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VOL. LI.
THE
COMMON SENSE
OF THE
EXACT SCIENCES
WITH 100 FIGURES
'For information commences with the senses; but the whole business
terminates in works. . . . The chief cause of failure in work (especially
after natures have been diligently investigated) is the ill determination
and measurement of the forces and actions of bodies. Now the forces
and actions of bodies are circumscribed and measured, either by distances
of space, or by moments of time, or by concentration of quantity, or by
predominance of virtue ; and unless these four things have been well
and carefully weighed, we shall have sciences, fair perhaps in theory,
but in practice inefficient. The four instances which are useful in this
point of view I class under one head as Mathematical Instances and
Instances of Measurement' — A'ovum Organum, Lib. ii, Aph. xliv
SECOND EDITION
LONDON
KEGAN PAUL, TRENCH, & CO., 1 PATERNOSTER SQUARE
1886
L.O
rights of translation and of reproduction are reserved )
PBEFACE.
IN March 1879 Clifford died at Madeira; six years
afterwards a posthumous work is for the first time
placed before the public. Some explanation of this
delay must be attempted in the present preface.1
The original work as planned by Clifford was to
have been entitled The First Principles of the Mathe
matical Sciences Explained to the Non-Mathematical, and
to have contained six chapters, on Number, Space,
Quantity, Position, Motion, and Mass respectively. Of
the projected work Clifford in the year 1875 dictated
the chapters on Number and Space completely, the
first portion of the chapter on Quantity, and somewhat
later nearly the entire chapter on Motion. The first
two chapters were afterwards seen by him in proof, but
never finally revised. Shortly before his death he ex
pressed a wish that the book should only be published
1 A still more serious delay seems likely to attend the publication of
the second and concluding part (Kinetic) of Clifford's Elements of Dynamic,
the manuscript of which was left in a completed state. I venture to
think the delay a calamity to the mathematical world.
VI PHEFACE.
after very careful revision, and that its title should be
changed to The Common Sense of the Exact Sciences.
Upon Clifford's death the labour of revision and
completion was entrusted to Mr. E. C. Eowe, then
Professor of Pure Mathematics at University College,
London. That Professor Eowe fully appreciated the
difficulty and at the same time the importance of the
task he had undertaken is very amply evidenced by the
time and care he devoted to the matter. Had he lived
to complete the labour of editing, the work as a whole
would have undoubtedly been better and more worthy
of Clifford than it at present stands. On the sad death
of Professor Eowe, in October 1884, 1 was requested by
Messrs. Kegan Paul, Trench, & Co. to take up the
task of editing, thus left incomplete. Tt was with no
light heart, but with a grave sense of responsibility that
I undertook to see through the press the labour of two
men for whom I held the highest scientific admiration
and personal respect. The reader will perhaps appre
ciate my difficulties better when I mention the exact
state of the work when it came into my hands.
Chapters I. and II., Space and Number ; half of Chapter
III., Quantity (then erroneously termed Chapter IV.) ;
and Chapter V., Notion, were in proof. With these
proofs I had only some half-dozen pages of the
corresponding manuscript, all the rest having un-
PREFACE. Vll
fortunately been considered of no further use, and
accordingly destroyed. How far the contents of the
later proofs may have represented what Clifford dictated
I have had no means of judging except from the few
pages of manuscript in my possession. In revising the
proofs of the first two chapters, which Clifford himself
had seen, I have made as little alteration as possible,
only adding an occasional foot-note where it seemed
necessary. From page 65 onwards, however, I am,
with three exceptions in Chapter V., responsible for
all the figures in the book.
After examining the work as it was placed in my
hands, and consulting Mrs. Clifford, I came to the
conclusion that the chapter on Quantity had been
misplaced, and that the real gaps in the work were from
the middle of Chapter III. to Chapter V., and again at
the end of Chapter V. As to the manner in which
these gaps were to be filled I had no definite information
whatever ; only after my work had been completed was
an early plan of Clifford's for the book discovered. It
came too late to be of use, but it at least confirmed our
rearrangement of the chapters.
For the latter half of Chapter III. and for the whole
of Chapter IV. (pp. 116-226) I am alone responsible.
Yet whatever there is in them of value I owe to Clifford ;
whatever is feeble or obscure is my own.
Vlll PREFACE.
With Chapter V. my task has been by no means I
light. It was written at a time when Clifford was
much occupied with his theory of ' Graphs/ and found ]
it impossible to concentrate his mind on anything else :
parts of it are clear and succinct, parts were such
as the author would never have allowed to go to
press. I felt it impossible to rewrite the whole without
depriving the work of its right to be called Clifford's,
and yet at the same time it was absolutely necessary
to make considerable changes. Hence it is that my
revision of this chapter has been much more extensive
than in the case of the first two. With the result I
fear many will be dissatisfied; they will, however, hardly
be more conscious of its deficiencies than I am. I can
but plead the conditions under which I have had to
work. One word more as to this chapter. Without
any notice of mass or force it seemed impossible to close
a discussion on motion ; something I felt must be added.
I have accordingly introduced a few pages on the laws
of motion. I have since found that Clifford intended
to write a concluding chapter on mass. How to express
the laws of motion in a form of which Clifford would
have approved was indeed an insoluble riddle to me,
because I was unaware of his having written anything
on the subject. I have accordingly expressed, although
with great hesitation, my own views on the subject;
PREFACE. IX
these may be concisely described as a strong desire to
see the terms matter and force, together -with the ideas
associated with them, entirely removed from scientific
terminology — to reduce, in fact, all dynamic to kine
matic. I should hardly have ventured to put forward
these views had I not recently discovered that they have
(allowing for certain minor differences) the weighty
authority of Professor Mach, of Prag.1 But since writing
these pages I have also been referred to a discourse
delivered by Clifford at the Eoyal Institution in 1873,
some account of which appeared in Nature, June 1 0,
1880. Therein it is stated that ' no mathematician
can give any meaning to the language about matter,
force, inertia used in current text-books of mechanics.' 2
This fragmentary account of the discourse undoubtedly
proves that Clifford held on the categories of matter
and force as clear and original ideas as on all subjects
of which he has treated ; only, alas ! they have not
been preserved.
In conclusion I must thank those friends who have
been ever ready with assistance and advice. Without
their aid I could not have accomplished the little that
1 See his recent book, Die MecJtanik in ihrer Entwickelung. Leipzig,
1883.
2 Mr. R. Tucker, -who has kindly searched Clifford's note-books for
anything on the subject, sends me a slip of paper with the following
words in Clifford's handwriting : ' Force is not a fact at all, but an idea
embodying what is approximately the fact.'
X PREFACE.
has been done. My sole desire has been to give to the H
public as soon as possible another work of one whose K
memory will be revered by all who have felt the MI
invigorating influence of his thought. Had this work I:
been published as a fragment, even as many of us I
wished, it would never have reached those for whom i
Clifford had intended it. Completed by another hand, I
we can only hope that it will perform some, if but a I
small part, of the service which it would undoubtedly
have fulfilled had the master lived to put it forth.
K. P.
UNITEBSITT COLLEGE, LONDON :
February 26.
CONTENTS.
CHAPTER I.
NUMBER.
PA OB
6KCTIOH
1. Number is Independent of the order of Count » .
1
2.
A Sum is Independent of the order of Adding
2
3.
A Product is Independent of the order oi Multiplying .
6
4.
The Distributive Law
. 14
5.
On Powers
. 16
6.
Square of a + 1
. 17
7.
On Powers of a + b .
. 19
8.
On the Number of Arrangements of a Group of Letters
. 21
9.
On a Theorem concerning any Power of a + b
• -7
10.
On Operations which appear to be without Meaning .
. 32
0 1
11.
Steps
. • J I
38
12.
13.
Addition and Multiplication of Operations
. 40
14.
Division of Operations
. 42
15.
General Results of our Extension of Terms .
. 45
CHAPTER II.
SPACE.
1.
. 47
2.
Lengths can be Moved without Change ....
. 52
3.
The Characteristics of Shape .
55
4.
The Characteristics of Surface Boundaries ....
. 63
5.
The Plane and the Straight Line
. M
6.
Properties of Triangles
. G9
7.
Properties of Circles ; Related Circles and Triangles
. 75
8.
The Conic Sections
. 81
9.
On Surfaces of the Second Order
87
10.
How to form Curves of the Third and Higher Orders .
. 91
Xll CONTENTS.
CHAPTER HI.
QUANTITY.
SECTION pAC(B
1. The Measurement of Quantities 95
2. The Addition and Subtraction of Quantities . .99
3. The Multiplication and Division of Quantities . . . 100
4. The Arithmetical Expression of Ratios . . 102
5. The Fourth Proportional . 105
6. Of Areas; Stretch and Squeeze .... . 113
7. Of Fractions .116
8. Of Areas; Shear ...... . 120
9. Of Circles and their Areas ... ... 123
10. Of the Area of Sectors of Curves .... 130
11. Extension of the Conception of Area 131
12. On the Area of a Closed Tangle 135
13. On the Volumes of Space-Figures . . . 138
14. On the Measurement of Angles . 141
15. On Fractional Powers . 144.
CHAPTER IV.
POSITION.
1. All Position is Relative ... . . 147
2. Position may be Determined by Directed Steps . . . 149
3. The Addition of Directed Steps or Vectors . . . .153
4. The Addition of Vectors obeys the Commutative Law . . 158
5. On Methods of Determining Position in a Plane . . . 159
6. Polar Co-ordinates .164
7. The Trigonometrical Ratios .... .166
8. Spirals jg-
9. The Equiangular Spiral 171
10. On the Nature of Logarithms ... 176
11. The Cartesian Method of Determining Position . . .181
12. Of Complex Numbers .... Igg
13. On the Operation which turns a Step through a given Angle . 192
14. Relation of the Spin to the Logarithmic Growth of Unit Step 195
15. On the Multiplication of Vectors . ... .198
16. Another Interpretation of the Product of Two Vectors . . 204
17. Position in Three-Dimensioned Space . . . .207
18. On Localised Vectors or Rotors . . t 210
19. On the Bending of Space .... .214
CONTENTS. XU1
CHAPTER V.
MOTION.
BICTIOS PAGE
1. On the Various Kinds of Motion 227
2. Translation and the Curve of Positions .... 230
3. Uniform Motion 235
4. Variable Motion 237
5. On the Tangent to a Curve 243
6. On the Determination of Variable Velocity . . . . 250
7. On the Method of Fluxions 253
8. Of the Relationship of Quantities, or Functions . . . . 255
9. Of Acceleration and the Hodograph 260
10. On the Laws of Motion 267
11. Of Mass and Force . . 269
THE
COMMON SEXSE
OF THE
EXACT SCIENCES.
CHAPTER I.
NUMBER.
§ 1. Number is Independent of the order of Count in a
THE word which stands at the head of this clir
contains six letters. In order to find out that there
are six, ' ;e count them ; n one, u two, m three, b four,
e five, if six. In this process we have taken the letters
one by one, and have put beside them six words whk-h
are the first six out of a series of words that we always
carry about with us, the names of numbers. After putting
these six words one to each of the letters of the word
number, we found that the last of the words was .<-•/ •• ; and
accordingly we called that set of letters by the name .six.
If we counted the letters in the word ' chapter : in
the same way, we should find that the last of the
numeral words thus used would be seven; and accor
dingly we say that there are seven letters.
But now a question arises. Let us suppose tha^ the
letters of the word number are printed upon separate
2 THE COMMON SENSE OF THE EXACT SCIENCES.
small pieces of wood belonging to a box of letters ; fhs.
we put these into a bag and shake them up and brin^
them out, putting them down in any other order, an I
then count them again ; we shall still find that thei
are six of tnem. For example, if they come out ij
the alphabetical order b e m n r u, and we put to eacl
of these one of the names of numbers that we havj
before used, we shall still find that the last name \vi|
be six. In the assertion that any group of things coi
sists of six things, it is implied that the word six wil
be the last of the ordinal words used, in whatevel
order we take up this group of things to count them!
That is to say, the number of any set of things is th\
same in whatever order we count them.
Upon this fact, which we have observed with regard
to the particular number six, and which is true of all
numbers whatever, the whole of the science of numbe]
is based. We shall now go on to examine somJ
theorems about numbers which may be deduced from itl
§ 2. A Sum is Independent of the order of Adding.
Suppose that we have two groups of things ; say th<
letters in the word ' number,' and the letters in th<
word ' chapter.' We may count these groups separately
and find that they come respectively to the numbers
six and seven. We may then put them all together, anc
we find in this case that the aggregate group which is
so formed consists of thirteen letters.
Now this operation of putting the things all togethe
may be conceived as taking place in two different ways
We may first of all take the six things and put them ir
a heap, and then we may add the seven things to then-
one by one. The process of counting, if it is performed
NUMBER. 3
in this order, amounts to counting seven more ordinal
words after the word six. We may however take the
seven things first and put them into a heap, and then
add the six things one by one to them. In this case the
process of counting amounts to counting six more
ordinal words after the word seven.
But from what we observed before, that if we count
any set of things we come to the same number in what
ever order we count them, it follows that the number
we arrive at, as belonging to the whole group of things,
must be the same whichever of these two processes we
use. This number is called the sum of the two numbers
6 and 7 ; and, as we have seen, we may arrive at it either
by the first process of adding 7 to 6, or by the second
process of adding 6 to 7.
The process of adding 7 to 6 is denoted by a short
hand symbol, which was first used by Leonardo da Vinci.
A little Maltese cross ( + ) stands for the Latin j>/7/.s>,
or the English increased by. Thus the words six increased
by seven are written in shorthand 6 + 7. Now we
have arriv 1 at the result that six increased by seven is
the same n amber as seven increased by six. To write this
wholly in shorthand, we require a symbol for the words,
is the same number as. The symbol for these is = ; it was
first used by an Englishman, Robert Recorde. Our
result then may be filially written in this way : —
6 + 7 = 7 + 6.
The proposition which we have written in this
symbolic form states that the sum of two numbers G
and 7 is independent of the order in which they are
added together. But this remark which we have madf
about two particular numbers is equally true of any
two numbers whatever, in consequence of our funda-
B 'A
4 THE COMMON SEXSE OF THE EXACT SCIENCES.
mental assumption that the number of things in anjl
group is independent of the order in which we count!
them. For by the sum of any two numbers we mean!
a number which is arrived at by taking a group on
tilings containing the first number of individuals, and
D O
adding to them one by one another group of things
containing the second number of individuals ; or, if we
like, by taking a group of things containing the second
number of individuals, and adding to them one by one
the group of things containing the first number of
individuals. Now, in virtue of our fundamental
assumption, the results of these two operations must be
the same. Thus we have a right to say, not only that
6 + 7 = 7 + 6, but also that 5 + 13 = 13 + 5, and
so on, whatever two numbers we like to take.
This we may represent by a method which is due to
Vieta, viz., by denoting each number by a letter of the
alphabet. If we write a in place of the first number
in either of these two cases, or in any other case, and b
in place of the second number, then our formula will
stand thus : —
a + b = b + a.
By means of this representation we have made a
statement which is not about two numbers in particu
lar, but about all numbers whatever. The letters a and
1) so used are something like the names which we give
to things, for example, the name horse. When we say
a horse has four legs, the statement will do for any
particular horse whatever. It says of that particular
horse that it has four legs. If we said ( a horse has as
many legs as an ass,' we should not be speaking of any
particular horse or of any particular ass, but of any
horse whatever and of any ass whatever. Just in the
same way, when we assert that a + b — b + a, we are
NUMBER. 5
mot speaking of any two particular numbers, but of all
numbers whatever.
We may extend this rule to more numbers than
two. Suppose we add to the sum a + b a third
number, c, then we shall have an aggregate group of
things which is formed by putting together three groups,
and the number of the aggregate group is got by adding
together the numbers of the three separate groups. This
number, in virtue of our fundamental assumption, is
the same in whatever order we add the three groups
together, because it is always the same set of things
that is counted. Whether we take the group of a
things first, and then add the group of 6 things to it
one by one, and then to this compound group of a + h
things add the group of c things one by one; or
whether we take the group of c things, and add to it
the group of b things, and then to the compound group
of c + b things add the group of a things, the sum
must in both cases be the same. We may write this
result in the ymbolic form a + & + c = c + b + a, or
we may state in words that the sum of three numbers /x
in<L'lii'n<lt:nt of the order in which they are added /</'/<'//c;/-.
This rule may be extended to the case of any
number of numbers. However many groups of things
we have, if we put them all together, the number of
things in the resulting aggregate group may be counted
in various ways. We may start with counting any one
of the original groups, then we may follow it with any
one of the others, following these by any one of those
left, and so on. In whatever order we have taken these
groups, the ultimate process is that of counting the
whole aggregate group of things ; and consequently
the numbers that we arrive at in these different ways
must all be the same.
6 TPIE COMMOX SEXSE OF THE EXACT SCIEXCES.
§ 3. A Product is Independent of the order of Multiply ing\
Now let us suppose that we take six groups oJ
things which all contain the same number, say 5, and!
that we want to count the aggregate group which is!
made by putting all these together. We may count!
the six groups of five things one after another, which]
amounts to the same thing as adding 5 five times over
to 5. Or if we like we may simply mix up the whole!
of the six groups, and count them without reference to)
their previous grouping. But it is convenient in this
case to consider the six groups of five things as arranged
in a particular way.
Let us suppose that all these things are dots which
are made upon paper, that every group of five things is
five dots arranged in a horizontal line, and that the
six groups are placed vertically under one another as in
the figure.
We then have the whole of the dots of these six
groups arranged in the form of an oblong which con
tains six rows of five dots each. Under each of the five
dots belonging to the top group there are five other dots
belonging to the remaining groups; that is to say, we
have not only six rows containing five dots each, but five
columns containing six dots each. Thus the whole set
NUMBER. 7
of dots can be arranged in five groups of six each, just
as well as in six groups of five each. The whole number
of things contained in six groups of five each, is called
six times five. We learn in this way therefore that
six times five is the same number as five times six.
As before, the remark that we have here made about
two particular numbers may be extended to the case of
any two numbers whatever. If we take any number of
groups of dots, containing all of them the same number
of dots, and arrange these as horizontal lines one under
the other, then the dots will be arranged not only in
lines but in columns ; and the number of dots in every
column will obviously be the same as the number of
groups, while the number of columns will be equal to
the number of dots in each group. Consequently the
number of things in a groups of b things each is equal
to the number of things in b groups of a things each,
no matter what the numbers a and b are.
The number of things in a groups of b things each
is called a time? b ; and we learn in this way that a
times b is equa'i to b times a. The number a times b
is denoted by writing the two letters a and b together,
a coming first ; so that we may express our result in the
symbolic form ab = ba.
Suppose now that we put together seven such com
pound groups arranged in the form of an oblong like
that we constructed just now. They cannot now be repre
sented on one sheet of paper, but we may suppose that
instead of dots we have little cubes which can be put
into an oblong box. On the floor of the box we shall
have six rows of five cubes each, or five columns of six
cubes each ; and there will be seven such layers, one on
the top of another. Upon every cube therefore which
is in the bottom of the box there will be a pile of six.
8 THE COMMON SENSE OF THE EXACT SCIENCES.
cubes, and we shall have altogether five times six such
piles. That is to say, we have five times six groups of
seven cubes each, as well as seven groups of five times
six cubes each. The whole number of cubes is indepen
dent of the order in which they are counted, and con
sequently we may say that seven times five times six is
the same thing as five times six times seven.
But it is here very important to notice that when we
say seven times five times six, what we mean is that
seven layers have been formed, each of which contains
five times six things ; but when we say five times six
times seven, we mean that five times six columns have
been formed, each of which contains seven things.
Here it is clear that in the one case we have first multi
plied the last two numbers, and then multiplied the result
by the first mentioned (seven times five times six = seven
times thirty), while in the other case it is the first two
numbers mentioned that are multiplied together, and
then the third multiplied by the result (five times six
times seven = thirty times seven). Now it is quite
evident that when the box is full of these cubes it may
be set upon any side or upon any end ; and in all cases
there will be a number of layers of cubes, either 5 or 6
or 7. And whatever is the number of layers of cubes,
that will also be the number of cubes in each pile.
Whether therefore we take seven layers containing
five times six cubes each, or six layers containing
seven times five cubes each, or five layers containing six
times seven cubes each, it comes to exactly the same
thing.
We may denote five times six by the symbol 5x6, and
then we may write five times six times seven, 5x6x7.
But now this form does not tell us whether we
are to multiply together 6 and 7 first, and then take 5
NUMBER. 9
times the result, or whether we are to multiply 5 and G
first, and take that number of sevens. The distinction
between these two operations may be pointed out by
means of parentheses or brackets ; thus, 5 x (G x 7)
means that the 6 and 7 must be first multiplied to
gether and 5 times the result taken, while (5 x 6) x 7
means that we are to multiply 5 and 6 and then take
the resulting number of sevens.
We may now state two facts that we have learned
about multiplication.
First, that the brackets make no difference in the
result, although they do make a difference in the pro
cess by which the result is attained; that is to say,
5x (Gx7) = (5xG)x7.
Secondly, that the product of these three numbers
is independent of the order in which they are multi
plied together.
The first of these statements is called the assocm-
tive law of multip1 cation, and the second the commuta
tive law.
Xow these remarks that we have made about the
[result of multiplying together the particular three
numbers, 5, 6, and 7, are equally applicable to any
three numbers whatever.
We may always suppose a box to be made whose
height, length, and breadth will hold any three num
bers of cubes. In that case the whole number of
cubes will clearly be independent of the position of the
box ; but however the box is set down it will contain a
certain number of layers, each layer containing a cer
tain number of rows, and each row containing a certain
number of cubes. The whole number of cubes in the
box will then be the product of these three numbers ;
and it will be got at by taking any two of the three
10 THE COMMON SENSE OF THE EXACT SCIENCES.
numbers, multiplying them together, and then multi
plying the result by the third number.
This property of any three numbers whatever may
now be stated symbolically.
In the first place it is true that a(bc) — (ab)c; that
is, it comes to the same thing whether we multiply the
product of the second and third numbers by the first,
or the third number by the product of the first and
second.
In the next place it is true that abc = acb = l>ca, &c.,
and we may say that the product of any three numbers
is independent of the order and of the mode of group
ing in which the multiplications are performed.
We have thus made some similar statements about
two numbers and three numbers respectively. This
naturally suggests to us that we should inquire if cor
responding statements can be made about four or five
numbers, and so on.
We have arrived at these two statements by con
sidering the whole group of things to be counted as
arranged in a layer and in a box respectively. Can
•we go any further, and so arrange a number of boxes as
to exhibit in this way the product of four numbers?
It is pretty clear that we cannot.
Let us therefore now see if we can find any other
sort of reason for believing that what we have seen to
be true in the case of three numbers — viz., that the re
sult of multiplying them together is independent of
the order of multiplying — is also true of four or more
numbers.
In the first place we will show that it is possible to
interchange the order of a pair of these numbers which
are next to one another in the process of multiplying,
without altering the product.
NUMBER. 11
Consider, for example, the product of four numbers,
rZ. We will endeavour to show that this is the same
;hing as the product acid. The symbol abed means
:liat we are to take c groups of d things and then 6
Croups like the aggregate so formed, and then finally a
groups of bed things.
Now, by what we have already proved, b groups of
\d things come to the same number as c groups of bd
brings. Consequently, a groups of bed things are the
lame as a groups of cbd things ; that is to say, abcd =
acbd.
It will be quite clear that this reasoning will hold
no matter how many letters come after d. Suppose,
for example, that we have a product of six numbers
abcdef. This means that we are to multiply / by e, the
result by cZ, then def by c, and so on.
Now in this case the product def simply takes the
place which the numb r d had before. And b groups of
times def things come to the same number as c groups
of b times def things, for this is only the product of three
numbers, b, c, and def. Since then this result is the
same in whatever order b and c are written, there can
no alteration made by multiplications coming after,
that is to say if we have to multiply by ever so
many more numbers after multiplying by a. It follows
therefore that no matter how many numbers are multi
plied together, we may change the places of any two of
them which are close together without altering the
product.
In the next place let us prove that we may change
the places of any two which are not close together.
For example, that dbcdef is the same thing as aecdbf,
where b and e have been interchanged. We may do
this by first making the e march backwards, changing
12 THE COMMON SEXSE OF THE EXACT SCIENCES.
places successively with d and c and b, when the
product is changed into aebcdf; and then making
march forwards so as to change places successively with
c and d, whereby we have now got e into the place of b.
Lastly, I say that by such interchanges as these
we can produce any alteration in the order that we like.
Suppose for example that I want to change abcdef into
dcfbea. Here I will first get d to the beginning; I
therefore interchange it with a, producing dbcaef.
Next, I must get c second ; I do this by interchanging
it with b, this gives dcbaef. I must now put / third
by interchanging it with b, giving dcfaeb, next put b
fourth by interchanging it with a, producing dcfbea.
This is the form required. By five such interchanges
at most, I can alter the order of six letters in any
way I please. It has now been proved that this alter
ation in the order may be produced by successive in
terchanges of two letters which are close together.
But these interchanges, as we have before shown, do
not alter the product ; consequently the product of six
numbers in any order is equal to the product of the
same six numbers in any other order ; and it is easy to
see how the same process will apply to any number o:
numbers.
But is not all this a great deal of trouble for the
sake of proving what we might have guessed before
hand ? It is true we might have guessed beforehand
that a product was independent of the order and group
ing of its factors ; and we might have done good work
by developing the consequences of this guess before we
were quite sure that it was true. Many beautiful
theorems have been guessed and widely used be
fore they were conclusively proved; there are some
even now in that state. But at some time or other the
NUMBER. ] 3
inquiry lias to be undertaken, and it always clears up
our ideas about the nature of the theorem, besides
giving us the right to say that it is true. And this is not
all ; for in most cases the same mode of proof or of in
vestigation can be applied to other subjects in such a
way as to increase our knowledge. This happens with
the proof we have just gone through ; but at present, as
we have only numbers to deal with, we can only go
backwards and not forwards in its application. We
have been reasoning about multiplication ; let us see if
the same reasoning can be applied to addition.
What we have proved amounts to this. A certain
result has been got out of certain things by taking
them in a definite order ; and it has been shown that
if u-e can interchange any two consecutive things without
altering the result, then we may make any change whatever
in the order without all ring the result. Let us apply
this to counting. The process of counting consists in
taking certain things in a definite order, and applying
them to our fingers one by one ; the result depends on
the last finger, and its name is called the number of the
things so counted. We learn then that this result will
be independent of the order of counting, provided only
that it remains unaltered when we interchange any two
consecutive things ; that is, provided that two adjacent
fingers can be crossed, so that each rests on the object
previously under the other, without employing any new
fingers or setting free any that are already employed.
With this assumption we can prove that the number of
any set of things is independent of the order of counting ;
a statement which, as we have seen, is the foundation
of the science of number.
14 THE COMMON SENSE OF THE EXACT SCIENCES.
§ 4. The Distributive Law.
There is another law of multiplication which is,
possible, still more important than the two we hav
already considered. Here is a particular case of it
the number 5 is the sum of 2 and 3, and 4 times 5 i
the sum of 4 times 2 and 4 times 3. We can make thi
visible by an arrangement of dots as follows : —
Here we have four rows of five dots each, and each row i
divided into two parts, containing respectively two dot
and three dots. It is clear that the whole number
dots may be counted in either of two ways; as fou
rows of five dots, or as four rows of two dots togethe
with four rows of three dots. By our general principl
the result is independent of the order of counting, an
therefore
4 x 5 = (4 x 2) + (4 x 3) ;
or, if we put in evidence that 5 = 2-4-3,
4 (2 + 3) = (4 x 2) + (4 x 3).
The process is clearly applicable to any three num
bers whatever, and not only to the particular numbers
4, 2, 3. We may construct an oblong containing a rows
of b + c dots ; and this may be divided by a vertical line
into a rows of 6 dots and a rows of c dots. Counted
in one way, the whole number of dots is a(b + c) ;
NUMBER. 1 5
counted in another way, it is ab + ac. Hence we must
always have
a (b + c) = ab -f ac.
This is the first form of the distributive law.
Now the result of multiplication is independent of
the order of the factors, and therefore
a (b + c) = (b + c) a,
ab = 6a,
ac = ca ;
so that our equation may be written in the form
(b + c) a = ba + ca.
This is called the second form of the distributive law.
Using the numbers of our previous example, we say that
since 5 is the sum of 2 and 3, 5 times 4 is the sum of 2
times 4 and 3 times 4. This form may be arrived at in
dependently and very simply as follows. We know that
2 things and 3 things make 5 things, whatever the things
are; let each of these things be a group of 4 things;
then 2 fours and 3 fours make 5 fours, or
(2x4) + (3x4) =5x4.
The rule may now be extended. It is clear that our
oblong may be divided by vertical lines into more parts
than two, and that the same reasoning will apply. This
• • » • o • •
tigure, for example, makes visible the fact that just as
2 and 3 and 4 make 9, so 4 times 2, and 4 times 3, and
4 times 4 make 4 times i). Or generally—
16 THE COMMON SEXSE OF THE EXACT SCIENCES.
a (b -f c + d) = ab + ac + ad,
(b + c + d) a — ba + ca + da ;
and the same reasoning applies to the addition of any
number of numbers and their subsequent multiplication.
§ 5. On Powers.
When a number is multiplied by itself it is said to
be squared. The reason of this is that if we arrange a
number of lines of equally distant dots in an oblong, the
number of lines being equal to the number of dots in
each line, the oblong will become a square.
If the square of a number is multiplied by the
number itself, the number is said to be cubed ; because if
we can fill a box with cubes whose height, length, and
breadth are all equal to one another, the shape of the
box will be itself a cube.
If we multiply together four numbers which are all
equal, we get what is called the fourth power of any one
of them ; thus if we multiply 4 3's we get 81, if we
multiply 4 2's we get 16.
If we multiph" together any number of equal num
bers, we get in the same way a power of one of them
which is called its fifth, or sixth, or seventh power, and
so on, according to the number of numbers multiplied
together.
Here is a table of the powers of 2 and 3 : —
Index 1234567 8
Powers of 2 ... 2 4 8 1C 32 64 128 2.36
„ 3 ... 3 9 27 81 243 729 2187 6561
The number of equal factors multiplied together is
called the index, and it is written as a small figure
* o
above the line on the right-hand side of the number
whose power is thus expressed. To write in shorthand
NUMBER. 17
the statement that if you multiply seven threes together
you get 2187, it is only needful to put down : —
3 = 2187.
It is to be observed that every number is its own
first power ; thus 2' = 2, 3' = 3, and in general a} — a.
§ 6. Square o/a + 1.
may illustrate the properties of square numbers
by means of a common arithmetical puzzle, in which
one person tells the number another has thought of by
means of the result of a round of calculations per
formed with it.
Think of a number .... say 3
Square it ...... 9
Add 1 to the original number ... 4
Square that . . . . . . Ib
Take the difference of the two squares . 7
This last is always an odd number, and the number
thought of is what we may call the less half of it ; viz.,
it is the half of the even number next below it. Thus,
the result being given as 7, we know that the number
thought of was the half of G, or 3.
We will now proceed to prove this rule. Suppose
that the square of 5 is given us, in the form of twenty-
five dots arranged in a square, how are we to form the
square of 6 from it? We may add five dots on the
right, and then five dots along the bottom, and then
one dot extra in the corner. That is, to get the square
of 6 from the square of 5, we must add one more than
twice 5 to it. Accordingly —
36 = 25 + 10 + 1.
c
18 THE COMMON SEXSE OF THE EXACT SCIENCES.
And, conversely, the number 5 is the less half of thel
difference between its square and the square of 6. I r;
The form of this reasoning- shows that it holds good
for any number whatever. Having given a square of
dots, we can make it into a square having one more
dot in each side by adding a column of dots on the
riffht, a row of dots at the bottom, and one more dot in
O ' J
the corner. That is, we must add one more than twice
the number of dots in a side of the original square.
If, therefore, this number is given to us, we have only
to take one from it and divide by 2, to have the num
ber of dots in the side of the original square.
We will now write down this result in. shorthand.
Let a be the original number ; then a+ 1 is the number
next above it ; and what we want to say is that the square
of a+1, that is (a + 1)2, is got from the square of a,
which is a2, by adding to it one more than twice a,
that is 2a + 1. Thus the shorthand expression is
(a + 1) 2 = a2 + 2a + 1.
This theorem is a particular case of a more general
one, which enables us to find the square of the sum of
NUMBER.
19
any two numbers in terms of the squares of the two
numbers and their product. We will first ilhistrate
this by means of the square of 5, which is the sum of
2 and 3.
The square of twenty-five dots is here divided into
two squares and two oblongs. The squares are respec
tively the squares of 3 and 2, and each oblong is tin1
product of 3 and 2. In order to make the square of >'>
into the square of 3 + 2, we must add two columns on
the right, two rows at the bottom, and then the square
of 2 in the corner. And in fact, 25 = 9 + 2x6 + 4.
§ 7. On Powers of a + b.
To generalise this, suppose that we have a square
with a dots in each side, and we want to increase it to
a square with a + b dots in each side. We must add b
columns on the right, b rows at the bottom, and then
the square of b in the corner. But each column and
each row contains a dots. Hence what we have to add
is twice ab together with b2, or in shorthand : —
(a + 6)2 = a2 + 2ab + b*.
The theorem we previously arrived at may be got from
this by making 6 = 1.
c 2
20 THE COMMON SENSE OF THE EXACT SCIENCES.
Now this is quite completely and satisfactorily
proved ; nevertheless we are going to prove it again in
another way. The reason is that we want to extend
the proposition still further ; we want to find an ex
pression not only for the square of (a + fy, but for any
other power of it, in terms of the powers and products
of powers of a and b. And for this purpose the mode
of proof we have hitherto adopted is unsuitable. We
might, it is true, find the cube of a-t-b by adding the
proper pieces to the cube of a ; but this would be some
what cumbrous, while for higher powers no such repre
sentation can be used. The proof to which we now pro
ceed depends on the distributive law of multiplication.
According to this law, in fact, we have
(a + 6)2 = (a + 6) (a + b] = a (a + 1} + I (a + b)
= aa + ab + ba + lib
= a? + 2ab + l>\
It will be instructive to write out this shorthand at
length. The square of the sum of two numbers means
that sum multiplied by itself. But this product is the
first number multiplied by the sum together with the
second number multiplied by the sum. Now the first
number multiplied by the sum is the same as the
first number multiplied by itself together with the first
number multiplied by the second number. And the
second number multiplied by the sum is the same as
the second number multiplied by the first number to
gether with the second number multiplied by itself.
Putting all these together, we find that the square of
the sum is equal to the sum of the squares of the two
numbers together with twice their product.
Two things may be observed on this comparison.
First, how very much the shorthand expression gains
NUMBER. 21
in clearness from its brevity. Secondly, that it is only
shorthand for something which is just straightforward
common sense and nothing else. We may always
depend upon it that algebra, which cannot be translated
into good English and sound common sense, is bad
algebra.
But now let us put this process into a graphical
shape which will enable us to extend it. We start
with two numbers, a and b, and we are to multiply each
of them by a and also by I, and to add all the results.
a + b
x\ x\
aa la ab bb
Let us put in each case the result of multiplying by a
to the left, and the result of multiplying by b to the
riLcht, under the number multiplied. The process is
then shown in the figure.
If we now want to multiply this by a + b again, so as
to make (a + b)3, we must multiply each part of the
lower line by a, and also by b, and add all the results,
thus : —
x\
aba bba aab bab
Here we have eight terms in the result. The first
and last are a3 and b3 respectively. Of the remaining
six, three are baa, aba, aab, containing two a's and one
b, and therefore each equal to a'*b ; and three are bb«,
bab, abb, containing one a and two b's, and therefore
each equal to ab2. Thus we have : —
(a + b)3 = a3 + 3a'2b + 3ab* + b3.
/ X
22 THE COMMON SENSE OF THE EXACT SCIENCES.
For example, II3 = 1331. Here a = 10, 6 = 1. and
for it is clear that any power of 1 is 1.
We shall carry this process one
step further, before making remarks
which will enable us to dispense
with it.
In this case there are sixteen
terms, the first and last being a4 and
&4 respectively. Of the rest, some
have three a's and one &, some two
a's and two 6's, and some one a and
three 6's. There are four of the
first kind, since the & may come first,
second, third, or fourth; so also
there are four of the third kind, for
the a occurs in each of the~same four
places ; the remaining six are of the
second kind. Thus we find that,
(a + iy = a* + 4a3& + Ga2&2 + 4a&3 + &4.
We might go on with this process
as long as we liked, and we should
get continually larger and larger
trees. But it is easy to see that the
process of classifying and counting
the terms in the last line would
become very troublesome. Let us
then try to save that trouble by
making some remarks upon the
process.
If we go down the tree last
figured, from a to a&aa, we shall find that the term
NUMBER. 23
abaa is built up from right to left as we descend. The
a that we begin with is the last letter of dbaa ; then in
descending we move to the right, and put another a
before it ; then we move to the left and put b before
that ; lastly we move to the right and put in the
first a. From this there are two conclusions to be
drawn.
First, the terms at the end are all different; for any
divergence in the path by which we descend the tree
makes a difference in some letter of the result.
Secondly, every possible arrangement of four A.7/>v-s
which are either a's or b's is produced. For if any Midi
arrangement be written down, say abab, we have only
to read it backwards, making a mean ' turn to the
left ' and b ' turn to the right,' and it will indicate
the path by which we must descend the tree to find
that arrangement at the end.
We may put these two remarks into one by saying
that evert/ such possible arrangement is produced once "/"/
once onli/.
Xow the problem before ns was to count the
number of terms which have a certain number of //s in
them. By the remark just made we have shown that
this is the same thing as to count the number of
possible arrangements having that number of b's.
Consider for example the terms containing one 1>.
When there are three letters to each term, the number
of possible arrangements is 3, for the b may be first,
second, or third, baa, aba, aab. So when there are four
letters the number is 4, for the b may be first, second,
third, or fourth; b<nn^ tilma, naha, aaab. And generally
it is clear that whatever be the number of letters in each
term, that is also the number of places in which the 6
can stand. Or, to state the same thing ill shorthand,
24 THE COMMON SENSE OP THE EXACT SCIENCES.
if n be the number of letters, there are n terms con
taining one b. And then, of course, there are n terms
containing one a and all the rest &'s.
And these are the terms which come at the beginning
and end of the nth power of a + b ; viz. we must have
(a + b) — a + nan~lb + other terms + nab ~l + bn.
The meaning of this shorthand is that we have n
(a + b)'s multiplied together, and that the result of that
multiplying is the sum of several numbers, four of
which we have written down. The first is the product of
n a's multiplied together, or an ; the next is n times the
product of b by (n— 1) a's, namely, nan~lb. The last
but one is n times the product of a by (n — 1) &'s, namely,
nabn~l ; and the last is the product of n b's multiplied
together, which is written 6".
O *
The problem that remains is to fill up this state
ment by finding what the ' other terms ' are, containing
each more than one a and more than one 6.
§ 8. On the Number of Arrangements of a Group of Letters.
This problem belongs to a very useful branch of
applied arithmetic called the theory of ' permutations
and combinations,' or of arrangement and selection.
The theory tells us how many arrangements may be
made with a given set of things, and how many selec
tions can be made from them. One of these questions
is made to depend on the other, so that there is an
advantage in counting the number of arrangements
first.
With two letters there are clearly two arrangements,
ab and ba. With three letters there are these six : —
I
NUMBER. 25
f namely, two with a at the beginning, two with 6 at the
beginning, and two with c at the beginning ; three
times two. It would not be much trouble to write
down all the arrangements that can be made with four
f letters abed. But we may count the number of them
without taking that trouble ; for if we write d before
\ each of the six arrangements of abc, we shall have six
arrangements of the four letters beginning with */,
and these are clearly all the arrangements which can
begin with d. Similarly, there must be six beginning
with a, six beginning with 6, and six beginning with c;
in all, four times six, or twenty -four.
Let us put these results together :
3
With two letters, number of arrangements is two = 2
„ three „ three times two . = 6
„ four „ four times three times two = 21-
Here we have at once a rule suggested. To find the
number of arrangements ivliich can be made with a (jiven
group of letters, multiply tor/ether the numbers two, three,
four, &c., up to the number of letters in the group. We
have found this rule to be right for two, three, and
four letters ; is it right for any number whatever of
letters ?
We will consider the next case of five letters, and
\ deal with it by a method which is applicable to all cases.
/ Any one of the five letters may be placed first ; there are
then five ways of disposing of the first place. For each
of these ways there are four ways of disposing of the
second place ; namely, any one of the remaining four
letters may be put second. This makes five times four
ways of disposing of the first two places. For each of
these there are three ways of disposing of the third
place, for any one of the remaining three letters may
26 THE COMMON SENSE OF THE EXACT SCIENCES.
be put third. This makes five times four times three
ways of disposing of the first three places. For each
of these there are two ways of disposing of the last
two places; in all, five times four times three times
two, or 120 ways of arranging the five letters.
Now this method of counting the arrangements will
clearly do for any number whatever of letters ; so that
our rule must be right for all numbers.
We may state it in shorthand thus : the number of
arrangements of n letters is 1 X 2 x 3 x ... X »; or
putting dots instead of the sign of multiplication, it is
1.2.3... n. The 1 which begins is of course not
wanted for the multiplication, but it is put in to in
clude the extreme case of there being only one letter,
in which case, of course, there is only one arrange
ment.
The product 1 . 2 . 3 ... n, or, as we may say, the
product of the first n natural numbers, occurs very often
in the exact sciences. It has therefore been found
convenient to have a special short sign for it, just as
a parliamentary reporter has a special sign for ' the t
remarks which the Honourable Member has thought
fit to make.' Different mathematicians, however, have
used different symbols for it. The symbol \n is very
much used in England, but it is difficult to print.
Some continental writers have used a note of admira- J
tion, thus, n ! Of this it has been truly remarked that ^
it has the air of pretending that you never saw it
before. I myself prefer a symbol which has the weighty
authority of Gauss, namely a Greek n (Pi), which may
be taken as short for product if we like, thus, Tin. We
may now state that —
111 = 1, 112 = 2, 113 = 6, 114 = 24, 05 = 120, IIG = 720,
and generally that
NUMBER. 27
n (n + 1) = (n + 1) IIw,
for the product of the first n+ 1 numbers is equal to the
product of the first n numbers multiplied by ?i + l.
§ 9. On a Theorem concerning any Power of a + b.
We will now apply this rule to the problem of
counting the terms in (a + &)"; and for clearness' sake,
as usual, we will begin with a particular case, namely
the case in which 91 = 0. We know that here there is
one term whose factors are all a's, and one whose
factors are all b's ; five terms which are the product of
four a's by one b, and five which are the product of one
a and four 6's. It remains only to count the number of
terms made by multiplying three a's by two I's, which
is naturally equal to the number made by multiplying
two a's by three b's. The question is, therefore, Itow
many different arrangements can lie made with three ars
and two b's ?
Here the three a's are all alike, and the two b's are
alike. To solve the problem we shall have to think of
them as different ; let us therefore replace them for the
present by capital letters and small ones. How many
different arrangements can be made with three capital
letters ABC and two small ones tie?
In this question the capital letters are to be con
sidered as equivalent to each other, and the small
letters as equivalent to each other; so that the arrange
ment A B C a" e counts for the same arrangement as
CABeo*. Every arrangement of capitals and smalls
is one of a group of 6 x 2 = 12 equivalent arrangements;
for the 3 capitals may be arranged among one
another in 113, = 6 ways, and the 2 smalls may be
arranged in 112, = 2. ways. Now it is clear that by
28 THE COMMON SENSE OF THE EXACT SCIENCES.
taking all the arrangements in respect of capital and
small letters, and then permuting the capitals among
themselves and the small letters among themselves, we
shall get the whole number of arrangements of the
five letters A B C d e ; namely IT5 or 1 20. But since
each arrangement in respect of capitals and smalls is
here repeated twelve times, and since 12 goes into 120 ten
times exactly, it appears that the number we require is
ten. Or the number of arrangements of three a's and
two &'s is 115 divided by 113 and 112.
