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DISCRETE
MATHEMATICS
AND
ITS APPLICATIONS
Series Editor
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Abstract Algebra Applications with Maple,
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An Atlas of The Smaller Maps in Orientable and Nonorientable Surfaces,
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Combinatorial Algorithms: Generation Enumeration and Search,
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Cryptography: Theory and Practice, Second Edition, Douglas R. Stinson
Design Theory, Charles C. Lindner and Christopher A. Rodgers
Frames and Resolvable Designs: Uses, Constructions, and Existence,
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Handbook of Applied Cryptography,
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Handbook of Discrete and Computational Geometry,
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Daryl D. Harms, Mirostav Kraetzl, Charles J. Co/bourn, and John S. Devitt
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ENUMERATIVE
COMBINATORICS
CHARALAMBOS A. CHARALAMBIDES
CHAPMAN & HALLICRC
A CRC Press Company
Boca Raton London NewYork Washington, D.C.

Library of Congress Cataloging-in-Publication Data
Charalambides, Ch. A.
Enumerative combinatorics I Charalambos A. Charalambides.
p. em. - (The CRC Press series on discrete mathematics and its applications)
Includes bibliographical references and index.
ISBN 1-58488-290-5
I. Combinatorial enumeration problems. I. Title. II. Series.
QAI64.8 .C48 2002
5ll'.62--dc21 2002019775
This book contains information obtained from authentic and highly regarded sources. Reprinted material
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Printed on acid-free paper

Preface
Combinatorics, at the early stages of its emergence, treated the enumera-
tion and properties of permutations, combinations and partitions of a finite
set under various conditions. Its advent coincides with that of discrete
probabilities in the 17th century. The advance of probability theory and
statistics, with an ever-increasing demand for more general configurations,
as well as the appearance and growth of computer science, have undoubt-
edly contributed to the rapid development of combinatorics during the last
decades. This subject was then expanded to cover enumeration and exam-
ination of properties as well as investigation of existence and construction
of configurations with specified properties.
This book provides a systematic coverage of the subject of enumeration
of configurations with specified properties. It is designed to serve as a
textbook for introductory or intermediate level enumerative combinatorics
courses usually given to undergraduate or first year graduate students in
mathematics, computer science, combinatorics, or mathematical statistics.
The broad field of applications of combinatorial methods renders this book
useful to anyone interested in operational research, physical and social sci-
ences.
In Chapter 1, the two basic counting principles of addition and multipli-
cation are introduced after a brief presentation of the necessary elements of
set theory. Along with these principles, the notion of a recurrence (recur-
sive) relation is introduced and its connection with a difference equation is
pointed out. Also, discrete probability is briefly introduced for use in the
subsequent chapters. In the last section, the symbols of summation and
product are also presented and some of their properties are discussed.
Chapter 2 is devoted to the enumeration and properties of permutations,
combinations, divisions and partitions of a finite set and also the enumera-
tion of integer solutions of a linear equation. Further, some basic elements
of enumeration of lattice paths are presented. This chapter concludes with
several classical applications in discrete probability theory and statistics.
Vandermonde's factorial formula, Newton's binomial formula and the

vi PREFACE
multinomial formula are presented in Chapter 3, after a suitable extension
of factorials and binomials. Stirling's approximation formula is also given.
In Chapter 4, in continuation to the counting principles of addition
and multiplication, the principle of inclusion and exclusion is extensively
treated. In addition, the Bonferroni inequalities are derived.
The famous problem of coincidences (probleme des recontres), which per-
haps constitute the first application of the inclusion and exclusion princi-
ple, is examined in Chapter 5 in the general framework of enumeration of
permutations with fixed points. The related problem of enumeration of
permutations with successions is examined in the same chapter.
Chapter 6 is devoted to a thorough presentation of generating functions,
which constitute an important means of unifying the treatment of enumer-
ative combinatorial problems.
In several enumeration problems, the number of configurations satisfying
specified conditions can only be expressed recursively. In Chapter 7 we
present the basic methods of solving linear recurrence relations.
Chapter 8 is devoted to an extensive treatment of the Stirling numbers
of the first and second kind, which are the coefficients of the expansion
of factorials into powers and of powers into factorials, respectively. In
addition, the coefficients of the expansion of generalized factorials into usual
factorials are examined.
The enumeration of distributions (of balls into urns) and occupancy (of
urns by balls) is closely related to the enumeration of permutations and
combinations. A formulation of this problem in a general framework and
its treatment from a different point of view warrants a separate chapter.
This treatment is the subject of Chapter 9.
A few elementary aspects of the combinatorial theory of partitions of
integers are discussed in Chapter 10. Specifically, after the introduction
of the basic concepts, recurrence relations and generating functions of the
numbers of partitions with summands of specified values are obtained. Also,
relations between the numbers of various partitions are concluded. The last
section of this chapter includes some interesting classical q-identities.
Chapter 11 deals with the partition polynomials in n variables. The coef-
ficient of the general term of these polynomials is the number of partitions
of a finite set of n elements in specified numbers of subsets of the same car-
dinality and the summation is extended over all partitions of the number
n. As particular cases, the partition polynomials include the exponential,
logarithmic and potential polynomials, which owe their particular names
to the form of their generating functions. The inversion of a power series
by using the potential polynomials is presented in the last section of this
chapter.
Enumeration problems emerging from the representation of a permuta-
tion as a product of cycles are treated in Chapter 12.

PREFACE vii
The problem of counting the number of equivalence classes of a finite set
under a group of its permutations is the subject of Chapter 13.
Finally, Chapter 14 considers the Eulerian and the Carlitz numbers. In
the last two sections of this chapter, these numbers are used to express the
number of permutations with a given number of ascending runs (or rises)
and the number of permutations with repetitions with a given number of
non-descending runs (or rises).
A distinctive feature of the presentation of the material covered in this
book is the comments (remarks) following most of the definitions and the-
orems. In these remarks the particular concept or result presented is dis-
cussed and extensions or generalizations of it are pointed out. In concluding
each chapter, brief bibliographic notes, mainly of historical interest, are in-
cluded.
At the end of each chapter, a rich collection of exercises is provided. Most
of these exercises, which are of varying difficulty, aim to the consolidation
of the concepts and results presented, while others complement, extend
or generalize some of the results. So, working these exercises must be
considered an integral part of this text. A few of the exercises are marked
with an asterisk, indicating that they are more challenging. Hints and
answers to the exercises are included at the end of the book. Before trying
to solve an exercise, the less experienced reader may first look up the hint
to its solution.
The material of this book has been presented several times to cla.'lses at
the department of mathematics of the University of Athens, Greece, since
1972. Its first Greek edition, containing only Chapters 1 to 7, was published
in 1984. Since then, the comments and suggestions communicated to me
by students and colleagues who used it as a textbook for an introductory
course in combinatorics contributed to improvements of certain points. The
need for a textbook for a second, more advanced course in combinatorics led
to its substantial expansion. The revised second Greek edition, expanded
by the addition of Chapters 8 to 14 was published in 1990. Thanks are
due to my colleagues Dr. M. Koutras and Dr. A. Kyriakoussis for their
comments and suggestions while I prepared this revision.
The preparation of this English edition was an opportunity for me to
clarify and improve several points. The comments of the reviewers and the
series editor were of great help and are gratefully acknowledged. Special
thanks are also due to Mrs. Rosa Garderi for the excellent typesetting of
the book.
Charalambos A. Charalambides
Athens, September 2001

The Author
Charalambos A. Charalambides, is professor of mathematics at the
University of Athens, Greece. Dr. Charalambides received a diploma in
mathematics (1969) and a Ph.D. in mathematical statistics (1972) from
the University of Athens. He was a visiting assistant professor at McGill
University, Canada (1972-73), a visiting associate professor at Temple Uni-
versity, Philadelphia (1985-86) and a visiting professor at the University
of Cyprus (1995-96). Since 1979 he has been an elected member of the
International Statistical Institute (lSI). Professor Charalambides' research
interests include enumerative combinatorics, discrete probability and para-
metric statistical estimation. He is an associate editor of Communications
in Statistics and co-edited Probability and Statistical Models with Applica-
tions, Chapman Hall/CRC Press.

Contents
Preface v
1 BASIC COUNTING PRINCIPLES 1
1.1 Introduction 0 1
1.2 Sets, relations and maps 3
1.201 Basic notions 3
1.202 Cartesian product 5
1.203 Relations 7
1.2.4 Maps 7
1.205 Countable and uncountable sets 9
1.206 Set operations 9
1.207 Divisions and partitions of a set 13
1.3 The principles of addition and multiplication 14
1.4 Discrete probability 24
1.5 Sums and products 27
1.6 Bibliographic notes 35
1.7 Exercises 36
2 PERMUTATIONS AND COMBINATIONS 39
201 Introduction 0 39
202 Permutations 40
203 Combinations 0 51
2.4 Divisions and partitions of a finite set 62
205 Integer solutions of a linear equation 68
206 Lattice paths 75
207 Probabilistic applications 82
20701 Classical problems in discrete probability 82
20702 Ordered and unordered samples 0 86
20703 Probability models in statistical mechanics 89
208 Bibliographic notes 90
209 Exercises 91
xnz

xiv CONTENTS
3 FACTORIALS, BINOMIAL AND MULTINOMIAL CO-
EFFICIENTS 103
3.1 Introduction . . . . . 103
3.2 Factorials . . . . . . 104
3.3 Binomial coefficients 110
3.4 Multinomial coefficients 123
3.5 Bibliographic notes . 124
3.6 Exercises . . . . . . . . 124
4 THE PRINCIPLE OF INCLUSION AND EXCLUSION 131
4.1 Introduction . . . . . . . . . . . . . . . . . . . 131
4.2 Number of elements in a union of sets . . . . 132
4.3 Number of elements in a given number of sets 144
4.4 Bonferroni inequalities . . . . . . . . 152
4.5 Number of elements of a given rank 155
4.6 Bibliographic notes 158
4.7 Exercises . . . . . . . . . . . . . . . 159
5 PERMUTATIONS WITH FIXED POINTS AND SUCCES-
SIONS 169
5.1 Introduction . . . . . . . . . . . 169
5.2 Permutations with fixed points 169
5.3 Ranks of permutations . . . . . 174
5.4 Permutations with successions . 176
5.5 Circular permutations with successions . 180
5.7 Exercises . . . . . . . . . . . . . . . . . 184
6 GENERATING FUNCTIONS
6.1 Introduction . . . . . . . . . .
6.2 Univariate generating functions .....
6.2.1 Definitions and basic properties .
6.2.2 Power, factorial and Lagrange series
6.3 Combinations and permutations
6.4 Moment generating functions ..
6.5 Multivariate generating functions
6.6 Bibliographic notes
6.7 Exercises . . . . . . . . . . .
7 RECURRENCE RELATIONS
7.1 Introduction .............. .
7.2 Basic notions ............. .
7.3 Recurrence relations of the first order
7.4 The method of characteristic roots ..
191
191
192
192
202
208
215
219
223
223
233
233
233
235
239

CONTENTS
7.5 The method of generating functions
7.6 Bibliographic notes .
7.7 Exercises . . . . . . .
8 STIRLING NUMBERS
8.1 Introduction . . . . . . . . . . . . . . . . . . .
8.2 Stirling numbers of the first and second kind
8.3 Explicit expressions and recurrence relations .
8.4 Generalized factorial coefficients . . . . .
8.5 Non-central Stirling and related numbers
8.7 Exercises . . . . . . . . . . . . . . . . . .
XV
250
264
264
277
277
278
289
301
314
319
321
9 DISTRIBUTIONS AND OCCUPANCY 339
9.1 Introduction . . . . . . . . . . . . . . . 339
9.2 Classical occupancy and modifications . . 340
9.3 Ordered distributions and occupancy . . . 348
9.4 Balls of general specification and distinguishable urns 350
9.5 Generating functions 353
9.7 Exercises . . . . . .
10 PARTITIONS OF INTEGERS
10.1 Introduction .......... .
10.2 Recurrence relations and generating functions
10.3 A universal generating function .....
10.4 Interrelations among partition numbers
10.5 Combinatorial identities
10.7 Exercises . . . . . . . .
11 PARTITION POLYNOMIALS
11.1 Introduction . . . . . . . . . . . . . . . .
11.2 Exponential Bell partition polynomials .
11.3 General partition polynomials ...
11.4 Logarithmic partition polynomials
11.5 Potential partition polynomials
11.6 Inversion of power series
11.7 Touchard polynomials
11.9 Exercises ...... .
359
369
369
370
376
383
391
396
396
411
411
412
419
424
428
433
442
447
448

xvi CONTENTS
12 CYCLES OF PERMUTATIONS 461
461
462
468
471
478
478
12.1 Introduction ........... .
12.2 Permutations with a given number of cycles
12.3 Even and odd permutations ........ .
12.4 Permutations with partially ordered cycles .
12.6 Exercises .................. .
13 EQUIVALENCE CLASSES
13.1 Introduction . . . . . . . . .
13.2 Cycle indicator of a permutation group .
13.3 Orbits of elements of a finite set
13.4 Models of colorings of a finite set
13.6 Exercises ............ .
487
487
488
493
499
507
507
14 RUNS OF PERMUTATIONS AND EULERIAN NUM-
BERS 513
14.1 Introduction . . . . 513
14.2 Eulerian numbers . 513
14.3 Carlitz numbers . . 522
14.4 Permutations with a given number of runs . 530
14.5 Permutations with repetition and a given number of runs 533
14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
HINTS AND ANSWERS TO EXERCISES 545
BIBLIOGRAPHY 591
INDEX 601

