p, q and r.
We have seen that in a parabolic function a value of y can
be found corresponding to any value of x that can be specified.
But the converse statement does not hold. Thus the " head
down " parabola
y=+l{x+5Y+7
of § 2 has no points lower than y = + 7, and the " head
up " parabola
y = - \{x - 10)2 - 8
no points higher than y = - S. That is, the function + \x'^
-t- fa; 4- 13 J has no values lower than -h 7 and the function
- \x^ + 5a? - 33 none higher than - 8. Similarly the
parabola
y = - 2(a; + 3)2 + 22
268 ALGEBRA
has no points above y = + 22, and the parabola
2/ = + S {x - 2)- - 15
none below - 15. Hence no value of x can give the function
- 2x'^ - 12x + 4 a higher value than + 22, or the function
+ Sx'^ - 12a? - 3 a lower value than - 15.
These results can easily be deduced algebraically. Con-
sider the function Sx- - 12a; - 3 or + S(x - 2)2 - 15. Let
X begin by being negative and very large, and let it approach
zero. Then x - 2 will be negative all the time, but {x - 2)^
will, of course, be positive. Both statements remain true as
X passes through zero and approaches + 2. At this point
x - 2 = and y = - 15. As x moves towards higher posi-
tive values X - 2 becomes positive and {x - 2)^ is, of course,
also positive. Thus (x - 2)^ if it has a value at all is always
positive. That is, it always makes the value of the function
higher than - 15. As ic approaches and passes through
+ 2, 2/ descends to - 15 and then ascends again. For this
reason - 15 is called the turning value of the function. In
this case it is a lower turning value, because the function,
after descending the number scale to - 15, begins to ascend.
Similar considerations show that, in the function
- 2(x + 3)2 + 22
{x + 3)2 is always positive, being a square, and that
— 2(x+ 3)2 is always negative. The value of the function
is, therefore, always below + 22 if a; 4- 3 has any value at
all. When x = - 3 it has no value and the function has
the upper turning value + 22. It is obvious that a function
whose graph is a "head up " parabola always has an upper
turning value and one whose graph is '' head down " always
a lower turning value. The turning value is simply the
distance the parabola has been moved up or down from what
may conveniently be called the ' ' standard position " in which
its vertex coincides with the origin. Start, for example, with
the head down parabola y = + 2-3a;2. Move it 3 '7 to the
right and 6*5 upwards and we have y = + 2'S{x - 3*7)2
4- 6-5. The function 2'S(x - 3*7)2 + 6*5 has a lower
turning value -H 6*5 when x = +3*7. Move the original
parabola 4*8 to the left and 9*3 downwards and we have
y = -i-2*3(a; + 4*8)2 - 9*3. The function 2*3(a; + 4*8)2 - 9*3
has a lower turning value of - 9*3 when x = - 4*8.
[Ex. XLII, B, can now be worked.]
HYPERBOLIC AND PARABOLIC FUNCTIONS 269
§ 4. The Method of Differences. — Given the graph of a
function it is easy to find whether it is parabolic and if it is
to determine its precise form. Draw any number of ordinates
to the curve separated by any constant distance h. Measure
their heights and set the results down in order in a column
under the heading '* ?/ "• Subtract the height of each ordinate
from that of the next one to the right and set the differences
down in another column. This column is called the "first
difiierences of y" and is usually headed by the symbol A^^.
Next subtract each first difference from the one below it in
the column and set these differences down in a third column.
The third set of numbers is called the " second differences of
y " and the column is headed A^y. Now it is easy to show
that if the curve is parabolic — (that is, if the function is of
the form px^ + qx + r) — the second differences should be
constant to the degree of exactness which the means of
measurement permit. Consider any three consecutive mem-
bers of the series of equidistant ordinates and call the abscissa
of the first x, so that the abscissae of the other two will be
respectively x + h and x + 2h. Then the numbers in the
three columns must be of the following form : —
y ^iv ^^y
p^x + qx + r p(2xh + h?) + qh 2ph'^
p(x + hf + q(x + h) -^ r p{2xh + 3h-) + qh
p(x + 2/i)2 + q{x + 2h) + r
But since the second difference, 'Uph^, does not contain x it
would obviously be the same whichever three consecutive
ordinates had been taken. It follows that however many
equidistant ordinates were measured the second differences
all along the curve would have the constant value 2ph^. It
is easy to see also that the second differences will not be
constant unless the function is parabolic. For suppose it to
be linear, that is of the form px + q. Then the height of the
ordinate to the right of the one whose abscissa is called x is
p{x + h) + q and the first difference is ph. Thus, if the
function is linear, the first differences are already independent
of X and therefore will be constant. It follows that the second
differences must be zero. These results are, of course, perfectly
obvious from the consideration of a sloping straight line. On
the other hand, if the function is of the third degree, that is,
if it has the form
px^ + qx^ + rx + s,
270 ALGEBRA
it is evident that the first differences will all contain the term
x^. For example, the difference between the ordinates whose
abscissae are x and x -v h, will begin with the difference
between p(aj + }if and 'px^. It follows from what was said
above that we cannot now possibly reach constancy until we
come to the third differences. A table, made on the same
plan as the one above but involving four consecutive ordinates,
will show that, as a matter of fact, the third differences are
in this case constant and have the value 6jp/i^. By the use
of Stifel's Table it is possible to show in exactly the same
way that if the function is of the wth degree (where n is any
given positive integer) the tith differences of the ordinates will
be the first to be constant.
The method illustrated in this article is called the method of
differejices.
[Ex. XLII, C, can now be worked.]
G. Qimdratic Equations (ch. xxiv., § 4 ; Ex. XLIII).
^ 1. Graphic Determination of Values of x. — The graph of
a parabolic function shows that although to every value of x
there corresponds only one value of y yet if a given value of
y be specified there are two values of x which will yield it.
In the graph these values are shown by drawing a horizontal
line across the curve at the proper height or depth. Eor
example, to find the values of x for which the function
^x^ - 6a; + 10 has the value + 4*5 we must draw across the
parabola 1 y = ^x^ - Qx + 10 the horizontal line y =- +4-5.
It will be found to cross the graph at the points where
a; = + 1 and x = +11.
To find the values of x for which the value of the function
becomes - 6 the line ?/ = - 6 must be drawn cutting the
graph where ic = + 4 and £c = + 8. As further examples
note that y = + 10 is given by ic = and x = + 12 ; and
that y = + 16-0 is given by a; = - 1 and x = +13.
§ 2. Solution by Calculation. — Can these results be reached
by calculation ? To this question Ex. XLII, No. 17, suggests
an answer. The values of x for which a parabolic function
has zero value were there found by expressing the function as
a product of two linear functions. In the present case, for
example, we have : —
1 Since y = ^x^ - 6x + 10 = ^ {x - Q)^ - S this is the parabola
y = ^^ moved 6 units to the right and 8 downwards.
HYPERBOLIC AND PARABOLIC FUNCTIONS 271
y = ^x^ - 6x + 10
= i{x^ - Ux + 20)
= mx - 6)2 - 16}
= Ux - 2)(x- 10)
It is now obvious that ^ = both ii x = + 2 and if ic = 4- 10.
This information can, then, always be obtained by calculation
as well as by consulting a graph.
The same method is easily extended to the problems of § 1.
For example, to find the values of x for which y = + 4*5 we
need only lower the parabola through 4*5 units making the
corresponding relation become
y = -^x^ - 6x + 5-5
= i{{x - 6)2 - 25}
= i{x -l){x- 11)
The former line y = +4-5 now coincides with the o^-axis ;
and the factorized function shows that it cuts the parabola
where x = •{■ 1 and x = +11. Since the movement of the
curve was entirely vertical the line y = + 4*5 must have cut
the original parabola in points having the same abscissae.
In other words the function ^x^ -6a:+10=+4-5 when
a; = + 1 and again when x = +11. Substitution (as well
as inspection of the graph) shows this conclusion to be
correct.
To find what values of x make y = —6 the parabola must
be raised 6 units so as to bring the line y = - 6 into coin-
cidence with the ir-axis. In the new position we have : —
y = -^x- - 6ir + 16
= i{{x - 6)2 - 4}
= Ux -4:){x- 8)
In this case the a;-axis cuts the curve where a; = + 4 and
a; = + 8. These are, then, the values of x for which the
original function has the value - 6.
The other cases can be dealt with similarly. For y = +10
we must lower the parabola 10 units, tor y = + 16*5, 16*5
units. We have, respectively : —
y = |aj2 _ g^ and y = -^x^ - 6x - 6*5
= ix(x - 12) = U{x - 6)2 - 49}
= i{x + l)(a; - 13)
We conclude that, in the original function, y = + 10 is given
by a; = and by a; = + 12, and y = 16*5 is given by a;= - 1
and by a; = + 13.
Next let us inquire for what values of x the function
272 ALGEBRA
- ^x^ + 6x - 10 has the value + 6. Throwing the function
into the form - ^{x -6)^ + 8 we see that its graph is the
former one inverted, for it is the parabola y = - ^x^ moved
6 units to the right and 8 units upwards. Two modes of
solution are, therefore, possible, (a) The parabola may be
lowered 6 units so that the line y = + 6 coincides with the
X-axis and the function becomes : —
y = - ^{x -6)2+2
' = - i(a; - 4) (a? - 8)
which gives y = ior x = + i and a; = + 8. {b) The
parabola may be revolved about the a;-axis. The correspond-
ing function then becomes y = ^{x -6)2-8 and the line
y = + 6 becomes y = - 6. The problem is thus made
identical with one solved above.
Lastly, if we ask (for instance) what values of x give
^x^ - 6ic + 10 the value - 8 and - ^x^ -i- 6a: - 10 the value
+ 8, we see that the former graph must be raised and the
latter lowered 8 units. In either case the vertex will now
touch the £C-axis at the point x = + 6. It seems, then, that
there is one exception to the rule that a given value of a para-
bolic function always corresponds to two distinct values of x.
But the algebraic solutions are : —
y = ^x^ - 6x+ IS y = - ^x- + 6x - 18
= U^ - 6)2 = - ^{x - 6)2
= i{x - 6){x - 6) = - ^{x - 6){x - 6)
That is, the parabolic functions are factorizable, as before,
into two linear functions, but these two are identical. It is
best, therefore, to say that in such cases two identical values
of X yield the given value of y.
§ 3. Quadratic Equations. — Consider the following batch
of questions. For what values of x does : —
(i) the function 2ic2 _ 12a; + 9 have the value + 23 ;
(ii) the function 2x^ - 12x - 5 have the value + 9 ;
(iii) the function 2a;2 - 12x - 17 have the value - 3 ;
(iv) the function - 2a:2 + i2ic -i- 27 have the value -f- 13 ;
(v) the function - 2a;2 + 12ic - 3 have the value - 17?
By § 2 the first three questions all reduce to the question :
" Where does the parabola y = 2a;2 - 1 2a; - 14 cross the a;-axis ? "
and the last two to the question : * ' Where does the parabola
y = - 2a;2 + 12a3 + 14 cross the a;-axis? " Moreover, since
the latter parabola can be turned into the former by revolving
it about the a;-axis, the answer to the last two questions must
HYPERBOLIC AND PARABOLIC FUNCTIONS 273
be the same as the answer to the first three. The answer to
all five is, in fact, given by the argument : —
2a;2 _ 12a; - 14 =
aj2 - 6a; - 7 =
{x - l)(x + 1) =
Therefore either a?- 7 = or a;+ 1 = 0;
that is, either x = + 1 or x = -1.
Such an argument may, then, be the means of solving any
number of problems of the kind stated at the beginning of
this article. For this reason it is important to become skilful
in carrying out the process indicated. Any statement which,
like the above, is of the form jpx'^ + qx + r = O'm called a
quadratic equation and the process just illustrated is called
" solving the equation ". The numbers which (like the - 1
and + 7 above) satisfy the equation are its roots. That is,
the roots of a quadratic equation jpx'^ + qx + r = Q are the
values of x for which the parabolic function px'^ -v qx + r has
zero value. To solve a quadratic equation we must express
the parabolic function as a product of two linear functions.
From these we form two simple equations. The roots of the
two simple equations are the two roots of the quadratic
equation. If the linear factors are identical the two roots
are the same.
In the examples given above the parabolic function was
generally factorized after being expressed as the difference
between two squares. If the factors can be seen by in-
spection this step may be omitted. For instance, it is obvi-
ous that a;2 - 7a; + 12 = can be expressed as
(a; - 3) (a; - 4) =
and that the roots are + 3 and + 4. The longer method is,
however, the best for difficult cases. It is also the only way
to make sure whether the equation has roots or not. Take,
for example, the equation x^ - 7a; + 20 = 0. Here we have
a;2 _ 7a; + 20 = (a; - J) 2 + 7|
but as the right-hand expression is the sum and not the
difference of two squares the function cannot be factorized.
The equation has, then, no roots. Consider, on the other
hand, a;^ - 7a; + 9 = 0. Here we have
a;2 - 7a; + 9 =
(a; - ^) 2 - i_3 =
T. 18
274 ALGEBRA
(•-D-(47=«
Therefore either
^ 7+71 3 _ 7 - Vl3 .
X ^ — = or £c ^ — =
That is, either
a: = i(7 + 713) or a; = ^(7 - JlS)
If these cases had not been dealt with in the way shown we
could not have been certain that there are no roots in the
first case and that there are roots, but complicated ones, in
the second case.
[Exs. XLIII and XLIV may now be worked.]
D. Inverse Functions (ch. xxiv., ^ 6 ; Ex. XLV).
§ 1. Changing the Subject of a Parabolic Formula. — A ball
is rolled up a smooth sloping board 75 inches long. Its
velocity is at first 20 ins. /sec. and falls off 4 ins./sec. every
second. Find a formula for its distance from the top end of
the board. The formula for the velocity is : —
V = 20 - 4:t
therefore, as in Ex. XXX, Nos. 22 et seq., we have, s being
the distance from the starting point : —
s = 20^ - 4 X 1^2
= 20t - 2^2
If we put S for 'the distance from the farther end of the
board we obtain: —
S = + 75 - 20^ + 2^2
= + 2^2 _ 20t + 75
= + 2 (^ - 5)2 + 25
The distance from the farther end of the board is, then, a
parabolic function of the time and has a lower turning value
of + 25. That is, the ball reaches a point 25 inches from
the top of the board and then begins to descend again.
Now suppose we want to change the subject of this para-
bolic formula to t — that is, to obtain a formula giving the
times at which the ball is at a given distance from the top of
the board. (There will, in general, be two times for each
HYPERBOLIC AND PARABOLIC FUNCTIONS 275
distance — one on the way up and one on the way down.)
An easy way to proceed is as follows. Take the graph cor-
responding to the formula S = + 2t^ and turn it into the
position (fig. 75) in which t is measured along the vertical
and S along the horizontal axis. It is now obvious that
if we change the subject of the formula to t we shall obtain
not one but two formulae. The first is ^ = + s/^J'2 corre-
sponding to the upper branch of the graph (OP, fig. 75) ;
while the second is t = - ^S/2 corresponding to the lower
branch (OP'). But to make the new formulae correspond
to the original formula S = 2(t - 5)^ + 25, i.e. S - 25 = 2
(t - 5)2, we must substitute S - 25 for S and ^ - 5 for t. In
Y ^^^
[_ _A
0" X
Fig. 75.
this way we arrive at the two formulae : —
i - 5 = + V(S - 25)/2
or t= +5+ V (S - 25)/2
and t= + 5 - J{S - 25)/2
The two formulae may be summarized in the form : —
i = + 5 ± V(S - 25)/2
The corresponding graph will be obtained by shifting fig. 75
to the right 25 units and 5 units upwards.
This combined formula shows once more that the ball
cannot come within 25 inches of the top of the board. For
if S were negative, or if, being positive, it were less than 25,
the number under the radical sign would be negative. But
18*
276 ALGEBRA
since a negative number has no square root there could be no
corresponding value of t. That is, there is no time when
the ball is within 25 inches of the top.
In general, if the original formula is : —
S = a (i - 6)2 + c
the formulae with t as subject are : —
t=: b± V(S - c)la
In the above problem a is positive. To obtain a problem
in which it is negative let us suppose that the distance of the
ball is measured from a point 10 inches up the board from
the bottom. Then we have : —
S = s - 10
= - 2*2 + 20* - 10
= - 2 (* - 5)2 + 40
This time there is an upper turning value of + 40 — a result
which agrees with the former lower turning value of + 25
measured to the top of the board. Beginning, as before, with
the graph in the standard position, we have : —
S = - 2*2
and t = ± s/^l{- 2). This result is not impossible as it
seems at first sight. It means simply that all the values of
S are negative. Moving the graph so that it represents the
relation S = - 2*2 + 20* + 10 we have the twin formulae : —
*= +5±V(S - 40)/(-2]
In order that the number under the radical sign may be posi-
tive the numerator must be negative. For this result S must
either be negative or, if positive, numerically not greater than
40. These conditions obviously mean, once more, that S has
an upper turning value of + 40.
In applying this method practically we proceed as follows : —
S = a (* - 6)2 + c
(t - hf = (S - c)la
t - b= ± J{ S - c)la
* = 6 ± V(S - c)la
If the original formula is given in the form
S = j9*2 + g* + r
we may, as an alternative method, write it
j9*2 + 5* + (r - S) =
and by the formula of Ex. X LIV, No. 1, ded uce at once that
t= {- q± Jq^ - ^p{r - S)}/2^
§ 2. Inverse Functions. — In the foregoing problem S was
HYPERBOLIC AND PARABOLIC FUNCTIONS 277
a parabolic function of t. In other problems about moving
bodies it might well be a linear function of the form S = a +pt
or a hyperbolic function of the form S = kl{t + a) + b. It
is instructive to set side by side with these formulae the corre-
sponding formulae in which t is the subject : —
^ = a+ bt ^ = (S - a)/b
S = k/{t + a)+ b t = &/(S - b) - a
S = a{t+ by + c t = - b± J{S - c)/a
Each of the right-hand formulae is said to be the inverse of
the corresponding direct formula in the left-hand column.
The relations between them are best brought out by replacing
the concrete variables by the abstract variables x B,ndL y. In
the direct formulae t is the independent variable (to be repre-
sented by x) ; in the inverse formulae the independent vari-
able is S. We have therefore : —
Direct Functions. Inverse Functions,
y = a + bx y = (x - a)lb
y = k/{x + a) + b y = hl(x - b) - a
y = a{x + by + c y = - b ± J{x - c)/a
The graphs of the inverse functions are most easily ob-
tained by revolving the graph of the direct function through
180° about the a;-axis and then turning it in its own plane,
anticlockwise, through 90°. What were before values of x
and y are thus converted into values of y and x. With the
direct and inverse graphs before us the following points at
once become clear : {a) Each of the direct functions is a single-
valued function of x ; that is, to every value of x there corre-
sponds one and only one value of the function. (6) The in-
verse linear and the inverse hyperbolic functions are also
single-valued, but the inverse parabolic function is two-valued :
to each value of x there correspond two values of the function.
In the direct and inverse linear functions the fields of x and
y are both unlimited ; that is, both variables are capable of
assuming all values whatsoever. The same is true of the
direct and inverse hyperbolic functions with the exception
that when x = - a in the former and x = + b m. the latter
there is not really any corresponding value of y (A, § 1).
But, in the direct parabolic function, while the field of x
is unlimited that of y is unlimited one way only. If a
is positive (that is, if the graph is "head down ") y'^ value
may be as high as you please but it can never be lower
than c ; if a is negative it may be as low as you please but
278 ALGEBRA
can never be higher than c. Finally, in the inverse para-
bolic function the opposite rule obtains. If a is positive
X - c must be positive so that x can never be lower than c ;
while if a is negative x can never be higher than c. On
the other hand, the value of the radical may be (numeric-
ally) anything from zero upwards. Hence y may have any
value as distant as we please above and below - b.
Consideration of the foregoing examples yields a simple rule
for obtaining the inverse of a given function of x : Take the
relation which expresses the original function ; change its
subject to a: ; in the result replace x by y and y hj x. The
relation so obtained is the expression of the inverse function.
[Exs. XLV and XLVI may now be worked.]
CHAPTER XXVII.
WALLIS'S LAW.
A. Area Functions (ch. xxiv., § 7 ; Ex. XLVII).
§ 1. Area of the Parabola by Wallis's Method. — The method
of ch. XIX., B, can be used to calculate an area such as 0PM in
fig. 76, the curve being half of the parabola y = Jcx'^. Let
m + 1 rectangles of equal breadth be set side by side, their
areas being 0"^, 1\ 2'^, . . . m?. Behind them suppose m + 1
other rectangles (shown dotted in fig. 76), each equal to the
largest of the former series, so that together they constitute a
rectangle AM whose area is (m -t- l)niK Then the increasing
V\
^
Fig. 76.
m
/
rectangles have a total area which is a certain fraction of AM,
namely :—
02 + 12 + 22 + 32 +
+ W2
(W + 1)^.2
The value of this fraction depends upon the value of w., lim
assumes in succession the values 1, 2, 3, . . . we have : —
0+1 ^ 1 0+1 + 4 ^ _5^
2 X 1 ~ 2' 3x4 ~ 12'
0+1 + 4 + 9 14 7
= 36 = 18' ^*''-
4x9
279
280 ALGEBRA
the subsequent values being, in order,
31113^1719^2325
8' 30' 36' 14' 48' 54' 20' 66' 72'
A glance at these results suggests that they may be
written : —
? A 7 ^ 1^ 13 15 17 19 21 23 25
6' 12' 18' 24' 30' 36' 42' 48' 54' 60' 66' 72' ®**^-
The law here followed is obviously given by the formula : —
-i:. .• 2w + 1
Fraction = — ji
6m
1 J^
~ 3 "'"em
It is difficult to suppose that this law, after holding good in
the first twelve cases, should not continue to hold good. As-
suming its truth for any number of rectangles we conclude
that by making them numerous enough their combined area
can be made to differ as little as we please from one-third of
the area of AM. But as the rectangles are made thinner the
area they cover will eventually become indistinguishable from
the area under the curve OP. We conclude, therefore, that
this area is — with an exactness more minute than can be
measured by any number of decimal places — one-third of the
rectangle AM. Since OM = x and PM = kx- this conclusion
may be expressed by the formula A = ^kx^.
It follows also that the area AOP = AM - 0PM = §AM,
and that the whole area of the parabola up to PP" is
f FM = §0A X PP'.
This result can be expressed in another way by turning the
original parabola into the form of fig. 75. The curve now
corresponds to the relation y = k' Jx (where k' = 1/ \/k)y and
we have : —
Area OPA = f OA x PA
= § X a? X k' Jx
= ^k'x sjx
§ 2. Wallis's Law. — The results of § 1 and of ch. xix., B,
can now be summarized as follows. Suppose an ordinate
to start from the origin and to move to the right. If it has a
constant height y = k it will, in moving through a distance
Xj trace out an area A = kx. If its height is at first zero but
increases in accordance with one of the laws y = kx, y = kx\
WALLIS'S LAW 281
y = h \/x, the area traced out will be given by the appropriate
one of the corresponding laws A = ^kx^, A = ^kx^, A = ^kx Jx.
Calling the function which gives the height of the ordinate
the ordinate-function, and the function which gives the area
traced out the area-function, the first three of these four results
can be reduced to a very simple rule. For to the ordinate-
functions : —
kx^ kx^ kx"^
correspond the area-functions : —
ikx^ Px2 p^s
That is, if the ordinate-function is kx"" ~ ^ the area-function is
-kx"". Now John Wallis who discovered this rule (Arith-
metica Infinitorum, 1655) was so struck with its simpli-
city that he sought to bring the fourth result under it also.
His argument was practically as follows. The indices of the
square roots of x^, x^, x^, x^, etc., are all formed by taking
one half of the indices 2, 4, 6, 8, etc. It is natural, therefore,
t'o inquire whether the square root of x^ cannot, in accordance
with the same rule, be expressed as x^. Consideration shows
that to do so would not contradict the law of indices. For,
according to that law, if the symbolism x^ can be permitted
at all, it must have such a meaning that x^ x x^ — x^^'^, i.e. = x.
But this is precisely what it was intended to mean. We may,
therefore, write x^ for Jx and may, further, write
xjx = X X x^ = x^.
All four of our results now fall under the one rule, for when
the ordinate-functions are : —
kx^ kx^ kx^ kx^
the corresponding area-functions are : —
]-kx^ ^cx^ ^kx^ \kx^
The rule that to an ordinate-function kx'^'^ there corre-
sponds an area-function ~kx^ may be called Wallis's Law in
memory of its discoverer. It has been shown to be true
when w is 0, ^, 1, 2. Later inquiries will determine whether
it holds good also for other values of n.
§ 3. Proof of the Theorem of § 1. — In § 1 it was assumed
that the fraction (^ m?)l{m + Ijm^ = 1/3 -f l/6m in all cases
because there seemed no reason why a law exemplified by
twelve successive numerators and denominators should not
282 ALGEBRA
continue to hold good. This kind of reasoning is called in-
duction and is obviously not completely satisfactory. A more
satisfactory treatment would be one which proves that the
law if true in one case will be true for all cases.
Let us suppose, then, that the rule holds when m has a certain
value p — that is, when there are _p + 1 rectangles of which
the largest has an area _p'^. Then we can show that it will
hold good also when m = p + 1 — that is, when there are
p + 2 rectangles with {p + 1)^ as the area of the largest.
By hypothesis, when there are p -h 1 rectangles the fraction
02 + 12 4. 22 + ... + j)2 ^ 2j9 + 1
{p + 1)^2 - 6^
Hence 0^ + l^ + 2^ + ... + p' = ^^ ^ x (^ + 1)/
= MP + 1)(2^ + 1)
Now let the number of rectangles be increased to^ + 1.
Then the numerator of the fraction will be
02+12+22+. . . +^2+(^+l)2_J^(^+l)(2^+l) + (^+l)2
= i(i'+l)(i^+2)(2^+3)
But the denominator of the new fraction should now be
{p + 2) (^ + 1)2. Dividing each side of the equality by this
product we have
02 + 12 ^ 22 + ... + {p+ ly _ 2ff + 3
{p + 2){p + 1)2 - 6(p + 1)
_ 1 1*
~ 3"^ 6(^ + 1)
We conclude that if the rule for the value of the fraction
holds good when m = ^ it holds good when m = p + 1.
But the rule has been proved to hold good when m is any
one of the numbers 1, 2, 3 . . . 12. Therefore it holds good
universally.
This kind of argument is called a jyroof by recurrence.^
[Ex. XL VII may now be worked.]
B. Differential Formulce (ch. xxiv., § 7 ; Exs. XL VIII,
XLIX).
§ 1. Differential Formulce. — In Ex. XLII, C, we saw
how to determine the nature of a given graph by the " method
^ Also mathematical induction — a bad term since it falsely sug-
gests a similarity with the induction of § 1.
WALLIS'S LAW 283
of differences ". A series of equidistant ordinates is drawn,
starting from any point on the graph. The heights of the
ordinates are naeasured and a table is made of their first,
second, third, . . . differences. We found that if the graph
is a straight line the first differences will be constant, if a
parabola, the second differences, while if it represents a
function of x of the third degree the third differences are con-
stant, and so on. We showed also that the converses of these
statements are true, so that by examining the differences of
the ordinates we can determine the " degree" of the function
which corresponds to the graph.
The formulae used in Ex. XLII, C, hold good if /t, the
distance between the ordinates, has any constant value. We
are now to inquire what these formulae become when h is
taken so small that only those terms need be retained in
which it appears in its lowest power.
It will be convenient to begin with the familiar case of the
parabola y = px^ + qx + r. With regard to the curves which
correspond to formulae of this form it was found (ch. xxvi., B,
§ 4) that
the 1st diff. oi y = p{2xh + h^) + qh
= (2px + q)h + phK . (i)
the 2nd diff. oi y = 2ph^ .... (ii)
Now in (i) the second term has to the first the ratio
ph?l{2px + q)h = phl{2px + q) . . (iii)
No matter, therefore, how large \p\'^ or how small \x\ and
(^1 may be, by taking h small enough this ratio may be
made smaller than any given fraction. That is, the error
produced by omitting the term ph^ may be made as unimport-
ant as we please. If we wish our formula to give results true
within c per cent, we have only to reduce h below the value
which makes the ratio in (iii) less than c/100. Provided that
the value of h is less than this it is called " small," no matter
what its absolute value may be. This rule is, in fact, to be
regarded as the definition of the word "small " when it is ap-
plied to numbers in mathematics.
When the distance between the ordinates is meant to be
" small " in this technical sense, it is convenient to denote the
fact by replacing fe by a special symbol. For this purpose
the symbol Sx is commonly employed. It must be carefully
noted that the Greek letter delta is here the " grammalogue "
1 The symbolism \p] means " the numerical value of p ".
284 ALGEBRA
not of the description of a number (as the x is) but of some
such phrase as " a small increment in the value of". For
brevity the whole symbolism 8x may be read " the differential
of Xy" the word "differential " being itself defined as meaning
" a small increment in the value of ". Similarly the difference
between two ordinates whose distance apart is Sx may be
symbolized by By — to be read "the differential of y'\ The
second difference between the ordinates may conveniently be
represented by S^y — to be read "the second differential of y,''
and so on. Finally we may note that although h^ (when h is
" small ") should properly be written {Bx)- it is usual to write
it in the more compact form 8ic'^. No doubt the correct inter-
pretation of 8x'^ should be " the differential of the square of
X ". If it should prove necessary to symbolize this phrase
the form o{x'^) may be employed.
We are now in a position to express relations (i) and (ii)
in the case when the distance between the ordinates is
" small ". The new formulae will be
y = px"^ + qx + r . . . A
Sy = {2px + q).8x . . . B
S'^y = 2p.8x^ . . . . C
Formulae like B and C will be called differential formulce.
B (whose subject is the first difference of y) will be called a
differential formula of the first order, (whose subject is the
second difference of y) one of the second order, and so on.
It is important to remember that in these formulae the
symbol " = " asserts not absolute, but only approximate
equality.^ One of the advantages of using the special symbol
8 in such a formula is that it may be taken as qualifying
the meaning of the " = ," and so making it unnecessary to
indicate in any other way that the statement may be true
only to a certain degree of accuracy (Ex. IX, note to No.
8). Formula A, which contains no differentials and asserts
exact equality, will be called (in relation to B and C) the
primitive formula.
§ 2. The Return from the Differential to the Primitive
Formula. — The investigation of § 1 illustrates the way in
which differential formulae may be derived from their primi-
tive. The results may now be used to investigate the con-
^ I.e. in some cases the equality may actually be exact but the
formula only guarantees a certain percentage of accuracy.
WALLIS'S LAW 285
verse problem — that of retracing the steps from a differential
formula towards its primitive. For this purpose the three
formulae may more conveniently be set down in the forms
y = px^ -\- qx + r . . A'
Sy/hx = 2px + q . . . B'
S'-ylSx'' = 2p . . . . C
Suppose, then, that we are given the relation C and are
asked to find B', how shall we proceed? The right-hand
side of B' is the sum of two terms, 2px and q. Of these the
former is easily obtained, for it is merely the product of (the
given) 2p by x. But whence are we to obtain the value
of q ? The answer is that from the information supplied
(namely that S'^yjSx'^ = 2p) we simply cannot tell what it is.
All we can do is to set down the symbol q to indicate that
the right-hand side of B' may contain some added constant
though we do not know what constant.
The source of this ambiguity may be made plain by
numerical instances. Take the four following primitive
formulae : —
y = Sx^ y = Sx^ + S ). .„
y = 3x^ + 5x+ 1 y = dx^ - 7x + 2 }
Sy/8x = 6x 8y/Sx = 6x I "R"
^/8x = 6x+ 6 ByjBx = 6a; - 7 I
B^y/8x^ = 6 h^y/Bx^ = 6
%/8ir2 = 6 B^y/Bx^ = 6
Now it is clear that, although the four parabolic functions in
A" agree only in having the same value iorp (namely 3) and
differ in respect of the values of q and r, these differences
disappear by the time we reach G". The formula B^y/Bx'^ = 6
is determined simply and solely by the fact that ^ = 3, and,
therefore, can give us information only about that constant.
Any other parabolic function would yield the same differential
formula of the second order if it also began with the term
3x^. In passing back from B'^y/Bx^ = 6 to the formula for
ByjBx, we can, therefore, only write By/Bx = 6x + q. This form
implies that there may be a constant but that there is no
evidence as to whether it is -i- 5 or - 7 or some other of an
endless range of possibilities.
Similarly if we are given B' (that is, Up and q have assigned
values) the passage back to A' is only partly determinate.
The term px"^ can be derived from 2px by multiplying by
ic/2, and the term qx from q by multiplying by x, but we have
C"
286 ALGEBRA
no means of deciding the value of the constant r. We can
only append the term + r to the expression px'^ + qx to in-
dicate that there may be such a constant although we do not
know what it is. Thus from the four differential formulae
B" we can derive only the information
2/=3a;2 + r, y = ^x^ + r, 2/ = 3a;2 + 5a? + r, y='^x'^ -Ix+r.
Finally, if we are given B^y/Sx'^ = 6 and are asked to find the
primitive, the introduction of an undeterminable constant will
occur at each of the two stages in the backward process.
From the given formula we derive in the first place the
differential formula of the first order
Sy/8x = 6x + q
and from this the primitive
y = Sx^ + qx + r.
Of the numbers q and r we can only say that they may be
zero or may have any values whatever.
§ 3. Geometrical Meaning of the Constants. — It is helpful
to consider from the graphic point of view this appearance of
an undeterminable constant at each step of the regress from
a differential formula to its primitive. Take the primitive
formula y = 2x + 7 which represents a certain straight line.
Substituting x + hior x and subtracting we have
the 1st difference oi y = 2h
and, therefore,^ By = 2 ,Bx or 8y/8x = 2.
In this case the differential formula is determined by the
coefficient of x in the primitive and by that alone. Any
primitive of the form y = 2x + q would yield the same differ-
ential formula 8y/Bx = 2. Conversely, from By/Sx = 2 we can
deduce the primitive only in the ambiguous form y = 2x + q
where q may have any value including zero. Expressed
graphically this means that from By/Bx = 2 we can deduce
only that the primitive represents a straight line making with
the ic-axis the angle whose tangent is 2 but cannot deduce the
position of the line. In other words, we can deduce that in
the " standard position," in which it passes through the origin,
the straight line would have y = 2x a,s its formula, but we
cannot deduce how far (if at all) the line has been raised or
lowered. The differential formula may, therefore, be regarded
^ Since there are no higher powers of h to neglect the equality
is, on this occasion, absolute (see p. 284, footnote).
WALLIS'S LAW 287
as a general description which includes in its reference y = ^^x
and all possible lines parallel to it.
Similarly from the differential formula of the second order
h^y/Sx^ = 6 we can deduce that the primitive describes a
parabola which in its standard position corresponds to
y = 3x^, but we cannot tell how much the parabola has been
(i) raised or lowered, (ii) moved to the right or left. The
differential formula is, therefore, a general description of all
parabolas which can be derived from y = Sx^ by these move-
ments. On the other hand from the differential formula of
the first order, By/8x = Gx + 5, we can deduce not only that
the parabola in its standard position would correspond to
y = dx^, but also that it has been moved ^ unit to the left.
What remains uncertain is whether or how much it has been
moved up or down. That is, SyjBx = 6x + 6 may be regarded
as a general description of all parabolas which can be derived
from y = S{x + |-)2 by vertical displacement.
§ 4. Wallis's Law applies to Differential FormulcB. — A
differential formula of given order can always be derived from
its primitive by the method of § 1. Nevertheless, it is profit-
able to inquire whether there are rules for writing down the
results of the process without actually carrying it out. Such
rules would be especially useful when we have to return from
a differential formula towards the primitive, for the solution
of this problem depends upon knowledge of the changes in the
terms which accompany the transition from the primitive to
the differential formula.
The results of § 1 suggest definite rules of this kind. In
the first place it will be observed that qx in A' becomes q in
B' and that '^px in B' becomes in like manner 2^ in C. These
results are similar because, although they appear at different
stages of the transition from the primitive, they are produced
by the same process — namely, that represented by the sym-
bolism h{x + h) - kx. We have, therefore," the general rule
(i) that in passing from a primitive to a differential formula
of the first order or from a differential formula of the first to
one of the second order, and so on, any term kx becomes
simply k. Conversely in taking any step backwards towards
the primitive a constant k must be replaced by kx.
Again the term px^ in A' becomes 2px in B' in virtue of the
process p{x + hy - px^ followed by suppression of the term
ph^. It is clear that a term of the same form will undergo
288 ALGEBRA
the same transformation at any stage of the downward move-
ment from the primitive formula. We have then the general
rule (ii) that kx^ becomes 2hx in a descending process, and that
conversely hx becomes \hx'^ in an ascending process.
Lastly, the disappearance of the constant terms r in A' and
g in B' in the descending process obviously exemplifies another
general rule. The converse of this rule is the addition of an
undetermined constant at each stage of the ascending process.
It will be noticed that rules (i) and (ii) are of the same form
as Wallis's Law. That is, in the descending process a term
of the form hx"" becomes nkx^~'^ ; in the ascending process
A;ic"~i becomes -kx"". If this form is followed in other cases
n
the rules for dealing with differential formulae will be extremely
simple. Let us then examine the case when w = 3. We
have
y — kx^
1st difference of ^ = k{x + hy - kx^
= k{3x% + Sxh^ + h^)
.'. Sy = Skx^ . Sx or Sy/Sx = Skx^.
If we were given that Sy/Sx = kx^ we should conclude by
exactly the same argument that S^y = Skx'^ . 8x^, or that
S^y/8x^ = Skx^, and similarly for any other step in the descend-
ing process. Thus this case is also covered by the form of
Wallis's Law.
Suppose now that we are given a differential formula
of the third order, of the form S^yjSx^ = p ; for example,
Py/Bx^ = 5. Then we have the following conclusions : —
S^yjSx^ = 5
S'^y/Sx^ = 5x + p
Sy/Sx = |ic2 + px + q
y = §x^ + ^px^ + qx + r
p, q, and r being constants which may have any values, in-
cluding zero.^ .
' The fact that three independent constants are introduced in the
ascending process indicates that the primitive cannot in this case be
simply a description of the curve y = %x^ after a horizontal and a
vertical movement. Ex. XLIV, Nos. 11-15, show why. By a suitable
horizontal movement the graph of a cubic function of x which
contains x^ can always be turned into the graph of one whidi does
not contain x^. For example, y = 2x^ + 6x^ + x - 2 becomes
y = 2x^ - 5x + 1 when moved one unit to the right. The constant term
WALLIS'S LAW 289
The question of the range of values of n over which Wallis's
Law holds good must be left (as in ch. xxvii., A) for future dis-
cussion. There is, however, one more case of such importance
in physical problems that it should be considered at once,
namely, when n = - 1. If the law is followed here kx~'^
ought to become - kx~^ in the descending process and kx~^
to become - kx~'^ in the ascending process. We have
y = k/x
1st diff. of 2/ = kl{x + h) - k/x
~ ' x(x + h)
xA 1 + h/xj
Now, however small the numerical value of x, the value of h
may be chosen so much smaller that the fraction h/x will
become as small as we please. If x is positive (h/x) /{I + h/x)
is less than h/x and, therefore, by what has just been said,
may also be made as small as we please. If x is negative
{h/x){l + h/x) is greater numerically than h/x, but, again, by
taking h small enough can be made as little so as we please.
Thus in either case when h is small enough we may ignore
the term {h/x)/{l + h/x) and write
k
hy = ^.hx ox 8y/Sx = - kx"^.
Conversely in the ascending process a term kx~^ must be re-
placed by - kx-'^.
We have now shown, therefore, that Wallis's Law holds
good in these descending and ascending transformations at
least in the cases where n = 1, 2, S and - 1.
§ 5. The Meaning of the Inverse Process. — We have post-
poned to the end what is in practice by far the most important
of the questions we have to face : namely, the exact signi-
can next be removed by moving the curve (in its new position) verti-
cally. Thus when y = 2x^ + Gx^ + x - 2 is moved one unit to the
right and one unit downwards it becomes y = 2x^ - 5x. But it is
not possible in this way to remove the term 5x without reintroduc-
ing x2 at the same time. That is to say, y = 2x^ and y = 2x^ - 5x
diflfer not only in position but also in shape. Thus from 8^yl8x^ = p
we cannot deduce even the shape of the curve. We can only deduce
that it is one of the curves whose shapes are given by the formula
y = ^x^/6 + qx where q may have any value.
T. 19
290 ALGEBRA
ficance of the inverse process by which a primitive is deduced
from a given differential formula. For example, supposing it
to be given that 8y = 2x . Bx what is the precise force of the
conclusion that y = x^ + p ?
To answer this question we must remember that we are
seeking what may be regarded as the unknown ordinate-
function of a certain curve. The data suppose that equi-
distant ordinates to this curve have been drawn and that the
first differences between these ordinates are described, to a
certain degree of accuracy, by the relation By = 2x . 8x. This
degree of accuracy is fixed by the statement that even if the
value of 8y were really 2x.Sx -h a{Sxy + b{Sxy + . . . (where
a, bj etc., either are constant numbers or involve powers of x)
the sum of the terms which follow the 2x . Sx would be smaller
than the smallest fraction of 2x . Sx of which cognisance is to
be taken. Let us suppose that H„, the height of one of
the ordinates, is known, and let its abscissa be x„. Then if
2x . Sx were the exact measure of the first difference the next
ordinate, H^ would be H^ + 2x„ . 8x. Similarly Hg would be
Ho + 2x . Bx + 2xi . Bx ; and, in general,
'R,=B.,+ 2x„.Bx + 2x^.Bx+ . . . + 2a;,. 8a; . D
Bx being the constant distance between the ordinates and
a?!, X2 . . . being the abscissae of H^,, Hg, etc. But the first
differences are not exactly 2Xo . Bx, 2x^ . Bx, Sa^g . Bx, etc., but
may depart from these numbers by any amount not greater
than (say) c per cent. It follows, therefore, that the values of
Hj, Hg, H3, etc., calculated by formula D are not necessarily
the exact heights of the ordinates but may differ from them
by not more than c per cent. Since, however, by reducing
Bx, c may be made as small as we please, H^, Hg, H3 . . .
can be calculated to any degree of accuracy.
There is to this procedure the obvious objection that if Bx
is small enough to secure a reasonable approximation to the
value of H^ the number of additions must be unmanageably
great. Consequently we are driven to inquire whether we
cannot deduce from By = 2x.Bxa> formula which will enable us
to calculate directly, if only approximately, any ordinate we
please. At this point we recollect that if the formula of a
curve were y = x^ + p the first difference would be given by
the formula
1st diff. of 2/ = 2xh + h^
or By = 2x.Bx + {Bxf exactly.
But by hypothesis Bx is so small that it is indifferent whether
WALLIS'S LAW 291
we equate By to 2x . Bx or to 2x.8x + {Sxy, for the added term
is less than c per cent of the original 2x . 8x. Consequently it
is legitimate to suppose that the first difference (which, by
hypothesis, may be only approximately 2x . Bx) is actually
2x.Sx + (Sxy. Upon this supposition it would follow at once
that the ordinate- formula or primitive is y = x^ + p.
To determine the amount of error to which this conclusion
is liable we consider the most unfavourable case possible.
By hypothesis, the true value of the difference may be c per
cent less than 2x . 8x and the value of 2x .Sx + {Bxy c per cent
more than 2x . Bx. Thus in assuming the first difference to be
2x .Bx + {Bxy we may be over-estimating it to the extent of
about 2c per cent of its true value. Consequently all the
ordinates calculated by the formula y = x^ + p may be about
2c per cent too high. But since the value of c depends merely
upon the size of Bx we can make it as small as we please by
supposing Bx to be small enough. That is to say, the formula
y = x^ + p gives, if not an absolutely exact measure of the
heights of the ordinates, at least as close an approximation
as anyone can require.
As a second example suppose it to be given that By/Bx
= - Ijx'^ or By — - Bx/x^. Here we must remember that
when y = Ijx the first difference of y is given by the formula
lstdifif.of2,= -4(l-^^)
^ x^\ 1 + hjxj
Bx {Bxf
"^^^^ -^^ + 0.3(1 V^g,/,) exactly. E
As before, the second term is, by hypothesis, less than the
smallest number of which cognisance is to be taken. To
assume, therefore, that By is given by the relation E does not
contradict the datum that it is approximately - Bxjx'^. On
the other hand, that assumption enables us to conclude that
the ordinates are given to any degree of accuracy anyone
chooses to name by the formula y = Ijx.
It will be seen that similar reasoning holds good of the
steps from the second differential formula to the first, etc.
The conclusion of the whole argument is, therefore, that when
a differential formula of any order holds good in the sense
described in § 1 the primitive deduced from it by such rules
as those of §§ 3, 4 also holds good to an unlimited degree of
accuracy.
[Exs. XLVIII, XLIX, may now be worked.]
19*
CHAPTBK XXVIII.
THE CALCULATION OF -n AND THE SINE TABLE.'
A. Calculation of -k (ch. xxiv., § 9 ; Ex. L).
§ 1. The Prijtciple to be used. — Hitherto nothing has been
said about the mode of calculating tt. Its value can, of course,
be determined only roughly by the measurement of actual
circles. As early as 250 b.c. the great Archimedes reached
by calculation the number 3y which we so frequently use,
and his method was essentially the one we shall follow,
although we will study it in a form due to Ludolph van
Ceulen (1586).
Consider two regular polygons with the same number of
sides respectively inscribed within and circumscribed without
any circle. It is assumed as obvious (i) that the ratio of the
perimeter of the inscribed polygon to the diameter of the circle
is less, and that of the perimeter of the circumscribed polygon
greater, than the ratio of the circumference of the circle to the
diameter ; (ii) that the difference between the first two of
these ratios can always be made smaller by increasing the
number of sides, but that neither of them can ever become
exactly equal to the third. ^ It follows that by taking an in-
scribed and a circumscribed polygon of a sufficiently large
number of sides and calculating the ratios of their perimeters
to the diameter we can find a pair of numbers, as close to one
another as we please, between which tt must lie (cf. p. 94).
Taking polygons containing 96 sides Archimedes showed in
this way that tt lies between Y- and ^23
^ No elementary student is likely to question these assumptions.
Strictly speaking, however, they are really a definition of what we
mean by " the length of circumference of the circle ". See Young,
Fundamental Concepts of Algebra and Geometry, p. 205.
292
THE CALCULATION OF it AND THE SINE TABLE 293
§ 2. Ludolph van Ceulen's Method.^ — Van Ceulen had
two advantages over Archimedes : a system of symbols by
which he could easily formulate the ratio corresponding to
any number of sides, and the decimal fractional notation ^ in
which the numerical value of a ratio could be expressed to
any required degree of exactness. Armed with these instru-
ments he sought in the following way to obtain a double
series of numbers, like those of p. 94, between the successive
pairs of which the value of tt may be imprisoned within
limits as narrow as the calculator pleases.
Let AB {Exercises, fig. 47) be any chord in a circle of unit
radius, and let AB' be a chord bisecting the arc AB. Let
C = the length of the chord AB, and C^ = the length of the
chord AB' ; also let OH = b, B'H = a, H being the foot of
the perpendicular from the centre upon AB. Then since
B'AC is a right angle, we have
Cj^ = a X B'C
= 2a
= 2(1-6)
= 2{1 - Jl- (iC)'^}
= 2 - V(4 - C2)
Hence C, = J{2 - ^(4 - C^)}
Next, bisect the arc AB' at B". Then putting C2 for the
length of the chord AB", we have by the foregoing formula,
C, = V{2 - V(4 - 0^2)}
= J:2- J: 2+ J(^- C^)
the notation " J :" being used ^ to signify that the radical
sign covers everything that follows it.
It is now possible to formulate the length of the chord
obtained after any number of bisections of AB. It will be,
in fact,
C, = J:2 - 7:2+ ... + J:2+ ^(4 - C^)
the symbolism " J :2" being repeated p times.
If C is the side of an inscribed regular polygon of 7t sides
then Cp is the side of one containing N = w x 2^ sides. The
^ De circulo et adscriptis liber, 1619 (Latin translation by Snell
of the Dutch original of 1596).
2 He actually expresses his results as vulgar fractions but the
denominators ■ are always powers of 10. It should, however, be re-
membered that the Greeks used a system of "sexagesimals " not
altogether unlike decimals (see Exercises, Pt. II, p. 97).
■^ Van Ceulen actually used " ^. " in this way.
294 ALGEBRA
perimeter will be NC^ and the ratio of this number to the
diameter, 2, that is ^ NC^, will be a number which constantly
approaches tt as _2? is increased. The formula for the ratio to
the diameter of the perimeter of the corresponding circum-
scribed polygon can be deduced from Eooercises, fig. 48. Let
PQ and AB be sides of the corresponding circumscribed and
inscribed polygons. Put AB = C^ and PQ = E^. Then, by
similar triangles,
Ep/C^ = OT/OH
= 1/6
whence E^ = CJb
= C,/V(l-iC/)
= 2C,/V(4 - V)
Whatever, then, be the length of the original chord C and
whatever the value of J9, it is certain that tt lies between
JNC, and iNE^ = NC,/ V(4 - C/).
§ 3. Actioal Calculation of TT. — For the actual calculation
of TT we may start with any convenient chord AB. For
example, if AB is the side of a square, then C = J2 and we
have
C,= V: 2- V: 2+ V: 2+. . .+ V(4-2)
= J: 2- J: 2+ J: 2 + . . .+ J2
If AB is the side of a hexagon C = 1 and the formula be-
comes
G,^ J: 2- J: 2+ J:2 + . . .+ J3
With inexhaustible patience van Ceulen calculated the value
of TT by both methods — and by still others — to an enormous
degree of minuteness. Thus starting with the inscribed square
he calculated the perimeters of inscribed polygons of 8, 16,
32, . . . up to 1,073,741,824 sides. In the case of the last
polygon he calculated also the perimeter of the corresponding
circumscribed figure. In this way he found that the value of
IT lies between the numbers
3-1415926535897959
and 3-1415926535897932
Starting now with the hexagon, he calculated the peri-
meters of the inscribed and circumscribed polygons of
6,442,450,944 sides and hence determined that tt lies between
the numbers
3-141592653589793238
and 3-141592653589793239
THE CALCULATION OF tt AND THE SINE TABLE 295
It would be useless to repeat the details of these prodigious
computations, but it is worth while to draw up a briefer
table giving the perimeters of the inscribed and circumscribed
polygons of 3, 6, 12, . . . sides until they are identical as
far as the (nearest) fourth place of decimals. We may call it
" Van Ceulen's Table " :—
N -^NO iNE
3 2-5981 6-1962
6 3-0000 3-4641
12 3-1058 3-2154
24 3-1326 3-1597
48 3-1293 3-1461
96 3-1410 3-1427
192 3-1415 3-1419
384 3-1416 3-1416
We conclude that the value of tt to four decimal places is
3-1416.
[Ex. L, A, can now be worked.]
B. The Calculation of the Sine-Table (ch. xxiv., § 9;
Ex. L).
§ 1. Preliminary. — The seaman and the surveyor, to say
nothing of the astronomer, require tables of sines, cosines,
etc., much more exact than those obtainable by measur-
ing triangles as in ch. xiii. We are to inquire, therefore,
how a table of these ratios, carried to any given number
of decimal places, can be computed. Note, first, that
cos a and tan a, and also cot a (that is, 1/tan a), sec a (that
is, 1/cos a) and cosec a (that is, 1/sin a) can all be de-
duced from sin a. Thus the calculation of sines only will
suffice. Moreover since sin(90° - a) = cos a and tan(90° - a)
= cot a we need actually calculate only the sines from 0° up
to 45° in order to have data for computing the whole of the
tables.
§ 2. The Calculation of Sines. — Ch. xiv., B, showed how
the sines could be calculated for angles of 60" and 45°. The
method of Section A of this chapter can be used to deduce
from these the sines of angles which are ■^, J, ^, etc. of 60° or
45°. For if, in Exercises, fig. 47, the angle AOH = a it is
evident that AB = 2AH = 2 sin a. In fact, the earlier
astronomers, such as Hipparchus (about 130 b.o.) and Ptolemy
(about 120 A.D.) used chords where we use sines. Sines,
296 ALGEBRA
cosines, and tangents appear only at the end of the Middle
Ages.
The values of C^ in van Ceulen's table (p. 295) are the
chords subtended by angles of 60°, 30°, 35°, etc., in a circle of
unit radius. The halves of these numbers are, therefore, the
sines of 30°, 15°, 7*5°, 3-75°, 1-875°, 0-9375°, 0-46875°. Note
that to four decimal places
sin 0-9375° = 00164 = 2 x 0-0082 - 2 sin 0-46875°.
That is, within the first degree the sine is, to four decimal
places, proportional to the angle. The value of sin 1° can,
therefore, be calculated by proportion : —
sin 1° = sin 0-9375° xi|gg
= 0-0174
Now we know sin 15°, sin 30°, sin 45°. If, then, we had
a formula for calculating sin (a + 1°) when sin a and sin 1°
are known we could, starting from either of these sines, and
working backwards and forwards, calculate all sines up to 45°.
Using sin 30' = ^ sin 1° = 0-008775 we could similarly
calculate the sines of the half -degrees, etc., etc. It is easy
to find a formula which, given the sines and cosines of two
angles, a and (B, will enable us to calculate sin (a + P).
Let the circle in Exercises, fig. 49, have unit radius. Let
AOB = 2a and BOG = 2yS. Then our problem may be ex-
pressed thus : Knowing the lengths of AB and BC how can
we calculate the length of AC ? We have
AB = 2 sin a, BG = 2 sin /g, and AG = 2 sin (a + yS).
Also, since AGB at the circumference and AOB at the centre
are on the same arc AB, AGB = ^ AOB = a. Similarly
BAG = y8. Thus
AG = AB cos ^ + BG cos a
That is 2 sin (a + y8) = 2 sin a cos /? + 2 sin /5 cos a
or sin (a + y8) = sin a cos y8 + cos a sin y8.
As an example take the calculation of sin 16° to four places.
We have
sin 15° = 0-2588 sin 1° = 0-0174
cos 15° = J{1 - sin215°) cos 1° = ^{1 - sin21°)
= 0-9659 = 0-9998
sin 15° cos 1° = 0-2587
cos 15° sin 1° = '0-0168
sin 16° = 0-2755
[Ex. L, B, can now be worked.]
SECTION III.
LOGAEITHMS.
THE EXERCISES OF SECTION III.
*^* The numbers in ordinary type refer to the pages of Exer-
cises in Algebra f Part I ; the numbers in heavy type to the pages
of this book.
BXERCISE PAGES
LI. Growth Factors 269, 302
LII. Growth Problems . . . . . . 272, 313
LIII. The Gunter Scale 277, 319
LIV. Logarithms and Antilogarithms . . . 281, 325
LV. The Base of Logarithms 283, 333
LVI. Common Logarithms 286, 335
LVII. The Use op Tables op Logarithms . . . 289, 337
LVIII. The Logarithmic and Antilogarithmic
Functions 292, 341
LIX. Nominal and Effective Growth Factors . 297, 346
. CHAPTER XXIX.
THE PKOGEAMME OF SECTION III.
§ 1. The Nature of Logarithms. — Logarithms can be looked
at from two distinct points of view. We may regard them
(as their inventor did) as " artificial numbers " intended to
facilitate computation, or we may regard them as the natural
mode of expression of an important kind of connexion between
variables. The usual method of teaching logarithms empha-
sizes the first point of view to the practical exclusion of the
second ; the method outlined in Section III attempts to com-
bine the two in a treatment consonant with the principles
followed in developing the previous sections. It begins with
the study of certain concrete problems — here called " growth
problems " — in which a particular kind of connexion between
the variables is exemplified. So long as we consider only
integral values of the independent variable (Ex. LI) these
can be solved by ordinary arithmetic, but other cases are
found to require a different method of solution. At first
(Ex. LII) we are contented to deal with these refractory cases
by means of graphs — here called " growth-curves " — but we
soon pass to the consideration of more economical and accur-
ate methods of procedure. After a transitional period (Ex.
LIII) in which calculating devices (the " Gunter scale " and the
** slide-rule ") derived from the " growth-curves " are con-
sidered, we finally reach the notion of logarithms (Exs. LIV-
LVI) and the use of tables (Ex. LVII). Then comes the usual
second stage of the inquiry (Ex. LVIII) in which we turn our
attention from the concrete problems and the methods needed
to solve them, and direct it to the properties of the function
which the behaviour of the concrete variables exemplifies.
In connexion with this part of our investigation we find oc-
casion for the first time to introduce the concept of an index
as a number which may have any value, positive or negative,
299
300 ALGEBRA
integral or fractional. The section ends with a further study
of practical problems from which there emerge the important
notions connected by mathematicians with the algebraic
symbol " e ".
We have already advanced (ch. v., § 4) some reasons for
ignoring both the historical accident that logarithms were in-
vented in order to lighten the labour of computing products
and quotients, and also the custom of deriving their theory
from the theory of fractional indices. In brief, the plan here
adopted of teaching logarithms as an alternative to the use of
growth-curves in the solution of certain problems presents
two advantages : in the first place it makes their practical
value much more evident, and in the second place it prepares
the way more effectively for the study of an extremely impor-
tant type of function. In connexion with these arguments it
is interesting to note that the calculation of logarithms was
conceived by Napier himself as a kind of " growth-prob-
FiG. 77.
lem ". This will be evident from the following summary of
his explanation of the principle underlying his method. ^ In
studying it the reader must remember that Napier was chiefly
concerned to facilitate evaluations of formulae in spherical
trigonometry. For this reason his logarithms were the logar-
ithms of sines. Moreover, the logarithm of the largest sine
( = 1) was taken as zero, and the logarithms increased as the
sine of the angle diminished. This device was adopted in
order that in computing spherical triangles sin 90° (which
occurs very often) might be ignored.
Let the line AB (fig. 77) be graduated uniformly from to
1. Distances measured along this line from A may be taken
as representing all the possible values assumed by the sine
of an angle as the latter increases from 0° to 90°. Next sup-
1 A translation (by W. R. Macdonald) of Napier's Mirifici
Logarithmorum Ganonis Constructio was published in 1889 by
Blackwoods but appears now to be out of print.
THE PROGRAMME OF SECTION III 301
pose a point to start from B and to move along the line
towards the right with a speed always proportional to its
distance from A. Let a, b, c, d, etc., mark points which it
reaches after 1, 2, 3, 4, etc., equal intervals of time. Then
it follows from the law of diminishing speed of the moving
point that both the distances Ba, ab, be, cd, etc., and the
distances BA, a A, &A, cA, etc., diminish in a constant ratio.
It is obvious that the moving point will never quite reach A.
While the first point is moving in the manner described along
the line BA, let another point, starting with the same speed,
move uniformly along the line L. Let a, b, c, d, on this line
represent the points reached by the second moving point at
the ends of the equal intervals of time which bring the first
moving point to a, b, c, d, etc., on BA. Since these points
will be equidistant the line which begins at L must be sup-
posed of indefinite length towards the right. Consider any
one of the points indicated on L — for example the fourth
point, d. Then the equal segments ha, ab, bo, cd, correspond
in succession to segments Ba, ab, be, cd on AB whose lengths
have equal ratios. If we take the lengths of the equal seg-
ments of L as unity, we can say that the length of the line
Lid gives us the number of the distances in equal ratio neces-
sary to bring the moving point from B to the corresponding
point d on the line AB. The same thing is true of any of
the points marked on L. For this reason Napier called the
lengths 0, ha, L6, Lc, etc., on the line L the logarithms
(Aoywj/ 'api^/xos == number of the ratios) of the lengths AB, Aa,
Ab, Ac, Ad, etc., on the line AB. But the points a, b, c, d,
etc., on AB mark the position of certain numbers on the scale
of sines. The corresponding lengths on the line L may then
be called the logarithms of the sines.
Now set the line BA at right angles to L so that B coin-
cides with L (fig. 78). Further, place the lengths Aa, Ab, Ac,
etc., at right angles to L at the points a, b, c, etc., in order.
Then we have a series of ordinates whose lengths fall off in
constant ratio as they recede uniformly from the origin. It
follows that if we draw a smooth curve through the upper
ends of the ordinates we obtain a "growth curve" of the
kind studied in Ex. LII.
Napier did not himself pursue his argument to the conse-
quence pointed out in the last paragraph. This step is said
to have been first taken by Edmund Gunter (1581-1626).
303
ALGEBRA
At any rate it was Gunter who first conceived (1623) the idea
of laying off the length of the ordinates of an exponential
curve along its line of abscissae and so produced a mechanical
instrument which became (through Wingate, 1627, Oughtred,
1627, Milborne, 1650, and Partridge, 1657) the ancestor of
the slide rule used so much by engineers to-day.
§ 2. Growth Problems (Exs. LI, LII ; ch. xxx.).— The
word " growth " as used in Section III scarcely needs formal
definition. It refers to the familiar fact that measurable
things of many kinds, when observed from time to time, are
found to increase or decrease in magnitude. In some cases
this growth is regular — that is, the magnitudes at different
times succeed one another in accordance with a single mathe-
matical law ; in other cases it is irregular — that is, the changes
during different periods follow different mathematical laws.
" Regular " growth may conceivably take place in an endless
number of different ways, but two types occur so frequently
in the phenomena of nature and society that they deserve
special consideration and study. The distinctive mark of the
first type is that in equal times, however small, the magnitude
of the grov^ng thing always shows equal increments or equal
decrements ; that of the second is that the magnitudes at the
beginning and the end of equal intervals of time are always
THE PROGRAMME OF SECTION III 3a3
in the same ratio. The former law is characteristic of some
of the simplest and most widespread phenomena of nature
— for example, it is exhibited by the increasing velocity of
a freely falling body or the decreasing velocity of a body
moving against a constant resistance. It is also the law of
" simple interest " and in this capacity plays an important
part in economic affairs. The latter law is also exemplified
in physical phenomena — for example, in the way in which the
temperature of a hot body falls when cooling under constant
conditions. But, just as physical phenomena give the most
impressive examples of the operation of the law of constant
increment or decrement, so biological and economic pheno-
mena exemplify most strikingly the law of constant ratio.
Not only is it the law of " compound interest " and in this
capacity the final arbiter of the world of finance. In addition
it is the law which, in its direct or derived forms, rules most
of the phenomena of vital growth and decay and so is at once
the law of life and of death. A method of dealing with the
theory of logarithms which is based upon a careful study of
this law seems sufficiently justified by that circumstance alone.
Ex. LI introduces the subject by examples in which growth
is seen to be measured much more appropriately by a
" growth-factor " than by the simple increment or decrement.
Thus it is found in No. 5 that the growth-factor of the height
of the average American girl ^ is almost constant for several
years, although the actual increment of her stature is never
the same for two years in succession. It is obvious that a
statement about the value of this constant factor gives much
more concise and illuminating information than a statement
about the values of the annual increments.
The examples of division A give practice in the application
of the idea of a constant growth-factor. Only integral time-
intervals are considered, and the investigation is summed up
in No. II in the formula Q = QqT'' where r is the growth-
factor and n may have any integral value, positive or nega-
tive. In these examples the growth-factor is always given ;
in those of division B the problem is to calculate the growth-
factor, given the magnitude of the growing thing at two
moments separated by a specified integral number of time
^ The table (Exercises, p. 269) is taken from Stanley Hall's
Adolescence.
304 ALGEBRA
intervals. These examples should play an important part in
preparing the student's mind for the arguments of the later
exercises.
The tables asked for in Nos. 13, 14 will be required in the
next exercise. They should, therefore, be computed carefully
and preserved.
Ch. XXX. begins with a review of the subject of regular
growth in which some of the ideas of the present article are
taught by simple examples. It goes on to develop a graphical
method of solving "growth-problems" when the time is not
an integral number of unit intervals. The argument is given
fully in ch. xxx., but the teacher may be reminded in con-
nexion with § 4 of the discussion in ch. iv., § 6.
The examples of Ex. LII are so framed that they can all
be solved by " growth-curves " in which the factor is either
1-1, 1-25, or 1-3. These three curves are represented in
Exercises, fig. 50, p. 273. The teacher may, of course, pre-
fer to make his pupils use their own graphs.
§ 3. The Gunter Scale ; the Slide Bule (Exs. LIU,
LIV, A ; chs. xxxi., xxxii., A). As was explained in § 1
the arguments of these exercises are intended to be a bridge
between the solution of growth-problems by a graph and the
use and theory of logarithms. The student who has worked
a number of the examples contained in them will have already
acquired the essential ideas underlying the use of logarithms.
Moreover, these ideas should have for him the vividness and
exactness which we have seen to be the peculiar gift of an
appropriate graphical method of approach to abstract notions
(ch. IV., § 1).
The special contribution of ch. xxxi. and Ex. LIU is the
idea that any growth-curve, if prolonged far enough, supplies
the means of solving any growth-problem, no matter what
special factor may be involved in it. It shows, further, that
the only thing actually needed is a record of the heights of
the ordinates of such a curve distributed at proper intervals
along a strip of paper. Such a graduated strip is here called
a Gunter scale (see p. 302). The argument (ch. xxxi., § 3) —
which shows that a curve need be used to graduate the Gunter
scale only from 1 to 10, and that the graduations can then
be extended indefinitely far to the right or left — is of great
importance. The teacher will see that it prepares the student
to recognize and understand the special value of logarithms
THE PROGRAMME OF SECTION III 305
to base 10. Similarly the method of § 4 is not only interest-
ing in itself but is also useful as a preparatory illumination of
the idea that logarithms may be calculated to any base. This
method may with advantage be demonstrated upon the squared
blackboard with the aid of a long Gunter scale. A suitable
scale is constructed by cutting a strip of " semi- logarithm
paper " (supplied by instrument makers at threepence or
fourpence a sheet) and sticking it parallel and near to an edge
of a large sheet of ordinary drawing paper. A line of suitable
length should be drawn upon the sheet at right angles to the
strip, meeting it at the end where the graduations begin.
Another straight line should be drawn to join the further ends
of the line and the strip. The line may now be graduated
similarly to the strip by the familiar method which consists
in drawing lines parallel to the hypothenuse of the triangle
from the graduations of the strip to meet the line at right
angles to it. In this way a Gunter strip of any required
length may be rapidly obtained.
Up to this point the properties of the growth-curve and the
Gunter scale have been used only for finding powers and
roots. In ch. xxxii., A, and Ex. LIV, A, the student learns
that they can also be used for abbreviating the calculation of
products and quotients. Once more an instrumental device
is introduced as the best means of ensuring a thorough under-
standing of the uses of the table of logarithms. It is shown
that multiplication and division may be performed with great
ease by employing a second Gunter scale. In this way the
student makes acquaintance with the slide rule. It will
be observed that the examples are intended not to give
practical skill in the use of the rule but to make the study of
the rule illuminate the properties of logarithms. Apart from
this pedagogical consideration there is no doubt some con-
venience in an early introduction of the slide rule. When its
principles have been mastered in Ex. LIV, practical skill in
using it may, at the option of the teacher of mathematics or
his scientific colleague, be given by special exercises in com-
putation. It must, however, be repeated that the rule, like
the Gunter scale, is introduced here as an illustrative device
intended to secure a really clear appreciation of the nature
and properties of logarithms. It may in this capacity have
full value for a student who never acquires facility in using
it for computative work. If it is used merely as a teaching
T. 20
306 ALGEBRA
aid there is no need to supplement the simple arrangements
described in ch. xxxii., A. If it is to be used for actual com-
putation it will be necessary to invoke the aid of the instru-
ment maker and the guidance of a special text-book such as
the excellent " Slide- Rule Notes " published in this series by
Colonel Dunlop and Mr. C. S. Jackson.
§ 4. Logarithms and Antilogarithms (Exs. LIV, B, LV ;
chs. XXXII., B, xxxni.). — The step from the Gunter scale to the
table of logarithms is very short. Both may be regarded as
derived from the growth-curve and as merely a convenient
means of applying its properties. In the scale the graduations
record the heights of the ordinates of the curve and the ab-
scissae are represented graphically by the distances of the
graduations from the beginning of the scale. In the table
the abscissae are themselves recorded numerically and con-
stitute the logarithms. The growth-factor of the curve from
which the table is constructed is called the "base" of the
logarithms.
This method of constructing a table of logarithms and the
uses of the table when constructed are explained in ch.
xxxii., B, and Ex. LIV, B. The uses of a table of antilogar-
ithms are illustrated at the same time. This table may be
regarded as giving the ordinates corresponding to an arith-
metic progression of abscissae, just as the former table gave
the abscissae corresponding to an arithmetical progression of
ordinates.
