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Linear and Convex Optimization 1st Edition Michael H.
Veatch Digital Instant Download
Author(s): Michael H. Veatch
ISBN(s): 9781119664048, 1119664047
Edition: 1
File Details: PDF, 4.73 MB
Year: 2021
Language: english
Linear and Convex Optimization
Linear and Convex Optimization

A Mathematical Approach

Michael H. Veatch
Gordon College
This edition first published 2021
© 2021 by John Wiley and Sons, Inc.

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Library of Congress Cataloging-in-Publication Data

Names: Veatch, Michael H., author. | John Wiley and Sons, Inc., publisher.
Title: Linear and convex optimization : a mathematical approach / Michael
Veatch, Gordon College.
Description: Hoboken, NJ : Wiley, 2021. | Includes index.
Identifiers: LCCN 2020025965 (print) | LCCN 2020025966 (ebook) | ISBN
9781119664048 (cloth) | ISBN 9781119664024 (adobe pdf) | ISBN
9781119664055 (epub)
Subjects: LCSH: Mathematical optimization. | Nonlinear programming. |
Convex functions.
Classification: LCC QA402.5 .V395 2021 (print) | LCC QA402.5 (ebook) |
DDC 519.6–dc23
LC record available at https://lccn.loc.gov/2020025965
LC ebook record available at https://lccn.loc.gov/2020025966

Cover Design: Wiley

Cover Image: © Hybrid_Graphics/Shutterstock

Set in 9.5/12.5pt STIXTwoText by SPi Global, Chennai, India

10 9 8 7 6 5 4 3 2 1
To Dad, who introduced me to operations research, and Christian and Jackie, who
were always curious how things worked
 vii

Contents

Preface xi
About the Companion Website xvii

1 Introduction to Optimization Modeling 1

1.1 Who Uses Optimization? 1
1.2 Sending Aid to a Disaster 3
1.3 Optimization Terminology 9
1.4 Classes of Mathematical Programs 11
Problems 16

2 Linear Programming Models 19

2.1 Resource Allocation 19
2.2 Purchasing and Blending 23
2.3 Workforce Scheduling 29
2.4 Multiperiod Problems 30
2.5 Modeling Constraints 34
2.6 Network Flow 36
Problems 44

3 Linear Programming Formulations 55

3.1 Changing Form 55
3.2 Linearization of Piecewise Linear Functions 57
3.3 Dynamic Programming 62
Problems 66

4 Integer Programming Models 71

4.1 Quantitative Variables and Fixed Costs 72
4.2 Set Covering 74
4.3 Logical Constraints and Piecewise Linear Functions 77
viii Contents

4.4 Additional Applications 81

4.5 Traveling Salesperson and Cutting Stock Problems 86
Problems 90

5 Iterative Search Algorithms 99

5.1 Iterative Search and Constructive Algorithms 100
5.2 Improving Directions and Optimality 106
5.3 Computational Complexity and Correctness 112
Problems 116

6 Convexity 121
6.1 Convex Sets 122
6.2 Convex and Concave Functions 127
Problems 131

7 Geometry and Algebra of LPs 133

7.1 Extreme Points and Basic Feasible Solutions 134
7.2 Optimality of Extreme Points 137
7.3 Linear Programs in Canonical Form 140
7.4 Optimality Conditions 145
7.5 Optimality for General Polyhedra 146
Problems 149

8 Duality Theory 153

8.1 Dual of a Linear Program 153
8.2 Duality Theorems 158
8.3 Complementary Slackness 162
8.4 Lagrangian Duality 164
8.5 Farkas’ Lemma and Optimality 167
Problems 170

9 Simplex Method 173

9.1 Simplex Method From a Known Feasible Solution 174
9.2 Degeneracy and Correctness 183
9.3 Finding an Initial Feasible Solution 186
9.4 Computational Strategies and Speed 192
Problems 200

10 Sensitivity Analysis 203

10.1 Graphical Sensitivity Analysis 204
10.2 Shadow Prices and Reduced Costs 208
 Contents ix

10.3 Economic Interpretation of the Dual 219

Problems 221

11 Algorithmic Applications of Duality 225

11.1 Dual Simplex Method 226
11.2 Network Simplex Method 234
11.3 Primal-Dual Interior Point Method 246
Problems 256

12 Integer Programming Theory 261

12.1 Linear Programming Relaxations 262
12.2 Strong Formulations 263
12.3 Unimodular Matrices 269
Problems 272

13 Integer Programming Algorithms 275

13.1 Branch and Bound Methods 275
13.2 Cutting Plane Methods 284
Problems 293

14 Convex Programming: Optimality Conditions 297

14.1 KKT Optimality Conditions 297
14.2 Lagrangian Duality 306
Problems 312

15 Convex Programming: Algorithms 317

15.1 Convex Optimization Models 320
15.2 Separable Programs 323
15.3 Unconstrained Optimization 325
15.4 Quadratic Programming 329
15.5 Primal-dual Interior Point Method 331
Problems 339

A Linear Algebra and Calculus Review 343

A.1 Sets and Other Notation 343
A.2 Matrix and Vector Notation 343
A.3 Matrix Operations 345
A.4 Matrix Inverses 347
A.5 Systems of Linear Equations 348
A.6 Linear Independence and Rank 350
x Contents

A.7 Quadratic Forms and Eigenvalues 351

A.8 Derivatives and Convexity 352

Bibliography 355
Index 361
 xi

Preface

This book is about optimization problems that arise in the field of operations
research, including linear optimization (continuous and discrete) and convex pro-
gramming. Linear programming plays a central role because these problems can
be solved very efficiently; it also has useful connections with discrete and convex
optimization. Convex optimization is not included in many books at this level.
However, in the past three decades new algorithms and many new applications
have increased interest in convex optimization. Like linear programming, large
applied problems can now be solved efficiently and reliably. Conceptually, convex
programming fits better with linear programming than with general nonlinear
programming.
These types of optimization are also appropriate for this book because they have
a clear theory and unifying mathematical principles, much of which is included.
The approach taken has three emphases.
1. Modeling is covered through a broad range of applications to fields such as
on-line marketing and inventory management, retail pricing, humanitarian
response and rural development, public sector planning, health delivery,
finance, manufacturing, service systems, and transportation. Many of these
tell the story of successful applications of operations research.
2. A mathematical approach is used to communicate in a concise, unified manner.
Matrix notation is introduced early and used extensively. Questions of correct-
ness are not glossed over; the mathematical issues are described and, where the
level is appropriate, proofs presented. Connections are made with some other
topics in undergraduate mathematics. This approach grew out of my 30 years
of teaching these topics to undergraduate mathematics students.
3. The reasoning behind algorithms is presented. Rather than introducing
algorithms as arbitrary procedures, whenever possible reasons are given for
why one might design such an algorithm. Enough analysis of algorithms is
presented to give a basic understanding of complexity of algorithms and what
makes an algorithm efficient. Algorithmic thinking is taught, not assumed.
xii Preface

