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A MATHEMATICAL

SOLUTION BOOK

CONTAINING

SYSTEMATIC SOLUTIONS OF MANY OF THE

MOST DIFFICULT PROBLEMS.

Taken from the Leading Authors on Arithmetic and Algebra, Many Prob-

lems and Solutions from Geometry, Trigonometry and Calculus,

Many Problems and Solutions from the Leading Math-

ematical Journals of the United States, and

Many Original Problems and Solutions.

WITH

NOTES AND EXPLANATIONS

BY

B. F. FINKEL, A. M., M. Sc.

ii

Member of the London Mathematical Society, Member of the American

Mathematical Society, Editor of the American Mathematical

Monthly, and Professor of Mathematics and

Physics in Drury College.

THIRD EDITION- REVISED.

KIBLER & COMPANY, PUBLISHERS,

Springfield, Mo.

COPYRIGHT, 1888,

BY

B. F. FINKEL,

IN THE OFFICE OF THE LIBRARIAN OF

CONGRESS,

WASHINGTON, D. C.

PREFACE.

This work is the outgrowth of eight years' experience in

teaching in the Public Schools, during which time I have ob-

of problems would be serviceable to both teachers and pupils.

It is not intended to serve as a key to any work on mathe-

matics ; but the object of its appearance is to present, for use in

the schoolroom, such an accurate and logical method of solving

problems as will best awaken the latent energies of pupils, and

teach them to be original investigators in the various branches of

science.

It will not be denied by any intelligent educator that the so-

called "Short Cuts" and "Lightning Methods" are positively in-

jurious to beginners in mathematics. All the "whys" are cut

out by these methods and the student robbed of the very object

for which he is studying mathematics ; viz., the devolpment of

the reasoning faculty and the power to express his thoughts in

a forcible and logical manner. By pursuing these methods,

mathematics is made a mere memory drill and when the memory

fails, all is lost ; whereas, it should be presented in such a way as

to develop the memory, the imagination, and the reasoning fac-

ulty. By following out the method pursued in this book, the

mind will be strengthened in these three powers, besides a taste

for neatness and a love of the beautiful will be cultivated.

Any one who can write out systematic solutions of problems

can resort to "Short Cuts" at pleasure ; but, on the other hand,

let a student who has done all his work in mathematics by form-

ulae, "Short Cuts," and "Lightning Methods" attempt to write

out a systematic solution one in which the work explains

itself and he will soon convince one of his inability to express

his thoughts in a logical manner. These so-called "Short Cuts"

should not be used at all, in the schoolroom. After pupils and

students have been drilled on the systematic method of solving

problems, they will be able to solve more problems by short

methods than they could by having been instructed in all the

"Short Cuts" and "Lightning Methods" extant.

It can not be denied that more time is given to, and more

time wasted in the study of arithmetic in the public schools than

2 PREFACE.

in any other branch of study ; and yet, as a rule, no better results

are obtained in this branch than in any other. The reason of

this, to my mind, is apparent. Pupils are allowed to combine

the numbers in such a way as "to get the answer" and that is all

that is required. They are not required to tell why they do this,

or why they do that, but, "did you get the answer?" is the

question. The art of "ciphering" is thus developed at the ex-

pense of the reasoning faculty.

The method of solving problems pursued in this book is

often called the "Step Method." But we might, with equal pro-

priety, call any orderly manner of doing any thing, the "Step

Method." There are only two methods of solving problems a

right method and a wrong method. That is the right method

which takes up, in logical order, link by link, the chain of rea-

soning and arrives at the correct result. Any other method is

wrong and hurtful when pursued by those who are beginners in

mathematics.

One solution, thoroughly analyzed and criticised by a class,

is worth more than a dozen solutions the difficulties of which are

seen through a cloud of obscurities.

This book can be used to a great advantage in the class-

room the problems at the end of each chapter affording ample

exercise for supplementary work.

Many of the Formulae in Mensuration have been obtained by

the aid of the Calculus, the operation alone being indicated. This

feature of the work will not detract any from its merits for those

persons who do not understand the Calculus ; for those who do-

understand the Calculus it will afford an excellent drill to work

out all the steps taken in obtaining the formulae. Many of the

formulae can be obtained by elementary geometry and algebra.

But the Calculus has been used for the sake of presenting the

beauty and accuracy of that powerful instrument of mathematics.

In cases in which the formulae lead to series, as in the case

of the circumference of the ellipse, the rule is given for a near

approximation.

It has been the aim to give a solution of every problem

presenting anything peculiar, and of those which go the rounds

of the country. Any which have been omitted will receive space

in future editions of this work. The limits of this book have

compelled me to omit much curious and valuable matter in

Higher Mathematics.

I have taken some problems and solutions from the School

Visitor, published by John S. Royer; the Mathematical Maga-

zine, and the Mathematical Visitor, published by Artemas Mar-

tin, A. M., Ph. D., LL. D.; and the Mathematical Messenger,

published by G. H. Harvill, by the kind permission of these

distinguished gentlemen.

PREFACE. 3

It remains to acknowledge my indebtedness to Prof. William

Hoover, A. M., Ph. D., of the Department of Mathematics and

Astronomy in the Ohio University at Athens, for critically read-

ing the manuscript of the part treating on Mensuration.

Hoping that the work will, in a measure, meet the object

for which it is written, I respectfully submit it to the use of

my fellow teachers and co-laborers in the field of mathematics.

Any correction or suggestion will be thankfully received by

communicating the same to rne.

THE AUTHOR.

1 -

In bringing out a second edition of this work, I am greatly

indebted to Dr. G. B. M. Zerr for critically reading the work

with a view to eliminating all errors.

THE AUTHOR.

Drury College, Feb. Jp, 1897.

PREFACE TO SECOND EDITION.

PREFACE TO THIRD EDITION.

The hearty reception accorded this book, as is attested by

the fact that two editions of 1,200 copies each have already been

sold, encouraged me to bring out this third edition.