Chapter 1
BASIC COUNTING PRINCIPLES
1.1 INTRODUCTION
Combinatorics, at the early stages of its emergence, was basically con-
cerned with the enumeration of permutations, combinations and partitions
of a finite set under various conditions. The demand for construction and
study of more general configurations expanded its subject. Combinatorics,
nowadays established as a branch of discrete mathematics, deals with the
existence, construction, enumeration and examination of properties of con-
figurations satisfying specified conditions.
The appearance of combinatorial problems may be traced back far into
time. In a letter considered to have been addressed by Archimedes to
Eratosthenes, it is proposed, subject to certain conditions, to "compute the
number of cattle of the Sun." This problem is one of the rare allusions in
antiquity to combinatorics and its confrontation depends on consideration
of the polygonal numbers of Pythagoras, Nicomachos and Diophantos.
Investigation of the existence or nonexistence of a configuration with cer-
tain specified properties constitutes the famous problem of "magic squares."
This is the problem of placing positive integers in a square of n rows and n
columns in such a way that the sum of the numbers in any row, column or
diagonal is the same. Two simple examples of magic squares for n = 3 and
n = 5 are given in Figure 1.1. The magic squares were known to Chinese
in antiquity. The "Grant Plan," which is described in one of the oldest
divinatory books in China, constitutes one such configuration, and legend
claims it was decorated upon the back of a divine tortoise that emerged
from the river Lo. Substituting the various sets of marks by positive inte-
gers, we obtain the left magic square of Figure 1.1. The magic squares were
also known to ancient Greeks, according to a reference to them by Theona
Smyrneou. The Indians and later the Arabs worked on these squares. The
Greek monk Emmanuel Moschopoulos gave the first method of constructing
magic squares in the 14th century. Euler and other great mathematicians

2
FIGURE 1.1
Magic squares
4
3
8
g 2
5 7
1 6
BASIC COUNTING PRINCIPLES
11 24 7 20 3
4 12 25 8 16
17 5 13 21 g
10 18 1 14 22
23 6 19 2 15
also showed interest in the same problem, in a more abstract setting, in the
construction of finite geometries.
For certain configurations that can be easily obtained, such as the com-
binations of n objects taken k at a time, it is natural to ask for their mul-
titude. In this way a new development in the evolution of combinatorics
began with deriving formulas for the number of configurations satisfying
specified properties. Of course, combinatorics has been greatly developed
in this direction as a result of the powerful influence of probability (with
the classical definition of Laplace) and statistics.
For a long time combinatorics was considered as the art of counting.
From this point of view, elements of combinatorics may be traced almost
everywhere. The majority of the classical formulas has been discovered and
rediscovered several times. Perhaps the oldest example is the binomial coef-
ficients which, in the 12th century, were known to the students of the Indian
mathematician Bhakra; according to a recent discovery, the recursive com-
putation of these coefficients was taught in 1265 by the Persian philosopher
Nasir-Ad-Din. Pascal and Fermat rediscovered the binomial coefficients as
a by-product of their study of games of chance. It is also known that Car-
dan, in 1560, proved that the number of subsets of a set of n elements is 2n.
In 1666, Leibniz published the first treatise in combinatorics "Dissertatio
de Arte Combinatoria." As the configurations became more complex, great
effort was put in counting techniques. In this direction, the most celebrated
discovery was Laplace's generating function technique.
In the 20th century several new applications of combinatorics appeared in
statistics (design of experiments), in coding theory (the problem of capacity
of a set of signals), in operational research (the traveling salesman problem),
etc. Polya proved his famous theorem on counting.
When the configurations under consideration become too complex and
the derivation of their exact number very difficult, the effort is focused on
asymptotic values, bounds, inequalities, etc. A particularly curious case
in this direction are the Ramsey numbers, which are very close to the
binomial coefficients; it is not even known how to calculate them when
their parameter values are greater than 7.

1.2. SETS, RELATIONS AND MAPS 3
In certain quite difficult problems the only method of dealing with them
remains that of listing all possible configurations. This method of reasoning
by exhaustion has been used in proving important results and in other
branches of mathematics.
This book concentrates on counting configurations satisfying specified
conditions. In this chapter, some necessary basic elements of set theory are
presented, in the next section, to pave the way for the presentation of the
basic counting principles of addition and multiplication. Combinatorics is
tightly connected with discrete probability theory; this connection greatly
influenced its rapid development. It is thus reasonable that a brief intro-
duction of the notion of probability and its connection to counting config-
urations follows the presentation of the basic counting principles. In order
to make the book self-contained, the symbols of summation and product
are presented and their basic properties are discussed.
1.2 SETS, RELATIONS AND MAPS
1.2.1 Basic notions
The concept of a set is a primitive notion in set theory like the concepts
of a point and a straight line in Euclidean geometry. For the presentation
of the basic concepts of combinatorics, only a few elements of set theory
are needed and not its axiomatic foundation. In this respect, it is sufficient
to consider a set as a (well-defined) collection of distinct objects. Sets are
usually designated by capital letters of the alphabet with or without sub-
scripts and their elements by lower case letters. The fact that the element
a belongs to (or is a member of> the set A is expressed by writing a E A.
The negation of this statement is expressed by writing a ~ A. In describing
which objects are contained in a set A we use the notation
which requires that the list of elements of the set A is known, or the notation
A= {a: a has property P},
where P is a characteristic property of the elements of the set A.
Two special sets are often of interest: the universal set, designated by
fl, which is the set of all objects under consideration, and the empty or
null set, designated by 0, which does not contain any of the objects under
consideration. It must be noted that these two sets may vary from case to
case. For example, in studying the real roots of polynomials, 'the set of real

4 BASIC COUNTING PRINCIPLES
numbers R is taken as the universal set. In this framework the set
{ x E R : x2
- 2x + 2 = 0}
is an empty set as it is known that the quadratic equation x2
- 2x + 2 = 0
has only complex roots and, more specifically, the roots x1 = 1 - i and
x2 = 1 + i, where i = A is the imaginary unit. Further, if the study is
extended to the complex roots of polynomials, the set of complex numbers
Cis taken as the universal set. In this case the set of roots of the polynomial
x2
- 2x + 2,
{x E C: x2
- 2x + 2 = 0} = {1- i, 1 + i},
is not empty.
A set B is called a subset of a set A if and only if for every b E B, we
have bE A (every element of B is also an element of A). This is indicated
by writing B ~ A (and is read as B is a subset of A or B is included in
A) or equivalently by writing A 2 B (and is read as A is a superset of A
or A includes B). If B ~ A and there exists a E A such that a fJ. B, then
B is said to be a proper subset of A and this is indicated by B C A or
equivalently by A ::::> B.
The fact that B ~ A does not exclude A ~ B; when both relations hold,
the sets A and B, consisting of the same elements, are called equal and this
is indicated by A= B.
The set of all subsets of a set A is called the power set of A, and is
denoted by P(A). In the sequel, all sets under consideration are considered
as subsets of a universal set fl, that is, all are elements of P(fl).
Schematic figures are frequently used for illustrating pictorially various
sets. The Venn diagrams are such figures in which the universal set fl is
defined by a closed area of the plane containing its elements; the elements
of fl are defined by geometrical points of this plane. The subsets of fl are
defined by subareas. Figure 1.2 represents the fact that B C A.
FIGURE 1.2
Subset of a set
.(J)

Note that these schematic figures are useful in verifying the validity of
theorems of the theory of sets and also in indicating their proofs. Naturally,
the proofs must exclusively be based only on the definitions of the notions.
1.2.2 Cartesian product
The concept of an ordered pair and, generally, of an ordered n-tuple is
needed for the definition of the Cartesian product of two and generally of
n sets. According to the definition of the equality of sets, the pair (two-
element set) {a, b} is equal to the pair {b, a}. Further, there are cases where
it matters which element is first and which is second; for example in analytic
geometry the pair of coordinates (a, b) of a point on the plane designates
its abscissa and ordinate, respectively. The necessity of this distinction of
the elements of a pair leads to the introduction of the concept of an ordered
pair.
A pair of elements a and b (not necessarily different) in which a is con-
sidered as the first and b as the second element is called an ordered pair
and is denoted by (a, b). According to this definition, two pairs (a, b) and
(c, d) are equal if and only if a= c and b = d.
The concept of an ordered n-tuple (a1 , a2 , ... , an) can be inductively
defined as follows. Thus, for n = 3, an ordered triple (a1 , a2 , a3 ) is defined
as
(a1,az,a3) = ((a1,az),a3),
an ordered pair with first element the ordered pair (a1, a2) and second the
element a3 . Generally, an ordered n-tuple (a1 , az, ... , an) is defined as
an ordered pair with first element the ordered (n-1)-tuple (a1 , az, ... ,an-d
and second the element an. In several cases it is convenient to adopt the
vector terminology where the first element a1 is called first coordinate, the
second element az is called second coordinate, etc.
After the introduction of the concept of an ordered pair and, generally,
of an ordered n-tuple, the Cartesian product can by defined as follows:
The Cartesian product of the sets A and B, denoted by A x B, is defined
as the set of ordered pairs in which the first coordinate is an element of the
set A and the second coordinate is an element of the set B, that is
Ax B ={(a, b): a E A, bE B}.
This definition can be extended to n sets A1 , A2 , •.. , An as follows:
A1 X Az X··· X An= {(al,az, ... ,an): a1 E A1,a2 E Az, ... ,an E An}·
Particularly, if A1 = Az = ··· = An = A, the Cartesian product is denoted
by An.

The Cartesian product A x B is geometrically represented by the points
(a, b) of the plane, with abscissa x = a taking values from the set A (x-
axis) and ordinate y = b taking values from the set B (y- axis). Generally,
the Cartesian product A 1 x A2 x · · · x An is represented by the points
(a1 , a2, ... , an) of the n-dimensional space with the k-coordinate taking
values from the set Ak (xk -axis), k =1, 2, ... , n.
Example 1.1 Routes
Suppose that there are three different roads a1 , a2 and a3 from town 0 to town
A, and four different roads b1, b2, b3 and b4 from town A to town B. Find the
different routes from town 0 to town B.
A route from town 0 to town B may be represented by an ordered pair (a, b),
where a is a road from the set A ={a1 , a2,a3} of roads from town 0 to town A,
and b is a road from the set B ={b1 , ~, b3 , b4 } of roads from town A to town B.
l'_hus, the set of routes from town 0 to town B is the Cartesian product:
Ax B = {(ai,bi),(ai,~),(ai,~),(ai,b4),(a2,bi),(a2,b2),
(a2, b3), (a2, b4), (a3, bi), (a3, b2), (a3, b3), (a3, b4) }.
This is geometrically represented in Figure 1.3. 0
FIGURE 1.3
Set of routes
b
b4 • • •
b3 • • •
b2 • • •
bl • • •
a

1.2.3 Relations
A binary relation from the set A to the set B is a subset R of the Cartesian
product A X B. The ordered pair (a, b) satisfies the relation n if and only
if (a, b) E R. This is usually denoted by aRb. If B = A, then R is called
a relation on A. Such relations are for example the equality relation and
the inclusion relation, which have already been introduced in Section 1.2.1.
These relations are defined on the set P(D) of the subsets of n. Note that
the equality relation on a set A is the diagonal DA of the Cartesian product
A2.
The above definition is very general. The most interesting relations are
those satisfying certain desirable properties. Such properties are the fol-
lowing:
A binary relation R on a set A is called:
(a) Reflexive, if and only if for every a E A, it holds aRa
(b) Symmetric, if and only if aRb implies bRa
(c) Antisymmetric, if and only if aRb and bRa imply a= b
(d) Transitive, if and only if aRb and bRc imply aRc
The inverse (or reciprocal) of a binary relation Ron A denoted by n-1
is defined as follows: an- 1
b, if and only if bRa.
An equivalence relation is a binary relation that is reflexive, symmetric
and transitive. Such relation is, for example, the equality relation. An order
relation is a binary relation that is reflexive, antisymmetric and transitive.
Such relation is, for example, the inclusion relation ~. The inverse of this
relation is the relation 2.
1.2.4 Maps
A subset F of the Cartesian product A x B is called a map (or function
or correspondence) of the set A into the set B if and only if, for every
a E A, there exists only one b such that (a, b) E F. Thus, if (a, b1 ) E F
and (a, b2) E F, then b1 = ~- In the ordered pair (a, b) E F, the element
a E A is called archetype and the element b E B is called the image of a by
F and is usually denoted by b = F(a). Consequently, a map F of the set
A into the set B associates to each element a E A only one element b E B,
the image of a by F. If, in addition, for every element bE B, there exists
at least one element a E A such that (a, b) E F, then F is called surjective
(onto) . A map F of the set A into the set B is called injective (one-to-one)
if and only if there exist at most one element a E A such that (a, b) E F.
Thus, if (all b) E F and (a2, b) E F, then a1 = a2. Finally, a map F is
called bijective (one-to-one and onto) ifF is both surjective and injective.