The arrangements are in fact —
bbaaa, babaa, baaba, baadb
abbaa, ababa, abaab
aabba, aabab
aaabb
The first line has a& at the beginning, and there are
four positions for the second b ; the next line has a b in
the second place, and there are three new positions for
the other b, and so on. We might of course have ar
rived at the number of arrangements in this particular
case by the far simpler process of direct counting,
which we have used as a verification ; but the advantage
* D
of our longer process is that it will give us a general
formula applicable to all cases whatever.
Let us stop to put on record the result just obtained ;
viz. we have found that
(a + 6)5 = a5 + 5a4& + 10a3Z>2 + 10a2&3 + 5a&4 + 65.
Observe that 1 + 5 + 10 + 10 + 5 + 1 = 32, that is, we
have accounted for the whole of the 32 terms which
would be in the last line of the tree appropriate to this
case.
We may now go on to the solution of our general
problem. Suppose that p is the number of a's and j is
NUMBER. 29
ihe number of 6's which, are multiplied together in a
3ertain term ; we want to find the number of possible
arrangements of these p a's and q 6's. Let us replace
;hem for the moment by p capital letters and q small
ones, making p + q letters altogether. Then any ar
rangement of these in respect of capital letters and
small ones is one of a group of equivalent arrangements
jot by permuting the capitals among themselves and
the small letters among themselves. Now by per
muting the capital letters we can make lip arrange
ments, and by permuting the small letters Uq ar
rangements. Hence every arrangement in respect of
capitals and smalls is one of a group of Tip x Yiq
equivalent arrangements. Now the whole number of
arrangements of the p + q letters is IT (p + q) ; and, as
we have seen, every arrangement in respect of capitals
and smalls is here repeated Yip x Tlq times. Conse
quently the number we are in search of is got by di
viding IT (p + q) by lip x Uq. This is written in the
form of a fraction, thus : —
n (p + g)
Up . 113 '
although it is not a fraction, for the denominator always
divides the numerator exactly. In iact, it would be
absurd to talk about half a quarter of a way of arranging
letters.
We have arrived then at this result, that the number
of ways of arranging p a's and q b's is
n (p + g]
Up . ilq '
This is also (otherwise expressed) the number of ways
of dividing p + q places into p of one sort and q of
30 THE COMMON SEXSE OF THE EXACT SCIENCES.
another ; or again, it is the number of ways of selectin"
p things out of p + q things.
Applying this now to the expression of (« + &)", \v(
find that each of our other terms is of the form
Up.Uq
10
where p + q = n; and that we shall get them all by|
giving to q successively the values 1, 2, 3, &c., and to]
p the values got by subtracting these from n. For
example, we shall find that
The calculation of the numbers may be considerably
shortened. Thus we have to divide 1.2.3.4.5.6 by
1.2.3.4; the result is of course 5 . 6. This has to
be further divided by 2, so that we finally get 5 . 3 or
15. Similarly, to calculate
IT6
113. Q3'
we have only to divide 4.5.6 by 1 . 2 . 3 or 6, and we
get simply 4 . 5 or 20.
To write down our expression for (a + b)n we re
quire another piece of shorthand. We have seen that
it consists of a number of terms which are all of the
form
Tin aP,q
Hp.Uq a
but which differ from one another in having for p and
q different pairs of numbers whose sum is n. Now just
NUMBER. 31
as we used the Greek letter II for a product so we use
the Greek letter S (Sigma) for a sum. Namely, the
sum of all such terms will be written down thus : —
^ J.-LTZ pi q
n „ TT , ' \-f -I -I *
Now we may very reasonably include the two
extreme terms a" and bn in the general shape of these
terms. For suppose we made p = n and q = 0, the
corresponding term would be : —
Tin
lln . Ho
and this will be simply a if 110 = 1 and 6° = 1. Of
course there is no sense in ' the product of the first no
numbers ' ; but if we consider the rule
n (n + 1) = (n + 1) Tin,
which holds good when n is any number, to be also
true when n stands for nothing, arid consequently
n + 1 = 1, it then becomes
ni = no,
and we have already seen reason to make 111 mean 1.
Next if we say that V1 means the result of multiplying
1 by b q times, then 6° must mean the result of multi
plying 1 by b no times, that is, of not multiplying it at
all ; and this result is 1.
Making the-n these conventional interpretations,
we may say that
it being understood that p is to take all values from n
down to 0, and q all values from 0 up to n.
This result is called the Binomial Theorem, and was
originally given by Sir Isaac Newton. An expression
32 THE COMMON SEXSE OF THE EXACT SCIEXCES.
containing two terms, like a + 6, is sometimes callec
binomial; and the name Binomial Theorem is an abbre-fci
viation for theorem concerning any power of a binomial}
expression.
. I
§ 10. On Operations winch appear to "be without Meaning.
We have so far considered the operations by which,!
when two numbers are given, two others can be deter- j
mined from them.
First, we can add the two numbers together and get '
their sum.
Secondly, we can multiply the two numbers together
and get their product.
To the questions what is the sum of these two
numbers, and what is the product of these two numbers,
there is always an answer. But we shall now consider
questions to which there is not always an answer.
Suppose that I ask what number added to 3 will
prodiice 7. I know, of course, that the answer to this
is 4, and Lhe operation of getting 4 is called subtracting
3 from 7, and we denote it by a sign and write it
7-3;= 4.
But if I ask, what number added to 7 will make 3,
although this question seems good English when ex
pressed in words, yet there is no answer to it ; and if I
write down in symbols the expression 3 — 7,1 am asking
a question to which there is no answer.
There is then an essential difference between adding
and subtracting, for two numbers always have a sum.
If I write down the expression 3 + 4, I can use it
as meaning something, because I know that there is
a number which is denoted by that expression. But if
I write down the expression 3 — 7, and then speak of it
NUMBER. 33
as meaning something, I shall be talking nonsense,
because I shall have put together symbols the realities
corresponding to which will not go together. To the
question, what is the result when one number is taken
From another, there is only an answer in the case
where the second number is greater than the first.
In the same way, when I multiply together two
numbers I know that there is always a product, and
am therefore free to use such a symbol as 4 x -5,
Because I know that there is some number that is
denoted by it. Cut I may now ask a question ; I
may say, What number is it which, being multiplied
Dy 4, produces 20 ? The answer I know in this case
.s 5, and the operation by which I get it is called
dividing 20 by 4. This is denoted again by a symbol,
20-T-4 = 5.
But suppose I say divide 21 by 4. To this there is
no answer. There is 110 number in the sense in which
we are at present using the word — that is to say, there
s no whole number — which being multiplied by 4 will
produce 21 : and if I take the expression 21-^4, and
speak of it as meaning something, I shall be talking
nonsense, because I shall have put together symbols
whose realities will not go together.
The things that we have observed here will occur
again and again in mathematics : for every operation
that we can invent amounts to asking a question,
and this question may or may not have an answer
according to circumstances.
If we write down the symbols for the answer to the
question in any of those cases where there is no answer
and then speak of them as if they meant something, we
shall talk nonsense. But this nonsense is not to be
thrown away as useless rubbish. We have learned by
34 THE COMMON SENSE OF THE EXACT SCIENCES.
very long and varied experience that nothing is mc
valuable than the nonsense which we get in this way
only i is to be recognised as nonsense, and by mean
of that recognition made into sense.
We turn the nonsense into sense by givino- a nev
meaning to the words or symbols which shall enable th,
ion to have an answer that previously had nc
answer.
Let us now consider in particular what meaninc, we
can glve to our symbols so as to make sense out of°the
.t present nonsensical expression, 3-7.
§ 11. Steps.
The operation of adding 3 to 5 is written 5 + 3
and the result is 8. We may here regard the +3 as
L way of stepping from 5 to 8, and the symbol +3
may be read in words, step forward three.
In the same way, if we subtract 3 from 5 and get 2
' write the process symbolically 5-3 = 2, and the
-3 may be regarded as a step from 5 to 2
the former step was forward this is backward, and
wemay accordingly read - 3 in words, step
A step is always supposed to be taken from -
number which is large enough to make sense of the
'It. This restriction does not affect steps forward,
because from any number we can step forward as far a
we like ; but backward a step can only be taken from
umbers which are larger than the step itself
The next thing we have to observe about steps is
hat when two steps are taken in succession from any
number, it does not matter which of them comes first
the two steps are taken in the same direction this is
!ear enough. +3 + 4, meaning step forward 3 and
NUMBER. 35
then step forward 4, directs us to step forward by
the number which is the sum of the numbers in the
two steps; and in the same way —3 — 4 directs us to
step backward the sum of 3 and 4, that is 7.
If the steps are in opposite directions, as, for
xample, +3 — 7, we have to step forward 3 and
then backward 7, and the result is that we must step
backwards 4. But the same result would have been
attained if we first stepped backward 7 and then
forward 3. The result, in fact, is always a step which
is in the direction of the greater of the twTo steps, and
is in magnitude equal to their difference.
"We thus see that when two steps are taken in suc
cession they are equivalent to one step, which is inde
pendent of the order in which they are taken.
We have now supplied a new meaning for our
symbols, which makes sense and not nonsense out of
;he symbol 3 — 7. The 3 must be taken to mean +3,
;hat is, step forward 3 ; the — 7 must be taken to mean
step backward 7, and the whole expression no longer
means take 7 from 3, but add 3 to and then subtract
7 from any number which is large enough to make
sense of the result. And accordingly we find that the
result of this operation is —4, or, as we may write it.
3-7 = -4.
From this it follows by a mode of pi'Oof precisely
nnulogousto that which we used in the case of multi-
)lication, that any number of steps taken in succession
lave a resultant which is independent of the order in
which they are taken, and we may regard this rule .is
an extension of the rule already proved for the addition
of numbers.
A step may be multiplied or taken a given number
of times, for example, 2( — 3) =— tJ ; that is to say,
36 THE COMMON SENSE OF THE EXACT SCIENCES.
if two backward steps of 3 be possible, they are equiva
lent to a step backwards of (5.
In this operation of multiplying a step it is cleai
that what we do is to multiply the number which is
stepped, and to retain the character of the step. On
multiplying a step forwards we still have a step for
wards, and on multiplying a step backwards we still
have a step backwards.
This multiplying may be regarded as an operation
by which we change one step into another. Thus in
the example we have just considered the multiplier 2
changes the step backwards 3 into the step backwards
6. But this operation, as we have observed, will onl;
change a step into another of the same kind, and th(
question naturally presents itself, Is it possible to fin<
an operation which shall change a step into one of
different kind ? Such an operation we should naturally
call reversal. We should say that a step forwards
reversed, when it is made into a step backwards ; and a
step backwards is reversed when it is made into a stej
forwards.
If we denote the operation of reversal by the letter
r, we can, by combining this with a multiplication
change —3 into +6, a step backwards 3 into a ste]
forwards 6 ; viz. we should have the expression
r2 ( — 3 ) = +6. Now the operation, which is performed on
one step to change it into another, may be of two kinds :
either it keeps a step in the direction which it originally
had, or it reverses it. If to make things symmetrical
we insert the letter k when a step is kept in its
original direction, we may write the equation &2( — 3)
= — 6 to express the operation of simply multiplying.
Of course it is possible to perform on any given
gtep a succession of these operations. If I take the
NUMBER. 37
tep + 4, treble it, and reverse it, I get —12. If I
Louble this and keep it, I get — 24, and this may
>e written, &2(r3)( + 4) = —24. But this is equal
o rG( + 4), which tells us that the two successive opera-
ions which we have performed on this step, trebling
ind reversing it, doubling and keeping it, are equiva-
ent to the single operation of multiplying by 0 and
e versing it. It is clear also that whatever step we had
aken the two first operations performed successively
ire always equivalent to the third, and we may thus
write the equation 7r2(r3) = rG.
Suppose however we take another step and treble
t and reverse it, and then double it and reverse it
.gain ; we should have the result of multiplying it by
ix and keeping its direction unchanged.
This may be written r2(r3) = k . G.
If we compare the last two formula} with those
•which we previously obtained, viz. /i2( — 3) = — G and
r2( — 3) = +6, we shall see that the two sets are alike,
except that in the one last obtained k and r are written
instead of -I- and — respectively.
The two sets however express entirely different
things. Thus, taking the second formula? of either set
on the one hand, the statement is, Double and reverse
the step backward 3, and you have a step forward G ;
on the other hand, Treble and reverse and then double
and reverse any step whatever, and you have the effect
of sextupling and keeping the step. We shall find that
this analogy holds good in general, that is, if we write
down the effect of any number of successive operations
performed upon a step, there will always be a correspond
ing statement in which this stepping is replaced by ;m
operation ; or we may say, any operation which convert s
one step into another will also convert one operation into
38 THE COMMON SENSE OF THE EXACT SCIENCES.
another where the converted operation is a multiplying
by the number expressing the step and a keeping 01
reversing according as th*e step is forward or backward
§ 12. Extension of the Meaning of Symbols.
We now proceed to do something which must appa
rently introduce the greatest confusion, but which, on
the other hand, increases enormously our powers.
Having two things which we have so far quite
rightly denoted by different symbols, and finding that
we arrive at results which are uniform and precisely
similar to one another except that in one of them one
set of symbols is used, in the other another set, we alter
the meaning of our symbols so as to see only one set
instead of two. We make the symbols + and — mean
for the future what we have here meant by k and r,
viz. keep and reverse. We give them these meanings
in addition to their former meanings, and leave it to the
context to show which is the right meaning in any
particular case. Thus, in the equation ( — 2) ( — .3) = + 6
there are two possible meanings ; the —3 and +6, may
both mean steps, in this case the statement is : Double
and reverse the step backwards of 3 and you get the
step forward 6. But the —3 and the +6 may also
mean not steps but operations, and in this case the
meaning is triple and reverse and then double and
reverse any step whatever, and you get the same result
as if you had sextupled and kept the step.
Let us now see what the reason is for saying that
these two meanings can always exist together. Let us
first of all take the second meaning, and frame a rule
for finding the result of any number of successive
operations.
NUMBER. 39
First, the number which is the multiplier in the
result must clearly be the product of all the numbers
n the successive operations.
Next, every pair of reversals cancel one another, so
ihat, if there is an even number of them, the result
must be an operation of retaining.
This then is the rule : Multiply together the
numbers in the several operations, prefixing- to them
if there is an even number of minus or reversing
operations, prefixing — if there is an odd number.
In the next place, suppose that many successive
operations are performed upon a step. The number
n the resulting step will clearly be the product of all
-he numbers in the operations and in the original step.
If there is an even number of reversing operations,
lie resulting step will be of the same kind as the
original one ; if an odd number, of the opposite
find. Now let us suppose that the original step
vere a step backwards ; then if there is an even number
of reversing operations, the resulting step will also be a
step backwards. But in this case the number of (— )
signs, reckoned independently of their meaning, will be
odd ; and so the rule coincides with the previous one.
If an odd number of reversing operations is per
formed on a negative step, the result is a positive step.
But here the whole number of ( — ) signs, irrespective
of their meaning, is an even number ; and the result
again agrees with the previous one.
In all cases therefore by using the same symbols
to mean either a 'forward' and a 'backward' step
respectively, or ' keep ' and ' reverse ' respectively, we
shall be able to give to every expression two interpreta
tions, and neither of these will ever be untrue.
In the process of examining this statement we have
40 THE COMMON SEXSE OF THE EXACT SCIENCES.
shown by the way that the result of any number of
successive operations on a step is independent of the
order of them. For it is always a step whose magnitude
is the product of the numbers in the original step and
in the operations, and whose character is determined
by the number of reversals.
§ 13. Addition and Multiplication of Operations.
We may now go on to find a rule which connects
together the multiplication and the addition of steps.
If I multiply separately the steps +3 and —7 by 4,
and then take the resultant of the two steps which I so
obtain, I shall get the same thing as if I had first
formed the resultant of +3 and —7, and then multi
plied it by 4. In fact, +12 - 28 '= --16. which is
4( — 4) . This is true in general, and it obviously
amounts to the original rule that a set of things comes
to the same number in whatever order we count them.
Only that now some of the counting has to be done
backwards and some again forwards.
But now, besides adding together steps, we may
also in a certain sense add together operations. It
seems natural to assume at once that by adding toge
ther -f 3 and — 7 regarded as operations, we must needs
get the operation —4. It is very important not to
assume anything without proof, and still more import
ant not to use words without attaching a definite
meaning to them.
The meaning is this. If I take any step whatever,
treble it without altering its character, and combine
the result with the result of multiplying the original
step by 7 and reversing it, then I shall get the same
result as if I had multiplied the original step by 4 and
NUMBER. 4 1
eversed it. This is perfectly true, and we may see it
o be true by, as it were, performing our operations in
he form of steps. Suppose I take the step + 5, and
rant to treble it and keep its chai-acter unchanged. I
;an do this by taking three steps of five numbers each in
he same direction (viz. the forward direction) as the
)riginal step was to be taken. Similarly, if I want to
nultiply it by — 7, this means that I must take 7 steps
>f five numbers each in the opposite or backward direc-
ion. Then finally, what I have to do is to take three
teps forwards and seven steps backwards, each of these
teps consisting of five numbers ; and it appears at once
hat the result is the same as that of taking 4 steps
>ackwards of five numbers each.
We have thus a definition of the sum of two
Derations ; and it appears from the way in which we
lave arrived at it that this sum is independent of the
order of the operations.
We may therefore now write the formuke : —
a + b = b + a
a (b + c) = ab + ac
(a + l)c — ac + be
ab = ba,
and consider the letters to signify operations performed
upon steps. In virtue of the truth of these laws the whole
of that reasoning which we applied to finding a power
of the sum of two numbers is applicable to the finding
of a power of the sum of two operations. If it did not
take too much time and space, we might go through it
again, giving to all the symbols their new meanings.
It is worth while, perhaps, by way of example, to
explain clearly what is meant by the square of the sum
of two operations.
42 THE COMMON SENSE OF THE EXACT SCIENCES.
We will take for example, +5 and —3.
The formula tells us that ( + 5 — 3)2 is equal to
( + 5)2 + (_3)2 + 2( + 5)(-3). This means that if AVC
apply to any step twice over the sum of the operations
-f 5 and — 3, that is to say, if we multiply it by 5 anci
keep its direction, and combine with this step the resull
of multiplying the original step by 3 and reversing it
and then apply the same process to the result so obtained
we shall get a step which might also have been arrived
at by combining together the following three steps : —
First, the original step twice multiplied by 5.
Secondly, the original step twice multiplied by £
and twice reversed ; that is to say, unaltered in
direction.
Thirdly, twice the result of tripling the original step
and reversing it, and then multiplying by 5 and retain
ing the direction.
§ 14. Division of Operations.
We have now seen what is meant by the multipli
cation of operations ; let us go on to consider whal
sort of question is asked by division.
Let. us take for example the symbolic statemenl
— 3( + 5) = — 15 ; and let us give it in the first place
the meaning that to triple and reverse the step forwarc
5 gives the step backward 15. We may ask two
questions upon this statement. First, What operation
is it which, being performed on the step forwards 5, wil
give the step backwards 15? The answer, of course,
is triple and reverse. Or we may ask this question
What step is that, which, being tripled and reversed,
will give the step backward 15? The answer is, Step
forwards 5. But we have only one word to describe
the process by which we get the answer in these two
NUMBER. 43
jases. In the first case we say that we divide the step
— 15 by the step +5; in the second case we say we
livide the step —15 by the operation —3.
The word divide thus gets two distinct meanings.
But it is very important to notice that symbolically the
inswer is the same in the two cases, although the
interpretation to be given to it is different.
The step —15 may be got in two ways ; by tripling
and reversing the forward step + 5, or by quintupling
the backward step —3. In symbols,
(-3) ( + 5) = (+5) (-3) = -15.
Hence the problem, Divide — 15 by —3 may moan
ither of these- two questions : What step is that which,
jeing tripled and reversed, gives the step —15? Or,
"What operation is that which, performed on the step
— 3, gives the step —15? The answer to the first
question is, the step -f 5 ; the answer to the second is
jhe operation of quintupling and retaining direction,
that is, the operation +5. So that although the word
divide, as we have said, gets two distinct meanings, yet
the two different results of division are expressed by
the same symbol.
In general we may say that the problem, Divide
the step a by the step b, means, Find the operation (if
any) which will convert b into a. But the problem,
Divide the step a by the operation b, means, Find the
step (if any) which b will convert into «. In both cases,
however, the process and the symbolic result are the
same. We must divide the number of a by the number
of b, and prefix to it + if the signs of a and b are alike,
— if they are different.
We may also give to our original equation
(-3) x ( + 5) = -15
44 THE COMMON SEXSE OF THE EXACT SCIENCES.
its other meaning, in which both —3 and +5 are ope
rations, and — 15 is the operation which is equivalent
to performing one of them after the other. In this case
the problem, Divide the operation —15 by the operation
— 3 means, Find the operation which, being succeeded
by the operation — 3, will be equivalent to the operation
— 15. Or generally, Divide the operation a by the
operation &, means, Find the operation which, being
succeeded by 6, will be equivalent to a.
Now it is worth noticing that the division of step
by step and the division of operation by operation, have
a certain likeness between them, and a common differ
ence from the division of step by operation. Namely,
the result of dividing a by b, or, as we may write it,
-, when a and 6 are both steps or both operations, is
an operation which converts 6 into a. This we may
write in shorthand,
a 7
- . i = «.
But when a is a step and 6 an operation, the result of
division is a step on which the operation 6 must be
performed to convert it into a ; or, in shorthand,
7 a
b . — = a.
o
The fact that the symbolic result is the same in the
two cases may be stated thus : —
and in this form we see that it is a case of the commu
tative law. So long, then, as the commutative law is
true, there is no occasion for distinguishing symboli
cally between the two meanings. But, as we shall see
NUMBER. 45
oy-and-by, there is occasion to deal with other kinds
of steps and operations in which the commutative law
•loes not hold ; and for these a convenient notation has
>een suggested by Professor Cayley. Namely, — means
:he operation wrhich makes b into a • but - - repre-
°\
sents that which the operation b will convert into a. So
fchat-
J . b = a, but b . —- = a.
I b }>
[t is however convenient to settle beforehand that when
ever the symbol - is used without warning it is to have
b
the first meaning— namely, the operation -which makes
!> into a,
§ 15. General Results of our Extension of Terms.
It will be noticed that we have hereby passed from
the consideration of mere numbers, with which we
gan, to the consideration first of steps of addition or
subtraction of number from number, and then of
operations of multiplying and keeping or multiplying
and reversing, performed 011 these steps ; and that we
have greatly widened the meaning of all the words that
we have employed.
To addition, which originally meant the addition of
two numbers, has been given the meaning of a combina
tion of steps to form a resultant step equivalent in effect
to taking them in succession.
To nntltijilii-iifinii, which was originally applied to
two numbers only, has been given the meaning of a
combination of operations upon steps to form a resultant
operation equivalent to their successive performance.
46 THE COMMON SENSE OF THE EXACT SCIENCES.
We have found that the same properties \vhicl
characterise the addition and multiplication of numbers
belong also to the addition and multiplication of steps
and of operations. And it was this very fact of the
similarity of properties which led us to use our ol(
words in a new sense. We shall find that this same
process is carried on in the consideration of those
other subjects which lie before us ; but that the precise
similarity which we have here observed in the pro
perties of more simple and more complex operations
will not in every case hold good ; so that while this
gradual extension of the meaning of terms is perhaps
the most powerful instrument of research which has
yet been used, it is always to be employed with a cau
tion proportionate to its importancet
47
CHAPTER II.
SPACE.
§ 1. Boundaries take up no Room.
GEOMETRY is a physical science. It deals with the
sizes and shapes and distances of things. Just as we
have studied the number of things by making a simple
and obvious observation, and then using this over and
over again to see where it would bring us ; so we shall
study the science of the shapes and distances of things
by making one or two very simple and obvious obser
vations, and then using these over and over again, to
see what we can get out of them.
The observations that we make are : —
First, that a thing may be moved about from one
place to another without altering its size or shape.
Secondly, that it is possible to have things of the
same shape but of different sizes.
Before we can use these observations to draw any
exact conclusions from them, it is necessary to consider
rather more precisely what they mean.
Things take up room. A table, for example, takes
up a certain part of the room where it is, and there is
another part of the room where it is not. The tiling
makes a difference between these two portions of space.
Between these two there is what we call the surface
of the table.
We may suppose that the space all round the table
48 THE COMMON SEXSE OF THE EXACT SCIENCES.
is filled with air. The surface of the table is thei
something just between the air and the wood, whicl
separates them from one another, and which is neithei
the one nor the other.
It is a mistake to suppose that the surface of th(
table is a very thin piece of wood on the outside of it
We can see that this is a mistake, because any reason
which led us .to say so, would lead us also to say that
the surface was a very thin layer of air close to the
table. The surface in fact is common to the wood and
to the air, and takes up itself no room whatever.1
Part of the surface of the table may be of one colou
and part may be of another.
On the surface of this sheet of paper there is drawn
a round black spot. We call the black part a circle
FIG. 1.
^Jt divides the surface into two parts, one where it is an<
one where it is not.
This circle takes up room on the surface, although
surface itself t&ke|*wp no room in space. We ar
5 led to considerTwrl^lifferent kinds of room ; space
room, in which solid bodies are, and in which the
'i move about ; an9.i^Afte-room, which may be regarde
•£kth
££***
•T*1'01
1 It is certain that however smooth a na.t.i,ral surface may appear to be,
it could be magnified to roughness. Hence, in the case of the surface of
the table and the air, it would seem probable that there is a layer in which
particles of wood and air are mingled. The boundary in this case of air
and table would not be what we ' see and feel ' (cf. p. 48), nor would it
correspond to the surface of the geometer. We are, I think, compelled to
consider the surface of the geometer as an 'idea or imaginary conception,'
drawn from the apparent (not leal) boundaries of physital objects, such as
the writer is describing. Strongly as I feel the ideal nature of geometrical
conceptions in the exact sciences, I have thought it unadvisable to alter
the text. The distinction is made by Clifford himself (Essays, I. pp. 306-
321).- K.P.
SPACE. 49
rom two different points of view. From one point of
iew it is the boundary between two adjacent portions
f space, and takes up no space-room whatever. From
lie other point of view it is itself also a kind of room
ehieh may be taken up by parts of it.
These parts in turn have their boundaries.
Between the black surface of the circle and the
srhite surface of the paper round it there is a line, the
ircmnference of the circle. This line is neither part of
he black nor part of the white, but is between the two.
t divides one from the other, and takes up no surface-
oom at all. The line is not a very thin strip of surface,
ny more than the surface is a very thin layer of solid.
Anything which led us to say that this line, the
oundary of the black spot, was a thin strip of black,
vould also lead us to say that it was a thin strip of white.
We may also divide a line into two parts. If the
aper with this black circle upon it were dipped into
ater so that part of the black circle were sub
merged, then the line surrounding it would be partly in
;he water and partly out.
The submerged part of the line takes up room on it.
[t goes a certain part of the way round the circum
ference. Thus we have to consider line-room as well as
space-room and surface-room. The line takes up
absolutely no room on the surface; it is merely the
boundary between two adjacent portions of it. Still
less does it take up any room in space. And yet it has
a certain room of its own, which may be divided
into parts, and taken up or tilled by those parts.
E
50 THE COMMON SENSE OF THE EXACT SCIENCES.
These parts again have boundaries. Between
submerged portion of the circumference and the othei
part there are two points, one at each end. These
points are neither in the water nor out of it. They are
in the surface of the water, just as they are in the sur
face of the paper, and on the boundary of the black spot.
Upon this line they take up absolutely no room at all.
A point is not a very small length of the line, any
more than the line is a very thin strip of surface. It is
a division between two parts of the line which are next
one another, and it takes up no room on the line at all.
The important thing to notice is that we are no1
here talking of ideas or imaginary conceptions, but
only making common-sense observations about matters
of every-day experience.
The surface of a thing is something that we con
stantly observe. We can see it and feel it, and it is a mere
common-sense observation to say that this surface is com
mon to the thing itself and to the space surrounding it.
A line on a surface which separates one part of the
surface from another is also a matter of every-day
experience. It is not an idea got at by supposing a
string to become indefinitely thin, but it is a thing
given directly by observation as belonging to both por
tions of the surface which it divides, and as being there
fore of absolutely no thickness at all. The same may be
said of a point. The point which divides the part of
our circumference which is in water from the part which
is out of water is an observed thing. It is not an idea
got at by supposing a small particle to become smaller
and smaller without any limit, but it is the boundary
between two adjacent parts of a line, which is the
boundary between two adjacent portions of a surface,
which is the boundary between two adjacent portions of
SPACE. 51
pace. A point is a thing which we can see and know,
tot an abstraction which we build up in our thoughts.
When we talk of drawing lines or points on a sheet
•f paper, we use the language of the draughtsman and
lot of the geometer. Here is a picture of a cube
•epresented by lines, in the draughtsman's sense.
Sach of these so-called ' lines ' is a black streak of
winter's ink, of varying breadth, taking up a certain
FIG. 3.
imount of room on the paper. By drawing such ' lines '
lufficiently close together, we might entirely cover up
is large a patch of paper as we liked. Each of these
itreaks has a line on each side of it, separating the
)lack surface from the white surface ; these are true
geometrical lines, taking up no surface-room whatever.
Millions of millions of them might be marked out
Between the two boundaries of one of our streaks, and
Between every two of these there would be room for
millions more.
Still, it is very convenient, in drawing geometrical
igures, to represent lines by black streaks. To avoid
ill possible misunderstanding in this matter, we shall
make a convention once for all about the sense in
which a black streak is to represent a line. When the
streak is vertical, or comes straight down the page, like
this | , the line represented by it is its riyld-liand boun
dary. In all other cases the line shall be the upptr
boundary of the streak.
So also in the case of a point. When we try to
represent a point by a dot on a sheet of paper, we
E 2
52 THE COMMON SENSE OF THE EXACT SCIENCES.
make a black patch of irregular shape. The boundary
of this black patch is a line. When one point of this if"
boundary is higher than all the other points, thai
highest point shall be the one represented by the dot.
When however several points of the boundary are ai
the same height, but none higher than these, so thai
the boundary has a flat piece at the top of it, then the
right-hand extremity of this flat piece shall be thej
point represented by the dot.
This determination of the meaning of our figures
is of no practical use. We "lay it down only that the
reader may not fall into the error of taking patches
and streaks for geometrical points and
§ 2. Lengths can be Moved without Change.
Let us now consider what is meant by the first of
our observations about space, viz., that a thing can be
moved about from one place to another without altering
its size or shape.
First as to the matter of size. We measure the size
of a thing by measuring the distances of various points
on it. For example, we should measure the size of a
table by measuring the distance from end to end, or the
distance across it, or the distance from the top to the
bottom. The measurement of distance is only possible
when we have something, say a yard measure or a piece
of tape, which we can carry about and which does not
alter its length while it is carried about. The measure
ment is then effected by holding this thing in the place
of the distance to be measured, and observing what
part of it coincides with this distance.
Two lengths or distances are said to be equal when
the same part of the measure will fit both of them.
SPACE. 53
Tims we should say that two tables are equally broad,
.f we marked the breadth of one of them on a piece of
rape, and then carried the tape over to the other table
ind found that its breadth came up to just the same
nark. Now the piece of tape, although convenient, is
not absolutely necessary to the finding out of this fact.
We might have turned one table up and put it on top
of the other, and so found out that the two breadths
were equal. Or we may say generally that two lengths
or distances of any kind are equal, when, one of them
>eirig brought up close to the other, they can be made
to fit without alteration. But the tape is a thing far
more easily carried about than the table, and so in prac
tice we should test the equality of the two breadths by
measuring both against the same piece of tape. We
ind that each of them is equal to the same length of
;ape ; and we assume thatfat'o lengths which are equal to
the same length are equal to each other. This is equiva-
ent to saying that if our piece of tape be carried
round any closed curve and brought back to its original
position, it will not have altered in length.
How so ? Let us assume that, when not used, our
riece of tape is kept stretched out on a board, with one
?nd against a fixed mark on the board. Then we know
what is meant by two lengths being equal which are
Doth measured along the tape from that end. Now take
;hree tables, A, B, C, and suppose we have measured
and found that the breadth of A is equal to that of B,
and the breadth of B is equal to that of C, then we
say that the breadth of A is equal to that of C. This
means that we. have marked off the breadth of A on
the tape, and then carried this length of tape to B, and
found it fit. Then we have carried the same length
from B to C, and found it fit. In saying that the
54 THE COMMON SENSE OF THE EXACT SCIENCES.
breadth of C is equal to that of A, we assert that 01
taking the tape from C to A, whether we go near B orj
not, it will be found to fit the breadth of A. That is,|
if we take our tape from A to B, then from B to C, andj
then back to A, it will still fit A if it did so at first.
These considerations lead us to a very singular con-j
elusion. The reader will probably have observed that
we have defined length or distance by means of a
measure which can be carried about without changing ]
its length. But how then is this property of the
measure to be tested? We may carry about a yard
measure in the form of a stick, to test our tape with;
but all we can prove in that way is that the two things
are always of the same length when they are in the
same place ; not that this length is unaltered.
The fact is that everything would go on quite as
well if we supposed that things did change in length
by mere travelling from place to place, provided that
(1) different things changed equally, and (2) anything
which was carried about and brought back to its original
position filled the same space.1 All that is wanted is
that two things which fit in one place should also fit in
another place, although brought there by different
paths ; unless, of course, there are other reasons to the
contrary. A piece of tape and a stick which fit one
another in London will also fit one another in New
York, although the stick may go there across the
Atlantic, and the tape via India and the Pacific. Of
course the stick may expand from damp and the tape
may shrink from dry ness ; such non-geometrical cir
cumstances would have to be allowed for. But so far
us the geometrical conditions alone are concerned — the
1 These remarks refer to the geometrical, and not necessarily to all the
physical properties of bodies. — K. P
SPACE.
55
•iere carrying about and change of place — two things
Haich fit in one place will fit in another.
Upon this fact are founded, as we have seen, the
Lotion of length as measured, and the axiom that
jngths which are equal to the same length are equal
one another.
Is it possible, however, that lengths do really
Change by mere moving about, without our knowing it ?
Whoever likes to meditate seriously upon this ques
tion will find that it is wholly devoid of meaning. But
'ihe time employed in arriving at that conclusion will
liot have been altogether thrown away.
§ 3. The Characteristics of Shape.
We have now seen what is meant by saying that a
[thing can be moved about without altering its size ;
namely, that any length Avhich fits a certain measure in
lone position will also fit that measure when both have
been moved by any paths to some other position. Let
rus now inquire what we mean by saying that a thing
can be moved about without altering its shape.
First let us observe that the shape of a thing
(depends only on its bounding surface, and not at all
iupon the inside of it. So that we may always speak
I of the shape of the surface, and we shall mean the
same thing as if we spoke of the shape of the thing.
Fio. 4.
Let us observe then some characteristics of the sur
face of things. Here are a cube, a cylinder, and a sphere.
56 THE COMMON SENSE OP THE EXACT SCIENCES.
The surface of the cube has six flat sides, with edgea
and corners. The cylinder has two flat ends and a
round surface between them ; the flat ends being
divided from the round part by two circular edges.
The sphere has a round smooth surface all over.
We observe at once a great distinction in shape be
tween smooth parts of the surface, and edges, and corners.
An edge being a line 011 the surface is not any part oi
it, in the sense of taking up surface room ; still less is
a corner, which is a mere point. But still we may divide
the points of the surface into those where it is smooth
(like all the points of the sphere, the round and flat parts
of the cylinder,- and the flat sides of the cube), into
points on an edge, and into corners. For convenience, let
us speak of these points respectively as smooth-points,
edge-points, and corner-points. We may also put the
edges and corners together, and call them rough-
points.
Now let us take the sphere, and put it upon a flat
face of the cube (fig. 5). The two bodies will be in con-
Fio. 5.
tact at one point ; that is to say, a certain point on the
surface of the sphere and a certain point on the surface
of the cube are made to coincide with one another and
to be the same point. And these are ooth smooth-points.
Now we cannot move the sphere ever so little without separ
ating thqse points, If we roll it a very little way on the
SPACE. 57
face of the cube, we shall find that a different point of
the sphere is in contact with a different point of the cube.
And the same thing is true if we place the sphere in
contact with a smooth-point on the cylinder (fig. 6).
Next let us put the round part of the cylinder on
the flat face of the cube. In this case there will be
:ontact all along a line. At any point of this line, a
certain point on the surface of the cylinder and a
:ertain point on the surface of the cube have been made
to coincide with one another and to be the same point.
And these are both smooth-points. It is just as true
as before, that we cannot move one of these bodies ever
so little relatively to the other without separating the
FIG. 7.
points of their surfaces which are in contact. If we
roll the cylinder a very little way on the face of the
cube, we shall find that a different line of the cylinder
is in contact with a different line of the cube. All the
points of contact are changed.
Now put the flat end of the cylinder 011 the face of
the cube. These two surfaces fit throughout and make
but one surface ; we have contact, not (as before) at a
point or along a line, but over a surface. Let us fix
58 THE COMMON SENSE OF THE EXACT SCIENCES.
our attention upon a particular point on the flat surface
of the cylinder and the point on the face of the cube
with which it now coincides ; these two being smooth-
points. We observe again, that it is impossible to move
one of these bodies ever so little relatively to the other
without separating these two points.'1
Here, however, something has happened which will
give us further instruction. We have all along sup
posed the flat face of the cylinder to be smaller than
the flat face of the cube. When these two are in con-
1 In all these cases (figs. 5-8) the relative motion spoken of must be
either motion of translation or of tilting; one body might have a spin
about a vertical axis without any separation of these two points. The true
distinction between the contact of smooth-points and of smooth and rough-
points seems to be this : in the former case without separating two points
'..here is only one degree of freedom— namely, spin about an axis normal to
the smooth surfaces at the points in question ; in the latter c=ise there are
at least two (edge-point or smooth-point) and may be an infinite number of
degrees of freedom — namely, spins about two or more axes passing through
the rough-point. The reader will understand these terms better after the
chapter on Motion. — K. P.
SPACE.
59
tact, let the cylinder stand on the middle of the cube,
as in fig. 8, the circle being wholly enclosed by the
square. Then when we tilt the cylinder over we shall
get it into the position of fig. 9. We have already
observed that in this case no smooth-points which were
previously in contact remain in contact. But there are
two points which remain in contact ; for in the tilted
position a point on the circular edge of the cylinder
rests on a point on the face of the cube ; and these two
points were in contact before. We may tilt the cylinder
as much or as little as we like— provided we tilt always
in the same direction, not rolling- the cylinder on its
edge — and these two points will remain in contact.
We learn therefore that when an edge-point is in contact
with a smooth-point, it may be possible to move one of the
two bodies relatively to the other witlwut separating those
two points.
The same thing may be observed if we put the
round or flat surface of the cylinder against an edge
of the cube, or if we put the sphere against an edge of
either of the other bodies. Holding either of them
fast, we may move the other so as to keep the same two
points in contact ; but in order to do this, we must
always tilt in the same direction.
If, however, we put a corner of the cube in contact
with a smooth point of the cylinder, as in tig. 10, we
Fio. 10.
shall find that we can keep these two points in contact
without any restriction on the direction of tilting. We
60 THE COMMON SENSE OF THE EXACT SCIENCES.
may tilt the cube any way we like, and still keep its
corner in contact with the smooth-point of the cylinder.
When we put two edge-points together, it makes a
difference whether the edges are in the same direction
at the point of contact or whether they cross one
another. In the former case we may be able to keep
the same two points in contact by tilting in a particular
direction ; in the latter case we may tilt in any direc
tion. So if a corner is in contact with an edge-point
there is no restriction on the direction of tilting, and
much more if a corner is in contact with a corner.
The upshot of all this is, that in a certain sense all
surfaces are of the same shape at all smooth-points ; for
when we put two smooth-points in contact, the surfaces
so fit one another at those points that we cannot move
one of them relatively to the other without separating
the points.1
It is possible for two edges to fit so that we cannot
move either of the bodies without separating the points
in contact. For this it is necessary that one of them
should be re-entrant (that is, should be a depression in
the surface, not a projection), as in fig. 11 ; and here
FIG. 11.
we can see the propriety of saying that the two surfaces
are of the same shape at a point where they fit in
this way. The body placed in contact with the cube
: See, however, the footnote, p. 58. — K. P.
SPACE. 61
is formed by joining together two spheres from which
pieces have been sliced off. If only very small pieces
aave been sliced off, the re-entrant edge will be very
sharp, and it will be impossible to bring the cube-edge
into contact with it (fig. 12) ; if nearly half of each
FIG. 12. FIG. 13. .b'iG. 14.
sphere has been cut off the re-entrant edge will be wide
open, and the cube will rock in it (fig. 13). There is
jlearly one intermediate form in which the two edges
will just fit (fig. 14) ; contact at the edge will be
Dossible, but no rocking. Now in this case, although
one edge sticks out arid the other is a dint, we may
still say that the two surfaces are of the same shape
at the edge. For if we suppose our twin-sphere
)ody to be made of wood, its surface is not only sur
face of the wood, but also surface of the surrounding
air. And that which is a dint or depression in the
wood is at the same time a projection in the air. In
just the same way, each of the projecting edges and
corners of the cube is at the same time a dint or
depression in the air. But the surface belongs to one
as much as the other ; it knows nothing of the differ
ence between inside and outside ; elevation and depres
sion are arbitrary terms to it. So in a thin piece of
embossed metal, elevation on one side means depression
on the other, and vice versa ; but it is purely arbitrary
62 THE COMMON SENSE OF THE EXACT SCIENCES.
which side we consider the right one. (Observe thai
the thin piece of metal is in no sense a representation
of a surface ; it is merely a thin solid whose two surfaces
are very nearly of the same shape.)
Thus we see that the edge of wood in our cube is
of the same shape as the edge of air in the twin-sphere
solid ; or, which is the same thing, that the two surfaces
are of the same shape at the edge.
Now this twin-sphere solid is a very convenient one
because we can so modify it as to make an edge of any
shape we like. Hitherto we have supposed the slices
cut off to be less than half of the spheres ; let us now
fasten together these pieces, and so form a solid with a
projecting edge, as in fig. 15. The two solids so formed,
one with a re-entrant edge from the larger pieces, the
other with a projecting edge from the smaller pieces,
will be found always to have their edges of the same
shape, or to fit one another at the edge in the sense
just explained.
FIG. 15.
Now suppose that we cut our spheres very nearly in
half. (Of course they must always be cut both alike,
or the flat faces would not fit together.) Then when
we join together the larger pieces and the smaller
pieces, we shall form solids with very wide open edges.
The projecting edge will be a very slight ridge, and the
re-entrant one a very slight depression.
If we now go a step further, and cut our spheres
actually in half, of course each of the new solids will
be again a sphere ; and there will be neither ridge nor
SPACE.
63
tij epression ; the surfaces will be smooth all over. But
tigj/e have arrived at this result by considering a project-
(vi) (vii)
FIG. 16.
ing edge as gradually widening out until the ridge dis
appears, or by considering a re-entrant edge as gradually
widening out until the dint disappears. Or we may
suppose the projecting edge to go on widening out till
it becomes smooth, and then to turn into a re-entrant
edge. We might represent this process to the eye by
putting into a wheel of life a succession of pictures like
that in fig. 16, and then rapidly turning the wheel. We
should see the two spheres, at first separate, coalesce
into a single solid in (ii) and (iii),then form one sphere as
at (iv), then contract into a smaller and smaller lens at
(v), (vi), (vii). The important thing to notice is that the
single sphere at (iv) is a step in the process ; or, what
is the same thing, that a smooth-point is a particular case
of an edge-point coming between the projecting and the re
entrant edges. As being this particular case of the
edge-point, we say that at all smooth-points the sur
faces are of the same shape.