Many introductory operations research textbooks emphasize models and algo-

rithms without justifications and without making use of mathematical language;
such books are ill-suited for mathematics students. On the other hand, texts that
treat the subject more mathematically tend to be too advanced and detailed, at the
expense of applications. This book seeks a middle ground.
The intended audience is junior and senior mathematics students in a course on
optimization or (deterministic) operations research. The background required is
a good knowledge of linear algebra and, in a few places, some calculus. These are
reviewed in the appendix. The coverage and approach is intentionally kept at an
undergraduate level. Material is often organized by depth, so that more advanced
topics or approaches appear at the end of sections and chapters and are not needed
for continuity. For example, the many ways to speed up the simplex method are
saved for the last section of Chapter 9.
In keeping with this audience, the number of different problem types and
algorithms is kept to a minimum, emphasizing instead unified approaches and
more general problems. In particular, heuristic algorithms are only mentioned
briefly. They are used for hard problems and use many different approaches,
while this book focuses on problems that have efficient algorithms or at least
unified approaches. The goal is to introduce students to optimization, not to be a
thorough reference, and to appeal to students who are curious about other uses
of mathematics. The many applications in the early chapters make the case that
optimization is useful. The latter chapters connect the solution of these problems
to the linear algebra and other mathematics that this audience is familiar with.
The book is also specifically written for the instructor who is mathematically
trained, not for a specialist in operations research and optimization. The care-
ful treatment of algorithmic thinking and the introduction to complexity of algo-
rithms are intended to assist these instructors. The mathematical style throughout
the book should be more accommodating to mathematics professors. It is also
intended to support learning objectives more likely to be found in a mathematics
department, including why the algorithms are correct and how they use theoret-
ical results such as convexity and duality. Being able to perform an algorithm by
hand is not a primary objective; it plays a supporting role to understanding the
notation and reasoning of the algorithm. Calculations that are well-understood by
mathematics students, such as solving a linear system or performing row opera-
tions, are not elaborated on. The somewhat more advanced material at the end of
sections or chapters is also intended to support instructors who are not specialists,
allowing them to extend their knowledge and explore the literature.
Chapters 1–4 are devoted to introducing optimization and optimization model-
ing. Convex models appear later with the other material on convex optimization.
In my experience teaching mathematics students, they find modeling challenging.
These chapters assume technology is available to solve problems, so that the focus
 Preface xiii

can stay on formulation, as well as interpreting solutions. They build steadily in

sophistication, starting with numerical instances but soon moving to algebraic for-
mulations to make clear the distinction between model structure and data. The
features of the models also build, particularly when using logical variables. In
contrast with the business case study approach, each model has a limited num-
ber of features and focuses on some novel feature. I have found that mathematics
students relate better to a succession of simpler models, from which they learn
different modeling principles, than to a long case study.
Chapters 5–8 discuss iterative algorithms, giving some easily explained
examples from discrete optimization, and the theoretical background for linear
programming. This includes a little computational complexity, convexity and the
study of polyhedra, optimality conditions for linear programming, and duality
theory for linear programming. It is unorthodox to cover all of these before
introducing the simplex method. However, conceptually these topics fit together
and do not depend on the simplex method; putting them together emphasizes this
fact. Chapter 8 on duality is independent of Chapter 9 on the simplex method,
so that they can be covered them in either order. I typically skip the topics in
Sections 5.3, 7.5, 8.4, and 8.5 to arrive at the simplex method about the middle of
the semester.
Chapters 9–11 present the simplex method, including sensitivity analysis, and
other algorithms for linear programming. A developmental approach is taken to
presenting the simplex method. Starting with geometric reasoning about why it
works, it is presented in the “naive” form first, where the so-called inverse basis
matrix is computed from scratch each iteration. While this form is computationally
inefficient, it is very easy to explain to a mathematics student, both for computing
and justifying that the algorithm works. When working examples, technology can
be used to invert and multiply matrices. After completing the picture of why the
method works (degeneracy, two-phase simplex), Section 9.4 takes up the issue of
making the simplex method more efficient, including the tableau form and revised
simplex method. Instructors who wish to start with the tableau can use the mate-
rial found here. Chapter 10 on sensitivity analysis, which depends Chapter 8, can
be skipped without loss of continuity; however, the interpretation of dual vari-
ables as shadow prices and partial derivatives is enriching, even in an age when
sensitivity analysis can be done quickly by solving modified linear programs. An
interpretation of strong duality in terms of average costs is also given in Section
10.3. Chapter 11 presents three more algorithms for linear programming, all of
which rely on duality: the dual simplex, transportation simplex, and a primal-dual,
or path following, interior point method. The transportation simplex method is
presented first as a minimum cost flow problem, then specialized to transportation
problems.
xiv Preface

Chapters 12 and 13 present integer programming algorithms. These relate to

the earlier material because of the importance of linear programming to establish
bounds when solving an integer program. Integer programming also has greater
modeling power, as demonstrated by the many applications in Chapter 4. Chapters
14 and 15 introduce convex programming, including some motivating applica-
tions. The earlier chapters are designed to prepare the reader to understand convex
programming more readily. The KKT optimality conditions and duality theorems
are a generalization of Lagrangian duality (Section 8.4). Necessary and sufficient
conditions for a global optimum follow from convexity theory, already applied to
linear programs in Chapter 6. Chapter 15 culminates in the primal-dual interior
point method, which was presented for linear programs in Section 11.3. Quadratic
programming is also introduced and the connection between the primal-dual inte-
rior point method and sequential quadratic programming is made.
Supplemental material will be available at the web site www.gordon.edu/
michaelveatch/optimization for the book. A full solution manual will be made
available to instructors.
The book contains the amount of material covered in a typical two-semester
sequence of undergraduate classes. A semester course focusing on linear program-
ming could cover Chapters 1, 2, Sections 3.1–3.2, 5, 6, Sections 7.1–7.4 and 8.1–8.3,
9, 10 plus some other topics from these chapters and Chapter 11. A course on lin-
ear and integer programming could cover Chapters 1, 2, Sections 3.1–3.2, 4, 5, 6,
Sections 7.1–7.4 and 8.1–8.3, 9, 12, and 13. A somewhat more advanced course
on linear and convex programming could cover Chapters 1–3, 5–7.4, 8, 9, Sections
11.1–11.3, 14, and 15.
Several more advanced or specialized topics have been included at the end of
chapters or sections that are optional and can be easily skipped. Section 3.3 shows
that a dynamic program can be solved as a linear program, an approach that relates
to machine learning. Section 5.3 on computational complexity, while not difficult,
is only occasional mentioned in the later chapters. Section 7.5 extends the optimal-
ity conditions needed to solve linear programs to general polyhedra. Section 8.4
introduces Lagrangian duality for linear programs and shows that it is equivalent
to the usual dual; it is only needed if convex programming (Chapter 14) is being
covered. Farkas’ lemma is presented in Section 8.5, providing another approach
to duality theorems. The computational strategies in Section 9.4 are important to
the simplex method but are not used in the sequel. The interior point algorithm in
Section 11.4 is computationally more involved. It is closely related to Section 15.5.
I want to express my deep appreciation to the many people who helped make
this book possible. First, I want to thank David Rader, Larry Leemis, and Susan
Martonosi for encouraging me to undertake the project. I am grateful to my for-
mer and current students Isaac Bleecker, Mackenzie Hinds, Joe Iriana, Stephen
Rizzo, and Michael Yee for reviewing portions of the draft. I also thank my friend
 Preface xv