The diagonal TA of the Cartesian product A2 ,
TA = { (a, b) E A2
: a = b},
is called identity map of the set A. Thus, TA associated to each element
a E A the same element TA(a) =a.
If F ~ A x B is a bijective map of the set A into the set B, then its
inverse map, denoted by F-1
~ B x A, is defined as follows: (b, a) E F-1
if and only if (a, b) E For, equivalently, a= p-l (b) if and only if b = F(a)
for every a E A and b E B.
Consider the set N = {1, 2, ... , n, ... } of natural numbers. A map of
the set N into a set A,
{(n,an): n E N,an E A},
which corresponds an element an E A to each natural number n E N, is
particularly called a sequence of elements of A. This sequence is usually
denoted as an E A, n = 1, 2, .... The element an is called the n-th term of
the sequence.
Consider, in general, an index set I. A map of the set I into a set A,
{(i,ai): i E I,ai E A},
which corresponds an element ai E A to each i E I, is called a family of
elements of A. This family is usually denoted as ai E A, i E I. The term
family is used instead of the term map when the interest is focused on the
elements ai E A and not on the map itself. Note that a sequence is a
particular case, I =N, of a family.
Example 1.2 Maps
Consider the sets A= {0, 1, ... ,9}, B = {0, 1, 2} and their Cartesian product
Ax B = {(a,b): a E {0, 1, ... ,9}, bE {0,1,2}}.
(a) The subset
{(0,0),(2,0),(4,0), (6,0),(8,0),(1, 1),(3i1), (5,1),(7,1),(9,1)},
of A x B, is a map F from the set A into the set B such that
F(O) =0, F(2) =0, F(4) =0, F(6) =0, F(8) =0
and
F(1) =1, F(3) = 1, F(5) =1, F(7) = 1, F(9) =1.
Note that the image b = F(a) of an element a E A is the remainder of the division
of a by 2. Further, this map F is not surjective since there is no element a E A
such that F(a) = 2.

1.2. SETS, RELATIONS AND MAPS
(b) The subset
{(0,0), (3,0),(6,0), (9,0), (1, 1),(4, 1), (7,1), (2,2),(5,2), (8,2)},
of A x B, is a surjective map G from the set A into the set B such that
G(O) = 0, G(3) = 0, G(6) = 0, G(9) = 0, G(1) = 1,
G(4) = 1, G(7) = 1, G(2) = 2, G(5) = 2, G(8) = 2.
9
This map corresponds to each element a E A the remainder b = G(a) of the
division of a by 3. 0
1.2.5 Countable and uncountable sets
Two sets A and B are called equivalent, and this is denoted by A ,...., B, if
and only if there exists a bijective map of the set A into the set B. For ex-
ample, the set A = {2, 4, ... , 2n} is equivalent to the set B = {1, 2, ... , n}
since the map F(a) = a/2 for every a E A is bijective. More generally, the
set A= {a1,a2, ... ,an} is equivalent to the set B = {1,2, ... ,n}, with
F(ak) = k for every k = 1, 2, ... , n, is a bijective map from A into B.
A set A is called finite, with n elements, if and only if it is equivalent to
the subset {1, 2, ... , n} of natural numbers. The empty set 0 is considered
finite with 0 elements. A set that is not finite is called infinite. A set
A is called infinitely countable if and only if it is equivalent to the set
N = {1, 2, ... , n, ... } of natural numbers. Denoting by ak the element of
A that corresponds to the natural number k, fork= 1, 2, ... , the set A can
be represented as
if it is finite with n elements, or as
if it is infinitely countable. A set A is called countable if it is finite or
infinitely countable. A set that is not countable is called uncountable.
1.2.6 Set operations
The union of two sets A and B is defined as the set that includes the
elements of{} belonging to A orB (the conjunction or being not exclusive)
and is denoted by A U B, that is
A U B = {w E {} : w E A or w E B}.
This definition is extended to n sets A1 , A2 , •.. , An:
A 1 U A2 U · · · U An = {w E {} : w E Ak for at least one k E {1, 2, ... , n}}

and more generally to a family of sets {Ai, i E J}:
UAi = {wE fl: wE Ai for at least one subscript i E J}.
iE/
The intersection of two sets A and B is defined as the set that includes
the common elements of the two sets and is denoted by A n B, that is
An B ={wE fl: wE A and wEB}.
For reasons of economy the notation AB instead of A n B is frequently
used. This definition is extended ton sets A1 , A2 , •.. , An:
Ar n A2 n ···nAn= {wE fl: wEAk for all subscripts k E {1, 2, ... ,n}}
and more generally to a family of sets {Ai, i E J}:
nAi = {wE fl: w E Ai for all subscripts i E 1}.
iE/
The union and intersection are pictorially illustrated by Venn diagrams
in Figure 1.4; the shaded area represents the set under consideration.
FIGURE 1.4
Union and intersection
The complement (with respect to the universal set fl) of a set A is defined
as the set that includes the elements of fl not belonging in A and is denoted
by A' or Ac orCA, that is
A'= {wE fl: w t/. A}.
The (set theoretic) difference of a set B from a set A is defined as the
set that includes the elements of A not belonging in B and is denoted by
A- B, that is
A- B ={wE fl: wE A, w t/. B}.
Note that
A' = n - A, A - B = An B'.

FIGURE 1.5
Complement and difference
The Venn diagrams of the difference of two sets and the complement of
a set are given in Figure 1.5.
The operations of union and intersection of sets share many similarities
and differences, with the operations of summation and multiplication of
real numbers, as follows from the next theorem.
THEOREM 1.1
For any sets A. B and C the following properties ofthe union, intersection and
complementation hold:
(a) Associativity ofthe union and intersection:
(Au B) u C =Au (B u C), (An B) n C =An (B n C).
(b) Distributivity ofthe intersection with respect to the union and ofthe union
with respect to the intersection:
An (B U C)= (An B) u (An C), Au (B n C)= (Au B) n (Au C).
(c) Commutativity ofthe union and intersection:
AU B = B u A, An B = B n A.
(d) The null set 0is the neutral elementfor the union:
Au0 = 0UA= A,
while the universal set n is the neutral elementfor the intersection
Ann= nnA= A.
(e) The complementation assigns to each set A, the set A' such that
AnA'= 0, AUA' = n.

(f)
!2' = 0, 0' = !2, (A')' = A
(g)
AUA=A, Auf2=f2, AnA=A, An0=0.
PROOF It can be easily verified that these properties are almost direct con-
sequences of the definitions of the union, intersection and complementation. The
proof of (b), which requires a somewhat longer series of steps than the others,
may be carried out as follows. Consider an element w E An (B U C). Then
w belongs to both A and B U C and thus it belongs to A and to at least one of
B and C. This implies that w belongs to both A and B or to both A and C and
hence it belongs to An B or to An C. Therefore w E (An B) u (An C)
and A n (B u C) ~ (A n B) u (A n C). Similarly, it can be shown that
(AnB)U(AnC) ~ An(BUC),whenceAn(BUC) = (AnB)U(AnC).
The proof of Au (B n C) = (Au B) n (Au C) is quite similar. I
REMARK 1.1 The set P(!2) of all subsets of !2, furnished with the operations
ofunion U, intersection nand the map C: P(f2) --+ P(f2), which to every element
A E P(f2) assigns its complement CA =A' E P(f2), is made a Boolean algebra
since, according to Theorem 1.1, properties (a) to (e), constituting the definition
of such an algebra, are satisfied. I
Two important interrelations between the operations of union, intersec-
tion and complementation are given in the next theorem.
THEOREM 1.2 De Morgan's formulae
Let A and B be subsets ofa universal set f2. Then
(AUB)'=A'nB', (AnB)'=A'uB'.
PROOF Consider an element w E (Au B)'. Then w ~ AU Band hence
w ~ A and w ~ B. This implies w E A' and w E B'. Thus w E A' n B' and
(AU B)' ~ A' n B'. Similarly it can be shown that A' n B' ~ (Au B)', whence
(A U B)' = A' n B'. The proof of the second formula can be similarly carried
out. I
REMARK 1.2
... ,An of !2:
De Morgan's formulae can be extended to n subsets AI, A2,
(AI U A2 U .. · U An)' = A; n A; n .. · n A~,
(AI n A2 n ···nAn)' = A; U A~ U · · · U A~,

and to a family of subsets {Ai, i E I} of [):
The derivation ofthese formulae follows verbatim the lines of the proofofTheorem
1.2. I
The distinction of two sets according to whether they have elements
in common or not will be useful in the sequel. In this respect, the next
definition is introduced.
Two sets A and B are said to be disjoint if they do not have elements
in common, that is, if An B =0. More generally, the sets A1 , A2, ... , An
are said to be pairwise or mutually disjoint if Ai n AJ = 0 for all pairs of
subscripts {i, j} with i -1- j, from the set of indices {1, 2, ... , n}. In such a
case, the operation of the union is designated by + or L instead of U.
Some useful properties of the Cartesian product are shown in the next
theorem.
THEOREM 1.3
For any sets A. B, C and D the following relations hold:
(Au B) X c =(A X C) u (B X C),
(An B) x (C n D)= (Ax C) n (B x D).
PROOF Consider an element u E (A U B) x C. Then u = (a, c) with
a E AU B, c E C and hence a E A or a E B and c E C, which implies
(a, c) E AxCor(a,c) E BxC. Thereforeu =(a, c) E (AxC)u(BxC)and
(AUB) xC ~ (A xC)u(Bx C). It can be shown, following the inverse procedure,
that (Ax C)u(B x C) ~ (AUB) x C and thus (Au B) x C =(Ax C)u(B x C).
The proof of the second formula is quite similar. I
1.2.7 Divisions and partitions of a set
An n-division of a set W (or a division of a set W in n subsets) is an
ordered n-tuple of sets (A1, A2, ... , An) that are pairwise disjoint subsets
of W and their union is W, that is:
Ai ~ W, i = 1, 2, ... , n, Ai n AJ = 0, i, j = 1, 2, ... , n, i -1- j,
Ar + A2 + ···+ An =W.

Note that, in a division of a set, the inclusion of one or more empty sets
is not excluded. For example, the ordered sequence (AI, A2 , A3, A4 ), with
AI = {w!}, A2 = {w2,w3,w4}, A3 = {w5} and A4 = 0, is a 4-division of
the set W ={wi,w2,w3,w4,w5}·
An n-partition of a set W (or a partition of a set W in n subsets) is a set
of n sets {AI, A2, ... , An} that are pairwise disjoint and not null subsets
of W and their union is W, that is:
Ai ~ W, Ai = 0, i = 1, 2, ... , n, Ai n Ai = 0, i,j = 1, 2, ... , n, i-::/= j,
Note that, in a partition, as opposed to a division of a set, no empty sets
are included and the sets constituting it are not ordered.
1.3 THE PRINCIPLES OF ADDITION AND
MULTIPLICATION
As has been noted in the introduction, counting configurations consti-
tutes a major part of combinatorics. The set of configurations is in any
case finite and so the problem of counting them is a problem of counting
the elements of a finite set.
The number of elements of a finite set A is denoted by N(A) or IAI
and is called the cardinal of it. In the case of a finite universal set n, its
cardinality is taken as N(n) =N. At this point, though clear from the
relevant definitions of the preceding section, it is worth noting explicitly
the following lemma.
LEMMAI.J
IfA and B are finite and equivalent sets, then
N(A) = N(B).
Thus, the cardinal of a finite set A may be deduced by determining a
finite set B, equivalent to A, with known cardinality.
Some basic properties of cardinality are proved in the next theorem.
THEOREM 1.4
(a) IfA and B are finite and disjoint sets, then
N(A +B) = N(A) + N(B).

1.3. THE PRINCIPLES OF ADDITION AND MULTIPLICATION 15
(b) IfA is a subset ofa finite universal set fl and A' its complement, then
N(A') = N- N(A).
(c) IfA and Bare finite sets, then
N(A- B)= N(A)- N(A n B)
and particularly for B ~ A,
N(A- B)= N(A)- N(B).
PROOF (a) Since An B = 0, any element of A+ B belongs either to A only
or to B only and thus
N(A +B)= N(A) + N(B).
(b) Note that the sets A and A' are disjoint and according to part (a),
N(A +A')= N(A) + N(A').
Further, A+ A' = fl, whence
N := N(D) = N(A) + N(A')
and
N(A') = N- N(A).
(c) Since
(An B') n (An B) = (An A) n (B' n B) = An 0 = 0,
the sets A n B' = A - B and A n B are disjoint and further
(An B') +(An B)= An (B' +B)= An fl =A.
Hence
N(A) = N((A n B') +(An B))= N(A n B') + N(A n B)
and
N(A- B)= N(A n B') = N(A)- N(A n B).
In particular, forB~ A, whence An B = B, it follows that
N(A- B)= N(A)- N(B)
and the proof of the theorem is completed. I