§ 4. The Characteristics of Surface Boundaries.
Remarks like these that we have made about solid
bodies or portions of space may be made also about
64 THE COMMON SEXSE OF THE EXACT SCIENCES.
portions of surface. Only we cannot now say that th*
shape of a piece of surface depends wholly on that of
the curve which bounds it. Still the only thing that
remains for us to consider is the shape of the boundary,!
because we have already discussed (so far as we profit-]
ably can at present) the shape of the included surface.
We shall find it useful to restrict ourselves still]
further, and only consider those boundaries which have
no rough points of the surface in them. Thus on the
surface of the cube we will only consider portions which
are entirely included in one of the plane faces ; on the
surface of the cylinder, only portions which are entirely
included in one of the flat faces, or in the curved part,
or which include one of the flat faces and part of the
curved portion.
This being so, the characteristics which we have to
remark in the boundaries of pieces of surface may be
sufficiently studied by means of figures drawn on paper.
We may bend the paper to assure ourselves that the
same general properties belong to figures on a cylinder,
and to make our ideas quite distinct it is worth while
to draw some on a sphere or other such surface.
In fig. 17 are some patches of surface; a square, a
three-cornered piece, and two overlapping circles. For
FIG. 17.
distinctness, the part where the circles overlap is left
white, the rest being made black.
Attending now specially to the boundary of these
patches, we observe that it consists of smooth parts and
of corners or angles. Some of these corners project
SPACE. 65
and some are re-entrant. The pieces of surface are not
solid moveable tilings like the portions of space we
considered before, but we can in a measure imitate our
previous experiments by cutting out the figures with a
penknife, so as to leave their previous positions marked
by the holes. We shall then find, on applying the cut
out pieces to one another, or to the holes, that at all
smooth-points the boundaries fit one another in a cer
tain sense. Namely, if we place two smooth-points in
contact we cannot roll one figure on the other without
separating these points ; whereas if we place a sharp-
point (or angle) on a smooth-point we can roll one figure
on the other without separating the points. If we
attempt to put two angles together without letting the
figures overlap, the same things may happen that we
found true in the case of the edges of solid bodies.
Suppose, for example, that we try to put an angle of the
square into one of the re-entrant angles of the figure
made by the two overlapping circles. If the re-entrant
angle is too sharp, we shall not be able to get it in at
all; this is the case of fig. 12. If it is wide enough,
the square will be able to rock in it ; this is the case of
fig. 13. Between these two there is an intermediate
case in which one angle just fits the other ; actual
, contact takes place, and no rocking is possible. In
' this case we say that the two angles are of the same
• shape, or that they are equal to one another.
From all this we are led to conclude that shape- is a
matter of angles, and that identity of shape depends on
equality of angle. We dealt with the sixe of a body by
considering a simple case of it, viz. length or distance,
and by measuring a sufficient number of lengths in dif
ferent directions could find out all that is to be known
about the size of a body. It is, indeed, also true that a
F
66 THE COMMON SENSE OF THE EXACT SCIENCES.
knowledge of all the lengths which can be measured
in a body would carry with it a knowledge of its shape;
but still length is not in itself an element of shape.
That which does the same for us in regard to shape
that length does with regard to size, is angle. In other
words, just as we say that two bodies are of the same
size if to any line that can be drawn in the one there
corresponds an exactly equal line in the other, so we say
that two bodies are of the same shape, if to every angle
that can be drawn on one of them there corresponds an
exactly equal angle on the other.
Just as we measured lengths by a stick or a piece of
tape so we measure angles with a pair of compasses ;
at'd two angles are said to be equal when they fit the
same opening of the compasses. And as before, the
statement that a thing can be moved about without
altering its shape maybe shown to amount only to this,
that two angles which fit in one place will fit also in
another, no matter how they have been brought from
the one place to the other.
§ 5. The Plane and the Straight Line.
We have now to describe a particular kind of surface
and a particular kind of line with which geometry is
very much concerned. These are the plane surface and
the straight line.
The plane surface may be defined as one which is of
the same shape all over and on both sides. This pro
perty of it is illustrated by the method which is practi
cally used to make such a surface. The method is to
take three surfaces and grind them down until any two
will tit one another all over. Suppose the three surfaces
to be A, B, c ; then, since A will fit B, it follows that the
SPACE. 67
space outside A is of the same shape as the space inside
B ; and because B will fit c, that the space inside B is of
the same shape as the space outside c. It follows there
fore that the space outside A is of the same shape as the
space outside c. But since A will fit o when we put
them together, the space inside A is of the same shape
as the space outside c. But the space outside c was
shown to be of the same shape as the space outside A ;
consequently the space outside A is of the same shape as
the space inside ; and so, if three surfaces are ground
together so that each pair of them will fit, each of them
becomes a surface which is of the same shape on both
sides : that is to say, if we take a body which is partly
bounded by a plane surface, we can slide it all over this
surface and it will fit everywhere, and we may also turn
it round and apply it to the other side of the surface
and it will fit there too. This property is sometimes
more technically expressed by saying that a plane is a
surface which divides space into two congruent regions.
A straight line may be defined in a similar way. It
is a division between two parts of a plane, which two
parts are, so far as the dividing line is concerned, of the
same shape ; or we may say what comes to the same
effect, that a straight line is a line of the same shape all
along and on both .sides.
A body may have two plane surfaces ; one part of it,
that is, may bo bounded by one plane and another part
by another. If these two plane surfaces have a common
edge, this edge, which is called their intersection, is a
straight line. We may then, if we like, take as our
definition of a straight line that it is the intersection of
two planes.
It must be understood that when a part of the sur
face of a body is plane, this plane may be conceived as
F 2
68 THE COMMON SEXSE OF THE EXACT SCIENCES.
extending beyond the body in all directions. For
instance, the upper surface of a table is plane and
horizontal. Now it is quite an intelligible question to
ask about a point which is anywhere in the room whether
it is higher or lower than the surface of the table. The
points which are higher will be divided from those which
are lower by an imaginary surface which is a continua
tion of the plane surface of the table. So then we are
at liberty to speak of the line of intersection of two
plane surfaces of a body whether these are adjacenl
portions of surface or not, and we may in every case
suppose them to meet one another and to be prolonged
across the edge in which they meet.
Leibniz, who was the first to give these definitions
of a plane and of a straight line, gave also another
definition of a straight line. If we fix two points of a
body, it will not be entirely fixed, but it will be able to
turn round. All points of it will then change their
position excepting those which are in the straight line
joining the two fixed points; and Leibniz accordingly
defined a straight line as being the aggregate of those
points of a body which are unmoved when it is turned
about with two points fixed. If we suppose the body to
have a plane face passing through the two fixed points,
this definition will fall back on the former one which
defines a straight line as the intersection of two planes.
It hardly needs any words to prove that the first
two definitions of a plane are equivalent; that is, that
two surfaces, each of which is of the same shape all over
and on both sides, will have for their intersection a line
which is of the same shape all along and on both sides.
For if we slide each plane upon itself it will, being of
the same shape all over, occupy as a whole the same
unchanging position (i.e. wherever there was part of
SPACE. 69
the planes before there will be part, though a different
part, of the planes now), so that their line of inter
section occupies the same position throughout (though
the part of the line occupying any particular position
is different). The line is therefore of the same shape
all along. And in a similar way we can, without
changing the position of the planes as a whole, move
them so that the right-hand part of each shall become
the left-hand part, and the upper part the lower; and
this will amount to changing the line of intersection
end for end. But this line is in the same place after
the change as before ; and it is therefore of the same
shape on both sides.
From the first definition we see that two straight
lines cannot coincide for a certain distance and then
diverge from one another. For since the plane surface
is of the same shape on the two sides of a straight line,
we may take up the surface on one side and turn it
over and it will fit the surface on the other side. If
this is true of one of our supposed straight lines, it is
quite clear that it cannot at the same time be true of the
other; for we must either be bringing over more to lit
less, or less to fit more.
§ 6. Properties of Triangles.
can now reduce to a more precise form our first
observation about space, that a body may be moved
about in it without altering its size or shape. Let us
suppose that our body has for one of its faces a tri«n<jl<>,
that is to say, the portion of a plane bounded by three
straight lines. We find that this triangle can be moved
into any new position that we like, while the lengths of
its sides and its angles remain the same ; or we may
70 THE COMMON SENSE OF THE EXACT SCIENCES.
put the statement into the form that when any triangl
is once drawn, another triangle of the same size an
shape can be drawn in any part of space.
From this it will follow that if there are two triangle
which have a side of the one equal to a side of the othei
and the angles at the ends of that side in the one equa
to the angles at the ends of the equal side in the othei
then the two triangles are merely the same triangle i:
different positions ; that is, they are of the same siz
and shape. For if we take the first triangle and so fa
put it into the position of the second that the two equa
sides coincide, then because the angles at the ends c
the one are respectively equal to those at the ends c
the other, the remaining two sides of the first triangl
will begin to coincide with the remaining two sides c
the second. But we have seen that straight lines canno
begin to coincide and then diverge ; and consequent!,
these sides will coincide throughout and the triangle
will entirely coincide.
Our second observation, that we may have thing
which are of the same shape but not of the same sizt
may also be made more precise by application to th
case of triangles. It tells us that any triangle miy b
magnified or diminished to any degree without alterin;
its angles, or that if a triangle be drawn, anothe
triangle having the same angles may be drawn of an
size in any part of space.
From this statement we are able to deduce two ver
important consequences. One is, that two straigh
lines cannot intersect in more points than one ; and th
other that, if two straight lines can be drawn in th
same plane so as not to intersect at all, the angles the,
make Avith any third line in their plane which meet
them, will be equal.
SPACE. 7 1
To prove the first of these, let AB and AC (fig. 18) be
two straight lines which meet at A. Draw a third line
BC, meeting both of them, and the three lines then form a
triangle. If we now make a point p travel along the line
AB it must, in virtue of our second observation, be always
possible to draw through this point a line which shall
meet AC in Q so as to make a triangle A p Q of the same
shape as ABC. But if the line AC were to meet AB in
some other point D besides A, then through this point
D it would clearly not be possible to draw a line so as
to make a triangle at all. It follows then that such
a point as D does not exist, and in fact that two
straight lines which have once met must go on diverg
ing from each other and can never meet again.1
To prove the second, suppose that the lines A c and
BD (fig. 19) are in the same plane, and are such as
7
FIG. 19.
never to meet at all (in which case they are called
parallel), while the line A B meets them both. If we
make a point p travel along B A towards A, and, as it
moves, draw through it always a line making the same
angle with B A that B D makes with B A, then this
1 This property might also be deduced from the first definition of a
straight line, by the method already used to show that two straight lines
cannot coincide for part of their length and then diverge.
72 THE COMMON SENSE OF THE EXACT SCIENCES.
moving line can never meet A c until it wholly coincides
with. it. For if it can, let p Q be such a position of
the moving line ; then it is possible to draw through
B a line which, with. A B and A c, shall form a tri
angle of the same shape as the triangle A p Q. But
for this to be the case the line drawn through B must
make the same angle with A B that p Q makes with it,
that is, it must be the line B D. And the three lines B D,
B A, A c cannot form a triangle, for B D and A c never
meet. Consequently there can be no such triangle as
A P Q, or the moveable line can never meet A c until it
entirely coincides with it. But since this line always
makes with B A the same angle that B D does, and in
one position coincides with A c, it follows that A c
makes with B A the same angle that B D does. This is
the famous proposition about parallel lines.1
The first of these deductions will now show us that
if two triangles have an angle of the one equal to an
angle of the other and the sides containing these angles
respectively equal, they must be equal in all particulars.
For if we take up one of the triangles and put it down
1 Two straight lines which cut one another form at the point where they
cross four angles which are equal in pairs. It is often necessarj7 to dis
tinguish between the two different angles which the lines make with one
another. This is done by the understanding that A B shall mean the line
0) (ii)
drawn from A to B, and B A the line drawn from B to A, so that the angle
between AB and CD (i) is the angle BOD, but the angle between BA and
C D (ii) is the angle D o A.
So the angle spoken of above as made by A c with B A is not the angle
CAB (which is clearly, in general, unequal to the angle DBA), but the
angle c A E, where E is a point in B A produced through A.
SPACE. 73
on the other so that these angles coincide and equal
sides are on the same side of them, then the con
taining sides will begin to coincide, and cannot there
fore afterwards diverge. But as they are of the same
length in the one triangle as they are in the other, the
ends of them belonging to the one triangle will rest
upon the ends belonging to the other, so that the re
maining sides of the two triangles will have their ends in
common and must therefore coincide altogether, since
otherwise two straight lines would meet in more points
than one. The one triangle will then exactly cover the
other ; that is to say, they are equal in all respects.
In the same way we may see that if two triangles
have two angles in the one equal to two angles in the
other, they are of the same shape. For one of them
can be magnified or diminished until the side joining
these two angles in it becomes of the same length as
the side joining the two corresponding angles in the
other ; and as no alteration is thereby made in the
shape of the triangle, it will be enough for us to prove
that the new triangle is of the same shape as the other
given triangle. But if we now compare these two, we
see that they have a pair of corresponding sides which
have been made equal, and the angles at the ends of
these sides equal also (for they were equal in the
original triangles, and have not been altered by the
change of size), so that we fall back on a case already
considered, in which it was shown that the third angles
are equal, and the triangles consequently of the same
shape.
If we apply these propositions not merely to two
different triangles but to the same triangle, we find
that if a triangle has two of its sides equal it will have
the two angles opposite to them also equal j and that,
74 THE COMMON SENSE OF THE EXACT SCIENCES.
conversely, if it has two angles equal it will have the two
sides opposite to them also equal ; for in each of these
cases the triangle may be turned over and made to fit
itself. Such a triangle is called isosceles.
The theorem about parallel lines which we deduced
from our second assumption about space leads Tery
easily to a theorem of especial importance, viz. that
the three angles of a triangle' are together equal to two
right angles.
If we draw through A, a corner of the triangle
ABC (fig. 20), a line DAE, making with the side A o
IE AD
FIG. 20.
the same angle as B c makes with it, this line will,
as we have proved, never meet B c, that is. it will be
parallel to it. It will consequently make with A B the
same angle as B c makes with it,1 so that the three
angles ABO, BAG, and BOA are respectively equal to
the angles E A B, BAG, and CAD, and these three make
up two right angles.
Another statement of this theorem is sometimes of
use.
If the sides of a triangle be produced, what are
called the exterior angles of the triangle are formed. If,
for example, the side B o of the triangle ABC (fig. 21)
is produced beyond c to D, A c D is an exterior angle of
the triangle, while of the interior angles of the triangle
A c B is said to be adjacent, and CAB and A B c to be
opposite to this exterior angle. It is clear that as
1 The convention mentioned in the last footnote must be remembered.
SPACE. 75
each side of the triangle may be produced in two
directions, any triangle has six exterior angles.
B CD
FIG. 21.
The other form into which our proposition may
be thrown is that either of the exterior angles of a
triangle is equal to the sum of the two interior angles
opposite to it. For, in the figure, the exterior angle
A c D, together with AGE, makes two right angles, and
it must therefore be equal to the sum of the two angles
•which also make up two right angles with A c B.
§ 7. Properties of Circles; Related Circles and Triangles.
We may now apply this proposition to prove an im
portant property of the circle, viz. that if we take two
fixed points on the circumference of a circle and join
them to a third point on the circle, the angle between
the joining lines will depend only upon the first two
points and not at all upon the third. If, for example,
we join the points A, B (fig. 22) to c we shall show that,
wherever on the circumference c may be, the angle
A c B is always one-half of A o B ; o being the centre of
the circle.
Let co produced meet the circumference in D.
Then since the triangle o A c is isosceles, the angles o A c
and OCA are equal, and so for a similar reason are the
angles o B c and o c B.
But we have just shown that the exterior angle
AO D is equal to the sum of the angles o A c and OCA;
76 THE COMMON SEXSE OF THE EXACT SCIENCES.
and since these are equal to one another it must be
double of either of them, say of OCA. Similarly the
angle B o D is double of o c B, and consequently A 0 B is
double of A c B.
In the case of the first figure (i) we have taken the
sum of two angles each of which is double of another,
and asserted that the sum of the first pair is twice the
sum of the second pair ; in the case of the second
figure (ii) we have taken the difference of two angles
FIG. 22.
each of which is double of another, and asserted that
the difference of the first pair is twice the difference of
the second pair.
Since therefore A c B is always half of A o B, wher
ever c may be placed in the upper of the two segments
into which the circle is divided by the straight line A B,
we see that the magnitude of this angle depends only
on the positions of A and B, and not on the position of
c. But now let us consider what will happen if c is in
the lower segment of the circle. As before, the tri
angles o A c and o B c (fig. 23) are isosceles, and the
angles DO A and DOB are respectively double of OCA
and o c B. Consequently, the whole angle A o B formed
by making o A turn round o into the position o B, so as
to pass through the position OD (in the way, that is,
SPACE. 77
in which the hands of a clock turn), this whole angle is
double of A c B.
By our previous reasoning the angle A D B, formed
by joining A and B to D, is one-half of the angle A o B,
which is made by turning o B towards o A as the hands
of a clock move. The sum of these two angles, each
of which we have denoted by A o B, is a complete re
volution about the point o ; in other words, is four
right angles. Hence the sum of the angles A D B, AGE,
which are the halves of these, is two right angles. Or
we may put the theorem otherwise, and say that the
opposite angles of a four-sided figure whose angles lie
on the circumference of a circle are together equal to
two right angles.
We appear therefore to have arrived at two dif
ferent statements according as the point c is in the
one or the other of the segments into which tin;
circle is divided by the straight line A B. But these
statements are really the same, and it is easy to include
them in one proposition. If AVO produce AC in the last
figure to E, the angles A c B and B c E are together equal
to two right angles ; and consequently B c E is equal to
A D B. This angle B c E is the angle through which c B
must be turned in the way the hands of a clock move,
78 THE COMMON SENSE OF THE EXACT SCIENCES.
so that its direction may coincide with that of A c. But
we may describe in precisely the same words the angle
A c B in fig. 22, where c was in the upper segment of the
circle ; so that we may always put the theorem in these
words :— If A and B are fixed points on the circumfer
ence of a circle, and o any other point on it, the angle
through which c B must be turned clockwise in order to
coincide with c A or AC, whichever happens first, is
equal to half the angle through which o B must be
turned clockwise in order to coincide with o A.
We shall now make use of this to prove another in
teresting proposition. If three points D, E, F (fig. 24)
FIG. 24.
be taken on the sides ot a triangle A B c, D being on B 0,
E on c A, F on A B, then three circles can be drawn
passing respectively through A F E, B D F, c E D. These
three circles can be shown to meet in the same point o.
For let o in the first place stand for the intersection of
the two circles A F E and B F D, then the angles F A E
and FOE make up two right angles, and so do the
angles D o F and DBF. But the three angles at o make
four right angles, and the three angles of the triangle
ABC make two right angles : and of these six angles
two pairs have been shown to make up two right
SPACE.
79
ingles each. Therefore the remaining pair, viz. the
ingles DOE and D c E, make up two right angles. It
follows that the circle which goes through the points
D E D will pass through o, that is, the three circles all
meet in this point.
There is no restriction imposed on the positions of
the points D, E, p,1 they may be taken either on the sides
FIG. 25.
of the triangle or on those sides produced, and in pnr-
ticular we may take them to lie on any fourth straight
line D E F ; and the theorem may be stated thus : — If
any four straight lines be taken (fig. 25), one of which
meets the triangle ABC formed by the other three in
the points D, E, F, then the circles through the points
1 If either of the points D, E, F, is taken on a side produced, the proof
given above will not apply literally; but the necessary changes are slight
and obvious.
80 THE COMMON" SEXSE OF THE EXACT SCIEXCES.
AFE, BDP, CED meet in a point. But there is ncjj
reason why we should not take A F E as the triangle!
formed by three lines, and the fourth line D c B as tha
line which cuts the sides of this triangle. The propo-l
sition is equally true in this case, and it follows thai]
the circles through ABC, BCD, FED will meet in onel
point. This must be the same point as before, sincel
two of the circles of this set are the same as two of the!
previous set; consequently all four circles meet in a|
point, and we can now state our proposition as follows :
Given four straight lines, there can be formed from
them four triangles by leaving out each in turn ; the
circles which circumscribe these four triangles meet in
a point.
This proposition is the third of a series.
If we take any two straight lines they determine a
point, viz. their point of intersection.
If we take three straight lines we get three such
points of intersection ; and these three determine a
circle, viz. the circle circumscribing the triangle formed
by the three lines.
Four straight lines determine four sets of three
lines by leaving out each in turn; and the four circles
belonging to these sets of three meet in a point.
In the same way five lines determine five sets of
four, and each of these sets of four gives rise, by the
proposition just proved, to a point. It has been shown
by Miquel, that these five points lie on the same circle.
And this series of theorems has been shown l to be
endless. Six straight lines determine six sets of five by
leaving them out one by one. Each set of five has, by
1 By Prof. Clifford himself in the Oxford, Cambridge, and Dublin
Messenger of Mathematics, vol. V. p. 124. See his Mathematical Paptrs, .
pp. 51-54.
SPACE. 81
Miquel's theorem, a circle belonging to it. These six
circles meet in the same point, and so on for ever. Any
even number (2n) of straight lines determines a point
as the intersection of the same number of circles. It
we take one line more, this odd number (2w + l) deter
mines as many sets of 2n lines, and to each of these
sets belongs a point ; these 2n + 1 points lie on a circle.
§ 8. The Conic Sections.
The shadow of a circle cast on a flat surface by a
luminous point may have three different shapes. These
are three curves of great historic interest, and of the
utmost importance in geometry and its applications.
The lines we have so far treated, viz. the straight line
ind circle, are special cases of these curves ; and we may
naturally at this point investigate a few of the properties
}f the more general forms.
If a circular disc be held in any position so that it
:s altogether below the flame of a candle, and its shadow
36 allowed to fall 011 the table, this shadow will be of
in oval form, except in two extreme cases, in one of
tfhich it also is a circle, and in the other is a straight
ine. The former of these cases happens when the disc
s held parallel to the table, and the latter when the
lisc is held edgewise to the candle ; or, in other words,
s so placed that the plane in which it lies passes
through the luminous point. The oval form which,
vith these two exceptions, the shadow presents is called
(tn ellipse (i) . The paths pursued by the planets round
ihe sun are of this form.
If the circular disc be now held so that its highest
I'oint is just on a level with the flame of the candle, the
hadow will as before be oval at the end near the candle;
G
82 THE COMMON SENSE OF THE EXACT SCIENCES.
but instead of closing up into another oval end as we
move away from the candle, the two sides of it will con
tinue to open out without any limit, tending however
to become more and more parallel. This, form of the
shadow is called a, parabola (ii). It is very nearly the
orbit of many comets, and is also nearly represented b]
the path of a stone thrown up obliquely. If there were
no atmosphere to retard the motion of the stone i
would exactly describe a parabola.
FIG. 26.
If we now hold the circular disc higher up still, S(
that a horizontal plane at the level of the candle flam
divides it into two parts, only one of these parts wil
cast any shadow at all, and that will be a curve sucl,
as is shown in the figure, the two sides of whicl
diverge in quite different directions, and do not, as iij
the case of the parabola, tend to become parallel (iii). ,
But although for physical purposes this curve is th
whole of the shadow, yet for geometrical purposes it i|
not the whole. We may suppose that instead of bein
a shadow our curve was formed by joining theluminoui
SPACE. 83
point by straight lines to points round the edge of the
disc, and producing these straight lines until they meet
the table.
This geometrical mode of construction will equally
pply to the part of the circle which is above the candle
ame, although that does not cast any shadow. If we
oin these points of the circle to the candle flame, and
rolong the joining lines beyond it, they will meet the
able on the other side of the candle, and will trace out
curve there which is exactly similar and equal to the
hysical shadow (iv). We may call this the anti-shadow
r geometrical shadow of the circle. It is found that for
eometrical purposes these two branches must be con-
idered as forming only one curve, which is called an
yperbola. There are two straight lines to which the
urve gets nearer and nearer the further away it goes
•oni their point of intersection, but which it never
ctually meets. For this reason they are called asymp-
otes, from a Greek word meaning ' not falling to
gether.' These lines are parallel to the two straight
mes which join the candle flame to the two points of
tie circle which are level with it.
We saw some time ago that a surface was formed
y the motion of a line. Now if a rigrht line in its
•/
notion always passes through one fixed point, the surface
Inch it traces out is called a cone, and the fixed point
8 called its vertex. And thus the three curves which we
lave just described are called conic sections, for they
nay be made by cutting a cone by a plane. In fact, it
s in this way that the shadow of the circle is formed ;
or if we consider the straight lines which join the
andle flame to all parts of the edge of the circle we see
hat they form a cone whose vertex is the candle flame
-nd whose base is the circle.
u 2
84 THE COMMON SEXSE OF THE EXACT SCIENCES.
We must suppose these lines not to end at the flame
but to be prolonged through it, and we shall so gel
what would commonly be called two cones with their
points together, but what in geometry is called one
conical surface having two sheets. The section of this
conical surface by the horizontal plane of the table is
the shadow of the circle ; the sheet in which the circle
lies gives us the ordinary physical shadow, the other
sheet (if the plane of section meets it) gives what we
have called the geometrical shadow.
The consideration of the shadows of curves is
method much used for finding out their properties, for
there are certain geometrical properties which are
always common to a figure and its shadow. For ex
ample, if we draw on a sheet of glass two curves which
cut one another, then the shadows of the two curves
cast through the sheet of glass on the table will also cui
one another. The shadow of a straight line is alway
a straight line, for all the rays of light from the flame
through various points of a straight line lie in a plane
and this plane meets the plane surface of the table in a
straight line which is the shadow. Consequently iJ
any curve is cut by a straight line in a certain numbei
of points, the shadow of the curve will be cut by the
shadow of the straight line in the same number of points
Since a circle is cut by a straight line in two points 01
in none at all, it follows that any shadow of a circle
must be cut by a straight line in two points or in none
at all.
When a straight line touches a circle the two pointi
of intersection coalesce into one point. We see thei
that this must also be the case with any shadow of the
circle. Again, from a point outside the circle it is pos
sible to draw two lines which touch the circle ; so frorr
I
SPACE. 85
a point outside either of the three curves which we have
just described, it is possible to draw two lines to touch
the curve. From a point inside the circle no tangent
can be drawn to it, and accordingly no tangent can be
drawn to any conic section from a point inside it.
This method of deriving the properties of one curve
from those of another of which it is the shadow, is
called the method of projection.
The particular case of it which is of the greatest
use is that in which we suppose the luminous point
by which the shadow is cast to be ever so far away.
Suppose, for example, that the shadow of a circle held
obliquely is cast on the table by a star situated directly
overhead, and at an indefinitely great distance. The
lines joining the star to all the points of the circle will
then be vertical lines, and they will no longer form a
cone but a cylinder. One of the chief advantages of
this kind of projection is that the shadows of two
parallel lines will remain parallel, which is not generally
the case in the other kind of projection. The shadow
of the circle which we obtain now is always an ellipse ;
and we are able to find out in this way some very
important properties of the curve, the corresponding
properties of the circle being for the most part evident
at a glance on account of the symmetry of the figure.
For instance, let us suppose that the circle whose
shadow we are examining is vertical, and let uy take a
vertical diameter of it, so that the tangents at its ends
are horizontal. It will be clear from the symmetry of
the figure that all horizontal lines in it are divided into
two equal parts by the vertical diameter, or we may s;iy
that the diameter of the circle bisects all chords parallel
to the tangents at its extremities. When the shadow
of this figure is cast by an infinitely distant star (which
86 THE COMMON SENSE OF THE EXACT SCIENCES.
we must not now suppose to be directly overhead, for
then the shadow would be merely a straight line), the
point of bisection of the shadow of any straight line
is the shadow of the middle point of that line, and thus
we learn that it is true of the ellipse that any line
which joins the points of contact of parallel tangents
bisects all chords parallel to those tangents. Such a
line is, as in the case of the circle, termed a diameter.
Since the shadow of a diameter of the circle is a dia
meter of the ellipse, it follows that all diameters of the
ellipse pass through one and the same point, namely,
the shadow of the centre of the circle ; this common
intersection of diameters is termed the centre also of the
ellipse.
Again, a horizontal diameter in the circle just con
sidered will bisect all vertical chords, and thus we
see that if one diameter bisects all chords parallel to a
second, the second will bisect all chords parallel to the
first.
The method of projection tells us that this is also
true of the ellipse. Such diameters ai*e called conjugate
diameters, but they are no longer at right angles in the
ellipse as they were in the case of the circle.
Since the shadow of a circle which is cast in this
way by an infinitely distant point is always an ellipse,
we cannot use the same method in order to obtain the
properties of the hyperbola. But it is found by other
methods that these same statements are true of the
hyperbola which we have just seen to be true of the
ellipse. There is however this great difference be
tween the two curves. The centre of the ellipse is
inside it, but the centre of the hyperbola is outside it.
Also all lines drawn through the centre of the ellipse
nieet the curve in two points, but it is only certain
SPACE. 87
ines through the centre of the hyperbola which meet
ihe curve at all. Of any two conjugate diameters of
ihe hyperbola one meets the curve and the other does
not. But it still remains true that each of them bisects
all chords parallel to the other.
§ 9. On Surfaces of the Second Order.
We began with the consideration of the simplest
dnd of line and the simplest kind of surface, the
straight line and the plane ; and we have since found
out some of the properties of four different curved lines
— the circle, the ellipse, the parabola, and the hyperbola,
jet us now consider some curved surfaces ; and first,
ihe surface analogous to the circle. This surface is the
ere. It is defined, as a circle is, by the property
ihat all its points are at the same distance from the
centre.
Perhaps the most important question to be asked
ibout a surface is, What are the shapes of the curved
anes in which it is met by other surfaces, especially
jn. the case when these other surfaces are planes ? Now a
Mane which cuts a sphere cuts it, as can easily be shown,
n a circle. This circle, as we move the plane further and
drther away from the centre of the sphere, will get
imaller and smaller, and will finally contract into a
joint. In this case the plane is said to touch the
iphere; and we notice a very obvious but important
?act, that the sphere then lies entirely on one side of
ihe plane. If the plane be moved still further away
'rom the centre it will not meet the sphere at all.
Again, if we take a point outside the sphere we can
Iraw a number of planes to pass through it and touch the
phere, and all the points in which they touch it lie on
88 THE COMMON SENSE OF THE EXACT SCIENCES.
a circle. Also a cone can be drawn whose vertex is
the point, and which touches the sphere all round the
circle in which these planes touch it. This is called
the tangent-cone of the point. It is clear that from a
point inside the sphere no tangent-cone can be drawn.
Similar properties belong also to certain other sur
faces which resemble the sphere in the fact that they
are met by a straight line in two points at most ; such
surfaces are on this account called of the second order.
Just as we may suppose an ellipse to be got from
a circle by pulling it out in one direction, so we may
get a spheroid from a sphere either by pulling it out so
as to make a thing like an egg, or by squeezing it so
as to make a thing like an orange. Each of these
forms is symmetrical about one diameter, but not about
all. A figure like an orange, for example, or like the
earth, has a diameter through its poles less than any
diameter in the plane of its equator, but all diameters in
its equator are equal. Again, a spheroid like an egg
has all the diameters through its equator equal to one
another, but the diameter through its poles is longer
than any other diameter.
If we now take an orange or an egg and make its
equator into an ellipse instead of a circle, say by pull
ing out the equator of the orange or squeezing the
equator of the egg, so that the surface has now three
diameters at right angles all unequal to one another,
we obtain what is called an ellipsoid. This surface
plays the same part in the geometry of surfaces that the
ellipse does in the geometry of curves. Just as every
plane which cuts a sphere cuts it in a circle, so every
plane which cuts an ellipsoid cuts it in an ellipse. It
is indeed possible to cut an ellipsoid by a plane so that
the section shall be a circle, but this must be regarded
SPACE. 89
as a particular kind of ellipse, viz. an ellipse with
two equal axes. Again, just as was the case with the
sphere, we can draw a set of planes through an exter
nal point all of which touch the ellipsoid. Their points
of contact lie on a certain ellipse, and a cone can be
drawn which has the external point for its vertex and
touches the ellipsoid all round this ellipse. The ellip
soid resembles a sphere in this respect also, that when
it is touched by a plane it lies wholly on one side of
:hat plane.
There are also surfaces which bear to the hyperbola
ind the parabola relations somewhat similar to those
aorne to the circle by the sphere, arid to the ellipse by
:he ellipsoid. We will now consider one of them, a
surface with many singular properties.
Let A BCD be a figure of card-board having four
jqual sides, and let it be half cut through all along B D,
90 THE COMMON SENSE OF THE EXACT SCIENCES.
so that the triangles A B D, c B D can turn about the line
B D. Then let holes be made along the four sides of it at
equal distances, and let these holes be joined by threads
of silk parallel to the sides. If now the figure be bent
about the line B D and the silks are pulled tight it will
present an appearance like that in fig. 27, resembling
a saddle, or the top of a mountain pass.
This surface is composed entirely of straight lines,
and there are two sets of these straight lines ; one set
which was originally parallel to A B, and the other set
which was originally parallel to A D.
A section of the figure through A c and the middle
point of B D will be a parabola with its concave side
turned upwards.
A section through B D and the middle point of A 0
will be another parabola with its concave side turned
downwards, the common vertex of these parabolas
being the summit of the pass.
The tangent plane at this point will cut the surface
in two straight lines, while part of the surface will be
above the tangent plane and part below it. We may
regard this tangent plane as a horizontal plane at the
top of a mountain pass. If we travel over the pass, we
come up on one side to the level of the plane and then
go down on the other. But if we go down from a
mountain on the right and go up the mountain on the
left, we shall always be above the horizontal plane. A
section by a horizontal plane a little above this tangent
plane will be a hyperbola whose asymptotes will be
parallel to the straight lines in which the tangent plane
meets the surface. A section by a horizontal plane a
little below will also be a hyperbola with its asymptotes
parallel to these lines, but it will be situated in the
other pair of angles formed by these asymptotes. If
SPACE. 9 1
suppose the cutting plane to move downwards from
a position above the tangent plane (remaining always
lorizontal), then we shall see the two branches of the
irst hyperbola approach one another and get sharper
and sharper until they meet and become simply two
crossing straight lines. These lines will then have
;heir corners rounded off and will be divided in the
other direction and open out into the second hyper-
x>la.
This leads us to suppose that a pair of intersecting
straight lines is only a particular case of a hyperbola,
and that we may consider the hyperbola as derived
roin the two crossing straight lines by dividing them
at their point of intersection and rounding off the
corners.
I 10. How to form Curves of the Third and Higher Orders.
The method of the preceding paragraph may be ex-
;ended so as to discover the forms of new curves by
Hitting known curves together. By a mode of expres
sion which sounds paradoxical, yet is found convenient,
a straight line is called a curve of the first order, because
.t can be met by another straight line in only one
K)int ; but two straight lines taken together are called
a. curve of the second order, because they can be met
by a straight line in two points. The circle, and its
hadows, the ellipse, parabola, and hyperbola, are also
sailed curves of the second order, because they can be
met by a straight line in two points, but not in more
than two points ; and we see that by this process of
rounding off the corners and the method of projection
we can derive all these curves of the second order from
pair of straight lines.
92 THE COMMON SEXSE OF THE EXACT SCIENCES.
A similar process enables us to draw curves of the
third order. An ellipse and a straight line taken to
gether form a curve of the third order. If now we
round off the corners at both the points where they
meet we obtain (fig. 28) a curve consisting of an oval
and a sinuous portion called a ' snake.' Now just as
when we move a plane which cuts a sphere away from
the centre, the curve of intersection shrinks up into a
H
. 28.
(i.) Full loop and snake.
(ii.) Shrunk loop and snake.
(iii.) The loop has shrunk to a point,
(iv.) Snake only.
point and then disappears, so we can vary our curve of
the third order so as to make the oval which belongs to
it shrink up into a point, s*nd then disappear altogether, I
leaving only the sinuous part, but no variation will get i
rid of the ' snake.'
We may, if we like, only round off the corners at
one of the intersections of the straight line and the
ellipse, and we then have a curve of the third order
crossing itself, having a knot or double point (fig. 29) ;
and we can further suppose this loop to shrink up, and
the curve will then be found to have a sharp point or
cusp.
SPACE.
93
It was shown by Newton that all curves of the third
order might be derived as shadows from the five forms
(ivr)
which we have just mentioned, viz. the oval and snake,
the point and snake, the snake alone, the form with a
mot, and the form with a cusp.
In the same way curves of the fourth order may be
jot by combining together two ellipses. If we suppose
FIG. 30.
them to cross each other in four points we may round
off all the corners at once and so obtain two different
forms, either four ovals all outside one another or an
94 THE COMMON SENSE OF THE EXACT SCIENCES.
oval with, four dints in it, and another oval inside it
(fig. 30).
But the number of forms of curves of the fourth
order is so great that it has never yet been completely
catalogued; and curves of higher orders are of still
more varied shapes.
95
CHAPTER III.
QUANTITY.
§ 1. The Measurement of Quantities.
WE considered at the beginning of the first chapter,
on Number, the process of counting things which are
separate from one another, such as letters or men or
sheep, and we found it to be a fundamental property of
this counting that the result was not affected by the
order in which the things to be counted were taken ;
that one of the things, that is, was as good as another
at any stage of the process.
We may also count things which are not separate
but all in one piece. For example, we may say that a
room is sixteen feet broad. And in order to count the
number of feet in the breadth of this room we should
probably take a foot rule and measure off first a foot
close to the wall, then another beginning where that
ended, and so on until we reached the opposite wall.
Now when these feet are thus marked off they may,
just like any other separate things, be counted in
whatever order we please, and the number of them
will always be sixteen.
But this is not all the variety in the process of
counting which is possible. For suppose that we take
a stick whose length is equal to the breadth of the
room. Then we may cut out a foot of it wherever we
please, and join the ends together. And if we tlu-n
'JO THE COMMON SENSE OF THE EXACT SCIENCES.
cut out another foot from any part of the remainder
and join the ends, and repeat the process fifteen times,
we shall find that there will always be a foot length
left when the last two ends are joined together. So,
when we are counting things that are all in one piece,
like the length of the stick or the breadth of the room,
not only is the order in which we count the feet im
material, but also the position of the actual feet which
we count.
Again, if we say that a packet contains a pound, or
sixteen ounces, of tea, we mean that if we take any
ounce of it out, then any other ounce out of what is
left, and so on until we have taken away fifteen ounces,
there will always be an ounce left.
If I say that I have been writing for fifteen minutes
it will of course have been impossible actually to count
these minutes except in the order in which they really
followed one another, but it will still be true that, if
any separate fourteen minutes had been marked off
during that interval of time, the remainder of it, made
np of the interstices between these minutes, would
amount on the whole to one minute.
In all these cases we have been counting things that
hang together in one piece ; and we find that we may
choose at will not only the order of counting but even
the things that we count without altering the result.
This process is called the measurement of quantities.
But now suppose that when we measure the breadth
of a room we find it to be not sixteen feet exactly, but
sixteen feet and something over. It may be sixteen
feet and five inches. And if so, in order to measure
the something over, we merely repeat the same pro
cess as before ; only that instead of counting feet we
count inches, which are smaller than feet. If the
QT'AXTITY. 97
breadth is found not to be an exact number of inches,
but that something is left beside the five inches, we
might measure that in eighths of an inch. There
might, for example, be three eighths of an inch over.
But there is no security that the process will end here ;
for the breadth of the room may not contain an exact
number of eighths of an inch. Still it may be said
hat nobody wants to know the breadth of a room more
xactly than to within an eighth of an inch.
Au'uin, when we measure a quantity of tea it may be
learly, but not exactly, sixteen ounces; there maybe
-omething over. This remainder we shall then measure
n grains. And here, as before, we are repeating \]\(.
ame process by which we count things which are all in
me piece; only we count grains, which are smaller
hings than ounces. There may still not be an exact
lumber of grains in the packet of tea, but then nobody
jvants to know the weight of a packet of tea so nea: iv
is to a grain.
And it is the same with time. A geological period
nay, if we are very accurate, be specified in hundreds
)f centuries ; the length of a war in years ; the time of
leparture of a train to within a minute ; the moment
)f an eclipse to a second ; our care being, in each ruse,
nerely to secure that the measurement is accurate
jnough for the purpose we have in hand.
To sum up. There is in common use a rouirh or
ipproximate way of describing quantities, which con
sists in saying how many times the quantity to be
lescribed contains a certain standard quantity, and in
leglecting whatever may remain. The smaller the
standard quantity is the more accurate is the pro.
}ut it is in general no better than an approximation.
If then we want to describe a quantity accurately
H
98 THE COMMON SENSE OP THE EXACT SCIENCES.
and not by a mere approximation, what are we to do?
There is no way of doing this in words ; the only pos
sible method is to carry about either the quantity itself
or some other quantity which shall serve to represent
it. For instance, to represent the exact length and
breadth of a room we may draw it upon a scale of, say,
one inch to a foot and carry this drawing about.
Here we are representing a length by means of
another length ; but it is not necessary to represent
weights by means of weights, or times by means of
times ; they are both in practice represented by lengths.
When a chemist, wishing to weigh with great delicacy,
has gone as near as he can with the drachms which he
puts into his scales, he hangs a little rider upon the
beam of the scale, and the distance of this rider from
the middle indicates how much weight there is over.
And, if we suppose the balance to be perfectly true,
and that no friction or other source of error has to be
taken into account, it indicates this weight with real
accuracy.
Here then is a case in which a weight is indicated
by a length, namely, the distance from the centre of the
scale to the rider. Again, we habitually represent time
by means of a clock, and in this case the minute hand
moves by a succession of small jerks, possibly twice a
second. Such a clock will only reckon time in half
seconds, and can tell us nothing about smaller intervals
than this. But we may easily conceive of a clock in
which the motion of the minute hand is steady, and not
made by jerks. In this case the interval of time since
the end of the last hour will be accurately represented
by the length round the outer circle of the clocls
measured from the top of it to the point of the minutf
hand. And we notice that here also the quantity
QUANTITY. 99
which is measured in this way by a length is probably
not the whole quantity which was to be estimated, but
only that which remains over after the greater part has
been counted by reference to some standard quantity.
We may thus describe weight and time, and indeed
uantities of any kind whatever, by means of the lengths
f lines ; and in what follows, therefore, we shall only
peak of quantities of length as completely representing
neasurable things of any sort.
§ 2. The Addition and Subtraction of Quantities.
For the addition of two lengths it is plainly sutHrient
o place them end to end in the same line. And we
must notice that, as was the case with counting, so now,
he possible variety in the mode of adding is far greater
n the case of two quantities than in the case of two
numbers. For either of the lengths, the aggregate
of which we wish to measure, may be cut up into any
lumber of parts, and these may be inserted at any
>oints we please of the other length, without any change
n the result of our addition.
Or the same may be seen, perhaps more clearly, by
reference to the idea of ' steps.' Suppose we Lave a
straight line with a mark upon it agreed on as a start
ing-point, and a series of marks ranged at equal distances
ilong the line and numbered 1, 2, :>, 4. . . . Then any
particular number is shown by making an index point to
he right place on the line. And to add or subtract
my other number from this, we have only to make the
ndex move forwards or backwards over the con-espoml-
ng number of divisions. But in the case of lengths we
ire not restricted to the places which are marked on the
scale. Any length is shown by carrying the index to a
u 2
TOO THE COMMON SENSE OF THE EXACT SCIENCES.
place whose distance from the starting-point is the
length in question (of which places there may be
as many as we please between any two points which
correspond to consecutive numbers), and another length
is added or subtracted by making the index take a
' step ' forwards or backwards of the necessary amount.