Some excellent teachers deprecate the introduction of the
antilogarithmic table and give reasons against it which cer-
tainly carry weight. The author favours its retention less on
practical grounds than because of its usefulness in developing
the theory of the logarithmic and exponential functions. This
usefulness appears very clearly in the discussion of ch. xxxiv.
Ch. XXXIII., A, and Ex. LV carry the theory of the subject
an important step forward, for they show how the notion of
logarithms can be freed from its dependence upon the growth
curve and become purely arithmetical. We have, that is,
now reached the concept of the logarithm of a number N as
it presented itself to Napier — namely, as a measure of the
" number of ratios " needed to bring us from 1 to N along a
certain geometric series. As an example suppose that I'l is
chosen to be the common ratio of the series of numbers. Then
if underneath each term we write the " number of ratios "
THE PROGRAMME OF SECTION III 307
from the beginning of the series up to that term this number
is, by definition, the logarithm of the term above it : —
1 11 1-21 1-331 1-464 1-611 1-772 1*949 2144 2358 2-594
0123 4 5 67 89 10
It is evident that upon this plan all logarithms must be
integers and that the difference between one table of logar-
ithms and another will consist simply in the fact that differ-
ent ratios are used in constructing the series. But without
changing the value of this ratio it is possible, by a slight
complication of the original idea, to vary the logarithms at
will. The complication consists in supposing the series 0, 1,
2, 3, . . .to measure not only the " number of ratios " in
the geometric series but also the number of steps in an auxili-
ary arithmetic series which may be supposed to be written
underneath it, term by term. If the name " logarithm " is —
with some violence to its original meaning — 'transferred to
the terms of the arithmetic series it becomes evident that any
term in one and the same geometric series may have an end-
less number of logarithms varying with the value of the
number taken as the common difference of the arithmetic
series. Thus if 0*1 is taken as the common difference the
logarithm of 1-611 will be 0*5 and that of 2-594 will be 1 ;
if the difference is 0-2 the logarithms of the same two num-
bers will be respectively 1 and 2 ; with a difference of 0-3 the
logarithms become 1-5 and 3 ; and so on endlessly.
So long as we consider only a few terms of the geometric
series the subsidiary arithmetic series appears to be an un-
necessary complication. But if we consider an actual table
of logarithms suitable for serious use its value becomes evi-
dent. To be of practical value the table of logarithms must
be constructed by means of a ratio very near to unity, so that
no number which can be used in a calculation lies far from
one of the terms of the geometric series. Ideally the terms
of the latter should be so close that, for practical purposes,
any given number may be considered identical with one of
them. But in this case, or in any case that approximates to
it, the " number of ratios " would soon become uncomfortably
large. It is convenient, therefore, to divide all the " numbers
of ratios " by some large number — preferably a high power
of 10 — so as to reduce them to an arithmetic series of deci-
mals of which any suitable number of places may be retained
20*
308 ALGEBRA
and the rest ignored. The most convenient way of describing
the relation between the geometric series and the arithmetic
series is to specify the value of the term of the former which
corresponds to the term 1 in the latter. This number is called
the " base " of the logarithms. In the first two of the three
cases considered above the bases are, in order, 2*594 and
1'611. In the third case the base does not appear directly in
the table. Since, however, the logarithm of 2*594 is 3 it is
evident that 1 must be the logarithm of the cube root of 2-594
— that is, of 1'374. This number is, then, the base.
§ 5. Napier's Logarithms. — In substance the foregoing
article is an exposition of Napier's theory of logarithms.
There is, however, an interesting subtlety in Napier's argu-
ment which is omitted because it tends to blur the clear
notion of the " number of ratios " as Napier seems at
first to have conceived it. The reader will remember that
in Napier's scheme the two points start moving from B and
L in fig. 77 with the same velocity, and that if the points
marked a are those reached at the end of the first unit interval
of time then La is the logarithm of Aa. Now since the speed
of the point in BA is constantly falling off La must be greater
than Ba. On the other hand, if B' be supposed to mark the
position of the upper point a unit interval before it reached
B, then La must for the same reason be less than B'B. Let
r be the common ratio of the lines or sines AB, Aa, Kb, etc.
Then, since by hypothesis AB is unity, Aa = r and Ba = 1 - r.
Also AB' = AB/r = 1/r, so that B'B = 1/r - 1 = (1 - r)lr.
Thus the logarithm La lies between 1 - r and (1 - r)lr.
In Napier's calculations r was taken to be 0-9999999, in which
case La lies between 0-0000001 and the same number divided
by 0-9999999, or 0-00000010000001. There will, therefore,
be little error in taking it to be the mean of these numbers or
0-000000100000005. This last number is adopted as the
common difference of the arithmetic series of logarithms while
0-9999999 is taken as the common ratio of the geometric
series of numbers or antilogarithms " adapted " to it. Thus
the logarithms of 0-99999980000001 and 0-99999970000003,
which are the next two numbers of the geometric series, must
be respectively 0-00000020000001 and 0-000000300000015.
The advantages of this procedure over the simpler one
described above are that it assigns theoretically a logarithm
to every number and indicates a method of calculating it
THE PROGRAMME OF SECTION III 309
within limits which may be made as small as we please.
According to the scheme of § 4 the only logarithms given
directly are those of the members of the geometrical series ;
the logarithm of another number is given onhr if, for practical
purposes, it may be considered as coinciding with a term of
that series. But Napier's concept of the two moving points
obviously assigns a position for the " logarithm point " on the
line L corresponding to every possible position of the ' ' number
point " on the line AB. Again Napier's method fixes the base
of the logarithms upon a definite principle. Suppose, for
instance, that in accordance with the method of § 4 the ratio
is taken to be 0*9 and the common difference of the logarithms
0-1. Then the base will be (0-9)i" or (1 - -^\y^. If a finer
scale of numbers is to be secured we may take 0*99 as the
ratio and 0*01 as the difference. The base will in this case
be (1 - T^o)^^^. Adopting the same general plan but aiming
at still closer geometric scales we may take 0*999, 0'9999, etc.,
as the ratio and 0"001, 0*0001, etc., as the common differ-
ence. Thus the successive bases will be (1 - toVo)^*^*^^'
(1 - YoFoo)^^*^*^*^j etc. These numbers have as their " limit "
the number which mathematicians denote by the symbol e~^
and which = 1/2*7182818 . . . ; but Napier's base, since it
corresponds to a geometrical scale constructed upon the same
plan but of infinite closeness, is the limit itself. It is easy to
see that this consequence follows from the hypothesis that
the two points of fig. 77 start their movements with the same
velocity. For if the point a be taken on AB so that Aa =
1 - (1/10)" then La is nearer to (1/10)" the greater the value
of n ; complete equality between Ba and ha being the limit
approached as n approaches infinity.
§ 6. Common Logarithms (Exs. LVI, LVII ; ch. xxxiii.,
B, C). — The next two exercises are devoted to illustrating the
special convenience of logarithms to base 10 and to giving
practice in the use of tables of such logarithms. The treat-
ment given in ch. xxxiii. is too full to need further elucidation,
but the teacher may be recommended to follow rather carefully
the methods suggested for determining the characteristic of a
logarithm or the number of figures before the decimal point
in an antilogarithm. After a little practice these methods
may, no doubt, be abbreviated to the more familiar rules
usually prefixed to books of mathematical tables ; but these
rules will be used with more intelligence if the students go
310 ALGEBRA
through a period of drill in which they are required to give
full analyses of their procedure. From this point of view it
is much to be desired that in tables of logarithms the " argu-
ment," and in tables of antilogarithms the logarithms, should
always be printed as a number in the " standard form " (see
p. 175). This useful practice is adopted in the logarithms and
antilogarithms published as a separate volume of this series.
It is also a feature of the excellent American tables of Prof.
E. V. Huntingdon.
§ 7. The Logarithmic and Antilogarithmic Functions.
Fractional Indices (Ex. LVIII; ch. xxxiv.). — The last few
exercises have been concerned entirely with the practical
aspect of logarithms. We now turn to the theoretical aspect
and study the connexion between a number and its logarithm
as a function possessed, like the parabolic or the hyperbolic
function, of definite and distinctive properties. In connexion
with the argument of ch. xxxiv., § 2, the teacher should con-
sult ch. XLi., p. 432.
Just as the practical problem of finding the logarithm for
any given number suggests the idea of the logarithmic func-
tion of the variable a?, so the inverse problem of finding the
number corresponding to any given logarithm leads to the
notion of the antilogarithmic function. This part of the
argument of ch. xxxiv. is of great importance. Its crux is
the question of the symbolic representation of the antilogar-
ithmic function. The first obvious mode of representation is
y = antilog„ic
and this would always be sufficient, though cumbersome.
But an examination of the properties of the function shows
that they are identical for all values of x with the properties
which the function a^ possesses for integral values of x. The
discovery of this fact has its natural sequel in the proposal to
extend the range of application of the symbolism a"" to include
fractional values of the variable. The consequences of the
extension are sufficiently described in § 4.
Division A of Ex. LVIII is intended to illustrate these
theoretical questions. Division B consists of examples which
bring out the usefulness of the index- notation for the anti-
logarithmic function. The method of investigating certain
physical laws illustrated by Nos. l6-2I is of considerable
interest and practical importance, especially to engineers.
THE PROGRAMME OF SECTION III 311
It is fully described by Prof. Perry in his Practical Mathe-
matics (Lecture IV).
Nos. 22-27 ^^J be regarded as illustrating in a simple
way the method by which Henry Briggs first calculated
logarithms to a given base (see the note, Exercises, p. 296).
Mr. Edwin Edser appears to have been the first person to
propose the method of graphic interpolation of No. 26.
The exercise ends with a few simple examples on the
manipulation of fractional indices — a subject to which too
much attention has been given in text-books and examina-
tions.
§ 8. The number " e " (Ex. LIX ; ch. xxxv.).— The ideas
connected with the symbol " e " are of such importance in
algebra that they should certainly find a place, if possible,
even in the elementary course. The general argument of
Section III makes it a comparatively easy matter to deal with
them in a way which brings out their real significance and
yet makes very moderate demands upon the student. Ab-
stracting from the concrete setting of the argument of ch.
xxxv. we see that it leads to the important conclusion that as
n increases the value of (1 + i/nY' grows constantly nearer to
a definite number which it never actually reaches, though it
comes and ever afterwards remains nearer to it than any
number that can be named, however small that may be. In
the language to be used at a later stage of the work, (1 + ^/n)"
is shown to have a definite " limit ". It is also shown that
for different values of i that limit can be expressed as e* where
e is the number related to (1 + 1/w)" in the same way as the
former number is related to (1 + i/ny. It will be noted that
the function of the geometrical argument is in the first place
to give concrete significance to the expression (1 + i/nY' and
in the second place to prove that it " tends to a limit ".
When this fact has been proved it is sufficient to obtain an
approximation to e by the method suggested in No. 22.
The objection that geometrical reasoning should not be
used in order to arrive at a conclusion in algebra will hardly
be thought serious in the present connexion, and at the stage
of the student's progress here in view. A simple and effective
argument need not be rejected upon the academic ground
that it mixes geometrical and arithmetical reasoning. It is,
however, important that the assumptions upon which its
cogency rests should be clearly understood, and for this
312 ALGEBRA
reason the teacher should not fail to point out that the argu-
ment assumes without proof that a " growth- curve " has at
each point a definite tangent. At a later stage the desire
of eliminating this assumption may be made the motive of
an attempt to discover a purely algebraic analysis. At the
present stage such an attempt would find little support in any
logical scruples on the part of the student.
CHAPTEK XXX.
THE GEAPHIC SOLUTION OF GROWTH PEOBLEMS.
(Ch. XXIX., § 2 ; Ex. LII.)
§ 1. Two Measures of Magnitude-Change. — The problems
of Ex. LI suggest that there are two distinct ways of
measuring the change in magnitude which a constantly in-
creasing or decreasing thing undergoes during a given interval.
Let the magnitudes at the beginning and the end of the
interval be m^ and m^. Then m.2 - m-^ gives one measure of
the change while mjm-^ gives the other. The first may be
called the growth-difference, the second the growth-factor.
If the thing has increased during the interval, the growth-
difference will be positive and the growth-factor greater than
one. If it has decreased, the growth-difference will be nega-
tive and the growth-factor less than one.
Of these two measures of change sometimes the one, some-
times the other, is the more important. Imagine, for example,
a vessel into which water is running. Let the amount in the
vessel at the beginning be 10 gallons and at the end of 1, 2,
3, 4, etc., minutes subsequently be 12, 14, 16, 18, etc., gallons.
Then the growth-differences for all these intervals are the
same, namely 2 gallons, but the growth-factors are different,
namely 1*20, 1-17, 1-14, 1-125, etc., approximately. It is
evident that in this case the statement about the growth-
differences is more useful and interesting than the statement
about the growth-factors. On the other hand, consider the
heights of the average girl between the ages of 8^ and 14^
{Exercises, p. 269). The successive growth-differences in this
case are 2-0, 2*0, 2-1, 2-3, 2-4, and 2*5 inches, while, as you
saw in Ex. LI, No. 5, the growth-factor has practically the
same value for each of these years — namely, 1'042. In this
case, then, the growth-factor is the more important measure
of change.
313
314 ALGEBRA
§ 2. Two Laws of Continuous Magnitude- Change. — What
we have said about these examples would be equally true if
the increase measured took place in " jerks " between the
measurements. As a matter of fact, however, both the filling
of a vessel and the growth of a girl are processes of con-
tinuous increase. From the facts that in the former case the
growth-differences, and in the latter the growth-factors, are
constant for measurements separated by a certain constant
interval, we cannot, of course, infer that the same regu-
larities would be exhibited if another interval were chosen.
Nevertheless, the examples suggest two types of continuous
increase in which these regularities would hold good respec-
tively for all equal intervals, however large or small. Let a
series of measurements of a growing thing be made at mo-
ments separated by equal intervals of time. Then if, no matter
what interval is chosen, each measurement is greater (or less)
than its predecessor by a constant amount the growth is an
instance of the former type ; if the ratio of each measurement
to its predecessor is constant it is an instance of the second
type.i
§ 3. Problems of the Second Type. — Suppose that the filling
of the vessel in § 1 followed the first type of growth. Let it
be given that at a certain moment t^ the vessel contained 20
gallons and that the growth-difference was 2 gallons/min.
Then the quantity of water at any other moment can, of
course, easily be calculated. For instance, 3^ minutes after
t^ the quantity will be 20 -I- 2 x 3|, 5J minutes before t^ it
must have been 20 - 2 x 5J. In making these calculations
we argue that the growth-difference for half a minute must
be one-half of 2 gallons, for a quarter of a minute one-quarter
of 2 gallons, etc. It is obvious that any other problem in-
volving this kind of growth could be solved by a similar
method.
Next suppose, on the other hand, that the growth of " the
average girl " is of the second type. Let it be given that at
11^ her height is 53-8 inches and that the yearly growth-factor
is 1-042. How shall we calculate her height, say at 14 years ?
The height at 13^ (i.e. after two years) is, by hypothesis,
given by the calculation
53-8 X 1-042 X 1-042 = 58-4 inches.
^ lb may, of course, belong to neither type.
THE GRAPHIC SOLUTION OF GROWTH PROBLEMS 315
The difficulty is to find how to deal with the extra half-
year. It will not do, of course, to find how much the girl
will grow during the next year and add one-half of this
amount to the above product. To act so would be to assume
that during this year her growth belongs to the first type —
which is contrary to our supposition. Nor can we argue that
the growth-factor for six months is one half of the yearly
factor. This argument would lead us to find the girl's height
at 14 by multiplying her height at 13| by 0-521 — that is, we
should conclude that in these six months her height is reduced
by nearly a half ! Ex. LI, Nos. 15-21, suggest the correct
method to follow. Let the growth-factor for six months be
called r. Then we have
height at 14 = 58'4 x r
height at 14^ = 58-4 xrxr
But, by hypothesis,
height at 14^ = 58-4 x 1-042
hence r^ = 1-042
and r = x/l-042
By a similar argument we could calculate the height, say
at lOJ. The height at 10^ is 53-8/1-042 = 51-6 inches. This
number must be divided by r, the growth-factor (in this case)
for a quarter of a year. To find r we note that four succes-
sive divisions by r should give us the height at 9^. But this
height could also be obtained by a single division by 1-042.
Hence r^ = 1-042 or r = ^1-042. The general rule is easily
derived from these examples. The growth-factor for ^ of a
year is always the nth root of the growth-factor for a year.
The foregoing rule can be applied without much difficulty in
simple cases but seems of little use in others. For instance,
how are we to find the growth-factor for f f of a year, so as to
calculate the " average girl's " height 200 days after the last
measurement? Even to calculate the growth-factor for a
month (taken as ^L of a year) we must go through the tedious
operation of finding the square root of the square root of the
cube root of 1*042.
§ 4. Graphic Solutions : the " Growth-Curve ". — To avoid
these difficulties we naturally fall back upon the graphic
method. This method could, if it were worth while, be used
to solve problems of the first type. Given one magnitude of
316
ALGEBRA
the growing thing and the growth-difference for a certain
interval we could at once draw the straight line whose points
represent all other magnitudes of the thing. In problems of
the second type the graph will not, of course, be a straight
line. But if we plot a number of points, representing the
magnitudes calculated by means of the growth-factor for a
given interval, we may reasonably expect that the smooth curve
through these points will give us the magnitudes which it is
tiresome or impossible to determine by calculation.
The results of Ex. LI, No. 14 (ii), set out in the fol-
lowing table, may conveniently be utilized in testing this
expectation. Imagine a continuously increasing quantity
subject to a constant growth-factor of 1*25 for the unit of time,
and let its present magnitude be unity. The second line of
the table gives the magnitude of the quantity 1, 2, ... 5 units
of time ago and 1, 2, ... 5 units hence. The third line of
figures gives, in a similar way, the history of a continuously
decreasing quantity, subject to a constant growth-factor of
0*8 for the unit of time. Since multiplication by 0-8 is the
same as division by 1-25 and vice versa, one of these rows is
simply the other row reversed.
lx(l-25)"
1 X (O-S)"
- 5
0-3280
3 052
2-441
2
4100-512
1-953
0-640
1-563
0-800
1-250
+ l! + 2
11-250
llo-800
1-563
0-640
+ 3
1 953
0-512
+ 4
2-441
+ 5
3-052
4100-328
The members of the class should undertake the plotting of
the curves in pairs. The first member of each pair should
plot the successive magnitudes given by a growth-factor of
1-25, and his companion those due to a growth-factor of 0-8.
Each should then draw a smooth curve through his points
with the greatest possible care. The curves should, of course,
be identical, except that corresponding parts lie on opposite
sides of the i/-axis.
The hypothesis which we have now to test is that these
curves are graphic expressions of the law exhibited in con-
tinuous growth of the second type. If this hypothesis is
correct the ratio between the heights of equidistant ordinates
should be the same in all parts of the curve, for these ratios
THE GRAPHIC SOLUTION OF GROWTH PROBLEMS 317
measure the growth-factors of the quantity during equal
intervals of time. Each pupil should draw in dififerent parts
of his curve three pairs of equidistant ordinates, the distance
between them being selected at random. He should determine
the ratios obtained by dividing the height of the right-hand
member of each pair of ordinates by that of the left-hand
member. If his curve is drawn finely and accurately he will
be able to satisfy himself that the three ratios are equal.
Since similar results are obtained, with different sets of ordin-
ates, by all the class, there can be no reasonable doubt that
the curves really do give the magnitude of the changing
quantity at all times within the scope of the graph.
§ 5. The Growth-factor necessarily Positive. — We may for
convenience refer to these graphs as "growth- curves," and to
the kind of continuous change which they represent as " uni-
form growth with constant growth-factor ". This expression
will distinguish growth of the second type from the "uniform
growth with constant growth-difference " which constitutes
the first type.
The graphs help to bring out certain important differences
between the two types. In the first type there is no lower or
upper limit to the magnitude of the quantity, for the graph is
a sloping straight line. In the second type, on the other hand,
while the magnitude may increase without limit it cannot
decrease without limit. A large number of divisions by a
factor greater than 1 or multiplications by a factor less than 1
will reduce its measure below any given positive number, but
can never make it zero or negative. The ic-axis is, therefore,
asymptotic to the growth- curve.
If the original magnitude is positive the whole of the growth-
curve will be above the £c-axis ; if negative, below. It is im-
portant to notice that the growth-factor must itself always be
positive in these problems. Thus, let a unit magnitude be
subject, if possible, to a growth-factor of - 2. Then its magni-
tudes at the end of successive units of time will be : 1 x ( - 2),
1 X (- 2)2, 1 X (- 2)3, etc., i.e. - 2, + 4, - 8, -|- 16, etc.
The corresponding past magnitudes will be - ^, -h J, - J,
+ Jg-, etc. Now it is, of course, conceivable that a changing
quantity should have these magnitudes at the end of succes-
sive equal intervals, but its "growth " would be very different
from the constantly increasing or decreasing magnitude which
we have been studying. It would be a " growth " which now
318 ALGEBRA
makes the thing larger and then smaller and then larger again,
etc., like the movement of the water up and down a sea-wall.
We shall find it convenient to ignore such kinds of " growth "
for the present ^ and to confine our attention to cases where
the growth-factor is positive.
[Ex. LII should now be worked.]
^ They will be considered in Exercises, Part II, Sections VI, VII.
CHAPTEE XXXI.
THE GUNTER SCALE.
(Ch. XXIX., § 3 ; Ex. LIII.)
§ 1. i4 Multiplicity of Curves unnecessary. — Problems in
Ex. LII which involved different growth- factors were solved by
means of different curves. When the growth-factor was 1'3
curve A of Exercises, fig. 50, was used, when it was 1-25
reference was made to curve B, and so on. If it were really-
necessary to have a new growth-curve for every problem
which involved a new growth-factor a large collection of such
curves would be required. Fortunately it can be shown that
no such necessity exists. Eor example, suppose that we had
only curve C (in which the growth-factor is 1 -1) and that we
wanted to solve problems involving a growth-factor of 1*3.
Inspection shows that an ordinate of height 1*3 is to be
found in this curve where the abscissa is 1*88. It follows,
from the fundamental property of growth -curves, that what-
ever pair of ordinates is taken at a distance from one another
* of 1*88, the ratio of the longer to the shorter will in every case
be 1-3. Thus the height of the ordinate at -f 3-76 is 1-69
(i.e. the height of the ordinate in curve A whose abscissa is
-t- 2), the ordinate at - 1-88 is 0*77 (i.e. that of the ordinate
in curve A whose abscissa is - 1). It is possible, then, to
determine the magnitude of an original unit after it has been
increasing or decreasing for a time t with a growth-factor of
1*3 by means of curve C as well as by curve A. The required
magnitude — which is the height of the ordinate whose abscissa
is t in curve A — will also be the height of the ordinate whose
abscissa is l'88t in curve C.
Again, suppose that we had only curve A and wished to
solve a problem in which the growth-factor is 1'2. The
ordinate which has this value is distant 0"61 from the origin.
The magnitude of the original unit after time t will, therefore,
319
320 ALGEBRA
be given by the ordinate whose abscissa is 0'61^. For ex-
ample, after 3| unit intervals its magnitude will be that of
the ordinate whose abscissa is (+ 0-61) x (+ 3|) ^ + 2*13,
i.e. 1-89. Similarly its magnitude 7 J intervals a.go was that
of the ordinate whose abscissa is ( + 0-61) x (- 7^) = - 4-27,
i.e. 0'38. Generalizing, we see that a problem involving
any growth-factor r, can be solved by means of any grovTth-
curve. Let p be the abscissa of the point whose ordinate
is r. Then the magnitude of an original unit at time t is the
height of the ordinate whose abscissa is pt.
§ 2. The Gunter Scale. — The preceding argument suggests
that it may be profitable instead of drawing a different growth-
curve for each growth-factor, to draw one such curve with
exceptional care and to use it for all growth-problems. Curve
A may be selected for this purpose on the ground that, since
it rises more rapidly than the others, a given length of base
offers a wider range of ordinates. Supposing it is to be
adopted as the working curve it is worth while to facilitate
its use by a simple device. This consists in recording at suit-
able points along the horizontal axis the heights of the
ordinates at those points. By 'this means the abscissa
corresponding to a given ordinate can be found much more
rapidly than by consulting the vertical scale.
The method is illustrated in Exercises, fig. 50. In order
to obtain a still wider range of ordinates the vertical axis is
supposed to be moved to the position of the dotted line, and the
vertical scale to be contracted. ^ The graduations at different
points of the line GG simply record, upon the new scale, the
heights of the ordinates immediately above them. Thus the
graduation "2 " is directly below the ordinate whose height
is 2 in the contracted scale, the graduation "7"3" directly
below the ordinate whose height is 7*3, and so on.
In graduating the line GG no notice has been taken of the
values of the abscissae of the various ordinates. It is, in fact
unnecessary to record them. Suppose, for example, that we
have a problem in which the growth-factor is 1*42. Applying
a centimetre rule to GG you will find that the ordinate
whose height is 1-4:2 is situated exactly 2 cms. from the origin
of the scale. The magnitude of an original unit after 2, 3,
4 . . . , time- intervals will therefore be given by the gradua-
^ If the size of the page had been unlimited it would, of course,
have been simpler and better to continue the curve to the right.
THE GUNTER SCALE 321
tions situated 4 cms., 6 cms., 8 cms., . . . along the line ; for
these graduations give the ordinates whose distance from the
dotted axis of the curve is 2, 3, 4 . . . times the distance of
the ordinate 1-4:2. Similarly, the magnitude of the unit after
1-2 intervals, 2*6 intervals, 3*05 intervals, etc., is given by the
graduations 2*4 cms., 5-2 cms., 6*1 cms., etc., from the origin
of the scale.
It is obvious from these examples that the scale might be
cut out and used without further reference to the curve from
which it has been constructed.
The first man to graduate a line in this way and to use it
for calculations was Edmund Gunter (1581-1626), an English
mathematician who invented many improvements in the arts
of navigation and surveying. Among these was the surveyor's
chain, still called " Gunter's chain ". A line graduated like
GG of fig. 50 is the essential part of a calculating apparatus
which has been much used by sailors and is called by them
" Gunter 's scale ".
^5 3. Extensions of the Scale. — The graduations of the
Gunter scale in Exercises, fig. 50, range only from 1 to 10.
It is, however, extremely easy to extend them both above 10
and below 1.
Imagine the scale on the dotted axis of Curve A to be
contracted 10 times — that is, let the figures 1, 2, 3, . . . be
replaced by the figures 10, 20, 30 ... It is obvious that each
graduation on the Gunter scale must now be multiplied by 10
in order to give the length of the ordinate above it. Similarly,
if the vertical scale be contracted 100 times each graduation
on the Gunter scale must be multiplied by 100. Thus to
construct a Gunter scale graduated from 1 to 1000, a line must
be taken three times as long as GG ; the mode of division of
the first or fundamental section of this line must be repeated
in each of the other two sections, but the graduations in the
second section must be 10 times, and those in the third section
ICO times as high as the corresponding graduations in the
first section. It is obvious that the only reason why the
graduations could not thus be continued indefinitely is that,
since the scale in each section is ten times as contracted as in
the preceding section, the subdivisions would soon become too
close together to be read with accuracy.
In order to obtain the graduations below 1 the scale on the
dotted vertical axis may be supposed to be expanded 10 times.
T. 21
322 ALGEBRA
The measure of each ordinate recorded on GG will thus be
reduced 10 times, so that the graduations will run from 0*1
to 1. If the vertical scale be expanded 100 times the gradua-
tions will be reduced to the range from 0*01 to 0*1, and so
on. Thus the Gunter line may be continued to the left of the
fundamental section in sections whose length is the same as
that of GG, the scale of the graduations in each section being
10 times as expanded as that in section to the right of it.
Theoretically, then, the graduation can be extended to the left
without limit. Practically, the subdivisions would soon become
too far apart to be used with convenience,
§ 4. The Gunter Scale in Growth-problems. — It was seen in
§ 2 that a Gunter scale can be used, in conjunction with a
centimetre or inch rule, to solve any growth-problem. The
operations can be made extremely simple by the method
illustrated in the following examples.
Example 1. — ^To find the magnitude of an original unit
after 5-2 years, the annual growth-factor being 1-4.
Take a strip of paper, lay it along the Gunter scale with
one end on the graduation "1," and mark on the edge the
point where the graduation "I'd " falls. Now lay the strip
across a sheet of squared paper, as shown in fig. 79.^ Place
the end of the strip anywhere on the vertical marked " " and
swing it round until the mark on the edge lies upon the vertical
through the graduation on the squared paper marked " 1 ".
Mark on the edge of the strip the point P, where it is crossed
by the vertical whose graduation is 5 "2. Once more lay the
strip along the Gunter scale. The reading against the mark
P is the magnitude required. It vnll be found to be 5*75.
Example 2. — An original unit increases to a magnitude of
33*6 in 7'5 units of time. What is the growth-factor?
By § 3 the graduation " 33*6 " occurs in the second section
of the Gunter scale in the position occupied by 3-36 in the
fundamental section. It is necessary, therefore, to mark along
the edge of the strip a distance equal to the whole length of
GG plus the length from the beginning to the graduation
''3-36". Lay the strip across the squared paper as before
and swing it round until the mark on the edge lies upon the
vertical whose graduation is "7'5". Mark the point where
the edge crosses the vertical graduated " 1 ". Once more lay
^ For clearness only the unit lines of the squared paper are shown
in the figure.
THE GUNTER SCALE
the strip along the Gunter scale. The reading against the
second mark gives the growth-factor. It should be 1'6.
Example 3. — The magnitude of a quantity decreases in 4*8
years from unity to 0'4:6. Find the growth-factor.
We are here concerned with the first section of the Gunter
scale to the left of the graduation " 1 ". In this section the
graduation " 0*46 " would occupy the same position as " 4*6 "
>A.
A
r
//
f
//
Y
V/
/
/
Va
/
/
V/
>
\
y
1
2
3
4
5
6
Fig. 79.
occupies in GG while the graduation ''1 " would occupy the
position of the present graduation "10". The beginning of
the strip must, therefore, be placed against the graduation
" 10 " in GG, and the point on the edge marked which lies
against the graduation "4-6 ". As before, the strip must be
placed with its beginning on the zero graduation of the squared
paper and must be swung round until the mark on the edge
21*
324 ALGEBRA
lies on the vertical whose graduation is "4-8". The edge
must be marked where it crosses the vertical graduated "1 ".
When the strip is again laid on the Gunter scale from right
to left (i.e. with its beginning on the graduation " 10 ") the
second mark lies against the graduation " 8*5 ". The growth-
factor must, therefore, be 0*85.
The risk of error involved in marking the auxiliary strip
could, of course, be avoided by using a strip of paper already
graduated as a Gunter scale. Such strips can be very easily
procured. Engineers make use of a kind of squared paper
(called " semi-logarithm paper ") in which the lines of one set
of parallels follow one another in exactly the same positions
as the graduations of the Gunter scale. A single sheet of this
paper can be cut into about 20 strips which need only to be
graduated like GG in Exercises^ fig. 50, to become Gunter
scales.
[Ex. LIII should now be worked.]
CHAPTEE XXXII.
LOGAEITHMS.
A. The Slide Bule (ch. xxix., § 3 ; Ex. LIV, A).
§ 1. Multiplication and Division by Means of Growth-
curves. — The curves of Exercises, fig. 50 (or the Gunter line de-
rived from them) v^ere used in solving the problems of Ex. LIU
as the only really practicable means of solution available.
It is now to be shovt^n that the property of growth-curves
which made them indispensable in these calculations can be
used to lessen in a remarkable way the labour of certain
calculations which are generally carried out by the ordinary
processes of arithmetic. These further applications all depend
upon the fact that the heights of two ordinates at a given dis-
tance apart, no matter in what region of the curve they are
taken, always have a fixed ratio.
This property can, in the first place, be used to facilitate
processes of multiplication. As an example consider how the
product 1*9 X 1-3 could be obtained by the aid of curve A
{Exercises, fig. 50). The first thing to do is to pick out the
ordinates whose heights are respectively 1"9 and 1*3. They
are found where the abscissae are respectively 2*45 and 1.
But, by the property of the curve, an ordinate situated 1 unit
to the right of any given ordinate will always be 1*3 times as
high as that ordinate. Thus the height of the ordinate whose
abscissa is 345 must be 1*9 x 1'3. But the height of that
ordinate is 247. It follows, that 1-9 x 1-3 = 247.
The same problem could have been solved by means of
curve B. The ordinates of height 1*9 and 1*3 have respec-
tively 2*9 and 1*2 for their abscissae. In this case, then, an
ordinate 1'3 times as high as a given ordinate will always be
found at a distance of 1'2 to the right of that ordinate. Thus
the product 1-9 x 1-3 is given by the height of the ordinate
whose abscissa is 2-9 + 1'2 = 4-1. The height of this
325
326 ALGEBRA
oidinate is 2 -48. The values for the product obtained from
the two curves differ by less than ^ per cent.
In general, then, we have the following method of de-
termining by means of any growth- curve the product of two
numbers P and Q. Find the abscissae, ^ and q, of the ordin-
ates whose heights are respectively P and Q. Then PQ is the
height of the ordinate whose abscissa is ^ -f q.
The result of a division process can obviously be obtained
by a similar method. The height of the ordinate situated q
to the left of a given ordinate will always be equal to the
height of the latter divided by Q. Hence the ordinate whose
height is P/Q is the ordinate whose abscissa is ]) - q. Por
example let P = 1-3 and Q = 1-9. Then in curve A, p = 1,
q = 2*45 and j9 - q = - 1"45. The ordinate whose abscissa
is - 1'4:5 gives, then, the value of the quotient. It will be seen
to be 0-68. In curve B, ^ = 1-2, ^ = 2-9 and j9 - g = - 1-72.
The ordinate with this abscissa has, again, a height of 0'68.
§ 2. Multiplication and Division by the Gunter Scale. — As
might be supposed, products and quotients can be obtained
still more readily by means of the Gunter scale. The validity
of the following rules is obvious from ch. xxxi.
To find P X Q mark off on a strip of paper the distance
from the beginning of the scale to the graduation " Q ".
Transfer the beginning of the strip to the graduation " P ".
Then the mark on the edge lies against the graduation which
measures PQ. To find P/Q place the mark on the edge
against the graduation " P," then the beginning of the strip
lies against the graduation which measures P/Q.
In two cases difficulty will arise, [a) The product PQ may
be greater than 10 so that the mark on the strip lies beyond
the scale on the right. The mark must in this case be sup-
posed to lie in the first section to the right of the fundamental
section and the graduation which would be against it must be
determined by the principle of ch. xxxi., § 3. Mark on the
strip the point against the graduation " 10 ". Carry this
point back to the graduation " 1 ". Then the former mark
on the strip lies against a graduation which is simply ten
times as small as the corresponding graduation in the second
section. Thus, if it lies against the graduation "2 "35," the
product in question is 23 '5.
An alternative way to overcome this difficulty is as follows.
Cut a strip of paper exactly as long as GG and, as before.
LOGARITHMS
327
mark on it the position of the graduation " Q ". Reverse the
strip and apply it to the Gunter Une so that what was the
beginning of the strip coincides with the graduation "10,"
while what was originally its right-hand end coincides with the
graduation " 1 ''. Now slide the strip to the right (fig. 80)
until the " Q " mark again coincides with the graduation " P ".
The present right-hand end of the strip now lies against what
would be the graduation " PQ " in the first section to the right
of GG if that section were there. But since the mark at the
(present) left-hand end of the strip has moved to the right
through the same distance as the other end it coincides with
the graduation in the fundamental section of the Gunter scale
which corresponds with the graduation " PQ" in the (hypo-
thetical) section to the right of it. The value of PQ is, there-
fore, the value of this graduation multiplied by 10.
PQ/IO
• ! ■ 2l 3l 4l si !6l 7| «| 9l.0|--1
I'o i 'I
Fia. 80.
-EL
Q
!
rXIO
q
Fig. 81.
WW
lonoDai:
(b) The second difficulty arises in determining P/Q where
Q is greater than P. In this case, when the mark on the strip
is made to coincide with the graduation " P," the beginning
of the strip, which should now coincide with the graduation
" P/Q," lies outside GG (fig. 81) in the (hypothetical) section
of the Gunter scale to the left of the graduation " 1 ". Its
position can, however, be determined by an argument similar
to the one employed above. The right-hand end of the strip
lies just as far to the left of the graduation " 10 " as the begin-
ning of the strip lies to the left of the graduation " 1 ". It
coincides, therefore, with a graduation which is exactly ten
times as great as P/Q. For instance, if that graduation is
8-7, P/Q = 0-87.
328 ALGEBRA
§ 3. The Slide Bute. — A very obvious improvement on these
methods is to employ, instead of the strip of paper, a second
Gunter Une graduated on the same scale as the first. A pair
of such scales, arranged so that one can slide backwards and
forwards beside the other, constitutes a Slide Eule — a calcu-
lating apparatus which is constantly used by engineers,
architects, etc. The easiest way to make a slide rule is to
cut two strips of "semi-logarithm paper,'' to graduate both
like GG in Exercises, fig. 50, and to paste them on to two strips
of cardboard or wood. One of these strips must be fastened
down to a cardboard or wooden base which must be wide
enough to accommodate also the second strip. (For tempor-
ary use it is sufficient to pin the fixed scale to a drawing-
board or the desk ; even the strips of cardboard or wood can
be dispensed with.)
The product and quotient of two numbers P and Q, each
of which is less than 10, can be determined precisely as in
§ 2. To obtain a product the beginning of the movable scale
must be set against the graduation "P" on the fixed scale.
The product PQ will then be the graduation on the fixed
scale which lies against the graduation " Q " on the sliding
scale. If the product is greater than 10, the sliding scale
must be reversed, the graduations " P " and " Q '' on the two
scales must be brought together, and the graduation of the
fixed scale against the graduation " 10 " of the sliding scale
must be noted. The required product is ten times this
graduation. To obtain the quotient P/Q set the reading " Q "
on the sliding scale against "P" on the fixed scale, both
scales being held so that their graduations increase to the
right. If Q is less than P the quotient is the graduation of
the fixed scale which lies against the graduation " 1 " on the
sliding scale. If Q is greater than P the quotient is ten times
less than the graduation against the " 10 " of the sliding scale.
If either P or Q is greater than 10 it must first be ex-
pressed in the " standard form ". The product or quotient of
the numbers less than 10 is then obtained and is afterwards
multiplied (or divided) by the appropriate power of 10. Thus
the quotient 7850/57 = (7-85 x 10=^)/(5-7 x 10) = (7 •85/5-7)
X 102. The quotient 7-85/5-7 is first obtained (it is 1-38)
and is then multiplied by 100 to give the (approximate)
quotient 138.
[Ex. LIV, A, should now be worked.]
LOGARITHMS
329
B. Logarithms (ch. xxix., § 4 ; Ex. LIV, B).
§1. Tables of '^ Logarithms '\ — Convenient as the slide
rule is as a "ready-reckoner" its use is subject to obvious
drawbacks. The accuracy obtainable depends upon the exact-
ness and fineness of the graduations and upon the correctness
with which they are read. Speedy and sure calculations can
be made only after considerable practice. Moreover a strong
and trustworthy slide rule is an expensive instrument. For
these reasons alone it would be worth while to examine
another way in which the properties of growth-curves can be
utilized in facilitating calculations.
As we have seen, a Gunter scale is a graphic record of the
heights and positions of the ordinates of some growth-curve.
The " ready-reckoner '' now to be studied is one in which the
same facts are recorded, not graphically, but in the form of
a table. Here is part of such a table, constructed from curve
A, Exercises, fig. 50.
Logarithms
TO Base
1-3.
n
log
n
log
n
log
10
0-00
1-6
1-79
2-2
3-00
11
0-36
1-7
2 00
2-3
317
1-2
0-69
1-8
2-24
2-4
3-33
1-3
1-00
1-9
2-44
2-5
3-49
1-4
1-28
2-0
2-64
2-6
3-64
1-5
1-55
21
2-83
2-7
3-78
In a table of this kind the "argument" or number which
the calculator has in mind when he refers to it is the height
of an ordinate of the growth-curve, while what he wants to
find out from the table is the abscissa of this ordinate. The
first column, therefore, headed "number'' (n), contains the
heights of ordinates while the second column gives the corre-
sponding abscissae. The abscissae are here called logarithms}
It is obvious that the logarithm of unity is zero in all growth-
curves, but that the logarithms of all other numbers will
depend upon the growth-factor of the curve. For example
^ This name was given to them by John Napier who invented
these aids to calculation about 1594. He also called them ' ' arti-
ficial numbers ".
330
ALGEBRA
the logarithm of 1-5 is 1*55 in curve A, 1-86 in curve B and
4-28 in curve 0. It is necessary, therefore, to specify the
base or growth-factor employed in constructing the table.
Thus the full description of the foregoing table will be : "A
table of logarithms of numbers from 1 to 2-7 to the base
13".
§ 2. The uses of Logarithms. — A table of logarithms can be
used for all the purposes for which the Gunter scale or slide
rule can be employed.
Example 1. —Find the fifth root of 2-7. The logarithm or
abscissa of the ordinate 2-7 is 3*78. By the property of the
growth-curve the abscissa of the ordinate ^2*7 is one-fifth of
this, or 0*756. This is not the logarithm of any number
given in the table, but it lies
between the logarithms of
1-2 and 1-3. . Fig. 82 shows
how its position between
them can be calculated ap-
proximately. PiNj and P^Ng
represent the ordinates 1'2
and 1-3, and N1N2 the por-
tion of the horizontal axis
between the abscissae 0*69
and 1 -0. For convenience the
abscissae are represented on
a larger scale than the ordi-
nates. It is seen that the
short length of curve between
the ordinates differs little from the (dotted) straight line P1P2.
Let pn be the ordinate of the straight line corresponding to
the abscissa 0'756. Then its height is approximately the same
as that of the ordinate to the curve which has the same ab-
scissa. The height of _pw is easily calculated by proportion : —
.o^a^ 0'756 - 0-69
pn = 1-2 -f 0-1 X
Fig. 82.
1-00 - 0-69
1-22
We conclude that 0*756 is approximately the logarithm of
1*22 so that ^2-7 = 1-22 approximately.
This method of determining the number corresponding to a
logarithm which lies between two logarithms given in the
table is called *' the method of proportional parts ". It can,
LOGARITHMS 331
of course, also be used to find the logarithm of a number
which lies between two of the numbers given in the table.
Example 2.— Divide 2-425 by 1-733. By the method of
proportional parts we have
log 2-425 = 3-33 + \ (3-49 - 3-33)
_ 3'37
log 1-733 = 2-00 + J (2-24 - 2-00)
= 2-08
That is, the ordinates whose heights are respectively 2-425 and
1-733 have as their abscissae 3-37 and 2-08. By the pro-
perty of the growth-curve the ordinate whose abscissa is
3-37 - 2-08 = 1-29 will have the height 2-425/1-733. Ee-
ference to the table shows that 1-29 is the logarithm of a
number which lies between 1-4 and 1-5 and is approxi-
mately : —
, , „ , 1-29 - 1-28 , ,„,
^•^ -^ Q-^ ^ 1-55 - 1-28 - ^'^Q^
We conclude that 2-425/1-733 = 1-40.
It will be found by arithmetic that the answers obtained
are (to two decimal places) exactly correct in the first example
and less than | per cent in excess in the second example.
Thus even the very simple table of p. 329 sufiQces to yield
results of considerable accuracy.
§ 3. Antilogarithms. — As we have seen, a table of logarithms
may be regarded as a list of the abscissas corresponding to
given ordinates of a given growth-curve. It would be
equally easy to tabulate the ordinates corresponding to given
abscissae. Such a table is called "a table of antilog-
arithms " — the " antilogarithms " being the ordinates or
" numbers " of the table of logarithms. Such a table is con-
venient when, at the end of a calculation conducted by means
of logarithms, it is necessary to know the number correspond-
ing to the logarithm finally obtained. The table on the
next page consists of antilogarithms obtained from curve A
of Exercises^ fig. 50, and corresponds to the first column of
the table on p. 329.
In § 2, example 1, the final logarithm was 0*756. The
corresponding number or antilogarithm lies, then, between
1-17 and 1-23. By the method of proportional parts it is
ALGEBRA
ANTHiOQARITHMS TO BaSE 1-3.
log.
antilog
(n).
log.
antilog
(n).
00
100
10
1-30
0-2
1-05
1-2
1-37
0-4
111
1-4
1-44
0-6
117
1-6
1-52
0-8
1-23
In example 2 the final logarithm was 1'29. The correspond-
ing antilogarithm is
1-37 + 0-07 X ^ = 1-405
These results^ agree with the former ones to two decimal
places.
[Ex. LIV, B, should now be worked.]
CHAPTEK XXXIII.
COMMON LOGAEITHMS.
A. Gunter's Scale and Logarithms obtained by Calculation
(ch. XXIX., § 4 ; Ex. LV).
§ 1. Logarithms apart from the Growth-curve. — In the last
three chapters Gunter's scale, the slide rule and tables of
logarithms have all been considered in connexion with growth-
curves. It is important to have a clear idea of- the nature of
this connexion. Suppose that a Gunter's scale were placed
in your hands, but that you were quite ignorant of growth-
curves and their properties ; what could you learn from an
examination of the scale by itself ? The answer is that the
scale is a line so divided that the graduations at equal
distances are always in the same ratio no matter what the
distances may be nor where they are taken. For example,
on the line GG of Exercises, fig. 50, any pair of graduations an
inch apart have a ratio of 1*56, any pair 0'8 cm. apart a ratio of
1*15, and so on. (It is to be understood that the ratio meant
is that of the right-hand number of the pair to the left-hand
number.)
Similarly, if a table of antilogarithms were placed in your
hands without any explanation of its mode of construction you
might easily discover that there is always a constant ratio
between pairs of numbers whose logarithms have a constant
difference. For example, in the table on p. 332, if pairs of
logarithms be selected whose difference is 1*0 (such as
and I'O, 0-4 and 1*4, 1-8 and 2 -8) the ratio of the second
number (or antilogarithm) to the former is always 1*3. If the
logarithms have a difference of 0-6 (e.g. 0*2 and 0'8, 1*4 and
2-0, 1-8 and 2-4) the ratio is always 1-17. If the difference
between the logarithms is 1-2 the ratio of the numbers is 1*37,
and so on.
A person who had discovered these facts could proceed to
834 ALGEBRA
make a Gunter's scale or a table of antilogarithms without
the assistance of a growth-curve. He could, for example,
make up his mind that the graduations which are separated
by one-tenth of an inch on the scale, or the numbers whose
logarithms differ by O'l, should have a ratio of I'Ol. By re-
peated multiplication by I'Ol, unity being the original multi-
plicand, he could determine the graduations to be placed on
the scale at points O'l inch, 0'2 inch, 0'3 inch, 0*4 inch,
etc., from the beginning, or the numbers to be placed in the
table against the logarithms 0*1, 0*2, 0*3, 0*4, etc. In order
to complete the Gunter scale it would be necessary to pick
out, by interpolation, the decimal graduations I'l, 1-2, 1*3,
etc., and to number them at convenient intervals as in
Exercises, fig. 50. Similarly, to form a table of antilogarithms
it would be necessary to calculate by the method of propor-
tional parts, the antilogarithms corresponding to a series of
equi-different logarithms and arrange them as on p. 332. By
another application of the same method, a table of the logar-
ithms corresponding to equidistant numbers could be calcu-
lated from the same figures.
It will now be seen that a growth-curve is by no means
essential to the construction of a Gunter scale or of tables of
logarithms and antilogarithms. The curve is useful merely
because it offers an easy method of determining scale gradua-
tions and numbers which could otherwise be determined only
by tedious and troublesome arithmetic. It will be understood,
on the other hand, that the arithmetical method, though
terribly laborious, can be carried out to any desired degree of
accuracy, while the use of the growth-curve gives, like all
graphical methods, results of strictly limited accuracy.
It should be noted that the factor by the constant repetition
of which the Gunter scale or the table of logarithms is con-
structed need not be greater than unity. As a matter of fact
in Napier's first table of logarithms it was less than unity. In
such a table, as the numbers increase the logarithms must
decrease. In the corresponding Gunter's scale the graduations
would decrease towards the right. The corresponding growth-
curve would slope downwards in the positive direction.
v^ 2. The Base of the Logarithms. — In § 1 nothing has been
said about the base of the logarithms. The reason is obvious.
When the logarithms are obtained graphically froni a f^-owth-"
curve the base is simply the growth-factor used in drawing
COMMON LOGARITHMS 335
the curve. It is selected, therefore, before the operation
begins. When the curve is completed the base is the ratio
between any two ordinates whose abscissae differ by 1. In
the table, then, the base is the ratio between any two num-
bers whose logarithms differ by 1. But if we begin by
selecting arbitrarily some number which shall be the ratio
between numbers whose logarithms differ by (say) O'l the
base is not at first known. It must be found by taking the
10th power of the selected ratio. If we chose to work with a
logarithm-difference of 0-01 the base would be the 100th power
of the selected factor. In general, if we decide that numbers
whose ratio is r shall have logarithms differing by 1/p of unity,
then the base is r^.
^ 3. Derivation of Logarithms from a Gunter Scale. — If we
have a Gunter scale constructed by either the graphical or the
arithmetical method it is easy to derive from it a table of
logarithms to any given base. For example let the chosen
base be 2. Lay the scale across squared paper as in fig. 79
so that the vertical graduated " 1 " meets the scale at the
graduation " 2 ". Then, since pairs of equidistant scale-
graduations all have the same ratio, the ratio of the gradua-
tions lying on any pair of verticals 1 inch apart will be
2. If, then, the graduations of the verticals be regarded as
logarithms and the scale-graduations as numbers we have the
materials for a table in which, by the foregoing definition, the
base will be 2. To compile the table we need only read off
the graduations of the verticals which strike the edge of the
scale at the points marked I'l, 1'2, 1*3, etc. Similarly, if a
table of logarithms is desired with 1*5 as base the Gunter
scale must be swung round until the vertical graduated " 1 "
meets its edge at the point marked " 1-5". The logarithms
can then be read off as before. Any other table can be con-
structed in the same way. To find the logarithms of numbers
above 10 or below 1 the Gunter scale must, of course, be
produced and the supplementary sections graduated in ac-
cordance with the principle studied in ch. xxxi., § 3.
[Ex. LV should now be worked.]
B. Common Logarithms (ch. xxix., § 6 ; Ex. LVI).
§ 1. The Advantage of Logarithms to Base 10. — Eeflexion
upon the principle used in A, § 3, suggests that the business of
constructing a table of logarithms can be greatly facilitated by
336 ALGEBRA
adopting 10 as the base. Turn the Gunter scale round until
the graduation " 10 " is on the vertical labelled " 1 ". Then,
as before, the logarithms of the numbers 1-1, 1*2 .. . 2*0,
2*1 . . . 9 '8, 9 -9, can be read off from the squared paper
graduations by interpolation. They will all be decimal frac-
tions between and 1. But in this case it is unnecessary to
produce the Gunter scale in order to read off the logarithms
of numbers above 10 and below 1. For we know that if the
scale were produced the graduation 100 would fall on the
vertical labelled "2," and that the graduations 11, 12 . . .
20, 21 . . . 98, 99, would divide the scale between the
verticals " 1 " and " 2 " in exactly the same way as the gradua-
tions 1-1, 1-2 .. . 2-0, 2-1 .. . 9-8, 9-9 divide the scale
between the verticals "0" and "1". It follows that if the
scale graduation n lies on the vertical whose graduation
is I the scale graduation lOn lies on the vertical whose
graduation is -f 1 -f Z. In other words, if the logarithm of
a number n is Z the logarithm of 10 times that number
is + 1 -I- Z. By the same principle the logarithm of 100/t will
be 4- 2 + Z ; of 1000?i, -f 3 + Z ; of -^j^n, - 1 -f- Z ; of y J^w,
- 2 -I- Z ; and so on. In general if p be any integer, posi-
tive or negative, the logarithm of n x 10^ is ^ -f Z.
§ 2. Practical Use of Logarithms to Base 10. — It follows
from the foregoing that if 10 be chosen as base the only log-
arithms that need be determined are those of numbers between
1 and 10. The logarithm of a number N, greater than 10
or smaller than 1, can be derived from one of them by the
following simple rule : (a) Express the number in the standard
form (ch. xvii., § 8), N = n x 10^. (b) Find from the table the
logarithm of n. This will be a decimal fraction Z, less than
1. (c) Then log^o N = _p -h Z. Thus to find the logarithm of
743-6 to base 10 we begin by throwing the number into the
form 7-436 x 10^. We then refer to a table and find that the
logarithm of 7*436 is (to four places) 0-8713. We conclude
that logio 743-6 = 2 + 0-8713 or 2-8713. If we required the
logarithm of 07436 we should proceed similarly: we
have 0-07436 = 7-436 x 10" 2; hence log^o 0-07436 is
- 2 -f- 0-8713. For compactness the logarithm is generally
written 2-8713, this form being adopted to indicate that the
minus belongs to the whole number only, and that the decimal
part of the logarithm is positive.
[Ex. LVI should now be worked.]
COMMON LOGARITHMS
337
C. The Use of Tables (ch. xxix., § 6 ; Ex. LVII).
^5 1. Tables of Logarithms. — For all usual calculations log-
arithms to base 10 are so much the most convenient that they
are universally used and are called " common " logarithms.
They are printed in tables in which the logarithms of numbers
between 1 and 10 are set out to various numbers of decimal
places to meet different requirements of accuracy. The tables
most used by engineers give the logarithms either to four or
to five places. Navigators and astronomers usually require
seven-figure logarithms. Seven-figure logarithms are neces-
sary and sufficient also for most problems in finance and
insurance, though for some such problems logarithms to twelve
places are required. Tables carrying the logarithms to twenty
and even sixty-one places have also been published, though
they are rarely needed.
In using a table of common logarithms it is important to
remember two things : {a) All the numbers given are to be
taken as lying between 1 and 10, even though they are not
generally so printed. Thus the numbers printed as 26, 782,
etc., are to read as if they were printed 2-6, 7 "82, et.c, the
decimal point being omitted to save space, (b) The log-
arithms are all decimal fractions less than 1. That is, the
symbols " 0' " (" nought point ") must be read in front of every
logarithm even if they are not printed, (c) In the logarithm
of a number which is>not between 1 and 10 the nought before
the decimal point is replaced by the index of the power of 10
used in expressing the given number in the standard form,
the minus sign being written above the index when it is
negative.
The decimal part of the logarithm of a number is called its
mantissa, the integral part its characteristic.
§ 2. The Arrangement of Tables, — The arrangement of the
logarithms in a four-figure table will be understood from the
following specimen : —
I
2
3
4
I
2
3
4
5
6
7
8
9
31
•4914
•4928
•4942
•4955
•4969
1
3
4
6
7
8
10
11
12
32
•5052
•5065
•5079
•5092
•5106
1
3
4
6
7
8
9
11
12
33
•5185
•5198
•5211
•5224
•5238
1
3
4
5
6
8
9
10
12
T.
22
338
ALGEBRA
The number 0'4914 in the row beginning 3-1 and the
column headed is the logarithm of 3 -10. The logarithm of
3-12 is in the same row but in the column headed 2, i.e.
04942. Similarly log 3-14 is 04969. In the actual table
the first set of columns is continued by others headed 5, 6, . . .
9. Thus the first complete row gives the logarithms of 3-10,
3-11, 3*12, . . . 3-19. The second set of (narrow) columns is
called the "difference columns". They contain numbers
which are in every case to be added to the last figure of the
logarithm just mentioned. Thus to find log 3-136 we must
take log 3-13, i.e. 0*4955 and add to it the number (8) in the
difference column headed 6 : —
log 3-136 = 0-4963
Similarly, to find log 3-245 we take log 3*24 = 0-5105 out of
the second row and add to the last figure the 7 out of the
difference column headed 5 : —
log 3-245 = 0-5112
Example.— Find the logarithms of 3307 and 0*6003122.
For log 3307 we have
3-307 X
0-5185
103
10-"
3307 =
log 3*30 =
log 3-307 = 0-5194
.-. log 3307 = 3-5194
For log 0-003122 we have
0*0003122 = 3-122 x
log 3-12 = 0-4942
log 3-122 = 0-4945
.-. log 0*0003122 = 4-4945
After a little practice this process can, of course, be abbreviated.
It must be remembered that in most printed tables the
decimal points given in the specimen are not inserted. The
student should, however, always supply them when he copies
the numbers and logarithms in calculations.
Five-figure logarithms are generally arranged upon a similar
plan — as in the following specimen in which the decimal
points are omitted.
1
2
3
4
I
2
3
4
5
6
7
8
9
31
49136
49276
49415
49554
49693
14
28
42
56
70
83
97
111
125
COMMON LOGARITHMS 339
Example 1. — Find log 3-146 to five places. Here we
have
log 3-14 = 0-49693
log 3-146 = 0-49776
83 being added to the last two figures.
Example 2. — Find log 3-1065. This time we have to
determine the difference corresponding to the last two figures,
65. If the number had been 3-106 we should have added the
number in the 6th difference column (83) ; if it had been
3-107 the number in the 7th difference column (97). We
must, therefore, actually add 83 + | (97 - 83) = 90. Thus
log 3-10 = 0-49136
log 3-1065 = 0-49226
§ 3. Antilogarithms. — These tables of logarithms to n places
can also be used to find to n significant figures the number or
antilogarithm corresponding to a given logarithm. Neverthe-
less tables of logarithms to four or five places are generally
accompanied by separate tables of antilogarithms. In such a
table the " argument " is the mantissa of the logarithms, and
is therefore generally printed as a decimal less than 1. The
numbers in the table are numbers between 1 and 10 and must
be read as if they had a decimal point after the foremost digit.
When the number corresponding to the mantissa of the
logarithm has been determined it must be multiplied by a
power of 10 whose index is the characteristic of the logarithm.
Thus if the given logarithm is 3-427 the table must be
entered with the mantissa -427. The antilogarithm or
number of which this is the logarithm is (to three places)
2*673. The number whose logarithm is 3-427 is, therefore,
2-673 X 10^ or 2673. Similarly the antilogarithm of 3-427
is 2-673 X 10-3 or 0-002673.
As far as concerns the difference columns, etc., the table of
antilogarithms is arranged like the corresponding table of log-
arithms and is used in the same way.
Example. — Find from a table of five-figure antilogarithms
the number whose logarithm is 2-31268.
The antilogarithm of -312 is found in the row beginning -31
and in the column headed 2. It is 2-0512. The next two
figures lie between 60 and 70. The numbers in the difference-
column headed 6 and 7 are 28 and 33 and their difference is 5.
22*
340 ALGEBRA
The difference for 68 (= 60 + ^ of 10) is therefore 28 + i of
5, or 32. Thus we have : —
antilog 0-31268 = 2-0512 + 0-0032
= 2-0544
.-. antilog 2-31268 = 2-0544 x IO-2
= 0-020544
[Ex. LVII should now be worked.]
CHAPTER XXXIV.
THE LOGAKITHMIC AND EXPONENTIAL FUNCTIONS.
(Ch. XXIX., § 7; Ex. LVIII.)
§ 1. Summary of Previous Work. — It will now be well to
restate the theory of logarithms in a form which contains
no reference to growth-curves. Choose any positive whole
number p, greater than unity, and let k = 1/p. Choose also
any positive number a, integral or fractional, and let h = ^a.
Now construct (1) the complete arithmetic sequence whose
starting term is zero and common difference k ; (2) the com-
plete geometric sequence whose starting term is 1 and common
ratio h. Arrange these sequences so that the starting and other
corresponding terms are against one another : —
. . . , - 3k, - 2k, - k, 0, + k, + 2k, . . . pk, . . .
[I]
. . . , h-\ h-\ h-\ 1, h\ h\ . . . h", . . .
Then each term of the arithmetic sequence is defined as the
logarithm of the corresponding term in the geometric sequence,
and a, the term in the geometric sequence which corresponds to
1 in the arithmetic sequence, is defined as the base of the log-
arithms.
Let mk and nk be any two terms of the arithmetic sequence
— so that m and n are integers, positive or negative. Then, by
the definition, mk = logji'"., nk = logji/". By the same defini-
tion {m ±n)k = \ogJV^-'\ Then since
(m ±n)k = mk ± nk, h"'+'' = h'" x h" and h""-" = /^'"//i"
it follows that the sum of the logarithms of the two numbers
h"' and h" is equal to the logarithm of their product, while
the difference of their logarithms is the logarithm of their
quotient. Similarly, if r be any integer, rm.^ = log„^"'', that
is mk X r = log,,{h"'Y. Hence the logarithm of the rth
341
342 ALGEBRA
power of the number /i"* is r times the logarithm of the
number. If w/r is also an integer, mkjr and /i"""" are cor-
responding terms of the two series, so that wfe/r = logji^'"';
that is the logarithm of the rth root of the number li^ is 1/rth
of the logarithm of the number. In this way the fundamental
properties of the logarithm are established.
§ 2. The Logarithmic Function. — Now by making p ex-
ceedingly large it is possible to make h so small and h so
little different from unity that the intervals between the terms
in any part of either sequence become smaller than any speci-
fied number. That is to say, either of the sequences can be
made to include, to any given closeness of approximation,
any number that anyone chooses to mention, while the term
1 in the arithmetic sequence still has a as its corresponding
term in the geometric sequence. Thus if a given number x is
practically identical with a term of the geometric sequence,
we can always find a term y corresponding to it in the arith-
metic sequence. In other words, given any value of x there
is always a y such that y = log„ x.
In this way we reach the idea of a new function of x — the
logarithmic function — whose properties are expressed by the
relations —
log„ x^ x^ = log„ a?! + log« x^
log„ {x^\x^ = log„ x^ - log. x^
log„ a;" = w log„ X
log« </ ^ = - loga ^
n
All these properties may be briefly summed up in the
following statements, (i) If a variable y depends on a vari-
able X in such a way that to two values of x with a fixed ratio
there always correspond two values of y with a fixed differ-
ence, and if 2/ = when a; = 1, then y is said to be the
logarithm of x. (ii) If, when the ratio between two values
of X is a, the difference between the corresponding values
of 2/ is 1, a is said to be the' base of the logarithms.
§ 3. The Antilogarithmic Function. — In the foregoing argu-
ment the term of the geometric sequence has been taken as the
independent variable. Instead of doing so we may regard the
term of the arithmetic sequence as the independent variable
— that is, we may put x = mk and y = /t"*. In that case we
can, of course, write y = antilog„ x. The function expressed
LOGARITHMIC AND EXPONENTIAL FUNCTIONS 343
by the symbolism antilogy x is obviously the inverse of the
logarithmic function with the same base. It may be called
the antilogarithmic function.
The properties of the antilogarithmic function can be found
by the method already applied to the logarithmic function.
Let mk and nk be any two values of x (just as in ^ 2 they
were values of y). Then the corresponding values of y are
h"" and ^" ; that is
h"^ = antilog„(mA;)
/t" = antilog„(MA;)
Now we have
antiloga (mA;) x s,nti\og^ {nk) = h"" x h""
= /i- + «
= antilogy (mfc + nk)
That is to say, if any two values of x are taken, x^ and rCg,
antilogaa^i x antilognCCg — antilog„(a;i + x^)
In the same way it can be shown that
antilogy a^i -^ antilogy iCg = antilogy (a^^ - x^)
§ 4. Fractional Indices. — The most notable thing about
the properties of the antilogarithmic function is that the
values of x — no matter whether they are integral or fractional,
positive or negative — are combined in multiplication and
division exactly as if they were indices or " exponents " of
powers of the' same number. For this reason the function
is often called the "exponential function," the symbolism
y = exp„a; being used instead oi y — antilogy x.
Now such symbolism as a""^ x a'^a = a^i+*2is both much
more easily written and has become, by use, much more
familiar than either
antilog„aJi x antilog^ajg = antilog„(iCi + x^
or exp„iCj X exp^ojg = exp„(£(:;i + x^
and is also more easily expressed in words. It would for
these reasons alone be worth while to inquire whether the
antilogarithmic (or exponential) function of x in which values
of X are certainly combined as if they were indices cannot be
written as if they were indices. The following argument
shows that they may be so written.
If from the arithmetic sequence on p. 341 each ^th term be
picked out, starting from and counting both ways, we obtain
the new arithmetic sequence, ... - 4, - 3, - 2, - 1, 0,
+ 1, + 2, 4- 3, + 4, . . . while the corresponding terms of
344 ALGEBRA
the original geometric sequence form the new geometric se-
quence . . . a~*, a~^, a~^, a~\ 1, a\ a^, a^, a* . . .
Remembering that y represents terms of the geometric
and X terms of the arithmetic sequence, we see at once that /or
these terms the function y = antilog„a; ory = exp„£c can be
expressed in the alternative form y = a''. It is, therefore,
a natural suggestion that the symbolism y = a" should be
used as an alternative way of expressing the relation between
all values of x and the corresponding values of y when y is
the antilogarithm of x.
If we adopt this plan we must, of course, abandon the old
definition of an index. The expressions a '^'^ and a ~ ^-^ cannot
mean that unity is repeatedly multiplied by a 3 "7 times in the
first case and divided by a 8 '2 times in the second, for these
operations are impossible. But such a consideration need
not prevent us from adopting the new symbolism if we wish
to do so. We can re-define W" as meaning " the antilogarithm
of X to the base a.". When a; is a whole number it will
have, in addition to this meaning, the old meaning of re-
peated multiplication or division. When it is not a whole
number it will have the new meaning only. There could, in
fact, be only one fataL objection to the proposed practice. We
could not adopt it if antilogarithms did not combine with one
another in accordance with the laws of indices. Since they
do so combine there is no reason why the symbolism should
not be used whenever it proves convenient.
The effect of these arguments is best understood by means
of examples. On p. 329 we read that log^.g 1*1 = 0-36 and
logpg 1-4 = 1-28. From these results, or from the table on
p. 332, we derive statements which can be expressed in three
equivalent ways : —
antUogj.g 0-36 = I'l antilog^.g 1-28 = 1-4
expi.g 0-36 = 1-1 expi.3 1-28 = 1-4
(1.3)0.3(5 _ 1.1 (1 •3)1-28 = 1-4
Employing the last form as the most convenient, we have
1-4 X 1-1 = (l-3)i-28 X (l-3)«-36
= (1 •3)1-64
= 1*54 by either of the tables.
Similarly 1-4 - 1-1 = (l-3)i-28 - (l-3)0'36
= (l-3)o-92
= 1^27 by either of the tables.
LOGARITHMIC AND EXPONENTIAL FUNCTIONS 345
Again (1-1)3 _ {(l.3)0-36p
= (l-3)i-i8
= 1-36
Finally ^ (1*4) = ^Hl'Sy''}
= (l-3)o-32
= 1-09
All these (approximate) results can be confirmed by arith-
metic.
§ 5. Graphs of the Functions. — It will now be useful to re-
turn to the growth-curve. As we saw long ago the ordinates of
this curve may be regarded as " numbers " or " antilogarithms "
and the corresponding abscissae as "logarithms". Given a
table of logarithms or of antilogarithms the curve might be
drawn without any reference to the growth -problems which
actually suggested it. In other words, it is the graph of the
antilogarithmic or exponential function. It may, therefore,
conveniently be called the antilogarithmic or exponential
curve. The second of these names is the one generally
employed.
When the growth -curve was first drawn it could not be
said to be the graph corresponding to any known formula.
Some of its points were given by the formula y = a", but only
those where x is integral. But by § 4 we can now say that
every point on the curve is given by the formula y = antilog^rc,
or y = exp„a^ or (lastly) y = a''. The last, as the most
concise and familiar, will henceforward be taken as the
standard formula of the curve.
To obtain the graph of the logarithmic function it is suffi-
cient (as in the case of all functions of which one is the in-
verse of the other) to turn the exponential curve (1) through
180° about the ic-axis, (2) through 90° anticlockwise about an
axis through the origin perpendicular to its plane.
[Ex. LVIII should now be worked.]
CHAPTER XXXV.
NOMINAL AND EFFECTIVE GROWTH-FACTOES.
(Oh. XXIX., § 8; Ex. LIX.)
§1. '' Nominai" aiid ''Effective'' Bates of Growth. — In
this chapter we are to study in detail one of the most im-
portant differences between the modes of uniform growth
contrasted in ch. li.
Suppose it to be known that a quantity, of present
magnitude Qq, is growing in accordance with the first law.