REMARK 1.3 The set function N (·),which corresponds to every set A E P(D)
its cardinal N(A), is (a) nonnegative: N(A) ~ 0 for every set A E P(D)
and (b) finitely additive: N(A + B) = N(A) + N(B) for any disjoint sets
A, BE P(D), according to part (a) of Theorem 1.4. These properties made N(·)
a finitely additive measure (or simply measure) on P(D). The most known
measures are also the length, area and volume in geometry, the mass in physics
and the probability (measure) in the theory of probability. This last measure is
introduced in the next section for the needs of the probabilistic applications of
combinatorics. I
As regards the cardinal of the union of more than two finite and pairwise
disjoint sets, the next corollary of the first part of Theorem 1.4 is shown.
COROLLARY 1.1
If A1 , A2, . .. , An are finite and pairwise disjoint sets, then
N(A1 + A2 +···+An)= N(AI) + N(A2) + · ·· + N(An)· (1.1)
PROOF Note first that, according to part (a) of Theorem 1.4, relation (1.1)
holds for n =2. Suppose that (1.1) holds for n - 1, that is
N(A1 + A2 +···+An-d =N(AI) + N(A2) + · · · + N(An-d·
It will be shown that (l.l) holds also for n. For this reason, set A= A1 + A2 +
···+An-I and B =An, whence
AnB =(A1+A2+· · ·+An_i)nAn =A1nAn+A2nAn+· · ·+An-lnAn =0,
A+ B = (A1 + A2 +···+An-d+ An= A1 + A2 +···+An-I+ An·
Thus, the sets A and Bare finite and disjoint and according to part (a) of Theorem
1.4, N(A +B) = N (A) + N(B), and the hypothesis that (1.1) holds for n- 1,
it follows that
N(A1 + A2 +···+An) = N(A1 + A2 +···+An-d+ N(An)
=N(A1) + N(A2) + · · · + N(An-d + N(An)·
Hence, according to the principle of mathematical induction, (1.1) holds for every
integer n ~ 2. I
REMARK 1.4 Relation (1.1) is often referred to as the addition principle and
can also be stated as follows: ifan element (object) wi can be selected in ki different
ways, for i = 1, 2, ... , n, and the selection of wi excludes the simultaneous
selection of w1, i, j = 1, 2, ... , n, i I j, then any of the elements (objects) w1 or
w2 or · · · or Wn can be selected in k1+ k2 + ·· · + kn ways. I

REMARK 1.5 For each division (A1 , A2 , ... , An), as well as for each parti-
tion {A1 , A2 , ... , An}. of a finite set W, the sets A1 , A2 , ... , An are finite and
pairwise disjoint. Thus, by Corollary 1.1, the following relation holds:
N(W) = N(A1 + A2 +···+An)= N(A1) + N(A2) + ···+ N(An),
for both a division and a partition of a finite set. I
The next theorem is concerned with the cardinality of the Cartesian
product of finite sets.
THEOREM 1.5
IfA and B are finite sets, then
N(A x B)= N(A)N(B). (1.2)
PROOF Let A= {a1, a2 , ... , ak} and B ={b1, b2 , •.. , br}. Then the set A
may be written in the form
A= A1 + A2 + ···+ Ak, A;= {a;}, i =1, 2, ... , k
and the Cartesian product A x B, according to Theorem 1.3, in the form
where the sets (Cartesian products) A1 x B, A2 x B, ... , Ak x Bare pairwise
disjoint. Hence
N(A X B)= N(Al X B)+ N(A2 X B)+ ... + N(Ak X B).
Noting that, for any i E {1, 2, ... , k}, the Cartesian product A; x B is the set ofthe
ordered pairs (a;, bj) with first element the only element ai of the set A; = {a;}
and second element any of the elements bj. j = 1, 2, ... , r, of the set B, it follows
that
N(A; X B)= N(B), i = 1,2, ... ,k.
Thus
N(A x B)= kN(B) = N(A)N(B)
and the proof of the theorem is completed. I
Example 1.3 Routes revisited
Suppose that there are three different roads a 1, a2 and a3 from town 0 to town
A, four different roads b1, b2 , b3 and b4 from town A to town B, and two different
roads c1 and c2 from town 0 directly to town B. Calculate the number of different
routes from town 0 to town B.

A route from town 0 to town B through town A may be represented by an
ordered pair (a, b), where a is a road from the set A = {a1 , a2,a3 } and b is a road
from the set B = {b1 , b2 , b3 , b4 }. Thus the set of routes from town 0 to town B
through town A is the Cartesian product Ax B = {(a, b) :a E A, bE B}. Also,
the set of routes from town 0 to town B directly is C = {c1, c2 }. Therefore, by
the addition principle and expression (1.2), the number of different routes from
town 0 to town B equals
N(A)N(B) + N(C) = 14. D
A subset S2 of the Cartesian product !!2
, of a finite universal set fl with
itself, cannot always be written as a Cartesian product A x B, with A ~ fl
and B ~ fl. Nevertheless, an expression similar to (1.2) may be obtained
for the number of elements of s2 when the number of selections for the first
coordinate and, for each of these selections, the number of selections for
the second coordinate, are known. Specifically, the next corollary, which is
readily deduced from Corollary 1.1 and Theorem 1.5, is concerned with the
number of elements of s2.
COROLLARY 1.2
lfS2 = (A 1 x B1 ) + (A2 x B2) + ···+ (Ak x Bk), where A 1 ,A2, . .. ,Ak are
finite and pairwise disjoint sets and B1 , B2 , ... , Bk are finite sets, then
In particular; if Ai = {ai} and Bi = {bi,l, bi,2, ... , bi,r}, i = 1, 2, ... , k,
whence s2 = {(ai,bi,j). i = 1,2, ... ,k, j = 1,2, ... ,r}, and introducing
A= {a1,a2, ... ,ak}, then
N(S2) = N(A)N(Bi) = kr. (1.4)
The cardinality of the Cartesian product of more than two finite sets is
inductively deduced from Theorem 1.5.
COROLLARY 1.3
IfA,, A2, ... , An are finite sets, then
N(A 1 X A2 X · · · X An)= N(AI)N(A2) · · · N(An)· (1.5)
PROOF Note first that, according to Theorem 1.5, relation (1.5) holds for
n = 2. Suppose that (1.5) holds for n - 1, that is,

It will be shown that (1.5) holds also for n. For this reason put A= At x A2 x
· · · x An-t and B =An, whence
A X B =(At X A2 X··· X An-t) X An= At X A2 X··· X An-t X An.
Thus, according to Theorem 1.5, N(A x B) = N(A)N(B), and the hypothesis
that (1.5) holds for n - 1, it follows that
N(A1 X A2 X··· X An)= N(At X A2 X··· X An-t)N(An)
= N(At)N(A2) · · · N(An_t)N(An)
and according to the principle of mathematical induction, (1.5) holds for every
integer n 2: 2. I
Example 1.4 Binary number system
In the binary number system each number is represented by a binary sequence
of Os and Is. For example, the numbers 5 and II, which are expressed in terms of
powers of 2 as 5 = 1· 22
+0 · 21
+1· 2° and 11 = 1· 23
+0 ·22
+1· 21
+1· 2°, are
represented by the binary sequences (1,0, 1) and (1,0, 1, 1), respectively. Note
that, with the exception of the number 0, which is represented by the one digit
sequence (0), all the other binary sequences start with digit 1. Calculate the
number of four-digit binary sequences.
The first digit of a four-digit binary sequence is necessarily l. Further, a four-
digit binary sequence (1, at, a2 , a3 ) uniquely corresponds to an ordered triple
(a1 ,a2,a3), with ai E Ai = {0, 1}, i = 1, 2,3. Thus, the set B 4 of four-digit
sequences is equivalent to the set At x A2 x A3, of ordered triples (at, a2, a3),
with ai E Ai = {0, 1}, i = 1, 2, 3 and by Lemma 1.1 and Corollary 1.3,
N(B4) = N(At X A2 X A3) = N(At)N(A2)N(A3) = 2
3.
Clearly, the 8 four-digit binary sequences are the following
(1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 0, 1, 1),
(1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0), (1, 1, 1, 1)
and represent the integers 8, 9, 10, 12, 13, 14 and 15. 0
Example 1.5 Factors of a positive integer
Evaluate the number of positive integers that are factors of the number 300.
Note first that the number 300 is expressed as a product ofprime numbers as
Each factor of this number is of the form

wherei1 E A1 = {0, 1, 2}, i2 E A2 = {0, 1} and i3 E A3 = {0, 1, 2}. Therefore,
by (1.5), the number of different factors of the number 300 equals
The set of these 18 factors is
{1,2,3,4,5,6,1o,12,15,2o,25,3o,5o,6o,75,1oo,15o,3oo}. D
An extension of expression (1.4) to a subset Sn of the n-fold Cartesian
product nn is given in the following corollary.
COROLLARY 1.4
Let Sn be a subset of elements (w1,w2, ... ,wn) of then-fold Cartesian prod-
uct fl'l, of a finite universal set Sl with itself Specifically, assume that the first
coordinate w1 can be selected from a set A = {a1, a2, ... , ak,} of k1 elements
and, for each selection w1 = ai1, the second coordinate w2 can be selected from
a set Ai1 = {ai,,1, ai,,2, ... ,ai1,k2 } of k2 elements, i1 = 1, 2, ... , k1, and so
on. Finally, assume that for each selection w1 = ai1 , w2 = ai 1,i2 , ... , Wn-1 =
a;1,i2,... ,in-1' the last coordinate Wn can be selected from a set Ai,,i2,... ,in-I =
{ ai, ,i2,··· ,in-1 ,1' a;, ,i2,··· ,in-! ,2, ... , ai, ,i2, .. ,in-1 ,kn} of kn elements, ir = 1, 2,
... , kr. r = 1, 2, ... , n - 1. Then
REMARK 1.6 Relation (1.6) is often referred to as the multiplication principle
and can be restated as follows: if an element (object) w1 can be selected in k1 ways
and for each of these ways an element w2 can be selected in k2 ways and so on
and for each of these ways an element Wn can be selected in kn ways, then all
the elements (objects) w1 and w2 and··· and Wn can be selected (sequentially) in
k1 k2 · · · kn ways.
In many applications the set of different selections of wi cannot be identified in
advance, but only after the selections ofw1, w2, ... , wi_1. This does not cause any
difficulty in the application of the multiplication principle since the only require-
ment is the cardinality of the set of selections of wi. This point is further clarified
in the next example. I
Example 1.6 Booking tickets
Suppose that there are n stations on a railway line. How many different kinds
of tickets have to be provided so that booking is possible from any station to any
other station?
Let us represent the route from station si to station si by the pair (si, si), i =/= j.
The set of different kinds of tickets to be provided is equal to the set 5 2 of ordered

pairs (a, b) of different stations. Note that the set ofdifferent selections of a is A =
{s1, s2, ... , sn}, while the set of different selections of bis not known in advance,
since bhas to be different from a. After the selection a =si from the set A, the set
of different selections of bis Bi = A - {si} = { s 1, s2, ... , si-I, si+1, ... , sn}.
Irrespective of the possibility of identifying the set Bi, its cardinality is in any case
equal to N(Bi) =n- 1 and hence formula (1.4) can be applied. Therefore
N(Sz) = N(A)N(Bi) = n(n- 1),
which is the required number of different kinds of tickets. 0
Example 1.7 The number of subsets of a finite set
Consider a finite set Wn ={WI, W2, ... ,Wn} and let An =P(Wn) be the set of
subsets of Wn. The number sn = N(An) may be determined as follows.
Note first that, to any subset U of Wn there corresponds an ordered n-tuple
(ai,a2 , •.. ,an) such that aj = 0 if WJ ¢ U, and aj = 1 if Wj E U for
j = 1, 2, ... , n. In this way, the null set 0 ~ Wn corresponds to the ordered
n-tuple (0, 0, ... , 0), with all components zero; the one-element set {WI} ~ Wn
corresponds to the ordered n-tuple (1, 0, ... , 0), with the first component equal to
one and the rest components zero; and the set Wn ~ Wn corresponds to the ordered
n-tuple (1, 1, ... , 1), with all components equal to one. This correspondence is
one to one and, according to Lemma 1.1, the number sn of subsets ofWn is equal to
the number ofordered n-tuples (aI, a2 , ... , an), with aj E Aj ={0, 1}, which, in
tum, is equal to the number of elements ofthe Cartesian product AI x A2 x · · · x An,
with A.J ={0, 1}, j =1, 2, ... , n. Then, according to formula (1.5), this number
is equal to Sn = 2n. 0
Example 1.8 Sum of terms of a geometric progression
The sum of the first n terms of a geometric progression with the first term equal
to 1 and proportion equal to 2 may be derived combinatorially as follows:
Let En be the set of nonempty subsets of a finite set Wn = {w1 , w2 , •.. , Wn}
and Ck the Set of SUbsets of the finite set Wk = {WJ, W2, ... ,Wk }, each of which
contains Wk, k = 1,2, ... ,n. Then cl,C2,··· ,Cn are pairwise disjoint and
C1 +C2 +···+Cn = En. Note that, to each subset of Wk containing the element
wk, corresponds a subset of Wk-I (without any restriction), from which it is
obtained by adding wk, k = 2, 3, ... , n. This correspondence is one to one and
hence the number N(Ck) is equal to the number of subsets of Wk-l, which, in
tum, according to Example 1.7, is equal to N(Ck) = 2k-l, k = 2, 3, ... , n.
Especially, the number of subsets of W1 = {WI} containing the element w1 is
equal to N(CI) = 1. Further, the number of nonempty subsets of Wn is equal to
N(En) =2n - 1. Therefore, on using (1.1 ), it follows that
1 + 2 + 22
+ ···+ 2n-I = 2n - 1,