It is seen at once that, for quantities in general as
well as for numbers, a succession of given steps may
be made in any order we please and the result will
always be the same.
§ 3. The Multiplication and Division of Quantities.
We have already considered cases in which a quan
tity is multiplied ; that is to say, in which a certain
number of equal quantities are added together, a process
called the multiplication of one of them by that number.
Thus the length sixteen feet is the result of multiplying
one foot by sixteen.
We may now ask the inverse question : Given two
lengths, what number must be used to multiply one of
them in order to produce the other ? And it has been
implied in what we have said about the measurement of
quantities that it is only in special cases that we can find
a number which will be the answer to this question. If
we ask, for example, by what number a foot must be
multiplied in order to produce fifteen inches, the word
' number' requires to have its meaning altered and ex
tended before we can give an answer. We know that
an inch must be multiplied by fifteen in order to become
fifteen inches. We may therefore first ask by what
a foot must be multiplied in order to produce an inch.
And the question seems at first absurd ; because an
inch must be multiplied by twelve in order to give a
QUANTITY. 101
foot, and a foot has to be, not multiplied at all, but
divided by twelve, in order to become an inch.
In order then to turn a foot into fifteen inches, we
must go through the following process ; we must divide
it into twelve equal parts and take fifteen of them ; or,
shortly, divide by twelve and multiply by fifteen. Or
we may produce the same result by performing the
steps of our process in the other order : we may first
multiply by fifteen, so that we get fifteen feet, and then
divide this length into twelve equal parts, each of which
will be fifteen inches.
Now if instead of inventing a new name for this
compound operation we choose to call it by the old name
of multiplication, we shall be able to speak of multiply
ing a foot so as to get fifteen inches. The operation of
multiplying by fifteen and dividing by twelve is written
thus: -j :"; ; and so, to change a foot into fifteen inches,
we multiply by the fraction |-f . Of this fraction the
upper number (15) is termed the numerator, the lower
12) the dunomliKitor.
Now it was explained in the first chapter, that
;he formula} of arithmetic and algebra are capable of a
louble interpretation. For instance, such a symbol
is 3 meant, in the first place, a number of letters or
lien, or any other things ; but afterwards was regarded
is meaning an operation, namely, that of trebling any-
;hing. And so now the symbol } ": may betaken either
meaning ' so much ' of a foot, or as meaning the
peration by which a foot is changed into fifteen inches.
The degree in which one quantity is greater or less
,,;|;han another; or, to put it more precisely, that amount.
f stretching or squeezing which must be applied to the
atter in order to produce the former, is called the rat in
f the two quantities. If a and b are any two lengths,
J02 THE COMMON SENSE OF THE EXACT SCIENCES.
the ratio of a to & is the operation of stretching or
squeezing which will make b into a; and this operation
can be always approximately, and sometimes exactly,
represented by means of numbers.
§ 4. The Arithmetical Expression of Ratios.
For the approximate expression of ratios there
are two methods in use. In each, as in measuring
quantities in general, we proceed by using standards
which are taken smaller and smaller as we go on. In
the first, these standards are chosen according to a fixed
law ; in the second, our choice is suggested by the par
ticular ratio which we are engaged in measuring.
The first method consists in using a series of stan
dards each of which is a tenth part of the preceding.
Thus to express the ratio of fifteen inches to a foot, we
proceed thus. The fifteen inches contain a foot once,
and there is a piece of length three inches, or a quar
ter of a foot, left over. This quarter of a foot is then
measured in tenths of a foot, and we find that it is
2-tenths, with a piece — which proves to be half a tenth
— over. So, if we chose to neglect this half-tenth we
should call the ratio 1 2-tenths, or as we write it
1-2. But if we do not neglect the half -tenth, it has to
be measured in hundredths of a foot ; of which it makes
5 exactly. So that the result is 125 hundredths, or
1'25, accurately.
Again we will try to express in this way the length
of the diagonal of a square in terms of a side. We find
at once that the diagonal contains the side once, with a
piece over : so that the ratio in question is 1 together
with some fraction. If we now measure this remaining
piece in tenth parts of a side we shall find that it contains
QUANTITY. 103
4 of them, with something left. Thus the ratio of the
diagonal to the side may be approximately expressed by
14- tenths, or 1*4. If we now measure the piece left over
in hundredth parts of the side we shall find that it con
tains one and a bit. Thus 141-hundredths, or 1'41 is a
more accurate description of the ratio. And this bit can
be shown to contain 4-thousandths of the side, and a
bit over ; so that we arrive at a still more accurate
value, 1414-thousandths, or 1'414. And this process
might be carried on to any degree of accuracy that was
required; but in the present case, unlike that con
sidered before, it would never end ; for the ratio of the
diagonal of a square to its side is one which cannot be
accurately expressed by means of numbers.
The other method of approximation differs from tlu>
one just explained in this respect — that the successively
smaller and smaller standard quantities in terms of
which we measure the successive remainders are not
fixed quantities, an inch, a tenth of an inch, u.
hundredth of an inch, and so on; but are suggested
to us in the course of the approximation itself.
We begin, as we did before, by finding how many
times the lesser quantity is contained in the greater,
say, the side of a square in its diagonal. The answer
in this case is, once and a piece over. Let the piece
left over be called a. We then go on to try how many
times this remainder, a, is contained in the side of the
square. It is contained twice, and there is a remainder,
say 6. We then find how many times b is contained in
a. Again twice, with a piece over, say c. And this
process is repeated as often as we please, or until no
remainder is left. It will, in the present case, be found
that each remainder is contained twice, with something
over, in the previous remainder.
104 THE COMMON SENSE OF THE EXACT SCIENCES.
Let us now inquire how this process enables us to
find successive approximations to the ratio of the
diagonal to the side of the square.
Suppose, first, that the piece a had been exactly half
the length of the side ; that is, that we may neglect the
remainder b. Then the diagonal would be equal to the
side together with half the side, that is, to three-halves
of the side.
Next let us include b in our approximation, but
neglect c ; that is, let us suppose that b is exactly one •
half of a. Then the side contains a twice, and half of
a ; that is to say, contains five-halves of a ; or a is two-
fifths of the side. But the diagonal contains the side
together with or, that is, contains the side and two-fifths
of the side, or seven-fifths of the side. The piece
neglected is here less than 6, and 6 is one-fifth of the
side of the square.
Again, let us include c in our approximation, and
suppose it to be exactly one half of 6. Then a, which
contains b twice with c over, will be five-halves Of 6,
that is b will be two-fifths of a. Hence the side will
contain twice a and two-fifths of a, that is, twelve-fifths -
of a ; so that a is five-twelfths of the side. And the
diagonal is equal to the side together with a; that is;
to seventeen-twelfths of the side. Also this approxi-.
nuition is closer than the preceding, for the piece
neglected is now less than c, which is one-half of b,
which is two-fifths of a, which is five-twelfths of the
side ; so that it is less than one-twelfth of the side.
By continuing this process we may find an approxi
mation of any required degree of accuracy.
The first method of approximation is called the
method of decimals; the second, that of continued frac--
tions.
QUANTITY. 105
§ 5. The Fourth Proportional.
One of the chief differences between quantities and
numbers is that, while the division of one number by
another is only possible when the first number happens
to be a multiple of the other, in the case of quantities
it appears, and we are indeed accustomed to assume,
that any quantity may be divided by any number wo
like; that is to say, any length — quantities of all kinds
being represented by lengths — may be divided into anv
given number of equal parts. And, if division is always
possible, that compound operation made up of multi
plication and division which we have called ' multiplv-
ing by a fraction ' must also be always possible ; for
example, we can find five-twelfths not only of a foot
but of any other length that we like.
The question now naturally arises whether that
general operation of stretching or squeezing which we
have'called a ratio can be applied to all quantities alike.
If we have three lengths, a, I, c, there is a certain
operation of stretching or squeezing which will convert
a into b. Can the same operation be performed upon c
with the re-suit of producing a fourth quantity <?, such
that the ratio of c to d shall be the same as the ratio of
a to b ? We assume that this quantity — tlae fourth
proportional, as it is called — does always exist; and
this assumption, as it really lies at the base of all
subsequent mathematics, is of so great importance as
to deserve further study.
We shall find that it is really included in the second
of the two assumptions that we made in the chapter
about space ; namely, that figures of the same shapu
may be constructed of different sixes. We found, in
106 THE COMMON SENSE OF THE EXACT SCIENCES.
considering this point, that it was sufficient to take the
case of triangles of different sizes of which the angles
were equal; and showed that one triangle might be
made into another of the same shape by the equal
magnifying of all its three sides ; that is to say, when
two triangles have the same angles, the three ratios of
either side of one to the corresponding side of the other
are equal. If this is true, it is clear that the problem of
finding the fourth proportionalisreduccd to that of draw
ing two triangles of the same shape. Thus, for example,
let A B and A c represent the first two given quantities,
and A D the third (fig. 31) ; and let it be required to
find that quantity which is got from A D by the same
operation of stretching as is required to turn A B into
A c. Suppose that we join B D, and draw the line c E
making the angle ACE equal to the angle A B D. The two
triangles ABD and ACE are now of the same shape, and
consequently ACE can be got from ABD by the equal
stretching of all its sides ; that is to say, the stretching
which makes A B into A c is the same as the stretching
which makes A D into AE. A E is therefore the fourth pro
portional required.
To render these matters clearer, it is well that we
should get a more exact notion of what we mean by the
fourth proportional. We have so far only described it
as something \\hich is got from A D by the same process
which makes A B into A c. In what way are we to tell
whether the process is the same? We might, if we
QUANTITY. 107
liked, give a geometrical definition of it, founded upon
the construction just explained ; and say that the ratio
of A D to A E shall be called ' equal ' to the ratio of A B to
A c, when triangles of the same shape can have for their
respective sides the lengths A B, A D, A c, and A E. But it
is better, if we can do it, to keep the science of quantity
distinct from the science of space, and to find some
definition of the fourth proportional which depends
upon quantity alone. Such a definition has been found,
and it is very important to notice the nature of it. For
shall find that similar definitions have to be given of
other quantities whose existence is assumed by what is
called the principle of continuity. This principle is
simply the assumption, which we have stated already,
that all quantities can be divided into any given number
of equal parts.
If we apply two different operations of stretching
to the same quantity, that which produces the greater
result is naturally looked upon as an operation which
under like circumstances will always produce a greater
effect. Now we will make our definition of the fourth
proportional depend upon the very natural assumption
that, if two processes of stretching are applied to two
different quantities, that process which produces the
greater result in the one case will also produce the
greater result in the other.
Suppose now that we have tried to approximate to
'the ratio which A c bears to A B, and that we have
[found that A c is between seventeen-twelfths and
ighteen-twelfths of A B, then we have two processes
')£ stretching which can be applied to A B, the process
jlenoted by -JJ- (that is, multiplying by 17 and dividing
j>y 12), and the process which makes A c of it. The
•esult of the former process is, by hypothesis, less than
108 THE COMMON SENSE OF THE EXACT SCIENCES.
the result of the latter, because A c is more than seven-
teen-twelfths of AB. Let us now apply these two
processes to AD. The former will produce seventeen-
twelfths of A D, the latter will produce the fourth pro
portional required. Consequently this fourth propor
tional must be greater than seventeen-twelfths of A D.
But we know further that A c is less than eighteen-
twelf ths of A B. Then the operation which makes A B
into A c gives a less result than the operation of multi
plying by 18 and dividing by 12. Let us now perform
both upon A D. It will follow that the fourth propor
tional required is less than eighteen-twelf'ths of A D.
The same thing will be true of any fractions we like to
take, and we may state our result in this general
form : —
According as A c is greater or is less than any speci
fied fraction of A B, so will the fourth proportional (if it
exists) be greater or be less than the same fraction of A D.
But we shall now show that this property is of
itself sufficient to define, without ambiguity, the fourth
FIG. 32.
proportional ; that is to say, we shall show that there
cannot be two different lengths satisfying this condition
at the same time.
If possible, let there be two lengths, A E and A E', each
of them a fourth proportional to A B, AC, AD (fig. 32).
Then by taking a sufficient number of lengths each1
QUANTITY.
qua! to E E', the sum of them can be made greater
han A D. Suppose for example that 500 of them
ust fell short of the length A D, and that 501 exceeded
t; then, if we divide AD into 501 equal parts, each of
hese parts will be less than E E'. Secondly, if we go
:>n marking off lengths from D towards E, each equal
o one of these small parts of A D, one of the points of
.ivision must fall between E and E' ; since E E' is
reater than the distance between two of them. Let
his point of division be at F. Then A F is got from
&.V by multiplying by some number or other and then
ividing by 501. If we apply this same process to A B
we shall arrive at a length A G, which must be either
O *
greater or less than AC. If it is less than A c, then the
Deration by which the length A B is made into A G is a
ess amount of stretching than the operation by which
A B is made into A c. Consequently the operation
which turns A D into A F is a less amount of stretching
hail that which gets A E, and also less than that which
jets A E' from A D. Therefore A F must be less than A E,
d also less than A E'. But this is impossible, because
lies between E and E'. And the argument would be
imilar if we had supposed A G greater than A c.
Thus we have proved that there is only one length
hat satisfies the condition that the process of making
^ D into it is greater than all the fractions which are
(ss than the process of making A B into AC, and less
han all the fractions which are greater than this same
>rocess.
Let us note more carefully the nature of this defi-
lition.
First of all we say that if any fraction whatever be
aken, and if it be greater than the ratio of A c to A B, it
also be greater than the ratio of A E to A D, and if
110 THE COMMON SENSE OF THE EXACT SCIENCES.
it be less than the one it will also be less than the
other.
This is a matter which can be tested in regard to
any particular fraction. If a length A E were given t
us as the fourth proportional we could find outwhethe
it obeyed the rule in respect of any one given fraction
But if there is a fourth proportional it must satisf
this rule in regard to all fractions whatever. We can
not directly test this ; but we may be able to give
proof that the quantity which is supposed to be a fourt
proportional obeys the rule for one particular fraction
which proof shall be applicable without change to an
other fraction. It will then be proved, for this case
not only that a fourth proportional exists, but that thi
particular quantity is the fourth proportional. Thi
is, in fact, just what we can do with the sides of siinila
triangles. If the length A B (fig. 33) is divided into airj
number of equal parts, and lines are drawn through the
points of division, making with A B the same angle thai
B D makes with it, they will divide A D into the saint!
number of equal parts.
If now we set off points of division at the sam(
distance from one another from B towards c, anc
through them draw lines making the same angl<
with the line AC that BD does, these lines will als<
cut off equal distances from D towards E. If any on<
of these lines starts from A c on the side of c toward
QUANTITY. 1 1 1
A, it will meet A E on the side of E towards A ; because
be triangle which it forms with the lines A c and A E
must have the same shape as A c E. So also any one of
aese lines which starts from A c on the side of c away
rom A will meet AE oil the side of E away from A.
Looking then at the various fractions of A B which
re now marked off, it is clear that, if one of them
3 less than A c, the corresponding fraction of A D is less
hau A E ; and if greater, greater. It follows, therefore,
hat the line AE which is given by this construction
atisfies, in the case of any fraction we choose, the con-
.ition which is necessary for the fourth proportional.
Consequently, if the second assumption which we made
,bout space be true, there always is a fourth propor-
ional, and this process will enable us to find it.
There is, however, still one objection to be made
against our definition of the fourth proportional, or
rather one point in which we can make it a firmer
ground- work for the study of ratios. For it assumes
hat quantities are continuous ; that is, that any quan-
ity can be divided into any number of equal parts,
his being implied in the process of taking any numer-
cal fraction of a quantity.
We say, for example, that if a, b, c, <7, are propor-
ionals, and if a is greater than three-fifths of b, c will
)e greater than three-fifths of d. Xow the process of
hiding three-fifths of b is one or other of the following
wo processes. Either we divide b into five equal parts
ind take three of them, or we multiply b by three and
livide the result into five equal parts. (We know of
:ourse that these two processes give us the same result.)
But it is assumed in both cases that we can divide ;i
jiven quantity into five equal parts.
Now in a definition it is desirable to assume as
112 THE COMMOX SEXSE OF THE EXACT SCIENCES.
little as possible ; and accordingly the Greek geometers
in defining proportion, or (which is really the same
thing) in denning the fourth proportional of three
given quantities, have tried to avoid this assumption.
Nor is it difficult to do this. For let us conside
the same example. We say that if a is greater thai
three-fifths of b, c will be greater than the same fractio
of d. Now let us multiply both the quantities a and
by five. Then for a to be greater than three-fifths of 6
the quantity which a has now become must be greate
than three-fifths of the quantity which b has become
that is, if the new b be divided into five equal parts th
new a must be greater than three of them. But each o
these five equal parts is the same as the original b ; an
so our statement as to the relative greatness of a and
is the same as this, that five times a is greater thai
three times b ; and similarly for c and d.
Now every fraction involves two numbers. It is
compound process made up of multiplying by on
number and dividing by another, and it is clear there
fore that we may, not only in this particular case o
three-fifths but in general, transform our rule for th
fourth proportional into this new form. According
m times a is greater or less than n times b, so is m times
c greater or less than n times d, where m and n are ani
whole numbers whatever.
This last form is the one in which the rule is givei
by the Greek geometers ; and it is clear that it doe
not depend on the continuity of the quantities con
sidered, for whether it be true or not that we cai
divide a number into any given number of equal parts
we can certainly take any multiple of it that w
like.
These fundamental ideas, of ratio, of the equality c
QUANTITY. US
r;iti»s, and of the nature of the fourth proportional
ire now established generally, and with reference to
quantities of any kind, not with regard to lengths alone ;
provided merely that it is always possible to take any
*iven multiple of any given quantity.
§ 6. Of Areas ; Stretch and Squeeze.
We shall now proceed to apply these ideas to areas,
>r quantities of surface, and in particular to plane areas,
phe simplest of these for the purposes of measurement
!? a rectangle. The finding of the area of a rectangle
5 in many cases the same process as numerical multi-
lication. For example, a rectangle which is 7 inches
)ng and 5 inches broad will contain 35 square inches,
nd this follows from our fundamental ideas about
multiplication of numbers. But this process, the
raltiplication of numbers, is only applicable to the
in which we know how many times each side of
rectangle contains the unit of length, and it then
us how many times the area of the rectangle con-
ins the square described upon the unit of length. It
mains to find a method which can always be used.
For this purpose we first of all observe that when
e side of a rectangle is lengthened or shortened in
ratio, the other side being kept of a fixed length,
.e area of the rectangle will be increased or diminished
exactly the same ratio.
In order then to make any rectangle OP K Q out of a
iare o A c B, we have first of all to stretch the side o A
til it becomes equal to o P, and thereby to stretch the
.ole square into the rectangle o D, which increases its
in the ratio of o A to o P. Then we must stretch
side o B of this figure until it is equal to o Q, and
i
114 THE COMMON SENSE OF THE EXACT SCIENCES.
thereby the figure o D becomes o E, and its area is in
creased in the ratio of o B to o Q. Or we may, if we
FIG. 34.
like, first stretch o B to the length o Q, whereby the
square o c becomes o E, and then stretch o A to 0 P, b;
which o E becomes o E.
Thus the whole operation of turning the square o c
into the rectangle o E is made up of two stretches ; or
as we have agreed to call them, ' multiplications ' ; viz
the square has to be multiplied by the ratio of o P tc
o A, and by the ratio of 0 Q to OB; and we may fine
from the result that the order of these two processes
is immaterial.
For let us represent the ratio of o P to o A by th<
letter a, and the ratio of OQ to OB by b. Then th<
ratio of the rectangle o D to the square o c is also a ; ii
other words, a times o c is equal to o D. And the rati«
of o E to 0 D is b, so that b times o D is equal to o E
that is, b times a times oc is equal to OE, or, as w
write it, b a times o c is o E.1
And in the same way b times o c is equal to 0
and a times 6 times o c is a times o E, which is o E.
1 It is a matter of convention which has grown up in consequence of o>|
ordinary habit of reading from left to right, that we always read tl|
symbols of a multiplication, or of any other operation, from riff hi to le,\
Thus a b times any quantity x, means a times b times x ; that is to say, i|
first multiply x by b, and then by a ; that operation being first perform'
vrhobe symbol comes last.
QUANTITY. 115
Consequently we have b a times 0 c giving the same
result as a 6 times o c ; or, as we write it
6 a = ab,
which means that the effect of multiplying first by the
ratio a and then by the ratio b is the same as that of
multiplying first by the ratio b and then by the
ratio a.
This proposition, that in multiplying by ratios we
may take them in any order we please without affecting
the result, can be put into another form.
Suppose that we have four quantities, a, 6, c, d,
then I can make a into d by two processes performed
in succession ; namely, by first multiplying by the ratio
of b to a, which turns it into b, and then by the ratio
of d to b. But I might have produced the same effect
on a by first multiplying it by the ratio of c to a, which
turns it into c, aud then multiplying by the ratio of d to
c. We are accustomed to write the ratio of 6 to a in
shorthand in any of the four following ways : —
b: a, b, 6-f-a, >/a,
Or
ind so the fact we have just stated may be written
ms : —
V. >< "/> = Va X "/,
Now let us assume that the four quantities, a, b,
d, are proportionals ; that is, that the ratios b/ a and
I1/,, are equal to one another. It follows then that the
Ratios c/0 and d/b are equal to one another.
This proposition may be otherwise stated in this
lorm ; that if a, 6, c, d are proportionals, then a,b, 6, d
nil also be proportionals : provided always that this
itter statement has any meaning, for it is quite possible
lat it should have no meaning at all. Suppose, for in-
ince, that a and 6 are two lengths, c and d two intervals
i 2
116 THE COMMON SENSE OF THE EXACT SCIENCES.
of time, then we understand what is meant by the ratio
of 6 to a, and the ratio of d to c, and these ratios may
very well be equal to one another ; but there is no such
thing- as a ratio of c to a, or of d to 6, because the
quantities compared are not of the same kind. When,
however, four quantities of the same kind are propor
tionals, they are also proportionals when taken alter
nately ; that is to say, when the two middle ones are
interchanged.
§ 7. Of Fractions.
We have seen in § 3, page 101, that a ratio may be
expressed in the form of a fraction. Thus, let a be
D 7*
represented by the fraction^- and b by the fraction ~,
q s
where p, q, r, s are numbers. Then the result on page
115 may be written —
p r r p
£_ x — X — »
q s s q
Let us examine a little more closely into the mean
ing of either side of this equation. Suppose we were
P R' P
) as
FIG. 35.
to take a rectangle o Q T s, of which one side, o Q, con
tained q units of length, and another, OS, s units.
Then this rectangle could be obtained from the unit
square by operating upon it with the two stretches g|
and s. Its area would thus contain q s square units, j
QUANTITY. 1 1 7
Now let us apply to this rectangle in succession the
fV\ A»
two stretches denoted by t and -. If we stretch the
q s
rectangle in the direction of the side o Q in the ratio of
?., we divide the side o Q into q equal parts, and then
1
take o P equal p times one of those parts. But each of
these parts will be equal to unity, hence o P contains
p units. We thus convert our rectangle o T into one
OP', of which one side, OP, contains p and the other,
os, s units. Now let us apply to this rectangle the
stretch - parallel to the side os (as the figure is drawn
s
r denotes a squeeze). We must divide o s into s equal
s
parts and take r such parts, or we must measure a
length o R along o s equal to r units. Thus this second
stretch converts the rectangle o p' into a rectangle
CJ O
o R', of which the side o P contains p and the side
OR contains r units of length, or into a rectangle
containing p r square units. Hence the two stretches
"P- and - applied in succession to the rectangle o T con-
q s
vert it into the rectangle o R'. Now this may be
written symbolically thus : —
*- x - . rectangle o T = rectangle o R
q s
= p r unit-rectangles.
Now unit-rectangle may obviously be obtained from
the rectangle o T by squeezing it first in the ratio in
the direction of OQ, and then in the ratio — in the di-
s
rection o S. Now this is simply saying that o T contains
118 THE COMMON SENSE OF THE EXACT SCIENCES.
q s unit-rectangles. Hence the operation ? x — applied
q s
to unit-rectangle must produce — of the result of its
qs
application to the rectangle o T. That is : —
± x - . unit-rectangle = — . p r unit-rectangle,
q s qs
or, in our notation, = ^— . unit-rectangle.
qs
Hence we may say that ±- x — operating upon unity
is equal to the operation denoted by ^1, or to multi
plying unity by p r and then dividing the result by q s.
This equivalence is termed the multiplication of frac
tions.
A special case of the multiplication of fractions
arises when s equals r. We then have —
P. x - = £T.
q r qr'
¥
But the operation - denotes that we are to divide unity
into r equal parts, and then take r of them ; in other
words, we perform a null operation on unity. The
symbol of operation may therefore be omitted, and we
read —
p pr
q ~^r'
This result is then expressed in words as follows :
Given a fraction, we do not alter its value by multiply
ing the numerator and denominator by equal quanti
ties.
From this last result we can easily interpret the
operation
QUANTITY. 119
2 s
For, by the preceding paragraph —
p ps -, r qr
±- =f__, and - = *-.
q qs s qs
Hence —
P. + t = PJ* + 1*.
q s qs qs
Or, to apply first the operation *- to unity and then to
add to this the result of the operation - is the same
s
thing as dividing unity into qs parts, taking ps of
those parts, and then adding to them q r more of the
like parts. But this is the same thing as to take at
once p s + q r of those parts. Thus we may write —
p r _ ps + qr
q s qs
This result is termed the addition of fractions. The
reader will find no difficulty in interpreting addition
graphically by a succession of stretches and squeezes of
the unit-rectangle.
We term division the operation by which we reverse
the result of multiplication. Hence when we ask the
meaning of dividing by the fraction *- we put the
question : What is the operation which, following on
the operation ^, just reverses its effect?
I -NT r p p r pr
Now, _ x i = i- x -= ^-— .
s q q s q s
Suppose we take r = q, s = p.
120 THE COMMON SENSE OF THE EXACT SCIENCES.
Then VXP=M.
p q qp
or, to multiply unity by ^, and then by 5, is to perform
the operation of dividing unity into qp parts and
then taking p q of them, or to leave unity unaltered.
Hence the stretch i completely reverses the stretch " ;\
P 2|
it is, in fact, a squeeze which just counteracts the
preceding stretch. Thus multiplying by ^ must be an
operation equivalent to dividing by £-. Or, to divide
by ^ is the same thing as to multiply by ^. This result
q p
is termed the division of fractions.
§ 8. Of Areas ; Shear.
Hitherto we have been concerned with stretching
or squeezing the sides of a rectangle. These opera
tions alter its area, but leave it still of rectangular
shape. We shall now describe an operation which
changes its angles, but leaves its area unaltered.
D F C
a A B 6
FIG. 36.
Let A B c D be a rectangle, and let A B E p be a j
parallelogram (or a four- sided figure whose opposite sides
are equal), having the same side, AB, as the rectangle,
but having the opposite side, E p (equal to A B, and
QUANTITY. 121
therefore to CD), somewhere in the same line as CD.
Then, since c D is equal to E p, the points E and p are
equally distant from c and D respectively, and it follows
that the triangles B c E and A D F are equal. Hence if
the triangle B c E were cut off the parallelogram along
B c and placed in the position A D p, we should have
converted the parallelogram into the rectangle without
changing its area. Thus the area of the parallelogram
is equal to that of the rectangle. Now the area of the
rectangle is the product of the numerical quantity which
represents the length of AD into that quantity which
represents the length of A B. A B is termed the bate
of the parallelogram, and A D, the perpendicular dis
tance between its base and the opposite side E F, is
termed its height. The area of the parallelogram is
then briefly said to be ' the product of its base into
its height.'
Suppose c D and A B were rigid rods capable of slid
ing along the parallel lines c d and a b. Let us imagine
them connected by a rectangular elastic membrane,
A B c D ; then as the rods were moved along a b and c d
the membrane would change its shape. It would, how
ever, always remain a parallelogram with a constant
base and height; hence its area would be unchanged,
it the rod A B be held fixed in position, and the rod
D pushed along c d to the position E F. Then any line,
H, in the membrane parallel and equal to A B will be
toved parallel to itself into the position I J, and will
tot change its length. The distance through which
has moved is c E, and the distance through which G
moved is G I. Since the triangles c B E and G B I
ive their sides parallel they are similar, and we have
ie ratio of c E to G i the same as that of B c to n G ;
>r. when the rectangle A B c D is converted into the
122 THE COMMON SENSE OF THE EXACT SCIENCES.
parallelogram A B E F, any line parallel to A B remains
unchanged in length, and is moved parallel to itself
through a distance proportional to its distance from A B.
Such a transformation of figure is termed a shear, and
we may consider either our rectangle as being sheared
into the parallelogram or the latter as being sheared
into the former. Thus the area of a parallelogram is
equal to that of a rectangle into which it may be
sheared.
The same process which converts the parallelogram
A B E F into the rectangle A B c D will convert the tri
angle ABE, the half of the former, into the triangle
FIQ. 37.
ABC, the half of the latter. Hence we may shear any
triangle into a right-angled triangle, and this will not
alter its area. Thus the area of any triangle is half I
the area of the rectangle on the same base, and with
height equal to the perpendicular upon the base from
the opposite angle. This height is also termed the
altitude, or height of the triangle, and we then briefly
say : The area of a triangle is half the product of its base
into its altitude.
A succession of shears will enable us to reduce any
figure bounded by straight lines to a triangle of equal
area, and thus to determine the area the figure encloses
by finally shearing this triangle into a right-angled
QUANTITY. 123
triangle. For example, let A B c D E be a portion of the
boundary of the figure. Suppose A c joined ; then
shear the triangle ABC so that its vertex B falls at B'
on D c produced. The area A B' c is equal to the area
ABC. Hence we may take A B' D E for the boundary of
our figure instead of A B c D E ; that is, we have reduced
be number of sides in our figure by one. By a suc-
ession of shears, therefore, we can reduce any figure
>ounded by straight lines to a triangle, and so find its
rea.
§ 9. Of Circles and their Areas.
One of the first areas bounded by a curved line which
suggests itself is that of a sector of a circle, or the
FIG. 38.
>ortion of a circle intercepted by two radii and the
,rc of the circumference between their extremities.
[Before we can consider the area of this sector it will
necessary to deduce some of the chief properties of
complete circle. Let us take a circle of unit
lius and suppose straight lines drawn at the extre-
lities of two diameters AB and c D at right angles ; then
circle will appear as if drawn inside a square (see
. 39). The sides of this square will be each 2 and
area 4.
Now suppose the figure composed of circle and
juare first to receive a stretch such that every line
124 THE COMMON SENSE OF THE EXACT SCIENCES.
parallel to the diameter A B is extended in the ratio of
a : 1, and then another stretch such that every line
parallel to c D is again extended in the ratio of a : 1.
Then it is obvious that we shall have stretched the
square of the first figure into a second square whose
sides will now he equal to 2 a.
FIG. 39.
It remains to be shown that we have stretched the
first circle into another circle. Let o p be any radius
and p M, PN perpendiculars on the diameters A B, c D/
As a result of the first stretch the equal lengths o M
and N P are extended into the equal lengths o' M' and"!
, ,, , o M N p 1 0. ..
N P, which are such that — - — - = ,—- = _. Similarly,
o M N p' a
as a result of the second stretch M p and o N, which
remained unaltered during the first stretch, are con-1
verted into M' p' and O'N'J so that °-~ = ?LZ = ?.
ON M p' a
During this second stretch o'M'and N'P' remain un-3
altered. Thus as the total outcome of the two stretches
we find that the triangle o p N has been changed into the
triangle o' P' N'. Now these two triangles are of the
same shape by what was said on p. 106, for the angles
at N and N' are equal, being both right angles, and we
have seen that —
QUANTITY. 1 'JO
N P 1 _ O M
N'P' " a O'M''
Thus it follows that the third side o P must be to
third side o' P' in the ratio of 1 to a ; or, since o P
s of unit length, o' P' must be equal to the constant
uantity a. Further, since the angles P o x, P' o' N'
re equal, o' P' is parallel to O P. Hence the circle of
nit radius has been stretched into a circle of radius <i.
n fact, the two equal stretches in directions at right
ngles, which we have given to ' the first figure, have
erformed just the same operation upon it, as if we
ad placed it under a magnifying glass which enlarged
i uniformly, and to such a degree that every line in it
as magnified in the ratio of a to 1.
It follows from this that the circumference of the
econd circle must be to that of the first as a is to 1.
r, the circumferences of circles are as their radii,
gain, if the arc P Q is stretched into the arc P' Q' — that
, if o' P', o' Q' are respectively parallel to o P, o Q — then
le arc P' Q' is to the arc P Q in the ratio of the radii of
he two circles. Since the arcs P Q, P' Q' are equal to
ny other arcs which subtend the same angles at the
mtres of their respective circles, we state generally
lat the arcs of two circles which subtend equal angles at
keir respective centres are in the ratio of the corre-
wnding radii.
Since the second figure is an uniformly magnified
nage of the first, every element of area in the first has
sen magnified at the same uniform rate in the second.
bw the square in the first figure contains four units
'area, and in the second figure it contains 4 a2 units
: area. Hence every element of area in the first
ure has been magnified in the second in the ratio of
to 1. Thus the area of the circle in the first figure
126 THE COMMON SENSE OF THE EXACT SCIENCES, i
must be to the area of the circle in the second figure
as 1 is to a2. Or : The areas of circles are as the squares
of their radii.
It is usual to represent the area of a circle of unit
radius by the quantity TT ; thus the area of a circle of
radius a will be represented by the quantity IT a2.
If, after stretching A B to A' B' in the ratio of a to 1,
we had stretched or squeezed CD to c' D' in the ratio of
6 to 1, where b is some quantity different from a, om
square would have become a rectangle, with sides equal
to 2 a and 2 b respectively. It may be shown that we
should have distorted our circle into the shape of that
shadow of a circle which we have termed an ellipse.
Furthermore, elements of area have now been stretched}
in the ratio of the product of a and b to 1 ; or, the area
of the ellipse is to the area of the circle of unit radius)
as a b is to 1 : whence it follows that the area of the I
ellipse is represented by Trab, where a and b are its
greatest and least radii respectively.
We shall now endeavour to connect the area of a
circle of unit radius, which we have written TT, with the
number of linear units in its circumference. Let us
QUANTITY.
127
take a number of points uniformly distributed round
the circumference of a circle, A B c D E F. Join them in
succession to each other and to o, the centre of the circle,
and draw the lines perpendicular to these radii (or the
tangents) at A B c D E F ; then we shall have constructed
two perfectly symmetrical figures, one of which is said
to be inscribed, the other circumscribed to the circle.
Now the areas of these two figures differ by the sum of
such triangles as A a B, and the area of the circle is
obviously greater than the area of the inscribed and
[ess than the area of the circumscribed figure. Thus
FIG. 41.
;he area of the circle must differ from that of the in-
cribed figure by something less than the sum of all the
ittle triangles A a B, B /9 c, &c. Now from symmetry all
ihese little triangles are equal, and their areas are
iherefore equal to one half the product of their heights,
>r a n, into their bases, or such quantities as A B. Hence
;he sum of their areas is equal to one half of the product
)f an into the sum of the sides of the inscribed figure,
w the sum of the sides of the inscribed figure is
lever greater than the circumference of the circle. If
re take, therefore, a great number of points uniformly
istributed round the circumference of our circle, A and
128 THE COMMON SENSE OF THE EXACT SCIENCES.
B may be brought as close as we please, and the nearer
\ve bring A to B, the smaller becomes a n. Hence, by
taking a sufficient number of points, we can make the
sum of the triangles A a B, B /3 c, &c. as small as we
please, or the areas of the inscribed and circumscribed
figures, together with the area of the circle which lies
between them, can be made to differ by less than any
assignable quantity. In the limit then we may say
that by taking an indefinite number of points we can
make these areas equal. Now the area of the inscribed
figure is the sum of the areas of all such triangles as
A o B, and the area of the triangle A o B is equal to
half the product of its height o n into its base A B ; or
if we write for the ' perimeter,' or sum of all the sides
A B, EC, &c. the quantity p, the area of the inscribed
figure will equal ^ p x on. Again if p' be the sum
of the sides a ft, ft 7, &c. of the circumscribed figure,
its area = | p' x OB.
Since the triangles o a B, o B n are of the same shape,
being right-angled and again equi-angled at o, we have
the ratio of B n to a B, or of their doubles A B to a /3, the
same as that of o n to o B. But p is obviously to p' in
the same ratio as A B to a ft ; hence p is to p' as o n to
o B. By taking a sufficient number of points we can
make o n as nearly equal to o B as we please ; thus we
can make p as nearly equal to p', and therefore either
of them as nearly equal to the circumference of the
circle (which lies between them),1 as we please. Hence
in the limit p will equal the circumference of the circle,
and o n its radius, and we may state that the areas of the
inscribed and circumscribed figures, which approach
nearer and nearer to the area of the circle as we in
crease the number of their sides, become ultimately
1 In the case of the circle the reader will recognise this intuitively.
QUANTITY. 129
equal to each other and to half the product of the cir
cumference of the circle into its radius. This must there
fore be the area of the circle. Hence we have the fol
lowing equality : — The area of a circle of radius a equals
one half its circumference x a. But it equals also ira2 ;
whence it follows that the circumference of a circle
equals TT . '2 a. We may express this result in two
different ways : —
(i) The ratio of the circumference of a circle to its
diameter (2 a) is a constant quantity TT.
(ii) The number of linear units (2 TT) in the cir
cumference of a circle of unit-radius is twice the
number of units of area (TT) contained by that circum
ference.
The value of TT, the ratio of the circumference of a
circle to its diameter, is found to be a quantity which,
like the ratio of the diagonal of a square to its side (f,v
p. 108), cannot be expressed accurately by numbers ;
its approximate value is 3*141 59.
We have now no difficulty in finding the area of
»/
the sector of a circle, for if we double the arc of a
sector we obviously double its area ; if we treble it, we
treble its area ; shortly, if we take any multiple of it,
we take the same multiple of its area. Hence it
follows by § 5, that two sectors are to each other
in the ratio of their arcs, or a sector must be to the
rhole circle in the ratio of its arc to the whole circum
ference.
If we represent by s the area of a sector of a circle
of which the arc contains s units of length and the
radius a units, we may write this relation symboli
cally —
R S
Tra2 2 TT a '
130 THE COMMON SENSE OF THE EXACT SCIENCES.
Thus we deduce s = ^ s x a ; or,
The area of a sector is half the product of the length of its\
arc into its radius.
§ 10. Of the Area of Sectors of Curves.
The knowledge of the area of a sector of a circle!
enables us to find as accurately as we please the area
of a sector whose arc is any curve whatever. Let the
arc P Q be divided into a number of smaller arcs P A, A B,
B c, c D, D Q. We shall suppose that P A subtends the
greatest angle at o of all these arcs. Further we shall
consider only the case where the line OP diminishes]
continuously if P be made to pass along the arc from p
to Q. If this be not the case, the sector QOP can
always be split up into smaller sectors, of which it shall
be true that a line drawn from the point o to the arc con
tinuously diminishes from one side of the sector to the
other, and then for the area of each of these sectors the
following investigation will hold. With o as centre de
scribe a circle of radius o p to meet o A produced in p'; with
the su,me centre and radius OA describe a circle to meet
QUANTITY
OB in A' and OP in a ; similarly circles with radius OB to
meet OA in b and oc in B', with radius oc to meet OB in
e and OD in c', with radius OD to meet oc in d andoQin
D', and finally with radius OQ to meet OD in e, OA in/,
and OP in Q'. Then the area of the sector obviously lie-i
between the areas of the figure bounded by OP, OD' and
the broken line PP' AA'BB'CC'DD', and of the figure
bounded by oa, OQ and the broken line axbttccdveq.
Hence it differs from either of them by less than their
difference or by less than the sum of the areas p'a, A'//,
B'C, c' d, D' e. Now since the angle at POP' is greater
than any of the other sectorial tingles at o, the sum of
all these areas must be less than that of the figure p P'/Q',
and the area of this figure can be made as small as wr
please by making the angle AOP sufficiently small. Tin*
can be achieved by taking a sufficient number of points
like ^,B,C,D, £c. We are thus able to find a series of
circular sectors, the sum of whose areas differs by as
small a quantity as we please from the area of the
sector POQ; in other words, we reduce the problem of
finding the area of anv h'u'uiv bounded by a curved lin^
\t CT1 J
to the problem already solved of finding the area of a
sector of a circle. The difficulties which then ari^e
are purely those of adding together a very great
number of quantities ; for, it may be necessary to take a
hvery great number of points such as A BCD . . . in
>rder to approach with sufficient accuracy to the mag-
itude of the area POQ.
§11. Extension of the Conception of Area.
Let ABCD be a closed curve or loop, and o a point
[.nside it. Then if a point p move round the perimeter
bf the loop, the line OP is sai 1 to trace out the area of
132 THE COMMON SENSE OF THE EXACT SCIENCES.
the loop ABCD. By this is meant that successive poofl
tions of the line o P, pair and pair, form together with!
the intervening elements of arc elementary sectors, the]
sura of the areas of which can, by taking the successive
FIG. 43.
positions sufficiently close, be made to differ as little ai
we please from the area bounded by the loop.
Now suppose the point o to be taken outside th<1
loop ABCD, and let us endeavour to find the area then
FIG. 44.
traced out by the line 0 P joining o to a point P whicl
moves round the loop. Let OB and OD be the extrem
positions of the line OP to the left and to the right a
p moves round the loop ABCD; then as P moves alonj
QUANTITY. 133
the portion of the loop DAB, OP moves counter-clock
wise from right to left and traces out the area bounded
by the arc DAB and the lines OD and OB. Further, as
p moves along the portion of the loop BCD, OP moves
clockwise from left to right and traces out the area
doubly shaded in our h'gure, or the area bounded by
the arc BCD and the lines OB and OD. It is the differ
ence, of these two areas which is the area of the loop
ABCD. If, then, we were to consider the latter area
OBCDO as negative, the line OP would still trace out the
area of the loop ABCD as p moves round its perimeter.
Now the characteristic difference in the method of de
scribing the areas ODABO and OBCDO is, that in the
former case OP moves counter-clockwise round o, in the
latter case it moves clockwise. Hence if we make a con
vention tbat areas traced out by OP when it is moving
counter-clockwise shall be considered positive, but areas
traced out by OP when it is moving clockwise shall be
considered negative, then wherever o may be inside or
outside the loop, the line OP will trace out its area pro
vided P move completely round its circumference.
But it must here be noted that p may describe the
loop in two different methods, either going round it
counter-clockwise in the order of points ABCD, or
clockwise in the order of points A D c B. In the former
case, according to our convention, the greater area
ODAI5O is positive, in the latter it is negative. Hence
we arrive at the conception that <in area men/ have a
sign; it will be considered positive or negative accord
ing as its perimeter is supposed traced out by a point
moving counter-clockwise or clockwise. This extended
conception of area, as having not only magnitude but
tense, is of fundamental importance, not only in many
134 THE COMMON SENSE OF THE EXACT SCIENCES.
brandies of the exact sciences, but also for its many
practical applications.1
Let a perpendicular o N be erected at o (which is,
as we have seen, any point in the plane of the loop)
to the plane of the loop, and let the length o N be
taken along it containing as many units of length as
there are units of area in the loop A B c D. Then o N
will represent the area of the loop in magnitude; it
will also represent it in sense, if we agree that ON shall
always be measured in such a direction from o, that to
a person standing with his feet at o and head at N the
point P shall always appear to move counter-clockwise.
