C.F.高斯著的《算術探究(英文版)》主要由七部
分組成：第一部分同餘數基本介紹，第二部分一次同
餘式，第三部分冪的乘餘，第四部分二次同餘數。第
五部分型和二次不定方程。第六部分是對之前討論的
各種應用介紹。第七部分定義圓截面方程。讀者對像
：從事理論學習的研究生和數學工作者。

Translator's Preface
Bibliographical Abbreviations
Dedication
Author's Preface
Section I. Congruent Numbers in General
Congruent numbers, moduli, residues, and nonresidues,
art. 1 ft.
Least residues, art. 4
Elementary propositions regarding congruences, art. 5
Certain applications, art. 12
Section II. Congruences of the First Degree
Preliminary theorems regarding prime numbers, factors, etc.,
art. 13
Solution of congruences of the first degree, art. 26
The method of finding a number congruent to given residues
relative to given moduli, art. 32
Linear congruences with several unknowns, art. 37
Various theorems, art. 38
Section III. Residues of Powers
The residues of the terms of a geometric progression which
begins with unity constitute a periodic series, art. 45
If the modulus = p (a prime number), the number of terms in
its period is a divisor of the number p - 1, art. 49
Fermat's theorem, art, 50
How many numbers correspond to a period in which the
number of terms is a given divisor of p - 1, art. 52
Primitive roots, bases, indices, art. 57
Computation with indices, art. 58
Roots of the congruence x" = A, art. 60
Connection between indices in different systems, art. 69
Bases adapted to special purposes, art. 72
Method of finding primitive roots, art. 73
Various theorems concerning periods and primitive roots, art. 75
A theorem of Wilson, art. 76
Moduli which are powers of prime numbers, art. 82
Moduli which are powers of the number 2, art. 90
Moduli composed of more than one prime number, art. 92
Section IV. Congruences of the Second Degree
Quadratic residues and nonresidues, art. 94
Whenever the modulus is a prime number, the number of
residues less than the modulus is equal to the number of
nonresidues, art. 96
The question whether a composite number is a residue or
nonresidue of a given prime number depends on the nature
of the factors, art. 98
Moduli which are composite numbers, art. 100
A general criterion whether a given number is a residue or a
nonresidue of a given prime number, art. 106
The investigation of prime numbers whose residues or non-residues are given numbers, art. 107
The residue - 1, art. 108
The residues + 2 and - 2, art. 112
The residues + 3 and - 3, art. 117
The residues +5 and -5, art. 121
The residues +7and -7, art. 124
Preparation for the general investigation, art. 125
By induction we support a general (fundamental) theorem
and draw conclusions from it, art. 130
A rigorous demonstration of the fundamental theorem,
art. 135
An analogous method of demonstrating the theorem of
art. 114, art. 145
Solution of the general problem, art. 146
Linear forms containing all prime numbers for which a given
number is a residue or nonresidue, art. 147
The work of other mathematicians concerning these in-
vestigations, art. 151
Nonpure congruences of the second degree, art. 152
Section V. Forms and Indeterminate Equations of the Second Degree
Plan of our investigation ; definition of forms and their notation,
art. 153
Representation of a number; the determinant, art. 154
Values of the expression (b2- ac) (mod. M) to which
belongs a representation of the number M by the form
(a, b, c), art. 155
One form implying another or contained in it; proper and
improper transformation, art. 157
Proper and improper equivalence, art. 158
Opposite forms, art. 159
Neighboring forms, art. 160
Common divisors of the coefficients of forms, art. 161
The connection between all similar transformations of a
given form into another given form, art. 162
Ambiguous forms, art. 163
Theorem concerning the case where one form is contained in
another both properly and improperly, art. 164
General considerations concerning representations of num-
bers by forms and their connection with transformations,
art. 166
Forms with a negative determinant, art. 171
Special applications for decomposing a number into two
squares, into a square and twice a square, into a square
and three times a square, art. 182
Forms with positive nonsquare determinant, art. 183
Forms with square determinant, art. 206
Forms contained in other forms to which, however, they are
not equivalent, art. 213
Forms with 0 determinant, art. 215
The general solution by integers of indeterminate equations
of the second degree with two unknowns, art. 216
Historical notes, art. 222
Distribution of forms with a given determinant into classes,
art. 223
Distribution of classes into orders, art. 226
The partition of orders into genera, art. 228
The composition of forms, art. 234
The composition of orders, art. 245
The composition of genera, art. 246
The composition of classes, art. 249
For a given determinant there are the same number of classes
in every genus of the same order, art. 252
Comparison of the number of classes contained in individual
genera of different orders, art. 253
The number of ambiguous classes, art. 257
Half of all the characters assignable for a given determinant
cannot belong to any properly primitive genus, art. 261
A second demonstration of the fundamental theorem and the
other theorems pertaining to the residues -1, +2, -2,
art. 262
A further investigation of that half of the characters which
cannot correspond to any genus, art. 263
A special method of decomposing prime numbers into two
squares, art. 265
A digression containing a treatment of ternary forms,
art. 266 ff.