John Sanderson for drawing the figures, my colleague Jonathan Senning for his
technical advice, and students Isaac Bleecker, Jessica Guan, Seth McKinney, and
Yi Zhou for their help with the exercises and figures.
Most of all, I am grateful for my wife Cindy’s confidence in me and acceptance
of my working odd hours on the project. Now we are both authors.

Wenham, MA Michael H. Veatch

March, 2020
 xvii

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/veatch/convexandlinearoptimization

The website includes the instructor solutions manual.

Other documents randomly have
different content
 CHAPTER III

THE EMPERORS AT CONSTANTINOPLE

476-527

Contrast between the fates of the Eastern and Western Empires—The East
recovers its strength—Leo I. and the Isaurians—The Emperor Zeno and the
rebellion against him—Wars of Zeno with the two Theodorics, 478-483—The
‘Henoticon’—Character of the Emperor Anastasius—Rebellion of the Isaurians
—War with Persia, 503-5—The ‘Blue and Green’ Factions—Rebellion of Vitalian
—Accession of Justin I.

At Rome the emperors of the third quarter of the fifth century—all

the ephemeral Caesars whose blood-stained annals fill the space
between the death of Valentinian III. and the usurpation of Odoacer
—had been the mere creatures of the barbarian, or semi-barbarian,
‘patricians’ and ‘masters of the soldiers,’ to whom they owed alike
their elevations and their untimely ends. The history of those
troubled years would be more logically arranged under the names of
the Caesar-makers, Ricimer, Gundobad, Orestes, than under those of
the unhappy puppets whom they manipulated.
But, when we turn our eyes eastward to Constantinople, we are
surprised to find how entirely different was the aspect of affairs. The
Western Empire was rapidly falling to pieces, province after province
dropping out of the power of the emperor, and becoming part of the
realm of some Gothic, Burgundian, or Vandal prince, who paid the
most shadowy homage, or no homage at all, to the ephemeral
 Contrast Caesar at Rome. The Eastern Empire, on the other
between hand, maintained its boundaries intact, and was
Eastern and slowly building up its strength for renewed activity in
Western
Empires. the next century. While nine emperors’ reigns filled no
more than twenty-one years at Rome (455-476), two
emperors were reigning for thirty-four years (457-491) on the
Bosphorus. And the character of the rulers of East and West was as
different as their fates: the short-lived Roman Caesars were either
impotent nobodies raised to the throne by the caprice of the
barbarian, or ambitious young soldiers who vainly dreamed that they
might yet redeem the evil day, and save the State. Their
contemporaries in the East, Leo, Zeno, and Anastasius, were three
elderly officials, men of experience, if not of great ability, who
followed each other in peaceable succession, and devoted their
declining years to a cautious defensive policy, with the result that
they left a full treasury, a strong and loyal army, and an intact realm
behind them.
At the beginning of the fifth century the eastern half of the Empire
had seemed no less likely than the western to fall under the
dominion of the barbarian, and crumble to pieces. The Goths were
cantoned all over Thrace, Moesia, and Asia Minor, and the Gothic
general Gainas had taken possession of the person and authority of
the Emperor Arcadius. Had he been a man of greater ability he
might have made and unmade emperors, as Ricimer afterwards did
in the West. But the schemes of Gainas were wrecked, and the
Empire saved by the great riot at Constantinople in 401, when the
Gothic foederati were massacred, and their leader chased away by
the infuriated populace, who thus saved not only their own homes,
but the whole East, from the danger of Gothic domination.
Though the European provinces of the Eastern Empire suffered
grievously from Teutonic ravages during the first eighty years of the
century, there was never again any danger that the barbarians
would get hold of the machinery of government, and subvert the
Empire from within. In the long reign of Theodosius II. (406-450), if
no progress was made in strengthening the realm, at least no
ground was lost.
Two external causes were, during this time, operating in favour of
the Eastern Empire. The first was the absolute impregnability of
Constantinople against any invader who could only assault it from
the land side: the town could not be starved out,—as Rome was
starved by Alaric,—and its walls could laugh to scorn all such siege
appliances as that age knew. Though Goth and Hun pushed their
ravages far and wide in the Balkan peninsula, they never seriously
attempted to molest the great central place of arms on which the
Importanc East-Roman power based itself. The Western Empire
e of had no such stronghold—capital, arsenal, harbour,
Constantin and centre of commerce all in one. Ravenna, where
ople.
the Western Caesars took refuge in times of storm
and stress, was in every way inferior to Constantinople as a base of
armed resistance to the invader. Though its marshes made it strong,
it did not cover or protect any considerable tract of country, and it
was just far enough from its harbour to allow of an enemy cutting
off its supplies.
The second great factor in the vitality of the Eastern Empire was
the prolonged freedom from foreign war enjoyed by its Asiatic
provinces. After the revolt of Gainas in 401, the Goths disappeared
from Asia Minor, and no other invaders made any serious breach into
that peninsula, into Syria, or into Egypt, for a hundred and forty
years. Two short Persian wars, in 420-421 and 502-505, led to
nothing worse than partial ravages on the Mesopotamian frontier. It
is true that the Asiatic provinces of the empire were not altogether
spared by the sword in the fifth century, but such troubles as they
suffered were due to native revolts, chiefly of the Isaurians among
the mountains of southern Asia Minor. These risings were local, and
led to no very widespread damage, nor was the fighting caused by
the revolts of the rebel-emperors Basiliscus and Leontius, in the
Prosperity reign of Zeno, much more destructive. On the whole,
of the the four oriental ‘dioceses’ of the Eastern Empire must
East. have enjoyed in the fifth century a far greater
measure of peace and prosperity than they had known, or were to
know, in the previous and the succeeding ages. It was their wealth,
duly garnered into the imperial treasury, that made the emperors
strong to defend their European possessions. We shall soon see that
their military resources also were to count in a most effective way in
the reorganisation of the East-Roman army.
But the strength of Constantinople and the wealth of Asia might
have proved of no avail had they fallen into the hands of a series of
