Edward Olney.

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Online LibraryEdward OlneyA university algebra → online text (page 2 of 28)
Font size Prob. 3.â€” To any required index (201) ; Cor. To put the coefficient
under the radical sign (202) 68

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XU CONTENTS.

*â–  PAGE

Prob. 4. â€” To a common index i

Prob. 5. â€” To rationalize a monomial denominator (204) 69

Prob. 6. â€” To rationalize a radical binomial denominator (205) 69

Prop. 1. â€” To rationalize any binomial radical (206) 69

Prop. 2.â€” To rationalize Va+ Vb+ Vc (207) ; Examples 70-72

SECTION IV.

Prob. 1.â€” To add or subtract (208) 72

Multiplication :

Prop. 1.â€” Productof like roots (209) 73

Prop. 2.â€” Similar Radicals (210) 73

Prob. 2. â€”To multiply Radicals (211) 73

Division :

Prop.â€” Quotient of like roots (212) 73

Prob. 3.â€” To divide Radicals (213) 73

Involution :

Prob. 4. â€” To raise to any power (214) ; Car, Index of power and
root alike (215) 73

Evolution :

Prob. 5. â€” To extract any root of a Monomial Radical (216). ...*... 74
Prob. 6. â€” ^To extract the square root of a Â± n Vb, or
mVa Â± nVb(^i1f); Examples 74-76

SECTION V.

Imaginary Quantitiks.

Definition (218) ; not unreal (219) ; a curious property of (220) 76

Prop.â€” Reduced to form m\/^{221); 8ch, The form w'/^ (222).. 77
Prob. â€” To add and subtract imaginary monomials of second degree (223);

Examples 77, 78

Prop. â€” Polynomial reduced to form a Â± hV^^ (224); Sch, Conjugate

Imaginaries, Modulus (225) ; Examples 79

Multiplication and Involution :

Prob. â€” To determine character of product (226) ; Examples 79, 80

Division of Imaginaries :

Prob. â€” To divide one imaginary by another (227) ; Examples 80, 81

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CONTENTS. xiii

PART ILâ€” ELEMENTARY COURSE IN ALGEBRA.

CHAPTER I.
SIMPLE EqXTATIONS.

SECTION I.

Equations with one Unknown Quantity.
Definitions : pags
Equation (1) ; Algebra (2) ; Members (3) ; Numerical Equation (4) ;
Literal Equation (5) ; Degree of an Equation (6) ; Simple Equa-
tion (7) ; Quadratic (8) ; Cubic (9) ; Higher Equations (10) 82, 83

Tbanspormations :

What (11, 12) ; Axioms (13) 83

Prob. â€” ^To clear of Fractions (14) ; Transposition (15) 83

Prob.â€” To transpose (16) 84

Solution op Simple Equations :

What (17) ; When an equation is satisfied (18) ; Verification (19). . . 84
Pbob. 1. â€” To solve a simple equation (20) ; 8ch. 1. Kinds of changes
which can be made (21) ; Cor, 1. Changing signs of both mem-
bers (22) ; 8ch. 2. Not always expedient to make the transforma-
tions in the same order (23) ; Sch, 3. Equations which become
simple by reduction (24) 85

Prob. 2. â€” To free an equation of Radicals (25, 26) 85

Summary op Practical Suggestions (27, 28) ; Examples 86-88

Applications to the Solution op Examples (29) ; Statement, Solu-
tion (30) i Knowledge required in making statement (31) ; Direc-
tions to guide in making statement (32) ; Not always best to use
X (33) ; Examples 89-92

SECTION II.

Independent, Simultaneous, Simple Equations with two Unknown

Quantities.
Dbpinitions :

Independent Equations (34) ; Simultaneous Equations (35) ; Elimi-
nation (36) ; Methods (37) 93-

Elimination :

Prob. 1.â€” By Comparison (38) 93

Prob. 2.â€” By Substitution (39) 94

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PAGB

Pkob. 3.â€” By Addition or Subtraction (40) 94

Prob. 4. â€” ^By Undetermined Multipliers (41) 95

Prob. 5. â€” By Division (42) ; Examples and Applications .^,.., 96-99

SECTION III.

Independent, Simultaneous, Simple Equations with more than two
Unknown Quantities.

Prob.â€” To solve (43) ; Examples and Applications 100-103

CHAPTEE II.
MATIO, PROPORTION, AND PROGRESSION.

SECTION I.
Ratio.