In doing so, I have availed myself of the opportunity of

making some important corrections, and such changes and im-

provements as experience and the suggestions of teachers using

the book have dictated. The very favorable comments on the

work by some of the most eminent mathematicians in this

country confirm the opinion that the book is a safe one to

put into the hands of teachers and students.

While mathematics is the exact science, yet not every book

that is written upon it treats of it as though it were such. In-

deed, until quite recently, there were very few books on Arith-

metic, Algebra, Geometry or Calculus that were not mere copies

of the works written a century ago, and in this way the method,

the spirit, the errors and the solecisms of the past two hundred

years were preserved and handed down to the present genera-

tion. At the present time the writers on these subjects are

breaking away from the beaten paths of tradition, and the re-

sult, though not wholly apparent, is a healthier and more vig-

orous mathematical philosophy. Within the last twenty-five

4 PREFACE.

years there has set in, in America, a reaction against the spirit

and the method of previous generations, so that C. A. Laisant,

in his La Mathematique Phi to sop hie Enseignement : , Paris, 1898,

says, "No country has made greater progress in mathematics

during the past twenty-five years than the United States. The

most of the text-books on Arithmetic, Algebra, Geometry, and

the Calculus, written within the last five years, are evidence of

this progress.

The reaction spoken of was brought about, to some ex-

tent, by the introduction into our- higher institutions of learn-

ing of courses of study in mathematics bearing on the wonder-

ful researches of Abel, Cauchy, Galois, Riemann, Weirstrass,

and others. This reaction, it may be said, started as early as

1832, the time when Benjamin Peirce, the first American worthy

to be ranked with Legendre, Wallis, Abel and the Bernouillis,

became professor of mathematics and natural philosophy at

Harvard University. Since that time the mathematical courses

in our leading Universities have been enlarged and strengthened

until now the opportunity for research work in mathematics

as offered, for example, at the University of Chicago, Harvard,

Yale, Cornell, Johns Hopkins, Princeton, Columbia and others,

is as good as is to be found anywhere in the world. For ex-

ample, the following are the subjects offered at Harvard for the

Academic year 1899-1900: Logarithms, Plane and Spherical

Trigonometry; Plane Analytical Geometry; Plane and Solid

Analytical Geometry ; Algebra ; Theory of Equations. Invar-

iants ; Differential and Integral Calculus ; Modern Methods in

Geometry. Determinants ; Elements of Mechanics ; Quater-

nions with application to Geometry and Mechanics ; Theory of

Curves and Surfaces; Dynamics of a Rigid Body; Trigonomet-

ric Series. Introduction to Spherical Harmonics. Potential

Function ; Hydrostatics. Hydrokinematics. Force Functions

and Velocity-Potential Functions and their uses. Hydroki-

netics ; Infinite Series and Products ; The Theory of Functions ;

Albegra. Galois's Theory of Equations; Lie's Theory as ap-

plied to Differential Equations ; Riemann's Theory of Func-

tions ; The Calculus of Variations ; Functions Defined by Linear

Differential Equations ; The Theory of Numbers ; The Theory

of Planetary Motions; Theory of Surfaces; Linear Associative

Algebra; the Algebra of Logic; the Plasticity of the Earth;

Elasticity; and the Elliptic and the Abelian Transcendants.

While the great activity and real progress in mathematics

is going on in our higher institutions of learning, a like degree

of activity is not yet being manifested in many of our colleges

and academies and the Public Schools in general. It is not

desirable that the quantity of mathematics studied in our Public

Schools be increased, but it is desirable that the quality of

PREFACE. 5

the teaching should be greatly improved. To bring about this

result is the aim of this book.

It does not follow, as is too often supposed, that any one

familiar with the multiplication table, and able, perhaps, to

solve a few problems, is quite competent to teach Arithmetic,

or "Mathematics," as arithmetic is popularly called. The very

first principles of the subject are of the utmost importance,

and unless the correct and refined! notions of these principles

are presented at the first, quite as much time is lost by the

student in unlearning and freeing himself from erroneous con-

ceptions as was required in acquiring them. Moreover, no ad-

vance in those higher modern developments in Mathematics is

possible by any one having false notions of its first principles.

As a branch for mental discipline, mathematics, when

properly taught, has no superior. Other subjects there are

that are equally beneficial, but none superior. The idea en-

tertained by many teachers, generally those who have pre-

pared themselves to teach other subjects, but teach mathematics

until an opportunity to teach in their special line presents itself

to them, that mathematics has only commercial value and

only so much of it should be studied as is needed by the student

in his business in after life, is pedagogically and psychologically

wrong. Mathematics has not only commercial value, but edu-

cational and ethical value as well, and that to a degree not

excelled by any other science. No other science offers such

rich opportunity for original investigation and discovery. So far

from being a perfected and complete body of doctrine "handed

down from heaven" and incapable of growth, as many sup-

pose, it is a subject which is being developed at such a mar-

velous rate that it is impossible for any but the best to keep

in sight of its ever-increasing and receding boundary. Because,

therefore, of the great importance of mathematics as an agent

in disciplining and developing the mind, in advancing the ma-

terial comforts of man by its application in every department

of art and invention, in improving ethical ideas, and in culti-

vating a love for the good; the beautiful, and the true, the

teachers of mathematics should have the best training possible.

If this book contributes to the end, that a more comprehensive

view be taken of mathematics, better services rendered in pre-

senting its first principles, and greater interest taken in its

study, I shall be amply rewarded for my labor in its prepa-

ration.

In this edition I have added a chapter on Longitude and

Time, the biographies of a few more mathematicians, several

hundred more problems for solution, an introduction to the

study of Geometry, and an introduction to the study of Algebra.

The list of biographies could have been extended indefi-

nitely, but the student who becomes interested in the lives of

6 PREFACE.

a class of men who have contributed much to the advancement

of civilization, will find a short sketch of the mathematicians

from the earliest times down to the present day in Cajori's

History of Mathematics or Ball's A Short History of Mathematics.