emperors like Honorius or Valentinian III. We must in common
fairness grant that the personal characters of the Emperors Leo I.,
Zeno, and Anastasius I. had also the most important influence on the
empire. These three cautious, persistent, and careful princes, who
neither endangered the empire by over-great enterprise and
ambition, nor let it fall to pieces by want of energy, were exactly the
men most fitted to tide over a time of transition.
Leo, the first of these three emperors, was already dead when
Romulus Augustulus was deposed in the West. He had left his mark
on Constantinopolitan history by his summary execution of Aspar,
the last of the great barbarian ‘masters of the soldiers,’ who rose to
a dangerous height of power in the East; and still more by his very
important scheme for reorganising the army, by enrolling a large
proportion of native-born subjects of the empire in its ranks.
Recognising the peril of trusting entirely to Teutonic mercenaries,—
the fatal error that had ruined the Western Empire,—Leo had
enlisted, in as great numbers as he could obtain, the hardy
mountaineers of Asia Minor, more especially the Isaurians. His
Leo and predecessors had distrusted their unruly and
the predatory habits, but Leo saw that they supplied good
Isaurians. and trustworthy fighting material, and dealt with them
as the elder Pitt dealt with the Highlanders after the rebellion of
1745, teaching them to use in the service of the government the
wild courage that had so often been turned against it. Leo had
indeed done all that he could for the Isaurians, and had at last
married his elder daughter Ariadne to Zeno, an Isaurian by birth,
and one of the chief officers of his court.
 It was this Zeno who was seated on the throne of the Eastern
realm at the moment that Odoacer made himself ruler of Italy, and
to him was addressed the celebrated petition of the Roman Senate
which besought him to allow East and West alike to repose under
the shadow of his name, but to confide the practical governance of
Italy to the patrician Odoacer. Zeno was neither so able nor so
respectable a sovereign as his father-in-law: two faults, a caution
which verged on actual cowardice and a taste for low debauchery,
The have blasted his reputation. His enemies were never
Emperor tired of taunting him with his Isaurian birth, and
Zeno, 475- recalling to memory that his real name was
491.
Tarakodissa, the son of Rusumbladeotus, for he had
only taken the Greek appellation of Zeno when he came to court.
But though he was by birth an obscure provincial, and by nature
something of a coward and a free liver, Zeno had his merits. He was
a mild and not an extortionate administrator, had a liberal hand, a
good eye for picking out able servants, was sanguine and
persevering in all that he undertook, and pursued in Church matters
a policy of moderation and conciliation, which may bring him credit
now, though in his own time it provoked many strictures from the
orthodox. The worst charges that can be laid to his account were
acts that were prompted by his timidity rather than by any other
motive,—two or three arbitrary executions of officers whom he
rightly or wrongly suspected of plotting against his life. After three
rebellions which came within an ace of success, it is not unnatural
that he grew somewhat nervous about his own safety.
Zeno’s reign was more troubled in this way than those of his
predecessor and successor. His well-known lack of daring tempted
men to conspire against him, but they reckoned without his cunning
and his perseverance, and in every case came to an evil end. Zeno
could count on the active support of his countrymen the Isaurians,
who now formed the most trustworthy part of the army, and on the
passive obedience, or at worst the neutrality, of the mercantile
classes and the bureaucracy, who disliked all change and disorder.
Hence it came to pass that court conspiracies, or local revolts of
divisions of the army, were not enough to shake his throne.
The first half of Zeno’s reign may be divided into three parts by
these three conspiracies. The emperor had hardly ascended the
throne when the first of them broke out: it was a palace intrigue
hatched by the Empress-Dowager Verina, who detested her son-in-
law. The conspirators took Zeno quite by surprise, they failed to
catch him, for he fled from Constantinople at the first alarm, but
they got possession of the capital, and proclaimed Basiliscus, the
Revolt of brother of Verina, as Augustus. The mob of the city,
Basiliscus, with whom Zeno was very unpopular, joined the
475-477. rising, and massacred the Isaurian troops who were
within the walls; their leader’s absence seems to have paralysed the
resistance of the soldiery. Zeno meanwhile escaped to his native
country, and raised an Isaurian army: Syria and the greater part of
Asia Minor remained faithful to him, and he prepared to make a fight
for his throne. Luckily for him, Basiliscus was a despicable creature,
—it was he who had wrecked the great expedition against the
Vandals which Leo I. had sent out seven years before. He soon
became far more hated by the Constantinopolitans than Zeno had
ever been; it is doubtful whether his arrogance, his financial
extortions, or his addiction to the Monophysite heresy made him
most detested. The army which he sent out against Zeno was
intrusted—very unwisely—to a general of Isaurian birth, the magister
militum Illus, who allowed himself to be moved by the prayers and
bribes of the legitimate emperor, and finally went over to him.
Having recovered all Asia Minor, Zeno then stirred up in Europe
Theodoric the Amal against his rival, and induced the Goth to beset
Constantinople from the West, while he himself blockaded it on the
Eastern side. The town threw open its gates, and Basiliscus, after a
reign of twenty months, was dragged from sanctuary and brought
before his nephew’s tribunal. Zeno promised him that his blood
should not be shed, but sent him and his sons to a desolate castle in
Cappadocia among the mountain-snows, where they were given
such scanty food and raiment in their solitary confinement, that ere
long they died of privation (477).
It was just after his triumph over Basiliscus that Zeno received the
ambassadors of Odoacer, and was saluted as Emperor of West and
East alike, in spite of his advice to the Romans to take back as their
Caesar their old ruler, Julius Nepos, who was still in possession of
part of Dalmatia, though he had lost Italy three years before.
Perhaps Zeno might have been tempted to interfere with something
more than advice in the affairs of the West, if his second batch of
troubles had not fallen upon him, in the form of his long Gothic war
with the two Theodorics—the sons of Theodemir and Triarius—which
began in the year following his restoration.

THE EASTERN EMPERORS, 457-518.