Depinitionb â€” Ratio (44) ; Sign (45) ; Cor, Effect of Multiplying or
Dividing the Terms (46) ; Direct and Reciprocal Ratio (47) ;
Greater and Less Inequality (48) ; Compound Ratio (49) ; Du-
plicate, Subduplicate, etc. (50) 104, 105

Examples 105, 106

SECTION II.

Proportion.

Oepinitions.â€” Proportion (51) ; Extremes and Means (52) ; Mean Pro-
portional (53) ; Third Proportional (54) ; Inversion (55) ; Alter-
nation (66) ; Composition (57) ; Division (58) ; Inversely Pro-
portional (59) ; Continued Proportion (60) 106, 107

Prop. 1. â€” Product of extremes equals product of means (61 ) ; Cor, 1.

Square of mean proportional (62); Cor, 2. Value of any term (63) 107

Prop. 2. â€” To convert an equation into a proportion (64) ; Cor, Taken

by alternation and inversion (65) 107, 108

Prop. 3. â€” What transformations can be made without destroying the

proportion (66, 67) 108

Prop. 4. â€” Products or Quotients of corresponding terms of two pro-
portions (68) ; Cor, Like powers or roots (69) 108

Prop. 5.â€” Two proportions with equal ratio in each (70) 108

Prop. 6. â€” Taken by composition and division (71) ; Cor. Series of equal
ratios as a continued proportion (72) ; Sch. Method of testing
any transformation (73) ; Examples and Applications 109-113

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OONTBNTS. XV

SECTION m.
Pboobbsbions.

^ PAQS

Definitions. â€” Progressioii, Arithmetical, GeometriciJ, Ascending, De-
scending, Common Difference, Ratio (74) ; Signs and Illustra-
tions (75) ; Arithmetical and Geometric Mean (76, 77) ; Five
things considered (78) 118, 114

Arithmetical Progression.â€” Prop. 1. To find the last term (79) ;
Prop. 2. To find the sum (80) ; Cor, 1. These formulas suffi-
cient (81) ; Cor. 2. To insert means (82) ; Formulae in Arith-
metical Progression (83) ; Examples 114-117

Geometrical Progression.â€” Prop. 1. To find the last term (84) ;
Prop. 2.â€” To find the sum (85) ; Cor, 1. These formulas suffi-
cient (86) ; Got, 2. Another formula for sum (87) ; Cor, 3. To
insert means (88) ; Got, 4. Sum of an infinite series (89) ; Geo-
metrical Formulae (90) ; Examples 117-122

SECTION IV.

Variation.

Definitions. â€” ^Variation, directly, inversely, jointly, directly as one
and inversely as another (91-93) ; Sign (94) ; Frcyp, Variation
expressed as Proportion (95) ; Exercises 128-125

SECTION V.

Harmonic Proportion and Progression.
Definitions. â€” Harmonic Proportion (96); Harmonic Mean (97);
Prop. Quantities in Harmonic Proportion, their Reciprocals in
Arithmetical (98) ; Harmonic Progression (99) ; Derivation of
the term Harmonic (100) ; Exercises 125, 126

CHAPTER III.

SECTION I.

DEFiNrriONS.â€” Quadratic (101); Kinds (102); Pure (103); Affected (104);

Root (105) 127

Resolution of a Pure Quadratic Equation (106) ; Gor. 1. A Pure

Quadratic has two roots (107) ; Gor. 2. Imaginary roots (108). . 127, 128
Examples and Applications 128-130

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SECTION II.

<

Definition (109) 130

Resolution. â€” Common Method (110) ; 8ch. 1. Cpmpleting the Square ;
Cor. 1. Two roots, character of (111) ; Cor. 2. To write the
roots of a;* +â€˘ ^ = g, without completing the square (112) ;
Cor. 3. Special methods (113, 114) ; Examples. 130-134

SECTION III.

Equations of other Degrees which may be Solved as QttADiiATics.

Prop. 1. â€” Any pure equation (115) 134

Prop. 2.^ Any equation containing one unknown quantity with only

two different exponents, one of which is twice the other (116). 135

Prop's 3-5.â€” Special Solutions (117-122) ; Examples 135-139

SECTION IV,

Simultaneous Equations of the Second Dbgrbb between two
Unknown Quantities.

Prop. 1. â€” One equation of the second degree and one of the first (123). 140
Prop. 2. â€” Two equations of the second degree usually involve one of

the fourth, after eliminating (124) 140

Prop. 3.â€” Homogenous quadratics (125, 126) 140, 141

Prop. 4.^ When the unknown quantities are similarly involved (127). . 141

Examples, Special Solutions, Applications 142-147

CHAPTER IV.