Suppose, also, that its changing magnitude has been observed
during 1/wth of the time-unit and that during this interval
each unit of magnitude changes by an amount a. Then the
growth-difference for a complete time-unit will, of course, be
a' X n. Putting a = a' x n the formula for the magnitude
of the quantity at time t becomes
Q = Qo (1 -f at)
Conversely, if we are given that the growth-difference of unit
magnitude during unit time is a, then we know that the
growth of a unit during any Ijnih. of this term will be ajn and
the change in the magnitude of the whole quantity QQa/n.
In the case of the second form of uniform continuous
growth there is no such simple connexion between the changes
of magnitude during the whole time-unit and one of its frac-
tions. The difference will be best brought out by an
example. Make the assumption that the population of a
certain town increases with a constant growth-factor, and
that on 1 January this year it was 40,000. Suppose, also,
that at the end of the first quarter the population was re-
counted and found to be 40,260. Then there has been
during this quarter an actual increase of 260 or 260/40 =6*5
per thousand of the population on 1 January. If the same
actual increase occurred in each quarter the growth-difference
346
NOMINAL AND EFFECTIVE GROWTH FACTORS 347
for the year would be 6-5 x 4 = 26 persons per thousand.
We may say, then, that during the first quarter the observed
increase was at the rate of 26 per thousand per annum.
But this number does not measure the actual yearly in-
crease in the population. An increase from 40,000 to
40,260 in one quarter means a quarterly growth-factor of
1-0065 and therefore an annual growth-factor ^ of (1*0065)'^ =
1-02625. Thus the actual annual increase would be
26-25 per thousand.
These facts may be shortly expressed by the statement that
while the nominal rate of increase during this quarter was
26 per thousand per annum, the effective rate of increase was
26-25 per thousand per annum.
It is obvious that in an actual town accidental causes —
immigration, emigration, etc. — are likely to affect the popu-
lation seriously in so long a time as a quarter. If we wanted
to know the natural rate of change of the original 40,000 it
would be necessary to consider the increase during a shorter
period. Suppose, then, that during the first fortnight of the
year the population increases from 40,000 to 40,040. This
is an increase of 1 per thousand for the fortnight or an in-
crease at the nominal rate of 26 per thousand per annum.
But though the nominal rate of increase is the same as before
the effective rate is different. For a fortnightly growth-factor
1-001 implies an annual growth-factor (1-001)2« = 1-02633.
That is, the effective rate of increase is 26*33 per thousand
per annum.
In general, if i is the observed increase per unit magnitude
during the first l/nth of the unit of time, then the growth-
factor for the complete unit of time is (1 -f- '?)". The nominal
increase of unit magnitude for the unit of time is j = ni.
The effective increase is
(1 -f- iy - 1 = (1 + j/ny - 1
§ 2. Financial Applications. — These ideas have important
financial applications. Suppose that a sum of £1 is deposited
in a bank which gives interest at the rate of j per pound per
annum. Suppose, also, that the interest is paid once a year
and is on each occasion of payment added to the deposited
principal. Then the original £1 will increase by annual
*' jerks " with a growth-factor of (1 + j). Now suppose that
the interest is paid quarterly. That is, suppose that at the
^ Calculated by means of seveu-figure logarithms.
348
ALGEBRA
end of the first quarter the banker adds ^ to your £1 and
gives interest during the second quarter on the whole sum
1 + j/4:. Then in this case your principal will increase by
quarterly " jerks " showing a growth-factor of 1 + j/i for the
quarter and (1 + j/4)^ for the year. Thus while the nominal
rate of interest is j per pound per annum the effective rate is
(1 + y/4:)^ - 1 per pound per annum. In general, if interest at
the nominal rate of j per pound per annum is added to the
principal n times a year the effective rate of interest is
i = (1 + y/w)'; - 1.
§ 3. Gi'aphic Determination of Bates. — It is easy to exhibit
graphically the effective rate which corresponds to a given
nominal rate of growth. Let AP (fig. 83) be any exponential
curve, and AO the ordinate of unit magnitude. Let j be the
nominal rate of increase per unit time, the actual increase
being jjn during the first Ijnih. of a time-unit. Find the
P
./
^^^-^^
C
P^^
v'
A^.^\
B'
B
Q
Fig. 83.
ordinate P'Q' whose height is 1 -f y, and draw AB'B
horizontally. In P'B' take By = 1/w of P'B' and draw 'p''p
horizontally to meet the curve in _p. Then
l^q = p'Q! = (1 + JIn)
Draw the secant A^C and let it cross the horizontal through
P' in C. Through C draw the ordinate PQ. Then in the
similar triangles, kph, ACB, CB = P'B' = ^B' x n = pb x n.
Hence AB = Ab x n, and (by the fundamental property of
the exponential curve)
PQ = tar = (1 +JM"
NOMINAL AND EFFECTIVE GROWTH-FACTORS 349
Thus PB = (1 + jjnY - 1 is the effective rate of increase
which corresponds to a nominal rate of increase P'B'.
§ 4. The Case of Continuous Increase. — The foregoing argu-
ment applies equally well to a continuously growing quantity
(e.g. rising temperature) whose increase is measured at the end
of the first nth. of. the time-unit, and a quantity (such as the
principal deposited in a bank) which actually acquires its
increments only at intervals of 1/wth of the time-unit. This
fact enables us to understand how the distinction between the
nominal and the efifective rate of interest can still be used
when growth by " jerks " passes into continuous growth.
Many important financial calculations depend upon the sup-
position that interest at the rate of j per pound per annum is
added to the principal as fast as it is earned. In such cases
the growth of the principal is represented graphically not by
a series of unequal steps (of which khp, fig. 83, is the first)
but by the exponential curve itself, every ordinate represent-
ing the amount of the principal at some moment. The transi-
tion from the case in which the interest is *' converted into
principal " at n distinct and equidistant times in the year to
the case in which the conversion takes place every moment
may be supposed to be effected by making n become larger
endlessly. Accompanying the increase in n the point p'
(fig. 83) will approach B' and the point y move along the
curve towards A. At the same time the secant Aj^C will
approach and eventually become indistinguishable from the
tangent AC (fig. 84), but it will never pass beyond it. Thus
when n is so large that the interest may be considered as
converted into principal every moment, the effective rate of
interest, PB (fig. 84), is obtained from the nominal rate P'B'
by a construction which differs from that of fig. 83 only by
the substitution of the tangent at A for the secant AjpC. We
may express the result symbolically by the statement that, if
FQ' (fig. 84) = 1 -1- j, then PQ = (1 + jinf when w = oo .
§ 5. The Meaning and Use of " e ". — When ?t is a definite
number the value of the effective rate which corresponds to a
given nominal rate can, of course, be calculated. It is simply
a question of computing (1 + jjnY by means of a table of
logarithms. But, when all we can say about n is that it is
endlessly great, computation in accordance with this formula
becomes impossible. Our last task is to seek some way of
calculating the effective rate in such cases.
350
ALGEBRA
In fig. 84 let FB' = ;* as before, and let R'D' = 1 ; that
is, let R'S' = 2. Then by § 4, if n = oo , the ordinate
RS = (1 + Ijuf and the ordinate PQ = (1 + jjny. Now
in the similar triangles AED, ACB, ED = R'D' = 1 and
CB = P'B' = 3. Hence AB = AD x j and PQ = (RS)^
That is to say
(1 + jlnf = {(1 + llnfY when w = 00 . (i)
It is most important to note here that the height of RS
must always be the same no matter what exponential curve
is used to find it. To assure yourself of this return to fig. 83
and suppose that P'B' = 1, and that ^'B' is 1/w, n being, of
course, a definite number. Then since PQ = (1 4- 1/w)"
where w is a definite number, its value is quite independent
of the growth-factor of the curve. Since this is true however
large n is, so long as it is a definite number, it must be true
when n has become so large that AC is henceforward indis-
tinguishable from the tangent. That is, it is true when n is
endlessly large or n = cd. It follows that although RS (fig.
84) would appear in different positions in different exponential
curves its height has the same value in all curves. In other
words, when n = 00 (1 + Ijiif has a definite numerical
value.
It is customary for conciseness to symbolize this value by
NOMINAL AND EFFECTIVE GROWTH-FACTORS 351
the letter e. The result (i) above can then be expressed in
the simple form
(l+i/^r=/ . . . (ii)
while the effective rate of increase in the case when increase
takes place every moment at a nominal rate of j per unit
time can be written i = e^ - 1. The value of e will be found, by
careful drawing and measurement, to be approximately 2 "72.
Algebraic methods, to be described later, show that it is
2-71828 . . . Like ir it is an endless decimal which has
been calculated to several hundreds of places.
[Ex. LIX may now be worked].
EXERCISES SUPPLEMENTARY
TO SECTIONS II AND III.
T. 23
SUPPLEMENTAEY EXEECISES.
■*^* The numbers in ordinary type refer to the pages of Exercises
in Algebra, Part T ; the numbers in heavy type to the pages of this
book.
EXERCISE PAGES
LX. The Use of Logarithms in Trigonometry 303, 355
LXI. Polar Co-ordinates 308, 356
LXII. Some Important Trigonometrical Identities 310, 359
LXIII. The Parabolic Function .... 317, 360
LXIV. Implicit Quadratic Functions (I) . . 320^ 361
LXV. Implicit Quadratic Functions (II) . . 325, 362
LXVI. Mean Position . . . . .. 331, 366
LXVII. Root-mean-square Deviation . . . 337,369
LXVIII. The Binomial Theorem .... 340, 372
LXIX. The Generalization of Wallis's Law . . 349, 374
CHAPTER XXXVI.
THE PROGEAMME OF EXERCISES LX-LXV.
§ 1. The Contents of the Section. — The exercises of this
section are supplementary in the sense that they introduce no
new algebraic notation and no new fundamental idea ; their
contents are in all cases fairly straightforward developments
of the subject-matter of previous exercises. It is chiefly for
this reason that the preliminary discussions which have
formed the bulk of previous sections of this book are now
discontinued, and the student left to work his way through
the exercises with the aid of a few " Notes". There is, how-
ever, another reason for this change in procedure. We have
now to deal with a student of some maturity who should soon
be called upon to acquire by unaided reading the information
he needs in order to do his work. The practice adopted in
this supplementary section aims at preparing him to exercise
(in Part II) this fuller responsibility for his own progress.
The nature of the section makes any division into subsec-
tions rather arbitrary. There is, however, a certain contin-
uity in Exs. LX-LXV, due to the fact that they all deal
directly or indirectly with the properties and uses of the
trigonometrical ratios. Similarly, Exs. LXVI-LXIX deal
more or less with applications and further developments of
Wallis's Law. There will, therefore, be a certain convenience
in considering these two groups of exercises in separate
chapters.
§ 2. The Use of Logarithms in Trigonometry (Ex. LX). —
The first exercise exemplifies the use of logarithms in evaluat-
ing trigonometrical formulae. As we have seen, the desire to
make these evaluations a less burdensome business was the
motive which led Napier to the invention of his "artificial
numbers ". One of the evil consequences of teaching trigono-
metry as a distinct subject has been that an excessive amount
355 23 *
356 ALGEBRA
of importance has been attached to these computations. The
single short exercise here given to them probably represents
the amount of consideration to which they are entitled in a
general, as opposed to a technical, course of mathematics. It
will be observed that no attempt is made to classify the dif-
ferent problems formally into " cases ".
Division A is intended simply to give practice in using
tables of logarithms of the functions. In connexion with these
examples the teacher is recommended to follow the growing
custom of using logarithms in which the characteristic has its
proper value instead of being increased by 10. For the ordin-
ary pupil and for ordinary purposes the intrusive 10 of the
" tabular logarithm " is simply an occasion of stumbling and
has nothing whatever to commend it. An incidental diffi-
culty in the use of logarithms of the trigonometrical functions
is that of dealing with negative values. Nos. 5, 6 are in-
tended specially to give practice in overcoming it, the proper
procedure in these cases being explained in a note prefixed
to No. o.
The examples of division B cover familiar ground. The
identities of Nos. IO-14 are developments of an argument
begun in Ex. LX and here to be repeated and brought to its
conclusion. The note before No. 16 suggests that the
student shall be contented to commit to memory only the
expression for calculating cos a in terms of the sides of the
triangle, this particular one of the three allied identities being
chosen simply because it is the easiest to remember. The
artificial questions which required the candidate to be ready
with all three in order to effect a solution by means of
logarithms selected by the examiner have fortunately disap-
peared from the modern examination paper.
Lastly, in division C, a few examples are given on the
formula usually called by continental writers •' Mollweide's
equation ". The geometrical argument suggested by the con-
struction in Exercises, fig. 51, is much easier for the average
student than the algebraic proof current in English textbooks.
Familiarity with the use of MoUweide's identity should be re-
garded as a luxury rather than a necessity but will probably
continue to be expected of candidates in examinations.
§ 3. Polar Co-ordinates (Ex. LXI). — The device of fixing
the position of a point by means of an angular measurement
and a distance has often been illustrated in the exercises from
THE PROGRAMME OF EXERCISES LX-LXV 357
Ex. II onwards. The present short exercise is intended to
show the power of the method as a means of geometrical de-
scription, and to prepare the student to use it freely in the
important arguments of Ex. LXV and in many later exercises.
The path of Halley's comet, No. 2, is really a very flat
ellipse, but the given formula, as is shown in Ex. LXIII, No.
13, describes the portion near the sun as a parabola. In
drawing the graphs in Nos. 3-9 different members of the class
should assign different values to the constants. Fig. 85
shows the limacon, No. 4, when a = 2 and 6 = 1. This figure
degenerates into the cardioid, No. 3, when Z> == 2 also. Fig.
86 shows the " three-leaved rose " described by the polar
Fig. 85. Fig. 86.
formula r = a sin 3a, No. 5. The one corresponding to
r = a cos 3a is of the same form rotated anticlockwise
through 90°. In drawing these graphs, as well as those
of No. 6, it must be remembered that the students have as
yet learnt to measure angles only up to 360°.' Thus the
greatest value which can be assumed by a in No. 5 is 120°.
It will be found, however, that this range of the variable is
sufficient to give the whole of the curve. For example, in
plotting r = a. sin 3a as a passes through the value 60°, 3a
passes through the value 180° and the radius vector begins to
be negative. Thus between a = 60° and a = 120° the lower
loop of the " rose " will be traced. The loop in the second
quadrant will be obtained by giving to a negative values from
0° to - 60°. As a passes through the values from - 60" to
~ 120°, 3a will move from - 180° to - 360°. Thus the
values of sin 3a and therefore of r will once more be positive
358
ALGEBRA
and the lower loop will be traced out a second time. There
is, of course, no objection to using this graph as a means of
introducing the idea of angles greater than 360°, but, on the
whole, it is better to delay this important step forward until
(as in Section YII) a more substantial justification for it ap-
pears.
Fig. 87 shows the conchoid, No. p, with a = 1, Z? -= 2. If
Z? = - 2 the figure will be reversed from right to left ; if
b = a the loop will degenerate into a point.
Fig. 87.
Fig.
No. 10, which illustrates the extraordinary movements of
which the Australian boomerang is capable, is likely to prove
very interesting to boys, if not to their sisters. The data
have been obtained by copying (roughly) a figure given by
Mr. G. T. Walker in his elaborate paper on the subject in the
Philosophical Transactions, vol. 190, A. Fig. 88 gives the
" bird's-eye " view of the flight of the boomerang ; fig. 89
shows how its movement would appear if watched by an ob-
server at a considerable distance on a line bisecting AJ at
right angles. The vertical scale is, of course, exaggerated.
THE PROGRAMME OF EXERCISES LX-LXV 359
To obtain fig. 89 perpendiculars Bb, Gc, Dd, etc., are drawn
to A J in fig. 88 and the distances Ab, be, etc., repeated along
the base line of fig. 89. At the points a, b, c, etc., lines
A'a, B'6, C'c, etc., are drawn to represent to scale the values
of h given in the table. A smooth curve is drawn through
A'B'C ....
§ 4. The Sum and Difference Formulce (Ex. LXII). — In
the discussion which preceded Ex. L (ch. xxviii., B, p. 295)
it was necessary to investigate a formula for calculating
sin (a + /S) when the sines and cosines of a and ^ are known.
The simple proof there given was adequate to the purpose,
but applies only in the cases where a + ^ is less than 90°.
Ex. LXII takes up the whole subject of the " sum and difference
formulae " for two angles, including all cases in which their
u tda paon
Fig. 89.
sum is not greater than 360". The method followed starts
from the notions of a " vector " and its resolution into " com-
ponents " at right angles — notions with which the student
made acquaintance as early as Ex. XIX. The treatment is
straightforward and needs no elucidation. The teacher will
observe that the mode of attack chosen makes it simpler
to deal with the factorization-formulae for cos a + cos /3,
etc., before the identities more specially known as the sum
and difference formulae. It will be noted, also, that only
sufficient examples are given to familiarize the student with
the results themselves and with the few transformations of
constant practical importance which they are needed to effect.
The formulae will shortly be put to an important use in Ex.
LXV.
§ 5. Quadratic Functions. — Exs. LXIII-LXV may be con-
360 ALGEBRA
veniently considered together since they all deal with functions
of the second degree and may be regarded as continuing the
discussions of Exs. XLII-XLVI.
Ex. LXIII is a short exercise dealing entirely with the
parabolic function. Division A illustrates a type of elimina-
tion-process which has often to be used in dynamics and
physics ; division B explores in a simple way the connexion
between the Cartesian and polar formulae which describe a
parabola.
In No. I the substitution of dll2 for t gives the formulae
h = 6d - d^d .... A
= 81 - J(27 - dy . . . B
d = 27 ± 3 7(81 - h) . . . C
From B we see that the greatest height reached above the
point of projection is 81 feet, or, above the ground, 86 feet.
Substitution of 30 (i.e. 35 - 5) for hinC gives d = 48f feet
and t = 48y/12 = 4^ seconds. These are respectively the
values of the horizontal range and the time of flight — the
former of these two terms meaning the distance between the
vertical through the hand of the thrower and the vertical
through the point where the ball hits the roof. For the second
part of the question we must substitute -14-5= -19
for h in C, whence d = bl feet and ^ = 4f seconds.
In No. 2 the formula for the line of greatest slope is evi-
dently
h = ^\d-5 . . . D
Substituting this value for /t in A we obtain the quadratic
equation
10^2 _ 537^ - 450 =
which gives d = 54-52. From D we now have h = - 3*18.
These are the required co-ordinates of the point where the
ball hits the ground. Let E be the range measured along the
slope of the hill. Then we have
R2 = d^ + h^
whence R = d{l + y/q^xt)
to a suflicient degree of accuracy. In the present case this
argument gives the result E = 54-55 ; that is, the slope of the
hill adds less than half an inch to the range of the ball.
When the ball is thrown downhill (No. 3) we must re-
place D by the formula
h= -^\d- 5 . . . E
THE PROGRAMME OP EXERCISES LX-LXV 361
which leads to the quadratic equation
lOd^ - 54:3d - 450 =
with 55*12 feet as the value of d. In this case R, determined
as before, is 55*15 ft.
In all these calculations only the positive values of the
roots have been retained. There is no difficulty in seeing that
the negative roots do not apply to the problems proposed.
In No. 13 the two formulae are seen to be equivalent by
the consideration that values of a differing by 180" would make
their denominators identical. In order, therefore, that the
same value of a may lead to the same point in each case the
two fractions must be prefixed by opposite signs. If p is posi-
tive the parabola will be " head down " ; if negative, " head
up ".
In all previous exercises functions of x have been " ex-
plicit " ; that is, they have been expressed by formulae in
which the subject is y. Exs. LXIV, LXV introduce the study
of ' ' implicit " functions of x and y and of the corresponding
graphs. The observations made in ch. iv., § 9, are here par-
ticularly relevant. Our business in these exercises is not to
study " the equations of the conic sections," but to use our
knowledge of these graphic forms to illuminate the properties
of implicit quadratic functions of x and y.
Throughout Ex. LXIV constant use is made of the prin-
ciple established in Ex. XXXIX, p. 218, namely, that if a
graph is moved horizontally through a distance a and verti-
cally through a distance b its formula must be amended by
the substitution oi x - a ior x and y - b ior y. Thus in
No. 2 the function of which the circle in its new position is
the graphic expression is
{x + 3)2 + {y - 4)2 = 16.
Conversely, since the function in No. 4 (iv) can be written
(x + If + {y - 4)2 = 64
its graph is a circle of radius 8 whose centre is at the point
(-7, +4). The same principle and its converse are, in
Nos. 22-4, applied to the eUipse, and the corresponding
functions.
Fig. 90 illustrates No. 12, the line PQ being the graph of
the equation
4a7 - 6^ - 13 = 0.
Since the points P and Q are on both circles their co-ordi-
nates must satisfy both the equations
362
ALGEBRA
x^ + 2/2 - 25 = and x^ + i/^ - 4a; + 6i/ - 12 =
and will therefore satisfy an equation in which the left-hand
side is the difference between the left-hand sides of these two.
That is, it will satisfy
do; - 61/ - 13 = 0.
It follows that the line which is the graph of this equation is
the "radical axis " or line through the intersections of the
two circles.
Division B is very important in view of later applications,
but requires little comment. All the examples should be
worked, special attention being given to Nos. 15-19. The
ellipse of No. 19 is, of course, simply that of No. 16 rotated
through a right angle about the origin.
Previous exercises have given numerous illustrations of the
fact that a single graph in different positions may correspond
to many different functions. Hitherto all the different posi-
tions of the graph have been derivable from one another by
vertical and horizontal movements and our repertory of func-
tions studied has been limited by this condition. In Ex. LXV
the restriction is removed, and we enter upon the study of
functions which correspond to the now familiar graphic forms
THE PROGRAMME OF EXERCISES LX-LXV 363
after they have been not only displaced vertically and hori-
zontally from their standard positions but in addition
rotated about the origin. In the course of the argument it
is shown that every possible function of the second degree
corresponds to one of the standard graphic forms in one of
the positions which these movements make possible. Thus
the study of these transformations gives occasion for an ex-
haustive review of quadratic functions, explicit and implicit.
It is well, however, to remind the student that the method
here employed does not possess universal validity ; we have
already seen (p. 288) that all cubic functions do not correspond
to the graph oi y = ax^ in the various positions which it may
be made to assume.
The geometrical principle upon which these transforma-
tions depend is explained in the note at the beginning of the
exercise and should be thoroughly understood. In No. I we
may suppose any (i.e. every) point of the graph to be joined
with the origin. It is then obvious that if one of the joining
lines is rotated through an angle a all will be, so that the
relation between the old and the new co-ordinates established
in the note will hold good for every point of the graph.
No. 2 gives an interesting and striking verification of this
conclusion.
The first important application of the new principle of trans-
formation occurs in No. 5. Here, substituting {x - y)/ J^
for X and {y + x)/ J2 for y in the standard rectangular
hyperbolic function, we obtain x^ - y^ = a^ as the formula
for a rectangular hyperbola in the position lettered P' in
fig. 91. The next few examples are intended to bring out
the striking and important analogies between this hyperbolic
function and the circular function x'^ + y^ = a^ which are to
reach their logical conclusion in the theory of the hyperbolic
sine and cosine in Section VII. Fig. 91 is the diagram
which the student is, in these examples, instructed to build
up. The curves drawn with firm lines are the circle and the
rectangular hyperbola which corresponds to it ; the broken
curves represent the ellipses which can be derived from the
circl9 and the hyperbolas derived in the same way (No. 12)
from the rectangular hyperbola.
The examples of division B give exercise in finding the
functions which correspond to the parabola, ellipse and hyper-
bola after the specified displacements, by translation and
364
ALGEBRA
rotation, from their standard positions. In No. 22 the
original relation is
in accordance with the note before No. 13, and the cosine and
sine of the angle whose tangent is J are respectively 3/^^10
and 1/^10. When the hyperbola is rotated anticlockwise
-•-.
\v
'/ y
vt
l/y
\
[/
z.
X' i
\s ^
if
4
\\ *^
,.-.—
' / >
V \^
y / 1
-w/
X
— '
y / /
"•--
--»'
X /
/ /
/ /
X t
X /
X '
X /
^ /
Y'
y
y
M
W
\
V
N
Fig. 91.
through this angle from its standard position it corresponds
to the relation
(3a; + yf _ (3?/ - xf ^ 25
10 10 4
that is
16a;2 ^ 24:xy - 16y^ = 125.
No. 26 is an example of the converse problem. If the
graph is rotated clockwise through an angle a we have (as in
No. 1) that it now corresponds to the implicit function
3(x cos a - y sin a)'^ + 8(0: cos a - y sin a){y cos a + ic sin a)
- 3{y cos a + X sin a)^ = 10
THE PROGRAMME OF EXERCISES LX-LXV 365
which can be thrown into the form
(x^ - y^){S cos^ a + 8 cos a sin a - 3 sin^ a)
+ xy {8 (cos^ a - sin^ a) - 12 cos a sin a} = 10.
But if the axis of the hyperbola is now coincident with the
ic-axis the term involving xy should disappear. The condi-
tion for this disappearance is
4(cos'^ a - sin^ a) = 3 X 2 cos a sin a
or
4 cos 2a = 3 sin 2a
whence tan 2a = 4/3 or 2a = 53° nearly.
In order to convert the implicit function of No. 30 into an
explicit function of x we express it as a quadratic equation
my:—
67/2 _ {x + l)y - 2(x^ - 1) =
of which the solution is
(^ + 7) + {7x + 1)
^ 12
Taking in succession the plus and minus we have that the
original function is equivalent to the two linear functions
2x - 3y + 2 == and x + 2y - 1 = 0.
CHAPTER XXXVII.
THE PKOGRAMME OF EXERCISES LXVI LXJX.
§ 1. Mean Position (Ex. LXVI). — Division A of this ex-
ercise may be regarded as supplenienting, by the introduction
of directed numbers, part of the argument of Ex. XXVI, D
(Statistics). In that group of examples the student learnt to
estimate the "mean deviation " of a series of measurements
or other numbers from the " median " or middle term of the
series. When the deviations of the individual terms of the
series are measured, not by signless, but by directed, numbers
their average may conveniently be called their " mean posi-
tion ". It is important to observe that whereas the mean
deviation of the series will be different if measured from
different terms — the mean deviation from the median being
its least value — the mean position is independent of the
point from which the individual deviations are estimated.
This property is meant to be brought out in Nos, 2, 3, but is
so essential that it should be emphasized by the teacher.
The following may be given as a formal proof. Let the n
numbers be, or be represented by, n points on a line, their
distances from the point O, from which the deviations are to
be reckoned, being £Cj, a^g, x^, . . . x^. In general some of
these numbers will be positive and some negative. By
definition the mean position of the numbers or terms with
respect to O is
{x^ + x^ -\- x^ + . . . + x,,)ln or (S x)ln
Now let the mean position be estimated with respect to a
point O' distant a from O. Then, whether a is positive or
negative, the distances of the individual points from O' will
be x^ - a, X2 - a, x^ - a, etc., and the mean position
{^x - na)ln = {^x)ln - a
Thus, though the meau position is represented by a different
number it is still (or ia gtill represented by) the sapa© point as
366
THE PROGRAMME OF EXERCISES LXVI-LXIX 367
before. For if (as in the note after No. 1) we put x for
{^x)ln, the distance of the mean position from O, the point
whose distance from O' is ;^ - a is obviously identical with
it. The practical consequence is that, in finding mean posi-
tions or " centroids," we may count the individual deviations
from any point or value which seems convenient.
These considerations show that the centroid or point of
mean position may be regarded as essentially a statistical
idea which has geometrical as well as arithmetical applications.
As the reader is aware, it has also important applications in
mechanics where it appears as (or, rather, underlies) the con-
ception of "centre of mass " or "centre of gravity". The
examples in divisions B and C do not go beyond the geo-
metrical applications which may be regarded as of universal
interest, but they will form a very useful introduction to
certain parts of mechanics in the case of students who take
up that study. The method followed is suggested imme-
diately by the argument of ch. xxvii., A, and the examples
may, as we have already said, be looked upon as illustrations
of the power and usefulness of Wallis's "arithmetic of in-
finites ".
In Nos. 7-10 since there are ^iSquares in the unit of length
there are 7ikh in AB and nkh x x/h = nkx in a row distant
X from 0. Again, there will be 7ih rows, or nh + 1 if we
count O itself as a row. In the pth row from O there will
be nkh x pl7ih = pk squares, the centres of which are at a
distance p/n from the line through O since the distance be-
tween consecutive rows of centres is Ijn. Thus the total
distance of the centres of the squares in the ^th row from
the line through O is pk x p/n = p^ x kjn. Since there are
{nh + 1) rows, the numbers of squares in which increase
in A. p. from to nkh, the total number of squares is
^nkh (nh -1-1). To find the mean distance of all the points
from the line through O we have then
x =
(02 4. 1'^ + 2^ -H 32 -f . . .+ n%^)k/n
\nkh{nh + 1)
02 + 12 4. 22 + 32 + . . . + {nhf
{nh + 1) {nhf ""
s
+ eil) " '»
368 ALGEBRA
by Wallis's theorem (p. 280). We conclude that when n is so
large that the squares may be considered points the mean
distance of the surface of the triangle from the line through
O is ^h.
No. 13 is to be done similarly, with the difference that we
must here find an expression for the mean distance of the
centres of a number of small cubes from the plane through
the apex of the cone parallel to the base. Let h be the
height of the cone and A the area of the base, and let the side
of each cube measure 1/w of a unit so that n^ of them would
stand on a square unit. Then there are nh layers of cubes
(or nh + 1 a the apex is counted as a layer), n'^A cubes in the
bottom layer and n^A x p'^jinhY = p^ x A/h'^ in the ^th layer
from the apex. The distance of the centre of each cube in
the pth layer from the plane through the apex is p/n and the
total distance of all their centres is
jo 9 A „ A
The total number of cubes in the whole cone will be
p(02 + 12 + 22 + . . . + n%') = p X lnh{nh + 1) {2nh + 1)
by the argument on p. 280. Also the total distance of the
centres of all cubes from the plane through O will be
(03 + 13 + 23 + . . . + n%^) . A.
The mean distance is therefore given by
jo3 + i3 + 2» + . . .+(«;i)»}A
03
\nh{nh -f
+ P + 23
■ 1) (2n/i
+ . . . +
+ !)■
{nhy
A
;
X
6nh^
(1
{nh +
1) (nhf
3h
2nh + 1
, . 1
^ "^ 2nh
When n is so large that the cubes may be considered points
making up the whole volume of the cone we have
X = ^h.
THE PROGRAMME OF EXERCISES LXVI-LXIX 369
In No. 15 the spherical shell is to be supposed divided
into a number of zones or bands by equidistant planes parallel
to the plane of section. It is supposed that the student knows
that these bands all have the same area — namely that of the
corresponding band on a cylinder circumscribing the ball
with its axis perpendicular to the plane of section. [This
proposition is proved in Part II, Ex. LXXXVI.] It follows
that the mean position of the spherical shell is identical with
that of the circumscribing cylinder of the same height — i.e.
that it is on the axis of the shell mid- way between the plane
of section and the remaining pole of the shell.
The examples in division C give apt and important illustra-
tions of the power residing in the notion of mean position ;
they are not likely to cause difficulty.
§ 2. Root-Mean- Square Deviation (Ex. LXVII).— The
reader is doubtless aware that in the analysis of many im-
portant physical phenomena (for example, those of rotation)
it is necessary to take account of the square of the distance
of each particle of a body from a certain point, axis or plane.
Just as calculations concerning " centres of mass " are best
regarded as special applications of the idea of mean position,
so calculations concerning " moments of inertia " and similar
physical entities are thought of most profitably as applications
of another general statistical notion — that of " root-mean-
square deviation ". As will be shown in Part II, Section IX,
this notion plays, in the general theory of statistics, a part
which makes it a very important subject of study quite apart
from its usefulness to the student of mechanics or physics.
It is introduced here partly on account of this far-reaching
significance and partly because it gives excellent opportunities
for applying Wallis's Law to simple and interesting problems.
The definition of the root-mean-square measure of the dis-
persion or degree of " scatter " of a series of measurements is
given in the note at the beginning of the exercise, together with
two reasons for its superiority to the simpler " mean devia-
tion " studied in Ex. XXVI, D. From the mathematical point
of view the former is the more important, and may be expressed
more fully as follows. The method of mean deviation is
simple and effective when applied to straightforward arith-
metical instances but leads to difficulties in an algebraic
treatment because it ignores the signs of the deviations. By
squaring the deviations we can at once avoid these difficulties
T. 24
370 ALGEBRA
and retain the advantages of dealing with numbers which have
all the same sign.^ For this reason and others Prof. Karl
Pearson has given to the root-mean-square measure the name
of the "standard deviation" by which it is now generally
known. The use of this term is reserved until the systematic
investigation of statistical methods is undertaken in Section
IX.
No. 3 is of great importance but is quite easy. If r is the
distance of one of the n points from the origin we have
r^ = x^ + y^
whence (^r^)ln = {%x^)/7i + {'^y^)ln.
Eepresenting the root-mean-square deviation of the points
from the two axes and the origin respectively by the symbols
D^ Dy and D, this relation can be expressed in the form
D2 = D.2 + D/
whence we have
D = J(DJ> + D/)
which is the required formula.
The solutions of Nos. 6-20 follow lines made familiar by
Ex. LXVI. In No. 6 the sum of the squares of the distances
of the centres of the beads from the centre of the end one is
(02 + 12 + 2^ -f ... +m2) d^
Since their number is (m + 1) the mean-square distance is
(02 + 12 + 22 + . .
. + m')dP-
m + 1
(02 + 12 + 22 + . .
. + m')d?m'
= (^ + ei)'
Noting that dm = I we conclude that
To obtain the answer in No. 7 we must suppose m so great
that its reciprocal may be neglected ; hence D = 1/ JS.
In No. 8 we assume that there are 2m + 1 beads whose
centres are d cms. apart and find the mean-square distance of
the centres from that of the middle one by the method just
illustrated. Then
^ See Yule, Theory of Statistics, pp. 134, 146.
THE PROGRAMME OF EXERCISES LXVI-LXIX 371
D2 = 2(0^ +1^+2^+ ... + m^)d^
2w + 1
= 0' + 1' + 2^ + . . . + m^ 2mH\m + 1)
{m + l)m^ ' 2m + 1
VS 6mJ \ 2m + 1/
since in this case md = 1/2. Making m an exceedingly large
number, we have
12
or D = 1/2 J 3. The same result can be obtained in a simpler
manner by applying No. 6 to the two halves of the line.
The result of No. 8 is easily seen to apply also to No. 9,
the beads being replaced by thin rods parallel to CG\ It is
less obvious that the two problems of No. ID have the same
solutions as Nos. 7 and 8. Remembering, however, that a
spherical surface may be supposed divided up into narrow
zones of equal depth whose areas will be equal to those of
corresponding belts of the circumscribed cylinder, we see that
the root-mean-square distances of the points of the bubble
from the two planes mentioned in the question will be the
same as the corresponding distances in the case of the
cylinder. But it is obvious that a hollow cylinder may be
treated in the same way as a line or a rectangle.
Nos. II and 15 require practically the same argument.
In the latter case suppose that the distances between the
circumferences of consecutive circles is d and that there are
p points or beads on the circumference of the circle of radius
d. Then the circle whose radius is (say) sd will have sp
beads. The sum of the squares of their distances from the
centre will be
(sd)^ X sp = s^ X pd^
Thus the sum of the iSquares of the distances of all the beads
from the centre is
(03 -f 13 + 23 4- ... -f m^)pd;'
The total number of beads is
(0-f-l + 2-H3-f- . . . + m)p = \m(m + l)p
We have, therefore, for the mean-square distance
24*
372 ALGEBRA
= 2r2
{m + l)m^
2m^d^
since r = ?w<i. Hence
D
41 + i)
When m is supposed so large that l/2?w can be suppressed we
have the answer to No. l6.
§ 3. The Binomial Theorem (Ex. LXVni).— In Ex.
XXXI the student learnt how to obtain the " expansion " of
(1 + a)""*"^ when the expansion of (1 + a)" is given, n being
any positive integer. As he then saw, the method is an
ancient one, going back at least to the German algebraist
Stifel (c. 1544). In 1665 Newton, by one of those " accidents "
which are resei-ved for men of genius, was led to make two
observations which converted a cumbrous rule of limited
application into a general theorem of fundamental import-
ance. The first was that the coefficients in the expansion of
(1 + a)" can be calculated directly for a given value of n
without reference to the coefficients belonging to any other
value ; the second that n, in his expansion-formula, need be
neither integral nor positive. The discovery was made in
the couree of an attempt to obtain a series for tt by a method
suggested, but not completed, by Wallis.^ But although we
are justified in commemorating the great Englishman's con-
nexion with the binomial theorem by the term " Newtonian
coefficients," it should be remembered that he never succeeded
in giving a general proof of its validity.
Tradition has assigned to the binomial theorem an impor-
tant place in the school course but it has done little to justify
it. The treatment of Ex. LXVIII is intended to show that
there are reasons for studying the expansion perhaps more
substantial than its " elegance," and more immediately con-
^ For details see the author's article in the Mathematical
Gazette for January, 1911.
THE PROGRAMME OF EXERCISES LXVI-LXIX 373
vincing than its usefulness in the advanced parts of the sub-
ject. It will be seen that the theorem is presented as an
" approximation-formula " which is a generalization of the
simple approximation-formulas of Exs. IX-XI, etc., and is
capable of numerous important applications. The starting-
point of the discussion — the approximate calculation of com-
pound interest — was suggested by a famous address of the
engineer George Bidder to the Institute of Civil Engineers. ^
The exposition given in the notes and examples of the
exercise is sufficiently full to make further explanation un-
necessary. The teacher should, however, take care to
emphasize the argument of Nos. 16-19. The aim here is to
find a means of estimating the exactness of the approximation
yielded by a given number of terms of the expansion of
(1 - iy where n is a positive integer. It is shown to be
impracticable to calculate exactly the value of the " comple-
ment " as was done in Ex. XXXV, but it is found to be pos-
sible, by an application of the theory of geometric series, to
name, in certain cases, a number which is certainly greater
than the complement. Whenever, therefore, it is possible
to use the expansion of (1 - i)~" or (1 -f i)~" as an approxi-
mation-formula it is also possible to find an outside estimate
of the error involved in limiting it to a given number of
terms.
The argument of division B may be regarded as a demon-
stration of the binomial theorem for a negative integral ex-
ponent. A demonstration in the case of a fractional exponent
is outside the scope of an elementary work and the student
must, like Newton himself, be contented with a verification
in certain instances. A simple and convincing form of veri-
fication, based upon the properties of the exponential curve,
is suggested in division C. The examples given to illustrate
the practical value of this case of the binomial expansion
regarded as an approximation-formula contain no diffi-
culties calling for comment.
^ Reported in the Minutes of Proceedings of the Institute for
1856. An interesting account of the paper, with extensive quota-
tions, is given by Mr. Branford in his Study of Mathematical
Education, ch. vii. , but the reader must consult the original for
Bidder's method of solving problems of compound interest. The
author owes his first acquaintance with the paper to Mr. Branford's
citations.
374 ALGEBRA
The short division D is intended to suggest a means of
calculating the value of " e," a number which has been esti-
mated hitherto only by a rather precarious graphical method.
The student is to see that as n increases the coefficients in
the expansion of (1 + 1/nY approach constantly nearer to
those of the formula given in No. 34. This formula may,
therefore, be adopted with fair confidence as the one to be
used in calculating the value of e to any required degree of
accuracy. The teacher should, of course, point out that the
argument must not be regarded as a sufficient demonstration.
For this the student must wait till Section VIII is reached.
§ 4. The Generalization of Wallis's Law (Ex. LXIX). —
With the binomial theorem in our hands it is easy to show
that Wallis's Law holds good for all values of the exponent
of X. So far as positive and negative integral values are
concerned the following proof may be regarded as satisfactory.
Let y = ax'" where m is either a positive integer n (No. l) or
a negative integer - n (No. 2). Then if (as in Ex. XLVIII)
a series of ordinates be drawn to the graph of the function at
equal distances h the first difference of y will have the value
a{x + h)"" - arc'" = a\ mx^-^h + ^'^ ~ — Ix'^-'^ . h?
(w-l)(m-2) ^ „ , 1 .
+ ^ ^^i ^x^-^ .h + ... V ... A.
Now if m is positive there is a definite number of terms in the
bracket in A and their sum will therefore have a definite
value when numbers are substituted for the symbols. Call
this sum S. Then the ratio to the term amx""% of the part
of the first difference which follows it is
amh'^ . S _ hS t»
amx"'-'^h ~ x'^-^
No matter how small x may be, h can be taken so much
smaller that the ratio may be reduced below any specifiable
number. In other words, when h is sufficiently small the
value of the first difference of y becomes equal to the term
amx"*'% within c per cent, c being as small a number as any
= amx'^-'^h + amh^\ ^ x""
THE PROGRAMME OF EXERCISES LXVI-LXIX 375
one pleases to name. If m is negative there is no definite
number of terms within the bracket in A and they have,
therefore, no definite sum. But the argument of Ex. LXVIII,
B, showed that if hjx is numerically small enough it is at
least possible to specify a number S which is greater than the
sum of any conceivable number of terms within the bracket.
Supposing S in B to be this number the rest of the argument
follows as before. But if all terms in A after the first may be
neglected h is the differential of x and the first difference is
the differential of y. We have, therefore, for all integral
values of m the differential formula
^x
When m is fractional the proof ceases to be so satisfactory
for we have not demonstrated the binomial theorem in this case
— we have only made it very probable. The note before
No. 7 gives an argument in which an attempt is made to
evade this difficulty. It is, perhaps, sufficiently good for this
stage of the student's development but its theoretical imper-
fection should not be overlooked. Assuming the foregoing
proofs that R and R' can be reduced below any given per-
centage of px and qx' by taking x and x small enough, it
follows, strictly speaking, only that
X' = P~x
?
to an indefinitely close degree of approximation. It is also
assumed that as x is reduced x' may be brought below unity.
The notes and examples of divisions B and C show how
differential formulae based' upon Wallis's Law can be used to
solve problems of area-evaluation and the determination of the
distance traversed by a point moving under specified condi-
tions. The arguments require careful consideration but are
set out very fully in the text. The teacher will doubtless
remember in connexion with them what was said in ch. xxiv. ,
§ 7, about the relation of these reasonings to those involving
the notion of a " limit ".
The exercise concludes with a few examples (division D)
illustrating the application of different differential formulae to
certain geometrical problems. These are not to be considered
of particular importance, but generally prove interesting to the
more able young mathematicians in the class.
376 ALGEBRA
It may be useful to add the solutions of a few typical
problems taken from the various divisions of the exercise.
In the second part of No. 9 it must be remembered that h
is always the increment of x. Thus when the plus before the
X becomes minus the sign before the h must also be changed.
It follows that the differential formula oi y = {a - x)"^ is
K^ = - m{a - x)'^-'^.
The results both in No. 9 and in No. 10 are proved very
simply by a graphic method which should be demonstrated
to the class. By Wallis's Law the gradient of the tangent at
a given point of the curve y = x"^ is, wa:'""^ x being the
abscissa of the point. Move the curve a distance a to the
left. Then the number previously symbolized by x must now
be symbolized hy a + x. That is, y = {a + x)'"' and the
gradient is m{a + ic)'""^. Again, it is easy to show from first
principles that if y ={- x)"' the gradient is - m(- ic)"*"^.
Hence by the previous argument the gradient of y={a- x)"" is
- m{a - ic)'"-i.
This result is applied in No. 13 (iii). Since y = SA/Sx
we have
8A
Sx
whence
= (1 - Sx)-^
= m - ^)}-'
The accuracy of this solution is seen when the result reached
in No. 9 is applied to the formula y = 2T^(i ~ x)~^.
The fact stated in No. 14 is of much theoretical and his-
torical importance.^ Wallis's Law breaks down when m =
because in this case the formula for the primitive, a;"*/w, takes
the meaningless form 1/0.
1 See Part II, Ex. LXXXIV, G. For the history see the author's
article in the Mathematical Gazette^ for December, 1910.
THE PROGRAMME OF EXERCISES LXVI-LXIX 377
No. l6 (iii) gives another illustration of the method of
No. 9. We have
I = 100(1 + 50-
whence
5 = ^.2(0-2 + ty + a
= 40(1 + 5i)i + a
a being the undetermined constant. To find the value of a
we note that, when f = 0, s = 0. Hence a = - 40. Thus
we have finally
5 = 40{(1 + Dtf - 1}.
In No. l8 (iv) we have
-^,= - 4(1 + 2^-^
= - ia + i)-'
whence
I = id + i)-' + «
= (1 + 2^-2 - 1
since 8s/8x = when t = 0. Again in No. 19 (iv) we have
whence
s = - i& + i)-' -t + ^
= - i(l + 2^-1 - t + ^
since, when i = 0, s = 0.
PART II
ALTERNATIVE SCHEMES OF STUDY.
(Cf. pp. 382, 398.;
A.
Section IV.
(Mainly Revision.)
i
(Trig.
Section V.
OP THE Sphere.)
Section VI.
(Complex Numbers.)
Section VII.
(Periodic Functions.)
Section VIII.
(Limits.)
Section IX.
(Statistics.)
Section VI.
Exs. xcii. and xciv.
Section IV.
Ex. LXXXIV.
Section IV.
Exs. LXXVIII. AND LXXIX.
Section VII.
Exs. xcix.-cni.
Section IX.
Exs. CXXII. AND cxxv., A, B.
GHAPTEE XXXVIII.
INTRODUCTION TO PART II.
§ 1. The Scope of Part 11. — The aim of Part I is to set
forth a course which, though Uberal in range, is confined to
fundamentals. That is to say, it contains no subject of which
a student leaving a secondary school should be permitted to
be ignorant, and treats no topic in a way relevant only to the
needs of the specialist. The scope of Part II goes beyond
fundamentals, at least in the sense that none of its contents
can be said to be an indispensable part of a universal mini-
mum course. On the other hand, it is still confined to them
in the sense that the choice of subject-matter and methods of
treatment are based upon the ideas set forth in ch. ii., § 1,
that is, upon the conception of mathematics as a cultural
rather than a technical subject. Text-books on " higher
algebra " are, as a rule, written professedly for the aspirant
to mathematical honours at the University ; they aim, there-
fore, at a rather advanced development of technique within a
somewhat limited field. The question whether it would not
be better in every case to reserve studies of this kind until
the student actually reaches the University or higher technical
institution is important but cannot be argued here. It is,
however, proper to urge that they do not offer the most
suitable course of instruction for the general body of students.
For the student who is to be a teacher or an engineer, or to
engage in higher industrial or administrative work, as well as
for the student who is continuing his mathematical studies
as part of a general education, the best course would seem
to be one which sets in clear relief the central aims and most
vital notions of the main branches of mathematics, supple-
ments exposition with sufficient practical exercise to give the
student a real training and the sense of mastery that comes
with training, and, in particular, illustrates vividly the essen-
381
382 ALGEBRA
tial part which mathematics plays in so many departments of
modern life and activity.
If there is room for individual choice and preference in the
details of the minimum course no specific programme for a
further course can claim to be much more than an expression
of the views and, perhaps, the temperament of the proposer.
Moreover, the variety in the conditions under which mathe-
matical teaching is carried on in this country makes the
actual adoption of a uniform programme impossible even if
it were desirable. For these reasons Exercises, Part II, is to
be regarded simply as setting forth the full range of topics
which the author would himself teach under favourable con-
ditions to pupils of fair ability. He recognizes that other
teachers who use the book will, for one reason or another,
utilize only a selection from its contents. To facilitate selec-
tion the various sections have been made as independent of
one another as the nature of the subject-matter permits, and
the exercises of each section have been divided into those
which are essential to the development of the subject and
those which may be omitted by the student for whom a shorter
course is prescribed.
The full course consists of six sections, numbered IV-IX in
continuation of those of Part I. Algebraic theory and practical
applications appear side by side in each ; but it is useful to note
that the development of theory is predominant in three of the
sections (IV, VI, VIII) while practical applications form the
central interest of the sections which alternate with them.
Scheme A on p. 380 shows the order in which the sections
are intended to be studied when the full course is taken.
The idea underlying the arrangement is that a " theoretical "
and a " practical " section shall always be studied at the same
time. Even if a selection is made from the full course it
would probably be best in most cases to preserve this feature
of the scheme. Scheme B suggests a minimum course in
which at least those subjects will be studied which are com-
monly required in examinations, such as the Intermediate
B.A. and B.Sc. Examination of the University of London,
the " Advanced " paper in the Board of Education's Examina-
tion of students in Training Colleges, and several professional
and technical examinations of a similar standard (see pp. 60
and 398). The aim and scope of the several sections will
now be indicated.
INTRODUCTION TO PART II 383
§ 2. Section IV. Mainly Bevision. — The title of this
section indicates that it looks backwards towards Part I
rather than forwards to the rest of Part II. As is explained
in the Introduction (Exercises, II, pp. 3, 4), its aim is not only
to consolidate and extend the knowledge gained in the earlier
part of the course but also to make explicit, and to bring into
scientific order, ideas about the nature of numbers, of arith-
metical processes, and of algebraic symbolism which are in
one way or another involved even in the first steps of the
student's mathematical progress, though he has not hitherto
made them the subjecb of systematic reflexion. The exercises
fall into three well-marked groups. Exs. LXX-LXXIV deal
with the nature of numbers, Exs. LXXV-LXXVIII with the
idea of a function and the properties of algebraic symbolism,
Exs. LXXIX-LXXXIII with certain more advanced applica-
tions of the ideas of Sections II and III. The supplementary
Ex. LXXXIV consists chiefly of revision papers upon the
subject-matter of Part I.
Some considerations about the properties of numbers and
arithmetical operations are a common feature of elementary
text-books on algebra and are usually summarized in the
" laws " of commutation, of association, etc. Too often these
appear at a point in the course where the pupil is not pre-
pared to appreciate their significance — namely, at the begin-
ning. .General pedagogical principles suggest that reflexions
upon the assumptions involved in a science should come
after the student has made some progress in the science by
the light of " intuition " and common sense. The postpone-
ment of these discussions about the foundations of algebra to
the place here assigned to them not only gives them a better
chance of being appreciated but also makes it possible to
expand them into a simple exposition of the main results of
the modern theory of the nature of numbers. ^ Numbers and
their relations to the great twin entities, space and time, have
always exercised an attraction upon thoughtful minds. Philo-
sophies have been based upon the properties of numbers ;
religions upon their power and mystery. In particular, the
seeming contradictions involved in the concept of " infinity "
^ This is, of course, an entirely difierent subject from the " theory
of numbers " which is one of the traditional topics of a course of
" higher algebra ''.
384 ALGEBRA
which provoked the acutest of the Greeks -to "arguments,
all imcaeasurably subtle and profound " ^ continue to "tease
out of thought " ordinary folks as well as poets. But in spite
of this universal interest in the subject the philosophy of
number could not, until comparatively recently, be considered
a suitable topic for elementary instruction. The labours of
modern philosophical mathematicians — especially the " al-
most unexampled lucidity " of. Cantor ^ — 'have, however, made
it possible to present and solve the main problems concerning
the nature of numbers in a way both interesting and profit-
able to the general student. Such a presentation is attempted
in Exs. LXX-LXXIV. Current opinion does not as yet
demand that these problems shall be discussed in schools, but
they have a character which appeals to the expanding in-
terests of the adolescent and are of profound importance
outside the boundaries of technical mathematics. The teacher
is recommended, therefore, to find a place for them in his
course. If he needs further persuasion to do so he is invited
to observe that without some consideration of the relations
between numbers and continuous quantities — such as lengths
and periods of time — many important conclusions and
practices in elementary algebra rest upon a very unsatisfactory
logical basis.
Exs. LXXV-LXXVIII stand less in need of defence. Exs.
LXXV-LXXVI are intended to revise and give further
illustrations of ideas about the nature of functions of a single
variable which have already been acquired in Part I. The
essentjials of the ideas connoted by the terms " indeterminate
value " and " singular points " find their place here. In Ex.
LXXVI an inquiry into the properties of a few functions
of two variables is made the occasion for extending the
method of rectangular coordinates to the analysis and de-
scription of curved surfaces. The investigation is undertaken
in the spirit of ch. iv., § 9 ; that is, the surfaces are treated
as tri-dimensional graphs to be studied not so much for their
own sake as for the light they throw upon the properties of
the functions of which they are the spatial expression. The
exercise should, however, prove a useful introduction to tri-
1 Russell, Prin. of Math., p. 347. The reference is to the para-
doxes of Zeno. (See Ex. LXXII, No. 4.)
^Russell, op. cit., p. 353. The author's acquaintance with the
subject is derived chiefly from Mr. Russell's masterly pages.
INTRODUCTION TO PART II 385
dimensional coordinate geometry in the proper sense of the
term. The last exercise of the group is given to a brief
review of the principles which govern the development of
algebraic symbolism in all its stages. The fact that it uses
the exponential notation as the chief illustration of its theme
makes it a suitable link between the two " theoretical "
subsections and the " practical " subsection which follows.
The exercises of this last group (Exs. LXXIX-LXXXIII)
are all concerned with the law of " growth with constant
growth-factor " and the exponential function which that form
of growth exemplifies. Exs. LXXIX-LXXXI are frankly in-
formative, being intended to exhibit the essentials of the theory
of public loans (and similar financial arrangements) and the
principles underlying the practice of life assurance. It should
be unnecessary to demonstrate the immense importance of
these parts of the " mathematics of citizenship " ; the common
neglect of them in schools is perhaps due only to an exag-
gerated estimate of their difficulty. This remark applies
especially to the theory of life assurance which is generally
supposed to require previous acquaintance with the theory of
probability. It will be seen that the typical problems of the
life office can be solved without the introduction either of the
term " probability " or of the ideas which the term is thought
to connote. In Ex. LXXXII the student turns from these
topics to a more formal study of the exponential function and
curve, theoretical conclusions being illustrated by references
to simple physical phenomena such as those exhibited by
cooling bodies. In Ex. LXXXIII he considers the differ-
ential formulae of the exponential and logarithmic functions
— a subject of great historical and practical importance.
Ex. LXXXIV is divided into six parts. The first five are
Revision Papers and consist almost entirely of straightforward
exercises on the subject-matter of Part I. The exception to
this statement is that in a few examples in division E the
simple theory of the decimal notation outlined in Ex. XXXI
is expanded into a brief treatment of " scales of notation " in
general. In the last division (F) the student follows the
steps that led Wallis, Mercator, and others to discover the
expansions by which logarithms may most conveniently be
calculated. With the exception of division F the whole of
this exercise is independent of the rest of the section and may
be given as tests at any convenient points.
T. 25
386 ALGEBRA
§ 3. Section V. The Trigonometry of the Sphere. — Per-
haps the most serious defect in the English curriculum in
mathematics is its neglect of what continental writers call
*' stereometry," that is, the elementary study of three-dimen-
sional space by the methods of geometry, algebra and trigo-
nometry. At various points of Part I examples have been
introduced with the object of freeing the pupil from intellectual
confinement to the plane ; these have been followed in Section
IV by an exercise (LXXVII) dealing specifically with graphic
surfaces and the solid forms they enclose. Section V is
devoted entirely to a systematic though elementary treatment
of stereometry in one of its most important branches — namely,
the trigonometry of the spherical surface. We have seen
(ch. II., § 3) that the problems of spherical trigonometry have
hitherto been excluded from the schoolroom — in spite of their
intrinsic interest and great practical importance — mainly
because they have been regarded as a distinct "subject"
whose place in the logical hierarchy comes after plane trigo-
nometry. The obvious way to escape from the consequences
of this pedagogical error is to select a group of spherical
problems of general rather than technical interest and to
investigate trigonometrical methods of dealing with them —
the subject being developed no farther and with no more
formality than the solution of these problems itself requires.
The increasing attention given to the study of geography in
schools suggests as the appropriate starting-point a simple
investigation of the theory of map-projections. Some con-
sideration of this subject must necessarily find a place in any
rational course of instruction in geography ; thus to begin
here is to build upon ideas with which the student has already
a certain familiarity. Moreover, some of the cartographic
projections which are found in every school atlas are based
upon principles of the highest interest and beauty from the
purely mathematical standpoint. Lastly, the study of these
concrete instances leads simply and naturally to the general
notion of a " transformation," which is one of the most signi-
ficant and vital of mathematical ideas.
The first thing which a student of this subject learns is that
the points of a spherical surface can be represented upon a
plane in an endless number of ways, and that, in practice,
the choice of a "projection" is always determined by the
particular purpose it is to subserve. One purpose of special
INTRODUCTION TO PART II 387
interest and importance is to assist the sailor in the task of
navigating his ship from port to port. The study of the
charts constructed for this end leads naturally to an investi-
gation of the trigonometrical formulae of which the navigator
has to make daily use. In this way the student reaches the
problems of spherical trigonometry in the- narrower sense of
the term.
The topics indicated in the two preceding paragraphs form
the subject-matter of the first group of exercises in Section V
—namely, Bxs. LXXXV-LXXXIX. Of these the first four
are given to the theory of the chief systems of map-projection
and to the graphical solution of the geographical problems
associated with each. The modest outfit of trigonometrical
formulae which suffices to solve these same problems by cal-
culation is all developed in the last exercise of the group
(Ex. LXXXIX).
The doctrine of the sphere learnt in the first group is
applied in another single exercise (XC) to a series of simple
astronomical problems. This part of the work has an obvious
connexion with the study of navigation but is justified less by
that circumstance than by its wider connexion with geographi-
cal theory and by its intrinsic interest. Some study of astro-
nomical phenomena is a necessary feature of every course
in geography. As in the parallel case of map-projections,
to make precise, to develop, and to apply the ideas thus
acquired by the pupil is at once the mathematical teacher's
plain duty and his valuable opportunity. Apart from
these utilitarian considerations, the neglect of the most
ancient branch of "Nature study" is an anomaly in our
school curriculum which ought no longer to be tolerated. If
a few lessons, desirable in themselves from the standpoint of
mathematical instruction, have the further happy result of
helping to remove that anomaly the case for including them
in any programme of moderately extensive scope is extremely
strong.
It should be clearly understood that this is not an argument
for the systematic study of astronomy in schools under the
name of " applied mathematics ". It is urged merely that
certain lessons in mathematics should have the incidental
result of teaching boys and girls to "lift up their eyes to the
heavens " in a spirit of intelligent inquiry. In other words,
the astronomical problems to be considered in the mathe-
25*
388 ALGEBRA
matical classroom must be limited to those whose solution
is simply a question of the straightforward mathematical state-
ment of facts accessible to ordinary observation. The con-
stitution of the solar system or the connexion between orbital
motion and "central forces," though excellent subjects for
special study in school, would be out of place in the general
mathematics course. After the exclusion of technical matters
of this kind there is still an abundance of interesting and im-
portant questions which our principle readily admits. They
include (i) the problem of fixing the positions of the stars by
co-ordinates, (ii) the related question of the diurnal revolution
of the heavens, (iii) the daily movements of the sun and moon,
(iv) the calculation of times of rising and setting, etc., (v) the
nautical problems of determining latitude and longitude, and,
lastly, (vi) the fascinating subject of dialling.
In the treatment of all these questions in Ex. XC the facts are
stated as they appear to direct observation. There is no mention
of the earth's daily rotation upon its axis or of its annual revolu-
tion around the sun. These are hypotheses invented to account
for the facts of observation and belong, therefore, to a techni-
cal study of astronomy. We are limited here to a precise
statement of the face-value of the facts which we observe.
According to that face- value the stars move round the sky
daily and the sun and moon move among them. Any attempt
to give a theory of stellar movements must begin with an
exact determination of the facts as they appear ; so that the
method pursued in Ex. XC is one which lays the foundation
indispensably necessary for any successful systematic study
of astronomy.
It cannot be assumed that the average student has a
knowledge of astronomical facts equal in scope to the geo-
graphical knowledge assumed in the treatment of problems of
navigation. It has been necessary, therefore, to summarize in
some detail what direct observation has to tell about stellar
and solar movements. A probable cause of the widespread
neglect of simple astronomical knowledge is the abstract and
technicalized form in which the systematic text-book presents
it. Happily in many schools this neglect is passing away,
and simple studies of solar shadows, etc., are being added
to the now customary programme of meteorological obser-
vations. Experience shows that there is no difficulty in
establishing at first hand, and without any instrumental
INTRODUCTION TO PART II 389
means more recondite than an upright pin and a home-
made sundial, all the facts of the diurnal and annual move-
ments of the sun and stars upon which rest the theories of
day and night and of the seasons. Experience also shows
that there are few branches of study which engage more
readily the interests of boys and girls, and none which gives
such direct aid to mathematical progress.^ It is not too much
to say that the general adoption of simple observations of this
kind, followed up by elementary discussions in the classroom,
would alone suffice to put an end to that neglect of tridi-
mensional geometry which we have signalized as perhaps the
most serious deficiency in current mathematical instruction.
It is not possible to describe here the practical exercises
which would satisfy the conditions mentioned above. The
most important are probably suggested with sufficient direct-
ness in the exposition of Ex. XC, and the tea,cher will
have no difficulty in giving practical effect to the suggestions
— particularly if he make common cause with a colleague
who teaches geography. The globe with the blackboard sur-
face, recommended for use in connexion with the exercises of
Section V, can easily be converted into an astronomical
globe by the addition of a cardboard horizon, and is then much
more useful than the globe of the makers, which contains too
much detail for the teacher's purpose. If a copy of that
excellent old text-book, Keith's Use of the Globes, can be ob-
tained, it will be found to give all the information which can
be needed. It must, however, be emphasized that the globe
should be employed as a means of summarizing and explaining
observations which have actually been made, and that to use
it as a substitute for first-hand observation is a thoroughly
unpedagogical practice.
The supplementary examples of this section (Ex. XCI) deal
partly with practical problems of greater difficulty and less
fundamental importance than those of the former exercises.
In addition they develop a little further some of the theoreti-
cal consequences of the earlier arguments. Thus in division
B the notion of a projection is generalized into the idea of
point-to-point correspondence which may subsist either be-
tween lines, surfaces, or volumes. The argument here paves
^ The author owes to a friend who speaks with authority the
observation that vitality in mathematical studies and interest in
astronomy have, in the past, constantly risen and fallen together.
390 ALGEBRA
the way for the study of functions of a complex variable which
is to be taken up in Section VI. Again, in divisions A and
D the doctrine of spherical triangles receives additions, some
of which (e.g. the study of " spherical excess ") are of almost
purely theoretical interest.
§ 4. Section VI. Complex Numbers. — In ch. xxiv., § 4,
it was stated that the " imaginary roots " of quadratic equa-
tions would be excluded from consideration in Part I to be
discussed later as part of a systematic doctrine of " com-
plex numbers ". At first sight this procedure may seem
in contradiction with the principle asserted with so much
emphasis in the preceding article and elsewhere. If it is an
error to hold back the methods of the calculus or of spherical
trigonometry until they can be taught as self-contained
subjects, how can it be right to pursue the opposite policy in
the case of *' imaginaries " ? The answer to this question has
already been indicated on p. 239. It is true that the doctrine
of complex numbers originated in the investigation of the
roots of equations. The roots which the earlier algebraists
(like the students of our Part I) rejected as " impossibles "
were recognized by Girard (1629) as specific solutions, and
were formally distinguished as " imaginary," in opposition to
"real," roots by Descartes in his Gcom^^rie (1637). Never-
theless, there are three good reasons for departing at this point
from the historical track. In the first place the admission of
" imaginary " roots would blur the clearness of the connexion
between the quadratic equation and the parabolic curve.
Secondly, "imaginary" roots are needed only in order that
every quadratic may be regarded as soluble ; and logical
completeness of this kind makes but a feeble appeal to the
immature mathematician. Thirdly, although the use of
" imaginary " numbers during the seventeenth and eighteenth
centuries produced some notable and fruitful results (for
example, De Moivre's Theorem, 1730), yet they were, and
continued to be, more or less of a mystery until Argand and
Gauss, at the beginning of the nineteenth century, reached
the views of their nature which are set forth in Section VI.
There seems, therefore, to be abundant reason to withhold all
consideration of the subject until the pupil is in a position to
study those views with profit.
A brief commentary, historical and philosophical, upon the
exposition of Section VI will be found in ch. xliv. It will be
INTRODUCTION TO PART II 391
sufficient here to indicate the ground covered by the exercises.
Exs. XCII, XCIII introduce the conception oi a + ib (where
a and b are any directed numbers) as a " complex number "
which can be used as a symbol either for the point whose
rectangular co-ordinates are a and b or for the straight line
joining this point to the origin. They also inquire in what
sense the operations of addition and multiplication can be
applied to these complex numbers. The most important
single result of this investigation is the discovery that the
manipulation of expressions of the form a + ib is immensely
simplified by treating the symbol i as if it were a number
whose square is - 1. The first fruit of this momentous dis-
covery is De Moivre's Theorem.
In Exs. XCIV and XCV the student sees that the adoption
of complex numbers leads to most important extensions of
the results of elementary algebra. The first of these (Ex.
XCIV) consists in the idea that an equation which has no
root in the simple arithmetical sense may yet be satisfied by
" complex " roots, and that, in other cases, ordinary or " real "
roots may be supplemented by complex roots. The second
(Ex. XCV) is the discovery that in the exponential function
y = ar% where r has been regarded hitherto as necessarily
non-directed or positive (ch. xxx., § 5), the introduction of
complex values of y enables us to remove the restriction and
to consider r as capable of any numerical value.
The force of these extensions may be expressed alternatively
as follows. Let y = f{x) be any function. Hitherto it has
been taken for granted that x and y represent numbers taken
from the complete one- dimensioned scale which runs from
- 00 through zero to -f- oo . We have learnt that a given
function will not, in general, assign a value to y for every
value of a; or a value to x for every value of y, but that it
connects the values of the variables only over a certain field.
We have learnt also that it is always possible to represent
this field of the function by a graph drawn upon a plane. In
Ex. XCIV we discover that, in some cases, the " real " values
of y which are not given by any " real " value of x yet cor-
respond to " complex " values of that variable, and that the
field of the function as enlarged by the admission of these
complex values of x can be represented by a three-dimensioned
graph. Similarly, it is found in Ex. XCV that, in the case
of other functions, where some "real " values of x have no
392 ALGEBRA
"real" values of y corresponding to them, the defect may
be made good by taking account of " complex " values of y,
and that the field of the function, thus enlarged, can again
be represented by a graphic line drawn in three-dimensional
space. The obvious completion of these ideas is the notion
that y = /(ic) may represent a form of connexion in which
both X and y may simultaneously have " complex " values.
The pursuit of this notion is begun in Ex. XCVI where it is
applied to very simple cases. It is observed that we now
have to do with two variables, say x = it + iv and 2/ = U + iV,
each of which is of two dimensions, so that three-dimensional
space no longer suffices for the graphic representation of the
field of the function. Following the example of Riemann we
turn to the projective method of Section V as the simplest
way of meeting the new need. Armed with this method we
attack, in Ex. XCVII, the especially important case of the
logarithmic function. The supplementary examples of Ex.
XCVIII derive from this investigation the doctrine of the
" exponential values " of the sine and cosine and apply it to
the discussion of the sine and cosine functions when the
independent variable is complex.
§ 5. Section VII. Periodic Functions. — The central
purpose of Section VII is the development of the idea of
periodic functions of a variable. A function such as
y = ax^+bx+coYy = a+ Jbx + c
is non-periodic. That is to say, although different values of
X may in some cases be associated with the same value
of y, or different values of y with the same value of x,
yet there is no regular repetition of a cycle of values of the
one variable as the other progresses through its scale of
possible values. The typical feature of a periodic function is,
on the other hand, that it does exhibit such a succession of
identical cycles. Periodicity in physical phenomena is among
the most familiar of our experiences — " seed time and
harvest, and cold and heat, and summer and winter, and day
and night " being only the most conspicuous instances.
Our task in this section is to find an algebraic language which
shall be appropriate to the analysis and description of such
phenomena and the numerical relations which they exem-
plify, just as the symbolism studied in previous sections is
appropriate to the analysis and description of phenomena
which do not exhibit periodicity.
INTRODUCTION TO PART II 393
The elements of such a language are readily obtained. We
have already found, in connexion with problems of mensura-
tion and surveying (Pt. I, Exs. XL and XLI), the need of
extending the idea of the sine and cosine of an angle to in-
clude " angles " measured up to 360° ; and this extension
has made us familiar with the repetition of the series of values
presented by the sines and cosines of angles between 0° and
90°. Our present task merely requires three further steps in
the same direction : (i) We must reach the idea of an angle
as a variable capable of an endless series of values, negative
as well as positive. (ii) To each of these values we must
assign a sine and cosine in such a way that all angles which
differ in value by a certain definite amount shall have the
same sine and cosine. (iii) Finally, just as in Section II
notions of the sine and cosine were made independent of
their original association with the triangle, so now they must
be made independent even of their associations with angles.