The Ostrogoths had never gone westward, like their kinsmen the
Visigoths. They had lingered on the Danube, first as members of the
vast empire of Attila the Hun, then as occupying Pannonia in their
own right. But, in the reign of Leo I., they had moved across the
Save into the territory of the Eastern Emperors, and had
permanently established themselves in Moesia. There they had
settled down and made terms with the Constantinopolitan
Government. But they were most unruly vassals, and, even in full
time of peace, could never be trusted to refrain from raids into
Early life Thrace and Macedonia. The main body of their tribe
of now acknowledged as its chief Theodoric the son of
Theodoric. Theodemir, the representative of the heaven-born
race of the Amals, the kings of the Goths from time immemorial.
Theodoric was now a young man of twenty-three, stirring and
ambitious, who had already won a great military reputation by
victories over the Bulgarians, the Sarmatians, and other tribes who
dwelt across the Danube. He had spent ten years of his boyhood as
a hostage at Constantinople, where he had learnt only too well the
weak as well as the strong points of the East-Roman Empire. His
after-life showed that he had there imbibed a deep respect for
Roman law, order, and administrative unity; but he had also come to
entertain a contempt for the timid Zeno, and a conviction that his
bold tribesmen were more than a match for the motley mercenary
army of the emperor, of which so large a proportion was still
composed of Goths and other Teutons, who could not be trusted to
fight with a good heart against their Ostrogothic kinsmen.
But Theodoric the Amal was not the only chief of his race in the
Balkan peninsula. He had a namesake, Theodoric the son of Triarius,
better known as Theodoric the One-eyed, who had long served as a
mercenary captain in the imperial army, and had headed the
Teutonic auxiliaries in the camp of the usurper Basiliscus. When
Basiliscus fell, Theodoric the One-eyed collected the wrecks of the
rebel forces, strengthened them with broken bands of various races,
many of whom were Ostrogoths, and kept the field against Zeno. He
retired into the Balkans, and occasionally descended to ravage the
Thracian plains; but meanwhile he sent an embassy to Zeno,
offering to submit if he were given the title of magister militum,
which he had held under Basiliscus, and taken with all his army into
the imperial pay.
Zeno indignantly refused to entertain such terms, and resolved to
The two take in hand the destruction of the rebel. He sent an
Theodorics. Asiatic army into Thrace to beset the son of Triarius
from the south, and bade his warlike vassal the son of
Theodemir to attack his namesake from the north, on the Moesian
side. The younger Theodoric eagerly consented, for he grudged to
see any other Gothic chief than himself powerful in the peninsula,
and looked down on the son of Triarius as a low-born upstart,
because he did not come like himself from the royal blood of the
Amals.[4]
4. By his name (Triarius) the father of Theodoric the One-eyed must have been
a Roman or a Romanised Goth, but the One-eyed had himself married a wife
who was close akin to Theodoric the Amal, for his son Recitach is called the
Amal’s cousin.
The campaign against Theodoric the One-eyed turned out
disastrously for the imperial forces. The Roman army in the south
missed the track of the rebel, whether by accident or design, while
Theodoric the Amal with his forces got entangled in the defiles of
the Balkans, and surrounded by the army of his rival. He had been
promised the co-operation of the army of Thrace, but no Romans
appeared, and his projects began to look dark. His one-eyed rival,
riding to within earshot of his camp, taunted him with his folly in
listening to the orders and promises of the emperor. ‘Madman,’ he
cried, ‘betrayer of your own race, do you not see that the Roman
plan is always to destroy Goths by Goths? Whichever of us falls,
they, not we, will be the stronger. They never will give you real help,
but send you out against me to perish here in the desert.’ Then all
the warriors of the Amal shouted that the One-eyed was right, and
that they would not fight against their brethren in the other camp.
The son of Theodemir bowed to their will and joined himself to the
son of Triarius. Uniting their armies, they moved down into the valley
of the Hebrus, and advanced toward Constantinople. They sent Zeno
an ultimatum, in which the Amal demanded more territory for his
tribe, and a supply of corn and money, while the One-eyed
stipulated for the post of magister militum, and an annual payment
of 2000 pounds of gold. Zeno, who was very anxious to keep the
younger Theodoric on his side, proffered him a great sum of money,
and the hand of the daughter of the patrician Olybrius, if he would
abandon his namesake the rebel. But the Amal refused to break the
oath that he had sworn to his ally, and marched westward to ravage
Macedonia up to the very gates of Thessalonica. Zeno sent his
troops into winter-quarters, as the season was late, and made one
final attempt to stave off the impending danger by offering terms to
Theodoric the One-eyed. Less true to his word than the Amal, the
elder Theodoric listened to the emperor’s offer, and, on being
promised the title of magister militum and all the revenues that he
had enjoyed under Basiliscus, led his troops over into the imperial
camp (479).
For the next two years the son of Theodemir ranged over the
whole Balkan peninsula from Dyrrhachium to the gates of
Constantinople, plundering and burning those parts of Macedonia
and Thrace which had hitherto escaped the ravages of the Huns of
Wars of Attila and the Ostrogoths of the previous generation.
Zeno and The generals of Zeno met with little good fortune in
Theodoric their attempts to check him, the only success they
the Amal.
obtained being a victory won by a certain Sabinianus
in 480, who cut off the rear-guard of Theodoric as it was crossing
the Albanian mountains, and captured 2000 waggons and 5000
Gothic warriors. But Sabinianus made himself too much feared by
Zeno, who, on a suspicion of treachery, had him executed in the
following year. It was not till 483 that the Amal, having wasted
Thrace and Macedon so fiercely that even his own army could no
longer find food, at last came to terms with Zeno, on being made
magister militum, and granted additional lands in Moesia and Dacia
for his tribesmen. The son of Triarius had died a year earlier: he had
again burst out into insurrection against the emperor, and was
mustering an army on the Thracian coast when he was slain in a
strange manner. A restive horse threw him against a spear which
was standing by the door of his tent, and he was pierced to the
heart. His son Recitach continued his rebellion, but Theodoric the
Amal, who wished to see no other Gothic chief but himself in the
Balkan peninsula, slew the young man, and incorporated his warriors
with the main body of the Ostrogoths.
The utter helplessness which Zeno showed in dealing with the two
Theodorics may be attributed in a large measure to his troubles at
home. In 479, the year when he had failed to support Theodoric the
Amal in the Balkans, his throne had nearly been overturned by a
rising in Constantinople. Marcianus and Procopius, the two sons of
Anthemius, the late emperor of the West, who were popular with the
citizens of the capital, formed a plot for overthrowing the emperor, in
which they enlisted many men of importance. They surprised the
palace and massacred the body-guard, but Zeno escaped, brought
over his faithful Isaurians from Asia, and crushed the rebellion after
a vigorous street fight. In 482-3 he had a prolonged
misunderstanding with his commander-in-chief Illus, the Isaurian
general who had put down the rebellion of Basiliscus five years
before. Zeno neither banished nor fully trusted him. He left him in
office, but was nervously on his guard, and always thwarting his
Minister. It is said that, with or without his consent, the Empress
Ariadne endeavoured to procure the assassination of Illus.
In 483, the year in which Theodoric the Amal made his peace with
Zeno, a certain Leontius raised a rebellion in Syria. Illus, who was
sent to put him down, had grown tired of serving his suspicious and
Revolt of ungrateful master, and joined in the revolt. He and
Leontius, Leontius seized Antioch, where the latter was
483. proclaimed emperor, and got possession of
Cappadocia, Cilicia, and north Syria. It is said that they designed to
re-establish paganism, a project which seems absolutely incredible in
the very end of the fifth century, when the heathen were no more
than a forlorn remnant scattered among a zealous Christian
population. The empress-dowager Verina, who was living in exile in
Cappadocia, joined herself to them, and adopted Leontius as her
son. But the rebels took more practical measures to support their
cause when they applied for aid to Odoacer the king in Italy, and to
the Persian monarch Balas. Both promised aid, but, before they
could send it, Zeno had put the rebellion down. He induced his late
enemy Theodoric to join his army, and the Goths and Isaurians
combined easily got the better of Leontius. Syria submitted, and the
rebel emperor and Illus, after a long and desperate defence in a
castle in Cappadocia, were taken and slain.[5]
5. This fort—it was called Castellum Papirii—is said to have held out for the
incredibly long period of four years after all the rest of the rebellious districts
had been subdued, and only to have fallen by treachery.
Zeno enjoyed comparative peace after Leontius’ rebellion had
been crushed, and was still more fortunate when, in 488, he induced
Theodoric the Amal to move his Ostrogoths out of Moesia and go