INEQUALITIES.

Definition (128); Fundamental Principle (129) ; Members (130) ; Same
transformations as equations (131) ; Same and opposite

sense (132) 148

Prop. â€” Sense of an inequality not changed (133) 148, 149

Pbop. â€” Sense of an inequality changed (134) ; Exercises 149, 150

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ooiprÂŁirrs. xvu

PART IILâ€” ADVANCED COURSE IN ALGEBRA.

CHAPTER I.
INFINITESIMAL ANALYSIS,

SECTION I.

DiPPBRENTIATION.

PAOK

Definitions.â€” Constant and Variable Quantities (135-137) ; Sch, Dis-
tinction between constant and variable, known and un-
known (138) ; Function (139) ; How represented (140) ; Inde-
pendent and Dependent Variable (141) ; Infinitesimal (142) ;
Consecutive values (143) ; Differential (144) ; Notation (145) ;
To differentiate (146) 151-153

BuLES FOR Differentiating:

Rule 1. â€” To differentiate a single variable (147) 154

Rule 2.â€” Constant factors (148) 154

Rule 3.â€” Constant terms (149) 154

Rule 4. â€” The sum of several variables (150) 155

Rule 5. â€” The product of two variables (151) 155

RuLEi 6. â€” The product of several variables (152) 155, 156

Rule 7. â€” Of a fraction with a variable numerator and denomina-
tor (153) ; Cor. With constant numerator (154) ; Sch. Constant

denominator (155) 156

Rule 8. â€” ^Of a variable affected with an exponent (156) ; Sch.

Rate of change (157); Examples 156-158

SECTION II.

Indeterminate Coefficientb.

Dbfinition (158) 159

Prop.â€” In A -v Bx + Ox^ -\- etc. = ul'+ B'x + C'a;' -i- etc., coefficients
of like powers of x equal to each other (159) ; Car. A, B, G,
etc., = 0(160) 159

Development of Functions (161) ; Examples 159-161

Decomposition of Fractions (163) ; Case 1. When the denominator
is resolvable into real and unequal factors of the first de-
gree (164) ; Case 2. Into real and equal factors of the first de-
gree (165) ; Case 3. Into real and quadratic factors (166) ; Sch.
Fonns combined (167) ; Examples 162-164

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XVlll CONTENTS.

SECTION III.
The Binomial Formula.

PAOB

Binomial Theorem (168) ; Cor. 1. The general tenn (169) ; Scale of
Relation (170) ; Cor. 2. Formula for scale of relation (171) ;
Examples 165-168

SECTION IV.
Logarithms.

Definitions.â€” Logarithms, Base (172); Cor. Logarithm of 1(173) ; Sys-
tem of Logarithms (174) ; Two in use (175) ; Cor. Quantities

that cannot be used as a base (176) 168, 169

Prop. 1. â€” ^Logarithm of product equals sum of logarithms (178) 169

Prop. 2. â€” Logarithm of quotient equals difference of logarithms (179). 170

Prop. 3. â€” Logarithm of a power (180) 170

Prop. 4.â€” Logarithm of a root (181) 170

Logarithms of most numbers not integral (182) 170

Characteristic and Mantissa (183) 170

Prop. â€” Mantissa of decimal fraction or mixed number (184) ; Cor. 1.
Characteristic of any number (185); Cor. 2. Logarithm of

0(186) 170,171

Computation op Logarithms:

Modulus (187) ; Prop. Differential of a logarithm (188) 172

Prob. â€” To produce the logarithmic series (189); Cor.X. Loga-
rithms of same number in two different systems, as moduli (190);
Cor. 2. To find logarithm of a number in any system know-
ing the modulus, and also to find modulus (191) 173, 174

Prob. â€” To obtain series for computing Napierian logarithms (192) 175
Prob. â€” To compute Napierian logarithms of natural num-
bers (193) 175

Prop. â€” Modulus of common system (194) 176

Tables op Logarithms.â€” What (195) 177

Prob.â€” To find the logarithm of a number (196, 197) 177, 178

Prob.â€” To find a number corresponding to a logarithm (198) ... 178

Prop.â€” The Napierian base (199) ; Examples 179-181

SECTION V.
Successive Differentiation and Differential Coefficients.