The biographies which have been added were taken from

the American Mathematical Mnothly. I have received much aid

in my remarks on Geometry from Study and Difficulties of

Mathematics, by Augustus De Morgan.

It yet remains for me to express my thanks to my colleague

and friend, Prof. F. A. Hall, of the Department of Greek, for

making corrections in the Greek terms used in this edition,

THE AUTHOR.

Drury College, July, 1899.

CONTENTS.

CHAPTER I.

DEFINITIONS.

Mathematics classified 11 |

CHAPTER II.

NUMERATION AND NOTATION.

PAGE

Definitions 11-14

Numeration defined 14

French Method defined 14

English Method defined 14

Periods of Notation 15

Notation defined

Arabic Notation defined

15

15

Roman Notation defined 15

Ordinal Numbers 15

Fractions 18

Irrational Numbers. 20

Examples 21

Addition defined

Subtraction defined

CHAPTER III.

ADDITION.

22 | Examples 23

CHAPTER IV.

SUBTRACTION.

23 | Examples 24

CHAPTER V.

MULTIPLICATION.

Multiplication defined 24 | Examples 25-26

CHAPTER VI.

DIVISION.

Division defined 26 | Examples 27

CHAPTER VII.

COMPOUND NUMBERS.

Definitions 28

Time Measure 29

Definitions in Time Measure . 23-31

Longitude and Time 31-34

Standard Time 34-35

The International Date Line . 36-37

Examples 37-36

Solutions 38-39

Examples 39-40

Divisor defined 41

Common Divisor defined 41

Multiple defined 42

Common Multiple defined 42

CHAPTER VIII.

GREATEST COMMON DIVISOR.

Greatest Com'n Divisor defined. 41

Examples 41-42

CHAPTER IX.

LEAST COMMON MULTIPLE.

Least Common Multiple defined 42

Examples 43-44

8

CONTENTS.

CHAPTER X.

FRACTIONS.

PAGB

Definitions 44-46

Fractions classified 44

Solutions of Problems.

Examples

PAGE

46-49

49-52

CHAPTER XI.

CIRCULATING DECIMALS.

54

55

55

IV.

I. Addition of Circulates

II. Subtraction of Circulates . .

III. Multiplication of Circulates

CHAPTER XII.

PERCENTAGE.

Division of Circulates . 56

Examples 56-57

.Definitions ....... 57

Solutions 57-69

II. Commission . 69

Definitions ...... .... 69

Solutions 69-71

Examples . 72

Trade Discount 72

Definitions 72

Solutions 73-76

Examples 76

Profit and Loss 77

III.

IV.

Definitions 77

Examples 83-84

V. Stocks and Bonds 84

Definitions . . . 84

Solutions 85-95

Examples 96-97

VI. Insurance 97

Definitions 97-98

Solutions . . 98-101

Examples 102

CHAPTER XIII.

INTEREST.

II.

III.

Simple Interest 103

Definitions 103

Solutions 103-106

True Discount . 106

Definitions 106

Solutions 106-107

Bank Discount 107

Definitions . . .107

Solutions 108-109

IV. Annual Interest 109

Annual Interest defined 109

Solutions 110-112

V, Compound Interest 112

Compound Int. defined. 112

Solutions 113-115

Definitions

Solutions .

CHAPTER XIV.

ANNUITIES.

. . . . 115 I Examples

.116-125

126

CHAPTER XV.

MISCELLANEOUS PROBLEMS.

Solutions . 127-139

CHAPTER XVI.

RATIO AND PROPORTION.

Definitions 139-141 I Problems

Solutions. ......... 141-144

... 144-146

Analysis defined. . .

Solutions . ... . .

CHAPTER XVII.

ANALYSIS.

. . . . 146 Problems

146-177

177-180

CONTENTS.

CHAPTER XVIII.

ALLIGATION.

PAGE.

I. Alligation Medial 181

II. Alligation Alternate. . 181

Solutions

PAGE

181-186

CHAPTER XIX.

SYSTEMS OF NOTATION.

Definitions 187

Names of Systems 187

Solutions 188-191

CHAPTER XX.

MENSURATION.

v

V,

Definitions 192-197

Geometrical Magnitudes class-

ified 192

I. Parallelogram . . 198-200

II. Triangles 200-204

III. Trapezoid 204-205

IV. Trapezium and Irregular

Polygons 205

V. Regular Polygons . . . 205-207

VI. Circles 207-210

VII. Rectification of Plane

Curves and Quadrature of

Plane Surfaces 210-213

II. Conic Sections 223

Definitions 223-224

1. Ellipse 224-227

2. Parabola 227-229

3. Hyperbola 229-232

IX. Higher Plane Curves ... 233

1. The Cissoid Diocles. 233-234

2. The Conchoid of Nicom-

edes 234-235

3. The Oval of Cassini 235

4. The Lemniscate of Ber-

nouilli 236

5. The Witch of Agnesi 236-237

6. The Limacon 237

7. The Quadratrix 238

8. The Catenary 238-239

9. The Tractrix 240

10. The Syntractrix 240

II. Roulettes 240

(a) Cycloids 240-243

(b) Prolate and Curtate

Cycloid 243-244

(c) Epitrochoid and Hy-

potrochoid 245-248

X. Plane Spiral 248

1. Spirals of Archimedes 249

2. The Reciprocal Spiral 249

3. The Lituus 250

4. The Logarithmic Spi-

ral 250

XI. Mensuration of Sol-

ids 251-254

1. Cylinder 254-255

2. CylindricUngulas 255-262

3. Pyramid and Cone 262-266

4. Conical Ungulas. 266-270

XII. Sphere 270-276

XIII. Spheroid 276-278

1. The Prolate Sphe-

roid 276-278

2. The Oblate Sphe-

roid 278-282

XIV. Conoids 282

1. The Parbolic Co-

noid .... 282-285

2. Hyperbolic Co-

noid 285-286

XV. Quadrature and Cuba-

ture of Surfaces and

Solids of Revolution 286

1. Cycloid 286-287

2. Cissoid 287-288

3. Spindles 288-289

4. Parabolic Spindle 289-290

XVI. Regular Solids 290

1. Tetrahedron .... 291-292

2. Octahedron 292*

3. Dodecahedron . . . 292-293

4. Icosahedron 293-294

XVII. Prismatoid 294-295

XVIII. Cylindric Rings ... 295-297

XIX. Miscellaneous Measure-

ments 297

1. Masons' and

Bricklayers' work 297

2. Gauging 297-298

3. Lumber Measure . 298

4. Grain and Hay. . , 298-299

10

CONTENTS.