which is the required expression. 0
Before concluding this section on the basic counting principles, it is worth
noting that, in several enumeration problems, the number of configurations
satisfying specified conditions can only be expressed recursively. In addi-
tion, even when the direct expression of this number in a closed form is
possible, a recurrence relation is useful at least for tabulation purposes.
The notion of a recurrence relation, which is extensively used in the se-
quel, is briefly introduced here. (The basic methods of solving recurrence
relations are presented in Chapter 7.)
Consider a sequence of numbers an, n = 0, 1, .... In combinatorics an
may represent the number of subsets of a finite set Wn = {WI, w2, ... ,wn}
that satisfy a set of specified conditions. In general, we may assume that
an+r = F(n, an, an+l, ... , an+r-1), n = 0, 1, ....
This equation, in which the term an+r is expressed as a function of the pre-
ceding r terms, an, an+I, ... , an+r-l, of the sequence, is called recurrence
relation of order r. The notion of a recurrence relation is also introduced
in the case of a double-index sequence an,k, n = 0, 1, ... , k = 0, 1, .... So,
the equation
is called recurrence relation of order (r, s). Notice that r is the order of
this recurrence relation with respect to the first index (variable) and s is
its order with respect to the second index (variable). In this case the term
an+r,k+s is expressed as a function of the (n + 1)(k + 1)- 1 =nk + n + k
terms an,k, an,k+l, an+l,k, ... , an+r-l ,k+s> an+r,k+s-l of the double-index
sequence. From the computation point of view, the tabulation of the num-
ber an, for n = 0, 1, ... , or the number an,k, for n = 0, 1, ... , k =0, 1, ... ,
can be done more easily step-by-step by using the corresponding recurrence
relation. For this purpose, the knowledge of the r initial conditions (val-
ues) a0 , a1 , ..• , ar-l, in the first case, and of the r + s initial conditions
(sequences) aok,alk,··· ,ar-lk and ano,anl,··· ,ans-I, in the second
1 I I 1 I I
case, are required. The initial conditions guarantee the uniqueness of the
solution of the recurrence relation. In this section only a very simple ex-
ample of a recurrence relation is discussed.
Closely connected with a recurrence relation is the (finite) difference equa-
tion, which is defined as follows. The first order finite difference, with
increment h, of a function y = f(x) denoted by .tJ.hf(x), is defined by
.tJ.hf(x) = f(x +h)- f(x). Recursively, the r-th order finite difference of
f(x) is defined by .:1/J(x) = .tJ.h[.tJ.~-I f(x)], r = 2, 3, .... Introducing the
displacement (shift) of f(x), which is defined by Ehf(x) = f(x +h) and
recursively by Ehf(x) =Eh[E~-l /(x)] = f(x +rh), r = 2, 3, ... , it follows

that !lhf(x) = Ehf(x)- If(x), where If(x) = f(x). An equation of the
form
Ef.f(x) = F(x, f(x), Ehf(x), ... , E~-I f(x)), x E R
or equivalently of the form
ll'f.f(x) = G(x, f(x), !lhf(x), ... , .:1~-I f(x)) = 0, x E R
is called difference equation of order r. In the particular case where the
function y = f(x) is defined only on a countable set of points {x0 , XI, •.. ,
Xn, ... } which, in the applications of the calculus of finite differences, are
usually equidistant, Xn = x0 + nh, n = 0, 1, ... , the transformation Zn =
(xn- x0 )/h, n = 0, 1, ... is used. So, the function f is transformed to the
function g, with g(n) = f(x0 + nh), which is defined on the set {0, 1, ... }
of nonnegative integers. In this particular case, using the sequence
Yn=g(n)=f(xo+nh), n=0,1, ... ,
and since E~(xn) = f(xn + jh) = g(n + j) = Yn+j, j = 1, 2, ... , the
difference equation, with x = Xn, may be written as
Yn+r = F(n,yn,Yn+I•··· ,Yn+r-I), n =0,1, ... ,
which is a recurrence relation of order r. Note that the method of solving
a finite difference equation is the same as that of solving the recurrence
relation even when the function y = f(x) is defined for every x E R. In the
last case an investigation of the solution is required.
Example 1.9 A recurrence relation for the number of subsets of a
finite set
As in Example 1.7, consider a finite set Wn = {w1 , w2 , . . . , Wn} and let An =
P(Wn) be the set of subsets of Wn. The number sn = N(An) may be deduced
recursively as follows.
The set An+I = P(Wn+I) of subsets of Wn+l = {w1, w2, ... ,Wn, Wn+l}
can be divided into the following two disjoint subsets: the set A of subsets of
Wn+1, each of which does not include the element Wn+ I and the set B of subsets
of Wn+I, each of which includes the element Wn+l· Hence An+I =A+ Band,
by the addition principle, N(An+d = N(A) + N(B). Clearly, A = An and
so N(A) = N(An). Further, to each subset B that belongs to B there uniquely
corresponds the subset A = B - {Wn+l} that belongs to An and so, by Lemma
l.I, N(B) = N(An)· Consequently N(An+I) = 2N(An) and
Bn+I = 2sn, n = 0, 1, ... ,
with so = 1, which is the number of subsets of the null set. This is a homogenous
recurrence relation of the first order with constant coefficients. Iterating (applying

repeatedly) it, we find
and, since so = 1, we conclude that Sn = 2n. 0
1.4 DISCRETE PROBABILITY
The theory of probability is concerned with the study of mathematical
models, known as stochastic models, which are used in explaining random
or stochastic phenomena or experiments. The basic characteristic of these
experiments is that the conditions under which they are performed and
the values of various quantities appearing in them do not predetermine the
outcome, but do predetermine the set of possible outcomes. The element of
randomness lies in the inability of predetermining the outcome of random
phenomena or experiments.
The set n of the possible outcomes of a random phenomenon or exper-
iment is called sample space and the elements w of n are called sample
points. It should be noted that it is possible to define more than one set
of possible outcomes for each random phenomenon or experiment and, ac-
cording to the requirements of the specific problem, the more appropriate
of these is chosen as the sample space. The inappropriate choice of the
sample space leads to many paradoxes. The sample space may be finite or
countably infinite or uncountable. For the cases of finite or more generally
countable sample spaces, on which this presentation concentrates, every
subset A of n is called an event. An event A = {w} containing only one
element of n is called an elementary or simple event.
The classical definition of probability was first expressed by De Moivre
(1711) as follows: the probability of the occurrence of an event is the frac-
tion whose numerator is the number of possibilities favorable for the oc-
currence of the event and the denominator is the number of all possibil-
ities, provided that all possibilities are equally probable. This condition
is essential because, otherwise, by considering the two possibilities of the
occurrence and the non-occurrence of an event, it can be concluded that
its probability is one half. This is not true in general since these two cases
are not always equally probable.
The concept of equally probable cases should be defined independently
of the notion of (the measure) of probability; otherwise, in the classical
definition of probability, there would be a vicious circle. This is achieved
by the adoption of the principle of the want of sufficient reason. So if,
according to all available data, no reason is known for regarding any of

1.4. DISCRETE PROBABILITY 25
the possibilities more or less probable than any other, then all possibilities
are regarded as equally probable. It should be noted that the classical
definition of probability applies only on finite sample spaces.
The foundation of the theory of probability based on the classical defini-
tion of probability is attributed to Laplace (1812). Consider a finite sample
space fl = {w1 ,w2 .•. ,wN }, whose elements (sample points, possibilities)
are, according to the principle of the want of sufficient reason, equally prob-
able and an event A E P(fl). The probability of A, denoted by P(A), is
given by the expression
P(A) = N~A)'
where N(A) is the number of elements of A and N =N(fl) is the number
of elements of the sample space fl. Note that the condition of equally
probable sample points (possibilities) is then expressed by
1
P({wi}) =P({w2 }) = · · · = P({wN}) = N"
According to the definition of the classical (uniform) probability, the cal-
culation of the probability P(A) of an event A in a finite sample space fl
whose elements are equally probable is a purely combinatorial problem of
counting the numbers N(A) and N of certain configurations.
REMARK 1.7 It is worth presenting the most important properties of the
classical probability that subsequently inspired the suitable choice of axioms in
the axiomatic foundation of the theory of probability.
The set function P(·), which, in the case of a finite sample space fl whose
elements are equally probable, assigns the number P(A) = N(A)/N to each
event A E P(fl) is (a) nonnegative: P(A) 2: 0 for every event A E P(fl), (b)
normalized: P(fl) = 1 and (c) finitely additive: P(A +B) = P(A) + P(B)
for any disjoint (mutually exclusive) events A, BE P(fl).
Properties (a) and (c), which follow directly from the definition of classical
probability and the corresponding properties of the set function N (·), made P(·) a
finitely additive measure on P(fl) (see Remark 1.3). Property (b), which is a direct
consequence of the definition of classical probability, distinguishes the probability
from other measures.
Note that, from (c) and the principle of mathematical induction, the next expres-
sion is deduced:
P(A1 + A2 +···+An) = P(AI) + P(A2) +. ·. + P(An),
for any pairwise disjoint events A1 ,A2 , ... ,An E P(fl). Also, if Wi, i
1, 2, ... , n are finite sample spaces, each with equally probable elements, and
Ai E P(Wi). i = 1, 2, ... , n, then from (1.5) it follows that
P(A1 X A2 X · · · X An) = P(AI)P(A2 ) · · · P(An),

provided the elements of the sample space fl = WI X w2 X ••• X Wn are also
equally probable. I
An extension of the classical definition of probability, in the case of a
finite sample space fl whose elements are not necessarily equally probable,
constitutes the next definition of probability:
A set function P(·) defined on the set of events P(fl), assuming real
values and satisfying properties (a), (b) and (c) is called probability (or
measure of probability). These properties are quite general and, for the
calculation of the probability of any event A E P(fl), the knowledge of the
probabilities Pi = P( {wi}) of the elementary events {wi}, i = 1, 2, ... , N
is required. Indeed, if A= {willWi2 , • •• ,w;k}, then, with Air = {w;r},
r = 1, 2, ... , k, it follows that
A= A;, + Ai2 + ···+ Aik, Air n Ai, = 0, r -::f. s,
and so
k
P(A) = LP({wir}).
r=O
In the special case in which P({w!}) = P({w2 }) =··· = P({wN}) = 1/N,
it reduces to
k
P(A) = N'
with k = N(A), which is the classical definition of probability.
Three useful properties of the probability that follow directly from the
classical definition of probability and Theorem 1.4 are quoted here for easy
reference in the following chapters.
The probability of the complementary event A' of A may be expressed
as
P(A') = 1 - P(A).
For any events A and B, the next relation holds
P(A- B) = P(A) - P(A n B)
and particularly for B ~ A,
P(A- B)= P(A)- P(B).
Examples with applications of the classical definition of probability are
given in the following chapters after the introduction of the combinatorial
concepts necessary for their presentation.

1.5. SUMS AND PRODUCTS 27
1.5 SUMS AND PRODUCTS
A finite sum of n terms, a1 , a2 , ... , an, is denoted by
or briefly by
n
Sn = Laj.
j=l
The number 1 is called the lower limit and the number n the upper limit
of the sum. Note that the lower limit can be any integer r < n. Clearly,
the value of the sum
n
Sr,n = Lai
j=r
is completely determined by the general term aj (j =r, r + 1, ... , n) and
the limits r and n and does not depend on the (bound) variable j even
though it occurs in its expression. Thus, the variable j can be replaced by
another variable i without any effect on the value of the sum:
n n
Sr,n = Laj =La;= ar +ar+l +···+an·
j=r i=r
More generally, the transformation i = j + m, with inverse j i - m,
where m is a given integer, may be used. In this case, the general term aj
(j = r, r + 1, ... , n) becomes b; =ai-m (i = r + m, r + m + 1, ... , n + m)
and the sum sr,n is transformed to
n n+m n+m
Sr,n = z=aj = L a;-m = 2:: b;
j=r i=r+m i=r+m
and particularly for m = -r
n n-r n-r
Sr,n = Laj = Lai+r = Lb;.
j=r i=O i=O
This brief notation of simple (with respect to one variable) sums is also
used to represent double (with respect to two variables) sums and generally
multiplesums. So,thesumofthetermsa;,j,j = 1,2, ... ,k,j = 1,2, ... ,n,
is denoted by
n k
Sk,n = LL a;,j
j=l i=l

and, more generally, the sum of the terms aJd2 , ... ,J,, jk
k = 1, 2, ... , r, is denoted by
nr n2 nt
Sn,,n2 ... ,n, = :2: ···:2: :2: aJ,,j2,···,Jr·
Jr=l }2=1 }J=l
1, 2, ... ,nk,
REMARK 1.8 If the set J of the values of the variable j and, more generally,
if the sets J1, Jz, ... , Jr of the values of the variables j 1, jz, ... ,ir, respectively,
are not sets of consecutive integers, the following notation of the sums is adopted:
s = :2:aj' s = :2: ... :2: :2: ajloi2o··· ,j,.
jEJ irEJr }2Eh j, EJ1
When the sets J and J1 , J2 , ... , Jr are defined by the conditions C and
C1 , C2 , ... , Cr. respectively, the following notation of the sums may be used:
According to the last notation, the sums
n n k
Sn = :2:aj' Sk,n = :2::2:ai,j
j=l j=l i=l
may, equivalently, be written as:
Sn = :2: aj, Sk,n = :2: :2: aj·
l~j~n l~j~n l~i~k
If the expression of the conditions C and C1 , C2 , ... , Cr is not very simple, then
it is preferable to avoid writing it underneath the summation sign. In this case
the conditions are written after the sum in the form of phrases like "where the
summation is extended over all j or j 1 , jz, . .. ,ir such that the conditions C or
cl' c2, ... 'Cr are satisfied." I
REMARK 1.9 If in the sum
the general term is constant (independent of the subscript j): ai a, j
1,2, ... ,n,then
n
sn = :2:a = a + a + ···+ a = na.
j=l