Thus, for a positive area, N will be above the plane ;
for a negative area, in the opposite direction or below
the plane. We are now able to represent any number
of areas by segments of straight lines or steps per
pendicular to their planes. The sum of any number of
areas lying in the same plane will then be obtained by
adding algebraically all the lines which represent these •
areas.
When the areas do not all lie in one plane the j
representative lines will not all be parallel. In this I
case there are two methods of adding areas. We may I
want to know the total amount of area , as, for example, !
when we wish to find the cost of painting or gilding ;
a many-sided solid. In this case we add all the repre- ^
sentative lines without regard to their direction.
In many other cases, however, we wish to find some
quantity so related to the sides of a solid that it can I
only be found by treating the lines which represent
their areas as directed magnitudes. Such cases, for
example, arise in the discussion of the shadows cast by
1 As in calculating the cost of levelling and embanking, in the indicator
diagram, &c. It was first introduced by Mobius.
QUANTITY. 135
the sun or of the pressure of gases upon the sides of
a containing vessel, &c. A method of combining
directed magnitudes will be fully discussed in the
following chapter. The conception of areas as directed
magnitudes is due to Hayward.
§ 12. On the Area of a Closed Tangle.
Hitherto we have supposed the areas we have talked
about to be bounded by a simple loop. It is easy,
however, to determine the area of a combination of
loops. Thus consider the figure of eight in fig. 45 which
has two loops : if we go round it continuously in the
direction indicated by the arrow-heads, one of these
loops will have a positive, the other a negative area, and
(therefore the total area will be their difference, or zero
if they be equal. When a closed curve, like a figure
of eight, cuts itself it is termed a tangle, and the points
where it cuts itself are called knots. Thus a figure of
3ight is a tangle of one knot. In tracing out'the area
)f a closed curve by means of a line drawn from a fixed
Doint to a point moving round the curve, the area may
rary according to the direction and the route by which
Ive suppose the curve to be described. If, however, we
suppose the curve to be sketched out by the moving
point, then its area will be perfectly definite for that
articular description of its perimeter.
We shall now show how the most complex tangle
| lay be split up into simple loops and its whole area
ijetermined from the areas of the simple loops. We
lhall suppose arrow-heads to denote the direction in
•j'hich the perimeter is to be taken. Consider either
If the accompanying figures. The moving line o P
ill trace out exactly the same area if we suppose it
136 THE COMMOX SEXSE OF THE EXACT SCIENCES.
not to cross at the knot A but first to trace out the
loop A c and then to trace out the loop A B, in both
cases going round these two loops in the direction
FIG. 45.
indicated by the arrow-heads. We are thus able in
all cases to convert one line cutting itself in a knot
into two lines, each bounding a separate loop, which
just touch at the point indicated by the former knot.
This dissolution of knots may be suggested to the
reader by leaving a vacant space where the boundaries
of the loops really meet. The two knots in the fol
lowing figure are shown dissolved in this fashion : —
FIG. 46.
The reader will now find no difficulty in separating
the most complex tangle into simple loops. The posi
tive or negative character of the areas of these loops
QUANTITY.
137
be sufficiently indicated by the arrow-heads on
their perimeters. We append an example : —
FIG. 47.
In this case the tangle reduces to a negative loop
a, and to a large positive loop b, within which are two
other positive loops c and d, the former of which con-
FIG. 48.
tains a fifth small positive loop e. The area of the
entire tangle then equals b + c + d + e— a. The
ispace marked s in the n'rst figure will be seen from the
second to be no part of the area of the tangle at all.
138 THE COMMON SENSE OF THE EXACT SCIENCES.
§ 13. On the Volumes of Space-Figures.
Let us consider first the space-figure bounded by
three pairs of parallel planes mutually at right angles.
Such a space-figure is technically termed a * rectangular
parallelepiped,' but might perhaps be more shortly
described as a ' right six-face.' We may first observe
that when one edge of such a right six-face is
lengthened or shortened in any ratio, the other non-
parallel edges being kept of a fixed length, the volume
/
/
l>
ii
-
/i/i
/
/I
!
/
a
/
A,
/
L
/
FIG. 49.
will be increased in precisely the same ratio. Hence,
in order to make any right six-face out of a cube we
have only to give the cube three stretches (or it may '
be squeezes), parallel respectively to its three sets of
parallel edges. Let o A, o B, o c be the three edges of
the cube which meet in a corner o. Let o A be
stretched to o A', so that the ratio of o A' to o A is
represented by a ; then if the figure is to remain right
all lines parallel to o A will be stretched in the same i
ratio. The figure has now become a six- face whose
section perpendicular to o A' only is a square. Now
stretch o B to o B', so that the ratio o B' to o B be
represented by 6, and let all lines parallel to o B be
QUANTITY.
139
increased in the same ratio ; the figure is now a right
six-face, only one set of edges of which are equal to the
edge of the original square. Finally stretch o c to 0 c',
so that o C and all lines parallel to it are increased in
the ratio of o c' to o c, -which we will represent by c.
By a process consisting of three stretches we have thus
converted our original cube into a right six-face. If
the cube had been of unit-volume, the volume of our
six-edge would obviously be abc, and we may show as
in the case of a rectangle (see p. 115) that abc = cba
= bac, &c. ; or the order of multiplying together three
ratios is indifferent. If we term the face A' c' of our
Fio. 50.
right six-face its base and OB' its height, ac will repre
sent the area of its base, and 6 its height, or the volume
of a right six-face is equal to the product of its base
ito its height.
Let us now suppose a right six-face OADCEBFG
receive a shear, or the face B E F G to be moved in its
)wn plane in such fashion that its sides remain parallel
their old positions, and B and E move respectively
ilong B F and E G. If B' E' G' F' be the new position of
face B E G F, it is easy to see that the two wedge-
jihaped figures B E E' B' o c and F G G' F' A D are exactly
jual; this follows from the equality of their corre
sponding faces. Hence the volume of the sheared
140 THE COMMON SENSE OF THE EXACT SCIENCES.
figure must be equal to the volume of the right six-face.
Now let us suppose in addition that the face B' E' G-' F'
is again moved in its own plane into the position B" E"
G" F", so that E' and E' move along B' E' and F' G'
respectively. Then the slant wedge-shaped figures
B' E" F" F' A o and E' E" G" F' D c will again be equal,
and the volume of the six-face B'' E" G" F" A D c o
obtained by this second shear will be equal to the
volume of the figure obtained by the first shear, and
therefore to the volume of the right six-face. But by
means of two shears we can move the face B E G F to
any position in its plane, B" E" G" F", in which its sides
remain parallel to their former position. Hence the
volume of a six- face will remain unchanged if, one of its
faces, o c D A, remaining fixed, the opposite face, B E G F,
be moved anywhere parallel to itself in its own plane.
We thus find that the volume of a six- face formed by
three pairs of parallel planes is equal to the product of
the area of one of its faces and the perpendicular
distance between that face and its parallel. For this
is the volume of the right six-face into which it may
be sheared ; and, as we have seen, shear does not alter
volume.
The knowledge thus gained of the volume of a six-
face bounded by three pairs of parallel faces, or of a
so-called parallelepiped, enables us to find the volume
of an oblique cylinder. A right cylinder is the figure
generated by any area moving parallel to itself in such
wise that any point p moves along a line p p' at right
angles to the area. The volume of a right cylinder is
the product of its height P P' and the generating area.
For we may suppose that volume to be the sum of a
number of elementary right six-faces whose bases, as
at P, may be taken so small that they will ultimately
QUANTITY.
141
completely fill the area ACBD, and whose heights are
all equal to P P'.
i
We obtain an oblique cylinder from the above right
cylinder by moving the face A'C'B'D' parallel to itself
anywhere in its own plane. But such a motion will
only shear the elementary right six-faces, such as P P'.
and so not change their volume. Hence the volume
of an oblique cylinder is equal to the product of its
base, and the perpendicular distance between its faces.
§ 14. On the Measurement of Angles.
Hitherto we have been concerned with quantities of
area and quantities of volume ; we must now turn to
quantities of angle. In our chapter on Space (p. 66)
we have noted one method of measuring angles ; but
that was a merely relative method, and did not lead us
to fix upon an absolute unit. We might, in fact, have
i taken any opening of the compasses for unit angle, and
'determined the magnitude of any other angle by its
I ratio to this anMe. But there is an absolute unit
142 THE COMMON SENSE OF THE EXACT SCIENCES.
which naturally suggests itself in our measurement of
angles, and one which we must consider here, as we
shall frequently have to make use of it in our chapter
on Position.
Let A o B be any angle, and let a circle of radius a
be described about o as centre to meet the sides of this
FIG. 52.
angle in A and B. Then if we were to double the angle
O O
A o B, we should double the arc A B ; if we were to treble
it, we should treble the arc ; shortly, if we were to take
any multiple of the angle, we should take the same
multiple of the arc. We may thus state that angles at
the centre of a circle vary as the arcs on which they
stand. Hence it' 9 and & be two angles, which are
subtended by arcs s and s' respectively, the ratio of 6 to
6' will be the same as that of s to s'. Now suppose 6'
to represent four right angles ; then s' will be the entire
circumference, or, in our previous notation, 2 TT a. We
have thus —
0 = 8
four right angles 2 TT a '
Now it is extremely convenient to choose a unit
angle which shall be independent of the circle upon
which we measure our arcs. We should obtain such
an independent unit if we took the arc subtended by it
QUANTITY. 143
equal to the radius of the circle or if we took s = a.
In this case our unit equals - - of four right angles,
1 7T
= — of two right angles, = '636 of a right angle
7T
approximately.
Thus we see that the angle subtended at the centre
of any circle by an arc equal to the radius is a constant
fraction of a right angle.
If this angle be chosen as the unit, we deduce from
the proportion 6 is to 6' as s is to s', that 6 must be to
unity as s is to the radius a ; or : —
s = a Q.
Thus, if we choose the above angle as our unit of
angle, the measure of any other angle will be the ratio
of the arc it subtends from the centre to the radius ;
but we have seen (p. 125) that the arcs subtended
from the centre in different circles by equal angles are
I in the ratio of the radii of the respective circles.
I Hence the above measurement of angle is independent
'of the radius of the circle upon vjhich we base our
^measurement. This is the primary property of the so-
illed circular measurement of angles, and it is this
rhich renders it of such great value.
The circular measure of any angle is thus the ratio of
the arc it subtends from the cer.tre of any circle to the
lius of the circle. It follows that the circular mea-
mre of four right angles is the ratio of the whole circum
ference to the radius, or equals " - ; that is, equals
!?r. The circular measure of two right angles will
len be TT, of one right angle -( , of three right
a
ijjles — and so on.
144 THE COMMON SENSE OF THE EXACT SCIENCES.
§15. On Fractional Powers.
Before we leave the subject of quantity it will be
necessary to refer once more to the subject of powers
which we touched upon in our chapter on Number
(p. 16).
We there used an as a symbol signifying the result
of multiplying a by itself n times. From this defini
tion we easily deduce the following identity : —
an x av x aq x ar =
For the left hand side denotes that we are first to
multiply a by itself n times, and then multiply this by
ap, or a multiplied by itself p times, and so on. Hence
we may write the left hand side —
(ax ax ax a, . . to n factors)
x (axaxaxa . . to p factors)
x (axaxaxa . . to q factors)
x (axaxaxa . . to r factors) .
But this is obviously equal to(axaxaxax ... to
n+p + q + r factors), or to an + p + q + r.
If b be such a quantity that bn=a, b is termed an nth
root of a, and this is written symbolically b = \/ a.
Thus, since 8 = 23, 2 is a 3rd, or cube root of 8. Or,
again, since 243 = 35, 3 is termed a 5th root of 243.
Now we have seen at the conclusion of our first
chapter that we can often learn a very great deal by
extending the meaning of our terms. Let us now see if
we cannot extend the meaning of the symbol a". Does
it cease to have a meaning when n is a fraction or I
negative ? Obviously we cannot multiply a quantity
by itself a fractional number of times, nor can we do \
QUANTITY. 145
so a negative number of times. Hence the old mean
ing of a", where n is a positive integer, becomes sheer
nonsense when we try to adapt it to the case of n
being fractional or negative. Is then an in this latter
case meaningless?
In an instance like this we are thrown back upon
the results of our definition, and we endeavour to give
to our symbol such a meaning that it will satisfy these
results. Now the fundamental result of our theory of
nteger powers is that —
It ' '==- Cl X CL X fl X Cl X . . .
This will obviously be true however many quantities,
n,p, q, r, AVC take. Now let us suppose we wish to inter-
i
pret G.~ where — is a fraction. We begin by as-
ra
uming it satisfies the above relation, and in order to
arrive at its meaning we suppose that n = p = a
= ?•=...= — , and that there are m such quantities.
Then
n+p+q+r = mx — = / ;
m
i i i
and we find a1 = a™ x am x a7 x . to m factors
-(•*)•
Thus am must be such a quantity that, multiplied by
.tself m times, it equals a1. But we have defined above
p. 144) an mth root of a' to be such a quantity that,
nultiplied m times by itself, it equals a1. Hence we
i_
">ay that a~ is equal to an ?/ith root of a' ; or, as it is
vritten for shortness, —
146 THE COMMON SEXSE OF THE EXACT SCIENCES.
We have thus found a meaning for an when n is a frac
tion from the fundamental theorem of powers.
We can with equal ease obtain from the same
theorem an intelligible meaning for an when n is a
negative quantity.
We have an x a" = an + . Now let us assume
p = — n in order to interpret a ~ n. We find a'1 x a~n
— an ~ n = a° = 1 (by p. 31). Or dividing by a",
.--I.,
a"'
that is to say, a ~ n is the quantity which, multi
plied by a", gives a product equal to unity. The former
quantity is termed the inverse of the latter, or we may
say that a ~ n is the inverse of an. For example, what
is the inverse of 4 ? Obviously 4 must be multiplied
by £ in order that the product may be unity. Hence
4 ~ 1 is equal to £. Or, again, since 4 = 22, we may say
that 2 ~ 2 is the inverse of 4, or 22.
The whole subject of powers — integer, fractional,
and negative — is termed the Theory of Indices, and is
of no small importance in the mathematical investiga
tion of symbolic quantity. Its discussion would, how
ever, lead us too far beyond our present limits. It has
been slightly considered here in order that the reader
may grasp that portion of the following chapter in
which fractional powers are made use of.
147
CHAPTEE IV.
POSITION.
§ 1. All Position is Relative.
THE reader can hardly fail to remember instances when
he has been accosted by a stranger with some such
question as : 'Can you tell me where the 'George ' Inn
lies? ' — ' How shall I get to the cathedral ? ' — ' Where
is the London Road ? ' The answer to the question,
however it may be expressed, can be summed up in the
one word — There. The answer points out the position.
of the building or street which is sought. Practically
the there is conveyed in some such phrase as the follow
ing : 'You must keep straight on and take the iirst
turning to the right, then the second to the left, and
you will find the ' George ' two hundred yards down
the street.'
Let us examine somewhat closely such a question
and answer. ' Where is the ' George ' ? ' We may ex
pand this into : ' How shall I get from liere ' (the point
at which the question is asked) 'to the 'George'?'
This is obviously the real meaning of the query. If the
stranger were told that the ' George ' lies three hundred
paces from the Town Hall down the High Street,
the information would be valueless to the questioner
unless he were acquainted with the position of the
Town Hall or at least of the High Street. Equally idle
i. 2
14S THE COMMON SENSE OF THE EXACT SCIEXCES.
would be the reply : ' The * George ' lies just past
the forty-second milestone on the London Road,' sup
posing- him ignorant of the whereabouts of the London
Road.
Yet both these statements are in a certain sense
answers to the question : ' Where is the ' George ' ? '
They would be the true method of pointing out the
there, if the question had been asked in sight of the
Town Hall or upon the London Road. We see, then,
that the query, Where ? admits of an infinite number
of answers according to the infinite number of posi
tions—or possible heres — of the questioner. The ivhere
always supposes a definite here, from which the desired
position is to be determined. The reader will at once
recognise that to ask, ' Where is the ' George ' ? '
without meaning, ' Where is it with regard to some
other place ? ' is a question which no more admits of an
answer than this one : ' How shall I get from the
'George' to anywhere?' meaning to nowhere in
particular.
This leads us to our first general statement with
regard to position. We can only describe the where
of a place or object by describing how we can get at it
from some other known place or object. We determine
its where relative to a here. This is shortly expressed
by saying that : All position is relative.
Just as the ' George ' has only position relative to
the other buildings in the town, or the town itself
relative to other towns, so a body in space has only
position relative to other bodies in space. To speak of
the position of the earth in space is meaningless unless
we are thinking at the same time of the Sun or of
Jupiter, or of a star — that is, of some one or other
of the celestial bodies. This result is sometimes
POSITION. 149
described as the ' sameness of space.' By this we only
mean that in space itself there is nothing perceptible to
the senses which can determine position.1 Space is, as
it were, a blank map into which we put our objects; it
is the coexistence of objects in this map which enables
us at any instant to distinguish one object from another.
This process of distinguishing, which supposes at least
two objects to be distinguished, is really determining
a this and a that, a here and a there; it involves the
conception of relativity of position.
§ 2. Position may be Determined by Directed Steps.
Let us turn from the question : ' Where is the
* George'?' to the answer: ' You must keep straight on
and take the first turning to the right, then the second
to the left, and you will find the ' George ' 200 yards
down the street.'
The instruction ' to keep straight on ' means to keep
in the street wherein the question has been asked, and
in a direction (' straight on ') suggested by the previous
motion of the questioner, or by a wave of the hand from
the questioned. Assuming for our present purpose
that the streets are not curved, this amounts to : Keep
a certain direction. How far? This is answered by the
second instruction : Take the first turning on the right.
More accurately we might say, if the first turning to the
right were 150 yards distant: Kt.-ep this direction for
150 yards. Let this be represented in our figure by the
step A B, where A is the position at which the question
is asked. At B the questioner is to turn to the right
and, according to the third instruction, he is to pass the
first turning to the left at c and take the second at D.
1 We shall return to thin point later.
150 THE COMMON SENSE OF THE EXACT SCIENCES.
More accurately we might state the distance B D to be,
say, J80 yards. Then we could combine our second and
third instructions by saying : From B go 180 yards in
a certain direction, namely, B D. To determine exactly
what this direction B D is with regard to the first direction
A B, we might use the following method. If the stranger
did not change his direction at B, but went straight on
for 180 yards, he would come to a point D'. Hence if
we measured the angle D'B D between the street in which
the question was asked and the first turning to the right,
FIG. 53.
we should know the direction of B D and the position of
D exactly. It would be determined by rotating B D'
about B through the measured angle D'B D. If we adopt
the same convention for the measurement of positive
angles as we adopted for positive areas on p. 133, the
angle D'B D is the angle greater than two right angles
through which B D' must be rotated counter-clockwise
in order to take it to the position B D. Let us term this
angle D'B D for shortness yS, then we may invent a new
symbol {/?} to denote the operation : Turn the direction
you are going in through an angle fi counter-clockwise.
POSITION 151
If we use the symbol 7r/2 to denote an angle equal to a
right angle, we have the following symbolic instructions :
{ 0 } = Keep straight on.
{ 7T/2 } = Turn at right angles to the left.
{ TT } = Turn right round and go back.
{3?r/2 } = Turn at right angles to the right.
Thus for a turning from A B to the left the angle of
our symbolic operation will be less, for a turning from
A B to the right greater, than two right angles.
If the directed person had gone to D' instead of to
D, he would have walked 150 yards to B and then 180
yards to D' ; he would thus have walked AB + BD', or
150 yards + 180 yards. In order to denote that he is
not to continue straight on at B we introduce the opera
tor of turning, namely {/3} , before the 180 yards, and
read 150 + {/3| 180 as the instruction: Go 150 yards
along some direction A B, and then, turning your direc
tion through an angle {3 counter-clockwise, go 180
yards along this new direction.
We are now able to complete the symbolic expression
of our instructions for finding the ' George.' The
fourth instruction runs : Take a turning at D to the
left and go 200 yards along the direction thus de
termined. Let D G' represent 200 yards measured
from D along B D produced, then we are to revolve D G'
through a certain angle G'D G counter-clockwise, till it
takes up the position D G. Then G will be the position
of the ' George.' Let the angle G'D G be represented
by 7. Our final instruction may be then expressed
symbolically by {7} 200.
Hence our total instruction may be written symboli
cally—
150 + {/3}180 + (7/200,
where the units are yards.
152 THE COMMON SENSE OF THE EXACT SCIENCES.
But we have not yet quite freed this symbolic in
struction from any suggestion of direction as determined
by streets ; the first 150 yards are still to be taken along
the street in which the question is asked. We can get
rid of this street by supposing its direction determined
by the angle which a clock-hand must revolve through
counter-clockwise, to reach that direction, starting from
some other fixed or chosen direction. For example,
suppose the stranger to have a compass with him, and
at A let AN be the direction of its needle. Then we
might fix the position of the street A B by describing it
as a direction so many degrees east of north, or still to
preserve our counter-clockwise method of reckoning
angles, we might determine it by the angle a which
the needle would have to describe through west and
south to reach the position A B. We should then in
terpret the notation {a.} 150 : Walk 150 yards along a
direction making an angle a with north measured
through west.
Our answer expressed symbolically is now entirely
cleared of any conception of streets. For,
(a) 150 + {/3}180 + {7} 200
is a definite instruction as to how to get from A to G
quite independent of any local characteristics. It ex
presses the position of G with regard to A in a purely
geometrical fashion, or by a series of directed steps.
Expanded into ordinary English our symbols read :
From a point A in a plane, take a step A B of 150 units
in a direction making an angle a with a fixed direction,
from B take a step B D of 180 units making an angle /3
with A B, and finally from D take a step D G of 200 units
making an angle 7 with B D. All the angles are to be
measured counter-clockwise in the fashion we have
described above.
POSITION. 153
§ 3. The Addition of Directed Steps or Vectors.
If we now compare our figure with the symbolical
instruction {a} 150 + {/8J180 + {7} 200, we see that
{a} 150 represents the step A B, when that step is
considered to have not merely magnitude but also
direction. Similarly B D and D G represent more than
linear expressions for number — they are also directed
steps. We shall then be at liberty to replace our
symbolically expressed instruction
{a} 150 + {/3J1SO 4- {7} 200
by the geometrical equivalent
AB + BD + D G,
provided we understand by the segments A B, B D, D G
and the symbol + something quite different to our
FIG. 54.
former conceptions. We give a new and extended
meaning to our quantity and to our addition.
AB + BD+DG no longer directs us to add the
number of units in B D to that in A B and to the sum of
these the number in D G, but it bids us take a step A B in
a certain direction, then a step B D from the finish of
the former step in another determined direction, and
finally from the finish D of this second step a third
154 THE COMMOX SEXSE OF THE EXACT SCIEXCES.
directed step, D G. The entire operation brings ua
from A to G. Now it is obvious that we should also
have got to G had we taken the directed step AG.
Hence, if we give an extended meaning to the word
'equal' and to its sign =, using them to mark the
equivalence of the results of two operations, we may
write
AG = AB + BD + DG,
and read this expression : — A G equals the sum of A B,
B D and D G.
Steps such as we considered in our chapter on
Quantity, which were magnitudes taken along any one
straight line, are termed scalar steps, because they have
relation only to some chosen scale of quantity. We
add or subtract scalar steps by placing them end to end
in any straight line (see § 2 of Chapter III.)
A step which has not only magnitude but direction
is termed a vector step, because it carries us from one
position in space to another. It is usual to mark by an
arrow-head the sense in which we are to take this
directed step. For example in fig. 54 we are to step
from A to B, and thus the arrow-head will point towards
B for the step A B. In letters this is denoted by writing
A before B. The method by which we have arrived at
the conception of vector steps shows us at once how to
add them.
Vector steps are added by placing them end to end
in such fashion that they retain their own peculiar
directions, and so that a point moving continuously
along the zigzag thus formed will always follow the
directions indicated by the arrow-heads. This may be
shortly expressed by saying the steps are to be arranged
in continuous sense. The sum of the vector steps is
then the single directed step which joins the start of
POSITION.
the zigzag thus formed to its finish. In fig. 55 let a I, c <7,
ef, and y h be directed steps. Then let A B be drawn
equal and parallel to a b ; from B draw B c equal and
parallel to cd, from c draw c D equal and parallel to ef,
and finally from D draw D E equal and parallel to g h.
We have drawn our zigzag so that the arrow-heads all
have ' a continuous sense.' Hence the directed step
A E is the sum of the four given vert < »rs. If, for example,
at c we had stepped c D', equal and parallel to ef, but on
the opposite side of EC to c D, and then taken D'E',
Fio. 55.
equal and parallel to gli, the reader will remark at once
that the arrow-heads in B c, c D' and B'F/ are not in
continuous sense, or we have not gone in the proper
direction at c.
Should the vector steps all have the same direction,
the zigzag evidently becomes a straight line ; in this
case the vector steps are added precisely like scalar
quantities; or, when vector steps may be looked upon
as scalar, our extended conception of addition takes the
ordinary arithmetical meaning.
We can now state a very important aspect of position
156 THE COMMON SENSE OF THE EXACT SCIENCES.
in a plane ; namely, if the position of G relative to A
be denoted by the directed step or vector A G, it may
also be expressed by the sum of any number of directed
steps, the start of the first of such steps being at A and
the finish of the last at G (see fig. 56). We may write
this result symbolically :—
AG = AB + EC + CD + D E + EP + FG.
It will be at once obvious that in our example as to
finding the 'George,' the stranger might have been
directed by an entirely different set of instructions to
FIG. 56.
his goal. In fact, he might have been led to make
extensive circuits in or about the town before he reached
the place he was seeking. But, however he might get
to G, the ultimate result of his wanderings would be
what he might have accomplished by the directed step
A G supposing no obstacles to have been in his way (or,
'as the crow flies'). Hence we see that with our
extended conception of addition any two zigzags of
directed steps, A B c D E F G and A B' c' D' E' F' G (which
may or may not contain the same number of com
ponent steps), both starting in A and finishing in G,
POSITION. 157
must be looked upon as equivalent instructions ; or, we
must take
AB + EC + CD + DE + EF + FG = AG =
AB' + B'C' + cV + D'E' -(- E'F' + F'G.
In other words, two sets of directed steps must be
held to have an equal sum, when, their starts being
the same, the steps of both sets will, added vector-wise
have the same finish.
Now let us suppose our stranger were unconsciously
standing in front of the ' George ' when he asked his
question as to its whereabouts, and further let us sup
pose that the person who directed him gave him a per
fectly correct instruction, but sent him by a properly
chosen set of right and left turnings a considerable
distance round the town before bringing him back to
the point A from which he had set out. In this case
we must suppose the ' George ' not to be at the point
G, but at the point A. The total result of the stranger's
wanderings having brought him back to the place from
which he started can be denoted by a zero step ; or
we must write (fig. 56) —
We may read this in words : The sum of vector steps
which form the successive sides of a closed zigzag is
zero. Now we have found above that—
Hence, in order that these two statements (i) and (ii)
•may be consistent, we must have — G A equal to A G, or
AG + G A = 0.
This is really no more than saying that if a step be
taken from A to G, followed by another from G to A, the
total operation will be a zero step. Yet the result is
158 THE COMMON SENSE OF THE EXACT SCIENCES.
interesting as showing that if we consider a step from
A to G as positive, a step from G to A must be considered
negative. It enables us also to reduce subtraction of
vectors to addition. For if we term the operation
denoted by A B — D c a subtraction of the vectors A TS
and DC, since D c + c D = 0, the operation indicated
amounts to adding the vectors A B and c D, or to
A B + c D. Hence, to subtract two vectors, we reverse
the sense of one of them and add.
TJ R P JS Q TV
FIG. 57.
The result A G 4- G A = 0 can at once be extended to
any number of points lying on a straight line. Thus, if
pQESTUvbea set of such points —
For starting from p and taking in succession the steps
indicated, we obviously come back to P, or have per
formed an operation whose result is equivalent to zero,
or to remaining where we started.
§ 4. The Addition of Vectors obeys the Commutative
Law.
We can now prove that the commutative law holds
for our extended addition (see p. 5). First, we can
show that any two successive steps may be interchanged.
Consider four successive steps, A B, B c, CD, and D E.
It' at B instead of taking the step B c we took a step
B H equal to C D in magnitude, sense and direction, we
could then get from H to D by taking the step H D.
Now let B D be joined ; then in the triangles B H D, D c B
the angles at B and D are equal, because they are formed
by the straight line B D falling on two parallel lines B H
rosmox. 159
and c D ; also the side B D is common, and B H is equal
to c D. Hence it follows (see p. 73) that these triangles
are of the same shape and size, or H D is equal to B c ;
and again the angles B D H and DBG are equal, or H D
and B c are parallel. Thus the step H D is equal to the
step B c in direction, magnitude and sense. We have
then from the two methods of reaching D from B,
EC + C D = B D = BH + HD
= C D f B C
by what we have just proved.
FIG. 58.
Hence any two successive steps may be inter
changed. By precisely the same reasoning as we have
used on p. 11 we can show that if we may inter
change any two successive steps of our zigzag we may
interchange any two steps whatever by a series of
changes of successive steps ; that is, the order in
which vectors are added is indifferent.
The importance of the geometry of vectors arises
from the fact that many physical quantities can be re
presented as directed steps. We shall see in the suc
ceeding chapter that velocities and accelerations are
quantities of this character.
§ 5. On Methods of Determining Position in a Plane.
It has been remarked (see p. 99) that scalar
quantities may be treated as steps measured along a
160 THE COMMON SENSE OF THE EXACT SCIENCES.
straight line. In this case we only require one point on
this line to be given, and we can determine the relative
position of any other by merely stating the magnitude
of the intervening step. A line is occasionally spoken
of as being a space of one dimension ; in one-dimensioned
space one point suffices to determine the relative posi
tion of all others.
When we consider however position in a plane, in
order to determine the whereabouts of a point p with
regard to another A we require to know not only the
magnitude but the direction of the step A P. Hence
what scalar steps are to one-dimensioned space, that
FIG. 59.
are vector steps to plane space. In order to deter
mine the direction of a step A p we must know at
least one other point B in the plane. Space which
requires two points to determine the position of a third
is usually termed space of two dimensions. There are
various methods in general use by which position in
two-dimensioned space is determined. We shall men
tion a few of them, confining our remarks however to
the plane, or to space of two dimensions which is of
the same shape on both sides.
(a) We may measure the distances between A and
p and between B and p. If these distances are of
POSITION. 101
scalar magnitude r and r' respectively, there will be
:wo points corresponding to any two given values of
r and r' ; namely P and p' the intersections of the two
jircles with centres at A and B and radii equal to r and
ft respectively. We may distinguish these points as
jeing one above, and the other below A B. Only in
:he case of the circles touching will the two points
coincide; if the circles do not meet, there will be no
point.
If p moves so that for each of its positions with re
gard to A and B the quantities /• and /•' satisfy some defi
nite relation, we shall obtain a continuous set of points
in the plane or a curved line of some sort. For example,
if we fasten the ends of a bit of string of length I to
FIG. 60.
pins stuck into the plane of the paper at A and B, and
:hen move a pencil about so that its point p always
remains on the paper, and at the same time always
xeeps the string A P B taut round its point, the pencil
will trace out that shadow of the circle which we have
called an ellipse.
In this case r + r' = AP + PB = I, the constant length
of the string. This relation r + r' = I is an equation
3etween the scalar quantities r,r' and /, which holds
for every point on the ellipse, and expresses a metric-
property of the curve with regard to the points A and B.
If on the other hand we cause P to move so that the
lifference of A P and B p is a constant length (r — / = /),
then p will trace out the curve we have termed the
M
1G2 THE COMMON SENSE OF THE EXACT SCIENCES.
hyperbola. We can cause p to move in this fashion by
means of a very simple bit of mechanism. Suppose a
rod B L capable of revolving about one of its ends B : let
a string of given length be fastened to the other end
L and to the fixed point A. Then if, as the rod is
moved round B, the string be held taut to the rod by a
Fio. 61.
pencil point p, the pencil will trace out the hyperbola.
For since LP + PA equals a constant length, namely
that of the string, and L P + p B equals a constant length,
namely that of the rod, their difference or PA — PB is
equal to the constant length which is the difference of
the string and the rod.
FIG. 62.
The points A and B are termed in the cases of both,
ellipse and hyperbola the foci. The name arises from
the following interesting property. Suppose a bit of
polished watch spring were bent into the form of an
ellipse so that its flat side was turned towards the foci
of the ellipse ; then if a hot body were placed at one
focus B, all the rays of heat or light radiated from I
POSITION.
1G3
which fell upon the spring would be collected, or, as it is
termed, ' f ocussed ' at A ; hence A wotild be a much
brighter and hotter point than any other within the
ellipse (B of course excepted). The name focus is
from the Latin, and means a fireplace or hearth.
This property of the arc of an ellipse or hyperbola, that
it collects rays radiating from one focus in the other,
depends upon the fact that A p and p B make equal
angles with the curve at P. This geometrical relation
corresponds to a physical property of rays of heat and
igdit; namely, that they make the same angle \\ifh •
reflecting surface when they reach it and when tliev
leave it.
A third remarkable curve, which is easily obtained
from this our first method of considering position, i ;
the lemniscate of James Bernoulli (from the Latin
lemniscus, a ribbon). It is traced out by a point p which
moves so that the rectangle under its distances from A
and B is always equal to the area of a given square
(r./ = c2). If the given square is greater than the
square on half A B, it is obvious that P can never
cross between A and B ; if it is equal to the square
on half A B, the lemniscate becomes a, figure of oigh' :
while if it is less, the curve breaks up into two loops
In our figure a series of lemniseates are represented.
A set of curves obtained by varying a constant, like the
M 2
1G4 THE COMMON SEN7SE OP THE EXACT SCIENCES.
given square in the case of the lemniscate, is termed a
family of curves. Such families of curves constantly
occur in the consideration of physical problems.
§ 6. Polar Co-ordinates.
(13} The points A and B determine a line whose
direction is A B. If we know the length A p and the
angle BAP, we shall have a means of finding the
position of P. Let r be the number of linear units in
A P and Q the number of angular units in BAP, where
r and 6 may of course be fractions. In measuring the
angle 6 we shall adopt the same convention as we have
employed in discussing areas (see p. 134) ; namely, if a
line at first coincident with A B were to start from
FIG. 64.
that position, and supposed pivoted at A to rotate
counter-clockwise till it coincided with A p, it would
trace out the angle 6. Angles traced out clockwise will
like areas be considered negative. Thus the angle BAP'
below AB would be obtained by a rotation clockwise
from AB to AP', and must therefore be treated as
negative. On the other hand, we might have caused a
line rotating about A to take tip the position A p' by
rotating it counter clockwise through an angle marked
in our figure by the dotted arc of a circle, Further we
POSITION. ] 65
might obviously have reached A p by a line rotating
about A clockwise, and might thus represent the position
of P by a negative angle. But even after we had got to
p we might cause our line to rotate about A a complete
n Timber of times either clockwise or counter-clockwise,
and we should still be at the end of any such number
of complete revolutions in the same position A p.
We have then the following four methods of rotating
~ r*
a line about A from coincidence with A B to coincidence
vith A P : —
(i) Counter-clockwise from A B to A P.
(ii) Clockwise from A B to A p.
(iii) The first of these combined with any number
of complete revolutions clockwise or counter
clockwise.
(iv) The second of these combined with any number
of complete revolutions clockwise or counter
clockwise.
The following terms have been adopted for this
method of determining position in space : —
The line A B from which we begin to rotate our line is
iermed the initial ('beginning') line; the length AP is
;ermed the null us vector (from two Latin words signify -
ng the carrying rod or spoke, because it carries the
ooint P to the required position); the angle BAP is
ermed the rectorial angle, because it is traced out by
lie radius vector in moving from A B to the required
position A P ; A is termed the pole, because it is the end
)f the axis about which we may suppose the spoke to
;urn. Finally A p (= r] and the angle B A p (= 0} are
•ermed the polar co-ordinates of the point p, because
hey regulate the position of P relative to the pole A and
he initial line AB.
166 THE COMMON SENSE OF THE EXACT SCIENCES.
§ 7. The Trigonometrical Ratios.
If p M be a perpendicular dropped from p on A B, the
ratios of the sides of the right-angled triangle PAM
have for the purpose of abbreviation been given the
following names : —
P Af
— , or the ratio of the perpendicular to the hypo-
A Jr
thenuse, is termed the sine of the angle BAP.
— , or the ratio of the base to the hypothenuse, is
termed the cosine of the angle BAP.
P M
— , or the ratio of the perpendicular to the base, is
termed the tangent of the angle BAP.
— , or the ratio of the base to the perpendicular, is
PM
termed the cotangent of the angle BAP.
If 6 be the scalar magnitude of the angle BAP these
ratios are written for shortness, sin6, cos6, tand, and
cotO, respectively. Let us take any other point Q on A P,
and drop QN perpendicular to A B, then the triangles
Q A N, PAM are of the same shape (see p. 106), and thus
the ratios of their corresponding sides are equal. It
follows from this that the ratios sine, cosine, tangent,
and cotangent for the triangles Q A N and PAM are the
same. Hence we see that sin#, cos#, tan#, and cot0
are independent of the position of p in AP; they are
ratios which depend only on the magnitude of the angle
B A P or 6. They are termed (from two Greek words
meaning triangle-measurement) the trigonometrical
ratios of the angle 6. The discussion of trigonometrical
ratios, or Trigonometry, forms an important element of
POSITION. ]G7
pure mathematics. The names of the trigonometrical
ratios themselves are derived from an older terminology
which connected these ratios with the figure supposed
to be presented by an archer whose bow string was
placed against his breast.1
§ 8. Spirals.
Let us suppose the spoke A p to revolve about the pole
A, and as it revolves let the point p move along the spoke
in such fashion that the magnitude r of A p is always de
finitely related in some chosen manner to the magnitude
6 of B A P. Then if p be taken as the point of a pencil
it will mark out a curved line on the plane of the paper.
FIG. 65.
Such a curved line is termed, a. polar curve or spiral,
the latter name from a Greek word denoting the coil,
as of a snake, to which some of these curves may be
considered to bear resemblance.
One of the most interesting of these spirals was
invented by Conon of Samos (/. B.C. 250), but its
1 In our figure the angle BAP has been taken /»>•.-• than a right
it may have any niairnitude whatever. It has been found u.-et'ul to establish
a convention with ivaranl to the signs of the perpendicular i' M and th-
AM. psiis considered positive when it falls above, but in-native when it
falls below the initial line AB; AM is considered pc>iiive wh^n M falls to
the right, but negative when it falls to the left of A. The reader will under
stand the value of this convention better after examining £§ 11, 12.
168 THE COMMON SENSE OF THE EXACT SCIENCES.
chief properties having been discussed by Archimedes,
it is usually called by his name. The spiral of Archi
medes is defined in the following simple manner. As
the spoke A p moves uniformly round the pole, the point
p moves uniformly along the spoke. Let c be the posi
tion of P when the spoke coincides with the starting
line A B, and let A C contain a units of length. Then if p
be the position of the pencil-point when the spoke has
described an angle BAP containing 6 units of angle, and
if A c' be measured along A P equal to A c, the point will
have described the distance c' P while the spoke was
turning through the angle CAP. But since the point
and spoke are moving uniformly, the distance c' p must
be proportional to the angle CAP, or their ratio must
be an unchangeable quantity for all distances and
angles. Let b be the distance traversed by the point
along the spoke while it turns through unit angle,
then c' P must be equal to the number of units in c A P
multiplied by b. Using r to denote the magnitude of
A P we have
c' p = b x 6, but c' P = r — a ;
Thus : r = a + b 6.
This relation between r and 6 is termed the polar equa
tion to the spiral.
The following easily constructed apparatus will
enable us to draw a spiral of Archimedes. D E F is a
circular disc of chosen radius ; upon the edge of this
disc is cut a groove. To the centre A of the disc is
attached a rod or spoke which can be revolved about A
as a pole ; at the other end of this rod is a small grooved
wheel or pulley G. A string is then fastened to some
point D in the groove of the disc, and passing round
the pulley G is attached to a small block p which holds
POSITION.
169
a pencil and is capable of sliding in a slot in the spoke.
If this block be fastened by a piece of elastic to A, the
string from p to G and then from G to the groove on the
disc will remain taut. Now supposing the disc to be
held firmly pressed against the paper, and the spoke
A c to be turned about A counter-clockwise, the pencil
P will describe the required spiral. For the string
touching the disc in the point T the figure GAT always
remains of the same size and shape as we turn the
spoke about the pole ; hence the length of string G T is
FIG. 66.
constant. Thus if a length of string represented by
the arc D T be wound on to the disc as we turn the
spoke from the position A B to the position A P, the
length P G (since the length G T always remains the
same) must lose a length equal to D T as P moves from
0 to P. But the amount of string D T wound on to the
disc is proportional to the angle through which the
spoke A P has been turned ; hence the point p must have
moved towards G through a distance proportional to
this angle, or it has described a spiral of Archimedes.
170 THE COMMON SENSE OF THE EXACT SCIENCES.
Once in possession of a good spiral of this kind we
can solve a problem which often occurs, namely to divide
an angle into any number of parts having given ratios.
Let the given angle be placed with its vertex at the
pole of the spiral and let the radii vectores A c and AP
be those which coincide with the legs of the angle.
About the pole A describe a circular arc with radius A c to
meet A p in c'. Now let us suppose the problem solved
and let the radii vectores A D, A E, A F be those which
divide the angle into the required proportional parts.
If these radii vectores meet the circular arc c c' in D', E',
FIG. 67.
F' respectively, then by the fundamental property of the
spiral we have at once the lines D'D, E'E, F'F, C'P in the
same ratio as the angles CAD, c A E, c A F, CAP. Thus
if we measure lengths A d, A e, A/ equal to A D, A E, A F
respectively along AP, c' P will be divided in definto
lengths which are proportional to the required angles.
Conversely, if we were to divide C'P into segments c'd, d e,
ef, and/p in the same ratio as the required angular
division, we should obtain lengths Ad, AS, A/, which
would be the radii of circles with a common centre
A cutting the spiral in the required points of angular
division. The spiral of Archimedes thus enables us to
POSITION. 171
1-educe the division of an angle in any fashion to the
like division of a line.
Now the division of a line in any fashion, that is,
into a set of segments in any given ratio, is at once
solved so soon as we have learnt by the aid of a pair of
compasses or a ' set square ' to draw parallel lines. Thus
suppose -\ve require to divide the line c'p into segments in
the ratio of 3 to 5 to 4 ; we have only to mark off along
any line through c', say C'Q, steps C'R, R s, s T placed end
to end and containing 3, 5, and 4 units of any kind respec
tively. If the finish of the last step T be joined to P
0,
and the parallels Rr, ss to T P through R and s be drawn
to meet c'p in r and s, then C'P will be divided in r and s
into segments in the required ratio of 3 to 5 to 4. This
follow.-; at once from our theory of triangles of the same
shape (see p. 10(5). For, since R c' r, s c'.s', and T C'P are
such triangles, they have their corresponding sides pro
portional, and the truth of the proposition is obvious.
A spiral of Archimedes accurately cut in a metal or
ivory plate is an extremely useful addition 1o the ordi
nary contents of a box of so-called mathematical instru
ments.
§ 9. The Equiangular Spiral.