When we have reached this point the symbols sin x and cos x
will mean simply numbers associated with a number x in
accordance with certain definite laws, and it will be no longer
necessary to regard x in the symbolism sin x as an angle
any more than it is necessary to regard x in the symbolism
ic^ as a length. It may stand for any variable which has
another variable depending upon it in accordance with the
special law of the function.
Thus conceived sin x and cos x are our first instances of a
periodic function. The next step is to find that by combining
sines and cosines we obtain formulae which describe periodic
relations of indefinite complexity. From this discovery it is
an easy passage to the converse idea that any periodic function
may be expressed by a formula built up of sines and cosines
upon the simple plan first laid down by the great French
mathematician, Fourier.
It is important that the teacher should keep closely before
him the main argument of the section. The incidental results
to which the argument leads have, however, much sub-
stantive importance and should be mentioned. Some are
particular methods, such as the measurement of angles in
radians, some are formulae of wide applicability, such as the
differential formulae for the sine and cosine, some are im-
portant expansions, such as Gregory's series for tt. The
most considerable, however, is the idea of the hyperbolic
394 ALGEBRA
functions. The treatment of this topic will, it is hoped, be
found simple enough to be followed without difficulty by the
average student. The main idea to be emphasized is that
these are functions whose properties are curiously analogous
to those of the ** circular " functions, but with the essential
difiference of being non-periodic. The student of physics will
know that for this reason the hyperbolic functions are most
useful in describing the behaviour of bodies whose movements
would have been periodic but for the presence of some re-
strictive condition — ^for example, friction.
Stated, as above, in abstract terms the programme of the
section may seem to threaten to be difficult and dry. If the
argument were developed abstractly it might easily deserve
both epithets. Fortunately it is possible to base the algebraic
analysis of periodicity upon a concrete foundation of most
attractive observations and experiments. Among these the
most important are the movements of vibrating bodies, waves,
and the fascinating phenomena of the tides.
The teacher may be reminded that the treatment of these
topics in the mathematics lesson need be in no sense an in-
trusion into the field of physics ; for the remark already made
with reference to the study of astronomical topics in Section
V applies also here. We are concerned only with those
features of vibrations, waves, etc., which are visible to the
eye of common sense, and our business is limited to the search
for an adequate mode of describing what anyone may see
who takes the trouble to look. The work of the physicist
begins where our inquiry leaves off, for it is his business to
bring to light the hidden conditions which determine the
behaviour of vibrating and undulating bodies.
The reader whom these arguments leave unconvinced is
invited to consider how greatly the whole subject of periodicity
gains in rationality, as well as in interest, when it starts from
a basis of organized observation. Consider, for example, the
first step of the argument — the concept of an angle of endless
magnitude. To the ordinary student there seems nothing more
arbitrary and unnecessary than the extension of the angle-
scale beyond 360°, and the resulting disturbance of the simple
arrangement which associates a single angle with each single
set of values of the ratios seems a purely gratuitous complica-
tion. But bring him face to face with the problem of find-
ing a clear and compact mode of describing (say) the behaviour
INTRODUCTION TO PART II 395
of a swinging lamp and the aspect of the matter is entirely
changed. He now sees that the extension of the angle-scale,
instead of being a complication, is actually a simplification of
the highest value.
This reason for the procedure adopted in Section VII
seems so cogent that there is, perhaps, only one objection
worthy of consideration. It may be protested that in spite
of the foregoing disclaimer the study of vibrations and
waves does as a matter of fact involve the introduction into
the classroom of elaborate physical apparatus and experi-
ments. The answer is that although when the teacher is a
physicist as well as a mathematician he may with some ad-
vantage draw upon the resources of the laboratory, yet such a
practice is by no means necessary, and, perhaps, on the whole,
not even advisable. His object in the mathematical lessons
is to instil mathematical ideas. It is claimed that these ideas
cannot be eifectively taught apart from the physical pheno-
mena which they interpret, but it is part of the same argument
that the phenomena should be studied in forms which are
already familiar to everybody. Thus it will be found that
where the teacher is advised to use a piece of apparatus it is
always of the nature of a tridimensional diagram intended
simply to help the pupil's analysis of the familiar phenomena
under consideration. For this reason models are described
which make the minimum demands upon the teacher's con-
structive ability. Moreover, since they are only diagrams they
are not indispensable.
The subdivisions into which the exercises naturally fall have
already been partly indicated. Bxs. XCIX-CI introduce the
notions of circular measure and of an angle as a quantity
which may have any magnitude, positive or negative ; they
also investigate the application of the fundamental trigono-
metrical formulae to angles of unlimited magnitude. Exs.
GII-CIII generalize the preceding argument into a formal
doctrine of the direct and inverse "circular functions".
Exs. GIV-CVI apply the new notions to the study of wave-
motion, including the composition and analysis of harmonic
wave-forms and functions. Ex. CVII takes up the important
question of the differential formulae of the sine and cosine.
Exs. CVIII-GIX treat of the hyperbolic functions and their
analogies with the circular functions. Ex. CX is a supple-
mentary exercise in which all these topics receive further
396 ALGEBRA
development, the most important topics being the calculation
of TT by series and the prediction of ocean tides.
§ 6. Section VIII. The Theory of Limits. — The special
aims of this section have been foreshadowed in ch. xxiv., § 6,
and are stated with some detail in the student's Introduction
{Exercises, II, p. 321). The numerous arguments in which
Wallis's Law and differential formulae have been established
or applied constitute, strictly speaking, only a " calculus
of approximations ". They are now to be placed upon
a more satisfactory logical basis. Thus our first task in
Section VIII is to convert the doctrine of approximations into
the differential and integral calculus (in the proper sense of
the terms) by means of the idea of a " limit ". The second
task is to develop a technique by which the methods of the
calculus can be readily applied to the problems in which they
are relevant. In view of former statements about the scope
of this book it need hardly be said that the development is
confined strictly to fundamentals, and that the range of the
problems is restricted to those which may be considered of
universal interest.
The general notion of a limit is explained in Ex. CXI and
is applied in Exs. CXII-CXIV to the derivation of the
standard formulae of differentiation and integration. These
exercises deal chiefly with the direct exemplification of first
principles ; secondary topics, such as partial differentiation,
and practical applications, such as the theory of curvature,
being reserved for the supplementary exercise. In Ex. CXV
the use of the " differential formula," regarded as an approxi-
mation formula, is improved into the " differential equation,"
and some of the simpler applications of differential equations
are illustrated. This part of the section is completed by a
simple discussion of partial differentiation and integration in
Ex. CXV.
Ex. CXVI is a connecting-link between the former and the
latter subdivisions of the section. It raises the important
theoretical question as to whether a function can always be
differentiated and shows (following Weierstrass) that un-
assisted common sense is not a safe guide in the search for an
answer. The important theorem called after Rolle finds its
natural place here. In Ex. CXVII we consider the nature of
expansions and ask whether it is possible to find a general
rule for developing a function in powers of the variable — a
INTRODUCTION TO PART II 397
rule which shall include a means of estimating the degree of
approximation to the true value of the function obtainable by
a given number of terms. The answer to this demand is an
investigation of the theorem generally known as " Taylor's ".
Ex. CXVIII supplements all the exercises of the section by
examples of wider scope and greater difficulty.
§ 7. Section IX. The theory of Statistics. — Examples
and discussions which are, essentially, contributions to a
theory of statistics have been met at various points of the
course— Exs. XXVI, D, LXVI, LXVII ; chs. iv., § 7,
xxxvii., §§ 1, 2. The results of these are now to be gathered
together and developed into a systematic doctrine embracing
the fundamental ideas of the subject and illustrating some
of its simpler and more important applications. Under the
names " permutations and combinations " and " probability "
certain parts of the doctrine of statistics have long had a
place in the mathematical curriculum. It must be confessed
that they hold it rather by hereditary privilege than by their
own merits.^ Another part — the theory of errors — is of
great practical importance and theoretical beauty, but has not
hitherto appeared in elementary text-books. In recent years,
however, the science has undergone such remarkable develop-
ments and has been applied so widely and to matters of such
immense importance that its claim to an honourable place
in the school curriculum can hardly be resisted. A certain
degree of cultivation of the " statistical sense " seems, in fact,
likely to become one of the essential qualifications for intelli-
gent citizenship. In any case it is no longer possible without
it to understand modern developments in sciences, such as
biology and psychology, which have hitherto been thought the
refuge of the mathematically destitute. The aim of Section
IX is to give in clear and simple outline an account of the
general notions and fundamental technical methods of modern
statistics. " Permutations and combinations " and *' proba-
bility " have their proper places in this account, and will (like
the periodic functions in Section VII) be found to gain very
greatly in rationality and interest by being absorbed in a
general doctrine of such far-reaching and substantial impor-
tance.
1 Some important exceptions must be made to this general com-
plaint. For example, Prof. Chrystal in his invaluable Text-Book
gave to his treatment of probability just the turn here advocated.
398 ALGEBRA
The argument of the section shows three well-marked stages.
The first (Exs. CXIX-CXXI) consists mainly in the revision
and extension of the methods of recording the " frequency-
distribution " of a series of measurements or other statistics.
In the second stage (Exs. CXXII-CXXIII) we have a simple
investigation of certain cases in which frequency- distributions
can be calculated either from a Knowledge of other frequency-
distributions or upon the basis of certain assumptions about
the nature of the things in question. At this point per-
mutations and combinations are studied as examples of
calculable frequency-distributions, and acquaintance is made
with some of the theorems of the calculus of probabilities.
In the last stage (Ex. CXXIV) we turn from the problem of
calculating the frequency-distribution of a single variable in
order to investigate the problem of " correlation," that is, the
problem of estimating the mode and degree of dependence
of one variable upon another. A supplementary exercise
(CXXV) gives further illustrations of all the chief topics
dealt with in the section.
§ 8. The Minimum Course. — Scheme B on p. 380 sets forth
a programme of exercises for a minimum course to be taken by
students who cannot work through the whole of the sections.
The principle followed in selecting these exercises is to choose
those which deal with subjects commonly required in public
examinations, together with as much of the other work as is
necessary to form a rational introduction to these subjects.
In Section IV nothing is retained except the theory of indices,
logarithms, and annuities, and the revision exercise. Section
V is omitted altogether. In Section VI the two exercises are
retained which contain enough material to provide a rational
explanation of the " imaginary " roots of equations. Since a
knowledge of these roots is (unfortunately) demanded at
present even in elementary examinations, the two exercises in
question are placed at the beginning of the scheme. In
Section VII Exs. XCIX-CIII are all retained since they con-
tain the theory of circular measure, of the trigonometrical
ratios of angles of unlimited magnitude, the " sum and differ-
ence " theorems for such angles, and the theory of inverse
circular functions — subjects which are found in the syllabuses
of many school examinations. Nothing is taken from Section
VIII, while Section IX contributes only Ex. CXXII and
INTRODUCTION TO PART II 399
Ex. CXXV, A, B, which deal with permutations, combina-
tions, and their appHcations to the binomial theorem. The
teacher who desires to do so will find no difficulty in filling
out this minimum course either by more detailed treatment
of the subjects of a single section or by adding exercises from
several.
SECTION IV.
MAINLY EEVISION.
T. 26
THE EXEECISES OF SECTION IV.
*^* The numbers in ordinary type refer tx) the pages of Exercises
in Algebra, Part II ; the numbers in heavy type to the pages of this
book.
BXKRCISK PAGES
LXX. Integers 5, 405
LXXI. Rational Numbers 9, 409
LXXII. Irrational Numbers 13, 410
LXXIII. Operations upon Numbers . , . .19, 420
LXXIV. The Complete Number Scheme . . .25, 421
LXXV. Functions of One Variable . . .32, 422
LXXVI. Some Peculiarities OF Functions . . 40, 425
LXXVII. Functions op Two Variables . . .47, 426
LXX VIII. The Development of Algebraic Symbolism 55, 428
LXXIX. Annuities-Certain 63, 432
LXXX. Contingent Annuities . . . .72, 432
LXXXI. Life Insurance 76, 432
LXXXII. The Exponential Function and Curve . 79, 433
LXXXin. Differential Formula . , . .83, 435
LXXXIV. Supplementary Examples
A-D. Revision papers 1-4 . . .90, 438
E.* Scales of notation . . . .96, 438
F. « Logarithmic approximation-series . 97, 438
CHAPTER XXXIX.
NUMBEE SYSTEMS.
§ 1. TheScopeofExs. LXX-LXXIV.— We ha.YQ seen {oh.
I., p. 5) that the object of an algebra is to develop a calculus,
that is, a system of symbols and rules for the manipulation of
the symbols, by means of which the investigation of some
definite "province of thought or of external experience" may
be facilitated. We have also seen that the nature of the
symbolism and the rules of manipulation must in each case
be determined by the special properties of the objects of
thought or experience to which the algebra is to be adapted.
In the case of ordinary algebra those objects are numbers.
Thus everything in ordinary algebra is, in the first instance,
a statement or a deduction about numbers or the operations
which can be performed upon numbers. Since, however,
numbers may, in one way or another, come into connexion
with everything in the universe, an algebraic statement may
refer, in the second instance, to any objects of thought or
experience whatsoever.
But this statement requires amplification in one important
particular, for it does not explain what is meant here by
" numbers ". The truth is that as we follow the development
of algebra, either in actual history or in the teaching of the
classroom, the meaning of the word " number " is itself
found to develop. At the outset it means simply the
ordinary integers and fractions of arithmetic supplemented,
perhaps, by the "irrational " numbers that make their first,
imperfectly understood, appearance as " surds ". Section I
of the present work was confined to the algebra of these
signless or "non-directed" numbers. But the progress
of the algebraic argument itself led to the introduction of a
new class of numbers — positive and negative, or " directed "
numbers. The investigation of the algebra based upon the
403 26*
404 ALGEBRA
properties of these new numbers was the subject-matter of
Sections II and III. At present the development has gone
no further, but in Section VI it is to make us acquainted with
yet another class of numerical entities — the "complex
numbers " of the form a + ih. Ordinary algebraic theory
has never found it necessary to go beyond these, and
the exploration of their main properties marks the limit of
the development of the idea of number in our course. It is,
however, instructive to note that there is another algebra —
the algebra of " vectors " — which is based upon yet another
extension of the number-concept, and has, accordingly, laws
differing in certain respects from those of ordinary algebra.
When we begin — as we propose to do in Section IV — to
submit these numerical concepts to critical examination, the
inquiry is seen at once to have a twofold character (cf. p. 16).
On the one hand, the various types of numbers either entered
historically into mathematical practice as means of furthering
man's attempts to elucidate the behaviour of the external
world or, having originated within the province of algebraic
theory, were afterwards found to have useful practical ap-
plications. On the other hand, regarded simply as concepts
or notions, they exhibit a definite logical development, frac-
tions being derived from integers, irrationals from rationals,
directed numbers from non-directed in accordance with
ascertainable principles. Thus to understand numbers
fully we must inquire both into their relations to the non-
numerical entities with which they are connected in mathe-
matical practice and into their logical relations to one another.
Exs. LXX-LXXIV pursue, in a simple way, both branches
of this inquiry, and are intended to present the most important
results reached by Dedekind (c. 1872), Georg Cantor (c.
1883), and Bertrand Eussell (c. 1903). The treatment is
based, in the main, upon Mr. Eussell's Principles of
Mathematics (1903), supplemented at certain points by Prof.
J. W. A. Young's Fundamental Concepts of Algebra and
Geometry (1911). The teacher who wishes to follow the
subject up should study those books, beginning with the
latter. If he is at ease with a French book he should consult
M. Louis Couturat's Les Principes des Mathematiques (1905),
in which the results of the most important modern writers,
including Mr. Bertrand Eussell, are described with character-
istic French lucidity. If he reads German he will do well to
NUMBER SYSTEMS 405
consult Dedekind's famous pamphlet Was sind und was
sollen die Zahlen ? (1888), a work not of forbidding
difficulty and one of the classics of the new logical move-
ment. Some of Cantor's most important papers have been
translated into French from the original German in Acta
Mathematica, vol. ii. The translations include the famous
Grundlagen einer allgemeinen Mannichfaltigkeitslehre (1883).
§ 2. Ex. LXX. Integers. — It is evident that the signless
integers must have been the first numbers to be used by
mankind, and also that they are the logical, as well as the
historical, basis of all other systems of numbers. Our in-
vestigation naturally begins, therefore, with a study of their
nature and properties. A given whole number, such as seven,
can be looked at from two points of view. Eegarded as a
" cardinal " number its individuality consists in its connexion
with all classes or collections of objects which have a certain
specifiable property. The following is a possible but fanciful
way of describing this property and so of discriminating from
all other collections those with which the number seven is
connected. On Sunday set aside or name or think of one
member of the collection under examination and one only.
Set aside or name or think of another on Monday, of another
on Tuesday, and so on until Saturday. If by this time the
whole collection is just exhausted it has the same cardinal
number as the days of the week ; if we call that number
" seven " then its number is seven. It is evident that this
definition by " one-to-one correspondence " gives to seven a
standing which is quite independent logically of all other
cardinal numbers. It would hold good if no one had ever
discovered any other collections than those containing seven
members. But, as a matter of fact, there are collections
with other numbers, and these collections can be derived from
one another by a definite process which may be endlessly re-
peated. To a collection with a certain number add another
member and you obtain a collection with another definite
number. When we take account in this way of the relations
of seven to other numbers we are regarding it ordinally. In
an elementary treatment it is not necessary to pursue this dis-
tinction very far. The important things to realize are (1) that
integers are based logically upon the fact of one-to-one
correspondence between collections, and (2) that they derive
from the ordinal relations of different collections the property
406 ALGEBRA
which enables us to use their symbols as " labels " of the
members of any sequence in which there is a definite order
corresponding to the order which subsists among countable
collections. In other words, we must realize that whereas
integers, regarded as cardinals, are necessarily connected with
things which have magnitude, regarded as ordinals they may
be connected with any things which can be arranged in a
sequence upon some definite principle, whether these things
have magnitude or not. Thus the magnitude of a cardinal
number is the magnitude of the collections which it describes,
and is a property which it possesses, so to speak, in its own
right. A number regarded ordinally may be said to be greater
than another only because it implies a greater number of
predecessors in the series of which it is a member. In view
of the errors which follow from confusing these two totally
different kinds of magnitude, it is best, except in special cir-
cumstances, to avoid speaking of ordinals as greater or less
than one another, and to speak of them only as coming before
or after one another in the sequence which they constitute.
This remark applies especially to positive and negative
numbers, for these are essentially ordinal (p. 160).
When the student has learnt that the notion of a cardinal
number is really based upon the idea of one-to-one corre-
spondence between the members of '* similar " or " equivalent "
collections, he is for the first time in a position to scrutinize
profitably the notion of an " infinite " number. The beginning
of the analysis is found in the recognition (§ 1, h) that an
infinite number cannot be exhausted by counting. The best
definition of an infinite collection is, however, supplied by
the arguments of Nos. 4-8. The essence of all these cases is
that part of a collection is found to correspond, term by term,
to the whole collection of which it is a part. It follows that
an infinite collection cannot be obtained by adding term to
term — nor be destroyed by taking away term after term ; but
this property is best regarded as a consequence of the former,
which is to be taken as the definition and test of an infinite
collection.
The objections which common sense suggests to this
startling notion of an infinite number are in part answered
by the last paragraph of the exercise. The natural man
starts out with the prejudice that all numbers must be like
the finite numbers whose properties can be explored by
NUMBER SYSTEMS 407
counting. Thus he finds it paradoxical to admit the exist-
ence of numbers some of whose properties are radically
different from those with which he is familiar. Faced with
incontrovertible arguments, like those of Nos. 4-8, he is apt
to turn philosopher and regard them as demonstrating noth-
ing but the inability of the human intellect to reach absolute
truth ! There are two ways of dealing with this pessimistic
conclusion. The first is to show that the rejection of in-
finite numbers in the sense defined leads common sense
itself to ridiculous conclusions. An example of this form of
reply is given in Ex. LXXII, No. 4. The second is to
show, as the modern mathematicians have shown abundantly,
that the assumption that in some collections the part is
actually equivalent to the whole leads to no results incon-
sistent with any other known truths. That is to say, un-
believers must be dealt with much as a mathematician who
lived in a world where only rectilinear figures were to be
seen might deal with persons who ridiculed his definition of
a new figure which he called a circle.
The method of one-to-one correspondence can be used to
show that there are different infinite, just as there are
different finite, numbers enjoying each its distinct individu-
ality. Thus it is possible to imagine any number of sequences
whose members have one-to-one correspondence with all the
integers, just as all collections whose number is seven have
one-to-one correspondence with the days of the week. The
days of a man who is born but never dies are an example of
such a sequence. If we may suppose that it is literally true
that " of the making of books there is no end," and that no
two books are ever finished at exactly the same time, we have
another example in the succession of books, beginning with
the first inscribed clay tablet but never ending. The charac-
teristics of all such sequences are (i) that there is a definite first
member of the sequence, (ii) that there is no last member, and
(iii) that every member has a definite successor. It is be-
cause they all possess these characteristics that the various
sequences of this kind exhibit one-to-one correspondence be-
tween their terms. It follows that they must all be supposed
to have the same infinite number. This number cannot, of
course, be expressed in digits but it may (like the finite num-
bers 6 and it) be referred to by a symbol. Cantor used the
Hebrew letter Aleph with zero as suffix to indicate that it is
408 ALGEBRA
the first of the infinite numbers. Other writers prefer the
symbol a^. Next, it is easy to point to other infinite col-
lections whose terms cannot be put into one-to-one corre-
spondence with the natural numbers. The points on a
straight line of finite length offer an example. They are
infinite in number, for a part of the line can be put into
point-to-point correspondence with the whole (Ex. LXXII,
No. 3). On the other hand, they diJEfer from the sequence
of integers in two respects : (i) there is a last member as
well as a first ; (ii) no member can be said to have a
definite successor, for between any two points, however
close, there is always a third. On account of these
differences the points cannot be put into one-to-one cor-
respondence with the integers. Nevertheless, Cantor has
shown ^ that there are at least two other collections with
which they can have one-to-one correspondence — namely the
points of a square and the points of a cube. Thus, just as
the collection of all the integers may be regarded as the
standard example of the infinite number a^, so the collection
of points on a straight line may be taken as the standard
instance of another infinite number — generally called the
" number of the continuum ". It can even be shown that the
two are connected by the relation
number of the continuum = 2°"
The analysis of the idea of infinite number leads to a
practical conclusion which the teacher should constantly
apply. The value of a variable is often said to be " infinite "
when it is greater than any finite number that can be named,
that is when it has no maximum. This is an improper use
of the term, for it does not follow from the fact that it has
no maximum that it is the number of a collection of which a
part is equivalent to the whole. All that can be said is that it
is indefinitely or endlessly great, and the symbol " oo " should
be understood to mean this and should never be read as " in-
finite ". It is still more desirable to avoid the too common
statement that the quotient of a finite number by zero is in-
finite. As we have seen at earlier points of the book (pp. 123,
^ Young, Concepts, etc.. Lectures VIII and XVI; Russell,
Principles, p. 311 ; Cantor, Acta Math., ii., " Une contribution 4 la
theorie des ensembles". See also Ex. CXVI, No. 20, and ch.
Lii., § 1.
NUMBER SYSTEMS 409
376) this statement is simply nonsense. In fact the discussion
of this article will have served a useful purpose if it convinces
any reader for the first time not only that the word " infinite "
is a numerical term capable of exact definition and therefore
of precisely limited application, but also that the occasions
for applying it in elementary mathematics are comparatively
rare. It is too apt to be used at other times merely as a
cloak to conceal absence of precise thought.
§ 3. Ex. LXXI. Bational Numbers. — All the other
numbers which appear in mathematics are based upon the
primitive series of integers. The first of these " artificial "
numbers are " fractions ". A fraction is, strictly speaking, a
pair'oi integers associated in accordance with a definite law.
This law enables us to substitute for each single integer a
pair of integers which can be taken as equivalent to it — the
pair being the given integer itseJf and the integer 1. (For
example 3/1 is equivalent to 3.) In this way we obtain an
infinite collection of numerical entities all of the same form.
They constitute the so-called " rational numbers ".
Nos. 1 and 2 are meant to bring out the differences be-
tween the sequence of integers and the sequence of rationals
arranged in their '* natural " order. In No. I we see that
there is an infinite number of rationals between any two
members of the series. This fact is evident from the con-
sideration that the sequence is "dense" (p. 10) so that its
terms cannot be exhausted by counting. In No. 2 (ii) the
number of rationals between 10 and 20 inclusive is infinite,
there is a first term (10) and a last term (20). It cannot be
said, however, that any term has either an immediate pre-
decessor or an immediate successor ; for example, it is im-
possible to name any definite fraction as coming next in
magnitude either before or after the fraction f . In No. 2
(iii) the rationals between 10 and 20 have no first term and
no last term ; for the rational 10/1 has no immediate suc-
cessor and the rational 20/1 no immediate predecessor. In
No. 3 every term has an immediate successor, so the sequence
cannot be dense. On the other hand, not every term has an
immediate predecessor ; for it is clear that (since there is no
limit to the value of r) the groups of terms preceding the
terms 2, 3, 4, . . . have no last member.
When we take the symbols of the rationals and, breaking
away from the " natural " order, arrange them in the order
410 ALGEBRA
indicated in Nos. 4, 5,^ we obtain a sequence which is
" ordinally similar " to the sequence of integers. That is to
say, it has a first but no last term, every term except the
first has an immediate predecessor and every term an im-
mediate successor. It follows (No. 6) that the rationals
arranged in this way can be brought into one-to-one corre-
spondence with the integers in their natural order, and there-
fore have the same infinite number, a„. To use Cantor's
term, they form a denumerable series. This result is a
striking instance of the power of the conception of one-to-
one correspondence.
§ 4. Ex. LXXII. Irrationals. Continuity. — The subject
of this exercise came before us at an early point of the course.
It was seen in ch. viii., B (p. 94), that, although a number
may always be obtained whose square is as nearly as we
please equal to a given number, yet in most cases there
seems no probability that an exact square root could be
found. Consequently, when we came in ch. xxiii. to identify
the complete sequence of positive and negative numbers with
points on an endless straight line, we were obliged to recog-
nize that certain points correspond to no number at all,
integral or fractional. For example, the point whose distance
from the origin measures the length of the side of a square
whose area is 2 units would be such a point. If this point
is to have a numerical label at all, we must invent a new one.
It was suggested (p. 231) that the best label for this purpose
would be the symbol " J 2 ".
Now the interesting thing about this suggestion is that the
symbol J2 had hitherto been used to represent any number
whose square was sufficiently near to 2 for the purpose in
view (p. 94). Thus, regarding the line OX in Exercises II,
fig. 62, as the positive half of the linear scale referred to above,
J2 would not be the label of the point P, but could be
attached ambiguously to any point sufficiently near to P. It
follows that the proposal to use J2 as a label for P itself is,
strictly speaking, a new departure. When examined more
carefully the proposal is seen to raise the following im-
portant question : Symbols such as 7 or f are not only
labels for specific points on the line OX but are also symbols
for numbers ; can it be said that J2, which we have now
^ The scheme in No. 5 is copied from Young, Concepts, etc., p. 74.
NUMBER SYSTEMS 411
adopted as the label for the point P, is also the symbol for a
number? The problem of Ex. LXXII is to answer this
question and the subsidiary questions which it includes.
The solution of the problem is very modern, but the problem
in its essence is of great antiquity. Pythagoras (c. 530 B.C.)
or his followers discovered that the diagonal of a square is
incommensurable with the side — that is, that the ratio of the
lengths of the lines cannot be expressed by any integral or
fractional number. This discovery was the starting-point of
investigations that culminated in Euclid's Book X, a masterly
treatise upon "irrational" ratios. ^ In accordance with a
remark made elsewhere (pp. 32, 70), Euclid's argument,
though essentially arithmetical, was necessarily conducted in
geometrical terms. A treatment arithmetical in form as well
as in substance was possible only after the invention of the
Arabic notation.
One of the earliest examples of such a treatment is to be
found in the second book of the Arithmetica Integra (1544)
of Michael Stifel, " pastor of the Church at Holtzdort ".^ It
is practically a restatement of Euclid's doctrine in the Arabic
notation, together with a running commentary upon it. In
the first chapter he comes at once to the question raised above :
namely, whether the so-called " irrational numbers " are
really numbers at all. As he subsequently points out (ch. ii.)
Euclid was clearly of opinion that they are not numbers.^
In this chapter, however, he discusses the matter, as we have
done, with reference to the problem of expressing " irrationals "
in the decimal notation. Some of his observations are worth
quoting as exemplifying the difficulties and uncertainties that
beset all early attempts to give a clear account of the funda-
mental notions of arithmetic and algebra (cf. p. 159). On the
^ See Sir T. L. Heath's edition, Vol. III. It is interesting to
n otethat Euclid's use of the terms '' rational " and ''irrational " is
not identical with the modern one. According to his definition a
line is irrational with regard to a given unit-line only if neither
the lengths of the lines nor the areas of the squares upon them
are commensurable.
2 The table of binomial coefficients given on p. 213 appears in
Book I of this work (folio 45).
'•^ Book X, Prop. V : " Commensurable magnitudes have to one
another the ratio which a number has to a number " ; Prop. VII :
" Incommensurable magnitudes have not to one another the ratio
which a number has to a number ".
412 ALGEBRA
one hand, argues Stifel, " since, in proving geometrical figures,
when rational numbers fail us irrational numbers take their
place and prove exactly those things which rational numbers
could not prove, ... we are moved and compelled to assert
that they truly are numbers, compelled, that is, by the
results which follow from their use — results which we per-
ceive to be real, certain, and constant ". '* On the other hand,
other considerations . . . compel us ,to deny that irrational
numbers are numbers at all. To wit, when we seek to sub-
ject them to numeration ... we find that they flee away
perpetually, so that not one of them can be apprehended
precisely in itself. . . . Now that cannot be called a true
number which is of such a nature that it lacks precision. . . .
Therefore, just as an infinite number is not a number, so an
irrational number is not a true number, but lies hidden in
a kind of cloud of infinity." " Again," he continues, " if ir-
rational numbers were real numbers they would be either
whole numbers or fractions." They are certainly not whole
numbers, for it is easily seen that each irrational falls between
A P B . X
i f i
Fig. 92.
two consecutive integers. Also they are not fractions with
definite numerators and denominators, for when a fraction is
multiplied by itself it can never yield a whole number, while
an irrational, if it is the square root of an integer, will always
do so. Thus they cannot be real numbers.
It is not easy to improve upon this statement of the
dilemma. On the one hand, irrationals must be real numbers ;
for calculations which make use of them are just as trust-
worthy as those which use only rationals. On the other
hand, they cannot be real numbers because they are neither
integers nor fractions. There is only one way of escape
from it ; some definition of the term ** number " must be
found which, while making rationals a special case of a
general concept, will at the same time leave a place for
irrationals as another special case.
The preliminary discussion in ch. xxiii. suggests two
lines of attack of which one or both may lead to a conquest
of this problem. Let OX (fig. 92) be a linear scale, beginning
with the point O but endless towards X. Imagine every
NUMBER SYSTEMS 413
point which corresponds to a rational number to be labelled
with the symbol of that number, integers being represented,
for the sake of homogeneity, in the form of fractions with 1
as the denominator. Since it is impossible actually to show
all these numerical labels in the diagram, we content our-
selves with inserting three as specimens (i, f, f). Let P
be any point on the scale selected at random; then, by
hypothesis, P is in every case to be regarded as the
representative of a number. If we happen to have hit
a point to which one of our labels is attached, there is,
of course, no difficulty; the number represented by P is
simply the number described by the label. Trouble arises
only if the point selected has no label ; for then we have
to ask how this point can be regarded as representing a
number iji the same sense as the point which does bear a label.
To answer this question we note that the point P can be
thought of in two ways : first, it marks a definite position on
the line OX between points on the left of it and points on the
right; secondly, it marks a definite length of the line OX,
starting from the origin. Both of these statements are true
of P whether it is or is not labelled with the symbol of a
rational number ; either of them may be taken, therefore, as
the principle underlying the new definition of the word
" number " which we are seeking. Since the new meaning
of "number" is to include more than "rational number" in-
cludes, it will be well to associate with the word a distinguish-
ing adjective, just as we added the adjective " rational " when
we wanted "number" to mean more than the original
sequence of integers. For historical reasons the term " real
number" must be adopted for this purpose in spite of its
misleading suggestions (ch. xlix.). Our immediate task,
then, is to find and to examine two alternative definitions of
" real numbers," both having the property that they include
two subclasses of numbers, "rationals" and "irrationals,"
upon an equal footing.
In seeking the first definition we follow the path taken in-
dependently by the German, Eichard Dedekind, and the
Frenchman, Paul Tannery. Let us begin by supposing the
point P (fig. 92) to be associated with a rational number — for
example f . Looking at this number isolated from others we
may regard it, as in § 3, simply as a combination of the integers
3 and 2. On the other hand, if we look at the symbol -| in
414 ALGEBRA
its place among the other symbols disposed along the line OX,
we see that, just as P may be regarded as a boundary separat-
ing points to the right of it from points to the left, so the
number symbolized by f may be thought of as a boundary
separating the numbers before it in the natural sequence from
the numbers that follow it. It may help to keep these two
aspects of the number apart if we adopt different ways of
printing its symbol to correspond to them respectively. Thus
we may print the symbol in ordinary type, f , when we think
of it simply as a combination of integers, and in heavy type,
f , when we think of it as a boundary between the pairs of
integers below and the pairs of integers above. In the second
capacity f is to be called a "real number ".
Fig. 93 illustrates this definition. The point P is shown
isolated, the two segments, OA and BX, which it separates,
being withdrawn to right and left, together, of course, with
the number-symbols attached to them. It is important to
observe that the segment OA can have no definite end-point
to the right, for in the original line it was impossible to say
Pig. 93.
that P had a definite point immediately to the left of it.
Similarly, the segment BX has no first point at the end
adjacent to P. Thus the point P may be regarded as a
"cut" {Schnitt, coupure) yNhich. divides the whole line into
a lower segment which has no end-point and an upper
segment which has no point of beginning.
An alternative way of expressing the same facts is to say
that P is the "upper limit" of the segment OA and the
" lower limit " of the segment BX, P being itself excluded
from both these segments. Correspondingly, we may say
that the " real number " f which is associated with P is a
"cut "in the complete sequence of rationals taken in their
natural order, a cut which divides them into a lower segment
without a last member and an upper segment without a first
member. Alternatively, we may say that it is the upper
limit of the rationals (i.e. of the pairs of integers) which come
before | and the lower limit of the rationals which follow |.
Now suppose that our random choice had lighted upon a
point P which is not associated with a label. Then we cannot
NUMBER SYSTEMS 415
this time say that the point corresponds to a number in the
sense that | is a number. But we may nevertheless think
of it as corresponding to a number in the sense that f is a
number ; for it marks in exactly the same way as before a
"cut" dividing the complete sequence of rationals into an
endless lower segment and a beginningless upper segment.
Since we have decided that the term "real number" shall
imply simply this aspect of being a "cut" or a "limit" in
the sequence of rationals, we are entitled to say that although
P does not now correspond to a rational number yet it still
corresponds to a real number. It remains only to assign to
this number (i.e. to this mode of section of the complete se-
quence of rationals) a suitable name and a suitable symbol.^
To sum up : Imagine fig. 93 completed by the addition of
the symbols of all the rational numbers each in its proper
place in the scale. Then the figure represents a division of
these numbers into a lower segment without a last term, and
an upper segment without a first term. Bach of the infinitely
numerous ways in which this division can be made constitutes
a " real number ". It may be that the point P itself cor-
responds to a rational number which is therefore excluded
from the two segments. In this case the mode of division is
a "rational real number" and is named from the excluded
rational represented by P. Again it may be that all the
rationals fall into either OA or BX. In that case the mode
of section is an " irrational real number," and must be named
in some suitable way. For example, if the rationals in the
lower section are all those whose squares are less than two,
and the rationals in the upper section all those whose squares
are greater than two, then the "real number" is most con-
veniently called "the square root of two". Lastly, a
rational number may be thought of either as merely a
rational number or as a rational real number ; an irrational
must always be thought of as a real number. ^
1 Compare .with this the argument on p. 182 which established
the validity of such expressions as 4 - 7, in which the second number
is greater than the first.
^ The terms " rational real number " and " irrational real number "
are used, for convenience only, to distinguish real numbers which
correspond to rational numbers from those which do not. It is
obvious that there is no difference in "rationality" between the
two subclasses.
416 ALGEBRA
We turn now to the definition of a real number suggested
by the consideration that the number attached to the point
P may be thought of as representing the length of the segment
OP. Here we follow Mr. Bertrand Eussell. Once more
divide the linear scale into two segments, but let P be the last
point of the lower segment (fig. 94). All the symbols of the
rationals will now always appear in one segment or the
other. There will, however, still be two cases. In the first
case the point P which ends the lower segment corresponds
to a rational number in the sense denoted by the symbol f .
In this case the corresponding " real number " represented by
f may be defined as the whole collection of rationals whose
symbols would have their places to the left of P. In the
second case the point P will represent no rational. But in
this case also it will be considered as corresponding to a real
number, and that real number will again be defined as the
collection of all the rationals whose places lie to the left of P.
Thus, according to Mr. Eussell's definition, every real number
is a collection or set of rationals taken in order from zero up-
A P B .X
Fig. 94.
f
wards. In some cases the set can be described adequately
by the statement that it consists of all the rationals which
are less than a certain rational N. In that case the real
number is a rational real number and may be named from
the rational N, though it would be well to emphasize the
difference between N and the real number named from it by
a distinctive method of printing the symbol (N). In other
cases there is no definite rational N from which the set may
be named. In that case the real number (i.e. the set of
rationals) is an irrational real number and must be named in
some suitable way. For example, if the set consists of all
rationals whose squares are less than two it may conveniently
be called the square root of two.
The Dedekind-Tannery definition has been explained fully,
partly because of its historical importance, partly because it
is at present the one most widely known, partly because it
involves ideas — such as a *' limit" — which will be of impor-
tance in the sequel. There can be no question, however, that
for a student who approaches the subject for the first time
NUMBER SYSTEMS 417
the Russell definition is incomparably the easier to grasp.
Moreover, it is logically superior — and this fact is without
doubt the source of its greater simplicity. For it will be seen
that rational and irrational numbers as defined by the former
method are not really homogeneous. When the limit of the
two segments is a rational number it is actually there ; when
it is not a rational number it is not there, but has, so to speak,
to be defined into existence. On the other hand, the sets
of rationals which constitute real numbers according to the
Russell definition are equally " there " whether they correspond
to rationals or not.
For these reasons the Russell definition is taken as the
basis of the treatment in Ex. LXXII. It need hardly be
said that the illustration of the boxes is no part of Mr.
Russell's argument, but is introduced merely for didactic
purposes and, in particular, to show how a " continuum " of
numbers can be built up from the ordered sequence of
integers (through the intermediate step of a " compact "
sequence of rationals) without reference to a line or any
other magnitude. If the teacher does not find the illustration
helpful, he can, of course, discard it and substitute a treatment
more on the lines of the present article. With regard to the
idea of continuity itself it would probably not be profitable
or even possible to carry the discussion much farther than it
is carried in Ex. LXXII. The results reached there are
(i) that the points of a line are a sequence obeying Dedekind's
Postulate, (ii) that the same is true of the real numbers, and
(iii) that for this reason both are said to possess continuity.
It should be noted that we do not prove that our scheme
provides a real number for every point on the line. Whether
it does so or not it is apparently impossible to say. What
can be said is, however, that mathematicians have never
found it necessary to postulate any points to which real
numbers do not correspond. It is worth while examining a
little the significance of this historical fact. Take one of the
imagined boxes of Ex. LXXII, § 2, and suppose it labelled
with an irrational symbol I. This label implies that the box
is reserved for a perfectly definite collection of rationals (con-
stituting the real number I) and that all other rationals are to
be placed elsewhere. Now suppose the label to be changed
to another irrational symbol V whose place is higher up the
linear scale. Then it is clear that the box now offers hos-
T. 27
418 ALGEBRA
pitality to rationals which were previously excluded ; for T
would not be a different real number from I if it consisted in
the same set of rationals. Thus we must suppose that certain
rationals which were previously outside are now to be found
inside the box. But it is clear that any one of these rationals
(say R) might have stopped on its way from the outside to the
inside and itself become the label of the box. The set of ra-
tionals implied by this new label would, of course, be the real
number R. It appears, therefore, that between any two real
numbers I and T which are not associated with rationals there
must be at least one real number which is so associated. Now
the same thing may be true of the points on a line ; in that
case real numbers, defined as collections of rationals, would
suffice for any calculation concerning points on the line. On
the other hand, the constitution of a line might be such that
two points unconnected with rationals could be found with
no point connected with a rational between them. In
that case real numbers would not be adequate for all cal-
culations about lines. But as a matter of fact no argument
has ever been produced which contradicts the common
assumption that the real numbers form a number scheme
adequate for all calculations involving space.
The teacher should not fail to point out that the results of
Ex. LXXII are needed to put much of the work of Part I
upon a proper logical basis. Thus it has constantly been
supposed that the graphs of functions such a,s y = ax,
y = a Jx, y = a"" are continuous lines, i.e. lines unbroken by
any gaps in which points could be inserted which were not
points of the original graph. Yet if the graphs are only
assemblages of the points which correspond to rational
values of x and y there must be an infinite number of such
gaps. The fact that they cannot be exhibited to the eye
does not touch the fact that reason shows them to be there.
Thus all through our elementary work we have been tacitly
assuming Dedekind's Postulate with regard to the number-
scale ; that is, we have taken it for granted that there is
always a number corresponding to any point upon a line.
We now know that this assumption is justifiable only if by
number we mean " real number ".
§ 5. The Nature of e and ir. — The special position of the
numbers e and tt demands some mention in a review of
numbers in general, but it must be restricted here to a brief
NUMBER SYSTEMS 419
note. The question whether it is possible to " square the
circle " has, of course, had a very long history ; that of the
value of e goes back only to the seventeenth century. In the
case of both these entities the inquiries of mathematicians have
gradually narrowed down to the question whether they are
"algebraic" or "transcendent" numbers. A number is
algebraic if it can be the root of an equation of any degree
Co + G^x + C,a;2 + C^x' + ... + C^ic" =
in which the coefficients Cq, C^, C2, etc., are rational numbers.
If it cannot be such a root it is transcendent. The solution
of the problem is quite modern. In 1873 Hermite succeeded
in proving that e is transcendent. In 1882 Lindemann
followed with a proof of the transcendence of tt based upon
the results of Hermite. The reader who wishes to know the
nature of these proofs should consult a paper by Prof. D. E.
Smith in Young's Monographs on Modern Mathematics.
The elementary student must be content to know that
although e and tt are not rational numbers their " irration-
ality "differs in an important respect from that of surds ; these
are algebraic numbers, those are not.
§ 6. The Paradoxes of Zeno. — We now turn to the ex-
amples of Ex. LXXII. In Nos. 2, 3 we simply meet
again the property that in an infinite collection a part can
have one-to-one correspondence with the whole. No. 4
states the best known of the paradoxes of Zeno. These
paradoxes have generally been thought to prove the incapacity
of the human mind to deal with the idea of the infinite, but
Mr. Russell has shown that they cease to be insoluble riddles
when we recognize that infinite numbers have the property
to which reference has just been made. The argument
underlying the paradox is as follows : At any moment of
time Achilles and the tortoise are each at some point of
their respective paths. Thus during any given period (that
is, in the course of a definite series of moments) each must
visit the same number of points. Hence the path of the
tortoise cannot be a part of the path of Achilles, for it would
in that case contain fewer points. The fallacy lies, of course,
in the last statement ; for although one path is a part of the
other the number of points in each is the same — namely, the
number of the ' ' real numbers ". Thus, even though Achilles
27*
420 ALGEBRA
should far outstrip his competitor, the points occupied by
each in his course can be correlated one by one.^
§ 7. Ex. LXXIII. Operations upon Numbers. — The aim
of this exercise is to find definitions of the arithmetical
operations which will apply to numbers of all kinds. The
analysis is probably carried far enough there for the average
pupil but may be completed here.
Addition of integers is explained in the exercise by the
combination of sub-collections into a whole; addition of
rationals by the combination of lengths. It will, however,
be seen that if the symbols are taken to represent real
numbers then the things combined are once more collections
— to wit, collections of rationals. The same would be true if
the symbols were irrational. Thus if we take " number " to
mean " real number " there is a single definition for addition
and (therefore) for subtraction.
Similarly for multiplication and with it division. It is
easily shown that ' ' repeated addition " is a definition which
holds good only for integers, but that the alternative definition
is applicable to all numbers if they are regarded as sets of
rationals — that is, it is applicable to all real numbers. The
argument of Ex. LXXIII, § 4, is directed to the case of
rationals, but it is obvious that the definition of the product
would apply equally if the factors were irrational.
In the case of the sum and product of rationals and
irrationals there is a second problem. It is not enough to
find a formal definition which brings them into line with the
sums and products of integers ; we must also find numbers to
express them. The case of rationals is dealt with in §§ 3, 4.
The principle followed is that the sum and product are to be
measured by rules that would give, in the case of rationals
which are also integers (e.g. 6/1, 17/1), the numbers which
we already recognize as the sum and product of those
integers. In the case of the sum of two irrationals the rule
gives us no assistance ; for example, if the irrationals are
Ja and Jb^ the sum can only be expressed in the form
J a + Jb. In the case of products it may, however, often
be applied with advantage. Thus let J a and Jb be any
two real numbers, rational or irrational, and let the rectangle
^ For this and the other paradoxes see Russell, Principles,
ch. XLII.
NUMBER SYSTEMS 421
be drawn the points of whose area represent the pairs of
rationals taken from the sets denoted by Ja and Jh re-
spectively. Then the top right hand point will be ( J a, Jh),
and our problem is to replace this pair of numbers by a
single number. Now in every case in which a and h are
squares the area will be measured by Jc, where c is the
product of ah by the rules for integers or rationals. By our
principle, therefore, Jc is also to be counted as the product
when Ja and Jh are irrational. This argument solves
No. 14. The teacher should have no difficulty in finding
another instructive proof based on the idea that the product
of J a and Jh must be represented by a number which comes
between (i.e. is the " limit " of) the products jpq and p'q^
where p and q are any rationals less than J a and Jh
respectively, and p and q^ any rationals greater respectively
than these irrationals.
§ 8. Ex. LXXIV. The Complete Numher Scale.— k good deal
of the argument of this exercise has already appeared in ch.
XVII., §§ 1-4, and in chs. xviii. and xxiii. To these the reader
should refer. Attention should be given to § 2 of the exercise,
for it expresses the essence of algebra regarded as a symbolic
logic : Algebraic operations simply carry us backwards and
forwards along the endless linear continuum of the numerical
symbols, but we can always interpret our results in terms of
things (numbers, magnitudes, etc.) which have independent
existence. In Section VI this view will be modified to the
extent that we shall find that the operations of algebra carry
us over a continuum of symbols of two dimensions — namely,
the field of "complex numbers" — but in essence the present
conclusion will stand.
In this exercise the student is introduced to the interesting
and important idea of a "group". The theory of groups is
a large department of pure mathematics which has very im-
portant applications in mathematical physics. The account
given in the exercise is based on Young, Concepts, etc.,
Lecture IX. ^
iProf. E. W. Hobson's little book, ''Squaring the Circle"
(Camb. Univ. Press), appeared too late to be mentioned in § 5.
The teacher will find it an admirable, as well as an authoritative,
introduction to its subject.
CHAPTER XL.
FUNCTIONS.
§ 1. Ex. LXXV. Functions of one Variable. — The first
two articles of this exercise are mainly a restatement in print
of what the student has learnt orally in Part I ; the third
gives a simple explanation, in graphic terms, of the meaning
of the adjectives "continuous" and "discontinuous" as
applied to functions. The teacher should remind the class
+ !•:
-2
-I
+1
Pig. 95.
that in speaking of 2/ as a continuous function of x we pre-
suppose that the values of x themselves form a " continuum " ;
i.e. that x means any and every real number within the range
in question. If this were not the case the graph could not
be considered unbroken even though the function were a
continuous one (p. 418).
Fig. 95 is the graph of No. 12. The dots on the left of
422
FUNCTIONS
423
the i/-axis correspond to ( - 4) - ^ ( - 3) " 3, ( - 2) " ^^ ( _ i) - i.
These are isolated values of the function, for it is impossible
to find a real number which gives the value of x"" when x is
both negative and non-integral. After a; = the function is
continuous, + 1 being the "limit" as x approaches from
the positive side. Figs. 96-8 are the graphs of the functions
Fig. 96.
\
S
+ 16
+ 14 +
+ 12
+ 10
+ 8
+ 6
+ 4
+ 2
^k -h -^ ~ ro
/
+1 +2 +3 +4
Fig. 97.
of No. 13 (cf. Hardy, Pure Mathematics, p. 47). Fig. 115,
p. 484* (omitting the broken line), is the graph of No. 14.
In division B the examples deal with the problem of
determining functions which have given properties. Inci-
dentally the student learns the use of the important notation
424
ALGEBRA
y\= f{x). The interesting examples Nos. 21-3 were sug-
gested to the author by Dr. L. Silberstein ; fig. 99 is the
solution of No. 23. It is obvious that the graph could be
continued in "waves" increasing indefinitely towards the
+ 2
^
+ 2 +4
-2
Fig. 98.
♦I
+4
Fig. 99.
right and decreasing indefinitely towards the left. The same
remark applies to the graph of No. 28. Note the parallelism
between Nos. 21-3 and Nos. 27-8. Nos. 31-2 are taken from
Mr. G. H. Hardy's Pure Mathematics. In No. 31 the graphs
approximate to fig. 100 A, in No. 32 to fig. 100 B.
FUNCTIONS
425
§ 2. Ex. LXXVI. Some Peculiarities of Functions. — The
first division of the exercise is again for revision. Careful
attention should be given to the argument in Nos. 8-1 1. In
No. II, putting X = 1 + h, we have
, m(m - 1), m(m - 1) (m - 2),» ,
y = m + ~>-^^ — ^h + -^ ^ ^h^ + . . .
whence it is obvious that as h approaches zero (i.e. as x
approaches unity) y approaches m.
Division B revises the notion of a " gradient " and
generalizes it into the idea of the rate of change of a function
as the value of the independent variable changes. The
teacher may find it profitable to treat this topic by means of
Newton's conception of " fluxions ". Newton thought of
A. B.
+1 ■
I o
-I
4-1 •
+1
-I
+1
Fig. 100.
both variables as changing with the time ; the rate of change
of the function is, upon this view, the ratio of the velocities
with which the two variables are increasing. It is possible
that this way of looking at the matter was suggested to
Newton by Napier's theory of logarithms. We have seen
(p. 300) that Napier thought of the sine and its logarithm as
represented by points (fig. 77) moving along two parallel
lines. This idea may be generalized and the points regarded
as representing the values of x and y in the case of any
function y = f{x). Then the speeds of the two points are
the "fluxions" of the variables, represented by Newton by
426 ALGEBRA
the notation x and y. The rate of change of the function
would thus be the ratio yjx. If the lines along which the
points are moving are set at right angles to one another the
successive positions of the points determine the graph of the
function as we understand it — just as the growth-curve of
fig. 78 (p. 302) is generated from fig. 77.
In dealing with this subject the teacher should bear in
mind what was said on pp. 246-50 about the relation of the
" calculus of approximations " to the doctrine, based upon
the theory of limits, to be studied in Section VIII.
Division C offers a simple treatment of the fascinating
topic of " singular values ". In No. 25 the graph must first
be shifted a unit place to the left so that the point (+ 1, 0)
may become coincident with the origin. The corresponding
expression for the function is
2/2 = [x + l)ic2
= aj3 + x^.
It is now seen that the two tangents at the origin are given
by y^ - flj'^ = ; that is, that they are the lines y = x and
y = - X,
§ 3. Ex. LXXVII. Functions of two Variables. — ^From
the technical point of view this exercise has great importance
as teaching the student to apply the methods of "coordinate
geometry " to tridimensional space. In accordance, however,
with the principle explained on p. 47 the argument is pre-
sented as a study of the graphic representation of functions
of two variables. The treatment is based upon an idea with
which the student may be presumed to be fairly familiar,
namely, that of the " contour lines " which are used to re-
present the relief of a district in a geographical map. The
teacher should note (i) that the general notion of a function
of two variables is approached through the study of concrete
cases (cf. pp. 109-10 and ch. xii.) ; and (ii) that constant use
is made of the two devices of shifting and rotating a graph in
order to obtain the functions to which it corresponds in its
various positions (see ch. xxvi., A, § 2 ; B, § 2 ; and Exercises,
I, p. 325).
The student should be taught to represent and to realize
solid forms by means of contour lines just as in the case of
the map of a mountainous country. It should be understood
that in order to obtain an adequate idea of a given form it is
sometimes necessary and always helpful to draw " contour
FUNCTIONS 427
maps " of it corresponding to sections parallel to each of the
three coordinate planes. These notions are suggested in
Nos. 9-13. It is also well worth while to construct at
least one solid from its contours in the way indicated in
No. 13. A simple method is to cut a series of contours out
in paper or thin card and to use them as " templates " in
cutting slabs of the same shapes out of a sheet of thick but
tractable material. The felt sold to fix under carpets serves
well for this purpose ; it is cheap, uniformly thick, and easily
manipulated. When the slabs have been fixed in the posi-
tions in which they build up the solid, the surface may be
smoothed by means of a sharp knife or razor. Clay or
plasticine models are, of course, better than those of felt but
are more troublesome to construct.
The student who has worked Part I, Exs. LXIV and
LXV, is not likely to find difficulty in the examples of the
present exercise. If Exs. LXIV and LXV have not been
worked they should be taken between Exs. LXXVI and
LXXVII.
CHAPTER XLI.
THE EXPONENTIAL FUNCTION AND CUEVE.
§ 1. Ex. LXXVIII. The Development of Algebraic Sym-
bolism. — The exercises reviewed in this chapter are not
very closely connected but present a certain degree of unity
through their relation to the theory of the logarithmic and
antilogarithmic or exponential functions.
Ex. LXXVIII aims at a brief exposition of the view of the
nature and development of algebraic symbolism which has
already been set out at some length in the first chapter of
this book. The teacher may, at his discretion, use the
material of that chapter to amplify the discussion. The
argument is illustrated (i) by a revision of the development
of the exponential notation as it actually occurred in the
course of Part I ; (ii) by a sketch of an alternative mode of
development which will be recognized as the one usually
given in textbooks. Something is no doubt to be gained by
showing that the evolution of algebraic ideas may proceed in
more than one way ; on the other hand, some teachers may
think the advantage of a wider view to be neutralized by the
risk of confusing the student by a double presentation. Truth
to tell, if prudence did not suggest that the demands of
examiners must not be ignored, the author would have
chosen to emphasize the argument of Section III rather than
to offer an alternative. Reasons for this attitude have been
given already (pp. 57, 300) ; they may be supplemented by
the contention that the method of Section III is much more
direct and arithmetical than the usual method, and is there-
fore more in accordance with the most fruitful tendencies of
modern mathematics. Consideration will show, in fact, that
the review in Ex. LXXIII of the fundamental operations on
numbers is incomplete through its omission of the operations
THE EXPONENTIAL FUNCTION AND CURVE 429
represented by the logarithmic and exponential notations ;
for the definition of the exponential operation as repeated
multiplication (or division), like the definition of multiplication
as repeated addition, breaks down when the operating number
is not an integer. Thus the symbolism al' presents a
theoretical problem of exactly the same kind as those solved
in the cases of the symbolisms a + 6 and ah. In all three
cases we have a pair of numbers (a, h) which is to be re-
placed by a single number c in accordance with certain
principles of equivalence, and the problem is to find a general
definition which will include as special cases the different
principles used according as the numbers a and h are in-
tegral, rational, or irrational.
In the case of equivalences of the form a^ = c the nearest
approach we have yet made to a universal definition is the
rule that if m and n are any values whatsoever of h, then a'"
and a"- must be subject to the relation
a'" X a** - a"' + ".
In other words, the principle of equivalence which enables us
to write a"" — c^ and a" = c.^ must also enable us to write
(j^m + n ^ ^^^^ Thus our task resolves itself into the search
for a definition of the symbolism a" from which this formal
rule will follow in the case of all numerical values of a and b.
The following definition (which is a paraphrase of the one
given by Cantor^) will be found to possess the required
generality. Let M and N be two classes or sets of elements
whose numbers are respectively a and b. Then, as we saw
in Ex. LXXII, ab may be defined, for all real values of a and
b, as the number belonging to the collection of pairs which
can be made by associating each element of M with each
element of N. According to the definition now to be studied,
a* is to mean another mode of association between the
elements of M and N — one in which we consider not pairs
but sets of pairs made up upon the following plan. Take
each of the elements of N and associate with it any one of
the elements of M, in such a way that a given element of M
may be either left unpaired or associated with 1, 2, 3, . . .
or all of the elements of N. Let c be the number of the sets
of pairs which can be formed in this way ; then, by definition,
c = a\ As an illustration let P, Q and R be the elements
^ See Russell, Principles, p. 308.
V
p
Q.
Q.
V
p
g.
3
V
9^
?
P
V
q
9.
p
430 ALGEBRA
of N and p, q the elements of M. Then a* is represented by
scheme I : —
I. II.
P Q R ST
P P P
9 q q
q p q
p q p
q
p
p
q
The rows below the top line exhibit the elements which
are combined with P, Q and R to give the successive sets.
The scheme represents the case in which a = 2,b = S, while
a* = 8, as it should do in accordance with the usual defini-
tion. Similarly, scheme II shows the case in which a = 2 as
before, b = 2 and a^ = 4.
In accordance with our definition scheme I represents a
class of 2^ elements, each element being one of the rows of
letters. Similarly scheme II represents a class of 2^ ele-
ments. If now we want to construct the class whose number
is the product of 2^ and 2^ we must, by the definition of
multiplication, associate each row of the one scheme with
each roiu of the other. In this way a more elaborate scheme
will be produced consisting of rows of letters associated with
the five letters P, Q, R, S, T. The following arrangement
shows a few of the rows : —
III.
P Q R S T
P P P P P
q q q p p
p p q p q
It is evident that in scheme III we have the two letters
p, q associated with the five letters P, Q, R, S, T in the manner
laid down in the definition under consideration. We conclude
that the number of terms in the product-scheme may be
represented as 2^ and therefore that
23 X 22 = 2^
THE EXPONENTIAL FUNCTION AND CURVE 431
Now it is clear in the first place that this argument could
be repeated to prove that
a^ X a" = a"'+'*
for all integral values of a, m, and n. It will further be seen
that it establishes the same relation when the numbers are
not necessarily integral but are any real numbers. For in
this case the elements ^, (7, . . . P, Q, K, . . . and
S, T, . . . may be taken to be the sets of rationals of which the
real numbers a, m, and n consist, and it is evident that,
although the infinite collections of elements now connoted by
these symbols cannot be set down on paper, yet the schemes
of association are in conception just as definite as before.
That is to say, we can conceive (though we cannot picture)
schemes of association in which the rationals contained in
the real numbers m and n play respectively the parts assigned
to P, Q, E and S, T in schemes I and II, while the rationals
contained in the real number a take up the r6le of 'p and q.
Further, we can conceive these modes of association combined
into a scheme whose relation to the schemes just described
is exactly that of III to I and II, the elements P, Q, R, S, T
being replaced by the rationals contained in the real number
m + n. We conclude that, the foregoing interpretation of
the exponential notation being adopted, the relation
a™ X a" = a" + "
holds good for any real numbers a, m, and n. Thus our new
definition of the equivalence a* = c has universal validity;
for it not only coincides with the original definition in the
case when the symbols represent integers but also leads to
the formal law with which any admissible enlargement of the
original use of the notation must comply.
In Ex. LXXIII, p. 23, we saw that after arriving at a
general definition of the equivalence ab = c we had still to
face the practical problem of determining a value for c when
a and b are given. Similarly, the foregoing definition of the
equivalence a** = C must be supplemented by a rule for
assigning to c a numerical value to correspond to given values
of a and b. For this rule we must return to the method of
ch. XXXIV., noting that, although the definition studied in the
present article applies with equal precision to integral, rational,
and irrational values of the variables, yet it is possible in the
last two cases to calculate only approximately the actual
arithmetical equivalences.
432 ALGEBRA
The teacher will observe that the foregoing argument carries
the analysis much farther than it is taken in Ex. LXXVII,
and he must decide for himself whether or not to supplement
the simpler discussion offered there. If he decides upon the
completer treatment he will find it convenient to introduce it
in connexion with No. 14 of the examples. For that example
is meant to bring out the fact that the definition hitherto
given of the functions
y = logaX and y = a""
fails to secure complete generality. The proof is very simple.
Eef erring to the scheme on p. 341 of this book we see that
the logarithm as there defined is always a multiple of a
rational number k{ = 1/p). Thus, in the function y = log^x,
y must always be rational, and in the function y == a", x
must always be rational. In the alternative definition of
Ex. LXXVIII, § 3, X, in the function y = a^, is necessarily
of the form piq where p and q are integers ; whence the
same results follow again. Thus the antilogarithm of an
irrational such as ^/2 (No. 15) can be defined only indirectly.
In the upper line of the scheme on p. 341 there will be
always one multiple of h which is less than J2 while the
next multiple above it is greater than ^2. In correspondence
with this fact the antilogarithm of ^2 will lie between a
certain power of h and the next higher power. The numerical
gaps between the two multiples and the two powers constantly
decrease as p increases and may be made less than any
assigned number by taking p sufficiently large. Thus anti-
log J^ may be defined as the number which always lies —
no matter how large jj may be — -between the two powers of h
which correspond to the two multiples of k between which
72 lies.
The teacher will see that these solutions of Nos. 14, 15
are independent of the rather abstruse argument of the former
part of this article, and are in themselves interesting and in-
structive.
§ 2. Exs. LXXIX-LXXXI. Annuities. Life Insurance. —
From the purely theoretical work of the previous exercises
we now turn to topics of an entirely practical nature. The
teacher who hesitates to demand from his students the
sustained abstract thinking required in Exs. LXX-LXXVIII
may prefer to begin at this point. The subject of compound
interest and annuities has already received some consideration
THE EXPONENTIAL FUNCTION AND CURVE 433
in Part I (Exs. XXXVI, C, LVII, B, LIX). The simple
treatment there given is here expanded into a doctrine which
deals with the essentials of the theory of private annuities,
public loans, and life insurance. These subjects are of great
interest from both the mathematical and the broader educa-
tional standpoints. It should be unnecessary to apologize
for directing the student's attention to the scientiJ&c principles
involved in matters of such immense private and public
concern. The author has attempted to make the treatment
as " real " as the necessity for brevity permits. The teacher
should ensure the success of this attempt by supplementing
the examples of the textbook by examples from actual life.
A few well-chosen advertisements in the columns of the
daily newspaper, the prospectus of a building society, and
the tables of benefits and premiums of an insurance company
may be used to secure, with an absurdly small expenditure
of effort, a great return in interest and understanding and in
enhanced respect for the social value of mathematics. The
equipment of technical knowledge needed to give profitable
instruction in this field is small and easily obtainable.
Whitaker^s Almanack or a similar publication is essential ;
most modern encyclopaedias and many inexpensive " business
handbooks " provide the necessary explanations and com-
mentaries. The reader who seeks a deeper knowledge of
the mathematical aspect of the subjects will find what he
needs in King's Theory of Finance and the exhaustive Text
Book of the Institute of Actuaries.^
§ 3. Ex. LXXXII. The Exponential Function and
Curve. — In this exercise we resume, extend, and generalize
the study of "growth-curves" and their corresponding
functions which occupied our attention in Section III. In
that section the growth-curve, whatever the growth-factor,
was assumed always to be in the position in which the or-
dinate at the origin is unity. In this " standard position "
it corresponds to the function y == r"" where r is the growth -
factor. We are now (i) to explore the functions which
correspond to other positions of the curve, and (ii) tp see
that a growth-curve can always be represented as correspond-
1 Part I of the Text Book deals with annuities -certain, public
loans, etc. — ground covered in a simpler but sufficient manner in
King's work. Part II deals with the theory of contingent annuities
and life insurance.
T. 28
434 ALGEBRA
ing to a function in which r is replaced by the " standard
growth-factor " e.
In accordance with the principle established in chs. xxv.
and XXVI. the graph of No. I, shifted through - a scale-
divisions horizontally, corresponds to the function ^ = r* + ".
But we have
y = 7-^ + «
= Ar"
where A = r". This result is independently evident from the
general property of the curve ; if the ordinate at the origin is
A the ordinate at distance x must be Ar"". The inverse func-
tion takes (No. 2) two corresponding alternative forms
y = log^ and y = log^(a7/A) = log^ - a.
The graph of the former inverse function is obtained from
that of y = f' by the usual two steps of (i) turning the
original graph in the plane of the paper through a clockwise
right angle, and (ii) revolving it about the a;-axis through 180°.
A further lowering through a distance a (corresponding to the
leftward shifting of the original graph) produces the graph of
the second inverse function. These principles are applied in
No. 3 and the following examples. Thus in No. 3 (i) we have
y = 3(10)^
=- (10)o-*77(io)-
= (10)^ + 0.477
the substitution being made, of course, by reference to the
table of logarithms. We conclude that the graph of the
given function is that of (10)"^ shifted (approximately) 0477
scale-units to the left. Similarly in (iv) we have
y = logio 27a;
= log ic + log 27
= log a; + 1-431.
That is, the graph of (10)"" has first been rotated in its own
plane, then out of its plane about the a;- axis, and then raised a
distance 1*431. In No. 4 (iii) the graph, after shifting,
corresponds to
y= (3-2)^ + 4 _ 13
= 104-86 (3-2)^ - 13.
In No. 10 we have
y = Ar'
THE EXPONENTIAL FUNCTION AND CURVE 435
where e^ == r. Applied in No. II (ii) this result gives
2/ = - 2(0-8)^
= _ 2e- 0-223^
since log, 0-8 = log, (1/1-25) = - 0-2231 by the table of
natural logarithms.
The examples in division B are important not only on
account of their physical applications but also (and especially)
in view of the argument of the next exercise. They do not,
however, offer any particular difficulty. For No. 17 (which
may be omitted) reference may be made to pp. 43-5 of this book.
§ 4. Ex. LXXXIII. Differential Formulce. — This exercise
deals with the differential formulae of the exponential and
logarithmic functions, and is very important both for that
reason and because it gives a good occasion for revising the
whole doctrine of differential formulae. The main argument
is based upon the properties of the exponential curve which
were brought to light in ch. xxxv. Even if the class has
worked the examples (Ex. LIX) which followed that chapter
it will no doubt be advisable to revise its conclusions before
passing to the present exercise. It will probably be well also to
revise the earlier work on differential formulae (ch. xxvii., B).
Nos. I-4 prepare the way for the main argument by
establishing certain properties of the "first differences" of
the ordinates of the exponential curve. No. I is quite
simple but of fundamental importance. We have
P'Q' _ ar^ + ^ - gr"
y ~ ar""
= r'' - 1
which is obviously constant so long as h, the distance between
the ordinates, is constant. The geometrical argument of the
note before No. 5 shows that when h is so small that it can
be treated as a differential then the constant r^ - 1 assumes
the value loger . hx. Since at the same time P'Q' becomes hy
the result of No. 1 is now the differential formula
^y = log,r .U or y = ar" . log,r.
y ^^
When r = e (No. 6) log r = 1, and this equivalence reduces
to the familiar result that when y = ae^
82/
= ae^.
bx
28
436 ALGEBRA
The remaining examples of division A are simple applications
of this extremely important relation. No. II needs a
reference to Exercises, I, p. 261. The solution of No. 12 is
as follows : —
Since ~ = lOe^
it follows that ^ = lOe" + 6.
To find the value of the undetermined constant b we note
that, when x = 0, 8y/8x = tan 42° = 0'9 nearly. Hence
0-9 =10 + 6
or b = - 9-1
and ^ = lOe^ - 9-1.
8x
From this relation again we deduce that
y = lOe^ - 9-la; +a
where a is a second constant to be determined by the con-
sideration that y = + 12-3 when x = 0. Making the sub-
stitutions we have
+ 12-3 = 10 + a
or a = + 2-3.