MENSURATION Concluded.

PAGE

XX. Solutions of Miscellaneous Problems 299-345

Problems 346-354

Examination Tests : . 354-360

Problems 361-366

GEOMETRY.

Definitions 367 (e) Assumption of the Sphere . . 380

On Geometric Reasoning 369 (/) of Motion 380

On the Advantages Derived from On Logic 380

the Study of Geometry and Laws of Thought 381

Mathematics in General 370 Law of Converse 383

Axioms 375 Methods of Reasoning 384

General Axioms 376 How to Prepare a Lesson in

Assumptions . 377 Geometry 388

(a) Assumption of Straight Line 377 Plane Geometry 390

(b) of the Plane 377 The Three Famous Problems of

(c) of Parallel Line 377 Antiquity 408

(d) of the Circle... 379

ALGEBRA.

Definitions 415 Arithmetical Fallacies 424

Solutions of Problems 416-419 Probability 425

The Quadratic Equation 419 Problems 428

Indeterminate forms 421

Biography of Prof. William Hoover 403

Probability Problems 434-435

Biography of Dr. Artemas Martin 436

Biography of Prof. E. B. Seitz 440

Biography of Rene 7 Descartes 442

Biography of Leonhard Euler 446

Biography of Spphus Lie 451

Biography of Simon Newcomb 454

Biography of George Bruce Halsted 457

Biography of Prof. Felix Klein 459

Biography of Benjamin Peirce 462

Biography of James Joseph Sylvester s 468

Biography of Arthur Cayley 475

Table I 473

II 478

478

J V 479

V 479

VI 480

VII 480

Example 4gl

CHAPTER I.

DEFINITIONS.

1. Mathematics (p.aftv)p.a.Ttxrj, science) is that science

which treats of quantity.

!(1.) Arithmetic.

(2.) Algebra...

*.j Geometry..

rl. Calculus

1 2. Quaternions.

El. Platonic Geometry..

2. Analytical Geometry.

3. Descriptive Geometry.

Differential.

Integral.

Calculus of Variations.

: a. Pure Geometry.

b. Conic Sections. i\. Plane Trigon'y.

c . Trigonometry.. <2. Analytical Trig.

(3. Spherical "

'(1.) Mensuration.

(2.) Surveying.

(3.) Navigation.

(4.) Mechanics.

(5.) Astronomy.

(6.) Optics.

(7.) Gunnery.

^(8.) &c., &c.

2>. Pure Mathematics treats of magnitude or quantity

without relation to matter.

3. Applied Mathematics treats of magnitude as subsist-

ing in material bodies.

4. Arithmetic (api^^nx^, from d^etf/io?, a number) is

the science of numbers and the art of computing by them.

5. Alyebra (Ar. al, the, and geber, philosopher) is that

method of mathematical computation in which letters and other

symbols are employed.

6. G-eOWietry (yajfj.Tpia, from yU){j.Tpiv to measure

land, from pa, yf h the earth, and p.Tp~iv, to measure) is the

science of position and extension.

7. Calculus ( Calculus, a pebble) is that branch of mathe-

matics which commands by one general method, the most diffi-

cult problems of geometry and physics.

12 FINKEL'S SOLUTION BOOK.

8. Differential Calculus is that branch of Calculus

which investigates mathematical questions by measuring the re-

lation of certain infinitely small quantities called differentials.

9. Integral Calculus is that branch of Calculus which

determines the functions from which a given differential has been

derived.

1C. Calculus of Variations is that branch of calculus

in which the laws of dependence which bind the variable quanti-

ties together are themselves subject to change.

11. Quaternions (quaternis, from quaterni four each,

from quator, four) is that branch of algebra which treats of the

relations of magnitude and position of lines or bodies in space by

means of the quotient of two direct lines in space, considered as

depending on a system of four geometrical elements, and as ex-

pressed by an algebraic symbol of quadrinominal form.

12. Platonic Geometry is that branch of geometry in

which the argument is carried forward by a direct inspection of

the figures themselves, delineated before the eye, or held in the

imagination.

13. Pure Geometry is that branch of Platonic geometry

in which the argument may be practically tested by the aid of

the compass and the square only.

14. Conic Sections is that branch of Platonic geometry

which treats of the curved lines formed by the intersection of a

cone and a plane.

15. Trigonometry (rptytovov, triangle, ptrpov, meas-

ure) is that branch of Platonic geometry which treats of the re-

lations of the angles and sides of triangles.

16. Plane Trigonometry is that branch of trigonom-

etry which treats of the relations of the angles and sides of plane

triangles.

17. Analytical Trigonometry is that branch of trig-

onometry which treats of the general properties and relations of

trigonometrical functions.

18. Spherical Trigonometry is that branch of trig-

onometry which treats of the solution of spherical triangles.

19. Analytical Geometry is that branch of geometry

in which the properties and relations of lines and surfaces are in-

vestigated by the aid of algebraic analysis.

20. Descriptive Geometry is that branch of geometry

which seeks the graphic solution of geometrical problems by

means of projections upon auxiliary planes.

DEFINITIONS. 13

21. Mensuration is that branch of applied mathematics

which treats of the measurment of geometrical'magnitudes.

22. Surveying is that branch of applied mathematics

which treats of the art of determining and representing distances,

areas, and the relative position of points upon the earth's surface.