Another important spiral was invented by Descartes,
land is termed from two of its chief properties either the
equiangular or the logarithmic
172 THE COMMON SENSE OF THE EXACT SCIENCES.
Let B o A be a triangle with a small angle at o, and
whose sides o A and o B are of any not very greatly differ
ent lengths. Upon o B and upon the opposite side of it
to A construct a triangle BOG ot the same shape as the
triangle A o B, and in such wise that the angles at B and
A are equal. Then upon o c place a triangle COD of the
same shape as either BOG or A 0 B ; upon o D a fourth
triangle DOE, again of the same shape ; upon o E a fifth
triangle, and so on. We thus ultimately form a figure
consisting of a number of triangles A o B, B o c, c 0 D,
V
FIG. 69.
DOE, &c., of the same shape, all placed with one of their
equal angles at o, and in such fashion that each pair
has a common side consisting of two non-corresponding
sides (that is, of sides not opposite to equal angles) . The
points A B c D E, &c., will form the angles of a polygonal
line, and if the angles at o are only taken small enough,
the sides of this polygon will appear to form a continuous
curved line. This curved line, to which we can approach
as closely as we please by taking the angles at o smaller
and smaller, is termed an equiangular spiral. It derives
its name from the following property, — A B, B c, c D, &c.,
POSITION. 1"3
being corresponding sides of triangles of the same shape,
make equal angles o B A, o c B, o D c, &c., with the cor
responding sides o B, o c, o D, &c. ; but when the angles
at o are taken very small A B, B c, c D, &c., will appear as
successive elements of the curved line or spiral. Hence
the arc of the spiral meets all rays from the pole o at
the same constant angle.
Let us now endeavour to find the relation between
any radius vector OP ( = r) and the vectorial angle A o P
0);
Since all our triangles A o B, BOG, COD, &c., are of
the same shape, their corresponding sides must be pro
portional (see p. 100) ; or,
OB _ OC OD _ OE _ OP o
OA OB 00 OD OE
Each of these equal ratios will therefore have the snme
scalar value ; let us denote that value by the symbol p.
Then we must have
O B = /i . O A ; O C = /i . O B ; O D = /4 . 0 C ; &C.
Or, OB = /A.OA;oo=/i2.OA;OD = /i3 . o A, and so on.
Hence if o N be the radius vector which occurs after n
equal angles are taken at o, we must have
ON = /j,n . o A.
Now let the very small angles at o be each taken
equal to some small part of the unit angle ; thus we
might take them yJ -^ or - (l\-- of the unit angle. We
will represent this fraction of the unit angle by 1/6,
where we may suppose b a whole number for greater
simplicity. Further let us use X to denote the 6th
power of fj., or A, = /A With the notation explained on
p. 144 we then term /a, a 6th root of X, and write
174 THE COMMON SEXSE OF THE EXACT SCIENCES.
Hence finally we have o N = o A . \n * 1/7), or in words :
The base of the nih equal-shaped triangle placed about
o is equal to the base of the first multiplied by a
certain quantity A, raised to the power of ft-times the
quantity 1/6 which expresses the magnitude of the
equal angles at o in units of angle.
Now let the spoke or ray o P fall within the angle
which is formed by the successive rays o N and o Q of
the system of equal-shaped triangles round o. Then o N
makes an angle n-times 1/6, and o Q an angle (n-t
times 1/6 with o A. Hence the angle A o P, or 0, must lie
in magnitude between n/b and (n + 1)/ 6. Similarly the
magnitude of o P must lie between those of o N and o Q.
Now by sufficiently decreasing the angles at o we can
approach nearer and nearer to the form of the spiral,
and the ray o P must always lie between two successive
rays of our system of triangles. The angle 6, which will
thus always lie between njb and (u-fl)/6, can only
differ from either of them by a quantity less than ] / 6.
If then 6 be taken large enough, or the equal angles at
o small enough fractions of the unit angle, this dif
ference 1/6 can be made vanishingly small. In this case
we may say that in the limit the angle 6 becomes equal to
njb and the ray o P equal to o N or o Q, which will tnus
be ultimately equal. Hence o p = o A . \n'b — o A . Xe, or in
words : If a ray o P of the equiangular spiral make
an angle A o P with another ray o A, the ratio of o P to
o A is equal to a certain number A, raised to the power
of the quantity Q which expresses the magnitude of the
angle A o P in units of angle.
If a and r be the numbers which express the
magnitudes of OA and OP, we have r = a \9. This is
termed the polar equation of the spiral,
We proceed to draw some important results from a
POSITION. 175
consideration of tins spiral. The reader will at once
observe that the ratio of any pair of rays o P and 0 Q is
equal to the ratio of any other pair which include an
equal angle, for the ratio of any pair of rays depends
only on the included angle. Further, if we wanted to
multiply the ratio of any two quantities p and q by the
ratio of two other quantities /• and .•>• we might proceed,
as follows : Find rays of the equiangular spiral o P, o Q,
O R, o S containing the same number of linear units as
) fb r> 6> '-oiitain units of quantity (see p. (J(J), and let
6 be the angle between the first pair, </> the angle
between the second pair.
Then
OP on
whence it follows that — - x -- = X9 x X* =
OP OK
or is equal to the ratio of any pair of rays which
include an angle $+(/>. Thus if the angle QOT be
taken equal to $, and o T be the corresponding ray of
O I1
the spiral, - = X9+*, and is a ratio equal to the pro
duct of the given ratios. Hence to find the product
of ratios we have only to add the angles between pairs
I of rays in the given ratios, and the ratio of any two
rays including an angle equal to the sum will be equal
176 THE COMMON SENSE OF THE EXACT SCIENCES.
to the required product. Thus the equiangular spiral
enables us to replace multiplication by addition. This is
an extremely valuable substitution, as it is much easier
to add than to multiply.
Since — - divided by — = V divided by \*= V~
OP J on
it is obvious that we may in like fashion replace the
division of two ratios by the subtraction of two
angles. A set of quantities like the angles at the pole
of an equiangular spiral which enables us to replace
multiplication and division by addition and subtrac
tion is termed a table of logarithms. Since the equi
angular spiral acts as a graphical table of logarithms,
it is frequently termed the logarithmic spiral.
§ 10. On the Nature of Logarithms.
Since in the logarithmic spiral o p = o A x \8, where
6 is equal to the angle A o p, we note that as 0 increases,
or as the ray o p revolves round o, o P is equally mul
tiplied during equal increments of the vectorial angle
A o P. When one quantity depends upon another inj
such fashion that the first is equally multiplied "fori
equal increments of the second, it is said to grow ati
logarithmic rate. This logarithmic rate is measured byi
the ratio of the growth of the first quantity for uniti
increment of the second quantity to the magnitude of
the first quantity before it started this growth.
Let us endeavour to apply this to our equiangular!]
spiral. Suppose A o B, B o c, c o D &c. to be as before11
the triangles by means of which we construct it (seeij
fig. 69), the angles at o being all equal and very small.
Along o B measure a length o A' equal to o A ; along o c,
a length o B' equal to OB; along o D, a length o c'
POSITION. 177
iqual to oc, and so on. Then A'B, B'C, C'D, &c., will
>e the successive growths as a ray is turned succes
sively from o A to 0 B, from o B to o c, and so on. Join
A A', B B', c c', &c. Now the triangles A o B, BOG,
0 D, Sec., are all of the same shape ; so too are the
sosceles triangles A o A', B o B', c o c', &c. Hence the
differences of the corresponding members of these sets,
A A'B, B B'C, c C'D, &c., must also be of equal shape, and
;hus their corresponding sides proportional. It follows
;hen that the lengths
A'B, B'C, C'D, &c., are in the same ratio as the lengths
A'A, B'B, c'c, &c., or again as the lengths
OA, OB, oc, &c.
Whence we deduce that
A'B B'C C'D „
— _ - - = - - =&c.
o A OB oc
the growth A'B is always in a constant ratio to
;he growing quantity OA.
Now, if the angles at o be very small, the line A A'
practically coincide with the arc of a circle with
:entre o and radius equal to o A. Hence (see p. 143)
IA' will ultimately equal o A. x the angle A o A', while
;he angle at A' will ultimately be equal to a right
ingle.
Further, the ratio of A'B to A A' remains the same
'or all the little triangles A A'B, BB'C, CC'D, &c. It is in
pach case the ratio of the base to the perpendicular when
look upon these triangles with regard to the equal
Angles ABA', B c B', CD c', &c. Now these are the
.ngles of the triangles which give the spiral its name.
!jet any one of them, and therefore all of them, be equal
a. By definition the cotangent of an angle (see p. 1 GO)
equal to the ratio of the base to the perpendicular.
N
178 THE COMMON SENSE OF THE EXACT SCIENCES.
Hence
A'B A'B
cota =
A A' o A x angle A o A''
A'B ,
or — — angle A o A x cota.
OA
Now A B denotes the growth for an angle A o A
supposed very small ; whence it follows that the loga
rithmic rate, or the ratio of the growth to the growing
quantity for unit angle, is equal to cota. Thus the
logarithmic rate for the growth of the ray of the equi
angular or logarithmic spiral, as it describes equa!
angles about the pole, is equal to the cotangent of the
angle which gives its name to the spiral.
Let us suppose o A to be unit of length, then, since
0 P = 0 A x \9, the result OP of revolving the ray OA
through an angle 0 equal to unity will be X, or X is the
result of making unity grow at logarithmic rate cota.
Now let us denote by the symbol e the result oi
making unity grow at logarithmic rate unity during
the description of unit angle. Then e will have some
definite numerical value. This value is found, by a pro
cess of calculation into which we cannot enter hei*e, to be
nearly equal to 2*718. This means that, if while unri
ray were turned through unit angle it grew at loga
rithmic rate unity, its total growth (1'718) would lie
between eight and nine-fifths of its initial length. Since
e is the result of turning unit ray through unit angle
and since the ray is equally multiplied for equal multi
ples of angle, ei must represent the result of turning unil
ray through 7 unit angles. Hitherto we have beer
concerned with unit ray growing at logarithmic rate
unity ; now let us suppose unity to grow at logarithmic
rate 7 ; then it grows 7 times as much as if it grew al
log.iiithmic rate unity, or the result of turning unit raj
POSITIOX. 179
througli unit angle, while it grows at logarithmic rate
7, must be the same as if we spread 1/7 of this rate
of growth over 7 unit angles ; that is, as if we caused
unity to grow at logarithmic unity for 7 unit angles,
or ey. .Hence ey denotes the result of making unit ray
grow at logarithmic rate unity while it describes 7 unit
angles, or again of making unit ray grow at loga
rithmic rate 7 while it describes a unit of angle.
Let us inquire what is the meaning of ey when 7 is
a commensurable fraction equal to ,s/i, s and t bring
integers. Let a- be the as yet unknown result of turn
ing unit ray througli an angle equal to 7 while it
grows at unit logarithmic rate ; then <c' will be the
result of turning unit ray through t angles equal to
7 while it grows at unit rate ; but t angles equal to 7
form an angle containing s units, or this result must
be the same as the result of turning unity through an
angle s while it grows- at logarithmic rate t. Thus we
have x' = es. That is, x is a £-th root of e*, or, as we write
it, equal to e" = eY. Thus ey, if 7 be a commensurable
fraction, is the result of causing unit rav to errow at
•/
logarithmic rate unity through an angle equal to 7, or
as we have seen at logarithmic rate 7 through unit
angle.
Now let us suppose it possible to find a commen
surable fraction 7 equal to cota ; then the result of
making unity grow at logarithmic rate cota as it is
turned through unit angle must be &. But we have
seen (see p. 1V8) that it is- equal to X. Hence
\ = &f.
Further, the result of making unity grow at loga
rithmic rate cota as it is turned through an an^le #
zj O
is \e ; or,
N 2
ISO THE COMMON SENSE OF THE EXACT SCIENCES.
Tims we may write OP = OA.Xfl = OA. ey°,
or with our previous symbols,
r = a . ey9.
This is therefore the equation to our equiangular
spiral expressed in terms of the quantity e.
If we take a spiral in which a is the unit of length,
and in which cota or 7 is also unity, we find
r = ee.
The symbol ee is then read the exponential of 6, and 6
is termed the natural logarithm of r. It is denoted
symbolically thus : —
0 = loger.
The quantity e is termed the base of the natural system
of logarithms. Our spiral would in this case form a
graphical table of natural logarithms.
Returning to the equation
r = a . e"*9,
let us suppose 7 so chosen that e?=10 ; then 7 will re
present the angle through which unit ray must be
turned in order that, growing at unit logarithmic rate,
it may increase to ten units. Again taking a to be
of unit length we find r = eYfl=10e. 0 is in this case
termed the logarithm of r to the base 10, and this is
symbolically expressed thus : —
0 = Iog10 r.
The spiral obtained in this case would form a graphical I T
table of logarithms to the base 10. Such logarithms | f
are those which are usually adopted for the purposes of
practical calculation.
Natural logarithms were first devised by John
Napier, who published his invention in 1614.1 Loga-
1 Logarithmorum Canonis Descriptio. 4to. Edinburgh, 1614.
POSITION.
181
ritlims to the base 10 are now used in all but the
simplest numerical calculations which it is needful to
make in the exact sciences ; their value arises solely
from the fact that addition and subtraction are easier
operations than multiplication and division.
§ 11. The Cartesian Method of Determining Position.
(7) In order to determine the position of a point p,
in space of two dimensions, we may draw the line B A B',
joining the given points A B and another line c A c' at
right angles to this through A. These will divide the
plane into four equal portions termed <]_Ha<l,-<ii<t*. Let
P! M be a line drawn from the point PJ (the position of
B'-
M
which relative to A we wish to determine), parallel to
C A and meeting B'A B in M. Then we may state the
following rule to get from A to P, : Take a step A M
from A on the line B'A B, and then a step to the left at
fight angles to this equal to M P,. Now a step lik->
A M may be taken either forwards along A B or back
wards along AB'. Precisely as before (seep. 100) w?
1S2 Till-: COMMON SENSJS OF THE EXACT SCIENCES.
shall take + A M to mean a step fonvards along A B,
and —AM to mean a step AM' backwards along A B'
through the same distance A M. Let us use the letter
t?
i to denote the operation, which we have represented
by (?r/2) on p. 151. Thus applied to unit step it will
signify : Step forwards in the direction of the previous
step and from its finish unit distance, and then
rotate this unit distance through a right angle
counter-clockwise about the finish of the previous step.
The operator i placed before a step, thus i. MPI? will
then be interpreted as follows: Step from M in the
direction A B a distance equal to the length M p,, and
then rotate this step M PI about M counter-clockwise
through a right angle. We are thus able to express
symbolically the position of PI relative to A, or the step
A P15 by the relation
AP, = AM 4- i.M Pr
If we had to get to a point P4 in the quadrant B A c',
instead of to p,, we should have, instead of stepping for
wards from M, to step backwards a distance M P4, and
then rotate this through a right angle counter-clock
wise. The step backwards would be denoted by insert
ing a — sign as a reversing operation (see p. 39), and
we should have
A P4 = A M — i . M P4.
Next let us see how we should get to a point like P2
in the quadrant c A B', where P2 is at a perpendicular
distance P2 M' from A B'. First, we must take a step,
A M', backwards ; this is denoted by —AM'; secondly,
we must step forwards from M' a distance M' P2 ; since
this step is forwards, it will be towards A ; thirdly, by
applying the operation i to this step, we rotate it about
POSITION. 183
M' counter-clockwise through a right angle, and so
reach P2. Hence
A P2 = — A M' + i . M' P2.
Finally, if we wish to reach P3 in the quadrant
B'A c', we must step backwards A M', and then still
further backwards a step M' P3, and lastly rotate this
step counter-clockwise through a right angle. This
will be expressed by
A P3 = — A M' — i . M' P2.
Now let us suppose pp P2, P3, P4, to be the four corners
of a rectangular figure whose centre is at A and whose
sides are parallel to BAB' and c A c'. L 't the number
of units in A M be x, and the number in M Pt be y, then
we may represent the four steps which determine the
positions of the p's relative to A as follows :—
AP, = x + iy A P2 = — x + iy
A P3 = — x — i y A P4 = x — i y.
Here x and y are mere numbers, but, when we
represent these numbers by steps on a line, the
y-numbers are to be taken on a certain line at right
angles to that line on which the ^-numbers are taken.
Thus the moment we represent our x and y numbers
by lengths, they give us a means of determining posi
tion.
The quantities x and y might thus be used to deter
mine the position of a point, if we supposed them to
carry with them proper signs. Our general rule would
then be to step forwards from A along A B a distance x,
and then from the end of a; a distance forwards equal
to y ; rotate this step y about the end of x counter
clockwise through a right angle, and the finish of //
will then be the point determined by the quantities x, y.
184 THE COMMON SENSE OF THE EXACT SCIENCES.
If x or y be negative, the corresponding forwards must
be read: Step forwards a negative quantity, that is,
step backwards. Thus : —
Pt, or position in the quadrant B A c is determined by x, y.
CAB
B'AC'
C' AB
-y.
The quantities x and y are termed the Cartesian co
ordinates of the point p, this method of determining the
u
I !
FIG. 72.
position of a point having been first used by Descartes.
BAB' and c A c' are termed the co-ordinate axes of x
and y respectively, while A is called the origin of co
ordinates. For example, let the Cartesian co-ordinates
of a point be ( — 3, 2). How shall we get at it from the
origin A ? If P be the point, we have A p = — 3 + i . 2.
Hence we must step backwards 3 units ; from this point
step forwards 2 and rotate this step 2 about the ex
tremity of the step 3 through a right angle counter
clockwise ; we shall then be at the required point.
If p be determined by its Cartesian co-ordinates x
and y, we might find a succession of points, p, by always
POSITION.
185
taking a step y related in a certain invariable fashion to
any step x which has been previously made.
Such a succession of points p, obtained by giving
* every possible value, will form a line or curve, and
the relation between x and y is termed its Cartesian
equation.
As an instance of this, suppose that for every step
«, we take a step y equal to the double of it. Then we
shall have for our relation y — 2 a;, and our instructions
LL
I
,
•
MM
FIG. 73.
to reach any point p of the series are x + i.2 x. Suppose
'the quadrant BAG divided into a number of'little squares
by lines parallel to the axes, and let us take the sides of
(these squares to be of unit length. Then if we take in
Succession x = l, 2, 3, &c., we can easily mark off our
steps. Thus : 1 along A B and then 2 to the left ; 2
(along A B and 4 to the left ; 3 along A B and then 6 to
the left ; 4 along A B and then 8 to the left ; 5 along
A B and then 10 to the left, and so on. It will be
iobvious (by p. 106) that our points all lie upon a
186 THE COMMON SEXSE OF THE EXACT SCIENCES.
straight line through A, and however many steps we
take along A B, followed by double steps perpendicular
to it, we shall always arrive at a point on the same
line. If we gave x negative values we should obtain
that part of the line which lies in the third quadrant
B'AC'. Hence we see that y = 2 x is the equation to
a straight line which passes through A.
Let us take another example. Suppose that the
rectangle contained by y and a length of 2 units,
always contains as many units of area as there are
square units in x2. Our relation in this case may be
expressed by 2 y = x2, and we have the following series
of steps from A to points of the series : —
4 + i.8, 5 + /i.2-j-5, C + 'i.lS, &c.
We can by means of our little squares easily follow
out the operations above indicated ; we thus find a series
of points like those in the quadrant B A c of the figure.
If however we had taken x equal to the negative
quantities — 1, —2, —3, — 4, —5, —6, &c., we should
have found precisely the same values for y, because we
have seen that ( — a) x ( — a)= a? =(4- a) x ( + a). These
negative values for x give us a series of points like those
in the quadrant B'AC of the figure. It is impossible
that any points of the series should lie below B A B',
because both negative and positive values for x give \
when squared a positive value for the step y, so that no |
possible £-step would give a negative y-step. The series |
of points obtained in this fashion are found to lie upon I
a curve which is one of those shadows of a circle which I
we have termed parabolas.
Hence we may say that 2y=x^ is the equation to a I
parabola.
rosmox.
1ST
This method of plotting out curves is of great value,
and is largely used in many branches of physical inves
tigation. For example, if the differences of successive
«-steps denote successive intervals of time, and i/-steps
the corresponding heights of the column of mercury
ill a barometer above some chosen mean position,
the series of points obtained will, if the intervals
of time be taken small enough, present the appear
ance of a curve. This curve gives a graphical repre
sentation of the variations of the barometer for the
whole period during which its heights have been plotted
out. Barometric curves for the preceding day are now
given in several of the morning papers. Heights cor
responding to each instant of time are in this case
188 THE COMMON SENSE OF THE EXACT SCIENCES.
generally registered automatically by means of a simple
photographic apparatus.
The plotting out of curves from their Cartesian
equations, usually termed curve tracing, forms an ex
tremely interesting portion of pure mathematics. It
may be shown that any relation, which does not in
volve higher powers of x and y than the second, is the
equation to some one of the forms taken by the shadow
of a circle.
§ 12. Of Complex Numbers.
We shall now return to our symbol of operation i,
and inquire a little closer into its meaning. Let the
point P be denoted as before by A M + i . M p, so that we
FIG. 75.
should read this result : Step from A to M along A B,
and from M to p' along the same line (where M p' = M p),
finally rotate M P' about M counter-clockwise through
a right angle ; M p' will then take up the position M P.
Now let M Q' be taken equal to A p', then A M + i . M Q' will
mean : Step from A to M and then from M perpendicular
POSITION. 189
to A M to the left through a distance, M Q', equal to A p'.
Since however MQ'==AP' = AM + MP=MP + PQ', PQ'
must be equal to A M and we can read our operation
A M + i . (M p + p Q'),
which denotes two successive steps at right angles to
A M, namely M p followed by the step P Q'. Suppose now
we wished to rotate this latter step through a right angle
counter-clockwise, we should have to introduce before
it the symbol i, and M p + i . p Q' would signify the step
M P followed by the step p Q at right angles to it to the
left. Now P Q' is equal to A M, and hence the result of
this operation must bring us to Q, a point on A c which,
might have been reached by the simple operation
0 + i '• • A Q. Thus we may put
0 + i . A Q = A M + i . (M P + i • P Q)
= AM + t.MP + 'i.i.PQ;
or, since the quantities A Q, AM, M p, and p Q here
merely denote numerical magnitudes, and since as such
A Q = M P and AM = P Q, we must have
0 = A M + i . i . A M,
or — A M = i . i . A M.
Thus the operation i is of such a character that
repeated twice it is equivalent to a mere reversor, or, as
we may express it symbolically,
- 1 = i\
This may be read in words : Turn a step counter
clockwise through a right angle, and then again
counter-clockwise through another right angle, and we
have the same result as if we had reversed the step.
Now we have seen (p. 1 44) that if ,t be such a quantity
that multiplied by itself it equals a, x is termed the
square root of a, and written Va. Hence since
i2= —1, we may write i=V - 1.
190 THE COMMON SENSE OF THE EXACT SCIENCES.
This symbol is completely unintelligible so far as
quantity is concerned; it can represent no quantity
conceivable, for the squares of all conceivable quantities
are positive quantities. For this reason V — 1 is some
times termed an imaginary quantity. Treated however
as a symbol of operation \/— 1 has a perfectly clear and
real meaning ; it is here an instruction to step forwards
a unit length and then rotate this length counter-clock
wise through a right angle.
Any expression of the form x + V — 1 y is termed a
complex number.
Let P be any point determined by the step A P =
AM + V — IMP, and let r, x, y be the numerical values
of the lengths A P, AM, and p M. It follows from the
right-angled triangle p A M that r2 = x2 + y2. The
quantity r is then termed the modulus of the complex
number x + V — 1 y.
Further let the angle MAP contain 6 units of
angle ; then
./, PMty -aAM X
sm0 = = -» cos# = — = -,
A p r A p r
or y = r sin<9, x = r cos#.
The angle 0 is termed the argument of the com
plex number. Here r and 6 are the polar co-ordinates
of P, and we are thus able to connect them with the Car
tesian co-ordinates ; they are respectively the modulus
and argument of the complex number which may be
formed from the Cartesian co-ordinates. Since r is
merely numerical we may write the complex number
x -f- V — ly in the form r . (cos# + V — 1 sin0), or as
the product of its modulus and the operator
cos0 + V — 1 siu#,
rosmox.
191
Hence we
•which depends solely on its argument 6.
may interpret the step
A p = r . (cos# + V— 1 sin#)
as obtained in the following fashion: Rotate unit length
from A B through an angle 0, and then stretch it in
the ratio of r : 1. The latter part of this operation
B
Fir.;. 76.
will be signified by the modulus r, the former by the
operator (cos# + V—l sinfl). Thus if AD be of unit
length and lying in A B, we may read —
A p = r . (cosO + V — 1 sin$) . A D,
and we look upon our complex number as a symbol
denoting the combination of two operations performed
on a unit step A D.
Starting then from the idea of a complex mimbev
as denoting position, we have been led to a new opera
tion represented by the symbol cos# + V — 1 sin#.
This is obviously a generalised form of our old symbol
V — l. The operator cos# + V — l sin# applied to
any step bids us turn the step through an angle 0.
"We shall see that this new conception has important
results.
192 THE COMMON SENSE OP THE EXACT SCIENCES.
§ 13. On the Operation which turns a Step through a
given Angle.
Suppose we apply the operator (cos# + V— 1 sin#)
twice to a unit step. Then the symbolic expression
for this operation will be
(cos0 + V^l sind) (cos0 + V^l sin0),
or (cos0 + V^l sintf)2.
But to turn a step first through an angle 6 and then
through another angle 0 is clearly the same operation
as turning it by one rotation through an angle 26, or
as applying the operator cos2# + V — 1 sin20. Hence
we are able to assert the equivalence of the operations
expressed by the equation —
(cos0 + V-I sin(9)2 = cos20 + V-l sin20.
In like manner the result of turning a step by n
operations through successive angles equal to 6 must
be identical with the result of turning it at once"
through an angle equal to n times 0, or we may write
(cos# + V — 1 sin#)w = cosnd + V—l sinnd.
This important equivalence of operations was first ex
pressed in the above symbolical form by De Moivre,
and it is usually called after him De Moivre's Theorem.
We are now able to consider the operation by means
of which a step A P can be transformed into another A Q.
\Ve must obviously turn AP about A counter-clockwise
till it coincides in position with A Q ; in this case p
will fall on p', so that A p' = A p. Then we must
stretch A p' into A Q ; this will be a process of multiply
ing it by some quantity p, which is equal to the ratio
of A Q to A P'.
[ POSITION. 193
Expressing this symbolically, if </> be the angle
p A Q, we have
(cos$ + V —I sine/)) . A P = A P'.
p . (cOSff) + V — 1 Sin<£) .AP=/3.AP' = AQ.
This last equation we can interpret in various ways :
(i) p . (cos</> + V — 1 sin</>) is a complex number of
which p is the modulus and </> the argument. Hence
we may say that to multiply a step by a complex number
is to turn the step through an angle equal to the argu
ment and to alter its length by a stretch represented
by the modulus.
(ii) Or, again, we may consider the step A p as itself
representing a complex number, x + V — I y, or if r be
FIG. 77.
the scalar value of A p and 6 the angle BAP, we may
put A P = r(cos# -f V — I sin#). Similarly A Q will be a
complex number, and its scalar magnitude (= p . A P'
= p r) will be its modulus, while the angle B A Q = 6 + <£
will be its argument. We have then the following
identity —
p (coS(/> + V — 1 sin<£) . r (cos^> + v — 1 sin$) =
/> r . (cos# + (f> + V — 1 cos(£ + ^) .
This may be read in two ways :
First, the product of two complex numbers is itself
a complex number, and has the product of the moduli
for its modulus, the sum of the arguments for its
argument.
o
194 THE COMMON SEXSE OF THE EXACT SCIENCES.
Or secondly, if we turn unit step through an
angle 0 and give a stretch r, and then turn the result
obtained through an angle <f> and give it a stretch p,
the result will be the same as turning unit step through
an angle 6 + <f> and giving it a stretch equal to p r.
Thus we see that any relation between complex
numbers may be treated either as an algebraical fact
relating to such numbers, or as a theorem concerning
operations of turning and stretching unit steps.
(iii) We may consider what answer the above identity
gives to the question : What is the ratio of two
directed steps A Q and A p ? Or, using the notation sug
gested on p. 45, we ask : What is the meaning of the
symbol , — -=-! ? A step like A p (or A Q) which has
| AP
magnitude, direction, and sense is, as we have noted,
termed a vector. We therefore ask : What is the ratio
of two vectors, or what operation will convert one
into the other ? The answer is : An operation which
is the product of a turning (or spin) and a stretch.
Now the stretch is a scalar quantity, a numerical
ratio by which the scalar magnitude of A p is con
nected with that of AQ. The stretch therefore is a
scalar operation. Further, the turning or spin converts
the direction of A p into that of A Q, and it obviously
takes place by spinning A p round an axis perpendi
cular to the plane of the paper in which both A p
and AQ lie. Thus the second part of the operation
by which we convert A p into A Q denotes a spin
(counter-clockwise) through a definite angle about a
certain axis. The amount of the spin might be
measured by a step taken along that axis. Thus, for
instance, if the spin were through 6 units of angle,
we might measure 6 units of length along the axis to
POSITION. 195
denote its amount. We may also agree to take this
length along one direction of the axis (' out from the
face of the clock ') if the spin be counter-clockwise, and
in the opposite direction (' behind the face of the clock ')
if the spin be clockwise. Thus we see that our spinning
operation may be denoted by a line or step having
magnitude, direction, and sense ; that is, by a vector.
We are now able to understand the nature of the ratio
of two vectors; it is an operation consisting of the pro
duct of a scalar and a vector. This product was termed
by Sir William Hamilton a '/""/<, •/<''<>», and made the
foundation of a very powerful calculus.
Thus a quaternion is primarily the operation which
converts one vector step into another. It does this by
means of a spin and a stretch. If we have three points
in plane space, the reader will now understand how
the position of the third with regard to the first can bo
made identical with that of the second by means of a
spin arid a stretch of the step joining the first to the
third, that is, by means of a quaternion.1
§ 14. Relation of the Spin to the Logarithmic Growth
of Unit Step.
Let us take a circle of unit radius and endeavour
to find how its radius grows in describing unit an^lo
about the centre. Hitherto we have treated of growth
only in the direction of length ; and hence it might be
supposed that the radius of a circle does not ' grow ' at
all as it revolves about the centre. But our method of
adding vector steps suggests at once an obvious extension
of our conception of growth. Let a step AP become
1 The term 'stretch' must be considered to include a squeeze or u
stretch denoted by a scalar quantity p less than unity.
o 2
1 06 THE COMMON SEXSE OF THE EXACT SCIENCES.
A Q as it rotates about A through the angle P A Q, then
if we marked off A Q a distance A P' equal to A P, P' Q
would be the scalar growth of A p ; that is, its growth
FIG. 78.
in the direction of its length. But if A p be treated
as a vector (see p. 153)
or the directed step p Q must be added to A P in order to
convert it into A Q ; p Q may be thus termed the directed
growth of A P. If we join p p', we shall have P Q equal
to the sum of P p' and P' Q. Now if the angle PAP' be
taken very small p p' will be ultimately perpendicular
to AP, and this part of the growth PQ might be
represented by V — 1. PP'. Hence we are led to
represent a growth perpendicular to a rotating line by
a scalar quantity multiplied by the symbol V— 1.
We can now consider the case of our circle of unit
radius. Let o P be a radius which has revolved through
FIG. 79.
an angle 6 from a fixed radius o A, and let o Q be an
adjacent position of o P such that the angle Q o P is very
small. Then p Q will be a small arc sensibly coincident
POSITION. 1 97
with the straight line p Q, and the line p Q will be to all
intents and purposes at right angles to o P. Hence to
obtain o Q we must take a step P Q at right angles to
OP. This we represent by V— 1 QP. Since the radius
of the circle is unity the arc Q P, which equals the
radius multiplied by the angle QOP (see p. 143), must
equal the numerical value of the angle QOP. Or the
growth of o P is given by V - 1 x angle QOP. Now
according to our definition of growing at logarithmic
rate (seep. 176), since OP is equally multiplied in de
scribing equal angles about o, it must be growing at
logarithmic rate. What is this logarithmic rate for
unit angle ?
It must equal — divided by the ratio of the an <rle
OP
o 0 P to unit angle = - - = v — 1 since o P
OP x angle QOP
is unity. Thus o P is growing at logarithmic rate V — 1
as it describes unit angle ; that is to say, the result of
turning OP through unit angle might be symbolically
expressed by e"J~l. Hence the result of turning OP
through an angle 6 must be e^~l(). We may then write
OP = o A .ev"l°.
Drop p M perpendicular to o A and produce it to meet
the circle again in p', then by symmetry M P = M P', and
we have
OP = o M -t- V— 1 M P.
OP' = OM — V— 1 MP'.
Now since o P and o P' are of unit magnitude,
OM PM
cos0 =— = OM, 8in0 = Q-- = P M.
Also the angle P'OM equals the angle M o P, but, according
198 THE COMMON SENSE OF THE EXACT SCIENCES.
to our convention as to the measurement of angles,
it is of opposite sense, or equals — d. Thus we must
write
OP' = OA .e~v~ld
Substituting their values, we deduce the symbolical
results
'^1 sin 01
Further,
OP — o P' = 2 A/— 1 P M
OP + OP'= 2oM;
that is,
-'-ie -^- ~ nflj f..
)
These values for cos# and sin0 in terms of the ex
ponential e were first discovered by Euler. They are
meaningless in the form (ii) when cos$ and sin# are
interpreted as mere numerical ratios ; bat they have a
perfectly clear and definite meaning when we treat
each side of the equation in form (i) as a symbol of
operation. Thus cos(9 + V— 1 sin# applied to unit
step directs us to turn that step without altering its
length through an angle 6 ; on the other hand, e ^~lti
applied to the same step causes it to grow at logarith
mic rate unity perpendicular to itself, while it is turned
through the angle 0. The two processes give the same
result.
§ 15. On the Multiplication of Vectors.
We have discussed how vector steps are to be
added, and proved that the order of addition is in
different ; we have also examined the operation denoted
POSITIOX. 199
by the ratio of two vectors. The reader will naturally
ask : Can no meaning be given to the product of two
vectors ?
If both the vectors be treated as complex numbers,
or as denoting operations, we have interpreted their
product (seep. 193) as another complex number or as a
resultant operation. Or again we have interpreted
the product of two vectors when one denotes an ope
ration and the other a step of position ; the product
in this case is a direction to spin the step through ;i
certain angle and then stretch it in a certain ratio.
But neither of these cases explains what we are to
understand by the product of two steps of position.
Let A P, A Q be two such steps : What is the meaning
of the product AP . AQ ? Were A P and A Q merely
FIG. 80.
scalar quantities then their product would be purely
scalar, and we should have no difficulty in interpreting
the result A P . P Q as another scalar quantity. But
when we consider the steps A P, P Q to possess not only
A
Fio. 81.
magnitude but direction, the meaning of their product
is by no means so obvious.
If A Q were at right angles to A P (see fig. 81), we
should naturally interpret the product A P . A Q as the
200 THE COMMON SENSE OF THE EXACT SCIENCES.
area of the rectangle on A Q and A p, or as the area of
the figure Q A p E. Now let us see how this area might
be generated. Were we to move the step A Q parallel
to itself and so that its end A always remained in the
step A P, it would describe the rectangle Q A p E while its
foot A described the step A p. Hence if A P and A Q are
at right angles we might interpret their product as
follows :
The product A p . A Q bids us move the step A Q
parallel to itself so that its end A traverses the step A p ;
the area traced out by A Q during this motion is the
value of the product A p . A Q.
It will be noted at once that this interpretation,
although suggested by the case of the angle Q A p being
a right angle, is entirely independent of what that angle
may be. If Q A P be not a right angle the area traced
out according to the above rule would be the parallelo
gram on A P, A Q as sides. Hence the interpretation we
have discovered for the product A p . A Q gives us an
intelligible meaning, whatever be the angle Q A p.
There is, however, a difficulty which we have not yet
solved. An area is a directed quantity (see p. 134), and
its direction depends on how we go round its perimeter.
Now the area Q A p R will be positive if we go round its
perimeter counter-clockwise, or from A to p ; that is, in
FIG. 82.
the direction of the first step of the product or in the
direction of motion of the second or moving step. Thus
the product A P . A Q will be the area Q A p E taken with
the sign suggested by the step A p. The product A Q. AP
POSITION. 201
will be formed by causing the step A p to move
parallel to itself along A Q, and it is therefore also the
area of the parallelogram on A Q and A p ; but it is to be
taken with the sign suggested by A Q, or it is the area
PAQE.
By our convention as to the sign of areas,
PAQE = — QAPE,
Or A Q . A P = — A P . A Q.
Hence we see that, with the above interpretation, the
/product of two vectors does not follow the commutative
law (see p. 45).
If we suppose the angle Q A P to vanish, and the
vector A Q to become identical with A p, the area of
the enclosed parallelogram will obviously vanish also.
Thus, if a vector step be multiplied by itself, the product
is zero ; that is,
A P . AP = (A p)2 = 0.
If we take a series of vector steps, a, /3, y, 8, &c.
then relations of the following types will hold among
them :
a2 = 0, /32 = 0, 72 = 0, S2 = 0, &c.
a/3 = — /3a, ay = — y a, /3 y = — y /3,
87= — 7 S, &c.
A series of quantities for which these relations hold
was first made use of by Grassmann, and termed by
him. alternate units.
The reader will at once observe that alternate units
have an algebra of their own. They dispense with
the commutative law, or rather replace it by another
in which the sign of a product is made to alternate with
the alternation of its components. Their consideration
will suggest to the reader tluit the rules of arithmetic,
202 THE COMMON SEXSE OF THE EXACT SCIENCES.
which he is perhaps accustomed to assume as neces
sarily true for all forms of symbolic quantity, have only
the comparatively small field of application to scalar
magnitudes. It becomes necessary to consider them as
mere conventions, or even to lay them aside entirely as
we proceed step by step to enlarge the meaning of the
symbols we are employing.
Although 2 x 2 = 0 and 2x3= —3x2 may be sheer
nonsense when 2 and 3 are treated as mere numbers, it
yet becomes downright common sense when 2 and 3 are
treated as directed steps in a plane.
Let us take two alternate units a, /3 and interpret
the quantity a a + 6 /3, where a and 6 are merely scalar
r'
FIG. 83.
magnitudes. If OA be the vector a, a a signifies that
we are to stretch o A to o A' in the ratio of 1 to a. To
this o A' we are to add the vector o R' derived from o B
by giving it the stretch &. Hence if A' p = o R' the1
vector o P represents the quantity a a + 6/3, which is
termed an alternate number. Let o Q represent a second
alternate number a' a + b' @, obtained by adding the
results of applying two other stretches a' and &' to the
POSITION. 203
alternate units a and ft. In the same way we might
obtain, by adding the results of stretching three alternate
units (a, ft,y), alternate numbers with three terms (of the
form a a + b ft + 07), and so on. If we take the^ro-
duct of as many alternate numbers as we have used
alternate units in their composition, we obtain a
quantity called a determinant, which plays a great part
in the modern theory of quantity. We shall confine
ourselves hei'e to the consideration of a determinant
formed from two alternate units. Such a determinant
will be represented by the product o P . o Q, which
according to our convention as to the multiplication of
vectors equals the area of the parallelogram on o P,
OQ as sides, or (by p. 122) twice the triangle Q o p.
Through Q draw c Q A" parallel to OB, and D Q B"
parallel to o A, then o A" = a' a and o B" = b' ft. Join
B'Q, then twice the triangle B'Q i> equals the parallelo
gram B" p. Hence, adding to both these the parallelo
gram A' B" we have the parallelogram A' B" together
with twice the triangle B'Q p equal to the parallelogram
B'A', or to twice the triangle B'O p. Hut the triangle
B'O P equals the sum of the triangles <>Q B', B'Q P, and
OPQ. It follows then that the parallelogram A' B"
must equal twice the triangle OPQ together with twice
the triangle OQ B'. Now twice the latter equals B' A".
Hence the difference of the parallelograms A' B" and
B'A" is equal to twice OPQ. The parallelogram A' B"
is obtained from the parallelogram A B by giving it two
j| stretches a and b' parallel to its sides, and therefore its
area equals a b' times the area A B. Similarly B'A"
equals b a' times the area A B ; but the area A B itself is
aft. Thus we see that the identity
0 P . o' Q = A' B" — B' A"
204 THE COMMON SENSE OF THE EXACT SCIENCES.
may be read
(a a + 1/3) (of a + &'/3) = (aV -la'} a/3.
Or, the determinant is equal to the parallelogram on
the alternate units magnified in the ratio of 1 to
a 6' — & a'. It obviously vanishes if a V — & of = 0, or if
a/6 = of /&'. In this case p and Q lie, by the property of
similar triangles, on the same straight line through o,
and therefore, as we should expect, the determinant
o P . o Q is zero.
The reader will find little difficulty in discovering
like properties for a determinant formed from three
alternate units. In this case there will be a geometrical
relation between certain volumes, which may be ob
tained by stretches in the manner explained on p. 139.1
We have in this section arrived at a legitimate I
interpretation of the product of two directed steps or
vectors. We find that their product is an area, or ac
cording to our previous convention (see p. 134), also a
directed step or vector whose direction is perpendicular]
to the plane which contains both steps of the product.
§ 16. Another Interpretation of tlie Product of Two
Vectors.
The reader must remember, however, that the resulli
of the preceding paragraph has only been obtained
means of a convention ; namely, by adopting the area of
certain parallelogram as the interpretation of the vecto^
1 I have to thank my friend Mr. J. Eose-Innes for suggesting the int
duction of the above remarks as to determinants. I may, perhaps,
allowed to add that by treating the alternate units, like Grassmann, aj
points, and the alternate number as their loaded centroid, a determinarij
of the second order is represented geometrically by a length, and we thv
obtain for one of the fourth order a geometrical interpretation as a volumn
POSITION.
205
product. Only as long as we observe that convention
will our deductions with regard to the multiplication of
vectors be true. We might have adopted a different
convention, and should then have come to a different
result. It will be instructive to follow out the results
of adopting another convention, if only by so doing we
can impress the reader with the fact that the funda
mental axioms of any branch of exact science are based
rather upon conventions than upon universal truths.
Suppose then that in interpreting the product
A P . A Q we consider A p to be a directed step which
FIG. 84.
represents the area D E F G. This area will be perpen
dicular to the direction of A P, and we might assume as
our convention that the product A P . A Q shall mean the
volume traced out by the step A Q, moving parallel to
itself and in such wise that its end A takes up every
possible position in the plane D E p G. This volume will
be the portion of an oblique cylinder on the base D E F G
intercepted by a plane parallel to that base through Q.
We have seen (p. 141) that the volume of this cylinder
is the product of its base into its height, viz. the per
pendicular distance A H between the two planes. Now
let r and p be the scalar magnitudes of AP and AQ
206 THE COMMON SENSE OP THE EXACT SCIENCES.
respectively, and 0 = the angle PAQ. Then AH =
p cos#, and the volume = AP.AQ = r p cos$, for r re
presents the number of units of area in D E F G. Hence,
since a volume is a purely numerical quantity having
only magnitude and no direction, we find that with this
new convention the product of two vectors is a purely
scalar quantity, or our new convention leads to a totally
different result from the old.
Further, since r and p are merely numbers, r p = p r,
and thus A p . A Q = r p cos# = p r cos0 = A Q . A p, if
A Q be treated as the directed step which represents
an area containing p units of area. Thus in this case
the vector product obeys the commutative law, which
again differs from our previous result. We can then
treat the product of two vectors either as a vector and
\ as a quantity not obeying the commutative law, or as a
scalar and as a quantity obeying the commutative law.
We are at liberty to adopt either convention, provided
we maintain it consistently in our resulting investiga
tions.