Whence finally y = lOe" - 9-l(r + 2-3.
With No. 13 we turn to the differential formula of the
logarithmic function. Since the function y = log^ is the
inverse of 1/ = r^, it follows that the differential formula of
the former is the inverse of that of the latter. That is to
say, to find the differential formula of log^ we have only to
interchange x and y in the result of No. 5. When this
interchange is effected we have
- = log,r . 8y
X
or (since log^r = 1/log^e)
hx X
In Nos. 16, 17 this result is used to demonstrate in a more
satisfactory and comprehensive way than in Part I, Ex.
LXIX the universal validity of Wallis's Law. In No. 17
we have (since y = ic") log y = n log x, logarithms being taken
THE EXPONENTIAL FUNCTION AND CURVE 437
to base e. But if we put z =^ logy = nlogx we have (as in
No. 16)
hz 1 ^hz n
^ = - and ^ = -
8y y Sx X
whence, dividing the second result by the first, we have
Sx X
x"
= n—
X
The examples in divisions B and C are all important and
should be carefully worked and clearly understood. A weak-
ness in this part of the work will be a source of much trouble
at many subsequent points, and pains should be taken to
avoid it. No. 26 is of great historical importance and will
be used more than once in the sequel. The answer to the
last part of the question is, of course, that there is no
logarithm of zero; thus when zero is one of the values of x
the rule for the hyperbolic area breaks down. In other
words, there is, strictly speaking, no " area-function " cor-
responding to the " ordinate-function " y = a/x; that is,
there is no function which gives the whole area under the
curve right from the ?/-axis. The area may, however, be
calculated from an ordinate as near as we please to the
?/-axis provided that it is not actually coincident with or on the
other side of it. This same result, in modified forms, recurs
in Nos. 29, 30. We have previously met it (Part I, ' Ex.
LXIX, No. 14) in the form of the statement that Wallis's
Law breaks down when 71 = 0.
The examples of No. 30 imply a knowledge that the
differential formula of the function y = log^ {x + a) is
Sic X + a
The simplest proof is to consider the curve y = log^o; and to
note that the gradient at the point where the abscissa is x is
given by hyjhx = Ijx. Let the curve be shifted horizontally
to the left through a distance a. Then the number previously
denoted by x must now be symbolized hj x + a. Thus the
ordinate- function becomes y = log« {x + a) and the gradient-
function hyl^x = lj{x + a).
438 ALGEBRA
In division G the relation between a differential formula
and its primitive is employed to calculate the "volume-
function " of a solid whose parallel cross- sections exemplify
a definite " area-function ". These examples are not only
important as further illustrations of the power of the method
in mensuration ; in addition they are valuable as preparing
for the general ideas of a "derived function " and "integral "
which are to be reached in Section VIII.
The exercise ends with a few examples in which differential
formulae are used to solve simple problems in kinematics.
These examples are to be regarded as a continuation of those
of Part I, Exs. XXX, B, and LXIX, C. In connexion with
them the teacher should read again the remarks on p. 171 of
this book and the note on p. 353 of Exercises, Part I.
§ 5. Ex. LXXXIV. Supplementary Examples. — Revision
Papers 1-4 contain nothing but examples of types already
studied in Part I and require no commentary. Revision Paper 5
(division E) presents in a simple way the essentials of the
theory of " scales of notation ". These examples may be re-
garded as a continuation of those of Part I, Ex. XXXI, C.
Division F may be regarded as supplementary to Ex.
LXXXIII. In it the student is led, following the steps of
Lord Brouncker, Wallis, Mercator and Gregory, to a know-
ledge of some of the most important " series " in elementary
mathematics. These results were, historically, achievements
of the highest significance, not because they made it possible
to calculate logarithms by processes easier than those of
Napier and Briggs — for the work of calculating the tables
was already accomplished — but because they showed the
immense potentialities of Wallis 's method and opened
altogether new vistas in mathematics. The reader who
would understand how suggestive these pioneer investigations
proved should seek access to Francis Maseres' Scriptores
Logarithmici, a monumental work in six large volumes,
published in the last half of the eighteenth century and con-
taining reprints (and sometimes translations) of all the more
important memoirs which had their origin in the wonderfully
fruitful work of Napier and John Wallis.
SECTION V.
TRIGONOMETRY OP THE SPHERE.
THE EXERCISES OF SECTION V.
*»* The numbers in ordinary type refer tx) the pages in Exercises
in Algebra, Part II ; those in heavy type to the pages of this book.
BXEBCI8B PAGES
LXXXV. Map Projections ; Sanson's Net . 105, 442
LXXXVI. Cylindrical Projections . . . 110, 443
LXXXVII. Mercator Sailing 117, 445
LXXXVIIL Great Circle Sailing .... 124, 447
LXXXIX. Calculations on Great Circle Sailing . 133, 454
XC. Some Astronomical Problems . . 143, 457
XCI.
Map Projections ; Sanson's Net .
Cylindrical Projections
Mercator Sailing ....
Great Circle Sailing
Calculations on Great Circle Sailing
Some Astronomical Problems
Supplementary Examples
A. Stereographic projections
B. Projections in general .
C. Astronomical problems .
D. Supplemental triangles, etc. .
157, 449
162, 452
167, 462
169, 466
CHAPTER XLII.
PROJECTIONS.
§ 1. Projections. — The work of the section begins with
four exercises on the theory and use of map projections.
These serve a triple purpose. First they bring the student
into close quarters with the laws governing the spatial rela-
tions of points upon a spherical surface. There is, in fact,
no equally effective means of bringing out the striking
differences between these laws and those of plane geometry.
Secondly, they offer, probably, by far the simplest mode of
approach to the trigonometrical calculations by which prob-
lems of position on a sphere are solved. Thirdly, they are
excellent concrete instances of the general notion of a pro-
jection or "transformation" — a notion which must be con-
sidered as one of the fundamental ideas of mathematics.
Speaking generally, a projection ^ is any rule or device by
which, given a set of points A, B, C, D, etc. in a line, sur-
face or volume, we can obtain a second set A', B', C, D', etc.
in another line, surface or volume, corresponding to the first
set, point by point. A plan (e.g. of a field) drawn to scale is
a simple example of a projection, for the essence of plan-
making is that to every point in the original there shall
correspond a definite point in the drawing. In the case of a
plan the whole of the spatial relations of the original points
are reproduced ; only the metric scale is changed. In
projection in general the relations may all be transformed ;
it is sufficient if they are transformed in any regular way
which secures point-to-point correspondence between the
original and the representation.
The cartographer's problem is one of projection, for he has
to represent in a definite way upon a flat sheet of paper the
1 The term is here used to cover both projections proper and
other transformations. See Exercises, II, p. 162.
441
442 ALGEBRA
spatial relations of points on a sphere. In representing them
he is bound to transform them. This fact makes the funda-
mental difiference between a map and a plan. But it is
possible so to choose the mode of representation that some
given feature of the original space-relations is preserved.
Thus for the geographer it is important that a map should
represent correctly the relative sizes of land and sea surfaces.
The cartographer furnishes him, therefore, with an " equal
area projection ". The sailor demands a chart which shall
facilitate the task of navigation. He receives, therefore,
either a " Mercator," a " gnomonic " or a " stereographic "
projection. The point to make clear in teaching is that each
of these projections preserves something and that the choice
of a projection is determined by the purpose for which it is
to be used.
It should be added that many of the maps actually used
by geographers are not projections at all, in the strict sense,
for the positions of the representative points are not deter-
mined by a single law. Different laws are used in different
parts of the map. The explanation of this usage is simple
but should be given in the geography lesson. In the mathe-
matics lesson attention should be concentrated upon pro-
jections which have mathematical as well as practical value.
The most important from this point of view are those dealt
with in the exercises. For further information about them
and about the theory of map-projections in general, the
teacher may consult the following works : Ency. Brit., art.
" Maps " ; Hinks, Map Projections.
Finally the teacher is strongly urged to make a free use of
the globe in these lessons. A " blackboard surface " globe
is most useful. The geographical equipment of a secondary
school frequently includes a number of small globes of this
kind which can be put into the hands of individual
students.
§ 2. Ex. LXXXV. Sanson's Net. — Sanson's " sinusoidal "
net is chosen for the first exercise as a simple instance of an
equal area projection. The completed drawing is represented
in the frontispiece of Exercises, II (firm lines). Its repro-
duction is not too laborious an exercise for an evening's
homework, but may be lightened by the omission of alternate
meridians and parallels. When the curves are satisfactorily
drawn in pencil they should be redrawn in ink so that the
PROJECTIONS 443
pencil lines used in solving problems may be rubbed out
without damage to the net.
The solution of No. 2 is obvious. Between a given
meridian and the central meridian Z/360 of each parallel is
intercepted, I being the difference of longitude between the
meridians. But (by p. 131) the length of a parallel is
Ittt cos A where A. is the latitude or 27rr sin p if ^ is the
distance in degrees from the north or south pole (i.e. p =
90" - A). Hence the intercept of the parallel whose polar
distance is p is
tttI .
180"°^-
That is, the meridian is a sine curve whose " amplitude " is
7rW/180.
The circles of Nos. 6, 7, 8 are represented in the frontis-
piece of Exercises, II (firm lines). It is well to bring out
the point of the problems by cutting from a sheet of card a
circular hole whose radius is that of the circles in question.
That is, the diameter must be the chord between two points
on the equator of the globe separated by 45° — easily measured
by a pair of compasses or dividers. When the card is laid
on the globe with the centre of the hole at the prescribed
point the circumference of the hole passes through the specified
positions on the globe in each case.
§ 3. Ex. LXXXVI. Lambert's Net.— The importance (for
mathematics) of Lambert's projection is that it leads to a.simple
formula for the area of any belt of a sphere and therefore to the
formula A = 4:7rr^ for the whole sphere. The argument in § 1
contains a point worthy of special emphasis, since it involves
the principle upon which the whole treatment of the calculus
(Section VIII) is to be founded. It is assumed (i) that the
area of a sphere lies between the area of any system of coni-
cal surfaces which can be inscribed in it and any similar
system in which it can be inscribed ; (ii) that as h decreases
the areas of the inscribed and circumscribed surfaces continu-
ally approach the area of the sphere but can never coincide
with it. It is shown that corresponding to each value of h
there is (i) a cylinder, as high as the sphere but of smaller
radius, whose surface is equal to that of the inscribed conical
system ; (ii) and another cylinder, also of equal height with
the sphere but of larger radius, whose surface is equal to that
of the circumscribing conical system. It is also shown that
444 ALGEBRA
as h decreases these cylinders continually approach the
cylinder which circumscribes the sphere but can never coin-
cide with it. Thus we have two series : —
. . . E, E, E, . . . S . . . I, I, I, I, . . .
... e, e, e, . . . C . . . i, i, i, i, . . .
— a series of external and internal conical surfaces (E, I) and
external and internal cylinders (e, i). The Es are separated
from the Is by the sphere S; the es from the ^s by the
circumscribing cyHnder C. The members of the respective
series approach S and C endlessly. That is why no member
can be represented as lying next to S or C ; it would always
be possible to imagine another nearer still. But to every E
an e corresponds and to every I an i. It follows, therefore,
that S which divides the Es from the Is must correspond to
C which divides the es from the ^s. That is to say, the area
of the sphere is identical with that of the circumscribing
cylinder whose height is equal to the sphere's diameter.
It will be observed that this method of proof leads to exact
results. It is, therefore, greatly preferable to the proofs which
depend upon the assumption that a narrow belt of the sphere
can itself be regarded as a conical frustum. The student
always feels such a proof unsatisfactory. It is true that the
error, as regards a single belt, may be diminished without
limit by diminishing its breadth, but the number of the belts
are magnified in the same proportion. The total effect of the
accumulation of errors remains, therefore, uncertain. In any
case such a method gives the area of a system of conical
surfaces, not of the sphere.
The teacher who can command the use of skilled fingers
will find it useful to construct Exercises, II, figs. 67 and 68, in
wire. Rotated by a turning table, by a gyroscopic top or
simply by the fingers, the wire frame will give a cinemato-
graphic representation of the sphere with the system of
conical surfaces and the corresponding cylinder.
Lambert's net is represented by the broken lines of the
frontispiece of Exercises, II. The same figure also contains
the solution of No. 5.
Section B of this exercise gives an interesting anticipation
of the results of "differentiating" and "integrating" the sine
and cosine of an angle. A class which has not worked through
Part I will need an explanation of the terms '* area-function "
and " ordinate-function ". See ch. xxvii., A.
PROJECTIONS 445
The examples on the •* central cylindrical " net serve a
two-fold purpose. First, the net offers a useful contrast to
Lambert's. In the latter the distance of a parallel from the
equator is r sin \ and can, therefore, never exceed r. In the
former (No. 13 (ii)) it is r tan A. and therefore increases end-
lessly as the latitude increases. Thus it is theoretically im-
possible to show the north and south poles in a central
cylindrical projection, and practically inconvenient to show
much beyond the Arctic and Antarctic Circles. Secondly, the
net is our first example of a "geometrical projection" (Ex.
XCI, § 4). It is generally possible to purchase from an
ironmonger a wire frame of the kind used to protect naked
gas flames in workshops and theatres. The wires often give
a passable imitation of the lines of latitude and longitude on a
globe. By means of a lighted candle with its flame at the
centre the shadows of the wire may actually be cast on to a
paper cylinder held round the frame.
§ 4. Ex. LXXXVII. Mercator Sailing. — In this exercise
we resume the problems in navigation studied in Bxs. XIX
and XX. 1 When a sailor has in contemplation a voyage of
some hundreds of miles he cannot regard the surface of the sea
as a plane. It becomes essential, therefore, to have a method
by which to guide his ship from port to port. It will be seen
that there are two methods in actual use. The first (" Mer-
cator sailing ") aims at finding the " rhumb line " which will
carry the sailor from port to port without any change of direc-
tion. That is the subject of Ex. LXXXVII. The second
method aims at determining the shortest track between the two
ports. That is studied in the next exercise. The student is
usually greatly surprised to find that the " straight " course is
not also the shortest. It is well to keep the discovery back
until Ex. LXXXVII is finished.
The teacher will note in § 3 of this exercise a repetition in
essence of the argument emphasized above on p. 444. He
will also recognize that the graphic method of § 3 is a device
for obtaining a practical solution of the integral J sec . SO.
A theoretical solution is given at a later stage (Ex. CX, D).
By following the directions the class should arrive at the
following : —
^ It is not necessary to have worked these exercises in order to
proceed with the present one.
446
ALGEBRA
Table op Meridional Pasts. ^
Lat.
5° 10° 15°
20° 25° 30° 35°
40°
Pts.
501 10-05 1518
20-42 25-83 3147 3740
43-71
Lat.
45° 50° 55°
60° 65° 70° [75°]
[80°]
Pts.
50-50 57-91 6613
75-46 86-31 99-43 [11617]
[139-6]
The examples in division A of the exercise can be solved by
70
jB'
60°
^''
.-'
''
^
y"
^
^
/
^
x^
^
50
/
'
^
^
t
40'
A
30-
20"
lO'
0*
__
__
__
Sr
__
_^
__
'-T
,«.
_
30 45 60 75 30
Fig. 101.
a Mercator net which contains 180° of longitude and 70° of
latitude on one side of the equator. One half of such a net
is represented in fig. 101. In different problems the left-hand
^ The unit is the length of a degree along the equator. In the
larger tables (e.g. Chambers') the unit is the length of a minute
along the equator.
PROJECTIONS 447
edge of the net must be taken to represent different meridians.
In dealing with places south of the equator the net must be
inverted.
Taking the left-hand edge to be the meridian of 100°
W. the straight line A'B' represents the rhumb line of Ex.
LXXXVIII, No. 6. The curved line represents the great
circle track joining the same two points. The angle between
A'B' and the meridian through A' is the course which would
take a ship from A' to B' without change of direction. Call
it a, and let I be the difference of longitude between A' and B'
— that is the length of the parallel through B' measured in
equator-degree units. Finally let m^ and m^ be meridional parts
corresponding to the latitudes of A and B. That is let m^ and
Wg be the distances in equator- degree units of A' and B' from
the base line of the net. Then we have
tan a = Z/(m2 - w^).
This is the solution of No* 12.
To calculate the distance D between A and B we suppose
the track to be broken up into pieces so short that the whole
of a piece may be supposed to have the same latitude. If d
be the length of such a piece in miles then d cos a gives the
north and south distance in miles between its two ends.
Since a is the same for each piece D cos a is the distance in
miles between the parallels of latitude through A and B.
But each mile is a minute of a degree. Hence the difference
of latitude in degrees is D cos a/60. Thus, given the differ-
ence of latitude and the course, the rhumb-line distance can at
once be found (No. 14).
§ 6. Ex. LXXXVIII. Great Circle Sailing.— This exercise
is of great importance because the spherical trigonometry of
the next two exercises is based upon it. It should, however,
give no difficulty to teacher or student. The wire frame of
§ 3 above serves usefully as a means of demonstrating both
the polar and meridian gnomonic nets. Fig. 102 shows (on a
reduced scale) the polar gnomonic net of No. 2. In No. 3
the pole and the parallels are supposed to be south of the
equator, in No. 5 north. The straight line A'B' in fig. 102
gives the solution of No. 5, the curved line that of No. 7,
while the lines joining A'B' in fig. 101 are the solution of
No. 6.
The argument of § 3 of the exercise is usefully supplemented
by a model which can be constructed with little difficulty. A
448
ALGEBRA
child's india-rubber ball about 17 cms. in diameter is blackened.
A figure BAP consisting of a great circle track and two
meridian segments is drawn on it in white paint. The
sphere is placed in contact with an upright sheet of glass, the
line OP being horizontal. Exercises, II, fig. 72, is drawn on
stiff card, the arcs PA, AB, BP, having the radius of the ball.
The card is cut along the arcs and cut half through along the
lines A'P, PB', B'A'. The three truncated triangles are bent
down and bound together with tape along their edges AA' and
BB', and are placed so that the arcs PA, AB, AP, rest on the
Fig. 102.
painted arcs on the ball, while PA'B' rests against the glass.
Finally, narrow strips of paper are stuck on the other side of
the glass covering the edges of the cardboard. The mechanism
of the projection can in this way be made clear to a class.
The cardboard can be removed, leaving the triangle on the ball
and its projection on the glass. Indeed the cardboard itself,
without the ball and glass, makes a very useful model, since
it exhibits at the same time the spherical triangle and its
plane projection. Whether this model is used or not, each
student should make (and keep for future use) the simple
paper model of No. p.
PROJECTIONS
449
To solve No. 12 mark the positions of Honolulu (A') and
Yokohama (B') on the polar gnomonic net, transfer the
triangle PA'B' to a sheet of paper and complete the construc-
tion of Exercises, II, fig. 72. Bisect the angle A'OB' by OC
cutting A'B' in C. Mark the corresponding point C in the
gnomonic net and read its latitude and longitude.
Fig. 103 shows the meridian gnomonic net of No. 13. The
straight line across it gives the solution of No. 14. To meet
Fig. 103.
the requirements of the problem the central meridian, though
marked zero, is taken, to be the meridian of 30° W. If it had
been taken (for example) as the meridian of 40° W. or 25° W.
the actual position of the line would have been different in
the diagram, but the answers to the questions set would have
been the same. This statement should be verified.
Fig. 104 gives the diagram described in § 5. It is im-
portant in connexion with the formulae of right-angled spherical
triangles (Ex. LXXXIX, No. 25).
§ 6. Supplementary Work : The Stereographic Projection. —
T. 29
450
ALGEBRA
To division A of the supplementary Ex. XCI are relegated
two more projections. Of these the stereographic will well
repay study if the necessary time can be afforded. It has
two beautiful properties : (i) any circle on the sphere is
projected as a circle, and (ii) the angle at which the pro-
jections of any two curves cross one another is the same as
the angle at which the actual curves cross upon the surface
of the sphere. The arguments required to establish these
truths give excellent work for the student. Taken together
with the fact that it is easy to construct a net for the greater
^art of the globe they also make the projection a useful one
■~--. B
L^-O
Fig. 104.
in connexion with great circle sailing problems. Unlike the
gnomonic projection it shows the correct course at any
moment of a ship which is sailing along a great circle.
Fig. 105 shows the net described in No. 4 together with the
projections of the three circles mentioned in the examples.
It will be seen that they are also circles.
The stereographic projection also gives the easiest means
of studying the sum of the angles of a spherical triangle. It
is easy to use it to demonstrate that the sum must always
exceed two right angles (No. II). It is left to the teacher to
point out that the larger the sphere the greater will be the
diameter of the great circles on it, for sphere and great circles
have the same radius. Thus if, in the fig. of No. 11, AB',
AC and the angle B'AC remain of constant size but the
PROJECTIONS
451
sphere becomes larger, the diameter B'B" will increase and
the angle B'B"C' (which is half the spherical excess) will
diminish and approach zero. On the other hand, if AB'.
and AC increase, since the diameter of the great circle re-
mains constant, the spherical excess increases. We may
conclude, then, that the larger the sphere or the smaller the
Fig. 105.
triangle the smaller is the spherical excess ; and that the
smaller the sphere or the larger the triangle the greater the
spherical excess.
The teacher may, if he feels disposed, refer to the speculation
that points in what we think of as plane surfaces may have
really the spatial relations which characterize the surface of
a large sphere. Measurement can inform us only that the
sum of the angles of a triangle is approximately 180" ; it
29 *
452 ALGEBRA
cannot decide whether apparent deviations from that sum
are due to defective observation or a real curvature in space.
The only way to settle the question would be to measure the
angles of an extremely large triangle (such as that marked
out by the centres of three fixed stars) in which the spherical
excess (if it exists) might be expected to show itself. The
speculation can be continued further. We can suppose that
space has a property to be described metaphorically as a
negative curvative the effect of which would be to give every
triangle a " spherical defect " increasing in magnitude with
its size. This hypothesis is also one which cannot be tested
so long as we are confined to the relatively minute triangles
of earth, but may nevertheless be true.
The teacher who finds himself — and his pupils — attracted by
such ideas is advised to refer to Prof. Carslaw's translation of
Bonola's Non-Euclidian Geometry (Open Court Publ. Co.) ;
Clifford's Philosophy of the Pure Sciences (" Lectures and
Essays," Vol. I) ; Stallo's Concepts of Modern Physics (Int.
Scientific Series) may also be consulted. The subject has
now an enormous literature.
§ 7. Supplementary Work; Projections in General. — In
division B of the supplementary exercise the student is led
first to summarize the properties which he has found in map
projections and then to generalize the idea of projection.
Nothing substantial need be added to what is said in the
exercise and in § 1 of this chapter. Reference may, however,
be made to the brief discussion (in § 7 of the exercise) of one-
to-one correspondence between the points of a straight lin^
and a plane — a notion which is, perhaps, best approached
by means of the study of projection. It will be recognized
that the elaboration of this idea is far too difficult for an
elementary course ; nevertheless it is convenient to introduce
it if only to show that it is difficult and must, therefore, be
treated with respect. Moreover, taken with Ex. LXXI, the
discussion helps to illuminate the difficult but important idea
that there is a multiplicity of "infinite," just as there is of
"finite," numbers, each possessed of its own individuality
and properties (see p. 408).
The argument relies too much upon geometrical intuition
to be really satisfactory, but it may awaken interest in the
question.
The interesting point which emerges from the discussion
PROJECTIONS 453
is that although a surface contains an " infinite " number of
separate lines, each possessing an " infinite " number of
points, yet the points in any surface and in any line must
be considered to have the same number, for they can be
brought into one-to-one correspondence. The teacher who
wishes to follow the subject further should turn to Young's
Fundamental Concepts (Lect. XVI) or Russell's Principles
of Mathematics. Further brief references to it will also be
found in Ch. LII and in Ex. CXVI.i
^ The following title should be added to the works recommended
at the end of v^ 1 : Mary Adams, A Little Book on Map-Frojection
(Geo. Philip & Co.). The teacher will find this book (which
appeared too late for mention in the text) extremely lucid and
informative.
CHAPTER XLIII.
THE TKIGONOMETRY OF SPHERICAL TRIANGLES.
§ 1. The Programme. — Exs. LXXXIX and XC investigate
the application of trigonometrical formulae to problems con-
cerning the relations of points upon the surface of a sphere.
The problems which are of practical importance here fall into
two classes. In the first are those of the navigator and the
surveyor who have to deal with points upon an actual spheri-
cal surface. In the second are the problems of the astronomer
who finds it convenient to state his problems in the form of
problems about points upon a sphere, though the sphere is
only imaginary, or at most a model representing in an easily
intuitable form his observations upon the stars. A few ad-
ditional problems of almost purely geometrical interest are
relegated to the supplementary exercise (division D) and may,
without serious loss, be omitted.
It has already been remarked in ch. xxxviii., § 3 that the
technical apparatus needed for mastering everything of first-
rate importance in this field is quite small in extent. It does not
go beyond the fundamental formula of spherical triangles : —
cos a = cos h cos c + sin h sin c cos A . (1)
the three formulae of right-angled triangles : —
sin A = sin a/sin c . . • (2)
cos A = tan 5/tan c . . • (3)
tan A = tan a/sin h , . • (4)
and the derived formula : —
sin a sin h sin c ,^.
sin A ~" sin B "" sin C * * * V /
The " supplemental formulae " corresponding to these are
given (as a cheap luxury) in the supplementary exercise, but
" Napier's analogies," and other complications introduced in
order to obtain formulae adapted to logarithmic computation
are entirely beyond the requirements of the ordinary student
454
THE TRIGONOMETRY OF SPHERICAL TRIANGLES 455
and are therefore omitted altogether. The classification of
problems into six " cases " (including the " ambiguous case "
with its formidable array of sub-divisions) is also excluded as
an enterprise from which the ordinary student is likely to
gain very little profit.
§ 2. Ex. LXXXIX. Navigation Problems. — Division A of
this exercise is devoted to establishing the fundamental for-
mula (1) of § 1. The formula is reached by considering a
concrete problem in navigation — namely to find the great
circle distance between places whose latitudes and longitude
difference are given. This problem has already been solved
graphically in Ex. LXXXVIII. The solution by formula de-
mands nothing more than the application of simple trigono-
metry to Exercises, II, fig. 72. The student is told (No. l)
to take this figure (or the paper model which he made in
Ex. LXXXVIII, No. 9) and to enter against each side an ex-
pression for its length in terms of the data.
In the triangle OAT it is obvious that OP = r, PA' = r tan_pi
and OA' = r secp^. But OA' in this triangle is identical vnth
OA' in the triangle OA'B', for when the two triangles are
folded respectively about PA' and A'B' the similarly lettered
edges fall together. Thus in the triangle A'OB' we have OA'
= r sec p-^, and in the same way OB' = r sec ^2- ^^ *^®
triangle PA'B' we can now write : —
(A'B') 2 = r^ tan2_pj + r^ tan^^g - ^r^t^np^ianp^cosl
and in the triangle OA'B'
(A'B')^ = r2 sec^ p^ + r^ sec^^g - ^r^ secp-^ sec j?2 cosD.
Equating these expressions, cancelling the r^ throughout, and
remembering that sec^ a - tan^ a = 1, we reach the first
equivalence of No. 4 from which the formula for cos D follows
at once.
In the first instance this formula is proved only for cases
which can be represented by Exercises, II, fig. 72 — that is,
when the track AB does not cross the equator. In § 2 of the
exercise the formula is without difficulty shown to hold good
equally when this restriction is removed.
So far the student has dealt only with concrete problenas
about great circle distances, etc. In division C, § 3, he is
first taught to see that the methods used in these problems
are applicable whenever we are deaUng with a figure composed
of the arcs of three , great circles, and that the fundamental
formula can be apphed, under proper conditions, to deter-
466 ALGEBRA
mine any side or angle of such a figure. At this point the
term ** spherical triangle " is first introduced. The idea of
the angle between two " sides " of a spherical triangle needs
careful attention. The paper model can here be used again
with useful effect.
In § 4 the important special case of the right-angled
spherical triangle is considered. The method of treatment
depends upon the fact that the gnomonic projection can, in
this case, be made to preserve unchanged both the right angle
and one of the other angles of the figure on the sphere.
Thus, when C is the right angle and the plane of the projec-
tion makes contact with the sphere at A, we have AB' = r tan c,
AC = r tan h and (since the angle A is conserved)
cos A = tan 6/tan c.
If, in this projection, AB' and AC are regarded (like PA'
and PB' in Exercises, II, fig. 72) as meridians we have no
direct information about the length of B'C and can, therefore,
derive no expressions for sin A and tan A. If, however (as
suggested in No. 25), A be regarded as the crossing point of
the central meridian and the equator of fig. 103, AC as a part
of the equator and B'C as a meridian, we have a means of
supplying values to all the sides in the projection, and (since
the angles A and G are again conserved) can at once obtain
expressions for all the trigonometrical ratios of A. The figure
required is, of course, fig. 104.
In division D these formulae are used to solve various prob-
lems with regard to great circle paths on a sphere. Most of
these are couched in the form of questions in which the
latitude, longitude and course of a ship are to be expressed
with reference to the "vertex " of its track — that is, the point
where the ti'ack comes nearest to the pole or where the
perpendicular great circle from the pole meets it. The results
of these problems are generalized and summarized in No. 35.
The formulae indicated in this example play, with regard to a
great circle, the same part as the formula
y = ax + h
plays with regard to a straight line in a plane. That is, they
enable us to determine whether a given point is or is not
situated upon a given great circle. The variety of the formulae
which serve this purpose is due to the fact that the position
of a point on a great circle may be fixed by any two out of
three coordinates : namely, its latitude, its longitude, and its
THE TRIGONOMETRY OF SPHERICAL TRIANGLES 457
distance from the vertex. In No. 35 (i) the coordinates to
be used are the co-latitude and longitude. We are supposed to
know AB ( = ^), BC ( = P) and the angle at C (= 90°). The
angle at B is L - Z. Hence by formula (3) (p. 454) we have
cos (L - I) = tan P/tan p,
a formula which may be regarded as stating the relation
between the values of the variables I and p for all points along
the great circle whose vertex is at (L, P). Similarly in No.
35 (iv) we know BC ( = P) , AC ( = d) and the angle C ( = 90°)
and have to find a formula for AB {= p). In this case we
use the fundamental formula (1) which reduces to
cos p = cos P cos d
— an expression which states the relation between p and d for
all points in a great circle whose vertex is in latitude 90° - P.
Lastly, these formulae make it possible to determine the
direction of a great circle at any point — that is the angle at
which it crosses one of a given system of meridians. Thus in
No. 35 (v) we are to determine A (== a), given BC (= P),
AB (= _p) and C (= 90°). By formula (2) (p. 454) the re-
quired relation is
sin a = sin P/sin^.
§ 3. Ex. XC. Astronomical Problems. — The first two ex-
amples are intended to suggest the use of gnomonic pro-
jections in recording the positions of the stars at a given
moment. The lines inscribed upon the roof of the imaginary
cubical room will form a polar gnomonic net in which lines
of equal azimuth take the place of the meridians of Ex.
LXXXVIIT, and circles of equal altitude the place of the
parallels of latitude, while the pole of the former net becomes
the zenith. The walls will be inscribed with four meridian
gnomonic nets in which the horizon takes the place of the
equator. The complete net of No. 2 is represented in fig.
106.
If the cubical room is tipped about its northern edge
through an angle equal to the co-latitude of the place of
observation (§ 3, No. 3) the azimuth lines become lines of
equal right ascension or equal hour-angle, the altitude lines
become lines of equal declination, the horizon becomes the
" celestial equator " and the zenith the " celestial pole ''. If
" stars " are stuck on to the glass so as to represent the
complexion of the sky when the " first point of Aries " is on
the meridian (i.e. when the sidereal clock registers Oh. m.
458
ALGEBRA
s.), the room must be re-oriented to represent the sky at
another hour (No. 4). Thus when the sidereal clock registers
3 h. the roof must have revolved in its own plane through
an angle of 45° ( = 15° x 3). It should be noted that since
the sidereal clock gains about 4 m. a day upon the solar
clock the interval measured in units of solar time will be
3 h. - 4 m. X /^ = 2 h. 59 m. 30 s. For naked-eye ob-
servations this difference is, of course, negligible.
15° 330°345°S 15° 30° 45?^°
225° 210° 195° N 165° 150° 135^
Fig. 106.
A revolving cubical room is obviously a practical impossi-
bility, but it is easy to gain most of the advantages it has to
offer by enlarging the scope of the altitude circles inscribed
on the roof and suppressing the sides. We thus obtain the
perfectly practicable star-net of Nos. 5 and 6. Such a net
should certainly be made and mounted in the way described
in No. 6. The positions of a number of prominent stars
should be entered upon it.^ The card should be held (No. 6)
^ The following is a satisfactory minimum list : the Pole Star,
the seven bright stars of the Plough, the " W" of Cassiopeia, the
stars Vega and Capella. The R.A. and Decl. of these stars may be
THE TRIGONOMETRY OF SPHERICAL TRIANGLES 459
so that the line joining its centre to the eye is parallel to the
axis of the sky's rotation and has a length equal to the radius
assumed in constructing the net. The E.A. line whose
graduation is identical with the momentary reading of the
sidereal clock should point from the pole of the card towards
the zenith. The " stars " on the card will then be identical
in position with the stars in the sky.^ The position of the
horizon can be indicated permanently by a slight addition to
the apparatus. Fix a strip of cardboard or paper across the
base in such a way that the circular chart can turn round
beneath it. Arrange that its upper edge (where it is a chord
of the circular chart) shall touch the circle whose polar
distance is equal to the latitude (or whose declination is
equal to the co-latitude). If the chart is now held in the
position prescribed above and with the edge of the paper
strip horizontal, the latter as viewed by the eye will coincide
with the horizon. Any " star " which is below the edge of
the strip will be invisible. It is possible in this way to
determine the sidereal times of the rising and setting of any
star which is represented on the chart.
Nos. 9 to 14 illustrate the way in which observations
of the altitude of a star on crossing the meridian can be
used to determine latitude. They prepare the way for the
study of the way in which this same determination is con-
stantly made at sea by observations on the meridian altitude
of the sun.
§ 4. The use of Trigonometrical Formulce. — In § 4 the as-
tronomical globe is introduced as a convenient means of re-
presenting the observed facts of stellar movement. It then
becomes obvious that the formulae of spherical triangles can
be used to solve numerous problems by calculation. The
problems here considered almost all resolve themselves into
read from any star-map — for example from those contained in Ball's
Popular Guide to the Heavens (Geo. Philip & Son). Whitaker's
Almanack also gives the positions of most of them.
^To find the sidereal time at a given moment take from
Whitaker's Almanack the sidereal time at noon on the day in
question and add the time of the observation. Thus on 2 May,
1913, the sidereal time at noon was 2 h. 39 m. 10 s. The sidereal
time at 7.30 p.m. was, therefore, practically 10 h. 10 m. The
difference between sidereal and solar units ' is here ignored since
only approximate results are needed.
460 ALGEBRA
the determination of the altitude and azimuth of a star
whose hour-angle and declination are given — or into the con-
verse problem. The analogy of the problem of No. 15 with
that of calculating a great circle distance should be made
clear — preferably with the help of a globe. The zenith dis-
tance may be regarded as the great circle distance between
the zenith and the star. The other two sides of the spherical
triangle are the polar distances of the zenith and the star —
the former being, of course, a constant for a given place of
observation. The hour-angle takes the place of the differ-
ence of longitude between the two ends of the great circle
track. The method of determining the azimuth is explained
in No. 16.
The solution of No. 21 is as follows. As usual the formula
should be remembered in the form (1) but used in form (2) : —
a = 54° 23', . S = 17° 35' 36", X = 42° 18'
, sin a - sin S sin X
cos h = r r
cos 5 cos X
^ 2 sin a - cos (X + S) + cos (X - S)
cos (X + S) + cos (X -^)
= 0-8645
.\h = 30° 11'
= 2 h. m. 44 s.
That is, the observation took place 2 h. m. 44 s. after Saturn
crossed the meridian. Since this passage occurred at 8 h. 41 m.
27 s. by the sidereal clock the sidereal time of the observa-
tion must have been 5 h. 42 m. 11 s. But the sidereal time
at noon was 20 h. 20 m. 41 s. The observation was, therefore,
made
24 h. + 5 h. 42 m. 11 s. - 20 h. 20 m. 41 s. = 9 h. 21 m. 30 s.
after noon. The time is measured here in sidereal units and
is practically 9 J sidereal hours. To reduce it to solar hours
we must deduct
4 m. X ^ = 1 m. 33 s.
24
We conclude that (within a few seconds) the local time was
9.20 p.m.
§ 5. Calculations based upon Solar Observations. — Cal-
culations based upon observations of the sun are compli-
cated by the irregularity of the daily movements of that body.
For this reason they are postponed until the student has
THE TRIGONOMETRY OF SPHERICAL TRIANGLES 461
mastered the simpler problems connected with stellar posi-
tion. The main facts about the sun's movements in right
ascension are best studied in connexion with the sundial ;
consequently division C of the Exercise begins with the pro-
blem of graduating a dial when the base upon which the
shadow falls is horizontal. ^ This is, of course, not the
simplest form of dial. The simplest possible form consists
of a style (e.g. a hat-pin) thrust through a square of card-
board or wood at right angles to its plane. Upon this the
shadow of the style is to be received. The lower end of
the style must be driven into a wooden horizontal base at
an angle of inclination equal to the latitude. The edge of the
receiving square should rest on the same base. The angle
which this square makes with the base will, of course, be the
co-latitude. If the dial is placed so that the style is parallel
with the axis of the sky's rotation it will be found that at all
seasons of the year the shadow of the style moves uniformly
over the receiving surface at the rate of 15° per hour. This
is, perhaps, the best way to introduce the young observer to
the knowledge of the law of the sun's rotation. The calcula-
tions of Nos. 25-27 presuppose this law and are based upon it.
Whatever form of sundial is employed the main points
which emerge from a comparison of its readings with those of
a good clock are those covered by the term " equation of time ".
In accordance with the principle ch. xxxviii., § 3, they are
treated in § 5 of the exercise simply as facts of observation.
They are not difficult to explain but the explanation lies in the
province of the text-book on astronomy. Our present pur-
pose is limited to applying them in simple calculations of the
same type as those considered in the earlier examples.
The solution of the first part of Nos. 31 and 32 is as
follows : —
Since the longitude of Bristol is 2° 35' W., the local time
corresponding to 7.10^ a.m. Greenwich time is 7 a.m. At 7
a.m. on 28 May the sun is
3 m. + 7 s. X 2T = 3 °^- 1"^ s-
before the clock. It is therefore 4 h. 56 m. 58^ s. before noon
— the time being measured in solar units. In accordance
^ The article Dialling in the Ency. Brit, is very full and in-
structive. There are also special books on the subject.
462 ALGEBRA
with the explanation which precedes No. 31 the hour-angle
of the sun is the angular equivalent of this interval, namely
74° 14' 39". The sun's decHnation at 7 a.m. was
21° 24' 41" - 24-5" x 5 = 21° 22' 38-5"
To find the altitude (a) we have the formula
sin a = sin 8 sin X + cos 8 cos X cos h
= 0-44263
a = 26° 16' 18"
To find the azimuth we apply the law of sines (No. 16) in
the form : —
sin ^ = sin /i . cos 8/cos a
whence /3 = 88° 3'
The exercise concludes v^th examples (Nos. 35 and 36)
on the method by which the latitude and the local mean time
are determined simultaneously by observing the altitude of the
sun as it crosses the meridian. This is the standard method
of nautical astronomy. The longitude is determined by the
difference between the local mean time and the Greenwich
mean time recorded by the ship's chronometer.
In former days the occasional determination of Greenwich
mean time by astronomical observations was itself one of the
ordinary tasks of the navigator. Such observations are now
required much less frequently — the rapid progress of wireless
telegraphy have made them to a great extent unnecessary.
At the present day " wireless " time-signals are flashed out
daily from suitably placed stations all over the world and
are picked up and transmitted from ship to ship along all the
great trade-routes. A uniform system of signals was adopted
at an International Conference held in Paris in 1912. An
interesting account of it is given in Nature (London) for
20 March, 1913. An account of the " standard time zones "
accompanied by a map will be found in Ball's Popular Guide
to the Heavens.
§ 6. Ex. XGI, C. Supplementary Examples. — Supplement-
ary examples on spherical triangles are given in divisions
C and D of Ex. XGI. In division C those on " lunar dis-
tances " need no additional explanation. Nos. 33 and 34 give
point to the account of the sun's movements in the pre-
vious exercise by showing that time can be read systeni^-
THE TRIGONOMETRY OF SPHERICAL TRIANGLES 463
atically by means of the shadow of a vertical rod. The angle
p which the shadow makes with the north and south hne
(No. 33) is, of course, equal to the sun's azimuth. The sun's
altitude must first be calculated by the formula
sin a = sin 8 sin X + cos 8 cos X cos h
and the value of yS derived from that of a by the formula
sin y8 = sin /t . cos 8/cos a.
Taking the latitude of London as 51^° the highest possible
altitude when the hour-angle is 15° x 3 = 45° is given by
sin a = sin 23° 27' . sin 51^° + cos 23° 27' . cos 51^ . cos 45°
= 0-71528
= sin 45° 40'.
In finding the lowest possible altitude in the same circum-
stances we must remember that sin 8 was, for convenience,
substituted for cos P in the original spherical formula. When
the sun's declination is 23° 27' south, _p = 90° + 23° 27' and
cos p = - sin 23° 27'. Thus we have
sin a = - sin 23° 27' . sin 51^° + cos 23° 27'. cos 51^° . cos 45°
= 0-09238
= sin 5° 18'.
Substituting the two values of a successively in the second
formula we have
sin 45° . cos 23° 27'
sm^ =
cos
45"
'40'
^ =
68°
10'
for the first,
and
sin p =
sin
45°.
cos
, cos 23°
, 5° 18'
27'
)8 =
40°
39'
for the second. We conclude that during the course of the
year the direction of the shadow 3 hrs. before (or after) noon
will vary in London to the extent of 27° 30'.
Nos. 35 and 36 deal with the fascinating problem of
graduating a sundial which is to be erected upon a vertical
wall. There are two cases : (i) when the plane of the wall
lies east and west (No. 35) ; (ii) when it is inclined to the
east and west line (No. 36).
The solution of the first of these problems will be aided
by consideration of figs. 107 and 108. In these figures the
northern edge PQ of a horizontal dial is supposed to be
464
ALGEBRA
pressed against the bottom edge PQ of a vertical dial, and the
dials are supposed to have a common style 00'. Fig. 107
shows a meridian section of the two dials which passes
through the style. If I is the length of the style it is seen
from this jQgure that ON = I cos X and O'N = I sin X. In
fig. 108 the vertical dial is supposed turned about PQ into
the same plane with the horizontal dial. OA is the shadow
on the latter when the hour-angle is h. It is evident that
the shadow on the vertical dial must pass through O' and also
through A ; its position is, therefore, the line O'A. Let the
o
N Va
Fig. 107.
Fig. 108.
angle NOA = H and the angle NO'A = H'. Then we have
ON . tan H = NA = O'N . tan H'
ON
i.e. tan H' = tan H x ^^j^
= tan H . cot X.
But by Ex. XC, No. 25, tan H = sin X . tan h ; hence»
tan H' = cos X . tan h.
Next let the wall be inclined at an angle of w° to the EW
line (No. 36). In order that the line ON on the horizontal
dial may still be in the meridian when the edges of the dials
are brought together its base must be cut, as in fig. 109, along
the line PQ so that the angle PNE = w°. When the dial,
THE TRIGONOMETRY OF SPHERICAL TRIANGLES 466
thus modified, is -pressed against the wall fig. 107 still holds
good ; that is to say, the style still makes an angle 90" - A.
with the vertical line O'N in the plane of the meridian. But
the plane of the meridian instead of being perpendicular to
the wall now makes an angle of 90° - w with it. This con-
sideration determines the position of the style with respect
"to the wall.
To determine the graduations we suppose, as before, that
the vertical dial is brought into one plane with the horizontal
dial (fig. 109). To calculate NA in the triangle ON A we ob-
serve that
L NAO - 180" - H - -^ ANO
= 90° - H + t(;.
Whence we have
NA ^ ON
sin H "" sin NAO*
ON
and
NA - ON X
30
cos (H - w)
sin H
cos (H - w)
466 ALGEBRA
But, in the triangle O'NA, NA = O'N . tan H' as before.
Hence
^ T-,, ON sin H
tan H
O'N cos (H - w)
_ sin H . cot X
cos (H - wY
It is not possible this time to make a simple substitution for
H in terms of h. The hourly values of H must be calculated
separately and inserted in the formula.
§ 7. Ex. XCI, D. Supplementary Examples. — Ex. XCI,
D, deals with certain properties of spherical triangles which
are of great theoretical but only of secondary practical im-
portance. The most interesting point about the area of a
spherical triangle (§ 10, Nos. 37-42) is its connexion with the
"spherical excess" previously considered in Nos. II and 12
and discussed in ch. xlii., § 6. In § 11, N OS. 43-50, the
student is led to see the reciprocal connexions which exist
between the sides and angles of a spherical triangle and
the angles and sides of the related polar triangle, and to use
these relations to deduce additional formulae for the solution
of problems. To what is said in the exercise nothing need
be added except a repetition of the recommendation that de-
monstrations on the spherical blackboard should accompany
the teacher's exposition of all such subjects as this.
SECTION yi.
COMPLEX NUMBEBS.
30
THE EXEBGISES OF SECTION VI.
*^* The numbers in ordinary type refer to the pages of Exercises
in Algebra, Part II ; the numbers in heavy type to the pages of this
book.
KXBBOISK PAGES
XCII. The Nature op Complex Numbbbs . . 175, 474
XCIII. Products op Complex Numbers . . . 181, 475
XCIV. Complex Values op the Independent Variable 186, 478
XCV. Complex Values op a Function . . 191, 481
XCVI. The' Relations between two Complex
Variables 201, 486
. XCVII. The Logarithm op a Complex Number . 205, 493
XCVIII. Supplementary Examples
A. The "Exponential Values" of the
sine and cosine .... 213, 494
B. Circular functions of the complex
variable 214, 494
CHAPTEE XLIV.
THE NATUEE OF COMPLEX NUMBEES.
§ 1. The History of " Imaginary " Numbers. — It has
already been pointed out that the term " imaginary numbers "
is (like the term " irrational numbers ") simply a relic of a
past in which the real significance of these mathematical
entities had not yet been perceived. It was first used by
Descartes in his G^om^trie (1637) in connexion with the
solution of equations. It is instructive to observe that even
to so modern a writer positive roots are the only " true " or
'• real " solutions, negative roots being regarded as '* false "
on the ground that they claim to represent numbers which
are less than nothing. ^ By the process (which Descartes
discovered) of raising the values of the roots of equations it
is possible to turn these " false " into " real " roots. There
are, however, some roots which cannot be made " real " by
increase or diminution ; these Descartes, so to speak, aban-
doned as imaginary.'^ Thus the equation
x^ - 6a;2 + 13a; - 10 =
has only one " real " root, namely 2 ; the other two roots
are the " imaginary " numbers 2 + J(-l) and 2 - ^( - 1).
The mathematical logic of the seventeenth and eighteenth
centuries did not reach any satisfactory view of the nature of
expressions of the form a + h J(-l). From the time of De
Moivre (c. 1730) onwards such expressions were used by
mathematicians in intermediate stages of their arguments
and calculations, but there seems always to have been a
lingering doubt of the validity of results reached by such
mysterious means. This is the reason why textbooks of
trigonometry still supplement investigations in which im-
aginaries are enjployed by " proofs not involving the use of
^ Geometriey Bk. Ill, p. 78 in the edition of 1664.
^Ibid., p. 86.
469
470 ALGEBRA
J(-l)". Through the connexion which Descartes estab-
lished between algebra and geometry " imaginary " points,
lines, etc., gradually entered into the latter science, until in
modern times these strange terms have come to connote
some of its most important conceptions.
The long delayed rational interpretation of " imaginary
numbers " appeared almost simultaneously in three distinct
quarters at the beginning of the nineteenth centurj^ The
Dane, Caspar Wessel, has the honour of priority, his tract
On the Bepresentation of Direction having been published
at Copenhagen in 1799, seven years before the obscure
Genevan, J. E. Argand, printed in Paris his now famous
Essai sur une maniere de representer les quantites im-
aginaires dans les constructions geometriques. Lastly, it
seems probable that earlier than either of these writers the
great Gauss had reached the views which, published in 1831,
became the actual source of the modern doctrine of complex
numbers.
Argand 's essay, though it seems to have been little known
until it was republished in 1874,^ contains a wonderfully
clear and confident exposition of the true doctrine of im-
aginaries, and is still worth careful study. The author starts
from the sound philosophical position that an entity symbol-
ized by a + b J{-1) is not necessarily more " absurd " or
"imaginary" than one symbolized by a negative number.
Thus the result symbolized hy b - a when b is less than a
is certainly "imaginary" unless we add to the conception of
magnitude, which necessarily belongs to it as a number, the
further conception of direction. When numbers are thus
regarded as directed, operations and results which were
previously impossible at once become admissible. For ex-
ample, we can prolong any given arithmetic progression
indefinitely far both ways, the repeated subtractions which
carry the terms past zero being representable as equal steps
taken along an endless straight line. May it not be that the
so-called imaginary numbers simply carry a stage further the
process by which the negative number is produced (by addi-
tion of the idea of direction) from the number without sign ?
To answer this question Argand bids us consider the sequence
+ 1 X - ]
1 An English translation by Prof. A. S. Hardy was published in
1881 in Van Nostand's Science Series.
THE NATURE OF COMPLEX NUMBERS
471
The intention here is that - 1 shall bear the same relation to
ic as rr bears to + 1. That is, the operation which turns + 1
into X must also be supposed to turn x into - 1. Remember-
ing that + 1 and - 1 must be representable as points, such
as A and I in fig. 110, equally distant from a zero-point K,
we see that any operation which satisfies the condition laid
down must be representable geometrically by an operation
which, in two identical applications, would carry a point
from A to I. One such geo-
metrical operation would be
the movement of the point
along AI through the distance
AK. This is the operation
which would generate an
arithmetic sequence and is to
be represented algebraically
as the addition of - 1. It
would obviously give to x the
value zero. Two other pos-
sible geometrical operations
would consist in revolving
the line KA through a right
angle either in the direc-
tion of E or in the direction of N; for each of these
operations when repeated would bring the point A into coin-
cidence with I. Further double applications of the same
operation would in each case bring the point successively
back to A ( + 1), to I (- 1), to A again, and so on for ever.
Now there are two algebraic operations which, by definition,
must produce, when applied in this same way, the sequence
+ 1^-1,-1-1,-1, ...for ever. They are multipUcation
by + ^( 1 1) knd by - V( - !)• Thus for exactly the same
reason that we identify - 1 with a unit step taken along a
line in a certain direction, we may identify + J{-1) with a
revolution of a line through a right angle in one sense and
_ ^( - 1) with an equal revolution in the opposite sense. The
choice of senses being arbitrary we take + ^( - 1) to imply
anticlockwise rotation.^
The remainder of Argand's essay is devoted to the ex-
1 The reader should be warned that although the above descrip-
tion represents correctly the substance of Argand's argument the
expression has been somewhat expanded.
472 ALGEBRA
pansion of this cardinal idea and to practical applications of
it. The first step in the generalization of the method will
be to inquire how to represent lines such as KP, KQ, KB
(fig. Ill) of which the first two divide equally the angle AKB.
The preceding argument has indicated that the operation
which carries the line KA from its original position to KP,
and then to KQ and KB in equal swings must be represented
as multiplication by a constant factor. It is therefore easily
seen that the factor must be of the form which De Moivre
introduced into algebra — namely, cos a + ^ - 1 . sin a where
a is the constant angle between the rays drawn from K.
The argument is as follows. If the length of KA is not unity
but r, if there are m - 1 intermediate rays, and if the angle
Fig. 111. Pig. 112.
AKB is ^ (= wa), then the line KB can be represented by
the expression r [cos 6 + J{-1) .sm$]oY a + b J(- 1) where
a = r cos 6 and b = r sin 6; for we know already (by De
Moivre's theorem) that cos ma + ^ - 1 . sin ma = (cos a +
7 - 1 . sin a)"'. Argand called KB a " directed line " and
spoke of r as its " modulus ". The line of descent from the
"directed line" to the "vector" and thence to Hamilton's
" quaternions " is easy to trace.
It may be interesting to reproduce one of the applications
which Argand makes of his theory, and we choose his
derivation of the expansion for log {1 + x). In fig. 112 let
the n arcs AB, BC, CD, etc., be equal. Then the " directed
lines" KA, KB, KG, . . . KN, are to be identified with
numbers of the form a + J - 1 ,b which form a geometrical
progression. Since the arcs AB, AC, AD, ... AN are in
THE NATURE OF COMPLEX NUMBERS 473
arithmetic progression they may (by Napier's definition) be
taken as the logarithms of the terms of the geometric pro-
gression KA, KB, KG, . . . KN. Thus we may write
log KN = mAN
= mnAB
where m is an arbitrary constant whose value determines the
*' base " of the logarithms. Now let mn become great without
end so that AB may be itself considered as a " directed
line " at right angles to KA. Then we have
AB = AK + KB
= - KA + KB
= _ 1 + (KN)i'".
Lastly, put KN = 1 + x and it follows that
log {1 + x) = mn {- 1 + (1 + a?)!/"}
, x'^ x^ .
= m{x-^ + j- ...).
Gauss first published his views upon imaginary numbers in
the " second commentary " upon his Theoria Besiduorum
Biquadraticorum (1831). ^ Many of his statements recall
those of Argand. Thus he says that if + 1, - 1 and ^( - 1),
instead of being called positive, negative and imaginary units,
had from the first been called direct, inverse and lateral units
the confusion which darkened algebraic doctrine would
never have arisen. Equally striking is his declaration that
"the arithmetic of complex numbers is capable of complete
intuitable representation [anschaulichsten Versinnlichung] ".
But the term "complex number" which he introduces in
this discussion, shows that Gauss had reached a clearer
analysis than Argand of the logical standing of the expression
a ■\- J - 1 .h. To him it implies simply a " couple " of
real numbers, just as a rational number is a " couple " of
integers ; the radical difference between the new " complex
numbers " and the old ones being that while the latter
denote the positions of points upon a line the former denote
the positions of points in a plane. In virtue of these
geometrical applications we may say that real numbers
constitute a one-dimensional series while complex numbers
form a two-dimensional series.
"When complex numbers had ceased to be a mystery and
^ Werke, Vol. II.
474 ALGEBRA
could be viewed in the clear light of common sense, it began
to be seen that a -h b J{-1) 13 really the typical number of
algebra, and that " real " numbers should be regarded as
merely special cases in which b = 0. The argument is as
follows. Let y = f{x) denote any function of x. Then if
we are confined to real values of the variables we must admit
that in the case of most functions there are either values of
X to which no values of y correspond or values of y which
are not produced by any value of x. But if our variables are
complex numbers, these exceptions never occur ; to a value of
X of the form u + v J{-1) there corresponds, in the case of
every possible function, a value of y of the form U + V ^( - 1),
u, V, U, V, being themselves real numbers. Thus if we
understand the word "number" in algebra always to mean
" complex number," we can say with Cay ley that " numbers
form a universe complete in itself, such that, starting in it,
we are never led out of it ".
These observations have an obvious geometrical intrepre-
tation. Real numbers correspond to points on a straight line,
complex numbers to points in a plane. If we represent the
values of x by points on one line and those of y by points on
another we cannot say that every function y =f{x) establishes
one-to-one correspondence between all the points of the two
lines ; in most cases whole stretches of points will remain
outside the correspondence. But if we take two planes, and
represent the values of x by the points of one of them and
the values of y by the points of the other, then one-to-one
correspondence between all the points of both planes is
effected by every function. The perception of this remarkable
consequence of the nature of complex numbers found its clear-
est expression in Riemann's graphic method of representing
functions which is to be studied in Ex. XCVII.
§ 2. Ex. XCII. The Nature of Comj^lex Numbers.— The
considerations brought out by the foregoing historical sketch
are made the basis of the treatment in Ex. XCII and the
following exercises. There is much to be said for beginning
with quadratic equations and showing that, just as their
•' real " roots correspond to points on a line, so their " imagin-
ary " roots correspond to points in a plane. But study of the
exposition given in Exs. XCII, XCIII, will, it is hoped, show
that a method which starts off boldly from Gauss's conception
of the complex number as simply a couple or pair of real
THE NATURE OF COMPLEX NUMBERS 475
numbers, the coordinates of a point in a plane, is really
simpler and more interesting as it is certainly more in accord
with the spirit of modern mathematics. Thus in Ex. XCIl
the form a + ib is introduced as a convenient substitute for
the notation {a, h), which has hitherto been used to indicate
the position of a point in a plane. In this form the i is not
a number but is merely a symbol warning us that the
measurement h is to be taken at right angles to the direction
in which a is measured. That is, i is Gauss's " lateral unit ".
After practice has been given in the use of complex numbers
to represent points whose rectangular or polar coordinates
are given, we turn to the problems of adding and subtracting
them. To " add " two complex numbers a-^ + ih^ and a^ + ibi
is found to mean : to take the movement which would bring
a point from the origin to the point {a^ b-^) and to follow it
by a movement parallel and equal to the one which would
carry a point from the origin to {a^, b^). It is also found that
the solution of the problem can be reached with extreme ease
if we treat the " lateral unit " i just as if it were a number.
It is vital that the student should realize that this usage is
entirely arbitrary and is adopted purely for the sake of the
resulting convenience of manipulation. The teacher will see
that the convention is made much more patent by being intro-
duced before i has been identified with the quasi- numerical
entity J{-1).
§ 3. Ex. XGIIL. Products of Complex Numbers. — This
identification is the aim of the next exercise. The argument
is simple but the teacher should study it with care since (for
the reasons already given) it reverses the usual order of treat-
ment. We begin by observing that cos a + i sin a may be re-
garded as what HankeP called a "direction coefficient," that
is, a complex number which, when it multiplies another
number, produces a result which corresponds to the turning
of a line through the angle a. If we multiply the result by
another factor of the same form, cos (3 + i sin ^, it is reason-
able to suppose that the new product describes the original
line rotated through an angle a + /?. Scrutiny shows that
this result would follow from mere algebraic manipulation of
the expressions if, in addition to the assumption that i can be
treated as a number, we assume that it can be treated as if its
^ Theorie der Gomplexen Zahlen-Systeme.
476 ALGEBRA
square were - 1. As is pointed out (Exercises, II, p. 182)
it is not surprising to find that i is not to be regarded as
equivalent to any number already recognized as such. If it
were so its use could not correspond to a totally new idea.
If the "lateral unit" is to be represented numerically at all
the symbol must be one not hitherto thought to represent a
number, just as the representation of the " inverse unit " in-
volved symbolism which had not before been regarded as
corresponding to any possible number.
After this first identification of the lateral unit i with ^/( - 1)
it is necessary to make sure that the equivalence holds good
under the same conditions in all cases. This task is carried
out in the form of an investigation which establishes "De
Moivre's Theorem " for all rational exponents. If it is needed
to prove it also for irrational exponents the argument used
on p. 432 to deal with irrational logarithms can, with suitable
modification, be reapplied.
§ 4. Complex Numbers and Vector Algebra. — Any treat-
ment of complex numbers which follows up the fruitful sug-
gestions of Argand is bound to lead into the algebra of vectors.
Informal excursions into this fascinating province have been
taken at various points of the course from Part I, Ex. XIX,
onwards. The author's conviction is that a few lessons in
vector algebra form an excellent top storey to the edifice of
school mathematics ; for since the laws of this science depart
from those of the algebra of numbers — even when the num-
bers are complex — the study of them greatly illuminates the
principles of ordinary algebra. After some hesitation, how-
ever, he decided that the inclusion of a section on the
algebra of vectors would be too serious an addition to the
programme of this book. He must be contented, therefore,
with mentioning a few works which deal suitably with the
subject. Clerk Maxwell's Matter and Motion and Clifford's
Dynamic are classical introductions to it, and the former
can be read without difficulty by beginners. Henrici and
Turner in their Vectors and Botors have developed mainly
the geometrical aspect, and Hay ward in The Algebra of Co-
planar Vectors the algebraic side. Kelland and Tait's well-
known Introduction to Quaternions is a clear and useful text-
book for those who wish to make acquaintance with Hamil-
ton's powerful methods. Two books, recently published, will
be found of the greatest use, G. Goodwill's Elementary
THE NATURE OF COMPLEX NUMBERS 477
Mechanics (Clarendon Press) to those who have little or no
acquaintance with mechanics or physics, and L. Silber stein's
Vectorial Mechanics (Macmillan) to those who have already
some knowledge of those subjects and are able, therefore,
to appreciate the extraordinary powers of the vectorial
calculus.
CHAPTEE XLV.
EELATIONS BETWEEN A REAL AND A COMPLEX VARIABLE.
§ 1. The Aim of Exs. XGIV, XCF.— Let y = f{x) be any
function of ic, and let account be taken, in the first place,
only of real values of the variables. Then, as we have seen,
those values can be represented by points upon two straight
lines, and their connexion by a system of correspondences be-
tween the points. If the two lines are set at right angles
to one another as " axes " the correspondences determine a
" graph " of the familiar character, i.e. a straight or curved
line lying in the plane of the axes. Next let account be taken
of complex as well as of real values of x and y. Then in this
case, since the values of each variable require a plane for their
representation, there can be no graph in the ordinary sense.
All that can be done is to exhibit in some way the corre-
spondence which the function brings about between the points
of the two planes. Between these two extreme cases come
(i) those in which, while account is taken only of real values
of y, complex as well as real values of x are considered, and
(ii) those in which, while complex values of y are admitted, all
except real values of x are excluded. In such cases one of
the variables is a one-dimensioned number and the other a
two-dimensioned number; it is possible, therefore, to repre-
sent their connexion by a graphic line drawn in three- dimen-
sioned space. In the former class of cases values of y will
be represented by points on the y-axis, values of x by the
points of a plane ; and since the two dimensions of this plane
must be different from that already occupied by the 2/- axis it
must be the plane through the rr-axis perpendicular to the
paper. Similarly in the second case the values of x may be
represented by points upon the a;-axis and the values of y by
points in the plane through the y-a,x\s at right angles to the
paper.
478
REAL AND COMPLEX VARIABLES 479
Exs. XCIV and XCV are devoted successively to the study
of these two types of three-dimensioned graph and of the
functional connexions which correspond to them. The study
is important for other reasons besides the fact that it simplifies
the transition from the ordinary graph to the Riemann
method of exhibiting the relations between the variables when
both are complex. In Ex. XCIV the investigation introduces
complex roots of equations in a way which commends them to
common sense and makes them seem the natural complement
of the real roots studied in Part I. In Ex. XCV it leads
simply and convincingly to throwing open to directed
numbers the one position from which they have hitherto
remained excluded — namely, that of " base " in the logarithmic
and exponential functions.
It is possible that the methods illustrated in these exercises
are new ; that is to say, the author discovered them inde-
pendently and has not found them described elsewhere. It
is, however, tempting to speculate that they are among those
which Gauss had in his mind when he spoke of the possibility
of giving to the whole arithmetic of complex numbers die
anschaulichsten Versinnlichung. Whether this speculation
is or is not well founded the teacher will, it is hoped, find
that subjects which are usually rather dry and unprofitable
gain greatly in vitality and intelligibility by the treatment
here proposed for them.
§ 2. Ex. XCIV. Complex Values of the Independent
Variables. — The argument of this exercise is given too fully
to need additional explanation. In accordance with what
was said above the student learns in division A that a
quadratic equation which would have been declared in Part I
to have no roots, or a cubic equation which would have been
credited with only one root, may be conceived, after all, to
have the full number of roots to which its degree entitles it
— the supplementary roots being complex numbers that
represent points not on the a;- axis but on the plane through
that axis perpendicular to the paper.
It is shown that the existence of complex roots as repre-
sentable entities depends on the fact that the graph which
exhibits the real roots is completed by another curve whose
points lie outside the plane of the paper. The original curve
and the supplement form together the tri-dimensional graph
which represents the correspondence between the real values
480
ALGEBRA
of y and the complex values of x. Exercises, II, fig. 80, illus-
trates the character of this tri- dimensional graph when the
function is parabolic. Fig. 113 below is the one to which
reference is made in No. II. The firm line is the circle on
which lie all points which correspond to real values of both
variables. The broken curve, turned so that its plane is
perpendicular to the paper, is the hyperbola composed of the
points which bring into correspondence real values of y and
complex values of x.
The adequacy of the representation is best demonstrated as
follows. Since y is to be real only those values of x are
admissible which make 4 - a;^ positive or zero. If, there-
fore, X is real its value must not be above 4- 2 or below - 2.
Also since 2/^ + ic^ = 4, all points which correspond to these
real values of x lie on the circle of radius 2. On the other
hand, if x is complex it must be of the form iv.^ For this
assumption makes x'^ = - v^ and so makes 4 - a;^ positive
for all values of v. It v^l be seen that the assumption
^ I.e. + iv^ the *' real " part of the complex being zero.
REAL AND COMPLEX VARIABLES 481
X = u + iv would not work; for then x^ would also be
complex instead of being a real negative number. When
X = iv we have
2/2 _ 4 + -y^
y"^ - v^ = 4,.
Since v means a length measured from the origin at right
angles to the paper, and y, as usual, a length along the
2/-axis, it is evident that this relation describes a rectangular
hyperbola lying in the plane through the ^/-axis perpendicular
to the paper. The other examples of division B can be solved
similarly.
§ 3. Ex. XCV. Complex Values of a Function. — In this
exercise we consider cases in which real values of x are
associated with complex values of y. The case taken as the
basis of the investigation is of great interest and importance.
In Section III the "growth-factor" r in the function y = r'
was thought of as necessarily non-directed, or at least
necessarily positive. The same fact came before us in Ex.
LXXV, No. 11 (v), where the negative part of the graph of
y = x"" was found to degenerate into a few isolated dots
on account of the impossibility of assigning values to a
fractional power of a negative number (see fig. 95, p. 422).
In Ex. XCV this limitation to the process of exponentiation
is removed ; it is found that fractional powers of a negative
number do exist, but that they are complex. In other words,
it is possible to have logarithms to a negative base, but their
antilogarithms will, in general, be complex.
The argument leading to this conclusion is given fully in
the exercise. Fig. 114 shows the diagram to be constructed
in accordance with the directions in Nos. 2, 3. If a line
were drawn smoothly through the points P, Q' and continued
through the points marked on the circumferences of the other
circles, the resulting curve would represent a plan or pro-
jection of the spiral y = ( - ry 2^^ viewed along its axis.
The teacher is strongly recommended to supplement the
drawings by the construction of a model of the spiral in the
manner indicated in No. 5. The author's model was made
in accordance with the following plan : —
A dozen empty cotton reels are first obtained, of identical
shape and size and about an inch long ; also a wooden rod
which will just pass through the holes of the reels. Next,
upon a sheet of fairly stout cardboard, thirteen circles are
T. 31
482
ALGEBRA
drawn, the radius of the largest being 5 cms. and the radius
of each of the others 0*9 of its predecessor in the series.
Rectangles of diminishing size are then drawn round the
circles. The largest measures 13-5 cms. by 10-4 cms., the
smallest 10 cms. by 3 '4 cms. Roughly the same proportions
are observed in the intermediate rectangles. In each case
the centre of the circle is 8*3 cms. from the lower edge of
the card. (It is important that this dimension should be
Fia. 114.
accurate.) Through the centre of each circle a line is drawn
parallel to the longer sides of the rectangle. This line
answers to YY' in fig. 114. A mark is made on the circum-
ference of the larger circle at the point corresponding to P in
fig. 114, and is followed by others upon the other cu'cles
in succession in the positions indicated in fig. 114 and de-
scribed in No. 3. The radii to the points thus marked need
not be drawn, but it is obvious that they would make with
the lines through the centres angles increasing regularly by
60°. The thirteen rectangles are now cut out. A hole
REAL AND COMPLEX VARIABLES 483
just large enough to admit a piece of firm but flexible wire ^
is made through each card at the point marked on the cir-
cumference of the circle. Another hole, to admit the wooden
rod, is made near the bottom of each card. The distance of
this hole from the bottom edge must be the same in each
case, namely, a little greater than the radius of the cotton
reels. The materials are now prepared for the construction
of the model. One end of the rod is thrust through the hole
at the bottom of the largest card and is forced tightly into a
smaller hole in a cork or piece of wood held at the back of
the card. A reel is now placed on the rod in contact with
the face of the card. Thus the card now stands at right
angles to the rod, held between the cork (or piece of wood)
and the reel. The second card is next " threaded " on to the
rod from the farther end, care being taken that its back is
turned towards the face of the largest card. A second reel
follows the second card down the rod so as to fix that card
between two reels. The process is repeated with the suc-
cessive reels and cards until the smallest card follows the
twelfth reel. Reels and cards are now pressed tightly to-
gether and are fixed in position by a second closely fitting cork
or block of wood. Lastly the wire is threaded carefully
through the holes prepared for it, and the model is complete
and exhibits two complete turns of the spiral. When viewed
horizontally against a vertical background the spiral is seen
projected as the curve y = ar'^'''^ . cos Stt^t/X (No. Io).