forth to conquer Italy. How Theodoric fared in Italy we have already
related. His departure was of enormous benefit to the empire, and,
for the first time since his accession, Zeno was now able to exercise
a real authority over his European provinces. They were left to him
in a most fearful state of desolation: ten years of war, ranging over
the whole tract south of the Danube and north of Mount Olympus,
had reduced the land to a wilderness. Whole districts were stripped
bare of their inhabitants, and great gaps of waste territory were
inviting new enemies to enter the Balkan peninsula, and occupy the
deserted country-side. North of the Balkans the whole provincial
population seems to have been well-nigh exterminated. When the
State of Ostrogoths abandoned the country there was nothing
the Balkan left between the mountains and the Danube but a few
peninsula.
military posts and their garrisons, nor was the country
replenished with inhabitants till the Slavs spread over the land in the
succeeding age. Illyria and Macedonia had not fared so badly, but
the net result of the century of Gothic occupation in the Balkan
peninsula had been to thin down to a fearful extent the Latin-
speaking population of the Eastern Empire. All the inland of Thrace,
Moesia, and Illyricum had hitherto employed the Latin tongue: with
the thinning out of its inhabitants the empire became far more
Asiatic and Greek than it had before been.
 When the Ostrogoths migrated to Italy, the empire acquired a new
set of neighbours on its northern frontier, the nomad Ugrian horde of
the Bulgarians on the lower Danube, and the Teutonic tribes of the
Gepidae, Heruli, and Lombards on the middle Danube and the Theiss
and Save. Contrary to what might have been expected, none of
these races pushed past the barrier of Roman forts along the river to
occupy Moesia. They vexed the empire with nothing worse than
occasional raids, and did not come to settle within its limits.
Zeno’s ecclesiastical policy demands a word of notice. He was
himself orthodox, but not fanatical: the Church being at the moment
grievously divided by the Monophysite schism, to which the
Churches of Egypt and Palestine had attached themselves, he
thought it would be possible and expedient to lure the heretics back
within the fold by slightly modifying the Catholic statement of
doctrine. In 482, though he was in the midst of his struggle with
Theodoric the Amal, he found time to draft his ‘Henoticon,’ or Edict
of Comprehension. The Monophysites held that there was but one
nature in our Lord, as opposed to the orthodox view, that both the
human and the divine element were fully present in His person. Zeno
Zeno’s put into his ‘Henoticon’ a distinct statement that Christ
Henoticon. was both God and man, but did not insert the words
‘two natures,’ which formed the orthodox shibboleth.
But his well-meant scheme fell utterly flat. The heretics were not
satisfied, and refused to conform, while the Catholics held that it
was a weak concession to heterodoxy, and condemned Zeno for
playing with schism. The patriarch, Acacius, who had assisted him to
draft the ‘Henoticon,’ was excommunicated by the Bishop of Rome,
and the churches of Italy and Constantinople were out of
communion for more than thirty years, owing to an edict that had
been intended to unite and not to divide.
The last years of Zeno’s reign were far more undisturbed by war
and rebellion than its earlier part. He survived till 491, when he died
of epilepsy, leaving no heir to inherit his throne. He had had two
sons, named Leo and Zeno: the first had died, while still a child, in
474; the second killed himself by evil-living, when on the threshold
of manhood, long ere his father’s death.
The right of choosing Zeno’s successor fell nominally into the
hands of the Senate and people, really into those of the widowed
Empress Ariadne and the Imperial Guard. The daughter of Leo made
a wise choice in recommending to the suffrages of the army and
people Anastasius of Dyrrhachium, an officer of the silentiarii,[6] who
was universally esteemed for his piety and virtue.
6. A body-guard, whose duty it was to preserve silence around the emperor’s
private apartments.
Anastasius was a man of fifty-two or fifty-three, who had spent
most of his life in official work in the capital, and was specially well
known as an able and economical financier. He was sincerely
religious, and spent many of his leisure hours as a lay preacher in
the church of St. Sophia, till he was inhibited from giving instruction
by the Patriarch Euphemius, who detected Monophysitism in his
sermons. He had once proposed to take orders, and had been
spoken of as a candidate for the bishopric of Antioch. Yet, in spite of
his religious fervour, he was never accused of being unworldly or
Character unpractical. Anastasius was a man of blameless life,
of learned and laborious, slow to anger, a kind and
Anastasius.
liberal master, and absolutely just in all his dealings.
‘Reign as you have lived,’ was the cry of the people
when he first presented himself to them clad in the imperial purple.
Only two objections were ever made to him—the first, that he
leaned towards the Monophysite heresy; the second, that his court
was too staid and puritanical for the taste of the multitude, who had
loved the pomp and orgies of the dissolute Zeno. He earned
unpopularity by suppressing gladiatorial combats with wild beasts,
and licentious dances.
Six weeks after his accession the new emperor married the
Empress-Dowager Ariadne, who had been the chief instrument in his
election. She was a princess of blameless life, and had done much in
the previous reign to redeem the ill-repute of her first husband. It
was a great misfortune for the empire that she bore her second
spouse no heir to inherit his throne.
The commencement of the reign of Anastasius was troubled by a
rebellion of the Isaurians. Zeno had not only formed an Imperial
Guard of his countrymen, but had filled the civil service with them,
and encouraged them to settle as merchants and traders in
Constantinople. They had been much vexed when the sceptre
passed to the Illyrian Anastasius, and entered into a conspiracy to
seize his person, and proclaim Zeno’s brother, Longinus, as emperor.
A few months after his accession they rose in the capital and
obtained possession of part of the city near the palace, but the
majority of the people and army were against them, and they were
put down after a sharp street fight, in which the great Hippodrome
was burnt. Longinus was captured, and compelled to take orders. He
died long after as a priest in Egypt. Anastasius, after this riot,
dismissed all the Isaurian officers from the public service. They
returned to their homes in Asia Minor, and organised a rebellion in
their native hills. A second Longinus, who had been magister militum
in Thrace, put himself at the head of the insurrection, which lingered
on for five years (491-496), but was never a serious danger to the
Rebellions empire. The rebels were beaten whenever they
in Isauria, ventured into the plains, and only maintained
492-496. themselves so long by the aid of the mountain-castles
with which their rugged land was studded. In 496 their last
fastnesses were stormed, and their chief, the ex-magister, taken and
executed. Anastasius punished the communities which had been
most obstinate in the rebellion by transferring them to Thrace, and
settling them on the wasted lands under the Balkans, where he
trusted that these fearless mountaineers would prove an efficient
guard to keep the passes against the barbarians from beyond the
Danube.
The Asiatic provinces of the empire had no further troubles till
502, when a war broke out between Anastasius and Kobad king of
Persia. The Mesopotamian frontier had been singularly quiet for the
last century; there had been no serious war with the great Oriental
monarchy to the East since Julian’s unfortunate expedition in 362.
The same age which had seen the Teutonic migrations in Europe had
been marked in inner Asia by a great stirring of the Huns and other
Turanian tribes beyond the Caspian, and while the Roman emperors
had been busy on the Danube, the Sassanian kings had been hard
at work defending the frontier of the Oxus. In a respite from his
Eastern troubles Kobad made some demands for money on
Anastasius, which the emperor refused, and war soon followed. It
began with several disasters for the Romans, and Amida, the chief
War with fortress of Mesopotamia, was stormed in 503. Nisibis
Persia, fell later in the same year, and when Anastasius sent
503-505. reinforcements to the East he appointed so many
generals with independent authority that the whole Roman army
could never be united, and the commanders allowed themselves to
be taken in detail and defeated in succession. In 504, however, the
fortune of war turned, when the supreme authority in the field was
bestowed on Celer, the magister officiorum; he recovered Amida
after a long siege, and began to press forward beyond the Persian
frontier. Kobad was at the same time assailed by the Huns from
beyond the Oxus, and gladly made peace, on terms which restored
the frontier of both parties to the line it had occupied in 502.
Anastasius provided against future wars by building two new
fortresses of the first class on the Persian frontier, Daras in
Mesopotamia, and Theodosiopolis farther north on the borders of
Armenia. These places served to break the force of the Persian
attack thirty years later, when the successors of Kobad and
Anastasius again fell to blows. The Persian war, like the Isaurian, had
only afflicted a very limited district,—the province beyond the
Euphrates,—and no raids had penetrated so far as Syria. Indeed,