1.5. SUMS AND PRODUCTS 29
The special case a = 1 is particularly noted:
n
L1=n.
j=l
In general,
L1 =N(J)
jEJ
and
L ... L L 1 = N(J1 )N(J2 ) .•. N(Jr),
irEJr j,EJ, j,EJt
for any finite sets J, J1, )z, ... , Jr. I
Some basic properties of finite sums, which constitute simple extensions
of the corresponding properties of the sum and product of two real numbers,
are presented in the next theorem.
THEOREM 1.6
n n
LbaJ =bLaJ,
j=l j=l
n n n
Lai + Lb; = L(aJ + bJ),
j=l i=l j=l
n k n k k n
l:aJ Lb; = LLaJbi = LLaJb;,
j=l i=l j=l i=l i=l j=l
n k k n
LLai,J = LLai,J·
j=l i=l i=l j=l
REMARK 1.10 As regards the possibility of changing the order of summation
in double (or multiple) sums, special care should be given when the set of values
of one variable depends on the set of values of the other variable. A characteristic
example of this nature constitutes the sum
n j
LLai,J,
j=l i=l
where, for a given value of the variable j E J1 = {j : 1 ~ j ~ n}, the set
/1 = {i : 1 ~ i ~ j} of values of the variable i depends on the value j E J 1 . This

dependence can be reversed without altering the set K = { (i, j) : 1 ~ i ~ j ~ n},
of values of the pair (i, j) of variables in the double sum. Indeed, for a given value
of the variable i E I 2 = {i : 1 ~ i ~ n}, the set of values of the variable j is
h = {j : i ~ j ~ n}. Hence
n j n n
LLai,j = LLai,j = LLai,j·
j=l i=l (i,j)EK i=l j=i
As a second example, the relation
n n-j+l n n-i+l
'L 'L a;,j='L 'L
j=l i=l i=l j=l
can be similarly established. I
a·.
t,J
A finite product of n terms, a1 , a2 , ... , an, is denoted by
or briefly by
n
Pn =IIaj.
j=l
Clearly, all notational and other remarks previously made on finite sums
are applied, with the necessary changes, to finite products as well. The
brief notation of simple products is also used for the presentation of double
and more generally multiple products. So, the product of the terms ai,j,
i =1,2, ... ,k, j = 1,2, ... ,n, is denoted by
n k
Pk,n = IIIIa;,j
j=l i=l
and more generally, the product of the terms aj,J2 , ... ,J,, ik = 1,2, ... ,nk,
k =1, 2, ... , r, is denoted by
n1' n2 n1
Pn, ,n2,··· ,n, = II ... II II a)! ,j,, ... ,j,.
j,=l )2=1)1=1
It can be easily shown that
n k k n
IIIIai.j =II IIai,j,
J=l i=l i=l j=l

1.5. SUMS AND PRODUCTS
n j n n
IIIIai,j = IIIIai,j,
j=l i=l i=l j=i
n n-j+l n n-i+l
II II ai,j = II II ai,j.
j=l i=l i=l j=l
The union of n sets, A1 , A2 , ... , An, is briefly denoted by
n
UAi := A1 U A2 U · · · U An
i=l
and the intersection of these sets by
n
nAi := A1 n A2 n ···n An
i=l
31
and both share properties analogous to those of finite sums and products.
Example 1.10 Sum of terms of an arithmetic progression
The general term of an arithmetic progression is a1 = a+ jb, j = 1, 2, ... ,n.
The sum
n
sn(a,b) = L:aj, aj =a+jb, j =1,2, ... ,n,
j=l
may be evaluated as follows. Using the transformation i =n- j + 1, with inverse
j = n- i + 1, the general term aj = a+ jb (j = 1, 2, ... , n) takes the form
bi =an-i+l ={a+ (n + 1)b}- ib (i = 1, 2, ... , n). Therefore
n
sn(a,b) = Lbi, bi ={a+ (n + 1)b}- ib, i = 1,2, ... ,n
i=l
and
n n n
2sn(a, b)= L aj + L bi =L(aj + bJ)·
j=l i=l j=l
Since a1 + b1 = 2a + (n + 1)b, j = 1, 2, ... , n, it follows that
2sn(a,b) =n{2a+(n+1)b}
and
( b) _~( 'b)_ n{2a + (n + 1)b}
Sn a, - L a +J -
2
.
j=l

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screen. The assistants are supposed to be showing the dissolving views.

CHAPTER XXI.
LIGHT, OPTICS, AND OPTICAL INSTRUMENTS.
Fig. 246.
"The moon shines bright:—In such a night as this."—The Merchant of Venice.
"To gild refined gold, to paint the lily,
To throw a perfume on the violet,
To smooth the ice, or add another hue
Unto the rainbow, or with taper light
To seek the beauteous eye of heaven to garnish,
Is wasteful and ridiculous excess."
Perfection admits of no addition, and it is just this feeling that might
check the most eloquent speaker or brilliant writer who attempted to
offer in appropriate language, the praises due to that first great
creation of the Almighty, when the Spirit of God moved upon the face

of the waters and said, "Let there be light." If any poet might be
permitted to laud and glorify this transcendant gift, it should be the
inspired Milton; who having enjoyed the blessing of light, and
witnessed the varied and beautiful phenomena that accompany it,
could, when afflicted by blindness, speak rapturously of its creation, in
those sublime strains beginning with—
"'Let there be light,' said God, and forthwith light
Ethereal, first of things, quintessence pure,
Sprung from the deep: and from her native east
To journey through the airy gloom began,
Sphered in a radiant cloud, for yet the sun
Was not; she in a cloudy tabernacle
Sojourn'd the while. God saw the light was good,
And light from darkness by the hemisphere
Divided: light the day, and darkness night,
He named."
There cannot be a more glorious theme for the poet, than the vast
utility of light, or a more sublime spectacle, than the varied and
beautiful phenomena that accompany it. Ever since the divine
command went forth, has the sun continued to shine, and to remain,
"till time shall be no more," the great source of light to the world, to be
the means of disclosing to the eye of man all the beautiful and varied
hues of the organic and inorganic world. By the help of light we enjoy
the prismatic colours of the rainbow, the lovely and ever changing and
ever varied tints of the forest trees, the flowers, the birds, and the
insects; the different forms of the clouds, the lovely blue sky, the
refreshing green fields; or even the graceful adornment of "the fair,"
their beautiful dresses of exquisite patterns and colours. Light works
insensibly, and at all seasons, in promoting marvellous chemical
changes, and is now fairly engaged and used for man's industrial
purposes, in the pleasing art of photography; just as heat, electricity,
and magnetism, (all imponderable and invisible agents,) are employed
usefully in other ways.
The sources from whence light is derived are six in number. The first is
the sun, overwhelming us with its size, and destroying life, sometimes,

with his intense heat and light, when the piercing rays are not
obstructed by the friendly clouds and vapours, which temper and
mitigate their intensity, and prevent the too frequent recurrence of that
quick and dire enemy to man, the coup de soleil.
The body of the sun is supposed to be a habitable globe like our own,
and the heat and light are possibly thrown out from one of the
atmospheric strata surrounding it. There are probably three of these
strata, the one believed to envelope the body of the sun, and to be
directly in contact with it, is called the cloudy stratum; next to, and
above this, is the luminous stratum, and this is supposed to be the
source of heat and light; the third and last envelope is of a transparent
gaseous nature. These ideas have originated from astronomers who
have carefully watched the sun and discovered the presence of certain
black spots called Maculæ, which vary in diameter from a few hundreds
of miles to 40 or 50,000 miles and upwards. There is also a greyish
shade surrounding the black spots called the Penumbra, and likewise
other spots of a more luminous character termed Faculæ; indeed the
whole disc of the sun has a mottled appearance, and is stippled over
with minute shady dots. The cause of this is explained by supposing
that these various spots represent openings or breaks in the
atmospheric strata, through which the black body of the sun is
apparent or other portions of the three strata, just as if a black ball was
covered with red, then with yellow, and finally with blue silk: on cutting
through the blue the yellow is apparent; by snipping out pieces of the
blue and yellow, the red becomes visible; and by slicing away a portion
of the three silk coverings the black ball at last comes into view. On a
similar principle it is supposed that the variety of spots and eruptions
on the sun's face or disc may be explained. The evolution of light is
not, however, confined to the sun, and it emanates freely from
terrestrial matter by mechanical action, either by friction, or in some
cases by mere percussion. Thus the axles of railway carriages soon
become red hot by friction if the oil holes are stopped up; indeed hot
axles are very frequent in railway travelling, and when this happens, a
strong smell of burning oil is apparent, and flames come out of the axle
box. The knife-grinder offers a familiar example of the production of
light by the attrition of iron or steel against his dry grindstone.

The same result on a much grander scale is produced by the apparatus
invented by the late Jacob Perkins; the combustion of steel ensues
under the action, viz., the friction of a soft iron disc revolving with
great velocity against a file or other convenient piece of hardened
steel. (Fig. 247)
Fig. 247.
Instrument for the combustion of steel.
The stand has a disc of soft iron fixed upon an axis, which revolves on
two anti-friction wheels of brass. The disc, by means of a belt worked
over a wheel immediately below it, is made to perform 5000
revolutions per minute. If the hardest file is pressed against the edge
of the revolving disc, the velocity of the latter produces sufficient heat
by the great friction to melt that portion of the file which is brought in
contact with it, whilst some particles of the file are torn away with
violence, and being projected into the air, burn with that beautiful
effect so peculiar to steel. If the experiment is performed in a darkened
room, the periphery of the revolving disc will be observed to have
attained a luminous red heat. Thirty years ago every house was
provided with a "tinder-box" and matches to "strike a light." Since the
advent of prometheans and lucifers, the flint and steel, the tinder, and
the matches dipped in sulphur, have all disappeared, and now the box
might be deposited in any antiquarian museum under the portrait of

Guy Fawkes, and labelled, "an instrument for procuring a light,
extensively used in the early part of the nineteenth century." (Fig. 248.)
Fig. 248.
c. The steel. b. The flint. e. The tinder. d. The matches of the old-
fashioned tinder-box, a.
The rubbing of a piece of wood (hardened by fire, and cut to a point)
against another and softer kind, has been used from time immemorial
by savage nations to evoke heat and light; the wood is revolved in the
fashion of a drill with unerring dexterity by the hands of the savage,
and being surrounded with light chips, and gently aided by the breath,
the latent fire is by great and incessant labour at last procured. How
favourably the modern lucifers compare with these laborious efforts of
barbarous tribes! a child may now procure a light with a chemically
prepared metal, and great merit is due to that person who first devised
a method of mixing together phosphorus and chlorate of potash and so
adjusted these dangerous materials that they are as safe as the "old
tinder-box," and have now become one of our domestic necessaries.
Ignition, or the increase of heat in a solid body, is another source of
light, and is well illustrated in the production of illuminating power from
the combustion of tallow, oil, wax, camphine or coal gas. The term
ignition is derived from the Latin (ignis, fire), and is quite distinct, and
has a totally different meaning from that of combustion. If a glass jar is

filled with carbonic acid gas, and a little tray placed in it containing
some gun cotton, it will be found impossible to fire the latter with a
lighted taper, i.e. by combustion (comburo, to burn), because the gas
extinguishes flame which is dependent on a supply of oxygen; whereas
if a copper or other metallic wire is made red hot or ignited, the
carbonic acid has no effect upon the heat, and the red hot wire being
passed through the gas, the gun cotton is immediately fired.
Flame consists of three parts—viz., of an outer film, which comes
directly in contact with the air, and has little or no luminosity; also of a
second film, where carbon is deposited, and, first by ignition, and
finally by combustion, produces the light; and thirdly, of an interior
space containing unburnt gas, which is, as it were, waiting its turn to
reach the external air, and to be consumed in the ordinary manner.
(Fig. 249.)
Fig. 249.
A candle flame. 1. Outer flame. 2. Inner flame, which is badly supplied
with oxygen, and where the carbon is deposited and ignited. 3. The
interior, containing unburnt gas.
Chemical action and electricity have been so frequently mentioned in
this work as a source of heat and light, that it will be unnecessary to
do more than to mention them here, whilst phosphorescence (the sixth
source of light) in dead and living matter, a spontaneous production of
light, is well known and exemplified in the "glow-worm," the "fire-fly,"

the luminosity of the water of the ocean, or the decomposing remains
of certain fish, and even of human bodies. Phosphorescence is still
more curiously exemplified by holding a sheet of white paper, a
calcined oyster-shell, or even the hand, in the sun's rays, and then
retiring quickly to a darkened room, when they appear to be luminous,
and visible even after the light has ceased to fall upon them.
For the purpose of examining the temporary phosphorescence of
various bodies, M. Becquerel has invented a most ingenious
instrument, called the "phosphorescope." It consists of a cylinder of
wood one inch in diameter and seven inches long, placed in the angle
of a black box with the electric lamp inside, so that three-fourths of the
cylinder are visible outside, and the remaining fourth exposed to the
interior electric light.
By means of proper wheels the cylinder, covered with any substance
(such as Becquerel's phosphori), is made to revolve 300 times in a
second, and by using this or a lesser velocity, the various phosphori are
first exposed to a powerful light and then brought in view of the
spectator outside the box.
It is understood that light is produced by an emanation of rays from a
luminous body. If a stone is thrown from the hand, an arrow shot from
a bow, or a ball from a cannon, we perfectly understand how either of
them may be propelled a certain distance, and why they may travel
through space; but when we hear that light travels from the sun, which
is ninety-five millions of miles away from the earth, in about seven
minutes and a half, it is interesting to know what is the kind of force
that propels the light through that vast distance, and also what is
supposed to be the nature of the light itself.
There are two theories by which the nature of light, and its
propagation through space, are explained; they are named after the
celebrated men who proposed them, as also from the theoretical
mechanism of their respective modes of propulsion: thus we have the
Newtonian or corpuscular theory of light, and the Huyghenian or
undulatory theory; the first named after Sir Isaac Newton, and the
second after Huyghens, another most learned mathematician. Many

