An EQUATION-The First Member-The Second Member, (107).-For
what purposes Equations are employed-How applied to the Solution of
Questions, (108).-The Solution of an Equation-Verification of the Value
* found for the Unknown Quantity, (109).—A Simple Equation-A Quadratic
Equation-A Cubic Equation, (110).—A Numerical Equation—A Literal
Equation-An Identical Equation, (111).—Transformation of an Equation,
(112).—An Axiom-Axiom first, second, &c., (113).—How the Value of the
Unknown Quantity is found-Transformations necessary, (114).—How to
clear an equation of Fractions-How by means of Least Common Multiple
-Advantage of this Method, (115).-How any term may be Transposed from
one Side of an Equation to the other, (116).—Change of the Signs in an
Equation, (117).
RULE XVII. For the Solution of a Simple Equation containing but one
unknown quantity, (118.)
A Problem, and in what its Solution consists-General Method of form-
ing the Equation of a Problem, (119).—Solution of Problems with Two or
more Unknown Quantities-Independent Equations, (120).—General Method
of solving Two Equations, (121).-Elimination by Addition or Subtraction,
(122).—Elimination by Substitution, (123).—Elimination by Comparison,
(124).—Solution of Three Equations-Of Four Equations, (125).—Of
Problems in which there are Three or more Required Quantities, (126).
RATIO. PROPORTION.-VARIATION.-91.... 110.
Ratio of one Quantity to another, (127).-Sign of Ratio, (128).-How
the value of a Ratio may be represented, (129).—Direct and Inverse Ratio,
(130).-Compound Ratio, (131).—Ratio of the first to the last of any Num-
ber of Quantities, (132).-Duplicate and Triplicate Ratios, (133).-Equi-
multiples and Equisubmultiples, (134).-Ratio of Equimultiples and Equi-
submultiples, (135).-PROPORTION, (136).-Four Quantities in Proportion,
(137).—Three quantities in Proportion, (138).-Direct and Inverse Propor-
tion, (139).-Sign of Proportion, (140).-Inverse Converted into Direct Pro
portion, (141).-VARIATION-Variation direct-Variation inverse, (142).—
Product of Two Quantities varying inversely with each other, (143).—Varia-
tion, an Abbreviated Proportion, (144).—A Theorem-A Corollary, (145).—
Ratio of two Fractions having a common Term, (146).—How the value of a
Fraction varies, (147).—Corresponding Equalties and Inequalities between
the Antecedents and Consequents of a Proportion, (148).—A Proportion con-
verted into an Equation, (149).—A Fourth Proportional, how found, (150).
-Product of the Extremes when Three Quantities are in Proportion, (151).
-Mean Proportional, how found, (152).—An Equation converted into a Pro-
portion, (153).-Ratio of the first to the third of Three Proportional Quanti-
ties, (154)-Proportion by Inversion, (155).-Proportion by Alternation,
(156). What Multiplications may be made in a Proportion, (157).—What
Divisions may be made in a Proportion, (158).-Proportion by Composition,
(159).-Proportion by Division, (160).-Proportion between the Sum of two
or more Antecedents and that of their Consequents, (161).-A Proportion
derived from Two other Proportions in which there are common Terms,
(162).-Proportion between the Sums and Differences of the Antecedents
and Consequents, (163).-Products of the Corresponding Terms of Two or
more Proportions, (164).-Proportion between the Powers or Roots of Pro-
portional Quantities, (165).—Substitution of Factors in a Proportion, (166).
-General Solution of a Problem, (167).—Two Numbers found from their
Sum and Difference, (168).-An Algebraic Formula, (169).-Of a Propor-
tion occurring in the Solution of a Problem, (170).—Percentage-Ratio of
Percentage-Basis of Percentage, (171).—Amount of Percentage, how found,
(172).—Interest—The Principal-The Amount, (173).—Amount of Interest,
how found, (174).
ARITHMETICAL, HARMONICAL, AND GEOMETRICAL PROGRESSION.-111...120.
An Arithmetical PROGRESSION, (175).—Last Term of an Arithmetical
Progression, equal to what, (176).—Common Difference of the Terms, (177).
-Sum of the two Extremes, (178).—Arithmetical Mean, (179). Sum of
all the Terms, (180).—Formulas in Arithmetical Progression, (181).—An.
Harmonical PROGRESSION, (182).-An Harmonical converted into an Arith-
metical Progression, (183).-Harmonical Mean, (184).—A Geometerical PRO-
GRESSION, (185).—Last Term of a Geomitrical Progression, (186).-Power
of the ratio found from the First and last Terms, (187).—Product of the two
Extremes, (188).-Geometrical Mean, (189).-Sum of all the Terms, (190)
-Sum of an infinite number of Decreasing Terms, (191).-Formulas in Ge
ometrical Progression, (192).
PERMUTATIONS AND COMBINATIONS.-INVOLUTION.-BINOMIAL.-THEOREM.
-EVOLUTION.-121.... 146.
PERMUTATIONS, (193).-Number of Permutations, how found, (194).—
COMBINATIONS, (195).-Number of Combinations, how found, (196).-INVO-
LUTION, (197).—A Higher Power found from lower Powers of the same Quan-
tity, (198.)-Powers of unity, (199).-Powers of Monomials, how found,
(200).-Powers of Fractions, how found-Powers of a mixed Quantity, (201).