during the whole reign of Anastasius, the only serious trouble to
which the Asiatic half of the empire was exposed was a Hunnish raid
from beyond the Caucasus, which in 515 caused grave damage in
Pontus, Cappadocia, and Lycaonia. This invasion, however, was an
isolated misfortune, followed by no further incursions of the nomads
of the Northern Steppes.
 The European provinces—now as in the time of Zeno—had a far
harder lot. The Slavs and Bulgarians repeatedly crossed the Danube
and pressed over the desolated plains of Moesia to assail Thrace.
More than once the Bulgarians defeated a Roman army in the field,
and their ravages were at last pushed so far southward that
Anastasius built in 512 the celebrated wall which bears his name,
running from the Black Sea to Propontis, thirty-five miles west of
Constantinople. These lines, extending for more than fifty miles
across the eastern projection of Thrace, served to defend at least
the immediate neighbourhood of the capital against the restless
horsemen from beyond the Danube. Macedonia and Illyricum seem
to have suffered much less than Thrace during this period; the Slavs
who bordered on them were as yet not nearly such a dangerous
enemy as the Bulgarians, while the Ostrogoths of Italy, on
reconquering Pannonia, proved more restful neighbours to the
north-western provinces of the empire than they had been in the
previous century.
It was in the reign of Anastasius that one of the most
characteristic features in the social life of Constantinople is brought
forward into prominence for the first time. This was the growing
turbulence of the ‘Blues and Greens,’ the factions of the Circus. From
the very beginning of the Roman Empire these clubs had existed,
but it was only at Constantinople that they became institutions of
high political importance. There the rivalry of the Blues and Greens
was not confined to the races of the Circus, but was carried into
every sphere of life. Nor was it any longer only the young men of
sporting and fashionable proclivities that joined the ‘factions.’ They
served as clubs or political associations for all classes, from the
ministers of state down to the poorest mechanics, and formed bonds
of union between bodies of churchmen or supporters of dynastic
The Blues claims. It is hard for an Englishman to realise this
and extraordinary development of what had once been a
Greens. mere rivalry of the Hippodrome. To make a parallel to
it we should have to suppose that all who mount the light or the
dark blue on the day of the Oxford and Cambridge boat race were
bitterly jealous of each other—let us say, for example, that all Dark
Blues were Conservatives and Anglicans, and all Light Blues were
Radicals and Dissenters. If this were so, we can imagine that in
times of political stress every boat race might be followed by a
gigantic free-fight. This, however, was exactly what occurred at
Constantinople; the ‘Blue’ faction had become identified with
Orthodoxy, and with a dislike for the family of Anastasius. The
‘Green’ faction included all the Monophysites and other heterodox
sects, and was devoted to the person and dynasty of Anastasius. In
any time of trouble the celebration of games in the Hippodrome
ended with a fierce riot of the two factions. No wonder that the just
and peaceable emperor strove to suppress shows of all sorts, and in
especial showed a dislike for the disloyal ‘Blue’ faction.
The worst of Anastasius’ domestic troubles were due to the
suspicion of heterodoxy that clung to him. In 511 when he added to
the hymn called the Trisagion the line ὁ σταυρωθεὶς δι’ ἡμᾶς in a
context which seemed to refer to the whole Trinity, the orthodox
populace of Constantinople headed by the Blue faction burst out into
sedition. It was only quelled by the old Emperor presenting himself
before the people in the Hippodrome, without crown or robe, and
announcing his intention of abdicating. So great was the confidence
which his justice and moderation had inspired in all ranks and
classes, that the proposal filled the whole multitude with dismay, and
they rose unanimously to bid him resume his diadem.
But the grievance against the Monophysite tendencies of
Rebellion Anastasius was not destined to be forgotten. In 514
of Vitalian, an ambitious general named Vitalian, who held a
514. command in Moesia, rose in arms, alleging as the
cause of his rebellion, not only certain misdeeds committed in that
province by the emperor’s nephew Hypatius, but also the dangerous
heterodoxy of Anastasius’ religious opinions. When Hypatius was
removed from his office the greater part of Vitalian’s army returned
to its allegiance, and the rebel then showed how much importance
was to be attached to his religious scruples, by calling in the heathen
Bulgarians and Huns to his aid. At the head of an army composed of
these barbarians he maintained himself in Moesia for some time.
The emperor, somewhat unwisely, replaced his nephew Hypatius in
command, and sent him with a large army to put down the rebel;
but, while the Romans lay encamped on the sea-shore near Varna,
they were surprised by a night attack of the enemy and completely
scattered. Many thousand men were driven over the cliffs into the
sea and crushed or drowned, while Hypatius himself was taken
prisoner (514). The old emperor was driven, by concern for his
nephew’s life, to make peace. He ransomed Hypatius for 15,000 lbs.
of gold, and granted Vitalian the post of magister militum in Thrace.
The pardoned rebel for the remainder of Anastasius’ reign occupied
himself in strengthening his position on the Danube, being
determined to make a bold stroke for the imperial throne when old
age should remove the octogenarian ruler of Constantinople.
In spite of all his troubles with the two Longini, king Kobad and
Vitalian, Anastasius may be called a successful and prosperous ruler.
All these rebellions had been of mere local import, and for the whole
twenty-seven years of his reign the greater part of the empire had
enjoyed peace and plenty. The best testimony to his good
administration is the fact that at his accession he found the treasury
emptied by the wasteful Zeno, and that at his death he left it filled
with 320,000 lbs. weight of gold, or £15,000,000 in hard cash. This
was in spite of the fact that he was a merciful and lenient
administrator, and had actually abolished several imposts including
the odious Chrysargyron or income-tax. Nor was the money
collected at the cost of neglecting proper expenditure. Anastasius
had erected many military works,—in especial his great wall in
Thrace, and the strong fortress of Daras—and restored many ruined
cities. ‘He never sent away petitioners empty, whether they
represented a city, a fortress, or a seaport.’ He left an army of
150,000 men in a good state of discipline and composed for its
larger half of native troops, with a frontier intact alike on east and
west and north.
The good old man died in 518; his wife Ariadne had preceded him
to the grave three years before. He had refrained from appointing as
his colleague his nephew Hypatius, whom many had expected him to
adopt, and the empire was left absolutely masterless. The great
State officials, the Imperial Guard, and the Senate had the election
of a new Caesar thrown upon their hands. The most obvious
candidates for the throne were Hypatius, whom the Green faction
should have supported, and the magister militum Vitalian, who at
once took arms to march on the capital. But neither of them was
destined to succeed. The sinews of war lay in the hands of the
treasurer Amantius; he himself could not hope to reign, for he was a
eunuch, but he had a friend whom he wished to crown. Accordingly
Accession he sent for Justinus, the commander of the Imperial
of Justin I., Guard, and made over to him a great sum to buy the
518. aid of the soldiery. Justinus, an elderly and
respectable personage whom no one suspected of ambition, quietly
took the gold, distributed it in his own name, and was saluted as
Augustus by his delighted guardsmen. The Senate acquiesced in the
nomination, and he mounted the throne without a blow being
struck.
Justinus was an Illyrian by birth, and had spent fifty years in the
imperial army; he had won his promotion by good service in the
Isaurian and Persian wars. He was very illiterate—we are told that
he could barely sign his own name—and knew nothing outside his
tactics and his drill-book. He had the reputation of being quiet, well-
behaved, and upright; no one had anything to say against him, and
he was rigidly orthodox in matters of faith. He was sixty-eight years
of age, fifteen years older than even the elderly Anastasius had been
at the moment of his accession.
Justinus seated himself firmly on the throne; he executed the
treasurer Amantius, but made terms with the two men who might
have been his rivals. Hypatius remained a simple senator; Vitalian
was confirmed in his command in Moesia and given a consulship.
While holding this office and dwelling in the capital he was
assassinated; rumour ascribed the crime to the emperor’s nephew
Justinian, who thought the turbulent magister too near the throne.
 There is very little to record of the nine years of Justinus’ reign,
save that he healed the forty-years’ schism which had separated the
churches of Rome and Constantinople since the publication of Zeno’s
‘Henoticon.’ Being undisputedly orthodox, he withdrew that
document, and the schism disappeared with its cause. The only real
importance of Justinus is that he prepared the way for his famous
nephew and successor, Justinian, whom he adopted as colleague,
and intrusted with those matters of civil administration with which
he was himself incompetent to deal. He died and left the throne to
Justinian in A.D. 528.
 CHAPTER IV