years before Newton made his grand discovery of the composition of
light in the year 1672, mathematicians were in favour of the undulatory
theory, and it numbered amongst its supporters not only Huyghens,
but Descartes, Hook, Malebranche, and other learned men. Mankind
has always been glad to follow renowned leaders, it is so much easier,
and is in most cases perhaps the better course, to resign individual
opinion when more learned men than ourselves not only adopt but
insist upon the truth of their theories; and this was the case with the
corpuscular theory, which had been written upon systematically and
supported by Empedocles, a philosopher of Agrigentum in Sicily, who
lived some 444 years before the Christian era, and is said to have been
most learned and eloquent; he maintained that light consisted of
particles projected from luminous bodies, and that vision was
performed both by the effect of these particles on the eye, and by
means of a visual influence emitted by the eye itself. In course of time,
and at least 2000 years after this theory was advanced, philosophers
had gradually rejected the corpuscular theory, until the great Newton,
about the middle of the seventeenth century, advanced as a champion
to the rescue, and stamping the hypothesis with his approval, at once
led away the whole army of philosophers in its favour, so that till about
the beginning of the nineteenth century the whole of the phenomena
of light were explained upon this hypothesis.
The corpuscular theory, reduced to the briefest definition, supposes
light to be really a material agent, and requires the student to believe
that this agent consists of particles so inconceivably minute that they
could not be weighed, and of course do not gravitate; the corpuscles
are supposed to be given out bodily (like sparks of burning steel from a
gerb firework) from the sun, the fixed stars, and all luminous bodies; to
travel with enormous velocity, and therefore to possess the property of
inertia; and to excite the sensation of vision by striking bodily upon the
expanded nerve, the retina, the quasi-mind of the eye. Dr. Young
remarks, "that according to this projectile theory the force employed in
the free emission of light must be about a million million times as great
as the force of gravity at the earth's surface, and it must either act with
equal intensity on all the particles of light, or must impel some of them
through a greater space than others, if its action be more powerful,

since the velocity is the same in all cases—for example, if the projectile
force is weaker with respect to red light than with respect to violet
light, it must continue its action on the red rays to a greater distance
than on the violet rays. There is no instance in nature besides of a
simple projectile moving with a velocity uniform in all cases, whatever
may be its cause; and it is extremely difficult to imagine that such an
immense force of repulsion can reside in all substances capable of
becoming luminous, so that the light of decaying wood, or two pebbles
rubbed together, may be projected precisely with the same velocity as
the light emitted by iron burning in oxygen gas, or by the reservoir of
liquid fire on the surface of the sun." Now one of the most striking
circumstances respecting the propagation of light, is the uniformity of
its velocity in the same medium. These and other difficulties in the
application of the corpuscular theory aroused the attention of the late
Dr. Young, and in the year 1801 he again revived and supported the
neglected undulatory theory with such great ability that the attention of
many learned mathematicians was directed to the subject, and now it
may be said that the corpuscular theory is almost, if not entirely,
rejected, whilst the undulatory theory is once more, and deservedly,
used to explain the theory of light, and its propagation through space.
By this hypothesis it is assumed that the whole universe, including the
most minute pores of all matter, whether solid, fluid, or gaseous, are
filled with a highly elastic rare medium of a most attenuated nature,
called ether, possessing the property of inertia but not of gravitation.
This ether is not light, but light is produced in it by the excitation on
the part of luminous bodies of a vibratory motion, similar to the
undulation of water that produces waves, or the vibration of air
affording sound. Water set in motion produces waves. Air set in motion
produces waves of sound. Ether, i.e. the theoretical ether pervading all
matter, likewise set in motion, produces light. The nature of a vibratory
medium is indeed better understood by reference to that which we
know possesses the ordinary properties of matter—viz., the air; and by
tracing out the analogy between the propagation of sound and light,
the difficulties of the undulatory theory very quickly vanish. To illustrate
vibration it is only necessary to procure a finger glass, and having
supported a little ebony ball attached to a silk thread by a bent brass
wire directly over it, so that the ball may touch either the outside or the

inside of the glass, attention must be directed to the quiescence of the
ball when a violin bow is lightly moved over the edge of the glass
without producing sound, and to the contrary effect obtained by so
moving and pressing the bow that a sharp sound is emitted, when
immediately the little ball is thrown off from the edge, the repulsive
action being continued as long as the sound is produced by the
vibration of the glass. (Fig. 250.)
Fig. 250.
a. The finger glass. b. The violin bow. c. The ebony ball. The dotted ball
shows how it is repelled during the vibration of the glass.
Here the vibrations are first set up in the glass, and being
communicated to the surrounding air, a sound is produced; if the same
experiment could be performed in a vacuum, the glass might be
vibrated, but not being surrounded with air, no sound would be
produced. This fact is proved by first ringing a bell with proper
mechanism fixed under the receiver placed on the air-pump plate; the
sound of the bell is audible until the pump is put in motion and the
receiver gradually exhausted, when the ringing noise becomes fainter
and fainter, until it is perfectly inaudible. This experiment is made more
instructive by gradually admitting the air again into the exhausted
vessel, and at the same time ringing the bell, when the sound becomes
gradually louder, until it attains its full power. The sun and other
luminous bodies may be compared to the finger glass, and are
supposed to be endowed naturally with a vibratory motion (a sort of
perpetual ague), only instead of the air being set in motion, the ether
is supposed to be thrown into waves, which travel through space, and

convey the impression of light from the luminous object. Another
familiar example of an undulatory medium is shown by throwing a
stone into a pool of water; the former immediately forces down and
displaces a certain number of the particles of the latter, consequently
the surrounding molecules of water are heaped up above their level; by
the force of gravitation they again descend and throw up another
wave, this in subsiding raises another, until the force of the original and
loftier wave dies away at the edge of the pool into the faintest ripples.
It must however be understood that it is not the particles of water first
set in motion that travel and spread out in concentric circles; but the
force is propagated by the rising and falling of each separate particle of
water as it is disturbed by the momentum of the descending wave
before it. When standing at a pier-head, or on a rock against which the
sea dashes, it is usual to hear the observer cry out, if the weather is
stormy and the waves very high, "Oh! here comes a great wave!" as if
the water travelled bodily from the spot where it was first noticed,
whereas it is simply the force that travels, and is exerted finally on the
water nearest the rock. It is in fact a progressive action, just as the
wind sweeps over a wide field of corn, and bends down the ears one
after the other, giving them for the time the appearance of waves. The
principle of successive action is well shown by placing a number of
billiard balls in a row, and touching each other; if the first is struck the
motion is communicated through the rest, which remain immovable,
whilst the last only flies out of its place. The force travels through all
the balls, which simply act as carriers, their motion is limited, and the
last only changes its position. Progressive movement is also well
displayed by arranging six or eight magnetized needles on points in a
row, with all their north poles in one direction. (Fig. 252.)

Fig. 251.
Boy throwing stones into water and producing circular waves.
Fig. 252.
a b. Series of needles arranged as described. c. The bar magnet, with the
north pole n towards the needles. The dotted lines show the direction
gradually assumed by all the needles, commencing at d.
On approaching the north pole of a bar magnet to the same pole of
one end of the series of needles, it is very curious to see them turn in
the opposite direction progressively, one after the other, as the
repulsive power of the bar magnet gradually operates upon the similar
poles in the magnetic needles. The undulations of the waves of water
are also perfectly shown by using the apparatus consisting of the
trough with the glass bottom and screen above it, as described at page
10. The transmission of vibrations from one place to another is also
admirably displayed in Professor Wheatstone's Telephonic Concert (see
page picture), where the musical instruments, as at the Polytechnic,
were placed by the author in the basement, and the vibration only
conducted by wooden rods to the sounding-boards above, so that the
music was laid on like gas or water. These vibrations or undulations in

air, water, and the theoretical ether, have therefore been called waves
of water, waves of sound, and waves of light, just as if three clocks
were made of three different metals, the mechanism would remain the
same, though the material, or in this case the medium, be different in
each.
Any increase in the number of vibrations of the air produces acute,
whilst a decrease attends the grave sounds, and when the waves
succeed each other not less than sixteen times in a second, the lowest
sound is produced. Light and colours are supposed to be due to a
similar cause, and in order to produce the red ray, no less than 477
millions of millions of vibrations must occur in a second of time; the
orange, 506; yellow, 535; green, 577; blue, 622; indigo, 658; violet,
699; and white light, which is made up of these colours, numbers 541
millions of millions of undulations in a second.
Although light travels with such amazing rapidity, there is of course a
certain time occupied in its passage through space—there is no such
thing as instantaneity in nature. A certain period of time, however
small, must elapse in the performance of any act whatever, and it has
been proved by a careful observation of the time at which the eclipses
of the satellites of Jupiter are perceived, that light travels at the rate of
192,500 miles per second, and by the aberration of the fixed stars,
191,515, the mean of these two sets of observations would probably
afford the correct rate. Such a velocity is, however, somewhat difficult
to appreciate, and therefore, to assist our comprehension of their great
magnitude, Sir J. Herschel has given some very interesting comparative
calculations, and coming from such an authority we can readily believe
them to be correct.
"A cannon-ball moving uniformly at its greatest velocity would require
seventeen years to reach the sun. Light performs the same distance in
about seven minutes and a half.
"The swiftest bird, at its utmost speed, would require nearly three
weeks to make the tour of the earth, supposing it could proceed
without stopping to take food or rest. Light performs the same distance
in less time than is required for a single stroke of its wing."

Dismissing for the present the theory of undulations, it will be
necessary to examine the phenomena of light, regarding it as radiant
matter, without reference to either of the contending theories.
Light issues from the sun, passes through millions of miles to the earth,
and as it falls upon different substances, a variety of effects are
apparent. There is a certain class of bodies which obstruct the passage
of the rays of light, and where light is not, a shadow is cast, and the
substance producing the shadow is said to be opaque. Wood, stone,
the metals, charcoal, are all examples of opacity; whilst glass, talc, and
horn allow a certain number of the rays to travel through their
particles, and are therefore called transparent. Nature, however, never
indulges in sudden extremes, and as no substance is so opaque as not
(when reduced in thickness) to allow a certain amount of light to pass
through its substance, so, on the other hand, however transparent a
body may be, a greater or lesser number of the rays are always
stopped, and hence opacity and transparency are regarded as two
extremes of a long chain; being connected together by numerous
intermediate links, they pass by insensible gradations the one into the
other.
If a gold leaf, which is about the one two-hundredth part of an inch in
thickness, is fixed on a glass plate and held before a light, a green
colour is apparent, the gold appearing like a green, semi-transparent
substance. When plates of glass are laid one above the other, and the
flame of a candle observed through them, the light decreases
enormously as the number of glass-plates are increased. Even in the
air a considerable portion of light is intercepted. It has been estimated
that of the horizontal sunbeams passing through about two hundred
miles of air, one two-thousandth part only reaches us, and that no
sensible light can penetrate more than seven hundred feet deep into
the sea; consequently, the vast depths discovered in laying the Atlantic
telegraph must be in absolute darkness.
Light is thrown out on all sides from a luminous body like the spokes of
a cart-wheel, and in the absence of any obstruction, the rays are
distributed equally on all sides, diverging like the radii drawn from the
centre of a circle. As a natural consequence arising from the divergence

Fig. 253.
of each ray from the other, the intensity of light decreases as the
distance from the luminous source increases, and vice versâ. Perhaps
the best mechanical notion of this law is afforded by an ordinary fan;
the point from which the sticks radiate, and where they all meet, may
be termed the light; the sticks are the rays proceeding from it. (Fig.
253.)
The fan is held in one hand, and the first
finger of the other can be made to touch all
the sticks if placed sufficiently near to A; and
supposing the sticks are called rays of light,
the intensity must be great at that point,
because all the rays fall upon it; but if the
finger is removed towards the outer edge—
viz., to b, it now only touches some three or
four sticks; and pursuing the analogy, a very
few rays fall upon that point—hence the light
has decreased in intensity, or to speak
correctly, "Light decreases inversely as the
squares of the distance." This law has already
been illustrated at page 13; and as an
experiment, the rays from the oxy-hydrogen
lantern may be permitted to pass out of a
square hole (say two inches square), and
should be thrown on to a transparent screen
divided into squares by dark lines, so that the light at a certain distance
illuminates one of them; then it will be found that at twice the
distance, four may be illuminated, at three times nine, and so on. (Fig.
254.)