Prop. â€” Differentials not necessarily equal (200) ; Cor. dy a varia-
ble (201) ; Notation (202) ; dx constant (203) ; Second and Third
Differentials (204) ; Examples '. 181-183

Differential Coefficients :

First Differential Coefficient (205) ; Second Differential Coeffi-
cient (206) ; Examples ; Successive coefficients written by in-
spection (207) 183-185

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CONTENTS. XIX

SECTIC:^ VI.
Taylor's Formula.

PAOB

Definition (208); Partial Differential Coefficients (209); Lemma.

g and J? equal(210) 185,186

Prob. â€” To produce Tajlor*s Formula (211) ; Sch. First, second, etc.,
terms (212) ; To develop a function of a variable with an in-
crement (213) ; Examples 18^189

SECTION VII.

Indeterminatb Equations.

Definition, Nature, etc. (214-220) ; Examples 189-195

CHAPTER 11.

LOCI OF EQUATIONS,

Prop. â€” Every equation between two variables may represent a line (221) 1 96

Definitions. â€” Axes of Reference, Abscissa, Ordinate, Co-ordinates

(222) ; Locus, Constructing Locus (223) ; Examples 198-202

Prob. â€” To construct real roots of equations with one unknown quan-
tity (224) ; Examples 202, 203

CHAPTER III.
HIGHER EQUATIONS.

SECTION I.

Solution of Numerical Higher Equations having Commensurable
OR Rational Roots.

No general method of solution (226) ; Real, commensurable roots found

with little difficulty (227) 208

Prop. â€” Transforming an equation into the form ic* -f- Aod^-"^ + 5ajÂ»-'

+ Ca^-z X = 0(228); Examples 204, 205

Prop. â€” Roots of an equation factors of absolute term (230) ; If a is a

root, f{x) divisible by {x â€” a), and converse (231) 206

PROP.â€”What equation can have no fractional root (232, 233) 206, 207

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XX OONTWTB.

PAOB

Prop.â€” Equation of nth degree has n roots (234) ; Cor, 1. f(x) = (Â«â€” a)
(Â« â€” 6) (a; â€” c) - - - - (a? â€” n), when a, 6, c - - - - n are roots of
/(a;) = (236) ; Cor. 2. f(x) can have equal roots (236) ; Cor. 3.
Imaginary roots enter in pairs (237) ; Cor, 4. Number of real
roots in equations of odd and even degrees (238) ; Limits of
imaginary roots (239) ; 8cfi. 1. Proposition illustrated geome*
tricallj (240) ; Sch, 2. Imaginary roots entering in pairs illus-
trated (241) 207-209

Prop. â€” Method of finding equal roots (242) ; Sch. Sometimoi conve-
nient to apply process several times (248) 209, 210

Prop.â€” Change of sign In f(x) (244) ; How illustrated by loci (245) 211

Prop.â€” Changing signs of roots of /(a?) (246) ; Cor, Another method (247) 212

Prob.â€” To evaluate /(a?) for aj = a(248) 212,213

Prob. â€” To find commensurable roots of numerical higher equations

(249); Examples 213-216

To produce an equation from its roots (250) ; Examples 216

SECTION II.

Solution op Numerical Higher Equations having Real, Incommen-
surable, OR Irrational Roots.

Typical form of equation (251) ; Best general method (252). 216, 217

Sturm's Theorem and Method :

Definition and Object (253, 254) ; Sturmian Functions (255) ; No-
tation (256) ; Permanence and Variation (257) 217, 218

Sturm's Theorem (258) ; Cor. 1. To find the number of real roots
of f(x) (259) ; Cor. 2. To find the number of real roots of f(x)
between a and b (260) ; 8ch. Number of imaginary roots known
by implication (261) 219-221

Prob. â€” To compute the numerical values of /(a;), f'{x), fi(x\ etc.
(262) ; Sch. 2. Usually unnecessary to find /Â«(ajO) (263) ; Sch. 3.
When the equation has equal roots (264) ; Sch. 4 Generally
the change of sign in f(x) enables us to determine situation of
roots more easily than Sturm's Theorem (265) ; Sch. 5. Not ne-
cessary that the coefficients should be integral (266) ; Exam-
ples 221-228

Horner's Method op Solution : Object (267) 228

Prob. â€” To transform an equation into another with roots less by

a (268) ; Sch. Signification of result (269) 229

Prob. â€” ^To compute the numerical values of /(a), f'{a), ^f"(a\
etc. (270); Examples 229-233

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(X^rr^^rr^. xxi

PAO>

Prop.â€” Value of a?, , when a + aji Ih a root of f(z) = (271) 28&-2d5

HoRNBB's RuLB (272) ; /ScMiums (273-278) ; Examples 235-247

SECTION in.

