Calculus Made Easy: Being a Very-Simplest Introduction to those Beautiful Methods of Reckoning which are Generally called by the Terrifying names of the Differential Calculus and the Integral Calculus

259 Pages·1965·12.57 MB·English

This book is sold subject to the condition that it shall not. by way of
trade, be lent. re-sold. hired out. or otherwise disposed of without the
publisher's consent. in any form of binding or cover other than that
in which it is published.
CALCULUS MADE EASY:
BEING A VERY-SIMPLEST INTRODUCTION TO
THOSE BEAUTIFUL METHODS OF RECKONING
WHICH ARE GENERALLY CALLED BY THE
TERRIFYING NAMES OF THE
DIFFERENTIAL CALCULUS
AND THE
INTEGRAL CALCULUS
BY
SILVANUS P. THOMPSON, F.R.S.
THIRD EDITION
MACMILLAN EDUCATION
NEW YORK • ST MARTIN'S PRESS
What one fool can do, another can.
(Ancient Simian Proverb)
This book is oopyright in all countries which
are signatories to the Berne Conventi<m
First Edition 1910.
Reprinted 1911 (twice), 1912, 1913.
Second Edition 1914.
Reprinted 1915, 1917; with corrections 1918 (twice).
Revised and enlarged, 1919.
Reprinted 11120, 1921, 1922, 1924, 1927; 1931, 1936, 1940, 1942, 1943.
Third Edition 1946.
Reprinted 1948, 1950, 1952, 1957, 1961, 1965.
ISBN 978-0-333-07445-9 ISBN 978-1-349-00487-4 (eBook)
DOI 10.1007/978-1-349-00487-4
MACMILLAN AND COMPANY LIMITED
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also Bombay Calcutta Madras Melbourne
THE MACMILLAN COMPANY OF CANADA LIMITED
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ST MARTIN'S PRESS INC
175 Fifth Avenue New York 10010 NY
PUBLISHERS' NOTE ON THE THIRD EDITION
ONLY once in its long and useful life, in 1919, has this book
been enlarged and revised. But in twenty-four years much
progress can be made, and the methods of 1919 are not likely
to be the same as those of 1945. If, therefore, any book is to
maintain its usefulness, it is essential that it should be over
hauled occasionally so that it may be brought up-to-date
where possible, to keep pace with the forward march of
scientific development.
For the new edition the book has been reset, and the dia
grams modernised. Mr. F. G. W. Brown has been good enough
to revise the whole of the book, but he has taken great care
not to interfere with the original plan. Thus teachers and
students will still recognise their well-known guide to the
intricacies of the calculus. While the changes made are not
of a major kind, yet their significance may not be inconsider
able. There seems no reason now, even if one ever existed, for
excluding from the scope of the text those intensely practical
functions, known as the hyperbolic sine, cosine and tangent,
whose applications to the methods of integration are so potent
and manifold. These have, accordingly, been introduced and
applied, with the result that some of the long cumbersome
methods of integrating have been displaced, just as a ray of
sunshine dispels an obstructing cloud.
The introduction, too, of the very practical integrals :
J
f,Pt
sin kt . dt and cos kt . dt
1€>t
has eliminated some of the more ancient methods of " Finding
Solutions " (Chapter XXI). By their application, shorter and
more intelligible ones have grown up naturally instead.
iii
iv PUBLISHERS' NOTE ON THE THIRD EDITION
In the treatment of substitutions, the whole text has been
tidied up in order to render it methodically consistent. A few
examples have also been added where space permitted, while
the whole of the exercises and their answers have been care
fully revised, checked and corrected. Duplicated problems
have thus been removed and many hints provided in the
answers adapted to the newer and more modern methods
introduced.
It must, however, be emphatically stated that the plan of
the original author remains unchanged ; even in its more
modern form, the book still remains a monument to the skill
and the courage of the late Professor Silvanus P. Thompson.
All that the present reviser has attempted is to revitalize the
usefulness of the work by adapting its distinctive utilitarian
bias more closely in relation to present-day requirements.
CONTENTS
CHAPTER PAGE
PROLOGUE • vi
I. To DELIVER YOU FROM THE PRELIMINARY TERRORS 1
II. ON DIFFERENT DEGREES OF SMALLNESS 3
III. ON RELATIVE GROWINGS - 8
IV. SIMPLEST CASES- 15
v.
