PREFACE

The present volume may be considerd a sequel to the
() author’s Elements of the Theory of Integers. However this
= volume is largely independent of that work.
__In this work the author has incorporated a number of his
~ discoveries, publisht here for the first time, some of which he
=~ has indicated by their dates of discovery. For one of these,
2§ 283, he has described in § 387 the method by which it was
s discoverd. He believes that such notes are sometimes inter-
esting and valuable to the student. Of course the author is
aware that some of these discoveries of his may have been
found before, by others, possibly in better form.

Attention is called particularly to the thoroness with which
the method of mathematical induction is illustrated; to the
two formulas (corresponding to » = 1 and » —= 1 respectivly)
for the sum of the first #» terms of a geometric progression
(§ 34), the text-books generally disregarding the case when
7 = 1; to the remark (§ 36) about the impossibility of finding
the nth term of a series when only a finite number of terms
at the beginning are given; to the formula for the product of
the first #» terms of a geometric progression (§ 41); the large -
number of tabular forms in the book; the proof (§ 101) of the
radix theorem (§ 102); the definition of a quantic (§ 104); the
completeness of illustration of change of the scale of notation
(§ 123); the discussion as to the best radix (§§ 137-141); the
discussion of the proposals for a fixed calendar (§§ 149-151);
the generalization of the algorithm of Euclid in § 226; the

o tables of §§ 241, 255; the complete discussion of radix systems

: and congruences and the application of congruences to check-
*, ing arithmetical work (§§ 310-396), especially the rules for
. congruences to the moduli »” (§ 314), r» — 1 (§ 341), ™ + 1
o (8§ 364-366), of which many other rules are simple conse-
1+ quences, as for example the rules for divisibility by 3, 7, 9,
< 11, 13 in the decimal system; the complete treatment of inde-

3B

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