Gbnbral SonmoN op Cubic akd Biquadratic Equatioks.

Cardan'0 Solution op Cubic Equations :

Prob.â€” To resolve Â»â€˘ + |WJÂ» + Â§Â» 4- r = (27Â») 248. 249

Prop. â€” Solution satisfactory and unsatisfactory, when (280);
ScMmm. Apparently 9 roots (281); Examples 249-251

Pros.â€” To resolve a?* + aaj' + &Â»* + ^ + â‚¬ = (282) ; 8eh. In-
volves solution of a cubic (283) ... 251, 252

Becurrino Equations :

Definition (284) 252

Prop. 1. â€” ^The roots reciprocals of each other (285) ; Sch. Recip-
rocal Equations (286) ; Oor, 1. Corresponding coefficients with
like or unlike signs (287) ; Cor, 2. Reduced to form having

first coefficient unity (288) 252, 253

Prop. 2.â€” Of an odd degree have roots â€” 1, and + 1, when (289). 253, 254

Prop. 3. â€” Of an even degree have same roots, when (290) 254

Prop. 4 â€” Of an even degree above second reduced to one of half
that degree (291) ; Examples 254, 255

Binomial Equations and the Roots op Unitt :

Definition (292) ; Examples and Scholium (293) 255, 256

Exponential Equations :

Definition (294) .' 256

Prob. 1.â€” To solve a* = w (295) 256

Prob. 2.â€” To solve Â«* = w (296) ; Examples 256-260

CHAPTER IV.
DISCUSSION, OR INTEBPRETATION, OF EQUATIONS.

Definition (297) 260

Prop.â€” Statement of Principles (298) 260-262

Real Number or Quantity (299) 262

Imaginart Number (300) ; Examples 263-266

ARITHMETrCAL INTERPRETATION OP NEGATIVE AND IMAGINARY RE-
SULTS (301) ; Sch. Symbol ^ (302) ; Examples 267-271

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XXll CONTENTS.

APPENDIX.

PA08

SECTION I.

Series.

Depinitions.â€” Series, Tenn (303) ; Recurring Series, Scale of Rela-
tion (804) ; Infinite, Convergent, Divergent (306) ; To revert a
Series (806) ; Orders of Differences (307) ; Interpolation (308) ;
Enumeration of Problems (809) 272-274

Lemma. â€” First term of any order of differences (310) ; Cor, Number of

terms necessary (311) ; Examples 274, 275

Prob. 1. â€” To find scale of relation in recurring series (812) ; 8ch, De-
pendence on too many or too few terms (818) ; Examples 275-277

Prob. 2.â€” To find the nth term (814) ; Examples 277, 278

Prob. 3. â€” To determine whether a series is convergent or divergent

(315); Examples 278-280

Prob. 4.â€” To find sum of a series (316) ; Examples 280-285

Piling Balls and Shells (317) 285

Prop. â€” Number in triangular pile (318); Cor, Number of

courses (319) 285, 286

Prop. â€” Number in square pile (820); Got. Number of courses (321) 286

â€˘ Prop.â€” Number in oblong pile (322) ; Examples 286, 287

Reversion of Series :

Prob.â€” To revert a series (823) ; Examples 287, 288

Interpolation :

To interpolate between functions (324) ; 8ch, 1. Result correct
when (325) ; Bch. 2. Another formula (326) ; Sch, 3. Used in
Astronomy (327) ; Examples 288-291

SECTION II.

Permutations.

Definitions.â€” Combinations (328); Permutations (329); Arrange-
ments (330) 292

Prop. â€” Number of arrangements of m things n and n (331) ; Cor. 1.
Permutations of m things (332) ; Cor, 2. When p things are
alike, etc. (833) ; Cor, 3. Combinations of m things n and
71(334); Examples 292-294

Probabilities :

Mathematical Probability and Improbability (335) ; Examples.. 294-298

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INTRODUCTION.

SECTION /.
GENERAL DEFINITIONS, AND THE ALGEBRAIC NOTATION.

BRANCHES OF PURE ItATHEMATICS.

1. Fure Mathematics is a general term applied to several
branches of science, which have for their object the investigation of
the properties and relations of quantity^-comprehending number,
and magnitude as the result of extension â€” and of form.

^ 2. The Several Sranches of Pure Mathematics are Arith-
metic, Algebra, Calculus, and Geometry.