Some applications to the theory of binary forms, art. 286 IT.
How to find a form from whose duplication we get a given
binary form of a principal genus, art. 286
Except for those characters for which art. 263, 264 showed it
was impossible, all others will belong to some genus,
art. 287
The theory of the decomposition of numbers and binary
forms into three squares, art. 288
Demonstration of the theorems of Fermat which state that
any integer can be decomposed into three triangular numbers
or four squares, art. 293
Solution of the equation ax2 + by2 + cz2 = 0, art. 294
The method by which the illustrious Legendre treated the
fundamental theorem, art. 296
The representation of zero by ternary forms, art. 299
General solution by rational quantities of indeterminate
equations of the second degree in two unknowns, art. 300
The average number of genera, art. 301
The average number of classes, art. 302
A special algorithm for properly primitive classes; regular
and irregular determinants etc., art. 305
Section VI. Various Applications of the Preceding Discussions
The resolution of fractions into simpler ones, art. 309
The conversion of common fractions into decimals, art. 312
Solution of the congruence x2 = A by the method of exclusion, art. 319
Solution of the indeterminate equation mx2 + ny2 = A by
exclusions, art. 323
Another method of solving the congruence x2 - A for the
case where ,4 is negative, art. 327
Two methods for distinguishing composite numbers from
primes and for determining their factors, art. 329
Section VII. Equations Defining Sections of a Circle
The discussion is reduced to the simplest case in which the
number of parts into which the circle is cut is a prime
number, art. 336
Equations for trigonometric functions of arcs which are a
part or parts of the whole circumference; reduction of
trigonometric functions to the roots of the equation
xn - 1 = 0, art. 337
Theory of the roots of the'equation x" - I = 0 (where n
is assumed to be prime), art. 341 ft.
Except for the root 1, the remaining roots contained in (Ω)
are included in the equation X = xn-1 + xn-2 + etc.
+ x + 1 = 0; the function X cannot be decomposed into
factors in which all the coefficients are rational, art. 341
Declaration of the purpose of the following discussions,
art. 342
All the roots in (fl) are distributed into certain classes
(periods), art. 343
Various theorems concerning these periods, art. 344
The solution of the equation X = 0 as evolved from the
preceding discussions, art. 352
Examples for n = 19 where the operation is reduced to the
solution of two cubic and one quadratic equation, and
for n = 17 where the operation is reduced to the solution of
four quadratic equations, art. 353, 354
Further discussions concerning periods of roots, art. 355 ft.
Sums having an even number of terms are real quantities,
art. 355
The equation defining the distribution of the roots (Ω) into
two periods, art. 356
Demonstration of a theorem mentioned in Section IV,
art. 357
The equation for distributing the roots (Ω) into three periods,
art. 358
Reduction to pure equations of the equations by which the
roots (Ω) are found, art. 359
Application of the preceding tO trigonometric functions,
art. 361 ft.
Method of finding the angles corresponding to the individual
roots of (Ω), art. 361
Derivation of tangents, cotangents, secants, and cosecants
from sines and cosines without division, art. 362
Method of successively reducing the equations for trigonometric functions, art. 363
Sections of the circle which can be effected by means of
quadratic equations or by geometric constructions, art. 365
Additional Notes
Tables
Gauss' Handwritten Notes
List of Special Symbols
Directory of Terms