The method of varying our interpretation, which has
been exemplified in the case of the product of two
vectors, is peculiarly fruitful in the field of the exact
sciences. Each new interpretation may lead us to vary
our fundamental laws, and upon those varied funda
mental laws we can build up a new calculus (algebraic
or geometric as the case may be). The results of our
new calculus will then be necessarily true for those
quantities only for which we formulated our funda
mental laws. Thus those laws which were formulated
for pure number, and which, like the postulates of
Euclid with regard to space, have been frequently
supposed to be the only conceivable basis for a theory!
of quantity, are found to be true only within the limits
POSITION. 207
of scalar magnitude. When we extend our conception
of quantity and endow it with direction and position,
we find those laws are no longer valid. "We are com
pelled to suppose that one or more of those laws cease
to hold or are replaced by others of a different form.
In each case we vary the old form or adopt a new one
to suit the wider interpretation we are giving to quan
tity or its symbols.
§ 17. Position in Three-Dimensioned Space.
Hitherto we have been considering only position in
a plane ; very little alteration will enable us to consider
the position of a point p relative to a point A as deter
mined by a step A p taken in space.
We may first remark, however, that while two points
A and B are sufficient to determine in a plane the position
of any third point P, we shall require, in order to fix the
position of a point p in space, to be given three points
A, B, c not lying in one straight line. If we knew only
the distances of P from two points A and B, the point
p might be anywhere on a certain circle which has its
centre on the line AB and its plane perpendicular to
that line ; to determine the position of p on this circle,
we require to know its distance from a third point c.
Thus position in space requires us to have at least
three non-collinear points (or such geometrical figures
as are their equivalent) as basis for our determination
of position. Space in which we live is termed space of
three dimensions ; it differs from space of two dimen
sions in requiring us to have three and not two points
as a basis for determining position.
Three points will fix a plane, and hence if we are
given three points A, B, c in space, the plane through
THE COMMON SENSE OF THE EXACT SCIENCES.
them will be a definite plane separating all space into
two halves. In one of these any point p whose position
we require must lie. We may term one of these halves
below the plane and the other above the plane. Let P N
be the perpendicular from p upon the plane ; then if
we know how to find the point N in the plane ABC, the
position of p will be fully determined so soon as we
have settled whether the distance p N is to be measured
above or below the plane. We may settle by convention
that all distances above the plane shall be considered
positive, and all below negative. Further, the position
of the point N, upon which that of P depends, may be
FIG. 85.
determined by any of the methods we have employed
to fix position in a plane. Thus if N M be drawn
perpendicular to A B, we have the following instruction
to find the position of P : Take a step A M along A B,
containing, say, x units ; then take a step M N to the right
and perpendicular to A B, but still in its plane, contain
ing, say, y units ; finally step upwards from N the distance
N P perpendicular to the plane ABC, say, through z units.
We shall then have reached the same point P as if we
had taken the directed step A p. If a; had been negative
we should have had to step backwards from A ; if y had
been negative, perpendicular to A B only to the left ; if
z had been negative, perpendicular to the plane but
POSITIOX.
209
downwards. The reader will easily convince himself
that by observing these rules as to the sign of x, y, z
he could get from A to any point in space.
Let i denote unit step along A B, j unit step to the
right perpendicular to A B, but in the plane ABC, and
k unit step perpendicular to the plane ABC upwards,
from foot to head. Then we may write
A p = x . i + y .j + s . k,
where x, y, z are scalar quantities possessing only
magnitude and sign ; but i, j, k are vector steps in
three mutually rectangular directions.
FIG. 86.
The step A p may be regarded as the diagonal of a
solid rectangular figure (a riyht six-face, as we termed
it on p. J38), and thus we shall get to the same point
P by traversing any three of its non-parallel sides in
succession starting from A. But this is equivalent to
saying that the order in which we take the directed
Isteps x .i, y .j, and z .k is indifferent.
The reader will readily recognise that the sum of a
number of successive steps in space is the equivalent
ito the step which joins the start of the first to the
210 THE COMMON SENSE OF THE EXACT SCIENCES.
finish of the last ; and thus a number of propositions
concerning steps in space similar to those we have
proved for steps in a plane may be deduced. By
dividing all space into little cubes by three systems of
planes mutually at right angles, we may plot out sur
faces just as we plotted out curves. Thus we shall choose
any values we please for x and y, and suppose the
magnitude of the third step related in some constant
fashion to the previous steps. For example, if we take
the rectangle under z and some constant length a,
always equal to the differences of the squares on x and
y, or symbolically if we take a 2 = x^ — y-, we shall
reach P by taking the step
. . a;2 — 7/2 7
AP = x.i + y .y + . K-
a
The series of points which we should obtain in this
way would be found to lie upon a surface resembling
the saddle-back we have described on p. 89. The
above relation between z, x, and y will then be termed
the equation to a saddle-back surface.
We cannot, however, enter fully on the theory of
steps in space without far exceeding the limits of our
present enterprise.
§ 18. On Localised Vectors or Rotors.
Hitherto we have considered the position of a point
p relative to a point A, and compared it with the
position of another point Q relative to the same point
A. Thus we have considered the ratio and product of
two steps A P and A Q.
We have thereby assumed either that the two steps
we were considering had a common extremity A, or at
least were capable of being moved parallel to themselves
POSITION. 211
till they had such a common extremity. Such steps are,
as we have remarked, termed vector steps.
Suppose, however, that instead of comparing the
position of two points p and Q relative to the same
point A, we compared their positions relative to two
different points A and B. The position of p relative to
A will then be determined by the step A p and the
position of Q relative to B by the step B Q.
Now it will be noted that these steps A p and B Q have
not only direction and magnitude, bat have themselves
position in space. The step A P has itself position in
space relative to the step B Q. It is no longer a step
FIG. 87.
merely indicating the position of p with regard to A,
but taken as a whole it has itself attained position
when considered with regard to the step B Q. This
localising, not of a point P relative to a point A, but
of a step A P with regard to another step B Q, is a new
-and important conception. Such a localised vector is
termed a rotor from the part it plays in the theory of
rotating or spinning bodies.
Let us try and discover what operation will convert
[the rotor B Q into the rotor A P ; in other words : What
is the operation r — •' ? In order to convert B Q into
I BQ
p 2
212 THE COMMON SENSE OP THE EXACT SCIENCES.
A p we must make the magnitude and position of B Q
the same as that of A p. Its magnitude may be made
the same by means of a stretching operation which
stretches B Q to A p. This stretch, as we have seen in the
case of a quaternion (see p. 195), may be represented
by a numerical ratio or a mere scalar quantity. Next
let c D be the shortest distance between the rotors A p
/
/
/
I
3
_s.
^r
x>^. **
Q
FIG. 88.
and B Q; then c D will be perpendicular to both of them.1
B Q may then be made to coincide in position with A p
by the following process :
First turn B Q about the shortest distance, c D,
through some angle, Q D Q', till it takes up the posi
tion B'Q' parallel to A P ; then slide B'Q' along the
1 That the shortest distance between two lines is perpendicular to both
of them may be proved in the following manner. Let us suppose the lines
replaced by perfectly smooth and very thin rods, and let two rings, one on
either rod, be connected by a stretched elastic string. Obviously the rings
will slide along the rods till the elastic string takes up the position of the
shortest distance; for that will correspond to the least possible tension of
the string. Suppose that the string is then not at right angles to one of
the rods, say, at the point c. By holding the string firmly at E, we might
shift the ring at c along the rod to c', so that the angle E c'c should be a
right angle. Then since c' is a right angle c E would be greater than C'E,
being the side opposite the greatest angle of the triangle EC' c. Hence the
length of string C'E + ED is less than the length CD, or CD cannot be the
shortest distance which we have supposed it to be. Thus the shortest
distance must be at right angles to both lines.
POSITION. 213
shortest distance parallel to itself till its position coin
cides with A P. If we wished B'Q' to coincide point for
point with A p', we should further have to slide it along
A P till B' and A were one.
Now the two operations of turning a line about
another line at right angles to it, and moving it along
that line, are just akin to the operations which are
applied to the groove in the head of a screw when we
drive the screw into a block of wood ; or again to the
handle of a corkscreAv when we twist the screw into a
cork. The handle in the one case and the groove in the
other not only spin round, but go forward in the direc
tion of the screw axis. Such a movement along an
axis, and at the same time about it, is termed a tivist. The
ratio of the forward space described to the angle turned
through during its description by the head of the screw
is termed the pitch of the screw. This pitch will
remain constant for all forward spaces described if the
thread of the screw be uniform. Thus turn an ordinary
corkscrew twice round, and it will have advanced twice
as far through the cork as when it has been turned
only once round. Let us see whether we cannot apply
this conception of a screw to the operations by which we
bring the rotor B Q into the position of the rotor A p.
Upon a rod placed at c D, the shortest distance, suppose a
fiae screw cut with such a thread that its pitch equals
the ratio of c D to the angle Q D Q'. Then if we suppose
B Q attached to a nut upon this screw at D, when we
turn B Q through the angle Q D Q', the nut with B Q will
advance (owing to the pitch we have chosen for the
screw) through the distance DC. In other words, B Q
will have been brought up to A P and coincide with it
in position and direction.
Hence the operations by means of which B Q can be
214 THE COMMON SENSE OF THE EXACT SCIENCES.
made to coincide with A p are a stretch followed by a
twist along a certain screw. A screw involves direc
tion, position, and pitch ; a twist (as of a nut) about
this axis involves something additional, namely a
magnitude, viz. that of the angle through which the
nut is to be turned. Magnitude associated with a
screw has been termed by the author of the present
book a motor1 (since it expresses the most general
instantaneous motion of a rigid body). Hence the
operation by which one rotor is converted into another
may be described as a motor combined with a stretch.
This operation stands in the same relation to two rotors
as the quaternion to two vectors. The motor plays
such an important part in several branches of physical
inquiry that the reader will do well to familiarise him
self with the conception.
The sum of two vector steps is, as we have seen
(p. 153), a third vector ; but unlike vector steps the sum
of two rotors is in general a motor ; only in special
cases does it become either a rotor or a vector. The
geometry of rotors and motors, which we have only
here been able to hint at. forms the basis of the whole
modern theory of the relative rest (Static) and the rela
tive motion (Kinematic and Kinetic) of invariable
systems.
§ 19. On the Bending of Space.
The peculiar topic of this chapter has been position,
position namely of a point p relative to a point A.
This relative position led naturally to a consideration of
the geometry of steps. I proceeded on the hypothesis
1 ' Preliminary Sketch of Biquaternions,' Proceedings of the London
Mathematical Society, vol. iv. p. 383.
POSITIOX. 215
that all position is relative, and therefore to be deter
mined only by a stepping process. The relativity of
position was a postulate deduced from the customary
methods of determining position, such methods in fact
always giving relative position. Relativity of position
is thus a postulate derived from experience. The late
Professor Clerk-Maxwell fully expressed the weight of
this postulate in the following words : —
All our knowledge, both of time and place, is relative.
"\Yhen a man has acquired the habit of putting words together.
without troubling himself to form the thoughts which ought to
correspond to them, it is easy for him to frame an antithesis
between this relative knowledge and a so-called absolute know
ledge, and to point out our ignorance of the absolute position of
a point as an instance of the limitation of our faculties. Any
one, however, who will try to imagine the state of a mind con
scious of knowing the absolute position of a point will ever after
be content with our relative knowledge.1
It is of such great value to ascertain how far we can
be certain of the truth of our postulates in the exact
sciences that I shall ask the reader to return to our
conception of position albeit from a somewhat different
standpoint. I shall even ask him to attempt an exami
nation of that state of mind which Professor Clerk-
Maxwell hinted at in his last sentence.
Suppose we had a tube of exceedingly sr all bore
bent into a circular shape, and within this tube a worm
of length A B. Then in the limiting case when we
make the bore of the tube and the worm infinitely fine,
we shall be considering space of one dimension. For
so soon as we have fixed one point, c, 011 the tube, the
length of arc c A suffices to determine the position of
the worm. Assuming that the worm is incapable of
1 Matter and Motion, p. 20.
216 THE COMMON SENSE OF THE EXACT SCIENCES.
recognising anything outside its own tube-space, it
would still be able to draw certain inferences as to the
nature of the space in which it existed were it capable
of distinguishing some mark c on the side of its tube.
Thus it would notice when it returned to the point c,
and it would find that this return would continually
recur as it went round in the bore ; in other words, the
worm would readily postulate the finiteness of space.
Further, since the worm would always have the same
amount of bending, since all parts of a circle are of the
same shape, it might naturally assume the sameness of
all space, or that space possessed the same properties at
all points. This assumption is precisely akin to the one
we make when we assert that the postulates of Euclidian
geometry, which, experience teaches us, are practically
true for the space immediately about us, are also true
for all space ; we assume the sameness of our three-
dimensioned space. The worm would, however, have
better reason for its postulate than we have, because it
would have visited .every part of its own one-dimen
sioned space.
Besides the finiteness and sameness of its space the
worm might assert the relativity of position, and deter-
POSITION. 217
mine its position by the length of the arc between c
and A. Let us now make a variation in our problem
and suppose the worm incapable either of making or
of recognising any mark on the tube. Then it would
clearly be impossible for the worm to ascertain whether
its space were limited or not ; it would never know
when it had made a complete revolution in its tube. In
fact, since the worm would always possess the same
amount of bending, it would naturally associate tl«it
bending with its plnjsical constitution , and not with tlie
space which it was traversing. It might thus very
reasonably suppose its space was infinite, or that it was
moving in an infinitely long tube. If the worm thus
associated bending with its physical condition it would
find no difference between motion in space of constant
bend (a circle) and motion in what is termed homaloulal
or flat space (a straight line) ; if suddenly transferred
from one to the other it would attribute the feeling
arising from difference of bending to some change
which had taken place in its physical constitution.
Hence in one-dimensioned space of constant bend all
position is necessarily relative, and the finite or in
finite character of space will be postulated according as
it is possible or not to fix a point in it.1
Let us now suppose our worm moving in a different
sort of tube ; for example, that shadow of a circle \\ e
have called an ellipse. In such a tube the degree of
bending is not everywhere the same ; the worm as it
passes from the place of least bending c to the place of
most bending D, will pass through a succession of bend-
ings, and each point H between c and D will have its
1 This supposes the one-dimensioned space of constant bend to lie in a
plane; the argument does not apply to space like tli.it of a helix (or the
form of a, corkscrew), which is of constant bend, bat yet not finite.
218 THE COMMON SENSE OF THE EXACT SCIENCES.
own degree of bending. Hence there is something
quite apart from the position of H relative to c which
characterises the point H ; namely, associated with H is
a particular degree of bending, and the position of the
point H in c D is at once fixed if we know the degree of
bending there. Thus the worm might determine abso
lute position in its space by the degree of bending asso
ciated with its position. The worm is now able to
appreciate differences of bend, and might even form a
scale of bending rising by equal differences. The zero
of such scale might be anywhere the worm pleased, and
degrees of greater and less bend might be measured as
positive and negative quantities from that zero. This
zero might in fact be purely imaginary ; that is, represent
a, degree of bending non-existent in the worm's space ;
for example, in the case of an ellipse, absolute straight-
ness, a conception which the worm might form as a
limit to its experience of degrees of bend.1 Thus it
would seem that in space of ' varying bend,' or space
which is not same, position is not necessarily relative.
The relativity has ceased to belong to position in space ;
it has been transferred to the scale of bending formed
1 Physicists may be reminded of the absolute zero of temperature.
POSITION. 21 9
by the worm ; it has become a relativity of physical feel
ing. In the case of an elliptic tube there are owing to
its symmetry four points of equal bend, as H, E, F, and
G, but there is the following distinction between H, p
and E, G. If the worm be going round in the direction
indicated by the letters c H D E, at H or F it will be pass
ing from positions of less to positions of greater bending,
but at E or G from positions of greater to positions
of less bending. Thus the worm might easily draw a
distinction between H, F and E, G. It would only be
liable to suppose the points H and F identical because
FIG. 01.
they possess the same degree of bending. We mi<_rht
remove even this possible doubt by supposing the worm
to be moving in a pear-shaped tube, as in the accom
panying figure ; then there will only be two points of
equal bend, like H and G, which are readily distinguished
in the manne'r mentioned above.
We might thus conclude that in one-dimensioned
space of variable bend position is not necessarily •
relative. There is, however, one point to be noted with
regard to this statement. We have assumed that the
worm will associate change of bending with change of
position in its space, but the worm would be sensible of
220 THE COMMON SENSE OF THE EXACT SCIENCES.
it as a change of physical state or as a change of feeling.
Hence the worm might very readily be led into the
error of postulating the sameness of its space, and
attributing all the changes in its bend, really due to its
position in space, to some periodic (if it moves uniformly
round its tube) or irregular (if it moves in any fashion
backwards and forwards) changes to which its physical
constitution was subject. Similar results might also
arise if the worm were either moving in space of the
same bend, which bend could be changed by some ex
ternal agency as a whole, or if again its space were of
varying bend, which was also capable of changing in
any fashion with time. The reader can picture these
cases by supposing the tube made of flexible material.
The worm might either attribute change in its degree
of bend to change in the character of its space or to
change in its physical condition not arising from its
position in space. We conclude that the postulate of
the relativity of position is not necessarily true for one-
dimensioned space of varying bend.
When we proceed from one to two-dimensioned
space, we obtain results of an exactly similar character.
If we take perfectly even (so called homaloidal) space of
two dimensions, that is, a plane, then a perfectly flat
figure can be moved about anywhere in it without
altering its shape. If by analogy to an infinitely thin
worm we take an infinitely thin flat-fish, this fish
would be incapable of determining position could it
leave no landmarks in its plane space. So soon as it
had fixed two points in its plane it would be able to
determine relative position.
Now, suppose that instead of taking this homaloidal
space of two dimensions we were still to take a perfectly
same space but one of finite bend, that is, the surface
POSITION. 221
of a sphere. Then let us so stretch and bend onr flat
fish that it would fit on to some part of the sphere.
Since the surface of the sphere is everywhere space of
the same shape, the fish would then be capable of
moving about on the surface without in any way alter
ing the amount of bending and stretching which we
had found it necessary to apply to make the fish fit in
any one position.. Wore the fish incapable of leaving
landmarks on the surface of the sphere, it would be
totally unable to determine position; if it could leave
at least two landmarks it would be able to determine
relative position. Just as the worm in the circular tube,
the fish without landmarks might reasonably suppose
its space infinite, or even look upon it as perfectly flat
(homaloidal) and attribute the constant degree of bend
and stretch to its physical nature.
Let us now pass to some space of two dimensions
which is not same — to some space, for example, like the
saddle-back surface we have considered on page 89,
which has a varying bend. In this case the fish, if it
fitted at one part of the surface, would not necessarily
fit at another. If it moved about in its space, it would
be needful that a continual process of bending and
stretching should be carried on. Thus every part of
this two-dimensioned space would be defined by the
particular amount of bend and stretch necessary to
make the fish fit it, or, as it is usually termed, by the
curvature. In surfaces with some degree of symmetry
there would necessarily be parts of equal curvature, and
. some cases the fish might perhaps distinguish
;tween these points in the same fashion as the worm
stinguished between points of equal curvature in the
ise of an elliptic tube. In irregular surfaces, however,
is not necessary that such points of equal curvature
222 THE COMMON SENSE OF THE EXACT SCIENCES
should arise. We are thus led to conclusions like those
we have formed for one-dimensioned space, namely :
Position in space of two dimensions which is not' same
might be determined absolutely by means of the curva
ture. Our fish has only to carry about with it a scale
of degrees of bending and stretching corresponding to
various positions on the surface in order to determine
absolutely its position in its space. On the other hand,
the fish might very readily attribute all these changes
of bend and stretch to variations of its physical nature
in nowise dependent on its position in space. Thus it
might believe itself to have a most varied physical life,
a continual change of physical feeling quite indepen
dent of the geometrical character of the space in which
it dwelt. It might suppose that space to be perfectly
same, or even degrade it to the * dreary infinity of a
hornaloid.' l
As a result, then, of our consideration of one and two-
dimensioned space we find that, if these spaces be not
same (a fortiori not homaloidal), we should by reason
of their curvature have a means of determining absolute
position. But we see also that a being existing in
these dimensions would most probably attribute the
effects of curvature to changes in its own physical
condition in nowise connected with the geometrical
character of its space.
What lesson may we learn by analogy for the three-
dimensioned space in which we ourselves exist? To
begin with, we assume that all our space is perfectly
same, or that solid figures do not change their shape in
passing from one position in it to another. We base this
postulate of sameness upon the results of observation
1 In this case of two-dimensioned space assume it to be a plane. Cf.
Clifford's Lectures and Essays, vol. i. p. 323.
POSITION. 223
n that somewhat limited portion of space of which
we are cognisant.1 Supposing our observations to
correct, it by no means follows that because the
wrtion of space of which we are cognisant is for
practical purposes same, that therefore all space is
same.2 Such an assumption is a mere dogmatic ex-
;ension to the unknown of a postulate, which may
)erhaps be true for the space upon which we can ex
periment. To make such dogmatic assertions with
regard to the unknown is rather characteristic of the
mediaeval theologian than of the modern scientist. On
;he like basis with this postulate as to the sameness
of our space stands the further assumption that it is
tiomaloidal. When we assert that our space is every
where same, we suppose it of constant curvature (like
;he circle as one and the sphere as two-dimensioned
space) ; when we suppose it homaloidal we assume that
:his curvature is zero (like the line as one and the
plane as two-dimensioned space). This assumption
appears in our geometry under the form that two
parallel planes, or two parallel lines in the same plane —
1 It may he held by some that the postulate of the sameness of our
space is based upon the fact that no one has hitherto been ;ibl.. to form any
geometrical conception of space-curvature. Apart from the fact that man
kind habitually assumes many things of which it can form no guometrieal
conception (mathematicians the circular points at infinity, theologians
lansubstantiation), I may remark that we cannot expect any being to
Form a geometrical conception of the curvature of his space till he views it
From space of a higher dimension, that is, practically, never.
2 Yet it must be noted that, because a snliil li-uiv vjij'tar* to us to retain
the same shape when it is moved about in that pun inn of space with
'hich we are acquainted, it does not follow that the figure really does
retain its shape. The changes of shape may be either imperceptible for
th> -M- distances through which we are able to move the figure, or if they do
take place we may attribute them to ' physical causes' — to heat, light,
or magnetism — which may possibly be nitre names for variations in the
curvature of our space.
224 THE COMMON SEXSE OF THE EXACT SCIENCES.
that is, planes, or lines in the same plane, which how
ever far produced will never meet — have a real existence
in our space. This real existence, of which it is clearly
impossible for us to be cognisant, we postulate as a
result built upon our experience of what happens in
a limited portion of space. We may postulate that
the portion of space of which we are cognisant is
practically homaloidal, but we have clearly no right
to dogmatically extend this postulate to all space. A
constant curvature, imperceptible for that portion oi
space upon which we can experiment, or even a cur
vature which may vary in an almost imperceptible
manner with the time, would seem to satisfy all that
experience has taught us to be true of the space in
which we dwell.
But we may press our analogy a step further,
and ask, since our hypothetical worm and fish might
very readily attribute the effects of changes in the
bending of their spaces to changes in their own phy
sical condition, whether we may not in like fashion be
treating merely as physical variations effects which are
really due to changes in the curvature of our space ;
whether, in fact, some or all of those causes which we
term physical may not be due to the geometrical con
struction of our space. There are three kinds of
variation in the curvature of our space which we ought
to consider as within the range of possibility.
(i) Our space is perhaps really possessed of a curva
ture varying from point to point, which we fail to appre
ciate because we are acquainted with only a small
portion of space, or because we disguise its small varia
tions under changes in our physical condition which we
do not connect with our change of position. The mind
that could recognise this varying curvature might be
POSITION. 225
assumed to know the absolute position of a point. For
such a mind the postulate of the relativity of position
would cease to have a meaning. It does not seem so
hard to conceive srfch a state of mind as the late
Professor Clerk-Maxwell would have had us believe.
It would be one capable of distinguishing those so-
called physical changes which are really geometrical
or due to a change of position in space.
(ii) Our space may be really same (of equal curva
ture), but its degree of curvature may change as a
whole with the time. In this way our geometry ba>>-<l
011 the sameness of space would still hold good for all
parts of space, but the change of curvature might
produce in space a succession of apparent physical
changes.
(iii) We may conceive our space to have everywhere
a nearly uniform curvature, but that slight variations of
the curvature may occur from point to point, and them
selves vary with the time. These variations of the
icurvature with the time may produce effects which we
not unnaturally attribute to physical causes indepen
dent of the geometry of our space. We might even ^-o
so far as to assign to this variation of the curvature <>l
|space ' what really happens in that phenomenon which
|we term the motion of matter.' l
1 This remarkable possibilit;/ seems first to have been suggested by
afessor Clifford in a paper prespnted to the Cambridge Philosophical
(Society in 1870 (Mathematical Papers, p. 21). I may add the following
emarks: The most notable physical quantities which vary with position
time are heat, light, and electro-magnetism. It is these that we ought
Jjeculiarly to consider when seeking for any physical changes, which may bo
lue to changes in the curvature of spaco. If we suppose the boundary of
ay arbitrary figure in spaco to be distorted by the variation of space-
rurvature, there would, by analogy from one and two dimensions, be no
Lhange in the volume of the figure arising from such distortion. Further,
If we assume as an axiom that space resists curvature with a resistance
226 THE COMMON SENSE OF THE EXACT SCIENCES.
We have introduced these considerations as to the
nature of our space to bring home to the reader the
character of the postulates we make in the exact
sciences. These postulates are* not, as too often
assumed, necessary and universal truths ; they are
merely axioms based on our experience of a, certain
limited region. Just as in any branch of physical
inquiry we start by making experiments, and basing on
our experiments a set of axioms which form the founda
tion of an exact science, so in geometry our axioms are
really, although less obviously, the result of experience.
On this ground geometry has been properly termed at
the commencement of Chapter II. a physical science.
The danger of asserting dogmatically that an axiom
based on the experience of a limited region holds
universally will now be to some extent apparent to the
reader. It ma}' lead us to entirely overlook, or when
suggested at once reject, a possible explanation o1
phenomena. The hypotheses that space is not homa-
loidal, and again, that its geometrical character may
change with the time, may or may not be destined to
play a great part in the physics of the future ; yet we
cannot refuse to consider them as possible explanations
of physical phenomena, because they may be opposed to
the popular dogmatic belief in the universality oi
certain geometrical axioms — a belief which has arisen'
from centuries of indiscriminating worship of the
genius of Euclid.
proportional to the change, \ve find that waves of ' space -displacement ' are
precisely similar to those of the elastic medium which we suppose to propa
gate light and heat. We also find that ' space-twist ' is a quantity exactly,
corresponding to magnetic induction, and satisfying relations similar tc
those which hold for the magnetic field. It is a question whether physicists
might not find it simpler to assume that space is capable of a varying
curvature, and of a resistance to that variation, than to suppose the exist-
•^nce of a subtle medium pervading an invariable homaloidal space.
227
CHAPTER V.
MOTION.
§ 1. On the Various Kinds of Motion.
WHILE the chapters on Space and Position considered
ihe sizes, the shapes, and the distances of things, the
>resent chapter on Motion will tivat of the changes in
;hese sizes, shapes, and distances, which take place from
;ime to time.
The difference between the ordinary meaning at-
ached to the word ' change ' in everyday life and the
meaning it has in the exact sciences is perhaps better
llustrated by the subject of this chapter than by any
other that we have yet studied. \Ve attained exactness
n the description of quantity and position by substitut
ing the method of representing them by straight liii'-s
Irawn on paper for the method of representing them by
means of numbers; though this, at lir.st sight, might
iasily seem to be a step backwards rather than a step
or wards, since it is more like a child's sign of opening
ts arms to show that its stick is so long, than a pro
)f scientific calculation.
It is, however, by no means an easy thing to give
in accurate description of motion, even although it is
tself as common and familiar a conception as quantity
>r position.
Let us take a simple case. Suppose that a man, on a
•ailway journey, is sitting at one end of a compartment
Q -1
228 THE COMMON SENSE OF THE EXACT SCIENCES.
with his face towards the engine ; and that, while th(
train is going along, he gets up and goes to the othei
end of the compartment and sits down with his back to
the engine. For ordinary purposes this description is
amply sufficient, but it is very far indeed from being
an exact description of the motion of the man during
that time. In the first place, the train was moving
and it is necessary to state in what direction, and how
fast it was going at every instant during the interva
considered. Next, we must describe the motion of the
man relatively to the train ; and, for this purpose, we
must neglect the motion of the train and consider how
the man would have moved if the train had been a
rest. First of all, he changes his position from one
corner of the compartment to the opposite corner
next, in doing this he turns round ; and, lastly, as he is
walking along or rising up or sitting down, the size am
shape of many of his muscles are altered. We shoulc
thus have to say, first, exactly how fast and in wha
direction he was moving at every instant, as we had to
do in the case of the train ; then, how quickly he was
turning round ; and, lastly, what changes of size or
shape were taking place in his muscles, and how fas
they were occurring.
It may be urged that this would be a very trouble
some operation, and that nobody wants to describe the
motion of the man so exactly. This is quite true ; the
case which has been taken for illustration is not one
which it is necessary to describe exactly, but we can
easily find another case which is very analogous
to this, and which it is most important to describe
exactly. The earth moves round the sun once in ever}
year ; it is also rotating on its own axis once every day ;
the floating parts of it — the ocean and the air — aw
MOTION. 229
constantly undergoing changes of shape and state
which we can observe and which it is of the utmost
mportance that we should be able to predict and
calculate ; even the solid nucleus of the earth is con
stantly subject to slight changes in size and shape,
which, however, are not large enough to admit of ac-
urate observation. Here, then, is a problem whose
complexity is quite as great as that of the former, ;nul
whose solution is of pressing practical importance.
The method which is adopted for attacking this
>roblem of the accurate description of motion is to begin
with the simplest cases. By the simplest cases we mean
;hose in which certain complicating circumstances do
not arise. We may first of all restrict ourselves to the
study of the motions of those bodies in which there is
no change of size or shape. A body which preserves
ts size and shape unaltered during the interval of tinn-
considered is called a rigid body. The word 'rigid ' is
lere used in a technical sense belonging to the science
of dynamic, and does not mean, as in ordinary lan-
juage, a body which resists alteration of size and shape,
>ut merely a body which, during a certain time,
lappens not to be altered in those respects. Then, as the
irst and simplest case, we should study that motion of
a rigid body in which there is no turning round, and
n which therefore every line in the bodv kf"j>s the
same direction (though of course not the same position)
ihroughout the motion. We state this by saying that
avery line 'rigidly connected' writh the body remains
>arallel to itself. Such a motion is called a motion of
'ranslation, or simply a translation ; and so the first and
amplest case we have to study is the translation of
igid bodies. After that we nmst proceed to consider
iheir turning round, or rotation ; and then we have to
230 THE COMMON SENSE OF THE EXACT SCIENCES.
describe the changes of size or shape which bodies may
undergo, these last changes being called strains.' The
study of motion therefore requires the further study of
translations, of rotations, and of strains, and further,
the art of combining these together. When we have
studied all this we shall be able to describe motions
exactly ; and then, but not till then, will it be possible
to state the exact circumstances under which motions
of a given kind occur. The exact circumstances under
which motions of a given kind occur we call a law of
nature.
§ 2. Translation and the Curve of Positions.
Let us talk, to begin with, of the translation of a
rigid body.
Suppose a table to be taken from the top to the
bottom of a house in such a manner that the surface of
it is always kept horizontal, and that its length is made
always to point due north and south ; it may be taken
down a staircase of any form, but it is not to be turned
round or tilted up. The table will then undergo a
translation. If we now consider a particular corner of
the table, or the end of one of its legs, or any other
point, this point will have described a certain curve ID
a certain manner ; that is to say, at every point of
curve it will have been going at a certain definite rateJ
Now the important property of a motion of translation'
which makes it more easy to deal with than any othei
motion, is that for all points of the body this curve if
the same in size and shape and mode of description
That this is so in the case of the table is at once seei j
from the fact that the table is never turned roun<
nor tilted up during the motion, so that the differen
points of it must at any instant be moving in the sam<
f MOTION. 231
direction and at the same rate. In order therefore to
describe this motion of the table it will be sufficient to
describe the motion of any point of it, say the end of
one of its legs. And so, in general, the problem of
describing the motion of translation of any rigid body
is reduced to the problem of describing the motion of
a point along a curve.
Now this is a very much easier t;isk than our
original problem of describing the motion of the e;irth
or the motion of the man in the train; but we shall
see that, by properly studying this, it will be easy to
^uild up out of it other more complicated cases. Still,
ven in this form our problem is not quite simple
nough to be directly attacked. What we have to do,
it must be remembered, is to state exactly where a
ertain point was, and how last it was going at every
instant of time during a certain interval. This would
require us first to describe exactly the shape of the
urve along which the point moved ; next, to say how
lar it had travelled along the curve from the beginning
up to any given instant; and lastlv, how fast it wa>
*oing at that instant. To deal with this problem we
must first take the very simplest case of it, that, namely,
In which the point moves along a straight line, and
.eave for the present out of account any description of
ihe rate of motion of the point ; so that we have only
say where the point was on a certain straight line
at every instant of time within a given interval.
But we have already considered what is the best wav
of describing the position of a point upon a straight
ine. It is described by means of the step which is
required to carry it to that position from a certain
tandard place, viz. a step from that place so far to the
right or to the left. To specify the length of the step,
232 THE COMMON SENSE OF THE EXACT SCIENCES.
if we are to describe it exactly, we must not make use
of any words or numbers, but must draw a line which
will represent the length corresponding to every instant
of time within a certain interval, so that we may
always be able to answer the question, Where was
the point at this particular instant ? But a question,
in order to be exactly answered, must first be exactly
asked ; and to do this it is necessary that the instant
of time about which the question is asked should be
accurately specified.
Now time, like length, is a continuous quantity
which cannot in general be described by words or
numbers, but can be by the drawing of a line which shall
represent it to a certain scale. .Suppose, then, that the
interval of time during which the motion of a point has
to be described is the interval from twelve o'clock to
one o'clock. We must mark on a straight line a point
to represent twelve o'clock and another point to repre
sent one o'clock; then every instant between twelve
o'clock and one o'clock will be represented by a point
which divides the distance between these two marked
points in the same ratio in which that instant divides
the interval between twelve o'clock and one o'clock.
Then for every one of these points it is necessary to
assign a certain length, representing (to some definite
scale) the distance which the point has travelled up to
that instant ; and the question arises, In what way shall
we mark down these lengths ?
Let us first of all observe the difficulty of answering
this question. If we could be content with an approxi
mate solution instead of an exact one, we might make
a table and put down in inches and decimals of an inch
the distances travelled, making an entry for every
minute, or even perhaps for every second during the
MOTION.
233
hour. Such tables are in fact constructed and pub
lished in the * Nautical Almanac ' for the positions of the
moon and of the planets. The labour of making this
table will evidently depend upon its degree of minute
ness ; it will of course take sixty times as long to make
a table showing the position of the point at every
second as to make one showing the position at every
minute, because there will be sixty times as many
values to calculate. But the problem of describing
exactly the motion of the point requires us to make a
table showing the position of the point at every instant ;
that is, a table in which are entered an infinite number
of values. These values moreover are to be shown, not
in inches and decimals of an inch, but by lengths drawn
upon paper. Yet we shall find that this pictorial mode of
constructing the table is in most cases very much easier
than the other. We have only to decide where we shall
put the straight lines which represent the distances
that the point has travelled at different instants.
FIG. 92.
Let a b be the length which represents the interval
lof time from twelve o'clock to one o'clock, and let ra be
[the point representing any intermediate instant. Then
lif we draw at m a line perpendicular to a b whose length
Ishall represent (to any scale that we may choose) the
[distance that the point has up to this instant travelled,
Ithen p, the extremity of this line, will correspond to
234 THE COMMON SENSE OP THE EXACT SCIENCES.
an entry in our table. But if such lines be drawn
perpendicular to a b from every point in it, all the
points p, which are the several extremities of these
lines, will lie upon some curve ; and this curve will re
present an infinite number of entries in our table. For,
when once the curve is drawn, if a question is asked :
What was the position of the point at any instant
between twelve o'clock and one o'clock ? (this instant
being specified in the right way by marking a point
between a and b which divides that line in the same
ratio as' the given instant divides the hour), then the
answer to this question is obtained simply by drawing
a line through the marked point perpendicular to a b
until it meets the curve ; and the length of that line
will represent, to the scale previously agreed upon, the
distance travelled by the point.
Such a curve is called the curve of positions for a
given motion of the point ; and we arrive at this result,
that the proper way of specifying exactly a translation
along a straight line is to draw the curve of positions.
We have now learned to specify, by means of a
curve, the positions of a body which has motion of
translation along a straight line ; and we have not
only represented an infinite number of positions in
stead of a finite number, which is all a numerical table
would admit, but have also represented each position
with absolute exactness instead of approximately. It
is important to notice that in this and in all similar
ca,ses the exactness is ideal and not practical ; it is
exactness of conception and not of actual measurement.
For though it is not possible to measure a given length
and to state that measure any more accurately by
drawing a line than it is by writing it down in inches
and decimals of an inch, yet the representation by
MOTION. 235
means of a line enables us to reason upon it with an
exactness which would be impossible if we were re
stricted to numerical measurement.
§ 3. Uniform ^lotion.
Hitherto we have supposed our point to be moving
along a straight line, but were it to move along a curve
the construction given for the curve of positions would
still hold good, only the distance traversed at any
instant must now be measured from some standard
position aloiuj the curve. Hence any motion of a point,
or any motion of translation whatever, can be specified
by a properly drawn curve of positions, and the problem
of comparing and classifying different motions is there
fore reduced to the problem of comparing and classi
fying curves. Here again it is advisable and even
necessary to begin with a simple case. Let us take
the case of uniform motion, in which ihe h-idy passes
over equal distances in equal times ; and then, as we
may easily see, the curve of positions is a straight line.
Uniform motion may also be described as that in \vhich
a body alwavs goes at the same rate, and not quicker
at one time and slower at another. It is obvious that
in this case any two equal distances would require equal
times for traversing them, so that the two descriptions
of uniform motion are equivalent.
It was shown by Archimedes ^the proof is an easy
one, depending upon the definition of tin- fourth pro
portional) that whenever equal distances are traversed
in equal times, different distances will be traversed in
times proportional to them. Assuming this proposition,
becomes clear that the curve of positions must be a
J straight line, for a straight line is the only curve which
236 THE COMMON SENSE OF THE EXACT SCIENCES.
has the property that the height of every point of it is
proportional to its horizontal distance from a fixed
straight line.
We may also see in the following manner the con
nection between the straight line and uniform motion.
Suppose we walk up a hill so as always to get over
a horizontal distance of four miles in an hour. The
rate at which we go up will clearly depend on the steep
ness of the hill ; and if the hill is a plane, i.e. is of the
same steepness all the way up, then our rate of ascent
will be the same at every instant, or our upward motion
will be uniform. If the hill be four miles long and
one mile high, then, since the four miles of horizontal
distance will be traversed in an hour, the one mile of
vertical distance will also be traversed in an hour, and
we shall be gaining height at the uniform rate of one
mile an hour. If the hill were two miles high, or, as we
say twice as steep, then we should have been gaining
height at the rate of two miles an hour. Bat now if
\ve suppose a hill of varying steepness, so that the out
line of it seen from one side is a curve, then it is clear
that the rate at which we go up will depend upon the
part of the hill where we are, assuming that the rate at
which we go forward horizontally remains always the
same. This ' elevation ' of the hill may be taken as the
curve of positions for our vertical motion ; for the
horizontal distance that we have gone over, being
always proportional to the time, may be taken to repre
sent the time, and then the curve will have been con
structed according to our rule, viz. a horizontal dis
tance will have been taken proportional to the time
elapsed, and from the end of this line a perpendicular j
will have been raised indicating the height which we]
have risen in that time. Uniform motion then has
MOTION. . 237
for its curve of positions a straight line, and the rate
of the motion depends on the steepness of the line.
Variable motion, on the other hand, has a curved line
for its curve of positions, and the rate of motion
depends upon its varying steepness.
In the case of uniform motion it is very easy indeed
to understand what we moan by the rate of the motion.
Thus, if a man walks uniformly six miles an hour,
we know that he walks a mile in ten minutes, and the
tenth part of a mile in one minute, and so on in propor
tion. It may not, however, be possible to specify this
rate by means of numbers ; that is to say, the man may
not walk any definite number of miles in the hour, and
the exact distance that he walks may not be capable of
representation in terms of miles and fractions of a mile.
In that case we shall have to represent the velocity or
rate at which the man walks in much the same way as
we have represented other continuous quantities. We
must draw to scale upon paper a line representing the
length that he has walked in an hour, or a minute, or
any other interval of time that we decide to select ;
thus, for example, a uniform rate of walking might be
specified by marking points corresponding to particular
hours upon an Ordnance map. The rate of motion, or
velocity, is then a continuous quantity which can be
exactly specified, as we specify other continuous quan
tities, but which can be only approximately described by
means of numbers.
§ 4. Variable Motion.
Let us now suppose that the motion is not uniform,
and inquire what is meant in that case by the rate at
which a body moves.
238 THE COMMON SEXSE OF THE EXACT SCIENCES.
A train, for example, starts from a station and in
the course of a few minutes gets up to a speed of 30
miles an hour. It began by being at rest, and it ends by
having this large velocity. What has happened to it in
the meantime ? We can understand already in a rough
sort of way what is meant by saying that at a certain
time between the two moments the train must have been
going at 15 miles an hour, or at any other intermediate
rate ; but let us endeavour to make this conception a
little more exact. Suppose, then, that a second train,
which is indefinitely long, is moving in the same direction
at a uniform rate of 15 miles an hour on a pair of rails
parallel to that on which the first train moves ; thus,
when our first train is at rest the second one will appear
to move past it at the rate of 15 miles an hour. When
the first train starts an observer seated in it will see
the second train going apparently rather more slowly
than before, but it will still seem to be moving forwards.
As the first train gets up its speed, this apparent
forward motion will gradually decrease until the second
train will appear to be going so slowly that conversation
may be held between the two ; this will take place when
the rate of the first train has amounted to something
nearly but not quite equal to 15 miles an hour, which
we supposed to be the constant rate of the second train.
But as the rate of the first train continues to increase
there will come a certain instant at which the second train
will appear to stop gaining upon the first and to begin
to lose. At that particular instant it will be neither
gaining nor losing, but will be going at the same rate;
at that particular instant, therefore, we must say that
the first train is going at the rate of 15 miles an
hour. And it is at that instant only, for the equality
of the rates does not last for any fraction of a second,
MOTION. 239
however small ; the very instant that the second train
appears to stop gaining it also appears to begin losing.
The two trains then run exactly together for no distance
at all, not even for the smallest fraction of an inch,
and yet we have to say that at one particular instant
our first train is going at the rate of 1 5 miles an hour,
although it does not continue to go at that rate during
the smallest portion of time. There is no way of
measuring this instantaneous velocity except that which
has just been described of comparing the motion with a
uniform motion having that particular velocity.
Upon this we have to make the very important
remark that the rate at which a body is going is a
property as purely instantaneous as is the precise
position which it has at that instant. Thus, if a stone
be let fall to the ground, at the moment that it hits
the ground it is going at a certain definite rate ; and yet
at any previous moment it was not going so fast, since
it does not move at that rate for the smallest fraction of
a second. This consideration is somewhat difficult to
grasp thoroughly, and in fact it has led many people to
reject altogether the hypothesis of continuity ; but still
we may be helped somewhat in understanding it by
means of our study of the curve of positions, wherein
we saw that to a uniform motion corresponds a straight
line and that the rate of the motion depends on the
steepness of the line.