NEXT STAGE. WHAT TO DO WITH CONSTANTS - 22
VI. SuMs, DIFFERENCES, PRoDUCTs, AND QuoTIENTS 30
VII. SUCCESSIVE DIFFERENTIATION 42
VIII. WHEN TIME VARIES - 45
IX. INTRODUCING A UsEFUL DoDGE 57
X. GEOMETRICAL MEANING OF DIFFERENTIATION 65
XI. MAXIMA AND MINIMA 78
XII. CURVATURE OF CURVES 93
XIII. OTHER USEFUL DoDGES 100
XIV. ON TRUE COMPOUND INTEREST AND THE LAW OF
ORGANIC GROWTH Ill
XV. How TO DEAL WITH SINES AND COSINES 137
XVI. pARTIAL DIFFERENTIATION· 146
XVII. INTEGRATION 152
XVIII. INTEGRATING AS THE REVERSE OF DIFFERENTIATING 159
XIX. ON FINDING AREAs BY INTEGRATING 172
XX. DODGES, PITFALLS, AND TRiuMPHS 188
XXI. FINDING SOLUTIONS • 195
XXII. A LITTLE MORE ABOUT CURVATURE OF CURVES· 208
XXIII. How TO FIND THE LENGTH OF AN ARc ON A CuRvE 222
EPILOGUE AND APOLOGUE • 236
Table of Standard Forms 238
ANSWERS TO EXERCISES 240
v
PROLOGUE
CoNSIDERINCt how many fools can calculate, it is surprising
that it should be thought either a difficult or a tedious task
for any other fool to learn how to master the same tricks.
Som-e calculus-tricks are quite easy. Some are enormously
difficult. The fools who write the text-books of advanced
mathematics-and they are mostly clever fools-seldom take
the trouble to show you how easy the easy calculations are.
On the contrary, they seem to desire to impress you with their
tremendous cleverness by going about it in the most difficult
way.
Being myself a remarkably stupid fellow, I have had to
unteach myself the difficulties, and now beg to present to
my fellow fools the parts that are not hard. Master these
thoroughly, and the rest will follow. What one fool can do,
another can.
COMMON GREEK LETTERS USED AS SYMBOLS
Capital Small English Name Capital Small English Name
A a Alpha A ..\ Lambda
B f3 Beta M Mu
fL
r y Gamma .!~:l g Xi
L1 0 Delta II Pi
7T
E € Epsilon p p Rho
H 'Y] :Eta .I; a Sigma
e
8 Theta <P Phi
c/>
K K Kappa Q w Omega
vi
CHAPTER I
TO DELIVER YOU FROM THE PRELIMINARY
TERRORS
THE preliminary terror, which chokes off most fifth-form
boys from even attempting to learn how to calculate, can be
abolished once for all by simply stating what is the meaning
-in common-sense terms-of the two principal symbols that
are used in calculating.
These dreadful symbols are:
(I) d which merely means" a little bit of".
Thus dx means a little bit of x ; or du means a little bit of
u. Ordinary mathematicians think it more polite to say " an
element of ", instead of " a little bit of ". Just as you please.
But you will find that these little bits (or elements) may be
considered to be indefinitely small.
J
(2) which is merely a long S, and may be called (if you
like) " the sum of ".
J J
Thus dx means the sum of all the little bits of x ; or dt
means the sum of all the little bits of t. Ordinary mathe
maticians call this symbol " the integral of ". Now any fool
can see that if x is considered as made up of a lot of little
bits, each of which is called dx, if you add them all up to
gether you get the sum of all the dx's (which is the same
thing as the whole of x). The word" integral" simply means
" the whole ". If you think of the duration of time for one
hour, you may (if you like) think of it as cut up into 3600
little bits called seconds. The whole of the 3600 little bits
added up together make one hour.
1
2 CALCULUS MADE EASY
When you see an expression that begins with this terrifying
symbol, you will henceforth know that it is put there merely
to give you instructions that you are now to perform the
operation (if you can) of totalling up all the little bits that are
indicated by the symbols that follow.
That's all.