SOLUTION BOOK

CONTAINING

SYSTEMATIC SOLUTIONS OF MANY OF THE

MOST DIFFICULT PROBLEMS.

Taken from the Leading Authors on Arithmetic and Algebra, Many Prob-

lems and Solutions from Geometry, Trigonometry and Calculus,

Many Problems and Solutions from the Leading Math-

ematical Journals of the United States, and

Many Original Problems and Solutions.

WITH

NOTES AND EXPLANATIONS

BY

B. F. FINKEL, A. M., M. Sc.

ii

Member of the London Mathematical Society, Member of the American

Mathematical Society, Editor of the American Mathematical

Monthly, and Professor of Mathematics and

Physics in Drury College.

THIRD EDITION- REVISED.

KIBLER & COMPANY, PUBLISHERS,

Springfield, Mo.

COPYRIGHT, 1888,

BY

B. F. FINKEL,

IN THE OFFICE OF THE LIBRARIAN OF

CONGRESS,

WASHINGTON, D. C.

PREFACE.

This work is the outgrowth of eight years' experience in

teaching in the Public Schools, during which time I have ob-

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served that a work presenting a systematic treatment of solutions of problems would be serviceable to both teachers and pupils.

It is not intended to serve as a key to any work on mathe-

matics ; but the object of its appearance is to present, for use in

the schoolroom, such an accurate and logical method of solving

problems as will best awaken the latent energies of pupils, and

teach them to be original investigators in the various branches of

science.

It will not be denied by any intelligent educator that the so-

called "Short Cuts" and "Lightning Methods" are positively in-

jurious to beginners in mathematics. All the "whys" are cut

out by these methods and the student robbed of the very object

for which he is studying mathematics ; viz., the devolpment of

the reasoning faculty and the power to express his thoughts in

a forcible and logical manner. By pursuing these methods,

mathematics is made a mere memory drill and when the memory

fails, all is lost ; whereas, it should be presented in such a way as

to develop the memory, the imagination, and the reasoning fac-

ulty. By following out the method pursued in this book, the

mind will be strengthened in these three powers, besides a taste

for neatness and a love of the beautiful will be cultivated.

Any one who can write out systematic solutions of problems

can resort to "Short Cuts" at pleasure ; but, on the other hand,

let a student who has done all his work in mathematics by form-

ulae, "Short Cuts," and "Lightning Methods" attempt to write

out a systematic solution one in which the work explains

itself and he will soon convince one of his inability to express

his thoughts in a logical manner. These so-called "Short Cuts"

should not be used at all, in the schoolroom. After pupils and

students have been drilled on the systematic method of solving

problems, they will be able to solve more problems by short

methods than they could by having been instructed in all the

"Short Cuts" and "Lightning Methods" extant.

It can not be denied that more time is given to, and more

time wasted in the study of arithmetic in the public schools than

2 PREFACE.

in any other branch of study ; and yet, as a rule, no better results

are obtained in this branch than in any other. The reason of

this, to my mind, is apparent. Pupils are allowed to combine

the numbers in such a way as "to get the answer" and that is all

that is required. They are not required to tell why they do this,

or why they do that, but, "did you get the answer?" is the

question. The art of "ciphering" is thus developed at the ex-

pense of the reasoning faculty.

The method of solving problems pursued in this book is

often called the "Step Method." But we might, with equal pro-

priety, call any orderly manner of doing any thing, the "Step

Method." There are only two methods of solving problems a

right method and a wrong method. That is the right method

which takes up, in logical order, link by link, the chain of rea-

soning and arrives at the correct result. Any other method is

wrong and hurtful when pursued by those who are beginners in

mathematics.

One solution, thoroughly analyzed and criticised by a class,

is worth more than a dozen solutions the difficulties of which are

seen through a cloud of obscurities.

This book can be used to a great advantage in the class-

room the problems at the end of each chapter affording ample

exercise for supplementary work.

Many of the Formulae in Mensuration have been obtained by

the aid of the Calculus, the operation alone being indicated. This

feature of the work will not detract any from its merits for those

persons who do not understand the Calculus ; for those who do-

understand the Calculus it will afford an excellent drill to work

out all the steps taken in obtaining the formulae. Many of the

formulae can be obtained by elementary geometry and algebra.

But the Calculus has been used for the sake of presenting the

beauty and accuracy of that powerful instrument of mathematics.

In cases in which the formulae lead to series, as in the case

of the circumference of the ellipse, the rule is given for a near

approximation.

It has been the aim to give a solution of every problem

presenting anything peculiar, and of those which go the rounds

of the country. Any which have been omitted will receive space

in future editions of this work. The limits of this book have

compelled me to omit much curious and valuable matter in

Higher Mathematics.

I have taken some problems and solutions from the School

Visitor, published by John S. Royer; the Mathematical Maga-

zine, and the Mathematical Visitor, published by Artemas Mar-

tin, A. M., Ph. D., LL. D.; and the Mathematical Messenger,

published by G. H. Harvill, by the kind permission of these

distinguished gentlemen.

PREFACE. 3

It remains to acknowledge my indebtedness to Prof. William

Hoover, A. M., Ph. D., of the Department of Mathematics and

Astronomy in the Ohio University at Athens, for critically read-

ing the manuscript of the part treating on Mensuration.

Hoping that the work will, in a measure, meet the object

for which it is written, I respectfully submit it to the use of

my fellow teachers and co-laborers in the field of mathematics.

Any correction or suggestion will be thankfully received by

communicating the same to rne.

THE AUTHOR.

1 -

In bringing out a second edition of this work, I am greatly

indebted to Dr. G. B. M. Zerr for critically reading the work

with a view to eliminating all errors.

THE AUTHOR.

Drury College, Feb. Jp, 1897.

PREFACE TO SECOND EDITION.

PREFACE TO THIRD EDITION.

The hearty reception accorded this book, as is attested by

the fact that two editions of 1,200 copies each have already been

sold, encouraged me to bring out this third edition.