Sign to be Prefixed to a Power, (202).—Powers of Binomials, or of any Po-
lynomials, (203).—BINOMIAL THEOREM-Exponents in any Power of (a±b)
-Coefficients-Signs, (204).-Formula for the development of (a+b)—At
what Term the development will terminate, (205).—EVOLUTION-Extract-
ing the Square Root, in what it consists-The Cube Root, (206).-Roots of
vnity, (207).—Roots of Monomials, how found, (208).—Roots of Fractions,
how found—Roots of a mixed Quantity, (209).—Of a Root whose Exponent
is resolvable into two Factors, (210).—What denoted by the Numerator and
Denominator of a Fractional Exponent, (211).-Root of a power of a Quantity,
(212).-Equivalent Exponents, (213).-Sign to be Prefixed to an odd Root
of a Quantity, (214).-Sign to be Prefixed to an even Root, (215).—Of an
even Root of a negative Quantity, (216).How the Roots of Polynomials
may be discovered, (217).
RULE XVIII. To Extract the Square Root of a Polynomial, (218.)
Principle for determining the Number of Figures in the Square Root of a
Number, (219).-Square of any two Parts into which a number may be di-
vided, (220).—Periods to be formed in Extracting the Square Root of a Deci-
imal Fraction -Why the last Period must be complete-Number of Decimal
Figures to be made in the Root, (221).
Principle for determining the Number of Figures in the Cube Root of a
Number, (223).—Cube of any two Parts into which a Number may be di-
vided, (224).
IRRATIONAL OR SURD QUANTITIES.-IMAGINARY QUANTITIES.-147...166.
Perfect and Imperfect Powers, (229).—A Rational Quantity-An Irrationa
or Surd Quantity-Radical Quantities, (230).—Radical Sign-How this
Sign may always be superseded, (231).—Similar and Dissimilar Surds, (232).
How a Rational Quantity may be expressed under the Form of a Surd, (233).
Transfer of an Exponent between Factors and Product, (234).—To what
the Exponent of a Quantity may be changed, (235).—Product of a Rational
and an Irrational Factor, (236).—How a Surd may be simplified, (237).—
How a Fractional may be reduced to an Integral Surd, (238).—Surds of
Different Roots reduced to the Same Root, (239).-How to find the Sum or
Difference of Similar Surds—of Dissimular Surds, (240).—How to find the
Product or Quotient of Surds of the same root-of different Roots-of any
two Roots of the Same Quantity, (241).-Expediency of rationalizing a Surd
Divisor or Denominator, (242).-How a Monomial Surd may be made to
produce a Rational Quantity—a binomial Surd—a trinomial Surd, (243).
-Of the Powers and Roots of Irrational Quantities, (244).—Of the Square
Root of a Numerical Binomial of the form a±√b (245).—Of Imaginary
Quantities-From what an Imaginary Quantity results, (246).—Of the Cal-
culus of Imaginary Quantities-By what means the Sign affecting the Pro-
duct of two Imaginaries may be determined, (247).—Resolution of an
Hmaginary Quantity, (248).-Product of two Imaginary Square Roots, (249)
QUADRATIC AND OTHER EQUATATIONS.-167 ... 204.
A Quadratic Equation-A Cubic Equation-A Biquadratic Equation,
(250).-A Pure Equation-An Affected and a Complete Equation, (251).—
A Root of an Equation, (252).—Principle for determining a Divisor of an
Equation, (253).-Principle for determining a Root of an Equation, (254).
Number of Roots of an Equation-Whether the several Roots of an Eqau-
tion are necessarily unequal, (255).
RULE XXII. For the Solution of a Pure Equation, (256).
How a Surd Quantity in an Equation may be rationalized, (257).—How
two Surds in an Equation may be rationalized, (258).—Of an Equation con-
taining a Fraction whose terms are both irrational, (259).
RULE XXIII. For the Solution of a Complete Equation of the Second Degree,
How any Binomial of the form ax2+bx may be made a Perfect Square (261).
RULE XXIV. To Reduce an Equation of the form ax2+bx to a Simple Equa-
tion, (262).
Of Equations having a Quadratic Form with reference to a Power or
oot of the Uuknown Quantity, (263).—Of the Solution of two Equations,
one or both of the Second or a Higher Degree, Containing two Unknown
Quantities, (264).-Solutions by means of an Auxiliary Unknown Quantity,
(265).-Solutions by means of two Auxiliary Unknown Quantities, (266).
-General Law of the Coefficients of Equations, (267).-Determination of
the Integral Roots of Equations, (268).-Solution of Equations by Ap
proximation, (269).-General Method of Elimination, (270).
GENERAL DESCRIPTION OF PROBLEMS.-MISCELLANEOUS PROBLEMS.-205..
217.
A Determinate Problem-How a Determinate Problem is represented, (270).—
An Indeterminate Problem-How represented, (271).—An Impossible Problem-
How represented, (272).—Greatest Product of any two Parts into which any
Quantity may be divided, (273).—Signification of the Different Forms under which
the value of the Unknown Quantity may be found in an Equation, (274).—Inequa-
tions, (275).—For what purpose Inequations are employed, (276).—Inequalities in
the same, and in a contrary sense, (277).—Inequalities between negative quantities,
(278).—Transformation of Inequations, (279).
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