When viewed from above it is seen projected as the curve
y = ar'^^'f^ . sin ^ttx/X. In the author's model the spiral
completes a turn in IS'd cms. (i.e. A = 18*4), the radius of
the largest circle is 5 cms. (i.e. a = value of y when x is zero
= 5), and r is 0*9. Hence the formulae of the projections
are
2/ = 5 (0-9)^/9.2 ^ cos 7rxl9'2 and 2/ = 5 (0-9)^/9.2 , gin Trxl9-2.
Expressed in the " standard form " (p. 310) the formulae
become y = 5e~ ^-^^^^ cos TrxJ9-2 and y = 5e- ^'^ii* sin irx/Q-^.
The reader who is a physicist will know that these formulae
are of great importance in the theory of all vibratory
phenomena.
The method of representation studied in division A can be
1 The wire used by milliners in making the *' shapes " of hats is
much the best.
31*
484
ALGEBRA
applied in all cases where coasideration is confined to the
complex values of y produced by real values of x. Division
B contains several examples of this type in which the in-
vestigation is complementary to that of Ex. XCIV, Nos. 11-13.
No. 17 introduces a very interesting deduction ; namely, that
a function whose real values are discontinuous for real values
of X may yet be continuous if its complex values are taken
into account. Fig. 115 shows the graph of Ex. LXXXV, No.
14, completed in this way. The broken line represents the
" imaginary " values oi y \ to get it into its proper position it
must be twisted through a right angle about the a; -axis.
If, in the argument of division A, r is made unity the spiral
becomes one of constant radius and corresponds to the
function ^ = ( - l)""- The very obvious properties of this
REAL AND COMPLEX VARIABLES 485
spiral are made, in division G, to assist investigation into that
well-worn mathematical topic " the nth. roots of ± 1 ". There
is not much to be gained at this stage by an elaborate study
of the subject, but Nos. 18-32 may be regarded as interesting
" riders " to the general thesis of the exercise.
CHAPTER XLVI.
RELATIONS BETWEEN TWO COMPLEX VARIABLES.
§ 1. Ex. XCVI. The Belations between two Complex
Variables. — We have seen (ch. xliv., end of § 1) that complex
numbers must be regarded as the typical numbers of algebra
because they ** form a universe complete in itself, such that,
starting in it, we are never led out of it ". In Ex. XCVI we
begin a series of investigations which illustrate this important
statement. The statement itself does not occur in the pages
intended for the student ; the teacher should, however, at an
appropriate moment, direct attention to the facts which it
summarizes.
In division A the student is asked in several instances to
calculate values of y of the form U + iV, which correspond to
values of x of the form u + iv. In division B he attacks
the problem of representing graphically the correspondence
between those values.
In No. 9 we have
a^ - x^ = a^ - r2 (cos ^ + i sin Of
= (^2 _ ^2 cog 20) - ir^ sin 2^.
Assuming that the last expression is of the form
E(cos </) - i sin </>)
we have the equivalences
R cos (^ = ^2 - r2 cos 26, R sin <^ = r^ sin W
whence R = V {{a^ - r^ cos 26f + r* sin^ 2(9}
= ^ (^4 + ^4 _ 2aV2 cos 26) . . (1)
^ , ^ r2 sin 2(9 ,^.
and tan </> = —9— — -„ ^. . . . • (^j
^ a^ - r^ cos 26
In No. 10 we start by assuming
a^ - x'^ = R(cos <^ - -i sin <^)
486
RELATIONS BETWEEN TWO COMPLEX VARIABLES 487
where R and </> have the values just determined. It follows
that
y= J{a? -x^)
= JB, . (cos <^ - i sin <^)^
= ^R. (cos \<j) - i sin \<^) . . (3)
Given the values of r and 6 this expression can always be
turned into a numerical form by means of relations (1) and
(2). If y is to be real sin ^<^ must be zero. That is,
^<j) = Sir, where s is any whole number, and (^ = 2s7r. It
follows that tan <^ = 0, and, therefore, from (2) above, that
sin W = 0. Hence 20 = sir, where s is any whole number,
and 6 = S7r/2. Now if 26 = sir, cos 26 must be either + 1
or - 1 according to whether s is odd or even. Hence when
y is real we have from (1)
R = J{aJ^ + r* ± 2aV2)
= {a" ± r2)
no alternative signs being required before the bracket since R
is a "modulus" and therefore non-directed. Again, since
^<^ = STT, cos ^<^ = ± 1. So relation (3) finally takes the
form
y^± J (a' ± O . . . (4)
Comparing this result with the formula
y = J{a? - x^)
we see that a? = ± r when the sign before the r'^ in (4) is
minus, and a? = ± ir when the sign is 'plus. This conclusion
was obtained by much simpler considerations in Ex. XGIV,
No. 11 (see p. 480).
The principles underlying the graphic representation of the
relations between two complex variables are fully explained
in § 2 of the exercise. The method is due ultimately to
Riemann.^
Two details should be noted, (i) Since all the points of
each of the representative planes are involved in the corre-
spondence some special plan must be adopted of presenting
the scheme of relations to the eye. The principle already
applied in map projections and in the more general " trans-
formations " of Ex. XCI, B, is the one to which it is natural to
have recourse, (ii) It is convenient to have a special notation
to show when complex values of the variables are under con-
^ Grundlagen filr eine allgemeine Theorie der Functionen einer
verdndlichen complexen Grosse (1851).
488
ALGEBRA
sideration. (Strictly speaking, complex values should be re-
garded as normal and restriction to real values regarded as the
exception ; but in an elementary work it is more useful to con-
sider that the variables are real unless the contrary is indicated.)
For this purpose the notation x, y has been adopted. This
departure from the usual symbolism ^ is adopted in order to
emphasize the continuity of ideas between the cases in which
the variables are real and those in which they are complex.
The diacritic dot would more conveniently be placed above
the letter, but the symbols would then be identical with
Newton's notation for "fluxions," which is still employed
c
l' e
* I
►' c
^'c
I I
>'\
P
y (
I
. .. T»
/
/
+2
J
/
7
//'
/
7t
/
V.' -
4-
""
2
/
/
■♦•
Z
+
4 xjt,
7c'
/
-2
l'
/
/
>
>
/
-4.
\^
... ri'
P/
u'
<
«
Fia. 116.
by many writers and is likely to become more rather than
less usual.
Fig. 116 is the a;-net of Nos. II-13; fig. 117 the net into
which the z^-lines and -y-lines of fig. 116 are transformed in
accordance with the function y = \x'^. It will be observed
that corresponding positive and negative -y-lines (e.g. those
lettered a and a') transform into a single parabola — or rather
into two coincident parabolas of which one " begins," so to
speak, above and the other below the U-axis. A similar ob-
servation applies to the w-lines. In fig. 116 the broken line
1 Riemann used z = x + it/ f or the independent variable and
w = u + iv for the dependent variable.
RELATIONS BETWEEN TWO COMPLEX VARIABLES 489
pOp 'mv = 1-bu. It transforms in fig. 117 into a straight
line doubled upon itself at O. The parabola in fig. 116
passing through the points where u = JtJ^ transforms in fig.
117 into the cusped curve qO^.
e,c'
4<
dA
V n.jTh'
T»t,7n»
\
/
^
rsj,
/
y^
6,?>'.^^^^^
\
. \y^
^^iX
a,a!^_^.._^
\
V «
V
^__.
___-fc,le'
U.TX'
7w
\ 1
, I',!''
U' -8
-6
~K
-M
{] *2
•» 4
1.
-^6
+6 U
CL/t-
1
O
/ ^fC
/
■ 7c, V
h.h'-""'^^
XV
^c^
^^■^^--7,1'
^
y
\
"X^
\
/
\
^^\^
c,c'^
9'
<LA'
y- n.n'
TflyTIt'
Fig. 117.
The method of obtaining the graphic transformation is as
follows. We have
y = if
• = i (w + ivf
whence U = (z^^ - i;2)/4: and V = iwv/2.
490
ALGEBRA
Let u have the constant value + 1 and substitute for v in
succession 0, + 1, - 1, + 2, - 2, . . . In this way the
values are obtained of the co-ordinates (w, v) of the points in
fig. 117 which correspond to the points in fig. 116 where the
horizontal -y-lines cross the vertical u = +1. Next put
u = + 2 and repeat the former series of substitutions. The
results give the co-ordinates of the points in fig. 117 which
correspond to the points in fig. 116 where the v-lines cross
Fig. 118.
the vertical u = + 2. By continuing this process the co-
ordinates (U, V) are found of all the points into which the
crossing points in fig. 116 are transformed. When these are
inserted in their proper places on the graph- paper the two
sets of parabolas which represent the original w-lines and
-y-lines can be drawn at once. The transformations of the
lines pOp and qOq in fig. 116 are obtained by marking in fig.
117 the points which correspond to those in which the lines
cross the ■w-lines and ■u-lines. For this purpose the lines are
treated exactly like lines of latitude and longitude in map-
RELATIONS BETWEEN TWO COMPLEX VARIABLES 491
drawing. When the points have been inserted in fig. 117
the graphs are drawn through them in the usual way.
Figs. 118 and 119 give the solutions to Nos. 14-16. Fig.
118 shows the netting of the a;-plane, the ^-lines being
numbered from 1 to 12 and the r- circles lettered from a to d
for ease of identification in the transformed figure. In this
case we have
Fia. 119.
y = 0-la;3
= 0-lr3(cos e + isin 6f
= 0-lr3(cos 3(9 + i sin 3^).
Thus in the net of the 7/-plane the line corresponding in fig.
118 to a certain value of 6 is transformed into a line making
with the initial line an angle 3^, while a given r- circle is
transformed into a circle in which R = r^/10. In fig. 118
the points for which the relation r = 4 sin 6 holds good lie
upon the broken circle. To find the transformation — the
492
ALGEBRA
beautiful broken curve of fig. 119 — the points are marked in
the second figure which correspond to the points in the first
where the circle crosses the ^- lines and r- circles.
In No. l6 r is transformed into JB, and <^ into <^/2, the
symbols having the meanings assigned to them in No. 9.
By means of these relations the transformed co-ordinates must
be found for each point where the r-circles and ^-lines of
7»--4
Fia. 120.
fig. 118 intersect. The calculation is a useful one if it
can be shared among several computers ; otherwise it is too
laborious to be worth undertaking. As a specimen of the
work take the transformation of the point where the circle
of radius 2 crosses the line 6 = 30°. To determine <^/2 we
substitute = 30', r = 2 in formula (2) on p. 486. The
result is a double one ; the value of <;^/2 is either 53° 3' or 143° 3'.
To find ^R we substitute $ = 30° and r = 2 in formula (1)
RELATIONS BETWEEN TWO COMPLEX VARIABLES 493
on p. 486 and obtain ^R = 1-9. Thus two points, lying on
lines at right angles must be plotted in the t/-plane to repre-
sent this particular point in the a;- plane. Similar results
hold good for the other intersections. The consequence is
that each of the r-circles transforms into two curves (fig. 120)
of which one is the other rotated through a right angle.
Fig. 120 shows the transformations of the r-circles, the
transformations of the ^-lines being omitted for the sake of
greater clearness. ^
§ 2. Ex. XCVII. The Logarithm of a Complex Numher. —
When a function is expressible as a sum of a definite number
of powers of a? or a root of such a sum De Moivre's theorem
makes it evident that to every value of x of the form u + iv
there must correspond a value of y of the form IT + iV. This
fact has, perhaps, been sufl&ciently illustrated in the previous
exercises. The same thing can be seen, but not so easily, to
be true of more complicated functions expressed in terms of a
definite number of powers of x, root signs, etc. But when we
come to functions, such as the logarithm, the sine, etc., in
which y is not expressed in terms of powers of x, the matter
is very different ; it is not by any means evident that for every
complex value of x there exists a corresponding complex
value of y. The next two exercises are intended in the first
place to show that what is true of functions of the " rational "
and " algebraic " types is also true of the typical " trans-
cendental " functions. In this way the evidence in favour
of Cayley's generalization (p. 474) is greatly widened. In
the second place the exercises are intended to emphasize
certain incidental results of the analysis — for example, the
"exponential values" of the sine and cosine — which are
themselves of considerable importance.
The question of the logarithmic and exponential functions
of a complex variable is not an easy one, and it is probable
that the line of argument followed in Ex. XGVII may not
satisfy the more exacting mathematician. It is believed,
however, that it is not actually faulty from the logical stand-
^ By an error, detected too late for correction, the transformation
of r = is represented in fig. 120 as a circle of unit radius. When
r = 0, tan <^ = by (2), p. 486 ; hence (^/2 = S7r/2. Also from (1)
/.yR = 1. Hence the origin in fig. 119 transforms into four points
situated where the circle drawn by error in fig. 120 cuts the axes.
494 ALGEBRA
point, and that it will prove easy and interesting. The
teacher who seeks a more rigorous and complete treatment
should consult Mr. Hardy's Pure Mathematics, ch. x.
Nothing need be added to the full exposition of the exercise
except the remark that the assumption, h = 1, which is
professedly made in § 2 for the sake of convenience, is
justified in Ex. XGVIII, Nos. 7, 8.
In No. 9 if we put ttx = 6 vfe have x = O/tt and
cos ^ + * sin ^ = {- ly^^
— {20 In
29
whence log (cos ^ + * sin ^) = — log i.
TT
But if log (cos ■{■ .i sin 0) = iO also, then we have
— log * = zO
TT
or log i = i^.
In No. 10 we have, by our definition of the logarithm of
a complex number, that
log ib = log b (cos 7r/2 + i sin -;r/2)
= log 6 + i^
= log 6 + log i
by the result of No. 9. Since this identity in the results
could follow only from the assumption that
log (cos 6 + isin 6) = ik 6
it may be taken as a strong confirmation of that assumption
and therefore as a partial confirmation of the validity of
the assumption
log (cos ^ + * sin ^) = iO.
§ 3. Ex. XGVIII. Supplementary Examples. — The short
supplementary exercise is devoted to two related topics, the
''exponential values" of the sine and cosine and the question
of finding values for the sine and cosine of a complex
variable. It is to be noted that the exercise is not intended
to be taken until the student has learnt the exponential ex-
pressions for the hyperbolic functions. The corresponding
expressions for the circular functions are then welcomed for
the sake of their analogy with the former ones. Once more
RELATIONS BETWEEN TWO COMPLEX VARIABLES 495
the teacher is warned that a number of important subtleties
are ignored ; for these he is again referred to works such as
Mr. H.auYdj's Pure Mathematics or GhrystOiVs Algebra, Part II.
In No. 7 the differential formulae are (by Ex. LXXXIII)
and (ii) ?|= 1 (e^+e-^^).
For if i may be treated as if it were a number, e* may be re-
placed by a single symbol a, and we have e^"" = a*. Similarly
e~" may be written b"" where b = e~\ Thus we have in the
first of the two given cases
2/ = i (a* + b')
Sv
and ^ = ^ (a^ . log a + b' . log b)
= -l-(e--e--).
But the formula of No. 7 (i) is the exponential value of cos x
and the differential formula derived from it the exponential
expression for - sin x. Hence differentiation of the ex-
ponential value produces the result which theory teaches us
to expect. The same thing is true of the other formula. On
the other hand, when we seek the differential formula of
No. 8 we find that it has the form
|=iift(e--e-"').
Now the interest of this result is that it shows that the formula
of No. 8 cannot be a valid expression for cos x, for the
corresponding differential formula is not the exponential ex-
pression for - sin x. Thus we are entitled to conclude that
the assumption k = 1 made in Exercises, II, p. 208, was not
only permissible but actually necessary. No other assumption
would have guaranteed us against inconsistency in our
arguments.
Fig. 121 exhibits the transformations of the w-lines and
i;-lines of fig. 116 in accordance with the function (No. 14)
y = sin x.
496
ALGEBRA
The figure was constructed as in the former cases by calcu-
lating (by the equivalence proved in No. 13) the values of U
and V (i.e. sin u . cosh v and cos u . sinh v) for each of the
a=+6
it»+5
u,«-5
v=0.
U--*^2
intersections of the ?^-lines and t;- lines in fig. 116 and drawing
smooth curves through the points thus obtained. The various
lines can be identified by the numbers attached to them and
should be carefully studied for the sake of the remarkable
results which they bring out.
SECTION VII.
PERIODIC FUNCTIONS.
T. 32
THE EXERCISES OF SECTION VII.
*^* The numbers in ordinary type refer to the pages of Exercises
in Algebra, Part II ; those in heavy type to the pages of the present
volume.
EXERCISE PAGES
XCIX. CmcuLAB Measure 217, 499
C. Angles of Unlimited Magnitude . . . 224, 501
CI. Sum and Difference Formula . . . 235, 508
CII. Circular Functions 243, 512
CHI. Inverse Circular Functions .... 248, 514
CIV. Progressive Wave Motion .... 254, 515
CV. Stationary Wave Motion .... 264, 518
CVI. Harmonic Analysis 273, 520
CVII. Differential Formuljii for Periodic Func-
tions 285, 528
OVIII. Hyperbolic Sines and Cosines . . . 293, 531
CIX. Hyperbolic Functions . . . . . 301, 534
ex. Supplementary Examples —
A. Diflferential formulae and expansions of
the sine and cosine 306, 528
B. Differential formulae for tan x
C. The calculation of tt .
D. The Gudermannian functions
E. The prediction of Tides .
310, 529
311, 530
313, 536
315, 526
CHAPTEE XLVIl.
THE CIRCULAR FUNCTIONS.
§ 1. Ex. XCIX. Circular Measure. — The reason (and the
only reason) for measuring angles in radians rather than
degrees is precisely the same as the reason for preferring
logarithms to base e to common logarithms — namely, that
theoretical arguments become much simpler. For example,
the " differential coefficient " of sin is cos ^ if ^ is the
number of radians in an angle while it is :j-^ cos ^ if ^ is a
number of degrees. On this account the study of circular
measure has been deferred until the student is ready to make
use of this superiority, and is introduced by a method which
brings out its advantages at the outset. The essence of
the argument is to show (i) that when an angle is small its
circular measure may be substituted for its sine (or tangent)
in approximate calculations, and (ii) that when it is not
small the values of the sine and cosine may still be expressed
approximately in terms of its circular measure by means of
the simple formulae
sin (9 = - g-
and cos ^ = 1 - ^
[*a
This interesting result is reached by a simple graphic
method. The method does not prove the equivalence
sm = - n- +
but it suggests that formula as a simple way of expressing
approximately the value of the sine of an angle in which
^ > 1. Thus it prepares the student for the demonstration
499 32*
500 ALGEBllA
given at a later stage (Ex. CX, A). Meanwhile it enables
him to apply the notion of an expansion of the sine (or
cosine) to the solution of various problems, and so to realize
the great advantages of radian measure.
The teacher will note the proposal (Ex. XCIX, § 1) to
confine the symbols 6 and </> to circular measure and to use
a, ^ and y as symbols for measurements in degrees. The
distinction is a useful one and should be maintained syste-
matically.
Division A of the exercise offers no difficulties. The argu-
ment of division B, Nos. 18-20, runs as follows : —
No. l8. cos e= Jl - sin2^
= J{1 - [0 - ^j} approx.
= J{\ - e^(l -1 + |g)! approx.
= 1 - |'(l - ^ + . • .) + [Pt. I, pp. 72, 73.]
= 1-2-+...
Since subsequent terms will all involve 6^ or higher powers
they may be neglected.
Nos. 19, 20. By No. 18 cos 2 ^ = 1 - ^2 + .
and sin2 $ = 6^ - ~ +
3
sin2<9 + Goa^O = 1 - j^ +
Now we know that the equivalence for sin 2 ^ is correct as far
as it goes. It follows that the equivalence cos 2^=1-^2
+ $'^14: is not adequate as for the term involving ^*. To
make it so we must assume that the expression for cos $
contains a term such that, when the expression is squared,
the coefficient of 6'^ becomes J instead of J. Assume, there-
fore,
THE CIRCULAR FUNCTIONS 501
2 a
Then cos2^ = I - 0'^ + i- + ~W + .
ii^iy
2 11
and we must nave — \. ~ = ~
a 4: d
whence a = 24.
We conclude that the completer approximation required is
cos ^ = 1 - - + - .
§ 2. Ex. C. Angles of Unlimited Magnitude. — The pur-
pose of Ex. G is to introduce and illustrate the notions
(i) that an " angle " may be of any magnitude, positive or
negative, (ii) that to every angle there correspond a sine, a
cosine, a tangent, etc., and (iii) that the values of the ratios
connected with the angle are repeated endlessly in cycles as
the value of the angle rises or sinks. In divisions A and B
these notions are introduced and applied to the analysis
of simple vibratory motion, such as the swinging of a
pendulum. In divisions C and D their wider geometrical
applications are illustrated. Thus the exercise is one of
critical importance.
The reason for basing the extension of the angle-concept
upon a study of harmonic motion has already been given. It
is not necessary actually to carry out the experiment of
Ex. C, § 1, though it is well (and easy) to do so. Many
school laboratories possess, in the shape of a flat steel spring
and a Fletcher trolley, apparatus specially adapted for per-
forming it luxuriously. Much simpler contrivances will,
however, suffice. The text describes one which may be
simplified further by the omission of the vice as follows :
Take a lath long and thin enough to vibrate widely and
fairly slowly. (Nothing serves better than one of the flat iron
laths used to support beds before the era of the spring mattress.
They can generally be obtained at a second-hand furniture
shop.) Fasten the paint brush firmly across it at one end ;
place the other end upon a flat box or pile of books a few
inches high, resting on the teacher's desk ; add another box,
block or book and let a boy or girl press steadily upon it to
keep that end of the lath at rest. Pull the other end two or
502 ALGEBRA
three inches vertically up or down and release it. Hold the
drawing board with its plane vertical so that the tip of the
brush just plays over the surface of the paper. Move the
board steadily along the desk. In this way several *' periods "
of the curve of Exercises, II, fig. 87, can be obtained with
little loss of amplitude.
Two other methods may be indicated, partly on the ground
of their simplicity and partly because they bring out the fact
that " pendulum motion " is identical with harmonic motion
when the pendulum swings through a sufficiently small
angle. One of these is described in Ex. CI, § 2, p. 239, and
its use at this point would be preparatory for the further use
to be made of it when that exercise is reached. The other is
as follows : Pin a sheet of paper to a drawing board and
fasten two strings of equal length at the corners on one of
the longer sides of the board. Fasten the free ends of the
strings to a gas bracket, a map holder or in any other way so
that the board may hang and move freely. At worst a boy
or girl, standing on a chair, may hold them. Draw the board
carefully aside in its own plane. At the same time let some
one hold a brush (dipped in red ink) horizontally so that the
tip touches the paper. Now release the board and, as it
swings, let the brush be steadily raised or lowered with the
tip always just touching the paper. A few trials will give
sufficient expertness in deciding the proper length of swing
and rate of movement of the brush.
The harmonic curves obtained by any of these methods
should be measured so as to show that their ordinates follow,
at any rate roughly, the law
. 360° ^
= a sm —J— d-
It will be obvious that deviations from this law will necessarily
be caused (i) by the uncertainty introduced by the thickness
of the line, (ii) by uneven movement of the board or the brush,
and (iii) by the inevitable decay in the amplitude of the swings
in all cases. On the other hand it will be understood that
Exercises, II, fig. 87, is the ideal curve which would be obtained
if these disturbing circumstances could be eliminated.
The foregoing experiments supply in a clear form the data
needed for our investigation. The "wheel " imagined in § 2
of the exercise is a device for analysing these data. The
THE CIRCULAR FUNCTIONS 503
teacher who distrusts his executive powers may (as we have
said) omit the former experiments altogether. The experi-
ment of § 2 is more important because it illuminates so clearly
the idea of a continuous increase of the angle associated with
periodic recurrences of the cycle of sines. The simplest piece
of apparatus will suffice. Take a cork, bore it with a hole
passing through the centres of its flat ends. Thrust a lead
pencil through the hole to act as the axle of the wheel and
fix a hat-pin (4 to 6 inches long) into the cork so that it is
perpendicular to the pencil. Hold it above the head near a
wall (or in front of a sheet of cardboard fixed vertically upon
the desk) so that the pencil is horizontal and parallel with the
wall (or card). Twist the pencil uniformly between the
fingers, and the shadow of the head, cast by a distant light in
the proper position, will exhibit the required s.h.m.
The argument of §§ 1, 2 of the exercise and examples Nos.
1-10 are of extreme importance but require no further com-
ment. In No. II, the extended definition of the circular
measure is, of course, as follows : Let a line OP of unit length,
starting from the usual initial position along OX, rotate about
O any number of times either in the positive or the negative
direction, and finally come to rest in any position. Then it
shall be considered to have traced out an angle whose circular
measure, 0, is the distance through which the point P has
moved. If the direction of rotation was anti- clockwise B is
reckoned positive, if clockwise, negative.
It follows from this definition that there is no single angle
corresponding to a single position of the line OP. Every
position is associated with an endless series of angles, positive
and negative, consecutive members of which differ constantly
by 27r. If we want to remove this ambiguity we must use
some such device as that of the German mathematician
Eiemann (c. 1857). When OP has travelled completely
round the circle in the course of any one revolution, it may
be supposed to move during its next revolution, not over the
same surface as before, but over another surface which is
pressed down so that it is, to all intents, in the same place as
the former. Thus OP moves over an endless spiral surface
arranged around the perpendicular to the paper through
like the turns of a spiral staircase, but compressed from above
and below so that all the endless series of spires appears in the
plane of the paper. If we adopt this artificial idea we may say
604
ALGEBRA
that a single angle corresponds to every single position of OP,
for as OP revolves it will never pass for a second time over the
same surface.
The teacher is left to decide whether he will add this idea
to those given in the text.
§ 3. Ex. C, C. Spirals. — -The practical value of the idea
that an angle may have any magnitude is further illustrated in
divisions and D. In fig. 122 the firm line is an Archimedes'
spiral (No. 2l) in which a and 6 are positive, the dotted line
one in which a and are negative. Fig. 123 shows the first
positive and the first negative turns of the logarithmic spiral
of No. 22, and parts of the second turns. The tracing point
Fig. 122.
starts from A, where OA = a^ = 1. In accordance with
ch. XXX., p. 317, a must always be positive. The spiral of No.
23 will be identical with fig. 123 but inverted. Since r = a^ is
merely another way of writing log„r = ^ it is obvious that, by
means of a centimetre rule and a protractor, either spiral can
be used to give logarithms to base a.
The lines OP, OP', etc., added to fig. 122 correspond to the
argument of § 3. To answer No. 24 we note that the angle
OPT ( = <^) at which the curve cuts the radius vector is the
sum of Z.OPF and ^TPF. As Q comes nearer to P Z.OPP'
and zOPT approach, and can be made as little different
from a right angle as we please. Hence we have, to as close
a degree of approximation as we please,
THE CIRCULAR FUNCTIONS
505
and therefore
<^ = 2 + Z.TPF
cot <^ = - tan TPF
1 8r
In No. 25 we have r = aO, 8r/S0 = a. Hence
«otcA^--.^
_ _ 1
~ ~ 6'
In No. 26 Sr/8^ = a« . log«a = r . log,a [Ex. LXXXIII,
No. 5], so that
cot </) = - log,a.
To prove the area-formula of No. 27 we note that (fig. 122)
area of OPQ = ^OQ x PP'
more nearly the more nearly PP' is perpendicular to OQ.
When Z_POQ is so small that we may call it SO we can at the
same time write
OQ = r -4- S^
and area of OPQ = ^{r + SO) . rSO
= ir^ . SO
since, by the definition of a differential, {SO)^ may be omitted.
The hyperbolic spiral of No. 31 is shown in fig. 124. As
approaches zero the curve approaches the line A distant a
506
ALGEBRA
from the rr-axis. As in Archimedes' spiral, a and 6 must have
the same sign in order that r may always be positive. The
area swept out by the radius-vector between two positions 6^
and $2 is found from the differential formula
Y
x'-^^
Y
Fia. 124.
Y
Fig. 125.
SA = ir^ . se
se
2 *62
whence
by Wallis's Law.
The lituus (No. 32) is shown in fig. 125. For the area-
problem we have
SA = ir2 . SO
= 20'^^
whence A = ^ log (OJO^) [Ex. LXXXIII, No. 26.]
Fid- 126.
THE CIRCULAR FUNCTIONS
507
§ 4. Ex. G, D. Roulettes. — Fig. 126 exhibits one complete
period of the cycloid of No. 34 together with the construction
Fig. 128.
508 ALGEBRA
necessary to prove the formulae of No. 33. Figs. 127, 128
show the epicycloids of Nos. 37 and 38.
It is easy to verify the theory of the cycloid by the simple
method explained in Exercises Pt. I, Ex. II, No. 30. The
epicycloids of Nos. 37 and 38 are often seen as " caustics "
produced by the reflexion of light from a circular mirror.
Thus half the epicycloid in v^hich n = 2 is seen on the sur-
face of milk in a glass, or on the tablecloth within a polished
serviette ring whenever the illumination comes from a single
distant light. If the light (e.g. a match) is brought close up
to the edge of the glass or the ring the curve becomes the
epicycloid in which n = 1. (See Preston's Theory of Light,
ch. V.)
§ 5. Ex. CI, A. Sum and Difference Formulce. — Before
free use can be made of the sines and cosines of angles >27r
it is necessary to determine whether they follow the laws of
combination established in Part I, Ex. LXII, for angles <jc27r
whose sum does not exceed 27r. This question is investigated
in Ex. CI, division A. The method adopted depends upon the
facts (i) that when we know the sine and cosine of an angle &
in the first quadrant we can at once write down the sines and
cosines of the angles in the second, third, and fourth quadrants
which are obtained by adding one, two, or three right angles
to 6 ', (ii) that the addition of a further number of right angles
simply brings OP into one of its former positions. Now let 6'
and </)' be two angles each less than 7r/2 and therefore subject to
the laws of combination of Ex. LXII. Obtain a new angle, 9,
by adding any number of right angles to 0\ and another new
angle </> by adding any number of right angles to <;^'. Then
by means of the above-mentioned relations between sin 6 and
sin 6', cos 6 and cos 6', etc., it is possible to show that the laws
of combination apply to every possible case.
To establish this statement different members of the class
are first set to verify it for each of the possible sixteen cases
indicated in No. 3. (There are sixteen because each of the
four values of 6 can be taken with any one of the four values
of </>.) In writing out in an examination the proof of any one
of the equivalences it would, of course, be sufficient to indicate
the field of cases, to prove one, and to state that the others
could be proved in the same way. The demonstration must
then be completed, as in No. II, by showing that the equiva-
THE CIRCULAR FUNCTIONS 509
lence also holds good when 6 and ^ are negative — singly or
together.
In No. 8 (^ + <^) is of the form 2mr + (6' + <^') where 2?t
means simply " some even number ". Hence
cos (^ + <^) = cos {6' + <^')
= cos 6' cos <f> - sin 0' sin </>'
= cos ^ cos (^ - sin S sin <^.
For since 9 = 2w7r + 6\ cos 6 = cos 0', sin $ = sin ^', etc.
In No. 10 consider the equivalence for sin (9 - ^) when
e = {2n + ^) IT + 9' and <^ = (2/1 + 1) tt + <j>. In this case
sin 9 = cos ^', cos ^ = - sin ^',
sin ^ = - sin </>', cos (f> = - cos <^'.
Also ^ - <^ is of the form (2n + f )7r + (<9' - <^'),i so that
sin (9 - cf>) = - cos (^' - <l>')
= - cos ^' COS <f>' - sin ^' sin <^
= ( - sin ^) . ( - cos </)) - ( - cos ^) . ( - sin <f>)
= sin 9 cos <^ - cos 9 sin <^.
"l^he division ends with a few identities most of which are
frequently needed in future work.
§ 6. Ex. CI, B. Compound Harmonic Motion. — This
subject has not the same fundamental importance as simple
harmonic motion and may be omitted without serious loss.
It is included as ofifering interesting application for formulae
in which the angles may have any magnitude.
The curves produced by compounding rectangular har-
monic motions are called " Lissajou's curves " and are of
some importance in the theory of sound. The " compound
pendulum " produces many of them in a simple and suflft-
ciently effective way but cannot be used to compound s.h.
motions of the same frequency. The teacher who is in touch
with a physical laboratory should be able to obtain assistance
in illustrating this topic in a more elaborate way if he feels
disposed to do so.
The solution of No. 22 is, of course,
yjx = {h sin 9)1 {a sin 9) = hja.
That is, the resultant vibration is in a straight line making
with the rc-axis the angle whose tangent is hja. In No. 23
the line lies symmetrically on the other side of the ^/-axis.
In No. 24 we have
^ Remember that 2n in the expressions for 6 and <^ and ^ + <^
means " even number " but not necessarily the same even number.
510
ALGEBRA
x^ja'^ = sin2 ^, y^jh^ = cos2 9
• • a2 + 52 - ■^•
In No. 26 we have
sin e = xja, cos 6> = ^(1 - jrVa^) = ^(0^ - x'^)/a and
- = sm ^ cos <^ + cos ^ sin <^
CL
X ^ J(a^ _ a;2) . ^
= - . cos 6 + -^^ . sin (f»
a a ^
whence y - x coacf> = ^{0^ - x^) . sin <^
and a;2 - 2a;?/ cos (ft + y^ = a^ sin^ ^.
To turn the curve clockwise through 45° (No. 27) we substi-
tute for X (x - y)/ J2 and ior y (x + y)/ J 2 and so obtain
x^l - Goscf>) + 2/^(1 + cos<^) = a2 sin2 <^
or a;2 . sin2 f + V^ - cos2 ^ = ^ • sin2 </, = 2a2 sin2^ . cos2 t
that is an ellipse whose semi-axes are
J2 . cos I and J2 . sin |.
Fig. 129 shows how the form of the ellipse changes as (f> in-
creases. The curves marked (i), (ii), (iii), (iv) are those of
No. 29.
In No. 31 we have x = a sin 0, y = a sin 2$, in No. 32
THE CIRCULAR FUNCTIONS
511
1/ = a cos 20, in No. 33y = a cos 20. The elimination of
is simple in each case. The first two of the three curves
512 ALGEBRA
(No. 34) are shown in fig. 130. Fig. 131 gives the curves of
No. 35, the broken being the one in which the phase differ-
ence is 45°. Text-books on sound usually give these and the
more complicated forms which correspond to other ratios of
frequency.
§ 7. Ex. CII. Circular Functions. — In these exercises
the important transition is made from the idea of a sine or
cosine of an angle to that of the sine or cosine of any variable
X. Nothing need, perhaps, be added to the discussion in the
text.
The proof of No. I is, of course, that
sin fix •\- —A = sin {^x + Qtt) = sin^ic.
In No. 2 the formula becomes y = a -^ h ^v£i {x - c). It
has maxima at intervals of Qtt, beginning where x - c = irj^
or X = G + 7r/2. It has minima also at intervals of Stt,
beginning where x - c = - 7r/2 or a? = c - 7r/2.
To solve No. 5 assume y = a sin p {x + b). Then we
have a = 3-2 and (by No. 1) ^ = 27r/2-7. Also when x = 0,
y = 3-2 sin pb. Hence pb = ^r-= .b == — ^or&= - 0*9.
The formula is, therefore,
2/ = 3-2 sin 1^(0: -0-9)
when x = + 2*7, y = 3-2 sin ^tt = - 1-6 ^3.
Nos. 8-10 illustrate the practical importance of the sine
function. They are also anticipatory of the study of the
tides in Ex. GX, E.
Fig. 132 gives the graph oi y = a tan px (No. 12) with
a = 1-5 and^ = 0"9. Since a coipx = - a tan j? \^-o
can be converted into the graph oiy = a coi px (No. 1 4) by
(i) moving it to right or left through a distance 7r/2^ and then
(ii) inverting it. Both functions are discontinuous ; for tan px
has no value when px is any odd number of right angles
(positive or negative), while cot px has no value when^a; is zero
or any even number of right angles (Nos. 13, 14). Fig. 133
gives the graph ot y = a sec px with a = 1'6, p = Q'8. Since
a cosec px = - a aec p Ix + ^\
THE CIRCULAR FUNCTIONS
513
\)
J
♦8
+e
♦4
♦2
r
J
J
r/
r
-4
-6'
I-
+
f
1
\
Fig. 132.
— ( M I
33
514 ALGEBRA
it can be converted into y = a cosec px by (i) moving it 7r/2p
to the left and then (ii) inverting it. Both functions are dis-
continuous ; for sec px has no value when px is any odd
number of right angles (positive or negative) , while cosec px
has no value when px is zero or any even number of right
angles (No. 17).
,^ 8. Ex. CIII. Inverse Circular Functions. — This exercise
travels over familiar ground and requires, therefore, little
comment. The teacher is recommended to employ the con-
tinental notation arc sin x, arc tan x in preference to our insular
sin ~ 1 X, tan ~ ^ x, etc. Since, however, the latter form is still
generally used in examination papers the student should be
made acquainted with it.
Fig. 132 can be transformed, by the usual inversion and
rotation, into the graph (No. lO) of
1 X
y = - . arc tan — .
p a
The same treatment applied to fig. 133 converts it into the
graph of
1 X
y = - . arc sec -,
^ p a
a simple case of which is required in No. 15 (ii).
The identities of Nos. 29 and 30 are needed in connexion
with Gregory's series (Ex. CX, C).
CHAPTER XLVIII.
WAVE MOTION.
§ 1. Preliminary Remarks. — The general spirit in which
the subject of wave-motion is to be attacked has been indi-
cated in ch. XXXVIII., § 5. From the mathematical point
of view wave-motion is a specially interesting concrete ex-
ample of the dependence of one variable upon the simul-
taneous values of two others. Thus, in the familiar case of
water-waves, the height of the water above the level of the
undisturbed surface is a function both of the position of the
point and of the time ; that is to say it varies from point to
point at the same moment and from moment to moment at
the same point. In general, then, the study of waves plays,
in connexion with the doctrine of the variation of a dependent
variable with two independent variables, the part which the
study of a moving point plays in connexion with the doctrine
of dependence upon a single variable. In particular, since
waves are, in most instances, periodic phenomena, they also
afford excellent concrete instances of periodic functions. Apart
from the circumstance that the phenomena of waves offer so
powerful a stimulus and such effective guidance in an im-
portant department of mathematical thought they are, merely
as phenomena, so familiar and impressive that they invite
mathematical analysis and description as naturally as any
other performances of nature or of man. Further, no one can
be in a position to appreciate the most striking triumphs of
physical science who has not given some attention to the ele-
mentary grammar of wave-motion. When to these is added
the further consideration that the mathematical treatment re-
quired is, in its essential features, as simple as it is beautiful,
it must be recognized that the inquiry is one into which the
616 33 *
516 ALGEBRA
ordinary student should no longer be debarred from entering
merely because it does not form part of a somewhat amorphous
tradition.
The theory of the subject is developed in Exs. CIV, CV, and
C VI. The first is given to * ' progressive " and the second
to " stationary " waves — phenomena whose special features are
suflBiciently described in the text. In these exercises we
consider not only waves which can be described by a simple
function of x and t, but also those which correspond to the
formulae based on a. combination of single functions. In
the third exercise an attempt is made, by very simple methods,
to illustrate the famous process by which the great Fourier
taught mathematicians how to analyse a given system of
waves of any complexity. As an important addendum to
these exercises the last section of the supplementary Ex. CX
is devoted to the fascinating question of tidal prediction.
§ 2. Ex. CIV. Progressive Waves. — The chief phenomena
of progressive waves are so familiar that it is not necessary to
exhibit them in the class-room. Nevertheless the teacher
who has a gift for experimenting will find that the mathe-
matical analysis gains point and interest if prefaced by observa-
tions directed ad hoc. A length of narrow stair-carpet serves
ideally well for the exhibition both of a single pulse and of a
train of waves. Eough measurements of the velocity and (in
the case of a train) of the wave-length and frequency are easily
made and add to the clearness of the students' initial ideas. A
length of india-rubber gas-tubing (which can be filled with
sand) is more likely to be part of a laboratory equipment. A
length of rope (within limits, the heavier the better) will do
as a substitute.
As in ch. xlvii., § 2, the experiments intended to aid the
analysis of the motion are for our purpose more important
than those which exhibit it. The simple model described in
the exercise (§ 2) should certainly be made by the teacher
for exhibition, if not by each student. Its use brings out
vividly the relation of the movements at certain points to one
another and to the motion of the " wave ". The following
very easy device is also remarkably effective. Take a wooden
rod, rather more than a yard long.^ At intervals of an inch
', 1 A broomstick serves the purpose perfectly well — or a length of
1 inch ' ' dowell-rod ",
WAVE-MOTION 517
along its length stick into i^ a series of hat-pins about 4 inches
long, arranged in a uniform spiral. Thirty-seven pins should
be used and should be arranged so that the spiral makes three
complete turns. In this case the pins, if viewed along the
length of the rod, are separated from one another by a uni-
form interval of 30°. The model should be held horizontally,
at some distance from a light, in front of a screen of white
card or paper and turned rapidly between the fingers. (The
teacher with a constructive gift may devise a more elaborate
arrangement for producing rotation, but, with a little practice,
the use of the fingers is sufficient.) As the model is rotated
the shadows of the heads upon the screen will exhibit beauti-
ful progressive wave-motion in the manner described in the
text in the last paragraph of i:^ 2. It becomes quite evident
that the progressive harmonic wave is in this case constituted
by equal simple harmonic motions of the shadows of the heads,
each of which differs in phase from the motions of adjoining
shadows by a constant amount — here 30°.
The formulae of Nos. I and 2 are fundamental. They are
simply applications of the general principle that if a graph is
moved a distance d, parallel to the ic-axis, x- d must be sub-
stituted for X in the formula. In this case the shifting in time
t is vt. Hence the formulae y ■■= a sin p{x + vt).
No. 3 is also extremely important and every student must
master it. (The principle has, of course, already been studied
in the preceding exercises.) (i) Let A. be the wave-length;
then to increase x hj X must be to increase the " argument "
p{x - vt) by 27r. That is p{x + X - vt) = p{x - vt) + 2ir
or p = 27r/A. (ii) Let T be the periodic time of the wave ;
then since one individual point goes through a complete
cycle of movements while the wave-form advances through
a length X, we have T = X/v, or v/X = 1/T. (iii) But if the
time for a complete cycle is T seconds the frequency, or
number of complete cycles per second is 1/T = n.
We pass to the examples on compound harmonic waves in
division B. In No. 15 the two component forms are
. 27r
y = A cos (fy . sm —a?
n A • , • /27r ttN . . ^ 27r
and 2/ = A sm </> . sm ( —x + ^ j = A sm <^ . cos —x.
518
ALGEBRA
The resultant'
^wa^Ye-form is, therefore,
y =
A cos (f) . sm -r- ic + A sm
A
0.
, cos
27r
The resultant
A sin (—^ + ^)-
■wa.ye-motio7i is
y = Asm ^.~Y {^ - '^'0 + <^ f •
No. 24 sets the problem of combining waves of the same
length in its most general form. We have for the resultant
y = a sin px + b sin {px + <^)
= {a + b cos <^) smpx + b sin cfi . cospx.
Assume (as in No. 18) that
(a + b cos </)) = A cos </>' and 6 sin </> = A sin <^'
then
A = w' {a^ + 2 a6 cos <i) + bH and tan d)' = ^ — ^^,
^ ^ ^ ^ ^ a + 6 cos </)
and y = ^ cos </>' . sin px + k sin <^' . cos px
= A sin (_pa; + </>').
In No. 26 since — = =-^, X = 20. The unbroken thick
A lU
line in Exercises, II, fig. 97, represents one period of the re-
sultant — the vertical scale being considerably exaggerated.
The formulae of No. 27 substitute for 2 sin -zx (i) - 2 cos ~pX,
(ii) + 2 cos ^x. Exercises, II, fig. 98, shows one wave-length
in the latter case. Fig. 134 of this book shows one wave-length
of the resultant in No. 28. The component wave-lengths are
8 and 12 so that the resultant wave-length is their least
common multiple, 24.
§ 8. Ex. CV. Stationary Waves. — To produce stationary
waves the teacher may have recourse to the length of stair-
carpet or a substitute. A, strip of carpet, 10 or 12 feet long,
can very easily be made to vibrate with two internodal segments.
The effect is almost equally well produced with a length of rope
or tubing weighted with sand. A long narrow trough, fitted
with a glass side, is found in many laboratories and can be
used for experiments on stationary water-waves. It should
be used as described in No. 17. The velocity of water-waves
whose length is large compared with the depth is given by
WAVE-MOTION 519
the formula v = Jgh, where h is the depth in feet or centi-
metres and ^ = 32 in the first case and 981 in the second.
It is not difficult to verify this formula by counting the
frequency of the uni-nodal vibrations and using the relation
n = vj{2l).
For the analysis of stationary wave- motion pieces of thick
iron wire about 30 inches long, bent and used as described in
Nos. 3, 4, 5, give striking results — except, perhaps, for the
case y = a sin rrx/l. Consideration of the rotated wire leads
easily to the fundamental formula of No. 3. It is evident
that the point P^ moves in a circle of radius a sin ttx/I. If
the time is counted from a moment when the plane of the
wire is vertical the projection of this radius upon the wall is,
at time t,
y = a sin-x . cos 27rnt.
Similar arguments apply in Nos. 4, 5.
When the proper moment has come each student should
certainly perform or see the experiment described in No- 22.
Fia. 134.
Nos. II-16 are extremely important but should give no
difficulty. The difference between the two formulae of No.
16 is that in the former the two waves are supposed to
2r7r A 2r7r , ..
be 2/ = A sin -— (x± vt) and in the latter 2/ = A cos . (^ ± ^^)-
In the first case y is zero when x and t are zero; in the
second y has then its maximum value A. In other words
the difference is simply one of origin. The necessity for
520 ALGEBRA
the alternative formula comes into view in No. 17; for in
the experiment with the trough the ends are obviously the
places of most disturbance. As is seen by putting ir = 2-5,
the node is half way along the trough. A very interesting
example of a natural water-motion similar to that of No. 17
is afforded by the curious phenomena observed on Lake
Geneva and called seiches. The lake appears at times to
vibrate about a nodal line drawn across its length. Taking
the average depth to be 115 metres v = ^981 x 115 x 100
= 33-5 m./sec. == 2 kilometres per minute. Also I = 70
kilometres. In accordance with these data the vibrations of
the surface should have a period of 2llv = 70 minutes, and
should be describable by the formula
y^aoos-. cos -.
The formula is found to agree curiously well with the facts,
a varying from a few millimetres to as much as a metre or
more. Interesting details (with references to the original
sources) are given in G. H. Darwin's Tides and Kindred
Phenomena.
§ 4. Ex. CVI. Harmonic Analysis. — The first article
deals with the analogy between a series of integral powers of
X, regarded as the natural means of expressing a class of non-
periodic functions, and a " Fourier series " of sines (and co-
sines) as the symbolic idiom appropriate for the expression of
a periodic function. The analogy is, of course, not perfect ;
there are many important functions (e.g. y = Jx, y = log x),
which cannot be expressed as a series of integral powers of x.
Nevertheless, it is very instructive and is worth emphasis.
The rest of the exercise deals with the practical problem of
resolving a given periodic function (or curve) into its harmonic
components. The reader who is ,not already familiar with
Fourier's theorem should make himself quite clear as to what
it asserts. Let him turn, for example, to Exercises, 11, figs. 97
and 98. Each of these diagrams exhibits a pattern which
is supposed to be repeated innumerable times to the right
and to the left, each repetition occupying a length X. In
fig. 97 the dotted line and the line drawn in dots and
dashes show how the firm-lined curve was built up. The
two components were a " fundamental " sine-curve of a cer-
tain amplitude whose axis-length is identical with that of the
WAVE-MOTION 521
\^ve — that is \, and another sine- curve whose length is X/2
and whose ampHtude is one-half of that of the former curve.
It is evident that if there had been other components, sine-
curves of lengths A./3, A./4:, etc., the wave-form would have
shown modifications which could be made indefinitely varied
by varying the selection of the subsidiary sine-curves and
their amplitudes. Much the same can be said of fig. 98
with the important difference that the curve in that case is
the resultant of a simple sine-curve of length A. and a simple
cosme-curve of length X/2. Thus this figure suggests that
periodic curves of increasing complexity can be built up by
taking as the fundamental component either a sine-curve or
a cosine-curve of length A. and adding to it either sine-curves
or cosine-curves of lengths X/2, X/3, X/4, etc.
Now Fourier's theorem asserts two things. The first is
that the preceding statement can be made conversely ; that is,
any repeated pattern of length A. can be produced by adding
to a certain fundamental sine- or cosine-curve of that length
sine- or cosine-curves of the proper amplitudes whose lengths
are A/2, X/3, A/4, etc. Thus if y is the ordinate of any periodic
curve of length A or the value of any periodic function whose
period is A we have
1/ = ^Q + flj sin px + a.2 sin 2px + a^ sin Spx + , . .
+ bi cos px + b^ cos 2px + b^ cos 3px + . . .
where p = 27r/A. (The constant a^ must be inserted to meet
the case in which the x-axis is not identical with the common
axis of the harmonic curves.)
The second thing asserted by Fourier's theorem is that by
a certain process any of the unknown amplitudes can be
determined at will. It is usual to express and to demon-
strate the method by means of the integral calculus. As
examples Nos. I-14 show, such heavy weapons are unneces-
sary. The method can be made to depend simply upon the
obvious fact that the total area of either a complete sine-curve
or a complete cosine-curve is zero — because half of it lies on
each side of the axis. In Nos. 6-12 a very important conse-
quence is seen to follow from this property. Imagine a solid
whose " plan " is bounded by the curve sin rpx (or cos rpx)
and the axis, and whose " elevation " is bounded by the same
axis and the curve a sin spx (or a cos spx) r and s being any
two positive integers. Then — taking into account the ordinary
conventions about the sign of an area — it can at once be
522 ALGEBRA
shown that this volume is always zero if r and s are different
and always al/2 when they are the same- — I being the axis-
length of the curve under analysis. Division B of the exercise
shows how this striking property can be used to "pick out"
the amplitude of any specified component of the given
curve.
The student is directed in No. 6 and elsewhere to draw his
plans and elevations and to set them at right angles to one
another. The teacher will find it useful to carry these in-
structions further and actually to build up one or two of the
described solids in plasticine or other plastic material. The
solid based upon the curves a sin px and sin Spx is a suitable
one for the purpose. The curves are best cut out of thin sheet
zinc (though cardboard will suffice), any convenient value
being assigned to a. The two positive volumes should be
built up of blue plasticine and the negative volume of red.
The segments should then be removed and placed, the positive
segment upon one pan of a balance, the two negative segments
upon the other. If they have been carefully modelled it will
be found that the negative segment just balances the two
positive segments — that is, that the total volume is zero.
In calculating the amplitudes of the components it is well
to begin by making a table of the measured heights of a series
of equidistant ordinates of the given curve. In the examples
given it will be sufficient to take these so that they represent
phase-differences of 15° in the first component, 30° in the
second component, etc. This will imply the measurement of
thirteen ordinates in the length AB' of No. 15 and twenty-
five ordinates in AB of No. 18, both terminal ordinates being
included. The following table shows the data and the method
of working in No. 15. The first column gives the numbers
of the ordinates. The second column, headed "2/'" gives the
heights of the eleven intermediate, and one-half the height
of the initial and final, ordinates. In the present instance,
since both the terminal ordinates are zero, the halving does
not make itself apparent. In the third column we have
the product of each of the ordinates by sin 15n°, n being the
number of the ordinate. These products are the areas of the
successive sections (but the half areas of the end sections) of
the solid of which one half of the given curve is the elevation
and the curve y = sin Trxjl is the plan. The two columns
headed "2/ • sin 30^°" contain the products which measure the
WAVE-MOTION 523
successive cross- sectional areas (but half the end-areas) of the
solid in which the elevation is as before but the plan is the
curve y = sin Stt^j/Z. It should be noted that the positive and
the negative products are set down separately. All these pro-
ducts were obtained in a very few minutes by means of a
slide- rule. When each member of the class measures a single
ordinate and computes the corresponding products the work
is done very quickly.
n
y
y . sin 15n°.
y ■
sin 30n°.
mras.
+
-
0-0
0-00
0-00
1
130
3-46
6-50
2
23-0
11-50
19-93
3
29-5
20-90
29-50
4
31-5
27-40
27-40
5
29-5
28-60
14-75
6
24-0
24-00
0-000
7
17-5
16-90
8-750
8
10-3
9-18
8-925
9
5-0
3-54
5-000
10
2-0
1-00
3-714
11
1-0
0-26
0-500
12
0-0
0-00
0-000
+ 146-74
+ 98-08
- 24-89
- 24-89
+ 78-19
Calling either of these totals S we have by Simpson's Rule
that the volume V is SZ/12, for 1/12 is the common interval
between the sections. But the average area, A, is Y/l ; hence
A = S/12. Thus the rule, a = 2A, becomes in the present
instance a = S/6. We deduce, therefore, that aj = 2*45 cms.,
a.2 = 1-22, and that y = 2-4:5 sin irx/l + 1-22 sin 2 irx/L
These results are in practically perfect agreement with the
facts.
In No. l8, if the instructions given in the text are followed,
each student will make a table consisting of two columns
answering to the first two in the foregoing table and a pair of
columns answering to the last two. In the case of the
students who are seeking h^, the amplitude of the assumed
component bc^ cos 4:7rxl (I being now = AB = A.), the table
will be as follows : —
524
ALGEBRA
n
y
y . cos cOn°.
n
y
y . cos 30)1°
mms.
+
_
+
_
+ 6-25*
6-25
13
+ 5-0
4-33
1
17-00
14-85
14
- 5-0
2-50
2
18-5
9-25
15
16-0
0-00
3
17-2
0-00
16
26-0
13-00
4
U-8
7-80
17
32-0
27-75
5
12-2
10-52
18
35-5
35-50
6
11-9
11-90
19
34-5
29-80
7
12-2
10-52
20
27-6
13-80
8
14-9
7-45
21
20-0
0-00
9
17-3
0-00
22
7-3
6-32
10
18-4
9-20
23
0-0
0-00
11
17-8
15-42
24
+ 5-65*
5-65
12
13-5
13-50
+ 98-3
- 57-0
* Half the ordinate.
57-0
+ 141-3
In
this case V =
= SZ/24,
A =
Y/l = S/
24, and b, =
2A.
Hence b^ = S/12 = 11-8 mms. The correct result is 12 mms.
The slight difference is due partly to uncertainties in reading
the height, partly to the draughtsman's errors in reproducing
the graph from the original drawing.
No. 20 differs from the former examples in that the ordi-
nates can (and should) be determined by calculation. In
order to include the ordinate at P it will be best to suppose
the ordinates drawn at intervals of 4 inches — that is, so as to
give phase-differences of 12° in the case of the first com-
ponent. From A to P and from P to B the heights are given
respectively by the formulae
y = OA ^ ^ ^^^ y = -- X 2
X
24
X
12
36
10
18*
Including the cases x = and a; = 60 we have the following
sixteen ordinates of which the first and last may be supposed
to have been halved : —
01 2 1 11 12 016 14 11 10 8 2 4 2 f)
^f 'Si 'Sf ^' ^3' ^3f -^' ~9~> "9"' -'-35 '^^f ¥» ^» ¥» 9"> ^•
When these numbers are multiplied in order by sin 0"",
sin 12°, sin 24° ... sin 180°, we obtain the fourteen whole
areas and the two semi-areas needed to find by Simpson's
WAVE-MOTION
525
ftg = 2A,
"4 - —4
= - 0-106'.
Eule the volume of the solid by which the value of a^ is to be
determined. Proceeding as before we have Sj = + 12*09,
Vi = 12-09Z/15, Ai = 12-09/15 and a^ = 2Ai = 1-61.
When the same sixteen ordinates are multiplied in order by-
sin 0°, sin 24°, sin 48°, ... sin 360°, the total, S^,, is 1-89 and
Ag is 0-126. Hence
«2 = 2Aij
= 0-25.
For the third component the numbers must be multiplied
by sin 0°, sin 36°, sin 72° . . . sin 540°, and for the fourth
component by sin 0°, sin 48°, sin 96°, ... sin 720°. The
corresponding values of S3 and S4 are - 0-855 and - 0-799.
We have, therefore,
and a^ = 2A^
0-113
As far as our analysis has gone we have found, then, that
the formula of the curve is
y = 1-61 sin ttx/QO + 0-25 sin ttxJSO
- 0-11 sin 7ric/20 - 0-11 sin ttx/IS.
The formula asked for in No. 20 is
y = 1-61 sin 7ra;/60 . cos 27r^/T
+ 0-25 sin 7ra;/30 . cos 47ri/T
- 0-11 sin 77x120 . cos 67r^/T
- 0-11 sin ttxJW . cos 87r^/T
If we substitute ^ = we obtain the initial form of the string ;
if we put t = l/2v, the form when the fundamental component
Incfies
+2-
♦I-
-I
-2--
30
Fig. 135.
40
60
has completed half a vibration. The shapes which these
formulae assign to the cord No. 21 are shown in fig. 135, the
vertical displacements being exaggerated five times.
Fig. 136 is the graph of No. 22 (ii).
§ 5 The Tides. Ex. CX, E.— It is unfortunate that any-
526 ALGEBRA
thing like a full statement of the results of tidal analysis
would be utterly beyond the scope of an elementary book.
Nevertheless the subject is so interesting and important that
it could not be omitted from a section dealing with periodic
functions. Division E of the supplementary Ex. CX deals
with the one topic which is amenable to simple treatment
— namely, the dependence of the time and depth of high
water upon the angular distance between the sun and moon
when the latter body is on the meridian. The data of the
table on p. 317 are taken from the papers by Lubbock in Philo-
sophical Transactions, 1831-7. It must not be forgotten that
they are the means of hundreds of observations taken when
the sun and moon were occupying all the variety of positions
possible to them at different times in the cycle of their move-
ments. Thus predictions based upon the table may disagree,
sometimes to the extent of a quarter of an hour or so, with
Fig. 136.
the actual times of high water. Lubbock in the same papers
examined in great detail the way in which the means must
be corrected according to the declination of the moon and of
the sun, the moon's distance from the earth, etc.
The theoretical formulae of Nos. 49 and 50 were obtained
from the hypothesis that the tides are caused by the formation
of a spheroid of water whose axis follows the moon round and
round the sky. If the axis pointed directly to the moon the
tides would everywhere be high when the moon is on the meri-
dian. That, however, is not the case. The axis makes a certain
angle with the line joining the centres of the earth and moon
and this angle varies with the angular distance between the
sun and moon. The number in the table of p. 317 measures
the angle in time. It appears in the formula of No. 49 as 0.
The angle X in the same formula is the average value of the
angle for a given port.
The angle <^ which also appears in the formula is the
angular distance between the sun and moon, and is measured
by the interval between their transits. It increases at the
rate of about 48 m. or 12" a day. The fact that the con-
stant a is, for London, 2 h. or 30° is interpreted as meaning
WAVE-MOTION 527
that the tide which follows a certain transit of the moon is, in
reality, determined, not by the present position of the moon
with regard to the sun but by its position 30712° = 2-^ days
ago. This is, then, the time which the tidal wave raised in
the vast area of the Pacific Ocean takes to reach London.
The formula of No. 50 is derived from the same hypothesis
by theoretical considerations. It gives the height of the pole
of the spheroid above its mean surface.
The subject of the tides has a considerable literature much
of which can be consulted, for teaching purposes, with ad-
vantage. The original papers of Lubbock, together with the
extremely lucid contributions by Dr. Whewell, which ap-
peared in Philosophical Transactions about the same time,
will be found very illuminating by the teacher who has access
to them. They are much more within the scope of the mathe-
matician of ordinary powers and the ordinary amount of
leisure than the monumental papers of Sir G. H. Darwin.
On the other hand Darwin has published in The Tides and
Kindred Phenomena (John Murray) a book which, though
" popular," is full of original and interesting information upon
this subject. The article on the tides in the Encyclopcsdia
Britannica was also written by Darwin and gives an easily
accessible summary of the development of the subject. Lastly,
Lord Kelvin included important lectures upon waves, ripples,
and tides in the well-known volumes which should have a
place in every school library.
CHAPTEE XLIX.
DIFFERENTIAL FORMULAE OF THE CIRCULAR FUNCTIONS.
§ 1. Ex. CVII. Differential Formulce for Sine and Co-sine.
— This topic might have been considered at an earlier point
—for example, in connexion with simple harmonic motion. It
is withheld, partly in order not to interrupt the main stream
of the argument and partly because its introduction at the
point chosen helps to develop further the analogy between
non-periodic and periodic functions. Differential formulae for
sin rpx and cos rpx are now needed to answer questions
about periodic functions exactly in the same way as differ-
ential formulae for x"" were needed to answer questions about
the non-periodic functions of Part I. This is the point of
view from which the subject-matter of Ex. CVII is presented
in the introductory article.
In the actual deduction of the differential formulae (Nos.
6-9) the student is led to examine rather carefully the exact
significance of the various steps. This increasing " rigour "
is appropriate to a higher stage of mathematical experience ;
and it prepares the student for the later treatment based on
the concept of limit (Ex. CXII).
The examples in divisions A and B contain nothing that
calls for comment. In division C the newly found formulae
are applied to questions of simple harmonic motion. The
questions are quite simple but the answers have most im-
portant applications in physics.
The solution of No. 25 is : —
ox
-r- = 2Trna cos 2irnt
bt
= 27rn Ja^ (1 - sin2 limt)
= 27m V(a' - x^)'
628
DIFFERENTIAL FORMULA OF FUNCTIONS 529
No. 29 is of fundamental importance but still easier : —
8x
-^ = ^ima cos i2irnt - <t>)
ot
-o72 = - (2x^)2 a sin (^irnt - <^)
= - 4L7r'^n^ . X
§ 2. Ex. CX, A. Expansions of Sine and Cosine. — The
first division of the supplementary exercise begins with
examples in which differential formulae are applied to various
practical problems. In connexion with these the student has
to face a difficulty which sometimes gives trouble. The
differential of sin 6 is cos 6M only if is measured in radians.
If (as will commonly be the case in practical problems) he is
dealing with a sin a in which a is measured in degrees, he
cannot assume that the differential is cos a. 8a. It is first
necessary to convert the a degrees into 7ra/180 radians. We
then have
ira \ TT 7ra tt ^
^^^ isrj = 180" • GOS 180" • ^" = 180 • «50S «-^^-
Nos. 6, 7, 12 should be treated carefully. They are im-
portant as laying a concrete foundation for the study of the
derivative of a product of functions and of partial differentia-
tion in Section VIII. The " reason " expected in No. 6 is as
follows : The errors in a due to errors in /B, a and b, are, by
hypothesis, small. If another number used in calculating one
of these errors is itself subject to a small error it will produce
a small error in the estimation of the small error in question.
But by the very definition of " small " a small error in a
small error is unrecognizable. Hence the errors in a due to
faulty measurements in /3, a and b can be calculated inde-
pendently of one another and the total error will be their
sum.
In Nos. 17-20 the subject of the expansion of the sine and
cosine is resumed from Ex. XCIX, and completed upon a
more satisfactory logical basis. ^
§ 3. Ex. CX, B. Differential Formula for tan x.— This
differential formula is important chiefly because it is needed
in connexion with Gregory's series for tt. For this reason it
1 The author owes the beautiful central idea of the method to a
friend who tells him that it is current in Cambridge but has not
enabled him to attribute it to its inventor.
T. 34
530 ALGEBRA
is considered in the supplementary exercise. Nos. 21, 22
indicate a sound method of deduction. By making h small
enough tan h may be made equal to h to any desired degree
of approximation and so small that the factor (1 - tan x . tan h)
becomes as nearly equal to unity as we please. Since with
further diminution of h the value of tan h is simply propor-
tional to hy we may write
^ cos^aj
= sec^a: . Sx.
This result, in its simple form, holds good for tan only when
is measured in radians. If the angle is measured in degrees
we have 8 (tan a) = -^ sec^ a.Sa. This consideration has
to be applied in all the practical examples.
§ 4. Ex. CX, C. The Calculation of it. — This section con-
tains examples of the calculation of tt which have great his-
torical importance. The methods all had their origin in the
discussions which sprang from the publication of Wallis's
Arithmetica Infinitorum in 1655. Gregory communicated
his series for arc tan a; in a letter to his correspondent Collins
(1671) but gave no proof. There is little question that he
reached it by the process indicated in Nos. 3I-3.
The teacher who wishes to come to close quarters with the
minds of the men who laid the foundations of modern mathe-
matics will find an invaluable collection of documents in
Maseres' Scriptores Logarithmici to which reference has
already been made (p. 468). The statement about Machin's
series in No. 37 condenses into a sentence an interesting
paper by Hutton, the author of the famous introduction to
logarithmic tables.
CHAPTER L.
THE HYPERBOLIC FUNCTIONS.
§ 1. Vahie of the Subject. — The hyperbolic functions can-
not be said to be a subject of fundamental importance ;
nevertheless there are good reasons for giving them a not in-
conspicuous place in a general course. The chief of these is
the beautiful parallelism which exists between the circle and
the circular functions on the one hand, and the rectangular
hyperbola and the hyperbolic functions on the other. In the
next place, the discovery of simple algebraic expressions for
the values of sinh x and cosh x fortifies the student's notions
of sin X and cos x as functional relations which do not neces-
sarily imply any connexion with angular measurement.
Again, as was remarked before, the hyperbolic functions wait,
so to speak, upon the circular functions in many regions of
physics. Thus, the change of sign produced by increasing
one term in a differential formula relatively to another may
imply that a body's motion is transformed from periodic oscil-
lations described by a circular function into a non-periodic
movement described by a hyperbolic function. Finally, we
have seen that there is a useful analogy between a power-series
of X and a sine or cosine series — the one playing with regard to
many important non-periodic functions the part which the
other plays in connexion with periodic functions. The dis-
covery of the hyperbolic functions shows that the analogy can
be carried into a field — that of the exponential functions —
where the power-series ceases to be the natural instrument of
analysis and statement. Indeed it shows that the analogy
between periodic functions and the non-periodic functions of
this field is the closest of all.
§ 2. Ex. CVIII. The Hyperbolic Sine and Cosine. — The
argument of the exercise may be summarized as follows:
Pt. I, Ex. LXV, Nos. 7-12 brought out the fact that all ellipses
631 34 *
582 ALGEBRA
can be supposed derived from the circle and all hyperbolas from
the rectangular hyperbola by an identical process. Thus there
is a striking correspondence between the curves x^ + y'^ = a^
and x^ - ip' = a^, each taken as a whole. It is reasonable,
therefore, to inquire whether there is not an equally striking
correspondence between the individual points of the two
curves. In NOS. 7-12 two distinct principles of correspondence
are suggested and are found to lead to an identical conclusion :
namely, that the point (a sec 6, a tan 0) on the hyperbola may
be held to correspond to the point (a cos 0, a sin 6) on the
circle. In this way it is possible, then, to "pair off" any
point on the one curve with a point on the other — with the
exception (No. lO) that the points where the circle cuts the
2/-axis have no partners on the hyperbola.
The next question concerns the numbering of the corre-
sponding points. Consider Exercises, II, fig. 99. For P, a
point on the circle, 6, which measures the angle POA, is the
natural numerical label. The point P' on the hyperbola might
also be labelled in virtue of its correspondence with P, but
it would obviously be better to seek some principle of denumer-
ation which may be applied without direct reference to the
point on the circle. Two such principles readily offer them-
selves. The numerical value of ^ is a measure (i) of the arc
AP and also (ii) of the sectorial area AOP. It would be
natural, therefore, to label P' either (i) with a number which
measures the hyperbolic arc AP', or (ii) with one which
measures the hyperbolic area AOF. There is no a priori
reason for choosing one of these rather than the other but
there proves to be a strong practical reason for selecting the
latter. It is not possible ^ to express the length of the arc
AP' definitely in terms of familiar functions of 0, but it is
comparatively easy so to express the area AOP'. It is proved
in Nos, 13-16 that just as the area AOP can be written as
■^a^O so the area AOF can be written ^a'^u where the number
u is calculated from by the relation
u = log tan (^^ + ly
By this formula it is possible to obtain a numerical label for
any point F by considering merely its position on the hyper-
bola. For since its ordinate is y = a tan 0, we can determine
^The attempt to do so carries us into the diflficult region of
*' elliptic functions ".