In doing so, I have availed myself of the opportunity of

making some important corrections, and such changes and im-

provements as experience and the suggestions of teachers using

the book have dictated. The very favorable comments on the

work by some of the most eminent mathematicians in this

country confirm the opinion that the book is a safe one to

put into the hands of teachers and students.

While mathematics is the exact science, yet not every book

that is written upon it treats of it as though it were such. In-

deed, until quite recently, there were very few books on Arith-

metic, Algebra, Geometry or Calculus that were not mere copies

of the works written a century ago, and in this way the method,

the spirit, the errors and the solecisms of the past two hundred

years were preserved and handed down to the present genera-

tion. At the present time the writers on these subjects are

breaking away from the beaten paths of tradition, and the re-

sult, though not wholly apparent, is a healthier and more vig-

orous mathematical philosophy. Within the last twenty-five

4 PREFACE.

years there has set in, in America, a reaction against the spirit

and the method of previous generations, so that C. A. Laisant,

in his La Mathematique Phi to sop hie Enseignement : , Paris, 1898,

says, "No country has made greater progress in mathematics

during the past twenty-five years than the United States. The

most of the text-books on Arithmetic, Algebra, Geometry, and

the Calculus, written within the last five years, are evidence of

this progress.

The reaction spoken of was brought about, to some ex-

tent, by the introduction into our- higher institutions of learn-

ing of courses of study in mathematics bearing on the wonder-

ful researches of Abel, Cauchy, Galois, Riemann, Weirstrass,

and others. This reaction, it may be said, started as early as

1832, the time when Benjamin Peirce, the first American worthy

to be ranked with Legendre, Wallis, Abel and the Bernouillis,

became professor of mathematics and natural philosophy at

Harvard University. Since that time the mathematical courses

in our leading Universities have been enlarged and strengthened

until now the opportunity for research work in mathematics

as offered, for example, at the University of Chicago, Harvard,

Yale, Cornell, Johns Hopkins, Princeton, Columbia and others,

is as good as is to be found anywhere in the world. For ex-

ample, the following are the subjects offered at Harvard for the

Academic year 1899-1900: Logarithms, Plane and Spherical

Trigonometry; Plane Analytical Geometry; Plane and Solid

Analytical Geometry ; Algebra ; Theory of Equations. Invar-

iants ; Differential and Integral Calculus ; Modern Methods in

Geometry. Determinants ; Elements of Mechanics ; Quater-

nions with application to Geometry and Mechanics ; Theory of

Curves and Surfaces; Dynamics of a Rigid Body; Trigonomet-

ric Series. Introduction to Spherical Harmonics. Potential

Function ; Hydrostatics. Hydrokinematics. Force Functions

and Velocity-Potential Functions and their uses. Hydroki-

netics ; Infinite Series and Products ; The Theory of Functions ;

Albegra. Galois's Theory of Equations; Lie's Theory as ap-

plied to Differential Equations ; Riemann's Theory of Func-

tions ; The Calculus of Variations ; Functions Defined by Linear

Differential Equations ; The Theory of Numbers ; The Theory

of Planetary Motions; Theory of Surfaces; Linear Associative

Algebra; the Algebra of Logic; the Plasticity of the Earth;

Elasticity; and the Elliptic and the Abelian Transcendants.

While the great activity and real progress in mathematics

is going on in our higher institutions of learning, a like degree

of activity is not yet being manifested in many of our colleges

and academies and the Public Schools in general. It is not

desirable that the quantity of mathematics studied in our Public

Schools be increased, but it is desirable that the quality of

PREFACE. 5

the teaching should be greatly improved. To bring about this

result is the aim of this book.

It does not follow, as is too often supposed, that any one

familiar with the multiplication table, and able, perhaps, to

solve a few problems, is quite competent to teach Arithmetic,

or "Mathematics," as arithmetic is popularly called. The very

first principles of the subject are of the utmost importance,

and unless the correct and refined! notions of these principles

are presented at the first, quite as much time is lost by the

student in unlearning and freeing himself from erroneous con-

ceptions as was required in acquiring them. Moreover, no ad-

vance in those higher modern developments in Mathematics is

possible by any one having false notions of its first principles.

As a branch for mental discipline, mathematics, when

properly taught, has no superior. Other subjects there are

that are equally beneficial, but none superior. The idea en-

tertained by many teachers, generally those who have pre-

pared themselves to teach other subjects, but teach mathematics

until an opportunity to teach in their special line presents itself

to them, that mathematics has only commercial value and

only so much of it should be studied as is needed by the student

in his business in after life, is pedagogically and psychologically

wrong. Mathematics has not only commercial value, but edu-

cational and ethical value as well, and that to a degree not

excelled by any other science. No other science offers such

rich opportunity for original investigation and discovery. So far

from being a perfected and complete body of doctrine "handed

down from heaven" and incapable of growth, as many sup-

pose, it is a subject which is being developed at such a mar-

velous rate that it is impossible for any but the best to keep

in sight of its ever-increasing and receding boundary. Because,

therefore, of the great importance of mathematics as an agent

in disciplining and developing the mind, in advancing the ma-

terial comforts of man by its application in every department

of art and invention, in improving ethical ideas, and in culti-

vating a love for the good; the beautiful, and the true, the

teachers of mathematics should have the best training possible.

If this book contributes to the end, that a more comprehensive

view be taken of mathematics, better services rendered in pre-

senting its first principles, and greater interest taken in its

study, I shall be amply rewarded for my labor in its prepa-

ration.

In this edition I have added a chapter on Longitude and

Time, the biographies of a few more mathematicians, several

hundred more problems for solution, an introduction to the

study of Geometry, and an introduction to the study of Algebra.

The list of biographies could have been extended indefi-

nitely, but the student who becomes interested in the lives of

6 PREFACE.

a class of men who have contributed much to the advancement

of civilization, will find a short sketch of the mathematicians

from the earliest times down to the present day in Cajori's

History of Mathematics or Ball's A Short History of Mathematics.