3. Arithmetic, Algebra, and Calculus treat of number, and Geo-
metry treats of magnitude as the result of extension.

^ 4. Quantity is the amount or extent of that which may be
measured; it comprehends number and magnitude.

The term quantity is also conventionally applied to symbols used
to represent quantity. Thus 25, m, xi, etc., are called quantities,
although, strictly speaking, they are only representatives of quantities.

5. Number is quantity conceived as made up of parts, and
answers to the question, " How many ? "

\ 6. Number is of two kinds. Discontinuous and Continu-
ous.

^ 7. Discontinuous Number is number conceived as made
up of finite parts ; or it is number which passes from one state of
value to another by the successive additions or subtractions of finite
units ; L e., units of appreciable magnitude.

8. Continuous Number is number which is conceived as
composed of infinitesimal parts ; or it is number which passes from

1

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2 INTBODUCnON.

one state of value to another by passing through all intermediate
values, or states.

9. Arithmetic treats of Discontimwus NumbeVy â€” of

its nature and properties, of the various methods of combining and
resolving it, and of its application to practical affairs.

^ 10* Algebra treats of the Equatiouy and is chiefly occupied in
explaining its nature and the methods of transforming and reducing
it, and in exhibiting the manner of using it as an instrument for
mathematical investigation.*

11. Calculus treats of Continuous Number, and is chiefly
occupied in deducing the relations of the infinitesimal elements of
such number from given relations between finite values, and the con-
verse process, and also in pointing out the nature of such infinites-
imals and the ipethod of using them in mathematical investigation.

12. Geometry treats of magnitude and form as the result of
extension and position.

LOOICO-MATHEMATICAL TERMS.

"^ 13. A Froposition is a statement of something to be con-
sidered or done.

14. Propositions are distinguished as Axioms, Theorems^ Lemmas,
Corollaries, Postulates, and Problems.

^ 15. An Axiom is a proposition which states a principle that
is so simple, elementary, and evident as to require no proof.

^ 16. A Theorem is a proposition which states a real or supposed
fact, whose truth or falsity we are to determine by reasoning.

^ 11. A Demonstration is the course of reasoning by means
of which the truth or falsity of a theorem is made to appear. The
term is also applied to a logical statement of the reasons for the
processes of a rule. A solution tells liow a thing is done ; a demon-
stration tells loliy it is so done. A demonstration is often called proof.

* The common deAuition of Algebra, which m^kes its di8tin<?n!8bing features to be the literal
notation^ and the use of the Hgns^ is entirely at fault. When Algebra first appeared in Europe, it
possessed neither of these features I What was it then? On the otlier hand, the signs are
common to all branches of mathematics, and the literal notation is as prominent in the Calculus
as in Algebra, and is used, more or less, in common Arithmetic and Geometry.

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LOGIOO-MATHEMATIOAL TEBlfS. 3

18* A iJemma is a theorem demonstrated for the purpose of
using it in the demonstration of another theorem.

"^ 19Â» A Corollary is a subordinate theorem which is sug-
gested, or the truth of which is made evident, in the course of the
demonstration of a .more general theorem, or which is a direct
inference from a proposition, vâ€” v-cOvT "^ P v v\ ; ^

^i 20. A Postvlate is a proposition which states that something
can be done, and which is so evidently true as to require no process
of reasoning to show that it is possible to be done. We may or may
not know how to perform the operation.

^ 21. A Problem is a proposition to do some specified thing,
and is stated with reference to developing the method of doing it

^ 22. A Mule is a formal statement of the method of solving a
general problem, and is designed for practical application in solving
special examples of the same class. Of course a rule requires a
demonstration.

23. A Solution is the process of performing a problem or an
example. It should usually be accompanied by a demonstration of
the process.

\ 24, A Scholium is a remark made at the close of a discussion,
and designed to call attention to some particular feature or features

OI it. '^<>v-~ajT,^aVs,s^

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PART i;

LITERAL ARITHMETIC.t

OHAPTEE I.

FUNI>AMENTAL RULES.

SECTION L

NOTATION.

25. A System of Notation is a system of symbols by means
of which quantities, the relations between them, and the operations
to be performed upon them, can be more concisely expressed than
by the use of words.

Symbols of Quai^tity.

26. In Arithmetic, as usually studied, numbers are represented
by the characters, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, called Arabic figures, or,
simply, figures.

27. In other departments of mathematics than Arithmetic, num-
bers or quantities are more frequently represented by the common

Online LibraryEdward OlneyA university algebra → online text (page 2 of 28)