THE HYPERBOLIC FUNCTIONS
533
by the relation = arc tan (yja) and then proceed to cal-
culate u without referring directly to the circle x"^ + y^ => d^.
Now if we consider a point P on the circle which is deter-
mined by a certain value of 0, the abscissa and ordinate of P
are a cos and a sin 0. In virtue of the general analogy
between the curves it is appropriate to give the names hyper-
bolic cosine and hyperbolic sine of u to the functions which
give the coordinates of the point P' determined by a certain
value of 11. In virtue of these definitions we write
X = aseo 6 = a cosh u and y = a tan $ = a sinh u.
:I
zi
i
}
■:i
r'
■^n
Y
Fig. 137.
In the remaining examples the properties which flow from
these definitions are studied. The graph of No. 21 should
be drawn with some care since it may be used in later ex-
amples. It is represented in fig. 1 37. The graph brings out
(i) that 2A is a single- valued periodic function of B ; (ii) that it
is discontinuous, since there are no values of u when
6 = i^n + l)o j (iii) ^ is a many- valued function of u.
Fig. 138 shows the graphs of sinh n (S), cosh n (C) and
tanh n (T) (Nos. 22, 23).
534
ALGEBRA
§ 3. Ex. CIX. The Hyperbolic Functions. — Just as the
symbolism y = sin x may mean simply a connexion of a
certain kind between an endless series of pairs of numbers,
the original dependence of the function upon the properties
of triangles being dropped put of sight, so ?/ = sinha; and
y = cosh X may be regarded as symbolizing connexions be-
tween numbers without reference to the rectangular hyper-
bola. The freeing of the hyperbolic functions from their
dependence upon the properties of a particular curve is best
effected by showing that the values of sinh x and cosh x can
be calculated directly by means of the exponential expressions
{e' - e~ '')/2 and (e^^ + e " '^)/2. The demonstration of these
equivalences is the subject of division A of the exercise. The
consequent generalization of the idea of hyperbolic functions
is studied in division B.
The examples from No. 11 onwards extend the analogy be-
THE HYPERBOLIC FUNCTIONS 536
tween the circular and the hyperbolic functions to their dif-
ferentials. Thus in No. II we have : —
y = sinha? = ^
.-. ?|-= t±A^l =, cosh X, [Ex. LXXXIIL No. 6.]
Similarly in No. 13 : —
y = a sinh pa; = a . ^
.*. ^ = ap . ^ = ap . cosh|>a;
and ^ = ap"^ . ^ = ap^ . smh j?a;.
In the same way ii y = a cosh px
gj = ai)2 cosh;?aj.
From these results it follows (No. 18) that, if a point has
any one of the motions x = a sinh pt, x = b cosh pt or
X = a sinh^iJ + b cosh pt, its " acceleration " is subject to the
law
Thus we reach the striking and important conclusion that
if the acceleration of a moving point is proportional to its
distance from a fixed point the distance will itself be either a
circular or a hyperbolic function of the time. It will be the
former if the acceleration is - p'^x and, therefore, is always
directed back towards the origin. It will be the latter if the
acceleration is + p^x and is, therefore, always directed away
from the origin. In the former case the motion will be
periodic or vibratory, in the latter case non-periodic or non-
vibratory.
Nos. 22-4 bring out another striking and useful instance of
the general analogy between the two kinds of functions. In
No. 22 we have
y = a{cosh.px^ - sinh_pa;2)
Since the index of e is here necessarily negative the greatest
value of 2/ is a (corresponding to x = 0). The other values
of y fall off symmetrically on both sides of the 2/-axis but
536
ALGEBRA
never reach zero value. The graph is, therefore, the single
symmetrical " hump " depicted in the upper part of fig. 139.
In this figure a = 1 and p = O'l as in No. 24 (i). Now let
the hump move to the right with velocity v. Then we have
a single wave, involving all points of the axis, and describable
by the formula of No. 23, namely
y = a{cosh_p(ic - vty - sinh ^(a; - vty].
A combination of such forms will produce a complex single
wave which can be compared with the complex repeated
waves obtained by combining harmonic waves. The lower
part of fig. 139 shows, for example, the compound single wave
of No. 24 (i). Thus, just as the circular functions are the
idiom most appropriate to the description of a train of identi-
cal waves, so the hyperbolic functions offer the most natural
means of describing the single wave.
In Section IX the function e'^"" will be found to play a
most important part at every stage of the discussion. Nos.
22-4 may be regarded as preliminary to its use there.
§ 4. Ex. CX, D. The Gudermannian Functions. — When
the student has learnt that 8 (tan x) = sec% . hx he is in a
position to return to the study of the function of which the
correspondence between points on a circle and its related
rectangular hyperbola is the most important concrete instance.
This function, symbolized in the general form
y = log tan (^ + ^j,
can be regarded as the inverse of another function, or con-
nexion between numbers, of the form
THE HYPERBOLIC FUNCTIONS 537
y = 2 arc tan e" - '^.
The latter is called the " Gudermannian function oi x" and
may, therefore, be written concisely as
y = gdx.
The best corresponding symbolism^ for the inverse Guder-
mannian function will be
y = arg gd x
though English writers generally write it
2/ = gd - ^x.
By considering the correspondence between P and F in
Exercises, II, fig. 99, it is easy to show that iiy = arg gd x then
Sy = seG X . Sx (Nos. 42-3). The practical importance of this
discovery constitutes the chief claim of the Gudermannian
functions to a place in the course. The student is reminded
of the principle by which he sought in Ex. LXXXVII to fix
the positions of the parallels of latitude in the Mercator net.
To apply that principle he had formerly to be contented with
a graphic method. He is now able to see that y, the equatorial
distance of a given parallel in the Mercator net, is connected
with X, the equatorial distance of the parallel on the corre-
sponding geographical globe, by the relation
y
arg gd a:; or 2/ = log tan (^ + ^ j.
Thus it is possible to determine the positions of the parallels
by calculation.
^ Hardy, Pure Mathematics, p. 377. The meaning is " t/ is the
argument whose Gudermannian is x ".
SECTION VIII.
LIMITS.
THE EXEECISES OF SECTION VIII.
*»* The numbers in ordinary type refer to the pages in Exercises
in Algebra, Part II ; the numbers in heavy type to the pages of
this book.
EXERCISES
CXI. The Meaning of a " Limit
CXII. Differentiation (I)
CXIII. Differentiation (II) .
CXIV. Integration ....
CXV. Differential Equations
CXVI. Some Theoretical Considerations
CXVII. A General Formula for Expansions
CXVIII. Supplementary Examples —
A. DiflFerentiation
B. Integration .
C. Differential equations
D. Partial differentiation
E. Total differentials
F. Various practical applications
PAGES
324, 541
338, 549
346, 549
354, 553
364, 556
376, 557
389, 559
403, 560
405, 560
407, 560
413, 563
420, 563
422, 563
CHAPTER LI.
DIFFERENTIATION AND INTEGRATION.
§ 1. Ex. CXI. The Nature of a Limit— The doctrine of
the differential and integral calculus in the proper sense of
the term differs from the " calculus of approximations " studied
in previous sections by the fact that it is founded upon the
conception of a limit. The student has been prepared for
this fundamental idea at various points of his earlier work ;
Ex. CXI is devoied entirely to the important task of making
it as definite and clear as possible.
The remark that the real nature of many algebraic ideas
has long been obscured for the student by the misappre-
hensions of the writers who fixed the traditions of the text-
book and the classroom is especially true of the idea of a
limit. Even now these misapprehensions have not lost their
currency. It is particularly important, therefore, that the
subject should be represented in the clear outline to which
it has been reduced by modern critical mathematicians. Ex.
CXI attempts an exposition which shall give the essential
features of Cantor's doctrine free from the technical com-
plications which make it a rather formidable affair as it
appears in the writings of the more rigorous modern writers.
These essential features — which are really extremely
simple — stand out most clearly in instances such as Ex. CXI,
Nos. 10, 14, 15. Thus in No. 10 the numbers obtained by
giving to n, in the expression 2 - l/n, successively higher
integral values form a sequence of rationals which constantly
rise in value but have no last term. There is, however, a
certain rational number — namely 2 — which comes next after
all possible terms of this sequence. That is to say, if any
rational number be named less than 2 there will always be
some term of the sequence 2 - Ijn between it and the
number 2. This is what is meant by calling 2 the limit of
541
542 ALGEBRA
the sequence 2 - Ijn. Similarly in No. 14 the points to
the left of P form an ordered sequence which has no last
term towards P, while P is itself the first point of OX which
lies beyond all possible terms of the sequence. P is, there-
fore, the limit of the points which lie on its left. [See
ch. XXXIX., pp. 414-16.] Again, in No. 15, the rationals be-
tween 2 and 5 form a sequence which is endless both ways,
and the numbers 2 and 5 are themselves the first rationals
met with beyond the sequence. These numbers are, there-
fore, the lower and upper limits of the sequence. The
rationals from 3 (inclusive) up to but not including 10 form
a sequence which is endless towards 10 but has a definite
beginning in 3. Thus it has an upper limit — the number 10
which is the first rational beyond the sequence — but no
lower limit. Finally, since the integers between 100 and
200 have both a definite beginning (101) and a definite end
(199), they form a sequence which has neither an upper nor
a lower limit.
Consideration of these examples enables us to state rather
more formally the definition of a limit, (i) A limit is always
the limit of a sequence S' which is thought of as part of a
wider sequence S. (ii) The sequence S' must have either no
first or no last term, (iii) If S' has no first term let there be
a term L of S which is not a term of S' but is the last term of
S before all possible terms of S' ; or if S' has no last term let
there be a term U of S which is not a term of S' but is the
first term of S after all possible terms of S'. Then L and U
are respectively the lower and upper limits of S' in the
sequence S.
Two points in this definition require emphasis. The first
is that the limits L and U are not themselves members of
the sequence S' but are terms of S which lie outside S'. The
neglect of this point is responsible for most of the current
inaccuracies in the use of the idea of a limit. The second
point follows from the first. The sequence S' may in itself
be capable of having a limit — that is, it may be without a
first or a last term or without both — yet the question whether
or not it actually has a limit and what that limit is depends
upon the sequence S of which it is regarded as forming a
part. Both points are well illustrated by Nos. 17, 18. In
No. 17 S' is the sequence of rationals less than 3 and (since
there is no last rational less than 3) is obviously capable of
DIFFERENTIATION AND INTFXJRATION 543
an upper limit. If S, the wider sequence of which it is a
part, be taken to be the whole sequence of rationals from
zero upwards, then the upper limit is evidently 3, for this is
the first term of S beyond all possible terms of S'. But in
No. 17 (i) we are told to regard S as a sequence composed
of all the rationals less than 3 together with the rationals
from 4 upwards. Now 3 is not a member of this sequence,
so that it is not the limit of S'. As a matter of fact the first
term of the prescribed S which comes after all possible terms
of S' is the rational 4 ; thus 4 is in this case the upper limit
of the rationals less than 3. Again in the last part of the
same question we are invited to consider the same S' as part
of an S which consists of the rationals less than 3 together
with the rationals greater than 4. In -this case it is evident
not only that neither 3 nor 4 is a term of S but also that
there is no term of S which can be said to be the first after
all possible terms of S'. Thus the rationals less than 3 have,
as a section of this new sequence, no limit at all. In No. l8
the function has already been studied as Ex. LXXV, No. 13
(iii), and its graph is shown as fig. 98 (p. 424 of this book).
If we give x values from + 1 to + 1*5 the value of n is, by
definition, 2 ; the number under the radical sign will, there-
fore, be negative and the function has no real values. It has,
on the other hand, real values for an endless sequence of
values of x between zero and 1. It has again real values for
values of x from 2 (inclusive) up to (but not including) 3. It
follows that if we regard the numbers which measure the
abscissae as segments of an S which consists of all the real
numbers, then the limit of the abscissae less than 1*5 is 1.
On the other hand, if we regard any one of the isolated
segments of the graph in fig. 98 as an S' belonging to an S
which is simply the whole of the points which correspond to
real values of the function, then it is evident that the limit
of the points constituting the first upper segment on the right
of the 2/-axis is the first point in the second segment, that is
the point ( + 2, + J2). For the first segment (S') has, as
we have seen, no last term, and the first term of S beyond it
is the first point of the second segment. This example shows,
even better than No. 17, that it is a mistake to suppose that
the terms of a sequence necessarily approach "indefinitely
near " to their limit, and still more erroneous to say that they
" ultimately coincide " with the limit. As we have seen, the
544 ALGEBRA
latter statement is never true ; for the limit is always outside
the sequence of which it is the limit. Whether the former
statement is true or not depends upon the nature of the S of
which the S' is conceived as a segment. In most cases of
practical importance it is true, because the S is either the
sequence of real (or else of rational) numbers or some other
sequence whose terms have one-to-one correlation with the
number-sequence ; but it is not necessarily true.^
It is extremely important to note that in the foregoing
discussion S' and S need by no means be numbers ; the de-
finitions and arguments apply equally well to any sequence
in which the terms follow one another in accordance with a
definite rule of order. The points on a line and the ordinates
of a curve will occur to everybody as instances of such non-
numerical sequences ; it is, however, profitable to quote at
least one example outside the ordinary field of mathematical
discussion. Consider the notes which a skilled violinist can
elicit by bowing (say) the "A" string of his instrument at
the same time that he "stops" it at some point with his
finger. As he runs his finger up the string these notes will
form a sequence S' regularly ordered in respect of "pitch".
They will all be higher in pitch than the note A but they
will have no last term towards A. Again, this sequence of
notes is a segment of a wider sequence S which consists of
all possible notes producible by means of strings of all pos-
sible lengths. Finally, the note A is the first term of S below
the sequence S'. Thus it is the lower limit of S' in exactly
the same sense as 5 is the lower limit of the sequence of
numbers produced by giving to n in the expression 5 -f- 1/n
all possible positive integral values from unity upwards in
order.
It is also very important to be clear about a complication
which appears constantly in the case of a sequence of numbers,
and may appear in other cases — for example, sequences of
notes. This is the complication dealt with in Ex. CXI,
p. 330, and in the footnote. It emerges there in connexion
with the problem of determining the height of a certain ordin-
ate of a curve. The point is that while the ordinates of a
^ The reader who demands further authority for these state-
ments should consult an admirably simple and lucid article on the
subject published by Mr. Bertrand Russell in the philosophical
journal Mind for April, 1908.
DIFFERENTIATION AND INTEGRATION 545
curve are all different lines the numbers which measure their
heights are not necessarily all different numbers ; since ordin-
ates having the same height may occur in different parts of
the curve. Now the foregoing discussion of limits presup-
poses that the terms of S' and S are all different from one
another ; only in that case can we say that the limit L or U
is not a member of S'. The question arises, therefore, how
this condition is to be satisfied in the case of the numbers
which measure ordinates whose heights may recur. To
answer it we note that in the case of these numbers the S
which we have in mind — the wider sequence to which the
limit belongs — is always the whole sequence of real numbers
in their natural order. It follows that the S' must be simply
a segment of this sequence. In order, therefore, to apply
the notion of a limit we must select from the whole sequence
of numbers measuring the heights of the ordinates a segment
in which the numbers are all different and all either increas-
ing or decreasing in the natural order ; and this segment is
to be taken as the S' to which our reasonings apply. As is
shown in Ex. CXI, p. 330, such a segment can always be
found however " wavy " the curve may be, provided that the
waves have definite dimensions. Difficulty will, in fact, arise
in only three cases. (1) A curve may have the infinite
waviness of the " crinkly curves " of Ex. CXVI and their
congeners. In this case (discussed in the next chapter) there
actually is no Hmit. (2) The numbers may be all the same.
In this case the notion of a limit as here expounded cannot
be applied. On the other hand, as is shown in the footnote.
Exercises, II, p. 331, we can do perfectly well without it.
(3) The terms of the sequence may, as in the case discussed
in the Note after No. 12, appear alternately above and below
the limit. In this case we may regard them as constituting
two sequences in which the upper limit of one coincides with
the lower limit of the other.
It is only in the last case that the definitions of a limit
found in modern treatises ^ show a clear advantage over the
1 " A function f{x) has the limit L at a value a of its argument
X, when in the neighbourhood of a its values approximate to L
within every standard of approximation." (Whitehead, Introduc-
tion, p. 229.) " If a variable x represents any number of a sequence
aj, ttg, a.,, . . . , ttn, . . . , it is said to approach a number a as a
limit, provided that, corresponding to every positive number, there
T. 35
546 ALGEBRA
Cantorian deJ&nition of § 4. The reader may demur to this
remark and point out that the adoption of the usual definition
would also make it unnecessary to exclude the case in which
all terms of the sequence are identical. For example, the
definitions given in the footnote and other recognized equi-
valents of them would give 2 as the limit of the endless
sequence
2, 2, 2, 2, 2,
To this objection it is perhaps sufficient to reply (1) that none
of the ways of regarding limits which these definitions imply
is comparable in clearness and interest with the Cantorian
notion in its direct and simple form ; (2) that the gain in
generality obtained by adopting them is purchased, as far as
the beginner is concerned, at far too great a price, and (3)
that they are apt to obscure for the beginner the vital fact
that the use of limits in calculations leads to exact results,
and to leave him with the notion that the calculus after all
only gives approximate results, though these may have in-
definite closeness to the truth. These arguments are offered
as a defence of the policy of making the view of limits pre-
sented to the beginner in Ex. CXI the standard definition
to be used in all problems that meet him at the present stage
of his studies. The usual definition is introduced in the Note
on No. 12 mainly to prevent confusion if the student should
come across it in his mathematical reading.
It is not superfluous to add — for it is frequently forgotten
— that the definitions of the footnote above are specially in-
tended to supply a necessary and sufficient test for a limit in
the case of a function ; ^ that is, in a case where the S is
necessarily the whole sequence of real numbers. Thus (as
Nos. 17, 18 show) it is not really so general as the definition
adopted as our standard, and is prone to give the student too
limited a view of the scope of the idea of a limit.
§ 2. Ex. CXI. The Practical Uses of Limits. — It was con-
venient to begin this chapter with a theoretical discussion of
the nature of limits. In accordance, however, with our uni-
versal pedagogical principle, Ex. CXI itself begins with the dis-
exisfcs a number m such that the numerical value of the diflference
a -an is less than m, provided only that wis greater than m."
(Young, Concepts, p. 204.)
^ Russell, Principles, p. 327, and references.
DIFFERENTIATION AND INTEGRATION 547
cussion of problems intended to prepare the student for the
notion of a limit by demonstrating its usefulness. Speaking
broadly, that usefulness falls under two heads which may be
briefly considered.
(1) The first head is illustrated by the use of a limit to
define the '* velocity of a moving point at a given moment ".
The discussion of Ex. CXI shows that, if we define velocity
as the quotient of a distance travelled by the time in which
it is traversed, then the " velocity at a given moment " is not
a velocity at all. On the other hand, if we consider the dis-
tances travelled by the point during a series of constantly de-
creasing intervals of time and divide each distance by the length
of the corresponding interval, we shall again fail, as a rule, to
obtain anything that can be called "the" velocity of the
point, for all the results will be different except in the special
case of " uniform " motion. But if the sequence of " average
velocities " thus calculated follows some definite law of suc-
cession as the interval is taken smaller, then it will generally
have a definite limit as the interval approaches zero. Thus
the limit is a perfectly definite number associated in a per-
fectly unambiguous way both with the given moment and
with the endless sequence of different average velocities.
Moreover, for small intervals of time the average velocities
are sensibly equal to the limit, the differences being of
theoretical rather than of practical importance. It follows
that, although the " velocity at the given moment " is not
really a velocity at all, it is quite the most useful number to
quote in order to describe the behaviour of the moving point
while it is in the neighbourhood of the place which it oc-
cupies at the given moment. A similar statement explains
the practical value of the derivative or " differential co-
efficient '' of a function in other cases. If, for instance, the
value of X in the function sin x is increased to x + h the
ratio {sin {x + h) - sin x\lli is quite ambiguous in value ; for
it depends upon the value of h. But as h approaches zero
the ratio has a limit, cos x^ and although this limit is not any
one of the ratios yet it is connected in an unambiguous and
unique way with the sequence of actual ratios. It may,
therefore, be regarded as representing them just as the so-
called " velocity at a given moment " represents the endless
sequence of average velocities during intervals of time which
succeed that moment.
35*
548 ALGEBRA
(2) The second practical use of limits comes into view
when we employ them to determine a magnitude which
cannot be evaluated directly. We have already had im-
portant instances of this kind of calculation, the most striking,
perhaps, being the calculation of the area of a spherical belt
in Ex. LXXXVI. The method was analysed on p. 444 of
this book, and it is unnecessary to repeat the analysis here ;
the teacher should, however, make sure that he appreciates
the argument as set forth in the first two paragraphs of that
page. It is necessary to add only (1) that in most cases one
of the two sequences is the sequence of real numbers, the other
being a sequence containing the magnitude to be determined ;
(2) that the indirect calculation is most convincing, especially
to the beginner, if the magnitude (M) and the number (N)
which measures it can be exhibited, like the S and C of
p. 444, as filling corresponding gaps in sequences which extend
on both sides of them — that is, if M and N can be exhibited
as at once upper and lower limits of sequences known to
correspond to one another term by term; (3) but that the
argument is unchanged in principle, though less striking, if,
through necessity or for brevity's sake, M and N are regarded
as the limits (upper or lower) of sequences which lie only on
one side of them. As an illustration of the last remark
consider the example in Ex. CXI, § 2. Here M, the magni-
tude to be determined, is the height of PQ ; N is the number
which measures it. It is shown (1) that the ordinate whose
magnitude is in question lies between, and is the limit of,
a lower sequence consisting of ordinates pq and an upper
sequence consisting of ordinates p'q, (2) that N lies similarly
between, and is the limit of, the sequences of numbers re-
presented respectively by
{3ic'^ - h{^x - h)}a and {^x^ + h(Sx + h)}a
and (3) that the latter sequences correspond to the former,
term by term. From these premises it is a very convincing
deduction that the height of PQ is exactly Sax'^ ; for PQ is
the only line between the two former sequences and Sax^ the
only number between the two latter. On the other hand, the
argument would have been perfectly sound if we had con-
tented ourselves with pointing out that the ordinates p'q and
the numbers {Sx^ + h{Sx + h)}a are sequences corresponding
term by term, that PQ is the limit of the former sequence
and Sax'^ of the latter, and that therefore they correspond.
DIFFERENTIATION AND INTEGRATION 549
As is pointed out in Ex. CXI at the end of § 4 this briefer
argument is always sufficient unless the sequence to which
M belongs is discontinuous so that there are, for certain
values of the variable, two limits corresponding to the two
ways in which that value may be approached.
§ 3. Exs. CXII, CXIII. Differentiation. — In the next two
exercises the preceding theory of limits is applied to the
technical problems of determining the derived functions or
derivatives of functions of given form. The teacher will
note that the idea of the derived function is introduced to
the student as a generalization of the familiar ideas of con-
nexions between area-functions and ordinate-functions, ordi-
nate-functions and gradient-functions, etc. No special
notations have hitherto been employed to symbolize these
different notions ; the need of them has not been felt and
their alDsence has probably been an aid rather than a hindrance
to clear thinking. But with the generahzation of these special
functional relations into the idea of a derivative a specific
notation becomes necessary. The practice of mathematicians
offers a choice from several alternative notations of which
D{y\ y and dyjdx are the most important. The student is
to be taught eventually to use each of these forms of sym-
bolism, but the first is introduced as the standard form for
reasons, positive and negative, that must be briefly indicated.
The negative reasons may be considered first. The no-
tation dyjdx, which undoubtedly has played and continues to
play the most important part in mathematical literature,
goes back, as is well known, to the mathematician and philo-
sopher Leibniz, who shares with his contemporary Newton
the credit of having invented the differential calculus as a
distinct branch of mathematics. It is not so well understood
that it expresses a view of the nature of a *' differential co-
efficient " which is quite out of harmony with modern ideas
and, in particular, conflicts with the doctrine of limits ex-
plained in §§ 1, 2. Briefly, the view was that any finite
value of the variables y and x is really the sum of a vast
number of " infinitesimal " values which, though immeasur-
ably small, have yet a definite magnitude — much as (to use
an illustration given by Leibniz in another context) the
sound of the sea is the sum of a vast number of sounds
which, though individually inaudible, must really exist and
have a defiaite degree of loudness. It was assumed that the
550 ALGEBRA
"infinitesimals" of a given variable all have (like the "in-
finitesimal " atoms of a given substance) the same magnitude.
Thus the differential coefficient dyjdx is simply the ratio of
the "infinitesimals" of the two variables in the case in
question — the ratio being finite and measurable, just as the
relative weights of atoms are measurable, in spite of the
smallness of the terms. It need hardly be said that this view
is no longer held.^ The student of the differential calculus
is always warned that dy and dx in the expression dyjdx are
7iot to be regarded as " infinitely small " numbers, or, indeed,
as numbers at all, and that dyjdx is not a ratio but only the
limit which the ratio of the increments of the variables ap-
proaches as the increment of x approaches zero. But in
spite of this warning the erroneous presuppositions of the
notation still produce a confusing effect upon the student's
mind. From the pedagogical point of view, to make the
student write the derivative as a fraction and at the same
time to forbid him to think of it as a fraction is a poor plan.
Moreover, any virtue which it may have is effectually de-
stroyed if, in spite of the protest, the student is taught to
treat dyjdx as a fraction in arguments of the following type : —
dyjdx = 3rc'^
dy = 3ic^ . dx
y = x^ + G.
It is scarcely a cause for wonder that few elementary students
have really clear ideas as to how the calculus "works," even
though familiarity with its applications may have convinced
them that in some mysterious way it produces trustworthy
results. There is only one way of avoiding this unsatisfactory
state of affairs — namely, to avoid the notation which creates it.
For these reasons the Leibnizian notation is to be with-
held until the student's grip of the logic of the calculus is
strong enough to withstand its misleading suggestions. For
the first stages of his progress we must use either D(y) or y'
as our symbol of differentiation. The second symbol has the
advantage of compactness but is not nearly so expressive as
the former. The notation D{y) constantly holds before the
student's mind the fact that the object of inquiry is a function
^ For criticisms of it see Russell, Principles, chs. xxxix.-xli.,
or the more popular account in Whitehead, Introduction, pp.
224-7.
DIFFERENTIATION AND INTEGRATION 551
which he is to derive from the given function y by means of
a definite rule of procedure. Since this relationship between
functions is the essence of the whole matter (see p. 248 of this
volume), the notation which so directly suggests it is incom-
parably the best for the beginner. Moreover, by the simple
device of inverting the D we have (as in Ex. GXIV) a
symbol which suggests in the clearest manner the fact that
in the process of integration we are merely tracing in the
reverse direction the relation between a function and its de-
rivative.
We turn now to the technical rules for differentiating a
given function. It should be noted that the student is already
acquainted with the results of differentiating most of the
standard forms. His business here is, first, to use the doctrine
of limits to place those results upon a proper logical basis
and, second, to apply them to the ready determination of
derivatives in the more complicated cases. For both purposes
it is necessary to estabUsh certain simple but most important
theorems about the sum, product, and quotient of limits.
Consider the problem of finding the derivative of sin x as it
is treated in the older textbooks. The argument runs as
follows : —
dy _ J . sin (x + h) - sin x
dx " A^o h
r 7 /nN sill (^/2)
= U, COS {x + /t/2) . — ^^j^
= cos x.
The important point is the step from the second to the third
line. The argument is that since the hmit of cos (x + h/^)
is cos X and the limit of sin (/i/2)//t/2 is unity the limit of
their product is the product of cos x and unity. But it is
clear that this conclusion only follows if the limit of the
product of two factors is identical with the product of the
limits of the factors. No process of differentiation which
implies the truth of this proposition can be logically satis-
factory until the proposition itself has been proved. In
Ex. CXII, No. 7, the student is, therefore, called upon to
prove it by the method already applied in § 2 of the exercise
to the simpler case of the sum of two functions. The proof
intended runs as follows : —
Any term of U can be expressed as L„ + p, and the corre-
552 ALGEBRA
spending term of V as L„ + q. Hence the corresponding
term of W may be written
L„ . L„ + gL„ + _pL„ + pq.
By hypothesis p and q represent numbers which both approach
zero in a sequence which has no last term. The correspond-
ing terms of W are therefore also a sequence with no last
term, for L„ and L„ are constant numbers. If we put p = q
= zero the corresponding term of W is L„ . L„, and is obvi-
ously the first term of W beyond the sequence just referred
to. That is, it is the limit of that sequence. We conclude
that as the terms of U and V approach respectively the limits
L„ and L,, the terms of W approach as their limit the product
This fundamental theorem about limits is applied in
Ex. CXII, A, to establish some of the simpler standard forms.
In Ex. CXII, B, it is used to justify the rule for differenti-
ating a function of a function. In Ex. CXIII it is once
more used to prove the familiar rules for differentiating the
product and the quotient of two functions. All these matters
are of well-recognized importance, and it is necessary only to
emphasize the necessity of securing that the logic involved in
them is clearly understood.
In Ex. GXIII, B, we return to Wallis's Law and prove it
for all rational exponents by a method independent both of
the binomial theorem (of which no general proof has yet been
given) and of the exponential curve. The new proof is based
upon an idea borrowed from Wallis's Arithmetica Infini-
torum. It is not purely algebraic but involves assumptions
much less serious than the assumption that the exponential
curve has at every point a definite tangent corresponding to a
definite limit of (a- + '^ - a')lh. [See Ex. CXVI, § 3.] The
proof is prepared for by three examples (Nos. 7, 8, 11). In
No. 7 we assume D(x"') = mx"^ ~ ^ and D(x) = 1. Hence
by the rule for differentiating a product, we have
Z)(a;- + i) = Dix'^.x)
= a;'" . D(x) -{• X . D{x"')
= X"" + WiC"*
= (w 4- l)a;'".
Similarly in No. 8 we have
D{af + *) = D{x'' . x^)
= x" . D(a;«) + x^ . D{x^)
= of .qx"-^ + x" . paf - ^
= (P + 2)a^ + '"^
DIFFERENTIATION AND INTEGRATION 553
No. II is established by a similar application of the rule for
differentiating a quotient.
§ 4. Ex. CXIV. Integration. — Current ideas about the
nature of an integral show, like those about the nature of a
differential coefficient, traces of the erroneous mathematical
philosophy of earlier days. Under the influence of Leibniz
the problems first systematically studied by Wallis came to
be regarded as having as their aim the summation of an
" infinite " number of "infinitesimals " dy of the form y . dx,
where dx symbolizes as before the minima indivisihiiia of
which any finite value x of the independent variable is com-
posed. This view is still represented not only by the usual
notation
1 = \y . dx
which (like dyjdx) was first introduced by Leibniz, but also
by the common statement that an integral is the sum of an
infinite number of infinitely small magnitudes. With the
rejection of the notion of an infinitesimal as a definite atomic
magnitude this statement and the notation which expresses
it have both become inadmissible and should certainly be
abandoned. If dx has any numerical significance at all it
stands for the increment h when h is zero. ' It follows that
the product y . dx is also zero for all values of y, and, there-
fore, that the problem represented by
\y .dx
is the summation of a series of zeros. To teach that any-
thing but zero can result from this process, even if the terms
are " infinite " in number, is simply to darken counsel.
Expressed in terms of sounder modern ideas the matter
may be stated as follows. Let y be any function of x and
let hx stand (as in the earlier sections of this work) for a
"small" constant increment of the independent variable.
Further, let the small product y .hx be the increment of
another function I (the integral). Let x assume in succession
values from zero up to a final value x - 8a; at intervals of Srr,
and let y in the products symbolized by y .hx assume in
succession the values corresponding to these. Then for
every assumed value of the interval hx the sum of the pro-
ducts will have a certain value. As the interval is shortened
and approaches zero this sum will (it is assumed) approach
a limit I. Then I may be defined as the integral of the
function y or of the product y . hx. Thus I is not the sum
554 ALGEBRA
of an infinite number of products ; it is simply the limit of the
sum of a finite number of products. Indeed there is neither
need nor warrant for introducing the term "infinite " at any
point of the definition or discussion. Substituting 8x for the
nonsensical dx we may still usefully retain the Leibnizian
mode of expression —
l = ly.Sx
but the symbol " J " must now be read : " hmit of the sum
as Sx approaches zero ".
As is well known there is a very striking and simple con-
nexion between the functions I and y : the function y is
simply the first derivative of I. Thus there are two alter-
native ways of regarding an integral of a function t/. We
may think of it either as the limit of the sum of products of
the form y . 8x or as the function from which y would be
obtained by the process of differentiation. Of these the first
way is undoubtedly the more important from the point of
view of practical applications ; in physics, in mensuration,
etc., the integral almost always appears as the limit of a sum.
But from the theoretical point of view it is much simpler to
regard the relation between a function and its integral as the
inverse of the relation between a function and its derivative.
The student of this book has been familiar from the middle
of Part I onwards with the ideas which find their logical
completion in the former, more complex notion. There
need, therefore, be no hesitation in choosing at this stage the
better theoretical mode of approach. Thus in Ex. CXIV the
integral is first treated as the inverse of the derivative ; the
demonstration that it may also be treated as the limit of a
certain sum is reserved till the end of the exercise.
A remark has already been made about the notation em-
ployed in the earlier part of the exercise. When writers on
formal logic employ a certain letter to symbolize a given
relation, they not infrequently invert the letter to symbolize
the inverse relation. That device is followed here. The
fact that a function I is the integral of a function y is sym-
bolized by the notation
I = a{y).
This use of the inverted D is believed to be an innovation.
The author also believes that it will be found to justify itself
by the directness with which it suggests to the student the
nature of the problem he is called upon to solve.
DIFFERENTIATION AND INTEGRATION 555
It is probably unnecessary to direct attention to any of the
examples, which are all limited to results common to ele-
mentary treatises on the calculus. It may, however, be
pointed out that the form of argument used in division C to
show that the integral may also be regarded as the limit of a
sum has been chosen chiefly because it seems a natural com-
pletion of the arguments of Wallis with which the student's
earlier work may be assumed to have made him familiar.
In § 3 a special case is treated as an illustration of the general
principle. Further concrete illustrations are suggested in
No. 21 and Nos. 24-26. The generalization is required in
No. 30. The proof that
f(x) = U {¥{x + h) - ¥{x)]lh
demands, of course, nothing but the repetition of the argument
of Ex. CXI, § 2, in a generalized form. The second part of
the argument may run as follows : —
Select any value of h which is an exact submultiple of
X - a, and erect between the points on the ic-axis where
X = a and where x has its final value x a series of rectangles
of width h, the heights being in succession f{a), f{a + h),
f(a + 2/t), . . , , f(x - h). Across these rectangles draw a
smooth curve in such a way that the area under the curve
above any of the segments h of the base is equal to the
rectangle standing on that segment. Then it is clear (1) thai,
if X be now taken to mean the abscissa of the left-hand side
of any rectangle, the curve will cut the top of that rectangle
and so have f{x) for its ordinate at some point whose abscissa
lies between x and x + h; (2) that as h approaches zero the
limit of this state of affairs is one in which f{x) is the ordinate
where the abscissa is x for all values of x between the ends
of the curve. It is also evident (3) that the total area under
the curve is, for all values of h,
Lt ^'h'' f{x).h
and (4) that as h approaches zero this sum has as its limit
the area under the curve whose ordinate-f unction is f{x).
But by hypothesis that area is F(ic) - F(a). It follows,
therefore, that
ht '^i~'f{x) .h = F{x) - F(a).
556 ALGEBRA
Now, by definition,
the relation
f(x) = L^ {^{x + h) -
may be expressed by the notation
F{x) = af{x)
and the relation
Lt
by the notation
^ f{x) . h
X = a
= F{x)
¥(x)\/h
F(a)
J,
f{x) . 8x = F(cc) - F(a).
We conclude, then, that in all cases the functions intended
by the notations (If(a) and [f{x) . Sx are identical.^
§ 5. Ex. CXV. Differential Equatiojis. — Under the head-
ing of "differential formulae" the student has already faced
the fundamental questions involved in the solution of differ-
ential equations. He should be able, therefore, without much
difficulty, to acquire the modest amount of technique de-
manded by the examples of Ex. CXV. These are limited
almost entirely to differential equations which play a part
of fundamental importance in the theories of mechanics,
physics, and engineering.
The treatises on the Differential and Integral Calculus by
Granville (Ginn & Co.) and by Hulburt (Longmans) both
contain chapters dealing with the subject in a simple way.
The teacher who desires more information than is given in the
text or in ch. lii., § 4, will probably find what he wants in
either of them. Prof. Perry's well-known Calculus for
Engineers (Macmillan) contains a number of interesting ap-
plications rather too technical in character for inclusion here.
1 The reader may notice an imperfection in this argument :
namely, the assumption that the ordinates are equidistant. For
complete rigour it should be shown that 2/(x) . h tends to the same
limit however the ordinates are distributed.
CHAPTER LII.
EXPANSIONS. SUPPLEMENTAKY EXAMPLES.
§ 1. Ex. CXVI. Some Theoretical Considerations. — Ex-
pansions, regarded as approximation-formulae, have engaged
our attention from time to time since the earliest exercises of
the course. The final substantive exercise in this section is
to be devoted to the quest for a generalization that shall in-
clude all the individual expansions in one comprehensive
formula and shall, at the same time, provide an universal
test of their usefulness and validity. This imperial formula
is, of course, Taylor's Theorem.
Taylor's Theorem is best regarded as a deduction from, or
an application of, the Theorem of Mean Value. The first
division, of Ex. CXVI is given, therefore, to the study of that
familiar proposition, including the necessary inquiry into the
conditions under which it holds good. Here is a favourable
opportunity for introducing the student to a topic which
illustrates, better almost than any other, the significance of
the "rigorist's " suspicion of proofs based upon "intuition "
— illustrates it by an instance in which it is strikingly justified.
•This topic is the fascinating one of tangentless curves. " If
we draw a curve with one stroke of a pencil from A to B so
that for one value of x there is only one value of y, and there
is no kink or sudden bend in the curve, we can easily satisfy
ourselves that there is always one point at least on the curve
the tangent at which is parallel to AB." Yet when the
student has made acquaintance with the cases, produced by
Weierstrass and others, in which this seductive conclusion is
demonstrably wrong, he will be readier to accept the rigorist's
contention that "you cannot prove anything by an appeal
to the eye " and to understand why he "tries to live up to
the sentence 'point d'images dans cet ouvrage' and to
state precisely the conditions that are vaguely implied in the
557
558 ALGEBRA
words ' drawn with one stroke of the pencil,' ' no kinks or
sudden bends,' and so on ".^
The question of tangentless curves is treated too fully in
the text to require commentary. Weierstrass's epoch-making
paper, ' ' Ueber continuirliche Functionem eines reelles Argu-
ments die fiir keinen Werth des letzteren bestimmten Differ-
entialquotienten besitzen," is reprinted in his collected
Werke, Vol. II. The argument is neither abstruse nor
very long, but cannot be summarized with enough brevity
for reproduction. Probably the graphic analysis given in
Exercises, II, figs. 112-13, will be sufiiciently illuminating.
The account of the " crinkly curves " of Moore and the
"space-filling curve" of Peano is taken from an interesting
paper by the former in the Trans. Amer.
Math. Soc, Vol. I, 1900. It will be
noted that Moore's two curves may be
regarded as giving respectively the ordin-
ates and abscissae of Peano's. Fig. 140
shows one-ninth of the abscissa-curve,
^ = <^ (i), at the stage described in No.
"^^ ' i6. Fig. 141 is the diagram asked for
■p ^.„ in No. 20, the abscissae of successive
nodes being the ordinates of the nodes
in fig. 140 and the ordinates being the ordinates of the nodes
in Exercises, II, fig. 115. The figure is, therefore, an early stage
in the development of Peano's curve towards the limit in which
it passes through every point of the square. Since every point
on Peano's curve is given by a distinct value of t and every
value of t gives a point on the curve, it is clear that the total
number of points in the square is the same as the number of
possible values of t. But each value of t corresponds to a
single point of the line OT ; hence the number of points in
the square is the same as the number of points in the line.
Cantor showed, by a dififerent line of argument, that this
result may be extended to the number of points in a cube.
Gosiewski's line composed of " condensed semi-circles "
always proves a very fascinating topic of discussion. The
author owes his knowledge of it to Dr. L. Silberstein.
^ The words placed between inverted commas are quoted from a
manuscript note kindly placed at the author's disposal by Mr. C S.
Jackson.
m
EXPANSIONS. SUPPLEMENTARY EXAMPLES 559
§ 2. Ex. CXVII. A General Formula for Expansions. —
As is well known, Brook Taylor, in his Methodus Incremen-
torum Directa et hiversa (1715) gave his expansion without
any formula for estimating the remainder. It is still usual
for text-books to begin by deriving the coefificients on the
assumption that the expansion is valid, and to keep the
question of the remainder for subsequent discussion on the
lines set by Lagrange or Cauchy. The proof given in the
text aims at developing at once the expansion with its re-
mainder-formula. The chief reason for adopting this plan
~yK — ^
d — -^^
B
Fig. 141.
is that, at the stage which the student has now reached, the
value of the theorem consists no more in its giving a general
expansion-formula than in supplying a general test of the
validity of such expansions. Moreover, it will probably be
found that the argument given, though it appears long, carries
more conviction to the student's mind than the ordinary
proof, because it gives a fuller insight into the reasons why
the formula should have such wide applicability. It will be
seen that it is based upon the theorem of Wallis of which so
much use has already been made and that it deals with the
remainder at each of the stages into which the derivation of
the whole expansion-formula is divided.
560 ALGEBRA
The usual method of deriving the theorem is given as an
exercise for the student in Nos. 24-6. The teacher who
prefers that plan may easily turn first to those examples and
consider the fuller proof later. In that case it would be
advisable to let the class use the remainder-formula without
proof in Exs. 1-23.
§ 3. Ex. CXVIII. Supplementary Examples, A, B. — In
division A of the supplementary examples the student is in-
troduced for the first time to the ordinary, or Leibnizian
notation for differentiation. The first application, in Nos.
1-5, is to the differentiation of an inverse function — a case
in which its usefulness is apparent. In Nos. 6-7 the
hyperbolic functions are considered for the first time in this
section, though their differential formulae were, it will be
remembered, deduced in Ex. CIX, p. 303. The method of
logarithmic differentiation is also explained. In No. 10 (vi),
(vii), for the functions gd x and arg gd x see Ex. CX, D. The
solution of No. 10 (viii) is as follows : —
?/ = 2 arc cos J[{x - h)l(a - h)]
X = b + {a - b) Gos^^y
dx/dy = - 2(a - &) cos 1^2/ • i sin ^y
= - J{{a - x)(x - b)].
Since dyjdx = htSy/Sx = IKLtSxjSy) = l(dxldy) the required
derivative is the reciprocal of the present result. No. 10 (ix)
is done similarly, with the exception that sinh ^y is to be
determined from cosh \y by the relation
cosh-'^2/ ~ sinh'^-|^ = 1.
The novelty in division B is the integration of a function
of a function, usually called "integration by substitution".
The name here used has the advantage of bringing out the
relation of the process to that of differentiating a function of
a function. The examples cover the more important forms
studied in an elementary course and should be done very
thoroughly, integrations found difl&cult at the first attempt
being written out until mastered.
§ 4. Ex. CXVIII, C. Differential Equations. — Division C
supplements the examples of Ex. CXV in one or two im-
portant respects. In Nos, 21-5 ^^® student learns the
general method of solving ^n equation of the first degree
EXPANSIONS. SUPPLEMENTARY EXAMPLES 561
with constant coefficients by the substitution y = e^'. The
teacher should refer here to Ex. CXV, Nos. 24-6, and show
how the use of the exponential expressions for the sine and
cosine makes possible a remarkable simplification of treat-
ment. In No. 23 this principle is to be combined with that
of Ex. CXV, § 3. For instance in (iii), to find the particular
integral we assume y = a cosh |ic + 6 sinh ^x, and obtain
{4(|)2a - |6 - Sajcosh |rr + {4(|)'^5 - fa - 5b\ sinh fic
= 2 cosh fa;
whence we derive the relations
Hifa - |6 - 5a = 2 and 4(|)2& - |a - 56 =
giving a = - 11/14 and b = + 3/14. So far the particular
integral. For the complementary function we assume that
y = e^'' satisfies
4:y" - y' - by =
and deduce the condition
4:p^ - p - 5 =
whence p = 5/4 or p = - 1. Thus the full solution of the
equation is
11 3
y = Ae^^^* -1- Be"'' - r-^ cosh |^ + t^ ^i^^ 1^
A and B being arbitrary constants.
In No. 25 (iii) we must assume for the particular integral,
in accordance with Ex. CXV, No. 31,
y = ax + b + ce~'^''.
Substitution in the equation gives
121ce-5^ + ax - 4:a + b = 2x + e"^^
whence c=l/121,a= + 2 and b = + 8. For the comple-
mentary function we assume y = e^* and obtain, on substitu-
tion, the quadratic
4j92 _ 4^ + 1 =
whose roots are both + |. Hence, in accordance with No.
24, the full solution is ?/ = (A -1- Bx)e^^ + e-"V121 + 2x + S.
In § 3 we attack the case, so important in physical and
other applications, where it is necessary to *' separate the
variables ". It should be noted that the customary practice
of separating the dy and dx is quite indefensible except upon
the understanding that, when separated, they are really used
instead of 8y and Sx. It seems to the interest of clear thinking
that ab first, at any rate, the student should actually make
the substitution of the differentials.
T. 36
562 ALGEBRA
No. 26 (vi)-(x) are, it will be noticed, all of a single type
which may be represented by the general equation
dyjdx = y(x + a)/(6o + b^x + b^x^).
Written in this form it plays an important part in Prof. Karl
Pearson's theory of " skew frequency-curves " to be considered
in Section IX, The method to be followed is essentially the
same in each case. Thus in (vii) we have, after separating
the variables,
Sy _ l.hx _ 3.8a;
y X x^
whence log ?/ --- C + 2 log x + 3/a;
or log y = log A + log x'^ + log e^ '"".
So that
y = Kx'^. ^ '"".
Similarly in No. 26 (x) we have
82/ _ (g? + 4) . hx
~y ~ {x - 1)2 + 52
{x - 1). 8a? b.^x
- (x - If + 52 "^ {X - 1)2 + 52
whence, by Nos. 12, 14,
log 2/ = + -^ log (a?2 - 2x + 26) + arc tan [{x - l)/5].
So that, finally, y = k{x^ - 2x + 26)* . e*'-'=**'^t(^-i>^5],
§ 4 gives a few simple instances of a process of much
mathematical interest and of real importance to the student
of physics. From the purely mathematical point of view their
interest consists in the demonstration that a new function may
be defined, and its properties explored, by means of a differ-
ential equation, even in the case when the function is not
expressible in terms of functions already known but only as
an endless power-series. The Bessel function is one of the
most important functions of this kind, and the one which the
ordinary student is most likely to meet in physics.^ From
the definition of a simple case of the function in No. 29 we
assume
y = Cq -h Cj^x + c^x^ -\- CJJC^ + . . .
and deduce that
X (2c2 + 3 • 2c^x + ...)+ (ci + 2c'2X + Sc^x- -1- . . .) +
+ P^i^O + ^1^ + <^2^^ + ^3^^ + • • •) = ^•
^ See Rayleigh, Theory of Sound, i., ch. ix., or Gray and
Mathews, Treatise on Bessel Functions.
EXPANSIONS. SUPPLEMENTARY EXAMPLES 563
From this relation we have c^ + p'^c^ = 0, etc., as stated, and
all the other coefficients become expressible in terms of Cq.
§ 5. Ex. CXVIII, D, E. Partial Differentiation, Total
Differentials. — Partial differentiation has been anticipated in
several of the examples of Ex. CX, A, B. The present ex-
amples put the topic upon its proper basis, and serve, in
particular, to extend the applications of the earlier exercises
to functions of more than one variable. For this reason alone
it would be well worth while giving to the subject the modest
amount of time it demands. Much higher claims can, how-
ever, be made for it ; the student who has no acquaintance
with partial differentiation is still too slightly equipped for
many important and interesting excursions in elementary
physics. The subject is treated fully in some of its more
fundamental aspects and is not likely to cause difficulty.
Special attention should be given (i) to the conditions for a
turning value in the case of a function of two variables —
namely, that the partial derivatives of the function with respect
to both variables must vanish, and (ii) to the method of finding
the derivative of an implicit function (No. 40).
The total differential is the natural complement of the
partial derivative and is, therefore, usefully studied beside it.
Speaking generally, the subject has not the wide importance
of partial differentiation, but the student of physics has to
have clear elementary views about it. Perhaps the treatment
given in division D is the most suitable for the purpose. In
most physical applications the magnitude of some entity, for
example potential energy, is a function only of the coordin-
ates of the points at which its value is considered. In that
case its values mark out surfaces in space over which they
are constant and the ideas of division D are directly applic-
able.
§ 6. Ex. CXVIII, F. Fourier's Theorem.— This has al-
ready been treated in a simple way in Ex. CVI and ch.
XLViii. of this book. The novelty is the solution of simple
problems of harmonic analysis by an algebraic instead of a
graphic process. It should be understood that such solutions
are possible only if the form to be analysed is expressed as a
definite function. In most of the practical applications of
Fourier's Theorem, for example in tidal prediction, this con-
dition is not fulfilled. In such cases the method of Ex. CVI
is, in principle, the one that has to be adopted.
36*
564 ALGEBRA
The solution of No. 47 includes that of No. 46. For the
first part of the -string we have y = 'axjb and for the second
part y = a{l - x)l(l - b), while the assumption is that the
whole shape of the string is given by a sum which may be
written for brevity
y = 2c^. sm ^.
To determine the value of a specified coefficient c, we
are to suppose a solid to be formed of which the elevation is
the triangle formed by the string and the base the curve
SttX
y = sin -y-. We then have as one expression for the volume
of the solid : —
^j ^ fax . sirx-] . ^^ rail
x) . S7rX~]
^ . sin — —
b • ^ Z J
a '^ [ . STTX^ al ^' r • sttx'I
a ^^ r . S7rx~\
y Z \2 . SttX ( I \ Sirxy-
( — ) . sm -1 \—\^' COS —^ I
STrrC / I \ SttX'^
sm —^ \~\ .X COS I
I \S7rJ I Jo
al r I STTxy
- , Y . — . COS -^
I - b Lstt I Jb
I - b LVstt/ I \S7rJ I J
/ Z \2 . s7rZ> fa a 1 / Z \ s-n-b T
al ab 1
■^ r^b " r^^j
/ Z \ r a^ aZ ~| a / Z \2 .
- y • ''°' ''- Lrn, - rrjj - r^b ■ y ^"^ ^'^
al .1^ . Sirb
sin
~ (stt)''^ .b{l -b) I
for the second and third terms vanish algebraically and the
last term vanishes because sin s;r = whenever s is integral.
But we have another expression for this same volume,
namely
EXPANSIONS. SUPPLEMENTARY EXAMPLES 565
V = a %Cp. sin ^-j- . sin ^- = ^(T ^c,, cos (p - s)-,
^ t L t
- cos (2? + S)-j .
Now unless p = s the integrals of all the cosine-functions
vanish. The expression for the volume reduces, therefore, as
in Ex. CVI to
v = ia c,.(i-cos?^)
Equating the two expressions for the volume we have
al . l^ . sirb
sm
' ' {sTrf.bil- b) I
2aP . S7rb
or c, = ■ . ... , ,j rr . sin —;-.
{siry ,b . {I - b) I
The case when Z = 60, a = 2, 6 = 24 was worked out by
our simpler method in Ex. CVI and the solution there obtained
is shown in fig. 135 of this book (p. 525).
In No. 48 we must make the most general assumption, i.e.
-^ = a^ + a^ sin X + a.^ sin 2x + ag sin 3a; + ...
Ad
+ b-^ cos X + b.j cos ^x + ^3 cos ^x ■\- ...
To find ^0 we integrate the equivalence, as it stands, from
to 27r and obtain
1 •27r
2a
1 47r2
""« ~ 2^ • 3 •
To find a, we multiply by sin sx ; to find b, by cos sx.
In this way the expansion given in the question is obtained
without difficulty.
§ 7. Ex. CXVITI, F. Curvature.— Thi^, like other topics
in the present exercise, is introduced mainly on account of its
importance in applied mathematics and physics. Nos. 5^-7
are, however, set as an interesting addendum to division D.
Nos. 58-60 are very important since they extend to a point
moving along any curve the property demonstrated in
Ex. CXV, Nos. 15-18, only for the case of the circle. In
-^ a {x^) = 2na,
566 ALGEBRA
Exercises, II, fig. 121, take as ?/-axis the momentary normal
at E, and for ic-axis any line at right angles to it. Then the
equivalence of No. 58 follows by putting dy/dx = in
No. 49. Next we have
dy dy dx
dt ~ dx' dt'
Therefore
dfdy\ _ dx d /dy\ dy d /dx\
dt\dt) ~ di ' diKdxJ "*" dx ' dt\dt)'
But the second term on the right vanishes since dyjdx is zero.
Also
d /dy\ _ d /dy\ dx
dt\dx) dx\dx) ' dt '
We have, therefore,
d^y d^y
acceleration = — ^^ = ^ . {dx/dty
dx'''
= v'lr by No. 58.
It may be noted that this is a case in which there would
be an advantage in using one of the former notations rather
than the Leibnizian. For instance, using the dot-notation for
^-derivatives and the D notation for o^-derivatives, the above
argument becomes
^' = D(2/) . X
y\^ x.D\y).x + T>{y) . x
— v^/r.
SECTION IX.
STATISTICS.
THE EXERCISES OF SECTION IX.
^^* The numbers in ordinary type refer to the pages of Exercises
in Algebra, Part II ; the numbers in heavy type to the pages of this
book.
EXERCISES
CXIX. Frequency-Distribution
CXX. Frequency-Curves , . . .
CXXI. Dispersion
CXXII. The Determination of Frequency by
Calculation
CXXIII. Probability
CXXIV. Correlation
CXXV. Supplementary Examples—
A. Permutations and Combinations
B. The Binomial CoeflBcients
C. Probability
D. The Normal Curve
E. Sampling .
F. Correlation
G. Partial Correlation
PAGES
433, 569
440, 574
449, 580
461, 584
474, 589
487, 602
501, 594
502, 595
503, 595
506, 596
508, 596
510, 608
511, 608
CHAPTER LIII.
FREQUENCY-DISTRIBUTION.
§ 1. Introductory. — Statistics constitute at once the oldest
and the newest branch of mathematics : the oldest, for their
practice, in some form, is one of the primary necessities of
ordered social life ; the newest, for their theory is, to a large
extent, a production of the present generation. For both
these reasons it is greatly to be desired that an elementary
study of the subject should come to be regarded as part of
the normal programme of secondary school mathematics.
On the one hand, the economic and social uses of statistics —
which go back, far beyond the Conqueror's Domesday Book,
to prehistoric " numberings of the people " on the banks of
the Nile or the Euphrates — have shown, in recent days, wide
and striking extensions which every educated person ought in
some measure to understand. On the other hand, modern
statisticians, seeking the most effective means of applying
their weapons, have built up a striking and beautiful system
of mathematical ideas. The result is that " statistics " need
no longer be regarded as a synonym for " dulness " ; there
are few branches of mathematics which have so much that
is attractive to offer to the beginner. The field has, in fact,
become so rich that the task of selection for the purposes of
a non-technical course is more than ordinarily difficult. The
author has, however, endeavoured to limit his topics to those
which are of fundamental importance and to develop them
just sufficiently to give to the general body of students a fair
idea of the scope and value of the subject, and to the future
specialist, whether in economics, the "higher commerce," or
the sciences which use statistical methods, a useful introduc-
tion to the technical methods employed in those departments.
The problems to be considered are indicated in ch. xxxviii.,
§ 7, and, in more detail, in the student's Introduction (p. 431).
The teacher who is interested in the subject and discovers
569
570 ALGEBRA
what excellent material it offers for mathematical study may
easily extend their range. With this purpose in view he will
naturally turn first to Mr. Udney Yule's Introduction to
the Theory of Statistics (Griffin &, Co.), an admirable text-
book to which the author is much indebted. Mr. Yule
gives references to all the works which it is important for the
serious student to consult. Among these the present writer
has drawn most inspiration from Prof. Karl Pearson's
masterly memoir (" Contributions to the Mathematical Theory
of Evolution ") in the Philosophical Transactions^ and
from his other works, among which the Grammar of
Science is widely known ; from the psychological papers of
Prof. C. Spearman who has developed another side of the
subject with great insight and skill ; and from some of the
writings of Prof. F. Y. Edge worth, whose articles in the
Encyclopcedia Britannica and delightful addresses to the
Statistical Society (particularly, perhaps, his Presidential
Address for 1912-13) are highly suggestive and stimulating.
It should be added that Mr. Palin Elderton has written a
book on Frequency Curves and Correlation (Layton) which
contains a clear and useful account of the mathematical
methods developed by Prof. Pearson in his Royal Society
papers, and that Dr. William Brown's little book on Mental
Measurement (Cambridge Univ. Press) is an eminently
modern summary, full of valuable material not easily
accessible to the general student of mathematics. The
Elements of Statistical Method (The Macmillan Co.) by
Mr. W. I. King is a semi-popular introduction which many
readers would find helpful.
§ 2. Ex. CXIX. Frequency -Distribution. — This exercise
is (i) to introduce the fundamental ideas of a frequency-
distribution and its representation by the frequency-table and
the frequency-diagram ; (ii) to familiarize the student with
the main forms of frequency-distribution met with in naturally
occurring statistics, and (iii) to supply materials to be used
further in the subsequent exercises. In view of (iii) it is
important that the graphs should be marked with reference
numbers and preserved. It is, perhaps, inadvisable that
every student should do each example ; it will, however, be
well for individual students to trace, through thin paper,
copies of their diagrams so that each member of the class
may have a complete collection.
FREQUENCY-DISTRIBUTION 571
With reference to (i) it should be made clear how a
frequency-diagram differs from the ordinary graphs of algebra
or physics. These represent the relations between two
variables, or the values of a function which correspond to
different values of a single variable. The purpose of a
frequency- diagram is simply to show how often each value
of the variable is met with in a given field. Thus if, in the
course of a morning's walk, I count the number of persons I
meet who have black hair, brown hair, etc., I have the
materials for a frequency-diagram in which the variable is
the colour of the hair. The only difference between such a
diagram and those to be drawn in Ex. CXIX lies in the fact
that in the latter, the variable is not merely qualitative but
is capable of quantitative definition. In either case any
" law " to which the graphic representation may point will
be simply a law of connexion or proportion among the
numbers of times each value of the variable is encountered.
This notion is of fundamental importance and must be
thoroughly understood.
With reference to (ii), the examples give typical instances
of the forms of frequency-distribution which occur most
widely in statistical practice. It will be seen (1) that they
are drawn from very diverse fields — from anthropometric
and biological measurements, from economics, from meteor-
ology, from physics, from medical records, and from records of
the workings of what, in our ignorance, we call pure chance ;
(2) that, nevertheless, they display resemblances that are often
most pronounced where the diversity of origin would seem to
be greatest. These resemblances are brought out by the
"frequency-curves" which the student is, in most cases,
instructed to add to his graph. It should here be noted that
the smooth curve is not, as in former uses, to preserve
scrupulously the original area of the columns across which it
passes. It is to represent the student's interpretation of
what may metaphorically be called the intention of the
distribution ; that is, the ideal distribution to which actual
samples might be expected to approach if they contained a
sufficient number of cases drawn from a field sufficiently
wide to be really representative. When frequency-curves
are drawn with this idea in view, they are found, as was said
above, to exemplify a few forms which constantly recur.
The student will learn in the next exercise that these forms
572
ALGEBRA
have been classified by Prof. Karl Pearson under seven
distinct types, of which two are symmetrical and the rest
asymmetrical or " skew ". The data for Nos. 4, 5, 10-14,
16 are taken from his papers on " Skew Frequency Distri-
bution " in Phil Trans., in vols. 185, 186, 191. No. 6 is
from Westergaard's Die Grundzilge der Theorie der Statistik ;
No. 7 was suggested by Prof. Edgeworth's Presidential
Address referred to above ; No. 8 is taken from a well-
known paper in the Philosophical Magazine; No. 15 is
drawn from Mr. Latter's article in Biometrika, Vol. IV.
Q
D
n 19 21 23 25
Fig. 142.
Fig. 143.
With regard to details. Fig. 142 is the histogram for the
lengths of the carrots of Nos. I-3 when the class- interval is
2 cms., fig. 143 when it is 3 cms. It is obvious that the
57 59 61 63 65 : 67: GBf. 71 73 75 77
Q, ^i Qj
Fig. 144.
latter brings out a "law" which is quite obscured in the
former. Fig. 144 shows both the actual histogram and the
ideal frequency- curve of No. 4. It is of great importance,
for it is an almost perfect instance of the " normal distribu-
FREQUENCY-DISTRIBUTION 573
tion " which has played so large a part in the evolution of
statistical theory. The curve in No. 5 is essentially of the
same type but obscured by a certain amount of " skewness ".
Nos. 6 and 7 are also, fundamentally, examples of normal
distribution, though the symmetry of the ideal curve is in
each case obscured by accidental irregularities due to the
smallness of the number of cases. They are especially in-
teresting since they exemplify the fact that pure chance,
working impartially in a given set of materials, produces
typically a normal distribution. This fact first came out in
the study (by Laplace, Legendre, Gauss, Bravais, etc.) of
the distribution of chance errors of observation. For this
historical reason the normal curve is still very commonly
spoken of as the graphic expression of the "law of error,"
even when it is actually representing things (such as statures)
which are not, to the plain mind, errors at all. It is, how-
ever, useful to note that the course of statistical theory has
been profoundly affected by the historical accident that it
began with the doctrine of errors of observation in astronomy
and surveying. Thus when the Belgian Quetelet (c. 1840)
began the modern study of anthropometric and meteoro-
logical distributions, he fell naturally into the way of thinking
of the mean of the distribution as the number (stature, chest
measurement, etc.) which represents Nature's aim or intention,
and deviations therefrom as her " errors ". The main effect
of this way of looking at natural statistics has been to pre-
judice inquirers unduly in favour of the normal distribution.
Only recently has it been recognized that, as a matter of fact,
ordinary distributions are not, even ideally, normal, and that
skewness is an essential feature in them. This last point is
well brought out by the distributions given in the present
exercise.
In many cases, perhaps most, the asymmetry or skewness
is moderate, as in the case of the head-breadths of No. 5 or
the barometer-readings of No. 13. Fig. 153 (p. 587) may
be regarded as showing the frequency-curve typical in such
cases. Nos. 8, 12 exhibit a much more markedly skew
distribution with a characteristic sharp rise on one side and
an essential " tail " on the other. It is remarkable that fig.
145 represents almost equally well the distribution of scarlet
fever in an epidemic and the way in which " alpha-particles "
are expelled from a radio-active metal, and it is diflBcult to resist
574
ALGEBRA
the suggestion that the two sets of phenomena, though so
different in outward character, are the expression of essentially
similar conditions.
The extremity of asymmetry is shown by the distributions
of Nos. 9, 10, and 11. Fig. 146, which actually represents
the data of No. II, would do, with little modification, for
each of the others. This distribution is typical of many
biological phenomena ; infant mortality and the incidence of
the death duties would be further instances.
so
60
Q/ Mz: Q3
Fig. 145.
m
No. 14 illustrates the rather rare " U -distribution
which the greatest frequencies occur at the two ends instead
of towards the middle.
The curves in Nos. 15, 16 appear to be almost perfectly
normal. That of No. 16 is the firm line in fig. 151 (p. 580).
It is shown in Ex. CXX, No. 32, to be really the sum of two
normal components. That of No. 15 is probably the resultant
of a still more complex system (see Exercises, II, p. 500).
§ 3. Ex. GXX. Frequency -curves. — Divisions A, B need
little comment. The mode (a term introduced into statistics
by Pearson) is a novelty, but the median and mean have been
treated before (see pp. 45, 115, 366, and Exercises, I, Exs.
XXVI, D ; XXVII, B, C ; LXVI).
In Nos. 3, 4 the median and quartiles are determined as
FREQUENCY-DISTRIBUTION
575
follows in accordance with the investigation of No. 2- Half
the total number of cases (= -^-N) is 4292-5 and 2n as far as
the class 66-7 includes 3589 cases. In the class 67-8 there
must be, then, 4292-5 - 3589 = 708-5 cases below the
median. Since this class contains 1329 cases we have for the
position of the median
= 67-52.
For the lower quartile we have JN = 2146 and ^n = 1376
cases down to the class 64-5, leaving 2146 - 1376 = 770 cases
Fig. 146.
to be taken from the 990 in the class 65-6. Thus we have
r^ an 1 ^70
Qi = 65 + 1 X gg^
= 65-78.
The position of the third quartile Q3 is found in the same
way, by working from the other end of the table. The posi-
tions of medians, quartiles, and means are marked in figs.
144-5.
In finding the mean the procedure illustrated in the follow-
ing example (in answer to No. 9) should be followed.
The first column, headed X, gives the mid-values of the
classes into which the lengths of the 24 carrots were grouped
576
ALGEBRA
in Ex. CXIX, No. 1. The column headed n states the fre-
quency, i.e. the number of carrots in each class. By guess,
the mean is about 19-5 ; the next two columns contain, there-
fore, the values of x^, the deviations of X from this number
(which is conveniently symbolized as MJ. It is best to
record x^^ in two columns headed respectively -f and - . The
next two contain the products nx^^, also separated according
to sign. The last (single) column contains the values of nx^^ ;
these are not needed at present but will be required in
Ex. XXII.
X.
n.
X
•
nx^.
nx^^.
+
_
+
_
15-5
5
4
20
80
17-5
3
2
6
12
18-5
3
1
3
3
19-5
3
20-5
4
1
4
4
21-5
3
2
6
12
22-5
2
3
6
18
25-5
1
6
6
36
24
22
29
165
The algebraic sum '^{nx{} = -h 22 - 29 = - 7 ; hence we
have
M = Ml -i- :S(wiCi)/N
== 19-5 - 7/24
= 19-2.
In division C the student faces the question of representing
by formulae the graphic forms which characterize so per-
sistently statistics drawn from the most varied sources. For
the present he is told simply to verify the correspondence of
certain given formulae with his frequency-curves ; the deriva-
tion of these curves from definite presuppositions is to be re-
ferred to later — in Ex. CXXII and the next chapter of this
book. All the formulae are taken from Prof. Pearson's papers
and represent the simpler of his " types ". We start with
the symmetrical curve given by the formula of No. 1 7. The
various cases set in No. 18 are represented in fig. 147 ; the
curves marked A, B, C correspond to the examples (i), (ii),
FREQUENCY-DISTRIBUTION 577
(iii) respectively. In No. 19 it is clear that the substitution
of - p for p would invert the curve and produce a sym-
metrical u.
The derivation in No. 20 of the normal curve y = y^e'^^^
from the symmetrical curve of No. Vj is of gi-eat importance.
The algebra goes as follows : —
y= U y,{l- x'lay
= U 2/0 (1 - x'la')
.2/^2\rt2D/a
if we put h^ = a/p. It is clear that, unless p is supposed to
Y
X'-6
+6 X
increase as well as a, the index pja would approach zero, and
the curve would degenerate into the y-B.xi8. The normal
curves to be drawn in No. 21 are those lettered A and B in
fig. 151. The ordinates may, of course, be calculated by
ordinary logarithms, or, if the teacher prefers, the table of
values of e""^ on Exercises, II, p. 473, may be used. The
student must enter this table not with the value of x but
with the value of x/h, i.e. ic/4'47 in (i) and £c/6-33 in (ii).
The curves will be needed in No. 32.
T. 37
578
ALGEBRA
The cui-ve of No. V] is Pearson's Type II; the normal
curve which we have derived from it is added by Elderton to
the original list and becomes Type VII. The formula of
No. 22 is the one with which Pearson actually starts in his
memoir and is, therefore, Type I. It is the curve of moderate
skewness typical of so many natural distributions. The two
cases of No. 23 are shown in fig. 148, A and B.