The biographies which have been added were taken from

the American Mathematical Mnothly. I have received much aid

in my remarks on Geometry from Study and Difficulties of

Mathematics, by Augustus De Morgan.

It yet remains for me to express my thanks to my colleague

and friend, Prof. F. A. Hall, of the Department of Greek, for

making corrections in the Greek terms used in this edition,

THE AUTHOR.

Drury College, July, 1899.

CONTENTS.

CHAPTER I.

DEFINITIONS.

Mathematics classified 11 |

CHAPTER II.

NUMERATION AND NOTATION.

PAGE

Definitions 11-14

Numeration defined 14

French Method defined 14

English Method defined 14

Periods of Notation 15

Notation defined

Arabic Notation defined

15

15

Roman Notation defined 15

Ordinal Numbers 15

Fractions 18

Irrational Numbers. 20

Examples 21

Addition defined

Subtraction defined

CHAPTER III.

ADDITION.

22 | Examples 23

CHAPTER IV.

SUBTRACTION.

23 | Examples 24

CHAPTER V.

MULTIPLICATION.

Multiplication defined 24 | Examples 25-26

CHAPTER VI.

DIVISION.

Division defined 26 | Examples 27

CHAPTER VII.

COMPOUND NUMBERS.

Definitions 28

Time Measure 29

Definitions in Time Measure . 23-31

Longitude and Time 31-34

Standard Time 34-35

The International Date Line . 36-37

Examples 37-36

Solutions 38-39

Examples 39-40

Divisor defined 41

Common Divisor defined 41

Multiple defined 42

Common Multiple defined 42

CHAPTER VIII.

GREATEST COMMON DIVISOR.

Greatest Com'n Divisor defined. 41

Examples 41-42

CHAPTER IX.

LEAST COMMON MULTIPLE.

Least Common Multiple defined 42

Examples 43-44

8

CONTENTS.

CHAPTER X.

FRACTIONS.

PAGB

Definitions 44-46

Fractions classified 44

Solutions of Problems.

Examples

PAGE

46-49

49-52

CHAPTER XI.

CIRCULATING DECIMALS.

54

55

55

IV.

I. Addition of Circulates

II. Subtraction of Circulates . .

III. Multiplication of Circulates

CHAPTER XII.

PERCENTAGE.

Division of Circulates . 56

Examples 56-57

.Definitions ....... 57

Solutions 57-69

II. Commission . 69

Definitions ...... .... 69

Solutions 69-71

Examples . 72

Trade Discount 72

Definitions 72

Solutions 73-76

Examples 76

Profit and Loss 77

III.

IV.

Definitions 77

Examples 83-84

V. Stocks and Bonds 84

Definitions . . . 84

Solutions 85-95

Examples 96-97

VI. Insurance 97

Definitions 97-98

Solutions . . 98-101

Examples 102

CHAPTER XIII.

INTEREST.

II.

III.

Simple Interest 103

Definitions 103

Solutions 103-106

True Discount . 106

Definitions 106

Solutions 106-107

Bank Discount 107

Definitions . . .107

Solutions 108-109

IV. Annual Interest 109

Annual Interest defined 109

Solutions 110-112

V, Compound Interest 112

Compound Int. defined. 112

Solutions 113-115

Definitions

Solutions .

CHAPTER XIV.

ANNUITIES.

. . . . 115 I Examples

.116-125

126

CHAPTER XV.

MISCELLANEOUS PROBLEMS.

Solutions . 127-139

CHAPTER XVI.

RATIO AND PROPORTION.

Definitions 139-141 I Problems

Solutions. ......... 141-144

... 144-146

Analysis defined. . .

Solutions . ... . .

CHAPTER XVII.

ANALYSIS.

. . . . 146 Problems

146-177

177-180

CONTENTS.

CHAPTER XVIII.

ALLIGATION.

PAGE.

I. Alligation Medial 181

II. Alligation Alternate. . 181

Solutions

PAGE

181-186

CHAPTER XIX.

SYSTEMS OF NOTATION.

Definitions 187

Names of Systems 187

Solutions 188-191

CHAPTER XX.

MENSURATION.

v

V,

Definitions 192-197

Geometrical Magnitudes class-

ified 192

I. Parallelogram . . 198-200

II. Triangles 200-204

III. Trapezoid 204-205

IV. Trapezium and Irregular

Polygons 205

V. Regular Polygons . . . 205-207

VI. Circles 207-210

VII. Rectification of Plane

Curves and Quadrature of

Plane Surfaces 210-213

II. Conic Sections 223

Definitions 223-224

1. Ellipse 224-227

2. Parabola 227-229

3. Hyperbola 229-232

IX. Higher Plane Curves ... 233

1. The Cissoid Diocles. 233-234

2. The Conchoid of Nicom-

edes 234-235

3. The Oval of Cassini 235

4. The Lemniscate of Ber-

nouilli 236

5. The Witch of Agnesi 236-237

6. The Limacon 237

7. The Quadratrix 238

8. The Catenary 238-239

9. The Tractrix 240

10. The Syntractrix 240

II. Roulettes 240

(a) Cycloids 240-243

(b) Prolate and Curtate

Cycloid 243-244

(c) Epitrochoid and Hy-

potrochoid 245-248

X. Plane Spiral 248

1. Spirals of Archimedes 249

2. The Reciprocal Spiral 249

3. The Lituus 250

4. The Logarithmic Spi-

ral 250

XI. Mensuration of Sol-

ids 251-254

1. Cylinder 254-255

2. CylindricUngulas 255-262

3. Pyramid and Cone 262-266

4. Conical Ungulas. 266-270

XII. Sphere 270-276

XIII. Spheroid 276-278

1. The Prolate Sphe-

roid 276-278

2. The Oblate Sphe-

roid 278-282

XIV. Conoids 282

1. The Parbolic Co-

noid .... 282-285

2. Hyperbolic Co-

noid 285-286

XV. Quadrature and Cuba-

ture of Surfaces and

Solids of Revolution 286

1. Cycloid 286-287

2. Cissoid 287-288

3. Spindles 288-289

4. Parabolic Spindle 289-290

XVI. Regular Solids 290

1. Tetrahedron .... 291-292

2. Octahedron 292*

3. Dodecahedron . . . 292-293

4. Icosahedron 293-294

XVII. Prismatoid 294-295

XVIII. Cylindric Rings ... 295-297

XIX. Miscellaneous Measure-

ments 297

1. Masons' and

Bricklayers' work 297

2. Gauging 297-298

3. Lumber Measure . 298

4. Grain and Hay. . , 298-299

10

CONTENTS.