Fig. 254.
Lantern at the three distances from the transparent screen, which is
divided into nine equal squares.
Upon this law is based the use of photometers, or instruments for
measuring light, and supposing it was required to estimate roughly the
illuminating power of any lamp, as compared with the light of a wax
candle six to the pound, the experiment should be conducted in a dark
room, from which every other light but that from the lamp and candle
under examination must be excluded.
The lamp, with the chimney only, is now placed say twelve feet from
the wall, and a stick or rod is placed upright and about two inches from
the latter, so that a shadow is cast on the wall; if the candle is now
lighted and allowed to burn up properly, two shadows of the stick will
be apparent, the one from the lamp being black and distinct, and the
other from the candle extremely faint, until it is approached nearer the
wall—say to within three feet—when the two shadows may be now
equal in blackness. (Fig. 255.) After this is apparent to one or more
persons, the distances of the lamp and candle from the wall are
carefully measured, and being squared, and the greater divided by the
lesser number, the quotient gives the illuminating power. For example:
The lamp was 12 feet from the wall12 × 12= 144.
The candle was 3 feet " 3 × 3 = 9.

9) 144
————
16
Therefore the illuminating power of the lamp is equal to 16 wax
candles six to the pound.
Fig. 255.
a. The lamp. b. The candle. c. The rod throwing the two shadows,
marked d and e, on a white wall or a sheet of paper.
There are other and more refined means of working out the same fact,
but for a rough approximation to the truth, the plan already described
will answer very fairly.
A most amusing effect can be produced on the principle that every
light casts its own shadow, called the "dance of death," or the "dance
of the witches;" either of these agreeable subjects are drawn, and the
outlines cut out of a sheet of cardboard. If a wet sheet is stretched or
hung on one side of a pair of folding doors partly open, and between
which the cardboard is tacked up, and the space left at the top and
bottom closed with a dark cloth, directly the room before the sheet is
darkened and a lighted candle held behind the figure cut out in the
cardboard, one shadow or image is thrown upon the sheet, and these
shadows may be increased according to the number of candles used,
and if they are held by two or three persons, and moved up and down,

or sideways, the shadows follow the direction of the candles, and
present the appearance of a dance. (Fig. 256.)
Fig. 256.
"Before the curtain."
Fig. 257.
"Behind the curtain."
Another very comic effect of shadow is that called "jumping up to the
ceiling," and when carried out on a large scale by the author on an
enormous sheet suspended in the centre transept of the Crystal Palace,
Sydenham, it had a most laughable effect, and caused the greatest
amusement to the children of all ages. (Fig. 258.)

Fig. 258.
The laughable effect of the shadows at the
Crystal Palace.
This very telling result is produced by placing an oxy-hydrogen light
some feet behind a large sheet, and of course if any one passes
between the two a shadow of the individual is cast upon the sheet,
then by walking towards the light the figure diminishes in size, and by
jumping over it the shadow appears to go up to the ceiling, and to
come down when the jump is made in the opposite direction over the
light and towards the sheet. The rationale of this experiment is very
simple, and is another proof of the distribution of light from a luminous
source being in every direction. By jumping over the light the radii
projected from the candle over the sheet are crossed, and the shadow
rises or falls as the figure passes upwards or downward. (Fig. 259.)

Fig. 259.
The rays of light marked a b c d e proceeding from a lighted candle or
oxy-hydrogen light. The arrow pointing to the right shows how these
rays are crossed in jumping up to the ceiling; and the second arrow,
pointing to the left, shows the reverse.
A beam of light is defined to be a collection of rays, and it is a
convenient definition, because it prevents confusion to speak only of
one ray in attempting to explain how light is disposed of under peculiar
circumstances.
The smallest portion of light which it is supposed can be separated is
therefore called a ray, and it will pass through any medium of the same
density in a perfectly straight line; but if it passes out of that medium
into another of a different density, or into any other solid, fluid, or
gaseous matter, it may be disposed of in four different ways, being
either reflected, refracted, polarized, or absorbed.
The reflection of light is the first property that will be considered, and it
will be found that every substance in nature possesses in a greater or
lesser degree the power of throwing off the rays of light which fall
upon them. Thus if we go into a room perfectly darkened, containing
every kind of work produced by nature or art, such as flowers, birds,
boxes of insects, rich carpets, hangings, pictures, statuary, jewellery,
&c., they cannot excite any pleasure because they are invisible, but
directly a lighted lamp is brought into the chamber, then the rays fall

upon all the surrounding objects, and being reflected from their
surfaces enter the eye, and there produce the phenomena of vision.
This connexion between luminous and non-luminous bodies becomes
very apparent when we consider that the sun would appear only as an
intense light in a dark background, if the earth was not surrounded
with the various strata of air, in which are placed clouds and vapours
that collectively reflect and scatter the light, so as to cause it to be
endurable to vision. It is when the sky is very clear during July or
August that the heat becomes so intense, directly clouds begin to form
and float about, the heat is then moderated.
Many years ago, Baron Alexander Funk, visiting some silver mines in
Sweden, observed, that in a clear day it was as dark as pitch
underground in the eye of the pit at sixty or seventy fathoms deep;
whereas, on a cloudy or rainy day he could even see to read at 106
fathoms deep. Inquiring of the miners, he was informed that this is
always the case, and reflecting upon it he imagined very properly that
it arose from this circumstance—that when the atmosphere is full of
clouds, light is reflected from them into the pit in all directions, so that
thereby a considerable proportion of the rays are reflected
perpendicularly upon the earth; whereas when the atmosphere is clear
there are no opaque bodies to reflect the light in this manner, at least,
in a sufficient quantity, and rays from the sun itself can never fall
perpendicularly in Sweden. The use of reflecting surfaces has now
become quite common in all crowded cities, and especially in London,
where even the rays of light are too few to be lost, and flat or
corrugated mirrors are placed at various angles, either to throw the
light from the outside on the white-washed ceiling within, and thus
obtain a better diffused light through the apartment, or it is reflected
bodily to some back room, or rather dark brick box, where perhaps for
half a century candles have been required at an early hour in the
afternoon. The brilliant cut in diamonds is such an arrangement of the
posterior facets, or cut faces of the jewel, that all light reaching them
shall be thrown back and reflected, and thus impart an extraordinary
brilliancy to the gem.

The intense glare of snow in the Alpine regions has long been noticed,
and the reflected light is so powerful, that philosophers were even
disposed to believe that snow possessed a natural or inherent
luminosity, and gave out its own light. Mr. Boyle, however, disproved
this notion by placing a quantity of snow in a room from which all
foreign light was excluded, and neither he nor his companion could
observe that any light was emitted, although, on the principle of
momentary phosphorescence, it is quite possible to conceive that if the
snow was suddenly brought into a darkened room after exposure to
the rays of the sun, that it would give out for a few seconds a
perceptible light. In trying such an experiment, one person should
expose the snow to the sun, and bring it into a perfectly darkened
room to a second person, whose eyes would be ready to receive the
faintest impression of light, and if any phosphorescence existed, it
must be apparent.
The property of reflection is also illustrated on a grand scale in the
illumination of our satellite, the moon, and the various planetary bodies
which shine by light reflected from the sun, and have no inherent self-
luminosity. Aristotle was well aware that it is the reflection of light from
the atmosphere which prevents total darkness after the sun sets, and
in places where the sun's rays do not actually fall during the daytime.
He was also of opinion that rainbows, halos, and mock suns, were all
occasioned by the reflection of the sunbeams in different
circumstances, by which an imperfect image of the sun was produced,
the colour only being exhibited, but not the proper figure.
The image, Aristotle says, is not single, as in a mirror, for each drop of
rain is too small to reflect a visible image, but the conjunction of all the
images is visible. Aristotle ascribed all these effects to the reflection of
light, and it will be noticed when we come to the consideration of the
refraction of light, that of course his views must be seriously modified.
The reflection of light is affected rather by the condition of the surface
than the whole body of a substance, as a piece of coal may be covered
with gold or silver leaf and caused to shine, whilst the brightest mirror
is dimmed by the thinnest film of moisture.

From whatever surface light is reflected, it always takes place in
obedience to two fixed laws.
First. The incident and reflected rays always lie in the same plane.
Second. The angle of incidence is equal to the angle of reflection.
With a single jointed two-foot rule, both of these laws are easily
illustrated. The rule may be held in the hand, and one end being
marked with a piece of white paper may be called the incident ray, i.e.,
the ray that falls upon the surface; and the other is the reflected ray,
the one cast off or thrown back. A perpendicular is raised by holding a
stick upright at the joint. (Fig. 260.)
Fig. 260.
a d. A two foot rule; the end a may be termed the incident ray, and the
end d the reflected ray. s. The stick held perpendicularly. The angle a b c
is equal to the angle d e f, and the whole may be moved in any direction
or plane, either horizontal or perpendicular, g g. The reflecting surface.
One of the most simple and pleasing delusions produced by the
reflection of light, is that afforded by cutting through the outline of a
vase, or statuette, or flower, drawn on cardboard, and if certain points
are left attached, so that the design may not fall out, all the effect of
solidity is given by bending back the edges of the cardboard, so that
the light from a candle placed behind it, may be reflected from the
back edge of one cardboard on to the design, which is bent back. The
light reflected from one surface on to the other, imparts a peculiarly

soft and marble-like appearance, and when the design is well drawn
and cut, and placed in a good position, the illusion is very perfect, and
it appears like a solid form instead of a mere design cut out of
cardboard. (Fig. 261.)
Fig. 261.
Cardboard design in frame, cut and bent back. The lighted candle is
behind.
The leaf at the side of the above picture is intended to give an idea of
the mode of cutting out the designs, and in this case the leaf would be
cut and bent back, and a small attachment slip of cardboard left to
prevent it falling out.
The cardboard design is always bent toward the light, which is placed
behind it. As a good illustration of the importance of reflected light and
its connexion with luminous bodies, a beam of light from the oxy-
hydrogen lantern may be allowed to pass above the surface of a table,
when it will be noticed that the latter is lighted up only when the beam
is reflected downward by a sheet of white paper.
By reference to the two laws of reflection already explained, it is easy
to trace out on paper, with the help of compasses and rule, the effect
of plane, concave, and convex surfaces on parallel, diverging, or
converging rays of light, and it may perhaps assist the memory if it is
remembered that a plane surface means one that is flat on both sides,
such as a looking-glass: a convex surface is represented by the outside

of a watch-glass; a concave surface, by the inside of a watch-glass;
parallel rays are like the straight lines in a copy-book; diverging and
converging rays, are like the sticks of a fan spread out as the sticks
separate or diverge; the sticks of the fan come together, or converge at
the handle.
The reflection of rays from a plane surface may be better understood
by reference to the annexed diagram. (Fig. 262.)
Fig. 262.
a i, a k. Two diverging rays incident on the plane surface, d. a d is
perpendicular, and is reflected back in the same direction. a i is
divergent, and is thrown off at i l. The incident and reflected rays
forming equal angles, as proved by the perpendicular, h. Any image
reflected in a plane mirror appears as far behind it as the object is
before it, and the dotted lines meeting at g show the apparent position
of the reflected image behind the glass, as seen at g. The same fact is
also shown in the second diagram, where the reflected picture, i m,
appears at the same distance behind the surface of the mirror as the
object, a b, is before it.
By the proper arrangement of plane mirrors, a number of amusing
delusions may be produced, one of which is sometimes to be met with
in the streets, and is called "the art of looking through a four-inch deal
board." The spectator is first requested to look into a tube, through
which he sees whatever may be passing the instrument at the time;
the operator then places a deal board across the middle of the tube,
which is cut away for that purpose, and to the astonishment of the

juveniles the view is not impaired, and the spectator still fancies he is
looking through a straight tube; this however is not the case, as the
deception is entirely carried out by reflection, and is explained in the
next cut. (Fig. 263.)
Fig. 263.
a a a a. The apertures through which the spectator first looks. b. The
piece of wood, four inches thick. c, d, e, f, are four pieces of looking-
glass, so placed that rays of light entering at one end of the tube are
reflected round to the other where the eye of the observer is placed.
During the siege of Sebastopol numbers of our best artillerymen were
continually picked off by the enemy's rifles, as well as by cannon shot,
and in order to put a stop to the foolhardiness and incautiousness of
the men, a very ingenious contrivance was invented by the Rev. Wm.
Taylor, the coadjutor of Mr. Denison in constructing the first "Big Ben"
bell. It was called the reflecting spy-glass, and by its simple
construction rendered the exposure of the sailors and soldiers, who
would look over the parapet or other parts of the works to observe the
effect of their shot, perfectly unnecessary; whilst another form was
constructed for the purpose of allowing the gunner to "lay" or aim his
gun in safety. The instruments were shown to Lord Panmure, who was
so convinced of the importance of the invention, that he immediately

commissioned the Rev. Wm. Taylor to have a number of these
telescopes constructed; and if the siege had not terminated just at the
time the invention was to have been used, no doubt a great saving of
the valuable lives of the skilled artillerymen would have been effected
in the allied armies. The principle of the reflecting spy-glass may be
comprehended by reference to the next cut. (Fig. 264.)
Fig. 264.
A picture of enemy's battery is supposed to be on the mirror, a, whence
it is reflected to b, and from that to the artilleryman at c.
By placing two mirrors at an angle of 45°, the reflected image of a
person gazing into one is thrown into the other, and of course the
effect is somewhat startling when a death's head and cross bones, or
other cheerful subject, is introduced opposite one mirror, whilst some
person who is unacquainted with the delusion is looking into the other.
Two adjoining rooms might have their looking-glasses arranged in that
manner, provided there is a passage running behind them. (Fig. 265.)

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Enumerative Combinatorics 1st Edition Charalambos A Charalambides

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