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8 INTRODUCTION

instances to lay emphasis upon the fact that the various
branches are in fact closely allied.

Number theory is the branch of mathematics which, be-
sides being the father of all mathematics, is also that nearest
to a non-technical public. This does not mean that the sub-
ject is necessarily of an elementary nature. Many of its more
advanced problems involve the use of extremely complicated
processes of logic, whilst some celebrated problems are still
no nearer to any solution whatever. Nevertheless, the greater
part of this book is confined to the less abstruse aspects of the
subject, and an attempt has been made to render all points
under consideration readily intelligible. Where proofs are
given or where algebraical notation is employed in any other
connexion, the notation has been kept to as simple a form as
possible.

In ancient times the philosophers ascribed mystical powers
to individual numbers, because of the apparently magical
quality observable in their construction and relationships.
Although these beliefs may have been swept away over the
centuries, the relationships upon which they were based still
exist and they well repay a little investigation.