Let us now suppose a motion in which a body goes
i at a very slow but uniform rate for the first second,
during the next second uniformly but somewhat faster,
[faster again during the third second, and so on. The
curve of positions will then be represented by a series
[of straight lines becoming steeper and steeper and form-
g part of a polygon. From a sufficient distance oft
240 THE COMMON SENSE OF THE EXACT SCIENCES.
this polygon will look like a curved line ; and if, instead
of taking intervals of a second during which the rates
of motion are severally considered uniform, we had
taken intervals of a tenth of a second, then the polygon
would look like a curved ]ine without our going so far*
away as before. For the shorter the lengths of the sides
of our polygon, the more will it look curved, and if the
intervals of time are reduced to one- tenth the sides
will be only one-tenth as long. The rate at which the
body under consideration is moving when it is in the
position to which any point of the polygon corresponds,
is obtained by prolonging that side of the polygon
which passes through the point; the rate will then
depend on the steepness of this line, since, where the
line is a side of the polygon, it represents the uniform
motion which the body has during a certain interval.
When the polygon looks like a curve the sides are very
short, and any side, being prolonged both ways, will
look like a tangent to the curve.
Now in considering the general case of varying
motion we should have, instead of the above polygon
which looks like a curve, an actual curve ; the difference
between them being that, if we look at the curve-like
polygon with a sufficiently strong microscope, we shall i
be able to see its angles, but however powerful a micro
scope we may apply to the curve it will always look like i
a curve. But there is this property in common, that if
we draw a tangent to the curve at any point, then, since j
the steepness of this tangent will be exactly the same as
the steepness of the curve at that particular point, it will j
give the rate for the motion represented by the curve, just!
as before the steepness of the prolonged small side of thej
polygon gave the rate for the motion represented by the
polygon. That is to say, the instantaneous velocity of
MOTION.
241
a body in any position may be learnt from its curve of
positions by drawing a tangent to this curve at the point
corresponding to the position; for the steepness of this
tangent will give us the velocity or rate which we want,
since the tangent itself corresponds to a uniform
motion of the same velocity as that belonging to the
given varying inotion at the particular instant. From
this means of representing the rate we can see how it
is that the instantaneous velocity of a body generally
belongs to it only at an instant and not for any length
of time however short ; for the steepness of the curve
is continually changing as we go from one part of it
to another, and the curve is not straight for any portion
of its length however small.
The problem of determining the instantaneous ve
locity in a given position is therefore reduced to the
problem of drawing a tangent to a given curve. We
have a sufficiently clear general notion of what is meant
by each of these things, but the notion which is suf
ficient for purposes of ordinary discourse is not sufficient
for the purposes of reasoning, and it must therefore be
i made exact. Just as we had to make our notion of
the ratio of two quantities exact by means of a definition
(of the fourth proportional, or of the equality of two
ratios which were expressed in terms of numbers, so
mere we shall have to make our idea of a velocity
exact by expressing it in terms of measurable quan-
Itities which do not change.
We have no means of measuring the instantaneous
[velocity of a moving body; the only thing that we can
leasure is the space which it traverses in a given interval
fof time. In the case in which a body is moving uni-
Iformly, its instantaneous velocity, being always tho same,
Is completely specified as soon as we know how far
242 THE COMMON SENSE OF THE EXACT SCIENCES.
the body has gone in a definite time. And, as we
have already observed, the result is the same whatever
this interval of time may be ; the rate of four miles an
hour is the same as eight miles in two hours, or two
miles in half an hour, or one mile in a quarter of an
hour. But if a body be moving with a velocity which
is continually changing, the knowledge of how far it has
gone in a given interval of time tells us nothing about
the instantaneous velocity for any position during that
interval. To say, for instance, that a man has
travelled a distance of four miles during an hour, does
not give us any information about the actual rate at
which he was going at any moment during the hour,
unless Are know that he has been going at a uniform
rate. Still we are accustomed to say that in such
a case he must have been going on an average at the
rate of four miles an hour; and, as we shall find it
useful to speak of this rate as an ' average velocity,'
its general definition may be given as follows : —
If a body has gone over a certain distance in a
certain time its mean or average velocity is that with
which, if it travelled uniformly, it would get over the]
same distance in the same time.
This mean velocity is very simply represented by]
the help of the curve of positions. Let a and 6 be two]
points on the curve of positions ; then the meanj
velocity between the position represented by a and that
represented by 6 is given by the steepness of the
straight line a b. This, moreover, enables us to make
some progress towards a method of calculating instan
taneous velocity, for we showed that the problem oJ; r;
finding the instantaneous velocity of a body is, in thd
above method of representation, the problem of draw-]
ing a tangent to a curve. Now the mean velocity of
MOTION.
243
body is defined in terms of quantities which we are
already able to measure, for it requires the measure
ment of an interval of time and of the distance
traversed during that interval ; and further the chord
of a curve, i.e. the line joining one point of it to another,
Flti. 93.
is a line which we are able to draw. If then we can
lind some means of passing from the chord of a curve
to the tangent, the representation we have adopted will
help us to pass from the mean to the instantaneous
velocity.
§ 5. On tJie Tangent to a Curve.
Now let us suppose the chord a b joining the
[points on the curve to turn round the point a, which
remains fixed; then b will travel along the curve
FIG. 94.
towards a ; and if we suppose b not to stop in this
[notion until it has got beyond a to a point such as 6
|>n the other side, the chord will have turned round
ito the position a V. Now, looking at the curve which
it '2
244 THE COMMON SENSE OF THE EXACT SCIENCES.
is drawn in the figure, we see that the tangent to the
curve at a obviously lies between a b and V a. Thus if
a b turn round a so as to move into the position a V it will
at some instant have to pass over the position of the
tangent. At the instant when it passes over this
position where is the point b ? We can at once see
from the figure that it cannot be anywhere else than at
a, and yet we cannot attach any definite meaning to a
line described as joining two coincident points. If we
could, the determination of the tangent would be very
easy, for in order to draw the tangent to the curve at
a, we should merely say, Take any other point b on the
curve ; join a b by a straight line ; then make b travel
along the curve towards a, and the position of the line
a b when b has got to a is that of the tangent at a.
Here however arises the difficulty which we have already
pointed out, namely, that we cannot form any distinct
conception of a line joining two coincident points;;
two separate points are necessary in order to fix a
straight line. But it is clear that, although it is not]
yet satisfactory, there is still something in the defini-l
tion that is useful and correct ; for if we make tl
chord turn from the position a b to the position of the
tangent at a, the point b does during this motion move
along the curve up to the point a.
This difficulty was first cleared up and its explana-l
tion made a matter of common sense by Newton. Th< j
nature of his explanation is as follows : — Let us
simplicity take the curve to be a circle. If a straighj
stick be taken and bent so as to become part of :|
circle, the size of this circle will depend upon thj
amount of bending. The stick may be bent complete!!
round until the ends meet, and then it will make a verl
. .,
small circle ; or it may be bent very slightly indeed, an|
•it ,,„
MOTION.
245
then it will become part of a very large circle. Now,
conversely, suppose that we begin with a small circle,
and, holding it fast at one point, make it get larger
and larger, so that the piece we have hold of gets less
and less bent ; then, as the circle becomes extremely
large, any small portion of it will more and more
nearly approximate to a straight line. Hence a circle
possesses this property, that the more it is magnified
the straightor it becomes ; this property likewise be
longs to all the curves which we require to consider.
It is sometimes expressed by saying that the curve is
straight in its elements, or in its smallest parts ; but
the statement must be understood to mean only this,
that the smaller the piece of a curve is taken the
straighter it will look when magnified to a given
length.
Now let us apply tins to the problem of determining
the position of a tangent. Let us suppose the tan
gent at of a circle to be already drawn, and that a
FIG. 95.
'certain convenient length is marked off upon it; from
I the end of this T let a perpendicular be drawn to meet
[the circle in B, and let a be joined to B by a straight
line. We have now to consider the motion of the point
[B along the circle as the chord a P> is turning round
la towards the position a T ; and the difficulty in our wTay
lis clearly that figures like a B T get small, as for ex-
I ample a 1 1, and continue to decrease until they cease
[to be large enough to be definitely observed. Newton
246 THE COMMON SENSE OF THE EXACT SCIENCES.
gets over this difficulty by supposing tliat the figure is
always magnified to a definite size ; so that instead
of considering the smaller figure a b t we magnify it
throughout until a t is equal to the original length a T.
But the portion a b of the circle with which we are now
concerned is less than the former portion a B ; conse
quently when it is magnified to the same length (or
nearly so) it must appear straighter. That is to say, in
the new figure a b' T, which is a b t magnified, the point b
will be nearer to the point T than B in the old one a B T ;
consequently, also, as b moves along to a the chord a b
will get nearer to the tangent a T, or, what is the same
thing, the angle tab will get smaller. This last result
is clear enough, because, as we previously supposed, the
chord a b is always turning round towards the position
at.
But now the important thing is that, by taking b
near enough to a, we can make the curve in the magni
fied figure as straight as we please ; that is to say, we
FIG. 96.
can make V approach as near as we like to T. If we
were to measure off from T perpendicularly to a T any
length, however small, say T d, then we can always
draw a circle which shall have a T for a tangent and
which shall pass between T and d ; and, further, if
we like to draw a line a d making a very small angle
with a T, then it will still be possible to make b go so
close to a that in the magnified figure the angle b'aT
shall be smaller than the angle d a T which we have
drawn.
Now mark what this process, which has been called •;
MOTION. 247
Newton's microscope, really means. While the figure
which we wish to study is getting smaller and smaller,
and finally disappears altogether, we suppose it to be
continually magnified, so as to retain a convenient size.
We have one point moving along a curve up towards
another point, and we want to consider what happens
to the line joining them when the two points approach
indefinitely near to one another. The result at which
we have arrived by means of our microscope is that,
by taking the points near enough together, the line
may be made to approach as near as we please to the
tangent to the curve at the point a. This, therefore,
gives us a definition of the tangent to a curve in
terms only of measurable quantities. If at a certain
point a of a curve there is a line a t possessing the
property that by taking b near enough to a on the curve
the line a b can be brought as near as we like to a t
(that is, the angle bat made less than any assigned
angle, however small), then a t is called the tangent to
the curve at the point a. Observe that all the things
supposed to be done in this definition are things which
we know can be done. A very small angle can be
assigned; then, this angle being drawn, a position of
,the point b can be found which is such that a b makes
with at an angle smaller than this. A supposition is
here made in terms of quantities which we already
know and can measure. We only suppose in addition
that, however small the assigned angle may be, the
point b can always be found ; and if this is possible,
then in the case in which the assigned angle is ex-
itremely small, the line a b or at (for they now coin-
|cide) is called a tangent.
It is worth while to observe the likeness between
[this definition and the one that we previously discussed
248 THE COMMON SENSE OF THE EXACT SCIENCES.
of the fourth proportional or of the equality of ratio.
In that definition we supposed that, a certain fraction
being assigned, if the first ratio were greater than this
fraction, so also was the second ratio, and if less, less ;
and the question whether these ratios were greater or less
is one that can be settled by measurement and com
parison. We then made the further supposition that
whatever fraction were assigned the same result would
hold good ; and we said that in that ca.se the ratios
were equal. Now in both of these definitions, applying
respectively to tangents and to ratios, the difficulty is
that we cause a particular supposition to be extended
so as to be general ; for we assume that a statement
which can be very easily tested and found true in any
one case is true in an infinite number of cases in which
it has not been tested. But although the test cannot
be applied individually to all these cases in a practical
way, yet, since it is true in any individual case, we know
on rational grounds that it must be satisfied in general ;
and therefore, justified by this knowledge, we are able
to reason generally about the equality of ratios and
about the tangents to curves.
Let us now translate the definition at which we
have thus arrived from the language of curves and tan
gents into the language of instantaneous and mean
velocities. The steepness of the chord of the curve of
positions indicates the mean velocity, while the steep
ness of the tangent to the curve at any point indicates
the instantaneous velocity at that point. The process
of making the point b move nearer and nearer to the
point a corresponds to taking for consideration a
smaller and smaller interval of time after that moment
at which the instantaneous velocity is wanted.
Suppose, then, the velocity of a body, viz. a railway
MOTION. 249
train, to be varying, and that we want to find what its value
is at a given instant. We might get a very rough approxi
mation to it, or in some cases no approximation at all, by
taking the mean velocity during the hour which follows
that instant. We should get a closer approximation by
taking the mean velocity during the minute succeeding
that instant, because the instantaneous velocity would
have less time to change. A still closer approximation
would be obtained were we to take the mean velocity
during the succeeding second. In all motions we should
have to consider that we could make the approximation
as close as we like by taking a sufficiently small interval.
That is to say, if we choose to name any very small
velocity, such as one with which a body going uniformly
would move only an inch in a century, then, by taking
the interval small enough, it will be possible to make the
mean velocity differ from the instantaneous velocity by
less than this amount. Thus, finally, we shall have the
following definition of instantaneous velocity : If there is
a certain velocity to which the mean velocity during the
interval succeeding a given instant can be made to
ipproach as near as we like by taking the interval small
enough, then that velocity is called the instantaneous
velocity of the body at the given instant.
In this way then we have reduced the problem of
inding the velocity of a moving body at any instant to
;he problem of drawing a tangent to its curve of posi-
ions at the corresponding point ; and what we have
ilready proved amounts to saying that, if the position of
:he body be given in terms of the time by means of a
3urve, then the velocity of the body will be given in
erms of the time by means of the tangent to this curve.
Now there are many curves to which we can draw
.an gents by simple geometrical methods, as, for example,
250 THE COMMON SENSE OF THE EXACT SCIENCES.
to the ellipse and the parabola ; so that, whenever the
curve of positions of a body happens to be one of these,
we are able to find by geometrical construction the
velocity of the body at any instant. Thus in the case
of a falling body the curve of positions is a parabola,
and we might find by the known properties of the tan
gent to a parabola that the velocity in this case is pro
portional to the time. But in the great majority of
cases the problem of drawing a tangent to the curve
of positions is just as difficult as the original problem
of determining the velocity of a moving body, and in
fact we do in many cases solve the former by means of
the latter.1
§ 6. On the Determination of Variable Velocity.
What is actually wanted in every case will be
apparent from the consideration of the problem we
have just mentioned — that of a body falling down
straight. We note, from the experience of Galilei, that
the whole distance which the body has fallen from rest
at any instant is proportional to the square of the time ;
in fact, to obtain this distance in feet we must multiply
the number of seconds by itself and the result by a
number a little greater than sixteen. Thus, for instance,
in five seconds the body will have fallen rather more
than twenty-five times sixteen feet, or 400 feet. Now
what we want is some direct process of proving that
when the distance traversed is proportional to the square
of the time the velocity is always proportional to the
time. In the present case we can find the velocity at
the end of a given number of seconds by multiplying
that number by thirty-two feet ; thus at the end of five
seconds the velocity of the body will be 160 feet per
1 [The method is due to Eobcrval (1602-1675).— K. P.]
MOTION. 2-51
second.1 Now as a matter of fact a process (of winch
there is a simple example in the footnote) has been
worked out, by which from any algebraical rule telling
us how to calculate the distance traversed iu terms of
the time we can find another algebraical rule which
will tell us how to calculate the velocity in terms of the
time. One case of the process is this : If the distance
traversed is at any instant a times the nth power of the
time, then the velocity at any instant will be na times
1 The following may be taken as a proof. Let a lie the distance from
rest moved over by the body in t' seconds, b that moved over l>y it. in t ~ t'
seconds, so that t' seconds is the interval we take to find out the mean
velocity. Now by our rule just quoted, since a feet are passed over in t
seconds, we have
and similarly b= I6(t + O2= 1G (t" + W + 1'-*).
Hence \ve have b — a = \G(t- + 2tC + t"-}- IGt-
giving the distance moved over in the interval H . But the mean velocity
during this interval is obtained by dividing the distance moved over by the
time taken to traverse it; hence the mean velocity in our ease for the
interval of t' seconds immediately succeeding the t seconds
= b-a
t'
t'
i! +
= 32t +16C.
Now if we look at this result, which we have obtained for the mean velocity,
we see that there are two terms in it. The first, viz. 3'2f. is quite hide-
^•pendent of the interval t' which we have'taken ; the second, vix lf>/', depends
^directly on it, and will therefore change when we change t he interval. Now
..—-he distance per second represented by IGf feet can be made as small as
we like by taking t' small enough ; so that the mean velocity during the
.^•interval t' seconds succeeding the given instant can be made to approach
;.-, ^32)' feet per second as near as we like by taking t' small enough. Recurring
our definition of instantaneous velocity, it is now evident that the instan-
velocity of our falling body at the end of t seconds is Z'lt feet per
'ond.
252 THE COMMON SENSE OF THE EXACT SCIENCES.
the (w-i)tli power of the time. It is by means of this
process of altering one algebraical rule so as to get
another from it that both of the problems which we
have shown to be equivalent to one another are s
in practice.
There is yet another problem of very great import
ance in the study of natural phenomena which can be
made to depend on these two. When a point moves
alono- a straight line the distance of it from some fixed
point in the line is a quantity which varies from time
to time. The rate of change of this distance is the
same thing as the velocity of the moving point; and
the rate of change of any continuous quantity can only
be properly represented by means of the velocity of a
point.
Thus, for instance, the height of the tide at a given
port will vary from time to time during the day, and it
may be indicated by a mark which goes up and down
on a stick. The rate at which the height of the tide
varies will obviously be the same thing as the velocity
with which this mark goes up and down. Again the
pressure of the atmosphere is indicated by means of the
height of a mercury barometer. The rate at which this
pressure changes is obviously the same thing as the velo-
citv with which the surface of the mercury moves up am
down. Now whenever we want to describe the changes
which take place in any quantity in terms of the time,
we may indeed roughly and approximately do so by
means of a table. But this is also the most troublesome
way ; the proper way of describing them is by drawing
a curve in which the abscissa, or horizontal distance, at
any point represents the time, while the height of 1
curve at that point represents the value of the quantity
at that time (see p. 184). Whenever this is done we
MOTION. 253
practically suppose the variation of the quantity to be
represented by the motion of the point on a curve.
The quantity can only be adequately represented by
marking ofT a length proportional to it on a line ; so
that if the quantity varies then the length marked off
will vary, and consequently the end of this length will
move along the curve. The rate at which the quantity
varies is the rate at which this point moves ; and when
the values of the quantity for different times are repre
sented by the perpendicular distances of points on a
curve from the line which represents the time, its rate
of variation is determined by the tangent to tliat
curve.
§ 7. On tlie Method of Fluxions.
Hence we have three problems which are practically
the same. First, to find the velocity of a moving point
when we know where it is at every instant ; secondly, to
draw a tangent to a curve at any point ; thirdly, to find
the rate of change of a quantity when we know how great
it is at every instant. And the solution of them all
depends upon that process by which, when we take the
algebraical rule for finding the quantity in terms of the
time, we deduce from it another rule for finding its
rate of change in terms of the time.
This particular process of deriving one algebraical
rule from another was first investigated by Newton.
He was accustomed to describe a varying quantity as
a fluent, and its rate of change he called the fluxion of
the quantity. On account of these names, the entire
method of solving these problems by means of the
process of deriving one algebraical rule from another
was termed the Method of Fluxions.
In general the rate of variation of a quantity will
254 THE COMMON SENSE OF THE EXACT SCIENCES.
itself change from time to time; but if we consider
only an interval very small as compared with that re
quired for a considerable variation of the quantity, we
may legitimately suppose that it has not altered much
during that interval. This is practically equivalent to
supposing that the law of change has been uniformly
true during that interval, and that the rate of change
does not differ very much from its mean value. Now
the mean rate of change of a quantity during an interval
of time is just the difference between the values of the
quantity at the beginning and at the end divided by
the interval. If any quantity increased by one inch in
a second, then, although it may not have been increas
ing uniformly, or even been increasing at all during the
whole of that second, yet during the second its mean
rate of increase was one inch per second. Now if the
rate of increase only changes slowly we may, as an
approximation, fairly suppose it to be constant during
the second, and therefore to be equal to the mean rate ;
and, as we know, the smaller the interval of time is, the
less is the error arising from this supposition. This is,
as a matter of fact, the way in which that process is
established by means of which a rule for calculating
position is altered into a rule for calculating velocity.
The difference between the distances of the moving
point from some fixed point on the line at two different
times is divided by the interval between the times, and
this gives the mean rate of change during that interval.
If we find that, by making the interval smaller and
smaller, this mean rate of change gets nearer and
nearer to a certain value, then we conclude that this
value is the actual rate of change when we suppose the
interval to shrink up into an instant, or that it is, as
we call it, the instantaneous rate of change.
MOTION. 255
Because two differences are used in the argument
which establishes the process for changing the one rule
into the other, this process was called, first in other
countries and then also in England, the Differential
Calculus. The name is an unfortunate one, because the
rate of change which is therein calculated has nothing
bo do with differences, the only connection with
differences being that they are mentioned in the argu
ment which is used to establish the process. However
bhis may be, the object of the differential calculus or of
the method of fluxions (whichever name we choose to
»ive it) is to find a rule for calculating the rate of
change of a quantity when we have a rule for calcu
lating the quantity itself; and we have seen that when
bhis can be done the problem of drawing a tangent to
a curve and that of finding the velocity of a moving
point are also solved.
§ 8. Of the Relationship of Quantities, or Function*.
But we not only have rules for calculating the value
of a quantity at any time, but also rules for calculating
the value of one quantity in terms of another quite in
dependently of the time. Of the former class of rules an
example is the one mentioned above for calculating the
rise of the tide. We may either write down a formula
which will enable us to calculate it at a given instant,
or we may draw a curve which shall represent its rise
at different times of the day. Of the second kind of
rule a good example is that in which the pressure of a
given quantity of gas is given in terms of its volume
when the temperature is supposed to be constant; the
algebraical statement of the rule giving the relation
between them is that the two things vary inversely as
one another, or that the product representing them is
256 THE COMMON SENSE OP THE EXACT SCIENCES.
constant. Thus if we compress a mass of air to one-
half of its natural volume the pressure will become twice
as great, or will be, as it is called, two ' atmospheres.'
And so if we compress it to one-fifth of the volume the
pressure will become five times as great, or five atmo
spheres.
If we like to represent this by a figure we shall
draw a curve in which the abscissa, or horizontal
distance from the starting point, will represent the
volume, and a vertical line drawn through the extremity
of this abscissa will represent the pressure. For any
particular temperature the curve traced out by the ex
tremity of the line representing the pressure will be
a hyperbola having one asymptote vertical and the
other horizontal; and lor different temperatures we
shall have different hyperbolas with the same asym
ptotes. Thus every point in the plane will represent a
particular state of the body, since some hyperbola can
be drawn through it ; the horizontal distance of the
point from the origin will represent the volume, and its
vertical distance the pressure, while the particular
hyperbola on which it lies will indicate the tempera
ture. We have here an example of the physical im
portance of a family of curves, to which reference was
made in the preceding chapter (see p. 163).
When the connection between two quantities has
to be found out by actual observation, this is done by
properly plotting down points on paper (as in § 11,
Chap. IV.) to represent successive observations. Thus
in the case of air the pressure would be observed for
different values of the volume. For each of these
observed pairs of values a point would be marked in
the plane ; and when a sufficient number had been
marked it would become obvious to the eye that,
MOTION 207
roughly speaking, the point lay on a hyperbolic curve.
But it is to be noticed that it is only roughly that this
result holds, because observations are never so accu-
•ate that the curve does not require to be drawn pretty
reely in passing through the points. But directly the
geometer has seen that the shape of the curve is hyper-
)olic he recognises the law that pressure varies inversely
as volume.
We have here the relation between two quantities
sxpressed by means of a curve. Whenever two quanti-
ies are related in some such way, so that one of them
>eing given the other can be calculated or found, each
s said to be a function of the other. Now a function
may be supposed to be given either by an algebraical
rule or by a curve. Thus to find the pressure corre
sponding to a given volume we might say that a certain
number was to be divided by the number representing
volume, and the result would be the number of units
)f pressure ; or we might say that from the given point
f the horizontal line which represented the volume a
>erpendicular was to be drawn and continued till it met
lie curve, and that the ordinate (or the part of this
)etween the horizontal line and the curve) represented
jthe pressure. We have thus a connection established
>etweeii the science of geometry and the science of
quantity, as, for example, the relation between the two
[uantities, volume and pressure, is expressed by means
f a certain curve.
Now every connection between two sciences is a
ielp to both of them. When such a connection is
;:stablished we may both use the known theorems
«-bout quantities in order to investigate the nature of
urves (and this is, in fact, the method of co-ordinates
troduced by Descartes), or we may make use of
s
258 THE COMMON SENSE OF THE EXACT SCIENCES.
known geometrical properties of curves in order to
find out theorems about the way in which quantities
depend upon one another. For the first purpose the
\ relation between the two quantities is regarded as an
equation. Thus, instead of saying that a pressure
varies inversely as a volume we should prefer to say
that the product of the pressure and the volume is
equal to a certain constant, the temperature being
supposed unaltered; or, paying attention only to the
geometrical way of expressing this, we should say that,
for points along the curve we are considering, the
product of the abscissa and the ordinate is equal to a
certain fixed quantity. This is written for shortness
and from such an equation all the properties of a hyper
bola may be deduced.
But we may also make use of the properties o
known curves in order to study the ways in whicl
quantities can depend on one another. Thus the per
FIG. 97.
pendicular distance P H from the point P of the circld
to a fixed diameter A 0 a is a quantity whose ratio ix
the radius OP depends in a certain definite way upoi
the magnitude of the angle POA, or, what is the samr
thing (p. 143), upon the length of the arc A P. Therati*
is in fact what we have termed the sine of the angle, or
MOTION. 259
as it is sometimes called, the sine of the arc. If the arc
AP is made proportional to the time, or, what is the
same thing, if P is made to move uniformly round the
circle, then the length of the line PM will represent
the distance from the centre 0 of a point Q oscillating
according to a law which is defined by this geometrical
construction. This particular kind of oscillation, which
is called simple harmonic motion, occurs when the air
is agitated by sound, or the ether by light, or when
any elastic body is set into a tremor. Relations such
as that which we have just mentioned between arcs of
a circle and straight lines drawn according to some
simple constructions in the circle give rise to what are
often termed circular functions. Thus the trigono
metrical ratios considered in § 7 of Chapter IV. art
functions of this kind. We have also hyperbolic func
tions, depending on the hyperbola in somewhat the
same way in which circular functions depend upon the
circle, and elliptic functions, so called because by means
of them the length of the arc of an ellipse can be cal
culated.
But the most valuable method of studying the
properties of functions is derived from the considera
tions of which we have been treating in this chapter,
/iz. considerations of the rate of change of quantities.
,Whenthe relation between two quantities is known, the
•elation between their rates of change can be found by
i known algebraical process ; and we have shown that
e problem of finding this relation ultimately comes
o the same thing as the problem of drawing a tangent
.r,j|o the curve which expresses the relation between the
o original quantities. Thus, in the case we pre-
'usly considered of two quantities whose product is
nstant or which vary inversely as one another, it is
s 2
260 THE COMMON SENSE OF THE EXACT SCIENCES.
clear that one must increase when the other decreases ;
it is found that the ratio of these rates of change
is equal to the ratio of the quantities themselves.
Thus the rate of change of the volume of a gas is
to the rate of change of the pressure (the temperature
being kept constant) as the volume is to the pres
sure, it being always remembered that an increase of
the one implies a decrease of the other.
The consideration of this ratio of the rates of change
is of great importance in determining one of the fun
damental changeable properties of a body, namely, its
elasticity. We define the elasticity of a gas as the
change of pressure which will produce a given contrac
tion ; where by the term contraction is meant the change
in the volume divided by the whole volume before change.
Thus if the volume of a gas diminished one per cent., it
would experience a contraction of y^^th. If then, in
accordance with our definition, we divide the pressure ;
necessary to produce this contraction by y^-, or, what i
is the same thing, multiply it by 100, we shall gel;;
what is called the elasticity. Now in our case the change c
of pressure divided by the whole pressure is equal to what fl
we have called the contraction, that is, to y^-; anc :
therefore the change of pressure is equal to y^th of the :.
whole pressure. But we have just proved that the elas- n
ticity is 100 times the change of pressure necessary to -
produce the contraction we have been considering, and i -
is therefore equal to the whole pressure. Consequent!; [•
the elasticity of a gas is measured by the pressure o'l e:
the gas.
§ 9. Of Acceleration and the Hodograph.
We may then consider the rate of change of an
measurable quantity as another quantity which we ca £ -
MOTION. 261
find ; and we have derived our notion of it from the
velocity of a moving point. In the simplest case,
when this point is moving along a straight line, the
rate at which it is going is the rate of change of
its distance from a point fixed in the line. But in
the general case, when the point is moving not on a
straight line, but along any sort of curve, we shall not
give a complete description of its state of motion if we
only say how fast it is going; it will be necessary to
say in addition in what direction it is going. Hence
we must not only measure the quantity of a velocity,
but also a certain quality of it, viz. the direction.
Now we do as a matter of fact contrive to study these
two things together, and the method by which we do so
is perhaps one of the most powerful instruments by
which the scope of the exact sciences has been extended
in recent times. Defining the velocity of a moving
point as the rate of change of its position, we are met
\>y the question, What is its position'?
This question has been answered in the preceding
chapter. The position of a moving point is determined
when we know the directed step or vector which con
nects it with a fixed point. If then the velocity of the
moving point means the rate of change of its position,
and if this position is determined by the vector which
would carry us from some fixed point to the moving-
point, in order to understand velocity we shall have to
get a clear conception of Avhat is meant by the rate of
change of a vector.
A
FIG. 98.
Let us <ro bank for a moment to the simpler case of
a point moving along a straight line ; its position is
262 THE COMMON SENSE OF THE EXACT SCIENCES.
determined by means of the step A P from the point A
fixed in the straight line to the moving point P. Now
this step alters with the motion of the point ; so that
if the point comes to P' the step is changed from A P
to A P'. How is this change made in the step ?
Clearly by adding to the original step A P the new
step P P', and we specify the velocity of P by saying
at what rate this addition is made.
Now let us resume the general case. We have the
fixed point A given ; and the position of the moving
point P is determined by means of the step A P. As
P moves about, this step gets altered, so that when
P comes to P' this step is A P' ; it is therefore obvious
that it is altered not only in magnitude but also in
direction. Now the change may be made by adding
to the original step A P the new step P P' ; and it is
quite clear that if we go from A to P and then from
P to P' the result is exactly the same as if we had gone
FIG. 99.
directly from A to P'. The question then is : At what
rate does this addition take place, or what step per
second is added to the position? The answer as before
is of the nature of a step or vector — that is, the
change of position of the moving point has not only
magnitude but direction. We shall therefore have
to say that the rate of change of a step or vector is
always so many feet per second in a certain direction.
To sum up, then, we state that the velocity of a
MOTION. 263
moving point is the rate of change of the step which
specifies the position ; and that in order to describe
accurately this velocity, we must draw a line of given
length in a given direction ; we observe also that
the rate of change of a directed quantity is itself a
directed quantity. This last remark is of the utmost
importance, and we shall now apply it to a considera
tion of the velocity itself.
If a point is moving uniformly in a straight line its
velocity is always the same in magnitude and the same
in direction ; and consequently a line drawn to re
present it would be unaltered during the motion. But
if a point moves uniformly round a circle its velocity,
although always the same in magnitude, will be con
stantly changing in direction, and the line which
specifies this velocity will thus be always of the same
length, but constantly turning round so as always
to keep parallel with the direction of motion of the
moving point. And so, generally, when a point is
moving along any kind of curve let us suppose that
through some other point, which is kept fixed, a line is
always drawn which represents the velocity of the
moving point both in magnitude and direction. Since
the velocity of the moving point will in general change,
this line will also change both in size and in direction,
and the end of it will trace out some sort of curve.
Thus in the case of the uniform circular motion, since
the velocity remains constant, it is clear that the end
of the line representing the velocity will trace out a
circle ; in the case of a body thrown into the air the
end of the corresponding line would be found to de
scribe a vertical straight line. This curve described
by the end of the line which represents the velocity at
any instant may be regarded as a map of the motion,
264 THE COMMON SENSE OF THE EXACT SCIENCES.
and was for that reason called by Hamilton the
hodograph. If we know the path of the moving point
and also the hodograph of the motion, we can find the
velocity of the moving point at any particular position
in its path. All we have to do is to draw through the
centre of reference of the hodograph a line parallel to
the tangent to the path at the given position ; the
length of this line will give the rate of motion, or the
velocity of the point as it passes through that position
in its path. Hamilton proved that in the case of the
planetary orbits described about the sun the hodo
graph is always a circle. In this case it possesses
other interesting properties, as, for example, that the
amount of light and heat received by the planet during
a given interval of time is proportional to the length of
the arc of the hodograph between the two points corre
sponding to the beginning and end of that interval.
But the great use of the hodograph is to give us a
clear conception of the rate of change of the velocity.
This rate of change is called the acceleration. Now, it
must not be supposed that acceleration always means
an increase of velocity, for in this case, as in many
others, mathematicians have adopted for use one word
to denote a change that may have many directions ;
thus a decrease of velocity is called a negative accelera
tion. This mode of speaking, although rather puzzling
at first, becomes a help instead of a confusion when
one is accustomed to it. Now a velocity may be
changed in magnitude without altering its direction —
that is to say, it may be changed by adding it to a
velocity parallel to itself. In this case we say that the
acceleration is in the direction of motion. But a
velocity may also be changed in direction without being
changed in magnitude, and we have seen that then the
MOTION. 205
hodograph is a circle. The velocity is altered by
adding to it a velocity perpendicular to itself, for the
tangent at any point to a circle is at right angles to
the radius drawn to that point, and in this case we
may say that the acceleration is at right angles to the
direction of motion. But in general both the magni
tude and the direction of the velocity will vary, and then
we shall see that the acceleration is neither in the
direction of motion nor at right angles to it, but that
it is in some intermediate direction.
If we consider the motion in the hodograph of the
end of the line representing the velocity, we observe the
motion of a point whose position is defined by the step
to it from the centre of the hodograph. N >w this step
is just the velocity of the point P in the original curve,
for the line 0 Q, is supposed to be drawn at every instant
FIG. 100.
to represent the velocity of P in magnitude and direc
tion. Now we saw that the rate of change of the step
from some fixed point A to P was the velocity of P.
BtHence, since the step OQ drawn from the fixed point
I '0 to Q defines the position of Q, it is obvious that the
I rate of change of the step 0 Q is the velocity of Q. Since
B'O Q represents the velocity of P, it follows that the velo
city of the point Q describing the hodograph is the rate
of change of the velocity of P ; that is to say, it is the
acceleration of the motion of P. This acceleration
2G6 THE COMMON SENSE OF THE EXACT SCIENCES.
being the velocity of Q, and a velocity being as we have
seen a vector, it at once follows that the acceleration
is a vector or directed quantity.
In changing the magnitude and direction of the
velocity of a moving point we may consider that we are
pouring in, as it were, velocity of a certain kind at a
certain rate. In the case of a stone thrown up
obliquely and allowed to fall again the path described
is a parabola, and the direction of motion, which ori
ginally pointed obliquely upwards, turns round and
becomes horizontal, and then gradually points more
and more downwards. But what has really been
happening the whole time is that velocity straight
downwards has been continually added at a uniform
rate during every second, so that the original velocity
of the stone is compounded with a velocity vertically
downwards, increasing uniformly at the rate of thirty-
two feet a second. In this case, then, we say that the
acceleration, or rate of change per second of the velocity
of the stone, is constant and equal to thirty-two feet a
second vertically downwards.
If we whirl anything round at the end of a string
we shall be continually pouring in velocity directed
towards the end of the string which is held in the hand ;
and since the velocity of the body which is being
whirled is perpendicular to the direction of the string,
the added velocity is always perpendicular to the exist
ing velocity of the body. And so also when a planet is
travelling round the sun there is a continual pouring in
of velocity towards the sun, or, as we say, the accelera
tion is always in the line joining the planet to the sun.
In addition it is in this case found to vary inversely
as the square of the distance from the sun.
MOTION. 267
§ 10. On the Laws of Motion.
These examples prepare us to understand that law
of motion which is the basis of all exact treatment of
physics. When a body is moving let us consider -what
it is that depends upon the circumstances, meaning
by the ' circumstances ' the instantaneous position
relative to it of other bodies as well as the instantane
ous state of the body itself irrespective of its motion.
We might at first be inclined to say that the velocity
of the body depends on the circumstances, but very
little reflection will show us that in the same cir
cumstances a body may be moving with very different
velocities. At a given height above the earth's surface,
for example, a stone may be moving upwards or down
wards, or horizontally, or at any inclination, and in any
of these modes with any velocity whatever ; and there is
nothing contrary to nature in supposing a motion of this
sort. Yet we should find that, no matter in what way
the stone may move through a given position, the rate
of change per second of its velocity will always be the
same, viz. it will be thirty-two feet per second vertically
downwards. When we push a chair along the ice, in
order to describe the circumstances we must state the
compression of those muscles which keep our hands
against the chair. Now the rate at which the chair
moves does not depend simply upon this compression ;
'or a given amount of push may be either starting the
;,lchair from rest or may be quickening it when it is
[going slowly, or may be keeping it up at a high rate.
What is it, then, Avhich does depend upon the cir-
[sumstances 'P In whichever of these ways, or in what-
sver other way this given amount of push is used, its
Result in every case is obviously to change the rate of
268 THE COMMON SENSE OF THE EXACT SCIENCES.
motion of the chair; and this change of the rate of
motion will vary with the amount of push. Hence it is
the rate of change of the velocity, or the acceleration of
the chair which depends upon the circumstances, and
these circumstances are partly the compression of our
muscles and partly the friction of the ice ; the one is
increasing and the other is diminishing the velocity in
the direction in which the chair is going.
The law of motion to which allusion has just been
made is this : — The acceleration of a body, or the rate of
change of its velocity depends at any moment upon the
position relative to it of the surrounding bodies, but
not upon the rate at which the body itself is going.
There are two different ways in which this dependence
takes place. In some cases, as when a hand is pushing
a chair, the rate of change of the velocity depends on
the state of compression of the bodies in contact ; in.
other cases, as in the motion of the planets about the
sun, the acceleration depends on the relative position
of bodies at a distance.
The acceleration produced in a body by a particular
set of surrounding circumstances must in each case be
determined by experiment, but we have learnt by ex
perience a general law which much simplifies the expe
riments which it is necessary to make. This law is as
follows : — If the presence of one body alone produces a
certain acceleration in the motion of a given body, and
the presence of a second body alone another accelera
tion ; then, if both bodies ai*e present at the same time,
the one has in general no effect upon the acceleration
produced by the other. That is, the total accelera
tion of the moving body. will be the combination of the
two simple accelerations ; or, since accelerations are
directed quantities, we have only to combine the simple
MOTION. 269
accelerations, as we did vector steps in § 3 of the pre
ceding chapter, in order to find the result of super
posing two sets of surrounding circumstances.
Now while this great law of nature simplifies ex
tremely our consideration of the motion of the same
body under different surrounding circumstances, it
does not enable us to state anything as to the motion
of different bodies under the same surrounding circum
stances. This case, however, is amply provided for by
another comprehensive law which experience also has
taught us. We may thus state this third all-important
law of motion : — The ratio of the accelerations which
any two bodies produce in each other by their mutual
influence is a constant quantity, quite independent of
the exact physical characteristics of that influence.
That is to say, however the two bodies influence one
another, whether they touch or are connected by a
thread or being at a distance still alter one another's
velocities, this ratio will remain in these and all other
cases the same.
§ 11. Of Mass and Force.
Let us see how we can apply this law. Suppose we
take some standard body P and any other Q, and note
the ratios of the accelerations they produce in each
other under any of the simplest possible circumstances
of mutual influence. Let the ratio determined by ex
periment be represented by ra, or ra expresses the ratio of
the acceleration of the standard body P to that of the
second body Q. This quantity m is termed the mass
of the body Q. Let m-' be the ratio of the accelerations
produced in the standard body P and a third body R by
their mutual influence. Now the law as it stands above
enables us to treat only of the ratio of the accelerations
270 THE COMMON SENSE OF THE EXACT SCIENCES.
of P and Q, or again of P and R under varied cir
cumstances of mutual influence. It does not tell us
anything about the ratio of the accelerations which Q,
and R might produce in each other. Experience, how
ever, again helps us out of our difficulties and tells us
that if Q and R mutually influence each other, the
ratio of the acceleration of Q to that of R will he in
versely as the ratio of m to mf. If then we choose to
term unity the mass of our standard body, we may
state generally that mutual accelerations are inversely as
masses. Hence, when we have once determined the
masses of bodies we are able to apply our knowledge
of the effect of any set of circumstances on one body,
to calculate the effect which the same circumstances
would produce upon any other body.
The reader will remark that mass as defined above
is a ratio of accelerations, or in other words a mere
numerical constant experimentally deducible for any
two bodies. It is found that for two bodies of the same
uniform substance, their masses are proportional to
their volumes. This relation of mass to volume has
given rise to much obscurity. An indescribable some
thing termed 'matter has been associated with bodies.
Bodies are supposed to consist of matter filling space,
and the mass of a body is defined as the amount of matter
in it. An additional conception termed force has been
introduced and is supposed to be in some way resident in
matter. The force of a body P on a body Q, of mass m
is a quantity proportional to the mass m of Q and to
the acceleration which the presence of P produces in
the motion of Q. It will be obvious to the reader that
this conception of force no more explains why the pre
sence of P tends to change the velocity of Q, than the
conception of matter explains why mutual accelerations
MOTION.
271
are inversely as masses. The custom of basing our
ideas of motion on these terms * matter ' and ' force '
has too often led to obscurity, not only in mathematical,
but in philosophical reasoning. We do not know why
the presence of one body tends to change the velocity
of another; to say that it arises from the force resident
in the first body acting upon the matter of the moving
body is only to slur over our ignorance. All that we
do know is that the presence of one body may tend
to change the velocity of another, and that, if it does,
the change can be ascertained from experiment, and
obeys the above laws.
To calculate by means of the laws of motion from the
observed effects on a simple body of a simple set of cir
cumstances the more complex effects of any combina
tion of circumstances on a complex body or system of
bodies is the special function of that branch of the exact
sciences which is termed Applied Mathematics.
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4 THE MERCHANT OF VENICE ACT t
Salar. My wind, cooling my broth,
Would blow me to an ague, when I thought
What harm a wind too great might do at sea.
I should not see the sandy hour-glass run
But I should think of shallows and of flats,
And see my wealthy Andrew, dock'd in sand,
Vailing her high-top lower than her ribs
To kiss her burial. Should I go to church
And see the holy edifice of stone,
And not bethink me straight of dangerous rocks,
Which touching but my gentle vessel's side,
Would scatter all her spices on the stream,
Enrobe the roaring waters with my silks,
And, in a word, but even now worth this,
And now worth nothing ? Shall I have the thought
To think on this, and shall I lack the thought
That such a thing bechanc'd would make me sad ?
But tell not me : I know Antonio
Is sad to think upon his merchandise.
Ant, Believe me, no : I thank my fortune for it,
My ventures are not in one bottom trusted,
Nor to one place ; nor is my whole estate
Upon the fortune of this present year :
Therefore my merchandise makes me not sad.
Salar. Why, then you are in love.
Ant. Fie, fie !
Salar. Not in love neither ? Then let us say you
are sad,
Because you are not merry ; and 'twere as easy
For you to laugh, and leap, and say you are merry.
Because you are not sad. Now, by two-headed
Janus,
Nature hath fram'cl strange fellows in her time :
Some that will evermore peep through their eyes
And laugh like parrots at a bag- pi per ;
And other of such vinegar aspect
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