MENSURATION Concluded.

PAGE

XX. Solutions of Miscellaneous Problems 299-345

Problems 346-354

Examination Tests : . 354-360

Problems 361-366

GEOMETRY.

Definitions 367 (e) Assumption of the Sphere . . 380

On Geometric Reasoning 369 (/) of Motion 380

On the Advantages Derived from On Logic 380

the Study of Geometry and Laws of Thought 381

Mathematics in General 370 Law of Converse 383

Axioms 375 Methods of Reasoning 384

General Axioms 376 How to Prepare a Lesson in

Assumptions . 377 Geometry 388

(a) Assumption of Straight Line 377 Plane Geometry 390

(b) of the Plane 377 The Three Famous Problems of

(c) of Parallel Line 377 Antiquity 408

(d) of the Circle... 379

ALGEBRA.

Definitions 415 Arithmetical Fallacies 424

Solutions of Problems 416-419 Probability 425

The Quadratic Equation 419 Problems 428

Indeterminate forms 421

Biography of Prof. William Hoover 403

Probability Problems 434-435

Biography of Dr. Artemas Martin 436

Biography of Prof. E. B. Seitz 440

Biography of Rene 7 Descartes 442

Biography of Leonhard Euler 446

Biography of Spphus Lie 451

Biography of Simon Newcomb 454

Biography of George Bruce Halsted 457

Biography of Prof. Felix Klein 459

Biography of Benjamin Peirce 462

Biography of James Joseph Sylvester s 468

Biography of Arthur Cayley 475

Table I 473

II 478

478

J V 479

V 479

VI 480

VII 480

Example 4gl

CHAPTER I.

DEFINITIONS.

1. Mathematics (p.aftv)p.a.Ttxrj, science) is that science

which treats of quantity.

!(1.) Arithmetic.

(2.) Algebra...

*.j Geometry..

rl. Calculus

1 2. Quaternions.

El. Platonic Geometry..

2. Analytical Geometry.

3. Descriptive Geometry.

Differential.

Integral.

Calculus of Variations.

: a. Pure Geometry.

b. Conic Sections. i\. Plane Trigon'y.

c . Trigonometry.. <2. Analytical Trig.

(3. Spherical "

'(1.) Mensuration.

(2.) Surveying.

(3.) Navigation.

(4.) Mechanics.

(5.) Astronomy.

(6.) Optics.

(7.) Gunnery.

^(8.) &c., &c.

2>. Pure Mathematics treats of magnitude or quantity

without relation to matter.

3. Applied Mathematics treats of magnitude as subsist-

ing in material bodies.

4. Arithmetic (api^^nx^, from d^etf/io?, a number) is

the science of numbers and the art of computing by them.

5. Alyebra (Ar. al, the, and geber, philosopher) is that

method of mathematical computation in which letters and other

symbols are employed.

6. G-eOWietry (yajfj.Tpia, from yU){j.Tpiv to measure

land, from pa, yf h the earth, and p.Tp~iv, to measure) is the

science of position and extension.

7. Calculus ( Calculus, a pebble) is that branch of mathe-

matics which commands by one general method, the most diffi-

cult problems of geometry and physics.

12 FINKEL'S SOLUTION BOOK.

8. Differential Calculus is that branch of Calculus

which investigates mathematical questions by measuring the re-

lation of certain infinitely small quantities called differentials.

9. Integral Calculus is that branch of Calculus which

determines the functions from which a given differential has been

derived.

1C. Calculus of Variations is that branch of calculus

in which the laws of dependence which bind the variable quanti-

ties together are themselves subject to change.

11. Quaternions (quaternis, from quaterni four each,

from quator, four) is that branch of algebra which treats of the

relations of magnitude and position of lines or bodies in space by

means of the quotient of two direct lines in space, considered as

depending on a system of four geometrical elements, and as ex-

pressed by an algebraic symbol of quadrinominal form.

12. Platonic Geometry is that branch of geometry in

which the argument is carried forward by a direct inspection of

the figures themselves, delineated before the eye, or held in the

imagination.

13. Pure Geometry is that branch of Platonic geometry

in which the argument may be practically tested by the aid of

the compass and the square only.

14. Conic Sections is that branch of Platonic geometry

which treats of the curved lines formed by the intersection of a

cone and a plane.

15. Trigonometry (rptytovov, triangle, ptrpov, meas-

ure) is that branch of Platonic geometry which treats of the re-

lations of the angles and sides of triangles.

16. Plane Trigonometry is that branch of trigonom-

etry which treats of the relations of the angles and sides of plane

triangles.

17. Analytical Trigonometry is that branch of trig-

onometry which treats of the general properties and relations of

trigonometrical functions.

18. Spherical Trigonometry is that branch of trig-

onometry which treats of the solution of spherical triangles.

19. Analytical Geometry is that branch of geometry

in which the properties and relations of lines and surfaces are in-

vestigated by the aid of algebraic analysis.

20. Descriptive Geometry is that branch of geometry

which seeks the graphic solution of geometrical problems by

means of projections upon auxiliary planes.

DEFINITIONS. 13

21. Mensuration is that branch of applied mathematics

which treats of the measurment of geometrical'magnitudes.

22. Surveying is that branch of applied mathematics

which treats of the art of determining and representing distances,

areas, and the relative position of points upon the earth's surface.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

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