CHLODOVECH AND THE FRANKS IN GAUL

481-511

The Franks in Northern Gaul—Their early conquests—State of Gaul in 481—

Chlodovech conquers Northern Gaul, 486—He subdues the Alamanni, 495-6—
Conversion of Chlodovech, 496—He conquers Aquitaine from the Visigoths,
507—He unites all the Frankish Kingdoms, 511.

While Odoacer was still reigning in Italy, and Theodoric the Amal had
not yet left the Balkans, or the banks of the Danube, the foundations
of a great kingdom were being laid upon the Scheldt and the Meuse.
Early in the fifth century the confederacy of marsh-tribes on the
Yssel and Lech who had taken the common name of Franks, had
moved southward into the territory of the Empire, and found
themselves new homes in the provinces which the Romans called
Belgica and Germania Inferior. For many years the hold of the
legions on this land had been growing weaker; and, long ere it
became a Frankish kingdom, it had been largely sprinkled with
Frankish colonists, whom the emperors had admitted as military
settlers on the waste lands within their border. In the lowlands of
Toxandria, which after-ages called Brabant and Guelders, there were
no large cities to be protected, no great fortresses to be maintained,
and, while the Romans still exerted themselves to hold Treveri and
Colonia Agrippina and Moguntiacum,[7] they allowed the plains more
to the north and west to slip out of their hands. By the second
 The Franks quarter of the fifth century the Franks were firmly
in Lower established on the Scheldt and Meuse and lower
Germany. Rhine, where the Roman garrisons never reappeared
after the usurper Constantine had carried off the northern frontier
legions to aid him in his attack on Italy (406). By this time, too,
Colonia Agrippina, first of the great Roman cities of the Rhineland,
seems to have already fallen into the hands of the Franks. Between
430 and 450 they continued to push forward as far as the Somme
and the Moselle, and when, at the time of Attila’s great invasion of
Gaul, the last imperial garrisons in the Rhineland were exterminated,
and the last governors driven forth by the Huns from Treveri and
Moguntiacum and Mettis, it was the Franks who profited. After the
Huns had rolled back again to the East, Frankish kings, not Roman
officials, took possession of the ravaged land along the Moselle and
Rhine, and the surviving provincials had for the future to obey a
Teutonic master near home, not a governor despatched from distant
Ravenna.
7. Trier, Köln, and Mainz.
The Franks were now divided into two main hordes; the Salians—
who took their name from Sala, the old name of the river Yssel—
dwelt from the Scheldt-mouth to the Somme, and from the Straits of
Dover to the Meuse. The Ripuarians, whose name is drawn from the
fact that they inhabited the bank (ripa) of the Rhine, lay along both
sides of the great river from its junction with the Lippe to its junction
with the Lahn, and extended as far east as the Meuse. Each of these
two tribes was ruled by many kings, all of whom claimed to descend
from the house of the Merovings, a line lost in obscurity, whose
original head may, perhaps, have been the chief who in the third
century first taught union to the various tribes who formed the
Frankish confederacy.
The Franks were one of the more backward of the Teutonic races,
in spite of their long contact with Roman civilisation along the Rhine.
Kings and people were still heathens. They had not learnt like the
Goths to wear armour or fight on horseback, but went to war half-
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