When the indices in this formula are negative the curve is
inverted and becomes the skew U of Ex. CXIX, No. 14.
Fig. 148.
The coefficients in No. 24 are so chosen as to give a curve
very similar to the graph there obtained. It may be interest-
ing to note that in the original memoir from which the formula
is taken it reads
y = 50-7505 (1 + aj/4-8109)-«-8^^4 (i _ x/d-llOby'^-^^K
These formidable figures will give some idea of the accuracy
with which Prof. Pearson and his assistants interpret their
data as well as of the incredible labour which they de-
vote to the task.
It is easy to show in No. 26 that the second factor in the
formula of No. 22 becomes exponential when a.2-> o:^ - The
values given for the constants are almost exactly those re-
quired to "fit" the barometric distribution of Ex. CXIX,
No. 13. The curve is Pearson's Type III.
FREQUENCY-DISTRIBUTION
579
The curves of Nos. 27-9 are of Type VI. The constants
in No. 29 will be found to give practically the curve of fig. 146.
+40
+20
00
16 X
Fig. 149.
The curve given in No. 30 is very nearly the frequency-
curve of Ex. CXIX, No. 10. It is shown in fig. 149.
Y
+6 +8
FiQ. 150.
The curve of No. 31 belongs to Type VI and is represented
in fig. 150. The constants are so chosen as to reproduce very
nearly the frequency-curve of Ex. CXIX, No. 12.
37*
580
ALGEBRA
Lastly, in fig. 151 the curve G is the resultant of the two
normal curves, A and B, drawn in No. 21 and now, in No.
32, to be combined. The constants are nearly those which
fit the distribution of Ex. CXIX, No. 16.
§ 4. Ex. CXXI. Dispersion. — The notions of quartile devia-
tion, mean deviation, and standard deviation (the last under
the name " root- mean-square " deviation) have already been
considered (pp. 115, 369, and Part I, Exs. XXVI, D, LXVII).
+ 5' +10 +15 +20
Fig. 151.
In Nos. 3, 4 what is required is to determine an ordinate
which cuts off one-quarter of the whole area of the frequency-
diagram or one-half of its semi-area. Let x be its distance
from the median. Then in No. 3 we have by similar tri-
angles that {a - xfla^ = -J, or a; = a (1 — 1/^2). In No.
4 the ordinates are given hy y = b . cos (7ra?/2a) and the area
from the median up to the ordinate whose abscissa is x is
h A cos (Trx/^a) . 8x = — . sin (Trx/2a).
Jo '^
The semi- area is ^ab/ir, so to find the abscissa of the quartile
we have the relation
sin (■ira?/2a) = i
= sin 30°.
^ o 30
Hence x = 'Aa. j^
= a/3.
FREQUENCY^DISTRIBUTION 581
In No. 6 we have
area of frequency-diagram = no. of cases/100.
Hence ^ablir = 10. Also Q = a/3 = 3|, so that a = 10 and
h = 7r/4:. Thus the ampUtude is 7r/4 cms. and the base-length
20 cms.
Nos. 7-14 are of extreme importance, but do not present
any particular difficulty. In No. 7 since xjh = u, x = hu
and Sx = h. Su. Since, also, the slices between two closely
situated ordinates have the same height in each figure, the
area of the one slice, y . Sx, is h times that of the other slice,
y . 821. Hence follows the result of Nos. 8, 9, 10.
The determination of the area under the normal curve is
too diflScult at this stage and is postponed to the Supple-
mentary Examples. In any case the determination of the
quartile deviation is possible only by some method of
approximation. In the present instance, to find the whole
area by Simpson's Rule we must add to half the ordinate
where u = the whole of the other ordinates of the table and
multiply the sum by the common interval in u, namely 0"1.
In No. 12 the student may either verify that the area up to
the ordinate u = 0-488 is one-half of the semi-area under the
curve, or, showing that the required abscissa is between 0*4
and 0*5, may find its length approximately by the method used
to find the quartile on p. 575. In both cases it must be re-
membered that Simpson's Rule requires the addition of only
half of the first and last ordinates.
The examples on mean deviation are not of great importance
and could be omitted if it were desired to shorten the course.
They have, however, the negative advantage of showing how
good are the reasons for which statisticians prefer to use the
standard deviation. The opportunity has also been taken to
use them as a means of applying some of the results learnt in
the parallel lessons in Section VIII. The method of calcu-
lating the mean deviation is in each case that indicated
in No. 24 in the relatively important case of a normal dis-
tribution.
On the other hand, division C, dealing with standard devia-
tion, is of the greatest importance and should be thoroughly
mastered. In Nos. 27-8, where standard deviations are to
be calculated from a frequency-table, the work should be set
down as in the table on p. 576. The last column contains
the products nXj^\ where the values of x^ are the deviations
582 ALGEBRA
from the trial mean Mj. The products are obtained by multi-
plying the values of nx-^^ in the fifth and sixth columns by the
values of x^ in the second or the third column. The total
'^(nx^^) is entered at the foot of the column. We then have
for the standard deviation
= 165/24 - 0-09
= 6-78
o- = 2-6
for in this case, as we saw on p. 576, the value of cZ is 0*3.
The result summed in the note after No. 32 is one of the
most frequently used in the theory of statistics and should be
known by heart. The full proof of the expression for the
ordinate at the origin is postponed to the Supplementary Ex-
amples. The equivalence stated in No. 34 is also one to
which reference will continually be made. The student should
be quite familiar with it. The equivalence in No. 35 is of
much less importance.
GHAPTEE LIV.
THE CALCULATION OF FREQUENCIES. PROBABILITY.
§ 1. Introductory. — Hitherto our work has been to record
and analyse frequencies actually given ; we are now to ex-
amine the possibility of predicting them among events that
have, perhaps, never been observed. It is from this point of
view and in connexion with this problem that those well-
established algebraic topics — combinations, permutations, and
probability — are best treated. Even if no other object were
in view than to " get up" these subjects in preparing for an
examination, some such method of approach as is here indi-
cated would be the most profitable. To present the theory of
combinations as a means by which the mysterious ways of
chance can be foreseen and unravelled is to make an appeal
to the young mathematician which he is very unlikely to
resist. The teacher is, therefore, strongly advised to accom-
pany this part of the course with select experiments intended
to awaken curiosity, to supply data for investigation, and to
give reality to the results of theory. By setting a class, as
part of their home-work or as a voluntary " extra," the task
of throwing sets of halfpennies and counting the heads, of
drawing numbered cards from a hat or differently coloured
marbles from a bag, of shooting at a mark with a primitive
dart, etc., it is easy to collect, at very little expense in time,
material of great interest and of more value than anything
that can be put into a set of printed examples.
It cannot be too carefully understood and remembered that
the calculation of probabilities is nothing more than the cal-
culation of frequencies. Students are so apt to have confused
ideas upon this point that, although there is no difference of
principle between the examples of Ex. CXXII and those of
Ex. CXXIII, yet the term " probability " has not been used
in the former but is reserved until the one and only notion
which the word should connote in mathematics — the notion
583
584 ALGEBRA
of " relative frequency " — has been illustrated by a consider-
able variety of examples.
The connexion in which the theory of combinations and
permutations is here introduced has necessarily affected the
character of the exercises by which it is illustrated. The
investigation of these subjects leads so easily and naturally
to problems of genuine importance in the theory and applica-
tions of statistics that there is little temptation to draw upon
the traditional problems of the text-book. Since, however,
the student who has not wrestled with these may at present
be at a disadvantage in a public examination, a selection
of the usual type is included among the Supplementary
Examples. The teacher who is confining his attention in
this section mainly to permutations and combinations should
take his class straight from the present exercise to the first
division of Ex. CXXV.
§ 2. Ex. CXXII. The Calculation of Frequency.— The
key to the comprehension of this part of the subject lies in a
clear understanding of the notion of "independent events".
The fall of a tossed coin is an independent event ; whether
it will fall " head " or " tail " the next time it is thrown depends
not at all upon how it fell last time or the last thousand times.
For instance, if there has been a "■ run '' of a hundred heads
the "chance" that the next result will be also a head is not
a whit the less than before. Assuming the coin to be un-
biassed, that chance is exactly ^ — a statement which means
that, if you collected a great many instances in which there
had been a run of a hundred heads, " head" would be found
to have been the result of the next toss also in almost exactly
half of them.
The Oxford and Cambridge Boat Eace may be taken as an
event which is not truly "independent," in spite of a good
deal of resemblance to the spinning of a coin. The victory
this year is generally dependent, in part, upon what happened
last year; a specially strong member of the last crew will
row again, a successful " coach " will repeat his services, the
previous victory will give greater confidence, etc. For these
reasons the record of the Boat Race shows a large proportion
of "runs" in favour of one side. Thus Oxford won con-
tinuously from 1861 to 1869 and Cambridge from 1870 to
1874.
But though a single race is not properly an independent
THE CALCULATION OF FREQUENCIES 585
event yet, in the long run, the series of annual races may-
still present the fundamental property of independent events
— namely, that, out of a great number of instances, they fall
out equally often in each of the ways possible to them. As
we have just seen, there are conditions which prevent the
result of one race from being entirely independent of its
predecessors ; but these conditions may themselves be " inde-
pendent " in the sense that they will, in the long run, connect
themselves just as often with the fortunes of one "blue '' as
with those of the other. In that case the record (which now
stands in favour of Oxford by 10 wins out of 68) will in time
show the equality in the distribution of the results which is
the typical characteristic of independent events.
The object of these remarks is to show how it is possible
that events undoubtedly moulded to some extent by men
" looking before and after " should yet, in the mass, conform
to the same laws as the spinning of coins and the drawing of
balls from urns. It is because experience shows this possi-
bility to be, on a large scale, an actuality that it is profitable
to consider in detail the dealings of the blind goddess with
this kind of material.
The next idea to be grasped is that frequency-predictions
are possible only in so far as the events predicted can be
regarded as compounded of independent elementary events
whose characteristic behaviour is already known. Thus,
knowing that the spin of a coin is an independent event
which will, in the long run, turn out heads and tails with
equal frequency, we can predict with confidence what will
happen (again in the long run) in the case of an event which
consists in the tossing (say) of a dozen coins.
In division A of Ex. CXXII the student is confronted with
simple examples to be solved directly by the principles to
which reference has here been made. In No. 8 he is called
upon to formulate the theorem which is the foundation of all
the more complicated predictions of the theory of probability.
No. 9 is meant to prepare the way for the theory of com-
binations. It is evident (i) that the two good jumps in
succession may occupy four different places in the series of
five, (ii) that, since each jump may be good or bad, the sum
of the possibilities is 2^, and (iii) that the required relative
frequency is, therefore, 4/2^ or 1/8.
In division B we turn to the theory of the combinations of
586
ALGEBRA
independent events. This subject should certainly be illus-
trated by experiments of the kind suggested in § 1. Apart
from the fact that the whole discussion is made to revolve
round the question of predicting frequencies instead of merely
enumerating combinations, there is nothing that calls for
comment until No. 1 3 is reached. That example must be
regarded as of fundamental importance ; for, as will be shown
in the next exercise, the properties of a normal distribution
can be deduced from it. In the example itself, as in the
complementary examples, Nos. 15, 16, there is no difficulty.
To take the general case presented in No. 16 : the relative
frequency of a compound event in which r specified con-
stituents turn out one way while the other n - r turn out the
opposite way is (by No. 8) jp'' . g" ~ ^ where p and q are the
relative frequencies of occurrence of those two ways when the
simple event is considered by itself. Now, out of the series
of n constituent simple events, r can be selected to be
"successes" (and the remaining % - r to be "failures") in
„C^ ways. Thus the total relative frequency of the event in
question is „C,. . p" . q''~'^', that is, it is a definite term in the
expansion of the binomial {p + q)".
It is important to note that from the condition p + q = 1
it follows that the
^ ^ total sum of the rel-
ative frequencies is
(as it should be)
unity. This condi-
tion, applied to the
frequency -graph, im-
plies that the total
area under the curve
is unity.
Fig. 152 shows
the frequency - dia-
gram of No. 14. It
is evident that the
smooth curve to
which the distribu-
■^^^- ^^^- tion points as its
ideal has all the appearance of being identical with the
normal curve. It is worth while to give to a specially skil-
ful and painstaking pupil the task of representing graphically
THE CALCULATION OF FREQUENCIES
587
the terms of the binomial (| + ^)" when n has a much larger
value than ten — for example, thirty. The results should be
graphed, not as a series of rectangles, but as a "point-
binomial " as described in Ex. CXXII, C. The curve should
be drawn through the points and the normal curve added in
which 0-2 = ^(n + 1)0^ where c is the^nterval between the
abscissae of the points. The approximation of the two curves
will be found very close.
Fig. 153.
Fig. 153 shows the graph of No. 14. It evidently points
to a " moderately skew " frequency-curve just as the former
distribution points to the normal curve. Here, again, it
would be profitable to have several curves drawn, varied
values being assigned to p and q but always so that p + q is
unity. It is interesting to note that the distribution repre-
sented by the point-binomial
(0-9 + 0-1)30
(the coefficients for which may have already been calculated
for the normal curve) is a fairly close approximation to the
distribution of barometric heights given in Ex. CXIX, No. 13.
The derivation of the normal curve from the point-binomial
is the subject of division C. The subject is of great import-
588
ALGEBRA
ance but the exposition should occasion little difficulty. The
teacher will note the justification for the assumption that
lo^n is finite although to^x is negligible ; it is merely another
way of saying that, however large x may be taken, the curve
must be supposed to extend so far beyond the point in ques-
tion that the ratio xjicn is negligible.
Another mode of deriving the normal curve from the point-
binomial is given in § 5 of this chapter.
In division D we turn from combinations to permutations.
There is nothing in the text or examples to attract attention
until we reach No. 34. This is intended to prepare the
student for the interesting result discussed in § 6 of the
exercise — a result for which the name ** Spearman's Theorem "
01 2 3 4 5;
Q/ M Ml Q3
Fig. 154.
is proposed on account of its connexion with Prof. C. Spear-
man's " foot-rule " method of measuring correlation to be de-
scribed in Ex. CXXIV. Spearman's own proof is given as
an appendix to a paper in the British Journal of Psychology,
Vol. II, pt. i. (1906).
The frequency-distribution for the total " loss of rank "
when, as in No. 40, m = 6 is shown in fig. 154. It will be
seen that, although the diagram is not symmetrical as a
whole, yet the central part does not depart very far from
symmetry while the left-hand half resembles that of a normal
distribution. Both these features become more marked as n
increases.
The theorem lends itself to easy and attractive experi-
mental illustration — illustration which will be found very
THE CALCULATION OF FREQUENCIES
589
useful in preparation for the discussion of correlation in
Ex. CXXIV. In an experiment performed by the author
seventeen small and equal cards were labelled A, B, . . . Q,
were shaken together in a hat, and drawn at random, the
order of appearance of each letter being noted and the total
loss of rank being estimated as described in the text. The
process was repeated 100 times. Fig. 155 is the frequency-
diagram for the different values of L thus obtained. It
differs from the diagram obtained theoretically for n = 6 in
26
38 ;42
0/
i^ ■■ 50
Fig. 155.
514. 58
62 66
After 10 20 30
L 48-5 48-2 48'!
that only the central part of the whole theoretical distribution
appeared in the experiment. By theory the average value of
L ( = L) should have been (17^ - l)/6 = 48 ; the actual
averages were as follows : —
40 50 60 70 80 90 100 drawings.
48-5 47-7 47-9 48-3 47-4 47*2 47-5
§ 3. Ex. CXXIII. Probability.— Diyision A introduces
formally the term probability and gives simple examples to
illustrate its use. Some are based on calculated, some on
observed, frequency-distributions. No. 5 (ii) is a typical
instance of the former kind. Each spin of the tee-to-tum
gives three possible events, two " blues " and one " red '.
The total number of possible cases for ten spins is, therefore,
590 ALGEBRA
3^^. The number of cases in which blue and red are each
obtained five times is
For, if you had ten tee-to-tums before you, you could choose
five of them to be " blue " in j^Cg ways ; and, taking any one
of the selected tee-to-tums you could make it give " blue" in
two ways, so that any selection of five gives 2^ different cases
of " all blue ". By a similar argument " three blues only "
is obtainable in i^Cg . 2^ ways. Thus, by the alternative
definition given in § 1 of the text
favourable cases
prob. =
all cases
,oC,.25 + ioC3.2«
310
^ 3008
" 19683
The same result could, of course, have been obtained directly
from the formula of Ex. CXXII, No. 15, by adding together
the two corresponding terms of the expansion
(f + ir;
for probability and relative frequency are synonymous terms.
Of the second class of problem No. 6 may be taken as
typical. By the frequency-table of Ex. CXIX, No. 4, out of
8585 British adults 336 are six feet high or more. The re-
quired probability is, therefore, 336/8585.
In No. 9 we turn from the use of the frequency-table to
that of the frequency- diagram or curve. Here we have to
remember simply that the area between any two ordinates
measures the number of cases whose magnitudes fall within
the limits indicated by the abscissae of those ordinates and
that the whole area of the diagram measures the total
number of cases. Hence the formula of No. 9. When the
diagram represents not absolute but relative frequencies its
total area A is, of course, unity.
Nos. II, 12 give an extremely important instance of the
application of this simple principle, and the main results of
the table in No. 12 will be used constantly. Their import-
ance consists in the fact that the relations which hold good
exactly for normal distribution hold good approximately for
many other distributions. Thus the great bulk of most
TJHE CALCULATION OF FREQUENCIES 591
distributions is found, in practice, to lie within the range 6Q
on either side of the median, and about 96 per cent of it
within the range 3Q. The student should check this state-
ment roughly by reference to his frequency- diagrams, not
failing to note some (e.g. that of Ex. CXIX, No. 14) in which
it fails entirely. No doubt there has been a tendency among
statisticians to use the principle too freely — a tendency
derived, once more, from the mistaken supposition that the
normal is also actually the usual form of natural distributions.
Still, used cautiously, the principle is of great service.
In division B we turn to the calculation of "compound
probabilities ". Here the fundamental principle of Ex.
CXXII, No. 8, is used in a more elaborate way for the
solution of more difficult problems, but there is no new
principle to be learnt. Nos. 13, 14 give a slight indication
of the way in which the theory of probability is applied by
actuaries to the calculation of premiums for benefits con-
ditioned in more complicated ways than the ordinary single
life assurance. (The teacher should, by the way, take a
suitable opportunity of pointing out that, in Exs. LXXX,
LXXXI, the use made of the Life Table was a simple
anticipation of these calculations of probability ; it is easy to
modify the arguments there used so as to bring in formally
the notion of relative frequency.) In No. 14 the probability
that the company will have to pay the £200 is the sum of
the probabilities (i) that both brothers will be alive in forty-
four years' time, (ii) that brother A will be alive and brother
B dead, (iii) that brother B will be alive and brother A dead ;
for in all these contingencies the money must be paid.
There remains only one possible case, namely, that both are
dead. In that case the company will not have to pay the
benefit. This is the easier probability to compute. From
the Life Table {Exercises, II, p. 73) it will be seen that the
probability of A's death before sixty-five is
-ifo=J-- = 0-53076.
The similar probability for B is (1 - 33234/71780) =
0-53687. The probability that both will be dead is the pro-
duct of these single probabilities, i.e. 0*28495, and the proba-
bility that one at least will be alive is, therefore, 1 - 0*28495
= 0*71505. If P is the premium paid in precisely similar
592
ALGEBRA
circumstances by (say) 100,000 uncles, the sum to be dis-
tributed by the company at the end of 44 years will be
£100,000P X 3-67 and the number of sums of £200 each to
be paid out of it will be 71,505. Hence
P = £200 X 0-71505/3-67
= ^39 nearly
From this example it is easy to deduce the rule followed by
actuaries in computing the premium for any given benefit :
premium = (present value of benefit) x (prob. that it will be
paid)
Nos. 16-18 are important for the reasons indicated in the
note that follows them and expanded in § 5 of this chapter.
To take the typical term of the general case, it is clear that
if, of the n balls, r are black and the rest white, the number of
possibilities of r black is ^,C^ and oi n - r white ,^,^C„_^. Thus
the total number of " favourable cases " or " successes " is the
product of these numbers. But the " all possible cases " are
evidently ^C„ ; hence the probability of the event is
It is easily verified that the successive terms of the hypergeo-
metric series give the values of
this probability as r assumes the
values w, w - 1, n - 2, . . . . , 0.
Fig. 156 exhibits the frequency-
diagram in the case of No. 17.
With division C we approach
some of the most useful and
attractive uses of the theory of
probability outside actuarial
practice. The exposition given
in the text is full enough to
make further explanation un-
necessary. The data for Nos.
21, 22 are taken from Merri-
man's well-known book, The
Method of Least Squares, to which the reader should refer
for full information about the reduction of observations.
Gauss's classical treatise (which was translated into French
by Bertrand) will also be found very illuminating and not
very difficult. The subject is too technical for treatment in a
general course.
Merriman gives the following for the theoretical numbers
Fig. 156.
THE CALCULATION OF FREQUENCIES 693
in No. 21, basing them upon a formula slightly different from
the one in the text : 3, 15, 50, 118,"197, 234, 197, 118, etc.
For No. 22 he gives (quoting Bessel) : 107, 87, 57, 30, 13, 5,
1, 0, 0, 0. It is to be noted that Bessel does not give the
signs of his errors ; we must assume, therefore, that positive
and negative errors of the same magnitude occurred with
equal frequency.
Here again is a very profitable field for original experiment.
The author has obtained very fair results by the following
two methods : (i) A dart was aimed a large number of
times at a point marked on a sheet of paper hung upon a
door. Concentric circles of radii 1 in., 2 in., 3 in., etc., were
drawn on the sheet and the number of shots falling within
each band counted. The usual assumption was made that all
the shots in a band lay upon its mid-line, the standard devia-
tion was calculated and the frequency formula deduced as in
No. 21. (ii) A line 10 cms. long was drawn upon a postcard.
The experimenter looked at it and then marked off on one of
the blue lines of a sheet of exercise paper, provided with a red
marginal line, the length which he judged to be 10 cms. He
covered over his first attempt, refreshed his memory of the
original line and made a second attempt. In this way a large
number of judgments of one individual may be obtained and
the frequency-formula deduced. Other experiments with the
same object will readily suggest themselves.
Careful attention should be given to the beautiful theorem
of the " standard error " of the mean in § 4. It is important
that the correct meaning of the unfortunate term " probable
error " should be appreciated. In this connexion the note in
No. 24 may be useful. In the proof of the theorem itself the
only difficulty likely to be felt lies in the assumption that
%{a,.a) is zero. When Ex. CXXIV has been read the student
will see that this is merely the assumption that the different
deviations that enter into a group of observations are not
" correlated " with one another in pairs ; that is, that the entry
of one into a given group is quite independent of the entry of
any other. In Ex. CXXIV it will be proved formally that in
that case %{a,a^ tends, as the number of cases increases, to
become zero.
In No. 25 we have, for the standard error of the mean
height of' 25 men, o- = 2-57/5 = 0-514. With the usual as-
sumption that different deviations of the mean of a sample
T. 38
594 ALGEBRA
from the mean of the whole follow a normal distribution, we
deduce that the probable error is 0*534 x 0*675 = 0*36. In
No. 26 the given mean of 68-^ in. represents a deviation from
the general mean of 68*5 - 67*46 = 1*04 in. Since Q = 0*36
this deviation is very nearly 3Q. By reference to the table
appended to No. 12 on p. 476 we see that the probability of
so great a deviation is about 1/24. It may be expected, that
is to say, once in each 20 to 25 batches of men.
An interesting experiment upon this subject will be to take
a table of logarithms, or square roots, etc., and select a number
of groups of last figures at random. For example, out of the
last figures of the 900 columns of an ordinary 4-place or
5-place table of logarithms or antilogarithms, groups of 9 may
be selected by one student, groups of 16 by a second, groups
of 25 by a third. The mode of selection does not matter
provided that it is truly a random selection and gives each
par>t of the table the same chance. Since the digits run from
to 9 and occur (no doubt) with practically equal frequency
among the original 900, we may take 4*5 as the mean value
of a digit. The sum of the squares of the deviations of these
900 digits from zero would be 100(0^ + 1^ + . . + 9^) =
28500. The mean square of the deviation from zero is,
therefore, 285/9 and the square of the deviation from 4*5 is
285/9 - 81/4 = 11-42. Thus for the whole collection of
digits or = 3*38. In accordance with the theorem the stand-
ard deviations of the different means obtained by random
selection of groups of 9, 16, 25 digits should be 3*38/3, 3*38/4,
etc. It will be understood that the closeness of the experi-
mental to the theoretical numbers depends upon the largeness
of the number of selections. The theory supposes an ex-
haustive number.
The " standard error of sampling," the subject of division
D, is also an extremely important matter that lends itself
easily to experimental illustration. The teacher should not
fail to have simple experiments carried out in accordance with
the suggestions given by the examples here and in Ex. GXXV.
§ 4. Ex. GXXV. Supplementary Examples. — The first five
groups of the supplementary examples may most profitably
be considered in this chapter. Division A has already been
mentioned ; it consists of additional problems in permutations
and combinations, all of a conventional character and of types
familiar from the text-books. The teacher who wishes to de-
THE CALCULATION OF FREQUENCIES
595
velop this subject further should consult the second volume
of Chrystal's Algebra or the well-known Choice and Chance
of W. A. Whitworth. The same remarks also apply to the
examples in division B. They are included almost solely
in view of the examinations in which they are occasionally set.
In division C a few examples are given upon the subject of
" local probability ". The assumption underlying the solution
of these problems is that a line or an area can be regarded as
made up of a very large number of equidistant points. This
assumption is, of course, arbitrary. For example, if the
number of points between any two given points on the line is
infinite, the distribution of probabilities of which the final ar-
FiG. 157.
rangement is the limit need not be uniform at all. Thence,
as the reader who has looked into the subject of probability
knows, many " paradoxes " have arisen. The simple ex-
amples of Ex. CXXV, G, avoid these and they must be sought
in special books of which Todhunter's History of the Theory
of Probability, Williamson's Integral Calculus, Borel's
Elemejits de la Theorie des Probabilites (Paris, Herman),
and Czuber's Wahrscheinlichkeitsrechnung (Leipzig, Teubner),
are, perhaps, the best for the purpose.
The double frequency-diagram required for the solution of
No. 24 is shown in fig. 157. The rectangle gives, as ex-
plained in the text, " all possible cases " of the double event :
" first break at P, the second anywhere on the broken piece ".
The area AB,QRjr^r,^'B gives the " favourable cases ". This
area is composed of two parts ; the area of the rectangle R^Pj
is b (since ARq = 1) and that under the curve is the integral
38*
596
ALGEBRA
of h . 8x/x between the limits x = b and x = a. The result
given in the text follows at once.^
Fig. 158 is the double frequency-diagram by which the
student is directed in
No. 25 to solve Buffon's
famous problem. " All
cases " are represented
by the rectangle whose
area is 2a. The " suc-
cesses " are represented
by the area under the
curve, which is the integ-
ral of arc cos (x/a) . 8x
from a; = to x = b.
The ratio of the two
areas is 2a/7rb. Many persons have sought to ** determine tt
experimentally " by using this result, a rod (e.g. an uncut
lead pencil) being thrown a large number of times upon a
ruled floor and a record being kept of the proportionate
number of times it falls across a line.^
In division D the investigation of Ex. CXXI, G, is completed
by a proof that the area under the normal curve y = e~*^ is
Jtt. The proof given was in part suggested by a question
set by Drs. Bromwich and Forsyth in the B.A. Mathematical
Honours examination of the University of London in 1912.
From the result it follows by Ex. CXXI, No. 9, that the area
oiy = Vif''^^'^'^ is 2/0 . o- Ji^ir). If, then, the whole area under
the curve is to be equal to the number of cases whose fre-
quency-distribution it represents we must have
2/o.o-V(27r) = N
or 2/0= N/o-VCS't)
as we have so often assumed to be the case.
Division E gives further examples of the most interesting
and important subject of sampling errors. The quotations
from Prof. Bowley are made in part from his Presidential
1 The author has to acknowledge the kindness of Mr. C. S.
Jackson who pointed out the source of a subtle paradox in which he
became involved in a first attempt to solve this problem by a simple
method. The problem itself is taken from Williamson's Integral
Calculus; No. 23, from Borel.
^ Mr. B. Branford informs the author that his mathematical
class obtained a very satisfactory approximation by this method.
THE CALCULATION OF FREQUENCIES 597
Address to the Economics Section of the British Association
reported in the Journal of the Royal Statistical Society for
1906, partly from his Elementary Manual of Statistics.
As Prof. Bowley points out, the great importance of the theory
of sampUng is in connexion with inquiries into unemployment,
wages, the cost of family life, etc., which should embrace the
whole population in their scope. It is obvious that it is im-
possible in such cases actually to review the whole popula-
tion ; we are necessarily driven back upon sampling. The
theory becomes then of immense importance as a means of
estimating the value of the results obtained from a given
sample. It shows that, provided the sample is selected truly
at random from the field of the whole population, so that
every member has an equal chance of being taken, the con-
clusions drawn will apply with surprisingly little modification
to the whole. The difficulty will generally be in securing a
really random selection of the sample. To select the com-
panies for the investigation described in Nos. 32-3 Prof.
Bowley numbered all in the list ; he then read down the
last four figures of a table in the Nautical Almanack and
whenever he came upon a number less than 3878 took the
corresponding company to be an element of the random sample.
With the aid of a collection of statistics, such as those
contained in Whitakers Almanack, the Statistical Abstract
or the Statesman's Year-hook, the teacher will easily be
able to supply a number of interesting examples of sampling
of the same type. Examples similar to No. 31 can also be
contrived with little difficulty.
The proof of the interesting theorem in No. 37 is very
simple. Since q ~ 1 - p we have, when p is small,
jjq = p - p'^ = p approximately. Hence the standard error
for the number of successes becomes simply J{pn). But pn is,
by hypothesis, the number of successes S. Hence the standard
error is ^S. The important point to note here is that, if p
is small, the standard error can be determined without any
knowledge of the actual ratio of successes to the whole. Thus
in No. 39, if we had no other information than that 119 rail-
way passengers were killed in 1912 and that the number is
a very small proportion of the travelling public, we could still
obtain a fair value for the standard error. It will be ^^119
= 10-9 and the probable error 10-9 x 0-675 = 7'35. Since
156 - 119 = 37, i.e. rather more than five times the probable
598 ALGEBRA
error, a mortality roll of 160 passengers might (by the table
on p. 476) be expected to occur (with our present population
and conditions of traffic) not oftener than once in about three
thousand years.
If the teacher is a student of physics he will be especially
interested to note, in connexion with No. 40, that the standard
deviation in the rate at which the atoms of radio-active
metals discharge "alpha-particles" brings the phenomenon
into line with events like fatal railway accidents. No doubt
the discharge of an " alpha-particle " is, like death in a
railway collision, an incident of comparatively great rareness
in the society in which it occurs. We have here a striking
modern instance of the well-known principle that it is possible
by statistical theory to give an account of many physical
phenomena, the kinetic theory of gases being the most
notable form in which that principle has hitherto expressed
itself. In his Presidential Address to the Royal Statistical
Society (published in the Society's Journal for 1912) Prof.
Edgeworth has dwelt in a peculiarly illuminating way upon
this aspect of statistical science.
may conveniently end with a brief sketch of the method by
which Prof. Karl Pearson derives his different types of
frequency-curves.
Consider, in the first place, the point binomial (-J + -|)" and
the normal curve ^ = 2/o^ ~ "" '^'^^. Take any two adjacent
points, Qi, Q2, on the former and find the point P on the
latter where the tangent is parallel to the line QiQ.2- Also
let PM be the ordinate of P. Then the analogue of PM in
the point-binomial is neither the ordinate at Q^ nor that at
Q2 but the ordinate QN which is drawn from Q, the mid-
point of Q1Q2. Thus QN = ^{y^ -}- y,. ^ j), y,. being the term
n^r - 1 • {-kY i^ *^® binomial expansion. If, further, we put
x^ = re where c is the distance between the ordinates of
successive points of the figure, we have the relation
(^.+1 - yr)lc ^ c{n+2) - (x,^^ + x;)
UVr+i + Vr) Un + ly
Now take as ^/-axis a line ^{n + 2)c to the right of the present
origin ; then the new abscissa x'^ — x^. - \{n -f 2)c. Also
put ^{n -f 2)c^ = 2(T^. Then the denominator of the second
fraction becomes - {x\ ^ ^ + x\) and the above relation can be
written
8 5. The Derivation of Frequency -curves. — This chapter
THE CALCULATION OF FREQUENCIES 599
slope of Qi Q2 _ mean abscissa
mean ordinate ~ cr^ '
But in the normal curve we have the relation
^ dy X
y'dx~ (T^'
Thus there is between the normal curve and the polygon
formed by joining the points of the point binomial a geo-
metrical similarity that is quite independent of the value of n.
By a similar argument we may derive in the case of the
asymmetrical point binomial (p + g)" a relation of the form
slope of Qi Q2 _ _ ^'r+h
mean ordinate a + X'^^^
where XV+i is the mean abscissa measured from a con-
venient point. Hence it is seen that the skew point- binomial
is geometrically similar to the curve
y = y^{l + ic/a)r . g-v-
for the differential equation of this curve is
1 dy __ X
y'dx a + x
But the point-binomial cannot be made to supply analogues
for frequency- curves of all the types actually encountered.
For this reason Prof. Pearson turns to the analogy of the
urn, described in Ex. CXXIII, B, and seeks the differential
equation which shall correspond to the relation between the
slope of the line Q1Q2 joining the points given by the two
consecutive terms of the hypergeometric series to the ordinate
mid- way between them. It is not difficult to obtain the
relation in question in the form
slope of Qi Q2 _ X
mean ordinate 6q + b^x + h^x^
X being the mean abscissa measured from an appropriate
origin. For the corresponding continuous curve we have,
then, the differential equation
1 dy _ X
y ' dx ~~ bQ + b^x + b^^'
By giving different values to the constants the different types
may be obtained. Four of them have been set, with
numerical coefficients, as No. 26 (vi)-(x) of Ex. CXVIII.
Those omitted can be derived without difficulty from the
former.
§ 6. The " Fitting " of Frequency -curves. — Supposing a
600 ALGEBRA
given distribution to be recognized as of a certain type there
still remains the problem of finding for the formula the
coefficients which will make the curve " fit " the data as
closely as possible. Prof. Pearson's method may be made
clear in principle by considering the way in which the
formulae were derived in Ex. CXIII, Nos. 21, 22, where it
was assumed that the type was the normal distribution. In
this case there were two constants to determine — the constant
which fixes the origin and the standard deviation. The first
is found by applying the familiar condition that if the mean
is' taken as the origin then "^nx is zero, n being the number
of cases whose magnitude is x. In other words, if the mean
does not in the first instance coincide with the origin the cor-
responding constant in the formula will be derived from the
value of ^nx. Similarly the second constant, the standard
deviation, is found from the value of l,7ix'^. Now it is obvious
that in this case all the data are taken into account in deter-
mining the values of the two constants, so that the resulting
curve may fairly be claimed to be the one which represents
best their general voice. In the case of a skew curve these
two sums do not suffice to (give the constants. Prof. Pearson
accordingly suggested that the values of "^nx^, "^nx^, etc.,
should be used to find them ; for each of those sums will be
related to some integral derived from the formula of the curve
in the same way as the standard deviation is related to the
integral ^x^^y . Sx.
By analogy with mechanics Prof. Pearson calls the sums
'Xnx, %nx^, etc., the first, second, etc., " moments " of the
data. The calculation of the constants by their means is, as
may be supposed, a tedious business and far too complicated
for full discussion here. Reference should be made either
to Prof. Pearson's memoirs already quoted or to Mr. Elder-
ton's more accessible book on Frequency Curves in which the
method of moments is explained thoroughly and applied to
examples. Meanwhile the following simple investigation may
serve to exemplify its general character.
Let it be known that a given frequency-distribution is of
the type whose "ideal" is described by a differential equation
of the form
dx a ■\- hx
then our task is to find for the constants a and b values which
THE CALCULATION OF FREQUENCIES 601
will give due weight to all the measurements contained in
the distribution. Write the equation in the form
{a + hx)y' = yx
and multiply each side by x^ . Sx. We thus obtain the
relation
x''{a + bx)y' . Bx = x"'^'^y . Bx
or ax^'y' . hx + hx"^^y' . hx = x'^^^y . hx.
Next integrate the three products by the ordinary rule for
" integration by parts," taking as limits the values of x where
the curve cuts the a;- axis. In this way we obtain
{ax'' + 6ic"
)y\' - afV-i^/'S^ - ^fV^/.Src = [V+it/.&r.
But, by hypothesis, y is zero for both the limits, so the first
term disappears. Also the integrals that remain are, by
definition, the {n - l)th, ?ith and [n + l)th moments of the
distribution about the vertical throughout the origin. Thus
our result may be compactly expressed by the notation
- a/A„_i - 6/x„ = /x„+i.
Now let n = 1 ; then ^q becomes \y . 8x, i.e. the total area
under the curve. If the numerical data are relative fre-
quencies the area is unity. Also, if the origin s taken at the
mean value of the variable, the first moment /Aj = ^xy . Sx, is
zero. Thus the equation reduces to
a = - fx.^.
Similarly, if n = 2, since /x^ is zero we have
whence it follows that the differential equation of the fre-
quency-curve is
dy _ X
dx jji^ + fx^x'
Finally the constants are to be given numerical form by the
relations /x^ = ^{nx'-) and yug == '^(nx^), where the values of x
are those given in the frequency-table.
CHAPTER LV.
COREELATION.
§ 1. Introductory. — The measurement of correlation is
the latest, and in some ways the most important of the
achievements of modern statistical science. For the person
who is seeking to straighten out the tangle of facts and
relationships which meets and baffles the inquirer into heredity,
social and economic phenomena and psychology — to say
nothing of meteorology — the correlation coefficient is proving
an instrument of indispensable usefulness and unsuspected
power. Our course may appropriately conclude with a study
of the ideas underlying its theory and some simple examples
of its application. For further study the student must turn
where his special interests draw him : to the memoirs of
Prof. Pearson and his school if it be to biology and " eugenics,"
to the writings of Mr. Yule and others if it be to economics,
to the papers of Prof. Spearman and his pupils, of Dr.
William Brown and other Englishmen or to those of Prof.
Thorndike and other distinguished Americans if it be to
psychology. In all these directions he will find that the
introduction of the quantitative measurement of correlation is
justifying old Roger Bacon's prescient dictum that " mathe-
matics is the gate-way to all the sciences ".
§ 2. Ex. CXXIV. Spearman's Goefficie^it. — The choice of
a means of measuring correlation is to some extent arbitrary
and it is well that the student should see it to be so. That
is one reason for the study of Prof. Spearman's method
before the " standard " method is considered. A second
reason is that it is extremely simple to understand and to
apply, and is therefore of great use as an introduction to the
general notion of the measurement of correlation. It should
be noted that (as is implied by the name "foot-rule") its
inventor does not regard it as a rival to the standard method,
602
CORRELATION 603
but rather as a quick and ready means of determining whether
there is in a given instance a degree of correlation which
would warrant the application of further labour to the inter-
pretation of the data.
Another advantage which the method possesses, from the
teaching point of view, is the readiness with which original
material can be found in schools to illustrate its use. In-
vestigations of the correlation between the performances of a
class in different subjects, in the same subject in different
terms, in different examinations in the same subject, in school
performances which are not both " subjects," etc. : all these
would be useful from the mathematical standpoint and would
often give information of no mean value to the teacher and so
to his pupils. Particularly important in this connexion is
the use of the correlation coefficient as a measure of the
** reliability " of an examination or other test.
The student is not likely to fail to understand the exposi-
tion of division A in the light of the investigation of
Ex. CXXII, E. The greatest difficulty, the necessity for a
knowledge of the probable error, has also been anticipated in
Ex. CXXIII. The calculation in Nos. 4-12 of Spearman's
formula for this constant is not very elegant ; his own (given
as an appendix to the paper already quoted from the British
Journal of Psychology, Vol. II) is too difficult for reproduction
here. Since, however, it is impossible to give here the proof
for the probable error of the standard coefficient it seemed
advisable to find a simple, if indirect, method of establishing
the formula in the present case.
The illustrative example in § 2 comes from the paper just
referred to. No. I is taken from a paper in the Journal
of Psychology for December, 1911. The data for No. 16
are given by Prof. Spearman in his well-known paper on
" General Intelligence " in the American Journal of Psy-
chology for April, 1904. The data of No. 17 were kindly
placed at the author's disposal, together with a full calculation
of the related correlation coefficients, by Miss N. Carey.
§ 3. Ex. CXXIV, B, G. The Standard Method.— The
origin of the standard method of measuring correlation is to
be found in a remarkable paper by the French savant Bravais
published in 1846. Bravais's problem was to determine the
most probable position of a point when the errors in its
coordinates are not unconnected with one another. To solve
604 ALGEBRA
the problem it was necessary to obtain some measure of the
degree of connexion between the errors in x and y. In this
way Bravais was lead to the "■ product-moment " which, in
the hands of Prof. Pearson, has become the standard means
of measuring correlation.
Pearson, like Bravais and Galton who preceded him,
worked at first with the assumption that the distribution of
the correlated variables was in each case normal. Upon this
assumption Galton's " regression lines " would actually pass
through the means of the various arrays. The wider inter-
pretation of the regression-line given in the text is due to
Mr. Yule. Galton's choice of the term " regression " has
reference to the fact, which his graphic method brings out so
well, that (for example) the mean height of the sons of a very
tall or a very short man is nearer than his own height to the
" mediocrity " of the general mean.
The late Prof. Weldon invented, in illustration of the
Pearson method of measuring correlation, experiments of the
following character. (The data and some details of the ex-
periment are taken from an interesting paper by Mr. A. D.
Darbyshire in the Memoirs of the Lit. and Phil. Soc. of
Manchester for 1907.) Take a dozen dice, and make in
parallel columns a record of the scores of a large number of
fairs of throws. It is evident that between the two sets of
scores thus obtained there would be little or no correlation,
for there is no connexion between the two members of a
pair of scores except that which may be due to chance.
The case is like the drawing of labelled cards from a hat as
described on p. 589. But now mark a certain proportion of
the dice by dyeing them in red ink, and after throwing the
whole set and taking the first score, leave the red dice on the
table and throw the residue again. To make the second
score add to the number now obtained from these the number
shown by the red dice lying on the table. A treatment,
by the Bravais-Pearson method, of the pairs of scores thus
obtained should now reveal a distinct correlation ; for the
second score of a pair depends in part upon the first score in
every case. In one of the experiments performed for Mr.
Darbyshire, out of twelve dice four were dyed red. Five
hundred pairs of throws were made with the results shown
in the following table. This may be regarded as a typical
" correlation table " showing in clear form the vertical and
CORRELATION
605
horizontal arrays whose means, in the ideal case, lie in or
near the regression lines. The vertical arrays may be
regarded as first throws, the horizontal as second throws.
1
2
3
4
5
6
7
8
9
10
11
12
1
1
1
2
2
1
1
1
2
1
1
7
3
2
4
3
4
5
2
20
4
1
2
8
18
8
12
6
1
1
57
5
6
3
8
i6
16
19
21
7
5
2
97
1
1
5
19
25
25
20
12
2
110
7
8
—
—
2
1
6
22
17
32
17
12
3
112
5
6
16
18
14
7
2
68
9
3
2
6
5
2
18
10
3
2
2
7
11
1
1
2
12
1
1
10
21
59
97
96
113
64
31
8
500
Fig. 159 presents the results of this experiment graphically,
the small circles indicating the means of the vertical arrays,
the crosses those of the horizontal arrays. To find b^ and h,
the gradients of the regression lines, it is necessary to deter-
mine first the means of the whole of the first throws as set
out in the bottom line and of the whole of the second throws
as set out in the right-hand line of the correlation table.
The coefficients must then be calculated by the formulae
b, = %{xy)l%x'
and bi = %{xy)l%y\ x and y being respectively the deviations
of the two numbers of a pair of scores from the means of the
totals.
606
ALGEBRA
Let X be the value of a first throw and Y that of a second
throw. Then to find X the numbers at the head of each
array must be multipUed by the partial total at the foot of
that array, the products summed, and the sum divided by
500:
(1.1 + 2. 10_+ 3 . 21 + . . . + 10 . 8)/500 = 6-086.
Similarly to find Y we have
(1 . 2 + 2 . 7 + 3 . 20 + . . . + 11 . 2)/500 = 6-062
the multipliers being taken from the last column.
Fig. 159.
To find the mean of the vertical array which sets out all
the second scores that were obtained with a first score of
(say) 4, the numbers in that array must be multiplied, each
by the number at the beginning of its row, and the sum of
these products divided by the number at the foot of the
array : (1.1 + 2. 1 + 3. 3 + 4.8 + . . . +8. 5)/59.
Similarly to find the mean for the horizontal array which
CORRELATION 607
records the values of all the first scores associated with a
second score of 4, each number in the array must be multi-
plied by the number at the head of its column, and the sum
of the products divided by the number at the end of the
array : (2. 1 + 3. 2+4. 8+... + 10. l)/57.
To calculate b-^^, b^, and r we must know the values of
'^{x2j), ^{x'^), and %{y^). In practice it will be better to de-
termine these from the values of 2(XY), ^X% and ^(Y^).
We know that
5(0^2) ^ ;^(X2) - NX and :${y^) = :${Y^) - NY.
To determine %{xy) from :S(XY) we have the relation
^(XY) = ^(X + X) (Y + y)
= NXY + ^xy)
since '^x and 2^/ are both zero.
In finding ^(XY) the sixteen cases whose scores are re-
corded in the table in heavy type will contribute 4 x 5 x 16,
the 4 and 5 being the values of x and y and the 16 the number
of associations of those values. Similarly the twelve cases
recorded in heavy type contribute 9 x 7 x 12 to the total
sum. In this way :S(XY) will be found to be 18,835. 3X2
will be found by multiplying the squares of the numbers at
the top of the table, each by the corresponding total recorded
at the bottom : P . 1 + 2'^ . 10 + 3^ . 21 + . . . = 19,900.
In the same way
:S(Y2) = 12 . 2 + 22 . 7 + 32 . 20 + . . . + 112 . 2 = 20,747
the multipliers being the totals in the final column.
In addition to these numbers we have X2 = 37*038,
Y2 = 36-74, and XY - 36-88. Hence, by the above relations
it follows that
5(^2/) = 394, %{x'-) = 1381, :S(3/2) = 2373
whence b^ = 0-286, 62 = 0-166, and r = J{b^ . b.,) = 0-218.
When (as is usually the case) correlation is to be calculated^
without a correlation table from a pair of records such as
those headed C„ and 0^, in No. 17, the best mode of pro-
cedure is indicated by the specimen on the next page.^
The first two columns contain the data. The mean for
the first column is 43, that for the second 40. The two
columns headed " x" contain the difference between the
mean and the values of D„, positive differences being separated
from negative. The differences for the D^ data are entered
^ Copied from Miss Carey's notebook, with alterations.
608
ALGEBRA
similarly in the next two columns. The squares follow and
finally the products. The last three columns being summed
give all the information required for the calculation of ^^ and
hi (if they are needed) and of r.
The products xy can be obtained (when they are too
diflBicult to be found mentally) by reference to a multiplication
table. Thorndike's Mejital and Social Measurements con-
tains such a table up to 100 x 100. For elementary pur-
poses, however, a table is unnecessary. An alternative is to
add columns giving the values of {x + y), (x + y)'^, {x - y),
and {x - yY, the squares being taken from a table of squares,
Da.
D..
X.
y-
x".
1/^
xy.
4-
+
+
40
40
3
9
37
43
6
3
36
9
18
53
46
10
6
100
36
60
37
30
6
10
36
100
60
61
55
18
15
324
225
270
34
30
9
10
81
100
90
which is a commoner thing than a table of products. The
subtraction of {x - y)'^ from {x + yY gives ^xy. %{xy) is
then obtained by summing the ^xy column and dividing the
result by 4.
§ 4. Ex. CXXV, F, G. Supplementary Examjjles. — The
last two divisions of the supplementary exercise deal with
further important matters connected with the measurement of
correlation. The first is the means by which faulty data may
be used in combination to give a truer verdict than they could
give alone. Here we follow Prof. Spearman whose studies of
this important subject have been especially fruitful. The proof
given is, however, essentially that of Mr. Yule. Prof. Spear-
man's proof is given (in its more developed form) in the
Journal of Psychology for October, 1910. Lastly, comes
a short division devoted to the subject of Partial Correlation.
This vitally important contribution to the theory and practice
of correlation-measurements is due to Mr. Yule. He has
treated it in a very general way in his Introduction^ illus-
trating his exposition by an appeal to an instructive model.
CORRELATION 609
Consideration is restricted here to the case of three variables ;
it is, however, easily seen that the method can be extended
to any number. The researches of psychologists and statistical
economists will offer many instances of the use of Yule's
formula, of which that given in No. 50 ^^J be regarded as
typical.
39
INDEX.
Adams, M., 453.
Addition, algebraic, 54, 164, 184.
Aim in teaching mathematics, 16.
Algebra, nature of, 1 ; essentials
of course in, 51.
Angles ; see Sine and Cosine, Tan-
gent ; complementary, 128 ; in
spherical triangles, 454 ; of un-
limited magnitude, 60, 394, 501.
Annuities, 432.
Antilogarithms, 306, 331, 333, 339 ;
antilogarithmic function, 310,
342 ; antilogarithmic curve, 345 ;
antilogarithm of an irrational,
432.
Approximation-formulae (a + 6)'-^,
(a ± hf, 72.
Approximations ; see Calculus.
Archimedes, 169, 178, 256, 257,
292.
Area, calculation of, by methods of
calculus, 169, 203 ; directed areas,
207 ; area-functions, 251, 279 ;
area of surface of sphere, 443.
Argand, J. R., 390, 470.
Arithmetic, and algebra, 1 ; gene-
ralized, 25 ; of infinites, 170.
Arithmetical series, 168, 199.
Association, law of, 193.
Astronomy, 387, 457.
Ball, R., 469, 462.
Ball, W. W. R., 24.
Barnard, S., 60.
Bessel, 593 ; functions, 562.
Bidder, G., 373.
Binomial theorem, 211, 372.
Board of Education, 78.
BoNOLA, R., 452.
Boole, Mary, 39.
Borel, E., 595, 596.
Bosanquet, B., 4.
Bowley, a. L., 596.
Bradley, F. H., 3.
Branpord, B., 24, 373, 596.
Bravais, a., 573, 603, 604.
Briggs, H., 57, 311, 438.
Bromwich, T. J. I' A., 596.
Brouncker (Lord), 438.
Brown, W., 570.
BuFFON, G. L. Le C, 596.
Cajori, F., 24.
Calculus ; position of the subject,
19 ; calculation of areas and
volumes by constant difference
series, 169, 203; Wallis's Law,
ordinate and area-functions, 246,
251, 279; differential formulae,
282 ; generalization of Wallis's
Law, 374 ; differential formulae
of the exponential and log-
arithmic functions, 435 ; differ-
ential formulae with polar co-
ordinates, 504 ; differential
formulae of the circular functions,
528 ; differentiation by the
method of limits, 549 ; D nota-
tion, 549 ; Leibnizian, 549, 560,
566 ; dot notation, 56 15 ; integra-
tion by the method of limits, 553 ;
G notation, 554 ; Leibnizian, 553 ;
differential equations, 556, 560 ;
partial differentiation, 563 ; total
differentials, 563 ; curvature,
565.
Cantor, G., 17, 384, 404, 405, 407,
408, 410.
Carson, G. St. L., 24, 162.
Cauchy, a. L., 559.
Cavalieri, a., 169, 170.
Cayley, a., 474, 493.
611
39
612
INDEX
Centroids, 367.
Changing the subject of a formula,
77, 78, 104.
Child, J. M., 60.
Chrystal, G., 397, 495, 595.
Circular functions, 295, 499, 512 ;
inverse, 514.
Circular measure, 60, 499.
Clifford, W. K., 452, 476.
Column-graphs, 36, 571.
Combinations, 60, 397, 399, 583,
594.
Commensurable magnitudes, 411.
Commutation, law of, 197.
Complex numbers, 390, 391, 469;
products of, 475 ; complex and
real variables, 478 ; relations be-
tween complex variables, 486 ;
logarithm of a complex number,
493.
Component and resultant, 185.
Compound interest, 347.
Constant-difference series, 168, 199.
Constant-ratio series, 176, 224,
Constants in a formula, determina-
tion of, 136.
Continuum, number of the, 408.
Contour Hues, 426.
Correlation, 398, 589, 602; Spear-
man's coefl&cient, 602 ; Bra-
vais-Pearson coefficient, 603 ;
correlation table, 605 ; partial
correlation, 608.
COUTURAT, L., 404.
Cosine ; see Sine and Cosine.
Curriculum in mathematics, 17.
Curvature, 565.
Curve of pursuit, 38.
Curve of squares, 149.
"Cut" (Schnitt), 414.
Cycloids, 506.
CzuBER, E., 595.
Darbyshire, a. D,, 604.
Darwin, G. H., 520, 527.
Davis, K F., 148, 150.
Dedekind, K, 404, 405, 413, 416.
De Moivre, a., theorem of, 390,
391, 469, 472, 476.
Descartes, R., 390, 469.
Differential calculus ; see Calculus.
Differential formulse ; see Calculus.
Directed areas, 207.
Directed numbers, 54, 159, 162, 181,
183, 228, 230 ; addition and sub-
traction, 164 ; multiplication and
division, 168, 193.
Division, algebraic, 176, 193, 222.
DuNLOP, H. C, 306.
e, 58, 300, 311, 350, 418, 434, 483,
495, 534.
Edqeworth, F. Y., 570, 572, 598.
Edser, E., 311.
Elderton, p., 570, 578, 600.
Ellipse, transformations of, 363 ;
elliptic functions, 532.
Epicycloids, 507.
Equality ,'8ign of, 8.
Equations, simple, 77-9, 113 ; sim-
ultaneous, 233; quadratic, 80,
238, 270 ; further, 240 ; differen-
tial, 248.
Error, law of, 593 ; standard error,
593 ; probable error, 593 ; prob-
able error in correlation, 603.
Euclid, 4, 32, 70, 411.
Examinations, public, 19, 60, 382,
398.
Expansions, 72, 73, 174, 177, 372.
Exponential curve, 345, 433.
Exponential function, 343, 428, 433.
Exponential values of the sine and
cosine, 494 ; of the hyperbolic
sine and cosine, 534.
Factorization, 68, 75, 82, 209.
Fahrenheit, D. G., 228,
Fluxions, 425, 488.
Formulae, 6, 25, 63, 96; substitu-
tion in, 30, 67 ; changing the
subject of, 77, 78, 104 ; combin-
ing of, 113, 136 ; generalization
of, 119, 146 ; relation to graphs,
119 ; differential formulae, 436.
Forsyth, A. R., 596,
Fourier, J. B. J,, 393, 516; his
theorem, 520, 563.
Fractional indices, 281, 343,
Fractions, algebraic, 13, 75, 96, 222,
224, 231.
Frequency : distributions, 398, 569 ;
curves, 574 ; dispersion, 580 ;
determination of, by calculation.
INDEX
613
583 ; derivation of frequency
curves, 598 ; fitting of ditto, 600 ;
relative frequency, 584.
Functions, 46, 110 ; inverse, 243,
274 ; linear, 110, 235, 258, 259 ;
hyperbolic and parabolic, 236,
264; area, 279; ordinate, 280;
periodic, 392 ; continuous and
discontinuous, 422 ; of two vari-
ables, 426 ; singular values, 426.
Galton, F., 604.
Gauss, C. F., 17, 390, 470, 474, 479,
573, 592.
Generalization, types of, 2.
Generalized arithmetic, 25.
Geometric series, 176, 224.
GiRAED, A., 390.
Gnomonie nets, 447.
Goodwill, G., 237, 476.
GosiEwsKi, L., 558.
Gradient, 34, 252, 376, 425.
Granville, W. A., 556.
Gray, A., 562.
Graphs, 31, 418, 422, 486, 504, 507,
509, 513; interpolation, 34; col-
umn-graphs, 36, 571 ; principles
of method, 40 ; graphs in practical
work, 43 ; medians and quartiles,
45 ; graphs and co-ordinate geo-
metry, 47 ; three-dimensional
graphs, 47, 479 ; practical sugges-
tions, 48; relation to formulse,
31, 67, 119 ; see also Hyperbola,
Parabola, Straight Line.
Great circle sailing, 447.
Grkgory, J., 438; his series, 393,
514, 529, 530.
Growth problems, 58, 302, 313 ;
growth curves, 304, 317, 319 ;
growth-difference, 315 ; growth-
factor, 313 ; growth-factors,
nominal and effective, 346.
Gudermannian functions, 536.
GuNTER, E., 57, 301, 321.
Gunter-scale, 302, 304, 319, 326,
333.
Hall, Stanley, 303.
Hamilton, W. R., 476.
Hankel, H., 475.
Hardy, A. S., 470.
Hardy, G. H., 423, 424, 494, 495,
537.
Harmonic analysis, 620.
Harmonic motion, simple, 601, 517 ;
compound, 509, 517.
Haywabd, R. B., 476.
Heath, T. L., 32, 257, 411.
Henrici, 0., 476.
Hermite, C, 419.
HiNKS, A. R., 442.
HiPPARCHUs, 256, 295.
History of mathematics, 4, 9, 14,
16, 24, 32, 46, 57, 58, 70, 77, 169,
212, 256, 281, 292, 300, 301, 302,
311, 321, 372, 376, 390, 404, 411,
419, 469, 569, 595.
HoBSON, E. W., 421.
hoefler, a., 24.
Hulbert, L. S., 556.
Huntingdon, E. V., 310.
HuTTON, C, 257, 530.
Hyperbola, as symbol of inverse
proportion, 145 ; hyperbolic func-
tions, 236, 264, 531 ; rectangular,
147, 264 ; transformations of,
363 ; differentials of hyperbolic
functions, 535.
i, 391, 469, 475, 480, 486.
Imaginary numbers, 60, 238, 390,
391, 398, 469, 484.
Incommensurable magnitudes, 411.
Index notation for numbers, 214.
Indices, integral, 174; fractional,
281, 343 ; positive, 214 ; negative,
217.
Induction, 282.
" Infinitesimals," 553.
Insurance, life, 432, 591.
Integers ; see Number Systems.
Integral calculus ; see Calculus.
Interest, compound, 347.
Inverse functions, 243, 274.
Inverse proportion curve, 146.
Jackson, C. S., 306, 558.
James, W., 4.
Jevons, W. S., 3, 16.
JOURDAIN, P. E. B., 23.
Keith, T., 389.
Kelland, p., 476.
614
INDEX
Kelvin (Lord), 527.
King, G., 433.
King, W. I., 570.
Lagbange, J. L., 17, 559.
Lambert's net, 443.
Laplace, P. S., 573.
Latitude, circles of, 129 ; middle
latitude sailing, 129.
Latter, 0., 572.
Least squares, method of, 43, 592.
Leqendre, a. M., 573.
Leibniz, G. W., 549 ; notation for
differentiation, 549, 560, 566;
for integration, 553.
Limits, theory of, 171, 396, 541 ;
practical applications, 546.
Lindemann, F., 419.
Linear functions, 110. 235, 258,
259.
Lissajou's curves, 509.
Logarithms, 57, 299, 325, 333;
tables of, 329, 337 ; common,
309, 335, 337; Napierian, 308;
change of base, 306 ; use of, in
trigonometry, 355; logarithmic
function, 310, 342; theory of
logarithms stated algebraically,
341 ; logarithms as indices, 343 ;
logarithm of an irrational num-
ber, 432 ; of a complex number,
493.
Lubbock, J., 526, 527.
Machin's series, 530.
Map projections, 386, 441.
Masebes, F., 438, 530.
Mathews, G. B., 562.
Maxwell, J. Clebk, 17, 476.
Mean deviation, 115, 366, 580.
Mean position, 366.
Mean value, theorem of, 557.
Medians, 45, 115, 366, 574.
Mercator, N., 57, 385, 438.
Mercator net, 446, 537.
Mercator sailing, 132, 445.
Meridian gnomonic net, 447.
Merriman, M., 592.
Method of differences, 269, 282.
Milbobne, 302.
Milne, J. J., 148, 150.
Minus sign, meanings of, 181, 185.
Mitchell, U. G., 10.
Mode, 574.
MOLESWORTH, G. L., 67.
MoUweide's equation, 356.
Moments, theory of, 600.
Moore, E. H., 558.
Multiplication, algebraic, 172, 193,
207.
Napier, John, 57, 58, 300, 308, 329,
355, 425, 438.
Napierian logarithms, 308.
Nautical mile, 129.
Negative nimibers, 159, 183.
Newton, I., 4, 170, 372, 425, 488,
549.
Nominal and effective growth-
factors, 346.
Normal curve and distribution, 572,
574, 577, 587, 598 ; area of curve,
581, 586, 590.
Numbers, non-directed, 52, 61;
directed, 54, 159, 162, 181, 183,
228, 230; rational, 409, 413;
irrational, 410, 413, 432; real,
imaginary, complex, cardinal,
406 ; ordinal, 405 ; infinite, 406 ;
algebraic, 390-1, 413, 419 ; trans-
cendent, 419 ; see also Imaginary
and Complex Numbers.
Number of the continuum, 408.
Number-scale, the complete, 179,
228, 421.
Number systems, 403.
One-to-one Correspondence, 405,
407, 452.
Ordinate functions, 248, 281, 480.
OUQHTRED, W., 10, 302.
TT, 27 ; calculation of, 256, 292,
530 ; transcendental nature of,
418, 421.
Parabola, as symbol of direct pro-
portion to the square, 149 ; para-
bolic functions, 236, 266 ; para-
bolic formula, changing subject
of, 274 ; parabola, area of, 251,
279 ; transformations of, 363.
Partridge, S., 302.
Peano, G., 558.
INDEX
616
Pearson, Karl, 370, «62, 570, 572,
574, 576, 578, 600, 604, 608.
Periodic functions, 392, 515, 531.
Permutations, 60, 397, 399, 583,
594.
Perry, J., 24, 311, 556.
Physics and mathematics, relation
between, 394.
Plus sign, meanings of, 183, 185.
Polar co-ordinates, 356, 504.
Polar gnomonic net, 447.
" Point-binomial," 587, 598.
Positive and negative numbers, 54,
159, 162, 182, 228, 230.
Preston, T., 508.
Probability, 385, 397, 583, 589.
Probable error, 593 ; in correlation,
603.
" Product-moment," 604.
Projections, map, 386, 441.
Proportion, direct, 109, 117 ; in-
verse, 109, 145 ; direct proportion
to the square or square root, 149;
inverse proportion to the square
or square root, 152 ; combination
of types of proportion, 154.
Ptolemy, 32, 256, 295.
Pythagoras, 411.
Quadratic equations, 80, 238, 270.
Quadratic functions, 236, 266, 359.
Quartiles, 45, 574 ; quartile devia-
tion, 115, 580.
Quetelet, L. a. J., 573.
Rate of change of a variable, 34,
170, 425.
Rational numbers, 409, 413.
Rayleioh (Lord), 562.
Recurrence, proof by, 282.
Regiomontanus, J., 256.
Regression, 604.
Resultant and component, 185.
Rhumb lines, 445.
RiEMANN, G. F. B., 392, 474, 487,
488, 503.
RoLLE, M., 396.
Root-mean-square deviation, 115,
369, 580.
Roulettes, 507.
Round numbers, 214.
Russell, Bertrand, 23, 384, 404,
408, 416, 419, 420, 453, 644, 546,
550.
Sampling, 594.
Sanson's net, 442.
Sidereal time, 459.
Signs, rule of, 53, 173, 186, 197, 211.
Silberstein, L., 424, 477, 558.
Simple equations, 233.
Simpson's rule, application of, 523,
581.
Sine and cosine, 111, 124 ; relations
between, 133 ; relations with tan-
gent, 132; values for 45°, 30°,
60°, 134; directed, 255, 261;
calculation of, 295 ; sum and dif-
ference formulae, 296, 359, 508;
exponential values of, 494; ex-
tension to angles of unlimited
magnitude, 501; curves of, in
wave-motion, etc., 517; differen-
tial formulae for, 528 ; expansions
of, 529 ; hyperbolic, 531.
Sinusoidal net, 442.
Slide rule, 302, 304, .828, 523.
Smith, D. E., 24, 419.
" Space-filling curve," 558.
Spearman, C, 570, 588, 602.
Sphere, area of surface of, 443 ; see
also Trigonometry.
Spherical defect, 452.
Spherical excess, 451, 466.
Spherical triangles, 454.
Spirals, 504.
Square root, 68, 70, 73, 81, 90;
curve, 151.
Squares, curve of, 149.
Stallo, J. B., 452.
Standard deviation, 115, 370, 580.
Standard error, 593.
Standard form of numbers, 175, 810,
328.
Statistics, 115, 214, 397, 569.
Stereographio projection, 449.
Stifel, M., 212, 270, 372, 411.
Straight line, as symbol of direct
proportion, 109, 119.
Subtraction, algebraic, 64, 164, 184.
Sun-dials, 460.
Surds, 71, 93, 231.
Symbolism, 4, 9, 10, 63.
616
INDEX
Symbols, substitutes for words, 6;
manipulation of, 12 ; literal, in-
troduction of, 26.
Tait, p. G., 476.
Tangent, 111, 121; of 90°, 123;
tables of, 122, 256 ; relations with
sine and cosine, 132 ; values for
45^ 30°, 60°, 134 ; directed, 236,
259; extension to angles of un-
limited magnitude, 501 ; differen-
tial formula for, 529.
Tannery, P., 413, 416.
Taylor's Theorem, 397, 557, 559.
Theon of Alexandria, 70.
Tides, 525.
TODHUNTER, I., 595.
Triangles, solution of, without for-
mulsB, 112; relations between
sides and trigonometrical func-
tions of angles, 263 ; spherical,
454.
Trigonometry, position of the sub-
ject, 19 ; trigonometrical ratios :
tangent (height problems), 121 ;
sine and cosine (navigation prob-
lems), 124 ; latitude and longi-
tude, 129 ; relations between sine,
cosine, and tangent, 132 ; ratios
of 45°, 30°, 60°, 134 ; combination
of formulae, 143 ; the tangent,
directed, 255, 259 ; the sine and
cosine, directed, 255, 261 ; rela-
tions between the sides of a
triangle and the ratios of the
angles, 263 ; calculation of tt,
292 ; calculation of sines, 295 ;
use of logarithms in trigonometry,
355 ; sum and difference formulae
(angle-sum less than 360°), 359 ;
trigonometry of the sphere, map
projections and navigation pro-
blems, 386, 441, 537; trigono-
metry of spherical triangles,
astronomical problems, 387, 454 ;
exponential values of the sine
and cosine, 493 ; circular func-
tions : circular measure, angles of
unlimited magnitude, spirals,
roulettes, 499, etc. ; harmonic
motion, simple, 501, 517 ; com-
pound, 509, 517 ; sum and differ-
ence formulae for angles of un-
limited magnitude, 508 ; inverse
circular functions, 514 ; wave-
motion, 515.
TucKEY, C. O., 174.
Turner, G. C, 476.
Turning values, 267.
Van Ceulen, L., 256, 292 ; table of,
295.
Variation, 154.
Vectors, 112, 124, 128, 165, 185,
188 ; algebra of, 188, 404, 476.
Venn, J., 6.
Vieta, F., 9, 12.
Volumes, calculation of, by methods
of the calculus, 169, 203.
Von Wyss, C, 33.
Walker, G. T., 358.
Wallis, John, 2, 3, 9, 12, 21, 57, 169,
250, 257, 281, 367, 372, 385, 530.
Wallis's Law, 246, 279, 287, 374,
396, 436, 437, 552, 555 ; theorem,
559.
Waves, 515, 536.
Weierstrass, K., 396,558.
Weldon, W. F., 604.
Wessel, C, 470.
Westergaard, H., 572.
Whewell, W., 527.
Whitehead, A. N., 5, 23, 57, 77,
545, 550.
Whitworth, W. a., 595.
Williamson, B., 595, 596.
Young, J. W. A., 10, 23, 24, 160,
404, 408, 410, 419, 421, 453, 545.
Yule, G. U., 370, 570, 604, 608.
Zeno, 384 ; paradoxes of, 419.
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